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cleartomark

%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 173 /Omega put
dup 177 /ffi put
dup 178 /ffl put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 187 /cedilla put
dup 189 /ae put
dup 190 /oe put
dup 195 /suppress put
dup 109 /m put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueXX 5000768 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 173 /Omega put
dup 177 /delta put
dup 178 /epsilon1 put
dup 180 /eta put
dup 181 /theta put
dup 182 /iota put
dup 183 /kappa put
dup 184 /lambda put
dup 187 /xi put
dup 189 /rho put
dup 190 /sigma put
dup 195 /psi put
dup 82 /R put
dup 96 /lscript put
dup 110 /n put
dup 112 /p put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueXX 5087385 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 173 /Omega put
dup 174 /alpha put
dup 175 /beta put
dup 176 /gamma put
dup 177 /delta put
dup 178 /epsilon1 put
dup 179 /zeta put
dup 180 /eta put
dup 181 /theta put
dup 182 /iota put
dup 183 /kappa put
dup 184 /lambda put
dup 185 /mu put
dup 186 /nu put
dup 187 /xi put
dup 188 /pi put
dup 189 /rho put
dup 190 /sigma put
dup 191 /tau put
dup 194 /chi put
dup 195 /psi put
dup 33 /omega put
dup 34 /epsilon put
dup 39 /phi1 put
dup 44 /arrowhookleft put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 125 /weierstrass put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueXX 5087386 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /parenleftbig put
dup 162 /parenrightbig put
dup 163 /bracketleftbig put
dup 164 /bracketrightbig put
dup 167 /ceilingleftbig put
dup 169 /braceleftbig put
dup 170 /bracerightbig put
dup 173 /angbracketleftbig put
dup 174 /angbracketrightbig put
dup 175 /vextendsingle put
dup 176 /vextenddouble put
dup 177 /slashbig put
dup 178 /backslashbig put
dup 179 /parenleftBig put
dup 180 /parenrightBig put
dup 181 /parenleftbigg put
dup 182 /parenrightbigg put
dup 183 /bracketleftbigg put
dup 184 /bracketrightbigg put
dup 185 /floorleftbigg put
dup 186 /floorrightbigg put
dup 187 /ceilingleftbigg put
dup 188 /ceilingrightbigg put
dup 189 /braceleftbigg put
dup 190 /bracerightbigg put
dup 191 /angbracketleftbigg put
dup 194 /backslashbigg put
dup 195 /parenleftBigg put
dup 40 /braceleftBigg put
dup 56 /bracelefttp put
dup 58 /braceleftbt put
dup 60 /braceleftmid put
dup 62 /braceex put
dup 63 /arrowvertex put
dup 76 /circleplustext put
dup 77 /circleplusdisplay put
dup 80 /summationtext put
dup 81 /producttext put
dup 82 /integraltext put
dup 88 /summationdisplay put
dup 89 /productdisplay put
dup 90 /integraldisplay put
dup 91 /uniondisplay put
dup 110 /braceleftBig put
dup 111 /bracerightBig put
dup 112 /radicalbig put
dup 120 /arrowtp put
dup 121 /arrowbt put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 46 /period put
dup 105 /i put
dup 110 /n put
dup 115 /s put
dup 117 /u put
dup 118 /v put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueXX 5000826 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBX6
%!PS-AdobeFont-1.1: CMBX6 1.0
%%CreationDate: 1991 Aug 20 16:35:30

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 70 /F put
dup 81 /Q put
readonly def
/FontBBox{-49 -250 1367 753}readonly def
/UniqueXX 5000764 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: rsfs10
%!PS-AdobeFont-1.0: rsfs10 001.000
%%CreationDate: Sat Mar 21 18:47:14 1998
%%VMusage: 120000 150000
11 dict begin
/FontInfo 14 dict dup begin
/version (001.001) readonly def
/Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def
/Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def
/FullName (rsfs10) readonly def
/FamilyName (rsfs10) readonly def
/ItalicAngle -12 def
/isFixedPitch false def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
/Weight (Roman) readonly def
end readonly def
/FontName /rsfs10 def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put } for
dup 65 /A put
readonly def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/FontBBox {-2 -300 1240 728} readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 35 /numbersign put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 46 /period put
dup 49 /one put
dup 54 /six put
dup 57 /nine put
dup 58 /colon put
dup 61 /equal put
dup 65 /A put
dup 66 /B put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 82 /R put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 121 /y put
readonly def
/FontBBox{-33 -250 945 749}readonly def
/UniqueXX 5000795 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueXX 5000789 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /minus put
dup 162 /periodcentered put
dup 163 /multiply put
dup 164 /asteriskmath put
dup 167 /plusminus put
dup 169 /circleplus put
dup 170 /circleminus put
dup 173 /circlemultiply put
dup 174 /circledivide put
dup 175 /circledot put
dup 176 /circlecopyrt put
dup 177 /openbullet put
dup 178 /bullet put
dup 179 /equivasymptotic put
dup 180 /equivalence put
dup 181 /reflexsubset put
dup 182 /reflexsuperset put
dup 183 /lessequal put
dup 184 /greaterequal put
dup 185 /precedesequal put
dup 186 /followsequal put
dup 187 /similar put
dup 188 /approxequal put
dup 189 /propersubset put
dup 190 /propersuperset put
dup 191 /lessmuch put
dup 194 /follows put
dup 195 /arrowleft put
dup 48 /prime put
dup 49 /infinity put
dup 95 /logicalor put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueXX 5000816 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 33 /exclam put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 86 /V put
dup 87 /W put
dup 90 /Z put
dup 96 /quoteleft put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /alpha put
dup 175 /beta put
dup 176 /gamma put
dup 177 /delta put
dup 178 /epsilon1 put
dup 179 /zeta put
dup 180 /eta put
dup 181 /theta put
dup 182 /iota put
dup 183 /kappa put
dup 184 /lambda put
dup 185 /mu put
dup 186 /nu put
dup 187 /xi put
dup 188 /pi put
dup 189 /rho put
dup 190 /sigma put
dup 191 /tau put
dup 194 /chi put
dup 195 /psi put
dup 59 /comma put
dup 69 /E put
dup 77 /M put
dup 78 /N put
dup 82 /R put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 100 /d put
dup 102 /f put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 119 /w put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueXX 5087381 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /minus put
dup 162 /periodcentered put
dup 163 /multiply put
dup 164 /asteriskmath put
dup 167 /plusminus put
dup 169 /circleplus put
dup 170 /circleminus put
dup 173 /circlemultiply put
dup 174 /circledivide put
dup 175 /circledot put
dup 176 /circlecopyrt put
dup 177 /openbullet put
dup 178 /bullet put
dup 179 /equivasymptotic put
dup 180 /equivalence put
dup 181 /reflexsubset put
dup 182 /reflexsuperset put
dup 183 /lessequal put
dup 184 /greaterequal put
dup 185 /precedesequal put
dup 186 /followsequal put
dup 187 /similar put
dup 188 /approxequal put
dup 189 /propersubset put
dup 190 /propersuperset put
dup 191 /lessmuch put
dup 194 /follows put
dup 195 /arrowleft put
dup 33 /arrowright put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 63 /perpendicular put
dup 65 /A put
dup 72 /H put
dup 79 /O put
dup 91 /union put
dup 94 /logicaland put
dup 95 /logicalor put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 110 /backslash put
dup 112 /radical put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueXX 5000818 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBX8
%!PS-AdobeFont-1.1: CMBX8 1.0
%%CreationDate: 1991 Aug 20 16:36:07

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 65 /A put
dup 67 /C put
dup 70 /F put
dup 77 /M put
dup 81 /Q put
dup 82 /R put
dup 84 /T put
dup 90 /Z put
dup 109 /m put
readonly def
/FontBBox{-59 -250 1235 750}readonly def
/UniqueXX 5000766 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 35 /numbersign put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 58 /colon put
dup 61 /equal put
dup 63 /question put
dup 70 /F put
dup 71 /G put
dup 76 /L put
dup 86 /V put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 126 /tilde put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueXX 5000791 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /alpha put
dup 175 /beta put
dup 176 /gamma put
dup 177 /delta put
dup 178 /epsilon1 put
dup 179 /zeta put
dup 180 /eta put
dup 181 /theta put
dup 182 /iota put
dup 183 /kappa put
dup 184 /lambda put
dup 185 /mu put
dup 186 /nu put
dup 187 /xi put
dup 188 /pi put
dup 189 /rho put
dup 190 /sigma put
dup 191 /tau put
dup 194 /chi put
dup 195 /psi put
dup 34 /epsilon put
dup 39 /phi1 put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 86 /V put
dup 87 /W put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueXX 5087383 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMSL12
%!PS-AdobeFont-1.1: CMSL12 1.0
%%CreationDate: 1991 Aug 20 16:40:41

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSL12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -9.46 def
/isFixedPitch false def
end readonly def
/FontName /CMSL12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 88 /X put
dup 89 /Y put
readonly def
/FontBBox{-56 -251 1102 750}readonly def
/UniqueXX 5000799 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMTT12
%!PS-AdobeFont-1.1: CMTT12 1.0
%%CreationDate: 1991 Aug 20 16:45:46

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /arrowup put
dup 175 /arrowdown put
dup 176 /quotesingle put
dup 177 /exclamdown put
dup 178 /questiondown put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /visiblespace put
dup 46 /period put
dup 47 /slash put
dup 58 /colon put
dup 64 /at put
dup 67 /C put
dup 70 /F put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 119 /w put
dup 121 /y put
readonly def
/FontBBox{-1 -234 524 695}readonly def
/UniqueXX 5000833 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 61 /equal put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueXX 5000829 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05

% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 161 /Gamma put
dup 162 /Delta put
dup 163 /Theta put
dup 164 /Lambda put
dup 167 /Sigma put
dup 169 /Phi put
dup 170 /Psi put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 177 /ffi put
dup 178 /ffl put
dup 179 /dotlessi put
dup 180 /dotlessj put
dup 181 /grave put
dup 182 /acute put
dup 183 /caron put
dup 184 /breve put
dup 185 /macron put
dup 186 /ring put
dup 187 /cedilla put
dup 188 /germandbls put
dup 189 /ae put
dup 190 /oe put
dup 191 /oslash put
dup 194 /Oslash put
dup 195 /suppress put
dup 33 /exclam put
dup 34 /quotedblright put
dup 35 /numbersign put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 42 /asterisk put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 61 /equal put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 91 /bracketleft put
dup 92 /quotedblleft put
dup 93 /bracketright put
dup 94 /circumflex put
dup 96 /quoteleft put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
dup 124 /emdash put
dup 126 /tilde put
dup 196 /dieresis put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueXX 5000794 def
currentdict end
currentfile eexec
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1 0 bop 5356 23755 a FH(Hec)-60 b(k)g(e)693 b(Algebras)f(and)h(Mo)60
b(dular)692 b(F)-180 b(orms:)-3202 26522 y(Notes)692
b(deriv)-60 b(ed)693 b(from)g(Rib)60 b(et's)693 b(1996)g(Berk)-60
b(eley)693 b(grad.)933 b(course.)17481 32417 y FG(William)518
b(Stein)15508 36360 y(Septem)-43 b(b)43 b(er)519 b(24,)i(1998)p
eop
%%Page: 2 2
2 1 bop 1263 -6698 a FF(ii)p eop
%%Page: 3 3
3 2 bop -3718 5686 a FE(Preface)-3718 11719 y FD(Disclaimer:)1578
b FF(These)511 b(notes)g(record)g(some)h(of)g(what)f(I)g(sa)-36
b(w)512 b(in)f(Ken)g(Rib)36 b(et's)511 b(course)g(on)h(Mo)36
b(dular)-3718 13324 y(F)-108 b(orms)424 b(and)h(Hec)-36
b(k)g(e)426 b(Op)36 b(erators)424 b(giv)-36 b(en)426
b(at)f(U.C.)i(Berk)-36 b(eley)426 b(during)e(the)h(Spring)f(semester)h
(1996.)577 b(They)-3718 14929 y(are)550 b(still)g FC(very)571
b(r)-66 b(ough)550 b FF(as)g(I)g(wrote)g(them)f(during)g(m)-36
b(y)550 b(\257rst)f(semester)h(of)g(graduate)g(sc)-36
b(ho)36 b(ol)550 b(b)36 b(efore)550 b(I)-3718 16534 y(knew)434
b(an)-36 b(y)434 b(real)g(mathematics.)-1767 18140 y(The)583
b(participan)-36 b(ts)582 b(in)h(the)f(course)h(w)-36
b(ere:)878 b(Amo)36 b(d)583 b(Agashe,)621 b(Matt)582
b(Bak)-36 b(er,)622 b(Jim)583 b(Borger,)621 b(Kevin)-3718
19745 y(Buzzard,)406 b(Bruce)398 b(Cask)-36 b(el,)408
b(Rob)36 b(ert)399 b(Coleman,)407 b(Jan\266)-650 b(os)400
b(Csirik,)407 b(Annette)399 b(Hub)36 b(er,)405 b(Da)-36
b(vid)400 b(Jones,)407 b(Da)-36 b(vid)-3718 21350 y(Kohel,)497
b(Loic)485 b(Merel,)497 b(Da)-36 b(vid)485 b(Moulton,)497
b(Andrew)483 b(Ogg,)498 b(Arth)-36 b(ur)482 b(Ogus,)497
b(Jessica)485 b(P)-36 b(olito,)498 b(Ken)484 b(Rib)36
b(et,)-3718 22955 y(Saul)391 b(Sc)-36 b(hleimer,)399
b(La)-36 b(wren)391 b(Smithline,)399 b(William)393 b(Stein,)399
b(T)-108 b(ak)-72 b(ahashi,)401 b(W)-108 b(a)-36 b(yne)391
b(Whitney)-108 b(,)400 b(and)390 b(Hui)h(Zui.)-1767 24560
y(I)509 b(wish)g(to)g(thank)g(Da)-36 b(vid)510 b(Moulton,)528
b(Jo)36 b(e)509 b(W)-108 b(etherall,)529 b(and)508 b(Kevin)i(Buzzard)e
(who)h(help)36 b(ed)509 b(me)g(in)-3718 26165 y(preparing)579
b(these)g(notes,)616 b(Arth)-36 b(ur)578 b(Ogus)h(who)g(ask)-36
b(ed)580 b(a)g(lot)g(of)g(stim)-36 b(ulating)580 b(questions)f(during)g
(the)-3718 27770 y(class,)434 b(and)g(of)g(course)f(Ken)h(Rib)36
b(et)433 b(who)h(sees)g(clearly)-108 b(.)-1767 29375
y(William)435 b(Stein,)e(Spring)g(1996,)i(Berk)-36 b(eley)-108
b(,)435 b(CA,)f FB([email protected])21643 77755
y FF(iii)p eop
%%Page: 4 4
4 3 bop 1263 -6698 a FF(iv)44564 b FA(PREF)-145 b(A)-36
b(CE)p eop
%%Page: 5 5
5 4 bop -3718 5803 a FE(Con)-86 b(ten)g(ts)-3718 13370
y FD(Preface)45718 b(iii)-3718 16509 y(1)1204 b(In)-42
b(tro)42 b(duction)40885 b(1)-1767 18254 y FF(1.1)1330
b(Tw)-36 b(o)435 b(Dimensional)g(Galois)f(Represen)-36
b(tations)725 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h
(.)f(.)h(.)f(.)h(.)f(.)h(.)2265 b(1)1224 19999 y(1.1.1)1491
b(Finite)433 b(Fields)h(\(W)-108 b(eil,)434 b(T)-108
b(ate\))1308 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265 b(1)1224
21744 y(1.1.2)1491 b(Galois)434 b(Represen)-36 b(tations)433
b(\(T)-108 b(aniy)-36 b(ama,)435 b(Shim)-36 b(ura,)433
b(Mumford-T)-108 b(ate\))661 b(.)j(.)h(.)f(.)h(.)2265
b(2)-1767 23489 y(1.2)1330 b(Mo)36 b(dular)434 b(F)-108
b(orms)433 b(and)g(Galois)h(Represen)-36 b(tations)378
b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h
(.)f(.)h(.)2265 b(2)1224 25234 y(1.2.1)1491 b(Cusp)433
b(F)-108 b(orms)809 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)h(.)2265 b(2)1224 26979 y(1.2.2)1491 b(Hec)-36
b(k)g(e)434 b(Op)36 b(erators)433 b(\(Mordell\))585 b(.)664
b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)2265 b(2)-3718 30117 y FD(2)1204
b(Mo)42 b(dular)499 b(Represen)-42 b(tations)500 b(and)f(Curv)-42
b(es)24621 b(5)-1767 31862 y FF(2.1)1330 b(Arithmetic)434
b(of)g(Mo)36 b(dular)433 b(F)-108 b(orms)1089 b(.)665
b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265 b(5)-1767 33607 y(2.2)1330
b(Characters)521 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h
(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)
h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265 b(6)-1767 35352 y(2.3)1330
b(P)-36 b(arit)g(y)434 b(Conditions)667 b(.)d(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)
f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265 b(6)-1767 37097
y(2.4)1330 b(Conjectures)434 b(of)g(Serre)f(\(mo)36 b(d)434
b Fz(`)f FF(v)-36 b(ersion\))1223 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h
(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265
b(7)-1767 38842 y(2.5)1330 b(General)434 b(remarks)g(on)f(mo)36
b(d)434 b Fz(p)f FF(Galois)h(represen)-36 b(tations)1035
b(.)665 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265
b(7)-1767 40587 y(2.6)1330 b(Serre's)434 b(Conjecture)395
b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265
b(8)-1767 42332 y(2.7)1330 b(Wiles')435 b(P)-36 b(ersp)36
b(ectiv)-36 b(e)371 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)
f(.)h(.)f(.)h(.)2265 b(8)-3718 45471 y FD(3)1204 b(Mo)42
b(dular)499 b(F)-125 b(orms)38985 b(9)-1767 47216 y FF(3.1)1330
b(Cusp)434 b(F)-108 b(orms)869 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265 b(9)-1767
48961 y(3.2)1330 b(Lattices)1140 b(.)664 b(.)g(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265
b(9)-1767 50706 y(3.3)1330 b(Relationship)434 b(With)g(Elliptic)g(Curv)
-36 b(es)622 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)2265 b(9)-1767
52451 y(3.4)1330 b(Hec)-36 b(k)g(e)434 b(Op)36 b(erators)1335
b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)1615
b(10)-1767 54196 y(3.5)1330 b(Explicit)435 b(Description)f(of)g
(Sublattices)476 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h
(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)1615 b(11)-1767
55941 y(3.6)1330 b(Action)434 b(of)g(Hec)-36 b(k)g(e)434
b(Op)36 b(erators)434 b(on)f(Mo)36 b(dular)433 b(F)-108
b(orms)1026 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)1615 b(12)-3718 59080 y FD(4)1204 b(Em)-42
b(b)42 b(edding)499 b(Hec)-42 b(k)g(e)499 b(Op)42 b(erators)500
b(in)f(the)f(Dual)21581 b(15)-1767 60825 y FF(4.1)1330
b(The)434 b(Space)f(of)i(Mo)36 b(dular)433 b(F)-108 b(orms)1341
b(.)665 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)1615 b(15)-1767
62570 y(4.2)1330 b(Inner)433 b(Pro)36 b(duct)682 b(.)664
b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)1615
b(16)-1767 64315 y(4.3)1330 b(Eigenforms)1348 b(.)665
b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h
(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)
1615 b(17)-3718 67454 y FD(5)1204 b(Rationalit)-42 b(y)500
b(and)f(In)-42 b(tegralit)g(y)500 b(Questions)23943 b(19)-1767
69199 y FF(5.1)1330 b(Review)520 b(.)665 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h
(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)
f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)1615
b(19)-1767 70944 y(5.2)1330 b(In)-36 b(tegralit)g(y)746
b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)h(.)1615 b(19)-1767 72689 y(5.3)1330 b(Victor)434
b(Miller's)h(Thesis)518 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)h(.)1615 b(20)-1767 74434 y(5.4)1330 b(P)-36
b(etersson)434 b(Inner)e(Pro)36 b(duct)986 b(.)664 b(.)h(.)f(.)h(.)f(.)
g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)1615 b(20)21841 77755 y(v)p eop
%%Page: 6 6
6 5 bop 1263 -6698 a FF(vi)43371 b FA(CONTENTS)1263 -3169
y FD(6)1204 b(Mo)42 b(dular)499 b(Curv)-42 b(es)37706
b(23)3214 -1554 y FF(6.1)1331 b(Cusp)433 b(F)-108 b(orms)870
b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)
g(.)h(.)1615 b(23)3214 61 y(6.2)1331 b(Mo)36 b(dular)433
b(Curv)-36 b(es)563 b(.)664 b(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h
(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)
f(.)h(.)f(.)g(.)h(.)1615 b(23)3214 1676 y(6.3)1331 b(Classifying)436
b(\241\()p Fz(N)139 b FF(\)-structures)1242 b(.)664 b(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)g(.)h(.)1615 b(24)3214 3291 y(6.4)1331 b(More)433
b(on)h(In)-36 b(tegral)434 b(Hec)-36 b(k)g(e)434 b(Op)36
b(erators)556 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615 b(24)3214
4906 y(6.5)1331 b(Complex)434 b(Conjugation)395 b(.)665
b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615
b(25)3214 6522 y(6.6)1331 b(Isomorphism)433 b(in)h(the)f(Real)h(Case)
822 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615 b(25)3214
8137 y(6.7)1331 b(The)433 b(Eic)-36 b(hler-Shim)g(ura)432
b(Isomorphism)495 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h
(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615 b(25)3214
9752 y(6.8)1331 b(The)433 b(P)-36 b(etterson)433 b(Inner)g(Pro)36
b(duct)433 b(is)h(Hec)-36 b(k)g(e)434 b(Compatible)602
b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615
b(27)1263 12674 y FD(7)1204 b(Higher)500 b(W)-125 b(eigh)-42
b(t)499 b(Mo)42 b(dular)498 b(F)-125 b(orms)28156 b(29)3214
14289 y FF(7.1)1331 b(De\257nitions)433 b(of)i FD(T)661
b FF(.)j(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)
1615 b(29)3214 15905 y(7.2)1331 b(Double)433 b(Cosets)560
b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h
(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)
h(.)1615 b(29)3214 17520 y(7.3)1331 b(More)433 b(General)h(Congruence)f
(Subgroups)826 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)
f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615 b(30)3214 19135
y(7.4)1331 b(Explicit)434 b(F)-108 b(orm)-36 b(ulas)684
b(.)665 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h
(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615
b(31)3214 20750 y(7.5)1331 b(Old)433 b(and)g(New)h(F)-108
b(orms)1344 b(.)665 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)
1615 b(31)1263 23673 y FD(8)1204 b(New)499 b(F)-125 b(orms)40927
b(33)3214 25288 y FF(8.1)1331 b(Connection)433 b(With)h(Galois)g
(Represen)-36 b(tations)859 b(.)665 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615 b(34)3214
26903 y(8.2)1331 b(Semisimplicit)-36 b(y)434 b(of)g Fz(U)17248
27102 y Fy(p)19028 26903 y FF(.)665 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)
h(.)f(.)g(.)h(.)1615 b(34)3214 28518 y(8.3)1331 b(Shim)-36
b(ura's)433 b(Example)h(of)g(Nonsemisimple)g Fz(U)28950
28717 y Fy(p)30308 28518 y FF(.)665 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615 b(34)3214
30133 y(8.4)1331 b(An)433 b(In)-36 b(teresting)433 b(Dualit)-36
b(y)963 b(.)664 b(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615
b(35)3214 31748 y(8.5)1331 b(Observ)-72 b(ations)433
b(on)g Fz(T)16556 31947 y Fy(n)18003 31748 y FF(.)664
b(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615
b(36)1263 34671 y FD(9)1204 b(Some)499 b(Explicit)g(Gen)-42
b(us)499 b(Computations)25005 b(37)3214 36286 y FF(9.1)1331
b(Computing)433 b(the)g(Dimension)h(of)g Fz(S)23939 36485
y Fy(k)24508 36286 y FF(\(\241\))899 b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)1615
b(37)3214 37901 y(9.2)1331 b(Application)434 b(of)g(Riemann-Hurwitz)351
b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
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b(erators)1295 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)
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b(75)-1767 22487 y(13.3)680 b(Finite)434 b(Flat)g(Group)e(Sc)-36
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h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h
(.)f(.)h(.)1615 b(81)-1767 36958 y(14.3)680 b(W)-108
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y(14.6)680 b(In)-36 b(tro)36 b(duction)577 b(.)665 b(.)f(.)h(.)f(.)h(.)
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b(84)-1767 43531 y(14.7)680 b(Adelic)434 b(Represen)-36
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b(88)1224 50104 y(14.8.3)841 b(T)-108 b(ate)433 b(Curv)-36
b(es)727 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h
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b(eigh)-42 b(t)500 b(and)f(Serre's)g(Conjectures)24626
b(91)-1767 54716 y FF(15.1)680 b(In)-36 b(tro)36 b(duction)577
b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h
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b(of)g(the)e Fz(\270)p FF(-adic)g(case)361 b(.)664 b(.)h(.)f(.)h(.)f(.)
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h(.)f(.)h(.)1615 b(92)-1767 61289 y(15.4)680 b(Serre's)434
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b(bac)-36 b(kground)433 b(p)36 b(oin)-36 b(ts)593 b(.)665
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y(15.5)680 b(The)434 b(w)-36 b(eigh)g(t)434 b(and)f(fundamen)-36
b(tal)433 b(c)-36 b(haracters)488 b(.)665 b(.)f(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)1615
b(94)-1767 66218 y(15.6)680 b(The)434 b(w)-36 b(eigh)g(t)434
b(in)f(Serre's)g(conjectures)h(on)f(mo)36 b(dular)434
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1615 b(98)1224 67862 y(15.6.1)841 b Fz(\265)36 b FF(-series)433
b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
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h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)964
b(100)-1767 71148 y(15.7)680 b(The)434 b(extra)g(assumption)343
b(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)964
b(100)1224 72791 y(15.7.1)841 b(Companion)433 b(F)-108
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b(102)-1767 74434 y(15.8)680 b(The)434 b(exceptional)g(lev)-36
b(el)435 b(1)f(case)1057 b(.)664 b(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)964
b(103)p eop
%%Page: 8 8
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b(106)1263 2947 y FD(17)457 b(Deformations)38839 b(109)3214
4552 y FF(17.1)681 b(In)-36 b(tro)36 b(duction)577 b(.)664
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b(The)433 b(case)h Fz(p)369 b Fw(6)p FF(=)g Fz(`)1311
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h(.)965 b(112)3214 14182 y(17.4)681 b(Wiles')435 b(Hec)-36
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h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h
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b(Hec)-42 b(k)g(e)499 b(Algebra)h Fz(T)17064 17287 y
Fx(\247)50827 17088 y FD(115)3214 18693 y FF(18.1)681
b(The)433 b(Hec)-36 b(k)g(e)434 b(Algebra)924 b(.)664
b(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
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b(115)3214 20298 y(18.2)681 b(The)433 b(maximal)i(ideal)f(in)g
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b(117)6206 21903 y(18.2.1)840 b(Strip)432 b(a)-36 b(w)g(a)g(y)435
b(certain)f(Euler)f(factors)790 b(.)664 b(.)h(.)f(.)h(.)f(.)g(.)h(.)f
(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)965
b(117)6206 23508 y(18.2.2)840 b(Mak)-36 b(e)434 b(in)-36
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25113 y(18.3)681 b(The)433 b(Galois)i(Represen)-36 b(tation)718
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26718 y(18.3.1)840 b(The)433 b(structure)g(of)h FD(T)21133
26917 y Fv(m)23130 26718 y FF(.)664 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
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(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)965
b(120)6206 29929 y(18.3.3)840 b(Massage)434 b Fz(\275)758
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h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)965
b(120)6206 31534 y(18.3.4)840 b(Massage)434 b Fz(\275)16219
31052 y Fu(0)16977 31534 y FF(.)665 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
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b(Represen)-36 b(tations)433 b(from)h(mo)36 b(dular)433
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h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)965 b(124)3214
41164 y(18.7)681 b(Wiles)434 b(Main)g(Conjecture)727
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42769 y FF(is)434 b(a)g(complete)g(in)-36 b(tersection)1311
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44374 y(18.9)681 b(The)433 b(inequalit)-36 b(y)435 b(#)p
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Fz(T)10805 73028 y Fy(`)11245 72829 y FF(\()p Fz(E)78
b FF(\))334 b Fw(\255)h FD(Q)16122 73028 y Fy(`)17054
72829 y FF(is)492 b(a)g(t)-36 b(w)g(o)492 b(dimensional)g(v)-36
b(ector)492 b(space)g(o)-36 b(v)g(er)492 b FD(Q)40752
73028 y Fy(`)41192 72829 y FF(.)753 b(This)492 b(giv)-36
b(es)-3718 74434 y(the)433 b(\257rst)g(non)-36 b(trivial)434
b(example)g(of)h Fz(`)p FF(-adic)397 b(\266)-614 b(etale)434
b(cohomology)-108 b(.)21859 77755 y(1)p eop
%%Page: 2 10
2 9 bop 1263 -6698 a FF(2)31524 b FA(CHAPTER)434 b(1.)1012
b(INTR)-36 b(ODUCTION)1263 -3169 y Fq(1.1.2)1793 b(Galois)398
b(Represen)-50 b(tations)401 b(\(T)-149 b(aniy)-50 b(ama,)439
b(Shim)-50 b(ura,)440 b(Mumford-T)-149 b(ate\))1263 -706
y FF(Let)476 b Fz(E)78 b(=)p FD(Q)477 b FF(b)36 b(e)476
b(an)g(elliptic)i(curv)-36 b(e)476 b(and)g Fz(G)442 b
FF(=)f Fw(G)79 b Fz(al)29 b FF(\()p 26373 -1777 1123
54 v FD(Q)q Fz(=)p FD(Q)p FF(\).)707 b(Then)476 b Fz(E)78
b FF([)p Fz(n)p FF(])443 b(=)e Fw(f)p Fz(P)623 b Fw(2)442
b Fz(E)78 b FF(\()p 43716 -1777 V FD(Q)o FF(\))442 b(:)g
Fz(nP)624 b FF(=)441 b(0)p Fw(g)52036 -1075 y(\273)52046
-651 y FF(=)1263 899 y(\()p FD(Z)p Fz(=n)p FD(Z)p FF(\))5527
417 y Fx(2)6482 899 y FF(is)428 b(acted)g(on)g(b)-36
b(y)429 b Fz(G)f FF(and)f(this)h(action)h(resp)36 b(ects)427
b(the)h(group)g(op)36 b(eration)428 b(so)h(w)-36 b(e)428
b(ha)-36 b(v)g(e)428 b(a)h(Galois)1263 2504 y(represen)-36
b(tation)18576 5428 y Fz(G)20363 4621 y Fy(\275)19971
5428 y Fw(\241)-885 b(!)159 b FF(Aut)o(\()p Fz(E)78 b
FF([)p Fz(n)p FF(]\))27730 5059 y Fw(\273)27741 5483
y FF(=)29132 5428 y(GL)30965 5627 y Fx(2)31491 5428 y
FF(\()p FD(Z)p Fz(=n)p FD(Z)p FF(\))1263 8352 y(Let)498
b Fz(K)594 b FF(b)36 b(e)498 b(the)h(\257xed)f(\257eld)g(of)h(k)-36
b(er)222 b Fz(\275)499 b FF(\(note)f(that)g Fz(K)594
b FF(is)499 b(a)f(n)-36 b(um)g(b)36 b(er)498 b(\257eld\),)514
b(then)498 b(since)g Fw(G)79 b Fz(al)29 b FF(\()p Fz(K)23
b(=)p FD(Q)p FF(\))52036 7984 y Fw(\273)52046 8408 y
FF(=)1263 9958 y Fz(G=)221 b FF(k)-36 b(er)222 b Fz(\275)6191
9589 y Fw(\273)6202 10013 y FF(=)7629 9958 y(Im)f Fz(\275)404
b Fw(\265)g FF(GL)13748 10157 y Fx(2)14274 9958 y FF(\()p
FD(Z)p Fz(=n)p FD(Z)p FF(\))455 b(w)-36 b(e)455 b(obtain)f(man)-36
b(y)454 b(subgroups)f(of)i(GL)38069 10157 y Fx(2)38595
9958 y FF(\()p FD(Z)p Fz(=n)p FD(Z)p FF(\))g(as)g(Galois)g(groups.)1263
11563 y(Shim)-36 b(ura)433 b(sho)-36 b(w)g(ed)433 b(that)g(if)h(w)-36
b(e)434 b(start)g(with)f(the)g(elliptic)i(curv)-36 b(e)20748
14487 y Fz(E)447 b FF(:)1670 b Fz(y)24871 13939 y Fx(2)25692
14487 y FF(+)295 b Fz(y)417 b FF(=)368 b Fz(x)30169 13939
y Fx(3)30990 14487 y Fw(\241)296 b Fz(x)33058 13939 y
Fx(2)1263 17411 y FF(then)403 b(the)h(image)g(of)h Fz(\275)f
FF(is)g(often)g(all)h(of)g(GL)22259 17610 y Fx(2)22785
17411 y FF(\()p FD(Z)p Fz(=n)p FD(Z)p FF(\))f(and)g(the)f(image)i(is)f
(\\most")h(of)f(GL)44912 17610 y Fx(2)45437 17411 y FF(\()p
FD(Z)p Fz(=n)p FD(Z)p FF(\))h(when)1263 19016 y Fz(E)511
b FF(do)36 b(es)434 b(not)f(ha)-36 b(v)g(e)434 b(complex)g(m)-36
b(ultiplication.)1263 23452 y Fs(1.2)2152 b(Mo)60 b(dular)716
b(F)-179 b(orms)717 b(and)f(Galois)h(Represen)-60 b(tations)1263
26760 y Fq(1.2.1)1793 b(Cusp)598 b(F)-149 b(orms)1263
29223 y FF(Let)642 b Fz(S)4602 29422 y Fy(k)5171 29223
y FF(\()p Fz(N)139 b FF(\))642 b(denote)g(the)g(space)h(of)g(cusp)f
(forms)i(of)f(w)-36 b(eigh)g(t)643 b Fz(k)688 b FF(and)642
b(lev)-36 b(el)644 b Fz(N)780 b FF(on)643 b(the)f(congruence)1263
30828 y(subgroup)515 b(\241)7802 31027 y Fx(1)8327 30828
y FF(\()p Fz(N)139 b FF(\))507 b(=)12547 29752 y Fr(\251\241)14152
30404 y Fy(a)402 b(b)14199 31204 y(c)k(d)15719 29752
y Fr(\242)16836 30828 y Fw(2)508 b FF(SL)19766 31027
y Fx(2)20291 30828 y FF(\()p FD(Z)p FF(\))g(:)h Fz(a)f
Fw(\264)h FF(1)590 b(\(mo)36 b(d)443 b Fz(N)139 b FF(\))p
Fz(;)221 b(c)508 b Fw(\264)g FF(0)591 b(\(mo)36 b(d)443
b Fz(N)139 b FF(\))p Fz(;)221 b(d)508 b Fw(\264)g FF(1)591
b(\(mo)36 b(d)442 b Fz(N)139 b FF(\))51933 29752 y Fr(\252)52707
30828 y FF(.)1263 32433 y(Th)-36 b(us)532 b Fz(S)5457
32632 y Fy(k)6026 32433 y FF(\()p Fz(N)139 b FF(\))532
b(is)h(the)f(\257nite)g(dimensional)h(v)-36 b(ector)533
b(space)g(consisting)g(of)h(all)f(holomorphic)g(functions)1263
34038 y Fz(f)142 b FF(\()p Fz(z)59 b FF(\))434 b(on)f
Fw(H)382 b FF(=)368 b Fw(f)p Fz(z)429 b Fw(2)369 b FD(C)g
FF(:)g(Im\()p Fz(z)59 b FF(\))369 b Fz(>)g FF(0)p Fw(g)434
b FF(v)-72 b(anishing)434 b(at)f Fw(1)h FF(and)f(satisfying)12500
37622 y Fz(f)142 b FF(\()13922 36723 y Fz(az)354 b FF(+)294
b Fz(b)p 13922 37316 3502 54 v 13922 38533 a(cz)354 b
FF(+)294 b Fz(d)17555 37622 y FF(\))369 b(=)g(\()p Fz(cz)354
b FF(+)295 b Fz(d)p FF(\))24324 37073 y Fy(k)24892 37622
y Fz(f)142 b FF(\()p Fz(z)59 b FF(\))1301 b(for)434 b(all)33312
36546 y Fr(\241)34142 37197 y Fy(a)402 b(b)34188 37997
y(c)407 b(d)35708 36546 y Fr(\242)36686 37622 y Fw(2)369
b FF(\241)38754 37821 y Fx(1)39279 37622 y FF(\()p Fz(N)139
b FF(\))p Fz(:)1263 41060 y FF(Since,)405 b(in)398 b(particular,)405
b Fz(f)142 b FF(\()p Fz(z)59 b FF(\))369 b(=)g Fz(f)142
b FF(\()p Fz(z)281 b FF(+)222 b(1\))398 b(w)-36 b(e)399
b(can)f(expand)f Fz(f)142 b FF(\()p Fz(z)59 b FF(\))398
b(as)g(a)h Fz(q)48 b FF(-series)397 b(\(this)g(requires)h(rigorous)1263
42666 y(justi\257cation\))22593 46602 y Fz(f)142 b FF(\()p
Fz(z)59 b FF(\))369 b(=)27290 44942 y Fu(1)26801 45340
y Fr(X)26873 48130 y Fy(n)p Fx(=1)28941 46602 y Fz(c)29501
46801 y Fy(n)30127 46602 y Fz(q)30752 46054 y Fy(n)31377
46602 y Fz(:)3214 50615 y FF(A)434 b(famous)g(example)g(is)17853
54478 y(\242)369 b(=)f Fz(q)21948 52817 y Fu(1)21570
53216 y Fr(Y)21532 56005 y Fy(n)p Fx(=1)23305 54478 y
FF(\(1)296 b Fw(\241)f Fz(q)26710 53929 y Fy(n)27336
54478 y FF(\))27842 53929 y Fx(24)29207 54478 y FF(=)31076
52817 y Fu(1)30587 53216 y Fr(X)30660 56005 y Fy(n)p
Fx(=1)32727 54478 y Fz(\277)148 b FF(\()p Fz(n)p FF(\))p
Fz(q)35853 53929 y Fy(n)1263 58491 y Fz(\277)654 b FF(is)505
b(called)h(the)e(Raman)-36 b(ujan)506 b(function.)793
b(One)504 b(no)-36 b(w)505 b(kno)-36 b(ws)506 b(that)f
Fz(\277)653 b FF(is)506 b(m)-36 b(ultiplicativ)g(e)506
b(and)f(satis\257es)1263 60096 y Fz(\277)148 b FF(\()p
Fz(p)3135 59614 y Fy(\272)3711 60096 y FF(\))369 b(=)f
Fz(\277)148 b FF(\()p Fz(p)p FF(\))p Fz(\277)g FF(\()p
Fz(p)10216 59614 y Fy(\272)10791 60096 y FF(\))295 b
Fw(\241)h Fz(p)13574 59614 y Fx(11)14570 60096 y Fz(\277)148
b FF(\()p Fz(p)16442 59614 y Fy(\272)58 b Fu(\241)p Fx(1)18220
60096 y FF(\).)578 b(\242)433 b(is)h(a)g(normalized)g(basis)g(for)g
Fz(S)36203 60295 y Fx(1)36729 60096 y FF(\(1\).)1263
63946 y Fq(1.2.2)1793 b(Hec)-50 b(k)g(e)599 b(Op)50 b(erators)599
b(\(Mordell\))1263 66409 y FF(Mordell)335 b(de\257ned,)354
b(for)336 b Fz(n)369 b Fw(\270)g FF(1,)355 b(op)36 b(erators)336
b Fz(T)22950 66608 y Fy(n)23911 66409 y FF(on)f Fz(S)26419
66608 y Fy(k)26988 66409 y FF(\()p Fz(N)139 b FF(\))334
b(called)i FC(He)-66 b(cke)372 b(op)-66 b(er)g(ators)p
FF(.)545 b(These)335 b(pro)-36 b(v)g(ed)335 b(v)-36 b(ery)1263
68014 y(fruitful.)598 b(The)440 b(set)g(of)h(suc)-36
b(h)439 b(op)36 b(erators)440 b(forms)g(an)g(\\almost")i(comm)-36
b(uting)439 b(family)j(of)f(endomorphisms)1263 69619
y(and)418 b(is)h(hence)f(\\almost")i(sim)-36 b(ultaneously)419
b(diagonalizable.)575 b(The)418 b(precise)h(meaning)f(of)h(\\almost")h
(and)1263 71224 y(the)414 b(actual)h(structure)e(of)i(the)e(Hec)-36
b(k)g(e)415 b(algebra)g FD(Q)p FF([)p Fz(T)27117 71423
y Fx(1)27644 71224 y Fz(;)221 b(:)g(:)g(:)445 b(;)221
b(T)31540 71423 y Fy(n)32167 71224 y Fz(;)g(:)g(:)g(:)j
FF(])415 b(will)g(b)36 b(e)414 b(studied)g(in)g(greater)g(detail)1263
72829 y(in)357 b(the)g(remainder)f(of)i(this)f(course.)553
b(Often)357 b(there)f(will)i(exist)g(a)g(basis)f(of)h(cusp)e(forms)i
Fz(f)511 b FF(=)45925 71833 y Fr(P)47328 72183 y Fu(1)47328
73217 y Fy(n)p Fx(=1)49378 72829 y Fz(c)49938 73028 y
Fy(n)50564 72829 y Fz(q)51189 72347 y Fy(n)52183 72829
y Fw(2)1263 74434 y Fz(S)2063 74633 y Fy(k)2632 74434
y FF(\()p Fz(N)139 b FF(\))374 b(so)i(that)e Fz(f)10138
74633 y Fy(n)11140 74434 y FF(is)h(a)h(sim)-36 b(ultaneous)374
b(eigen)-36 b(v)g(ector)376 b(for)g(all)g(of)g(the)e(Hec)-36
b(k)g(e)376 b(op)36 b(erators)375 b Fz(T)45271 74633
y Fy(n)46273 74434 y FF(and,)387 b(in)375 b(fact,)p eop
%%Page: 3 11
3 10 bop -3718 -6698 a FA(1.2.)1013 b(MODULAR)433 b(F)-36
b(ORMS)433 b(AND)g(GALOIS)g(REPRESENT)-108 b(A)g(TIONS)13850
b FF(3)-3718 -3169 y Fz(T)-2956 -2970 y Fy(n)-2330 -3169
y Fz(f)563 b FF(=)420 b Fz(c)866 -2970 y Fy(n)1492 -3169
y Fz(f)142 b FF(.)669 b(All)464 b(of)g(the)g Fz(c)9808
-2970 y Fy(n)10897 -3169 y FF(will)h(b)36 b(e)464 b(algebraic)h(in)-36
b(tegers)464 b(and)f(the)g(\257eld)g FD(Q)p FF(\()p Fz(c)35415
-2970 y Fx(1)35941 -3169 y Fz(;)221 b(c)37083 -2970 y
Fx(2)37609 -3169 y Fz(;)g(:)g(:)g(:)j FF(\))464 b(will)h(b)36
b(e)463 b(\257nite)-3718 -1564 y(o)-36 b(v)g(er)434 b
FD(Q)p FF(.)-1767 41 y(A)377 b(go)36 b(o)g(d)377 b(claim)h(can)f(b)36
b(e)377 b(made)g(that)f(the)g Fz(c)19077 240 y Fy(n)20080
41 y FF(are)h(often)g(in)-36 b(teresting)377 b(in)-36
b(tegers)377 b(b)36 b(ecause)377 b(they)f(exhibit)-3718
1646 y(remark)-72 b(able)370 b(prop)36 b(erties.)557
b(F)-108 b(or)370 b(example,)384 b Fz(\277)148 b FF(\()p
Fz(n)p FF(\))369 b Fw(\264)21444 650 y Fr(P)22846 2033
y Fy(d)p Fu(j)p Fy(n)24439 1646 y Fz(d)25115 1164 y Fx(11)26702
1646 y FF(\(mo)36 b(d)443 b(691\).)558 b(Ho)-36 b(w)370
b(can)g(w)-36 b(e)371 b(study)f(the)f Fz(c)46847 1845
y Fy(n)47473 1646 y FF(?)-3718 3251 y(Ho)-36 b(w)500
b(can)g(w)-36 b(e)501 b(in)-36 b(terpret)498 b(the)i
Fz(c)12059 3450 y Fy(n)12685 3251 y FF(?)777 b(W)-108
b(e)500 b(can)g(do)g(this)g(b)-36 b(y)499 b(studying)h(the)g
(connection)f(b)36 b(et)-36 b(w)g(een)500 b(Galois)-3718
4856 y(represen)-36 b(tations)543 b(and)h(mo)36 b(dular)544
b(forms.)910 b(In)544 b(1968)i(w)-36 b(ork)545 b(w)-36
b(as)544 b(originally)i(b)36 b(egun)544 b(on)g(this)g(b)-36
b(y)544 b(Serre,)-3718 6461 y(Shim)-36 b(ura,)433 b(Eic)-36
b(hler)433 b(and)g(Deligne.)p eop
%%Page: 4 12
4 11 bop 1263 -6698 a FF(4)31524 b FA(CHAPTER)434 b(1.)1012
b(INTR)-36 b(ODUCTION)p eop
%%Page: 5 13
5 12 bop -3718 5686 a FE(Chapter)1033 b(2)-3718 11221
y(Mo)86 b(dular)1034 b(Represen)-86 b(tations)1033 b(and)h(Curv)-86
b(es)-3718 18084 y Fs(2.1)2151 b(Arithmetic)717 b(of)g(Mo)60
b(dular)716 b(F)-179 b(orms)-3718 21005 y FF(Supp)36
b(ose)630 b Fz(f)846 b FF(=)4785 20008 y Fr(P)6187 20359
y Fu(1)6187 21392 y Fy(n)p Fx(=1)8237 21005 y Fz(a)8920
21204 y Fy(n)9546 21005 y Fz(q)10171 20523 y Fy(n)11428
21005 y FF(is)631 b(a)g(cusp)f(form)h(in)g Fz(S)23164
21204 y Fy(k)23733 21005 y FF(\()p Fz(N)139 b FF(\))630
b(whic)-36 b(h)630 b(is)i(an)e(eigenform)i(for)g(the)e(Hec)-36
b(k)g(e)-3718 22610 y(op)36 b(erators.)918 b(Then)546
b(the)g(Mellin)h(transform)g(asso)36 b(ciates)548 b(to)f
Fz(f)689 b FF(the)546 b Fz(L)p FF(-function)g Fz(L)p
FF(\()p Fz(f)70 b(;)221 b(z)59 b FF(\))562 b(=)42942
21614 y Fr(P)44344 21964 y Fu(1)44344 22997 y Fy(n)p
Fx(=1)46527 22087 y Fy(a)47027 22198 y Ft(n)p 46527 22304
1067 54 v 46553 23068 a Fy(n)47124 22816 y Ft(s)47726
22610 y FF(.)-3718 24215 y(Let)536 b Fz(K)638 b FF(=)544
b FD(Q)p FF(\()p Fz(a)4327 24414 y Fx(1)4852 24215 y
Fz(;)221 b(a)6117 24414 y Fx(2)6644 24215 y Fz(;)g(:)g(:)g(:)i
FF(\),)562 b(then)536 b(one)g(can)g(sho)-36 b(w)536 b(that)g(the)g
Fz(a)27715 24414 y Fy(n)28877 24215 y FF(are)g(algebraic)i(in)-36
b(tegers)536 b(and)f Fz(K)632 b FF(is)536 b(a)-3718 25820
y(n)-36 b(um)g(b)36 b(er)311 b(\257eld.)538 b(When)313
b Fz(k)414 b FF(=)369 b(2)313 b(Shim)-36 b(ura)312 b(asso)36
b(ciates)315 b(to)e Fz(f)455 b FF(an)314 b(ab)36 b(elian)313
b(v)-72 b(ariet)-36 b(y)314 b Fz(A)35995 26019 y Fy(f)36914
25820 y FF(o)-36 b(v)g(er)313 b FD(Q)h FF(of)g(dimension)-3718
27425 y([)p Fz(K)464 b FF(:)369 b FD(Q)p FF(])435 b(on)e(whic)-36
b(h)433 b Fz(K)529 b FF(acts)434 b(\(see)f(theorem)g(7.14)i(of)g([31)q
(]\).)-3718 29454 y FC(Example)465 b(2.1.1)f(\(Mo)-66
b(dular)466 b(El)66 b(liptic)464 b(Curves\).)651 b FF(When)617
b(all)h(of)g(the)f(co)36 b(e\261cien)-36 b(ts)618 b Fz(a)38022
29653 y Fy(n)39265 29454 y FF(of)h(the)d(mo)36 b(dular)-3718
31059 y(form)621 b Fz(f)763 b FF(lie)622 b(in)e FD(Q)i
FF(then)e([)p Fz(K)783 b FF(:)688 b FD(Q)p FF(])h(=)e(1)621
b(so)h Fz(A)20666 31258 y Fy(f)21892 31059 y FF(is)f(a)g(one)g
(dimensional)g(ab)36 b(elian)622 b(v)-72 b(ariet)-36
b(y)-108 b(.)1141 b(A)620 b(one)-3718 32665 y(dimensional)381
b(ab)36 b(elian)381 b(v)-72 b(ariet)-36 b(y)381 b(of)h(nonzero)e(gen)
-36 b(us)380 b(is)h(an)f(elliptic)h(curv)-36 b(e.)561
b(An)380 b(elliptic)h(curv)-36 b(e)381 b(isogenous)-3718
34270 y(to)434 b(one)f(arising)h(via)h(this)e(construction)g(is)h
(called)g FC(mo)-66 b(dular)p FF(.)-3718 36723 y FD(De\257nition)499
b(2.1.2.)652 b FF(Elliptic)417 b(curv)-36 b(es)416 b
Fz(E)16874 36922 y Fx(1)17816 36723 y FF(and)f Fz(E)21290
36922 y Fx(2)22232 36723 y FF(are)i FC(iso)-66 b(genous)416
b FF(if)g(there)g(is)h(a)f(morphism)g Fz(E)44007 36922
y Fx(1)44901 36723 y Fw(!)370 b Fz(E)47562 36922 y Fx(2)-3718
38328 y FF(of)434 b(algebraic)h(groups,)e(whic)-36 b(h)434
b(has)f(a)h(\257nite)f(k)-36 b(ernel.)-1767 40782 y(The)433
b(follo)-36 b(wing)436 b(conjecture)e(motiv)-72 b(ates)434
b(m)-36 b(uc)g(h)432 b(of)j(the)e(theory)-108 b(.)-3718
43235 y FD(Conjecture)499 b(2.1.3.)651 b FF(Ev)-36 b(ery)544
b(elliptic)g(curv)-36 b(e)543 b(o)-36 b(v)g(er)544 b
FD(Q)g FF(is)f(mo)36 b(dular,)571 b(that)543 b(is,)572
b(isogenous)543 b(to)h(a)g(curv)-36 b(e)-3718 44840 y(constructed)432
b(in)i(the)f(ab)36 b(o)-36 b(v)g(e)434 b(w)-36 b(a)g(y)-108
b(.)-1767 47294 y(F)g(or)622 b Fz(k)737 b Fw(\270)691
b FF(2)624 b(Serre)e(and)h(Deligne)h(found)e(a)h(w)-36
b(a)g(y)624 b(to)g(asso)36 b(ciate)624 b(to)f Fz(f)765
b FF(a)623 b(family)i(of)f Fz(`)p FF(-adic)e(rep-)-3718
48899 y(resen)-36 b(tations.)1080 b(Let)601 b Fz(`)g
FF(b)36 b(e)600 b(a)i(prime)e(n)-36 b(um)g(b)36 b(er)600
b(and)g Fz(K)696 b FF(b)36 b(e)601 b(as)g(ab)36 b(o)-36
b(v)g(e,)644 b(then)600 b(it)h(is)h(w)-36 b(ell)602 b(kno)-36
b(wn)601 b(that)-3718 50504 y Fz(K)390 b Fw(\255)-1188
50703 y Fv(Q)-22 50504 y FD(Q)1100 50703 y Fy(`)1909
50135 y Fw(\273)1919 50559 y FF(=)3311 49508 y Fr(Q)4565
50891 y Fy(\270)p Fu(j)p Fy(`)6037 50504 y Fz(K)7144
50703 y Fy(\270)7747 50504 y FF(.)579 b(One)433 b(can)g(asso)36
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73953 y Fz(")21859 77755 y FF(5)p eop
%%Page: 6 14
6 13 bop 1263 -6698 a FF(6)12705 b FA(CHAPTER)435 b(2.)1012
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eop
%%Page: 7 15
7 14 bop -3718 -6698 a FA(2.4.)1013 b(CONJECTURES)434
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y Fy(\270;f)13062 18411 y FF(is)h(irreducible)f(\(and)g(hence)g
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b(ector)474 b(space)f Fz(W)654 b FF(\(or)473 b(a)-3718
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%%Page: 8 16
8 15 bop 1263 -6698 a FF(8)12705 b FA(CHAPTER)435 b(2.)1012
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9990 y FF(=)694 b Fz(')42352 10189 y Fy(`)43416 9990
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b(dular.)p eop
%%Page: 9 17
9 16 bop -3718 5726 a FE(Chapter)1033 b(3)-3718 11300
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%%Page: 10 18
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%%Page: 11 19
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b FF(+)295 b Fz(b)p 19998 36760 3551 54 v 19998 37977
a(c\277)443 b FF(+)295 b Fz(d)23682 37066 y FF(\)\()p
Fz(c\277)442 b FF(+)295 b Fz(d)p FF(\))28750 36517 y
Fu(\241)p Fy(k)-3718 40030 y FF(for)434 b(all)75 38954
y Fr(\241)906 39605 y Fy(a)401 b(b)952 40405 y(c)407
b(d)2472 38954 y Fr(\242)3450 40030 y Fw(2)369 b FF(SL)6241
40229 y Fx(2)6766 40030 y FF(\()p FD(Z)p FF(\).)578 b(Using)434
b(a)g(F)-108 b(ourier)433 b(expansion)g(w)-36 b(e)434
b(write)16595 44091 y Fz(f)142 b FF(\()p Fz(\277)148
b FF(\))369 b(=)21341 42431 y Fu(1)20852 42829 y Fr(X)20925
45619 y Fy(n)p Fx(=0)22992 44091 y Fz(a)23675 44290 y
Fy(n)24301 44091 y Fz(e)24904 43542 y Fx(2)p Fy(\274)32
b(i\277)102 b(n)27413 44091 y Fz(;)-3718 48095 y FF(and)392
b(sa)-36 b(y)393 b Fz(f)535 b FF(is)393 b(a)f(cusp)g(form)h(if)h
Fz(a)12255 48294 y Fx(0)13149 48095 y FF(=)369 b(0.)565
b(There)392 b(is)h(a)g(corresp)36 b(ondence)392 b(b)36
b(et)-36 b(w)g(een)392 b(mo)36 b(dular)392 b(forms)h
Fz(f)535 b FF(and)-3718 49700 y(lattice)434 b(functions)f
Fz(F)615 b FF(satisfying)435 b Fz(F)181 b FF(\()p Fz(\270L)p
FF(\))368 b(=)g Fz(\270)19301 49218 y Fu(\241)p Fy(k)20602
49700 y Fz(F)181 b FF(\()p Fz(L)p FF(\))432 b(giv)-36
b(en)435 b(b)-36 b(y)433 b Fz(F)181 b FF(\()p FD(Z)p
Fz(\277)443 b FF(+)295 b FD(Z)p FF(\))369 b(=)g Fz(f)142
b FF(\()p Fz(\277)148 b FF(\).)-3718 54135 y Fs(3.5)2151
b(Explicit)717 b(Description)f(of)h(Sublattices)-3718
57055 y FF(The)433 b Fz(n)p FF(th)h(Hec)-36 b(k)g(e)434
b(op)36 b(erator)434 b Fz(T)11152 57254 y Fy(n)12212
57055 y FF(of)g(w)-36 b(eigh)g(t)434 b Fz(k)478 b FF(is)434
b(de\257ned)e(b)-36 b(y)15457 60130 y Fz(T)16219 60329
y Fy(n)16845 60130 y FF(\()p Fz(L)p FF(\))369 b(=)f Fz(n)21267
59581 y Fy(k)24 b Fu(\241)p Fx(1)24238 58868 y Fr(X)24042
61763 y Fy(L)24682 61451 y Fo(0)24981 61763 y Fu(\275)p
Fy(L)23260 62620 y Fx(\()p Fy(L)p Fx(:)p Fy(L)25167 62308
y Fo(0)25466 62620 y Fx(\)=)p Fy(n)27356 60130 y Fz(L)28241
59581 y Fu(0)28551 60130 y Fz(:)-3718 65392 y FF(What)433
b(are)h(the)f Fz(L)5223 64910 y Fu(0)5967 65392 y FF(explicitly?)580
b(Note)434 b(that)f Fz(L=L)20743 64910 y Fu(0)21488 65392
y FF(is)g(a)h(group)g(of)g(order)f Fz(n)h FF(and)14496
68308 y Fz(L)15381 67760 y Fu(0)15692 68308 y Fz(=nL)370
b Fw(\275)f Fz(L=nL)g FF(=)g(\()p FD(Z)p Fz(=n)p FD(Z)p
FF(\))28985 67760 y Fx(2)29512 68308 y Fz(:)-3718 71224
y FF(W)-108 b(rite)335 b Fz(L)369 b FF(=)f FD(Z)p Fz(!)4153
71423 y Fx(1)4773 71224 y FF(+)93 b FD(Z)p Fz(!)7601
71423 y Fx(2)8127 71224 y FF(,)354 b(let)335 b Fz(Y)11377
71423 y Fx(2)12238 71224 y FF(b)36 b(e)334 b(the)h(cyclic)h(subgroup)d
(of)j Fz(L=L)28876 70742 y Fu(0)29521 71224 y FF(generated)f(b)-36
b(y)335 b Fz(!)37866 71423 y Fx(2)38726 71224 y FF(and)f(let)h
Fz(d)369 b FF(=)g(#)p Fz(Y)47201 71423 y Fx(2)47726 71224
y FF(.)-3718 72829 y(Let)430 b Fz(Y)-636 73028 y Fx(1)258
72829 y FF(=)369 b(\()p Fz(L=L)4565 72347 y Fu(0)4876
72829 y FF(\))p Fz(=)72 b(Y)6859 73028 y Fx(2)7384 72829
y FF(,)432 b(then)d Fz(Y)11891 73028 y Fx(1)12847 72829
y FF(is)i(generated)f(b)-36 b(y)431 b(the)f(image)h(of)g
Fz(!)30157 73028 y Fx(1)31114 72829 y FF(so)f(it)h(is)g(a)g(cyclic)g
(group)g(of)g(order)-3718 74434 y Fz(a)444 b FF(=)h Fz(n=d)p
FF(.)712 b(W)-108 b(e)478 b(w)-36 b(an)g(t)478 b(to)g(exhibit)h(a)f
(basis)g(of)h Fz(L)20377 73952 y Fu(0)20688 74434 y FF(.)712
b(Let)477 b Fz(!)24993 73952 y Fu(0)24945 74763 y Fx(2)25916
74434 y FF(=)444 b Fz(d!)28858 74633 y Fx(2)29828 74434
y Fw(2)h Fz(L)32044 73952 y Fu(0)32832 74434 y FF(and)478
b(use)g(the)f(fact)i(that)f Fz(Y)46210 74633 y Fx(1)47213
74434 y FF(is)p eop
%%Page: 12 20
12 19 bop 1263 -6698 a FF(12)29447 b FA(CHAPTER)434 b(3.)1013
b(MODULAR)433 b(F)-36 b(ORMS)1263 -3169 y FF(generated)501
b(b)-36 b(y)500 b Fz(!)9939 -2970 y Fx(1)10966 -3169
y FF(to)h(write)g Fz(a!)17507 -2970 y Fx(1)18516 -3169
y FF(=)483 b Fz(!)20869 -3651 y Fu(0)20821 -2841 y Fx(1)21687
-3169 y FF(+)341 b Fz(b!)24403 -2970 y Fx(2)25429 -3169
y FF(for)502 b(some)f(in)-36 b(teger)501 b Fz(b)f FF(and)h(some)g
Fz(!)43011 -3651 y Fu(0)42963 -2841 y Fx(1)43972 -3169
y Fw(2)483 b Fz(L)46226 -3651 y Fu(0)46536 -3169 y FF(.)780
b(Since)501 b Fz(b)f FF(is)1263 -1564 y(only)434 b(w)-36
b(ell-de\257ned)433 b(mo)36 b(dulo)434 b Fz(d)f FF(w)-36
b(e)434 b(ma)-36 b(y)434 b(assume)g(0)369 b Fw(\267)g
Fz(b)g Fw(\267)g Fz(d)295 b Fw(\241)h FF(1.)578 b(Th)-36
b(us)21343 304 y Fr(\263)22136 965 y Fz(!)22994 483 y
Fu(0)22946 1294 y Fx(1)22136 2570 y Fz(!)22994 2088 y
Fu(0)22946 2899 y Fx(2)23472 304 y Fr(\264)24634 1779
y FF(=)26015 304 y Fr(\263)26808 965 y Fz(a)1168 b(b)26824
2570 y FF(0)1124 b Fz(d)29274 304 y Fr(\264\263)30860
965 y Fz(!)31670 1164 y Fx(1)30860 2570 y Fz(!)31670
2769 y Fx(2)32196 304 y Fr(\264)1263 5122 y FF(and)433
b(the)g(c)-36 b(hange)434 b(of)g(basis)g(matrix)g(has)g(determinen)-36
b(t)432 b Fz(ad)368 b FF(=)h Fz(n)p FF(.)579 b(Since)17380
7764 y FD(Z)p Fz(!)19151 7215 y Fu(0)19103 8092 y Fx(1)19924
7764 y FF(+)295 b FD(Z)p Fz(!)23002 7215 y Fu(0)22954
8092 y Fx(2)23849 7764 y Fw(\275)369 b Fz(L)26136 7215
y Fu(0)26815 7764 y Fw(\275)h Fz(L)f FF(=)f FD(Z)p Fz(!)32575
7963 y Fx(1)33396 7764 y FF(+)295 b FD(Z)p Fz(!)36426
7963 y Fx(2)1263 10406 y FF(and)531 b(\()p Fz(L)k FF(:)h
FD(Z)p Fz(!)8484 9924 y Fu(0)8436 10734 y Fx(1)9324 10406
y FF(+)361 b FD(Z)p Fz(!)12468 9924 y Fu(0)12420 10734
y Fx(2)12946 10406 y FF(\))536 b(=)f Fz(n)c FF(\(since)h(the)f(c)-36
b(hange)531 b(of)h(basis)g(matrix)g(has)f(determinen)-36
b(t)530 b Fz(n)p FF(\))h(and)g(\()p Fz(L)k FF(:)1263
12011 y Fz(L)2148 11529 y Fu(0)2459 12011 y FF(\))368
b(=)h Fz(n)434 b FF(w)-36 b(e)434 b(see)g(that)f Fz(L)13645
11529 y Fu(0)14324 12011 y FF(=)369 b FD(Z)p Fz(!)17476
11529 y Fu(0)17428 12339 y Fx(1)18249 12011 y FF(+)295
b FD(Z)p Fz(!)21327 11529 y Fu(0)21279 12339 y Fx(2)21805
12011 y FF(.)3214 13616 y(Th)-36 b(us)522 b(there)f(is)i(a)f
(one-to-one)g(corresp)36 b(ondence)521 b(b)36 b(et)-36
b(w)g(een)522 b(sublattices)g Fz(L)40721 13134 y Fu(0)41551
13616 y Fw(\275)e Fz(L)j FF(of)f(index)h Fz(n)f FF(and)1263
15221 y(matrices)6536 14145 y Fr(\241)7366 14796 y Fy(a)402
b(b)7381 15596 y Fx(0)376 b Fy(d)8933 14145 y Fr(\242)10038
15221 y FF(with)496 b Fz(ad)476 b FF(=)f Fz(n)497 b FF(and)f(0)476
b Fw(\267)g Fz(b)g Fw(\267)g Fz(d)338 b Fw(\241)g FF(1.)767
b(In)496 b(particular,)513 b(when)496 b Fz(n)476 b FF(=)g
Fz(p)496 b FF(is)g(prime)g(there)1263 16826 y Fz(p)361
b FF(+)g(1)531 b(of)h(these.)870 b(In)530 b(general,)556
b(the)530 b(n)-36 b(um)g(b)36 b(er)530 b(of)h(suc)-36
b(h)530 b(sublattices)h(equals)g(the)g(sum)f(of)i(the)e(p)36
b(ositiv)-36 b(e)1263 18431 y(divisors)435 b(of)f Fz(n)p
FF(.)1263 22825 y Fs(3.6)2152 b(Action)717 b(of)f(Hec)-60
b(k)g(e)718 b(Op)60 b(erators)716 b(on)h(Mo)60 b(dular)716
b(F)-179 b(orms)1263 25746 y FF(No)-36 b(w)475 b(assume)f
Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))439 b(=)13198
24750 y Fr(P)14600 25100 y Fu(1)14600 26133 y Fy(m)p
Fx(=0)16911 25746 y Fz(c)17471 25945 y Fy(m)18359 25746
y Fz(q)18984 25264 y Fy(m)20345 25746 y FF(is)475 b(a)f(mo)36
b(dular)475 b(form)f(with)h(corresp)36 b(onding)474 b(lattice)h
(function)f Fz(F)181 b FF(.)1263 27351 y(Ho)-36 b(w)434
b(can)g(w)-36 b(e)434 b(describ)36 b(e)433 b(the)g(action)h(of)g(the)f
(Hec)-36 b(k)g(e)434 b(op)36 b(erator)434 b Fz(T)33175
27550 y Fy(n)34235 27351 y FF(on)g Fz(f)142 b FF(\()p
Fz(\277)148 b FF(\)?)578 b(W)-108 b(e)433 b(ha)-36 b(v)g(e)12307
30225 y Fz(T)13069 30424 y Fy(n)13696 30225 y Fz(F)181
b FF(\()p FD(Z)p Fz(\277)443 b FF(+)294 b FD(Z)p FF(\))370
b(=)e Fz(n)22397 29677 y Fy(k)24 b Fu(\241)p Fx(1)24841
28963 y Fr(X)24846 31792 y Fy(a;b;d)24698 32699 y(ab)p
Fx(=)p Fy(n)24390 33451 y Fx(0)p Fu(\267)p Fy(b<d)27432
30225 y Fz(F)181 b FF(\(\()p Fz(a\277)443 b FF(+)295
b Fz(b)p FF(\))p FD(Z)g FF(+)g Fz(d)p FD(Z)p FF(\))20241
35997 y(=)368 b Fz(n)22397 35449 y Fy(k)24 b Fu(\241)p
Fx(1)24390 34736 y Fr(X)26530 35997 y Fz(d)27206 35449
y Fu(\241)p Fy(k)28507 35997 y Fz(F)181 b FF(\()30169
35099 y Fz(a\277)442 b FF(+)295 b Fz(b)p 30169 35692
3551 54 v 31606 36909 a(d)33852 35997 y FD(Z)g FF(+)g
FD(Z)p FF(\))20241 39173 y(=)368 b Fz(n)22397 38624 y
Fy(k)24 b Fu(\241)p Fx(1)24390 37911 y Fr(X)26530 39173
y Fz(d)27206 38624 y Fu(\241)p Fy(k)28507 39173 y Fz(f)142
b FF(\()29929 38274 y Fz(a\277)443 b FF(+)294 b Fz(b)p
29929 38867 V 31366 40084 a(d)33612 39173 y FF(\))20241
41921 y(=)368 b Fz(n)22397 41373 y Fy(k)24 b Fu(\241)p
Fx(1)24932 40659 y Fr(X)24390 43488 y Fy(a;d;b;m)27614
41921 y Fz(d)28290 41373 y Fu(\241)p Fy(k)29591 41921
y Fz(c)30151 42120 y Fy(m)31038 41921 y Fz(e)31641 41373
y Fx(2)p Fy(\274)32 b(i)p Fx(\()33502 40999 y Ft(a\277)77
b Fn(+)p Ft(b)p 33503 41171 1897 40 v 34238 41720 a(d)35532
41373 y Fx(\))p Fy(m)20241 46644 y FF(=)368 b Fz(n)22397
46096 y Fy(k)24 b Fu(\241)p Fx(1)24600 45382 y Fr(X)24390
48211 y Fy(a;d;m)26950 46644 y Fz(d)27626 46096 y Fx(1)p
Fu(\241)p Fy(k)29397 46644 y Fz(c)29957 46843 y Fy(m)30845
46644 y Fz(e)31581 45734 y Fn(2)p Ft(\274)g(iam\277)p
31581 45895 2847 40 v 32790 46443 a(d)34761 45746 y FF(1)p
34748 46339 676 54 v 34748 47556 a Fz(d)35894 44984 y
Fy(d)p Fu(\241)p Fx(1)35778 45382 y Fr(X)35935 48211
y Fy(b)p Fx(=0)37697 46644 y FF(\()p Fz(e)38939 45734
y Fn(2)p Ft(\274)g(im)p 38938 45895 1935 40 v 39692 46443
a(d)41061 46644 y FF(\))41567 46096 y Fy(b)20241 50354
y FF(=)368 b Fz(n)22397 49805 y Fy(k)24 b Fu(\241)p Fx(1)24597
49092 y Fr(X)24413 51921 y Fy(ad)p Fx(=)p Fy(n)24390
52722 y(m)25222 52409 y Fo(0)25521 52722 y Fu(\270)p
Fx(0)26944 50354 y Fz(d)27620 49805 y Fx(1)p Fu(\241)p
Fy(k)29391 50354 y Fz(c)29951 50553 y Fy(dm)31267 50301
y Fo(0)31621 50354 y Fz(e)32224 49805 y Fx(2)p Fy(\274)32
b(iam)34918 49493 y Fo(0)35217 49805 y Fy(\277)20241
54842 y FF(=)21828 53580 y Fr(X)21644 56409 y Fy(ad)p
Fx(=)p Fy(n)21621 57210 y(m)22453 56897 y Fo(0)22752
57210 y Fu(\270)p Fx(0)24175 54842 y Fz(a)24858 54293
y Fy(k)24 b Fu(\241)p Fx(1)26629 54842 y Fz(c)27189 55041
y Fy(dm)28505 54789 y Fo(0)28859 54842 y Fz(q)29484 54293
y Fy(am)30816 53981 y Fo(0)31169 54842 y Fz(:)1263 59613
y FF(In)500 b(the)g(second)g(to)h(the)e(last)i(expression)g(w)-36
b(e)501 b(let)f Fz(m)482 b FF(=)h Fz(dm)31209 59131 y
Fu(0)31519 59613 y FF(,)517 b Fz(m)33535 59131 y Fu(0)34328
59613 y Fw(\270)483 b FF(0,)518 b(then)499 b(used)h(the)f(fact)i(that)f
(the)1263 61219 y(sum)4156 60696 y Fx(1)p 4149 60913
485 54 v 4149 61677 a Fy(d)4987 60222 y Fr(P)6390 60573
y Fy(d)p Fu(\241)p Fx(1)6390 61606 y Fy(b)p Fx(=0)8131
61219 y FF(\()p Fz(e)9373 60366 y Fn(2)p Ft(\274)24 b(im)p
9373 60527 1935 40 v 10127 61076 a(d)11496 61219 y FF(\))12002
60737 y Fy(b)12893 61219 y FF(is)434 b(only)g(nonzero)g(if)g
Fz(d)p Fw(j)p Fz(m)p FF(.)3214 62824 y(Th)-36 b(us)19935
64429 y Fz(T)20697 64628 y Fy(n)21323 64429 y Fz(f)142
b FF(\()p Fz(q)48 b FF(\))368 b(=)25676 63167 y Fr(X)25492
65996 y Fy(ad)p Fx(=)p Fy(n)25618 66748 y(m)p Fu(\270)p
Fx(0)28000 64429 y Fz(a)28683 63880 y Fy(k)24 b Fu(\241)p
Fx(1)30454 64429 y Fz(c)31014 64628 y Fy(dm)32385 64429
y Fz(q)33010 63880 y Fy(am)1263 68658 y FF(and)433 b(if)i
Fz(\271)368 b Fw(\270)h FF(0)434 b(then)f(the)g(co)36
b(e\261cien)-36 b(t)434 b(of)g Fz(q)22001 68176 y Fy(\271)23055
68658 y FF(is)23743 70197 y Fr(X)24037 73075 y Fy(a)p
Fu(j)p Fy(n)24039 74080 y(a)p Fu(j)p Fy(\271)25883 71459
y Fz(a)26566 70910 y Fy(k)24 b Fu(\241)p Fx(1)28337 71459
y Fz(c)29030 71287 y Ft(n\271)p 29030 71496 1010 40 v
29080 72154 a(a)29528 71963 y Fn(2)30227 71459 y Fz(:)p
eop
%%Page: 13 21
13 20 bop -3718 -6698 a FA(3.6.)1013 b(A)-36 b(CTION)434
b(OF)f(HECKE)h(OPERA)-108 b(TORS)433 b(ON)g(MODULAR)g(F)-36
b(ORMS)11203 b FF(13)-3718 -3169 y FC(R)-66 b(emark)463
b(3.6.1.)650 b FF(When)385 b Fz(k)414 b Fw(\270)369 b
FF(1)386 b(the)f(co)36 b(e\261cien)-36 b(ts)386 b(of)g
Fz(q)22995 -3651 y Fy(\271)24002 -3169 y FF(for)g(all)h
Fz(\271)e FF(b)36 b(elong)386 b(to)f(the)g FD(Z)p FF(-mo)36
b(dule)386 b(generated)-3718 -1564 y(b)-36 b(y)433 b(the)g
Fz(c)888 -1365 y Fy(m)1776 -1564 y FF(.)-3718 594 y FC(R)-66
b(emark)463 b(3.6.2.)650 b FF(Setting)433 b Fz(\271)368
b FF(=)h(0)434 b(giv)-36 b(es)435 b(the)e(constan)-36
b(t)433 b(co)36 b(e\261cien)-36 b(t)433 b(of)i Fz(T)32019
793 y Fy(n)32645 594 y Fz(f)576 b FF(whic)-36 b(h)433
b(is)15597 2424 y Fr(X)15891 5303 y Fy(a)p Fu(j)p Fy(n)17737
3686 y Fz(a)18420 3138 y Fy(k)24 b Fu(\241)p Fx(1)20191
3686 y Fz(c)20751 3885 y Fx(0)21646 3686 y FF(=)369 b
Fz(\276)23766 3885 y Fy(k)24 b Fu(\241)p Fx(1)25537 3686
y FF(\()p Fz(n)p FF(\))p Fz(c)27885 3885 y Fx(0)28411
3686 y Fz(:)-3718 8092 y FF(Th)-36 b(us)469 b(if)h Fz(f)612
b FF(is)470 b(a)g(cusp)f(form)h(so)g(is)g Fz(T)14410
8291 y Fy(n)15037 8092 y Fz(f)142 b FF(.)687 b(\()p Fz(T)18136
8291 y Fy(n)18762 8092 y Fz(f)612 b FF(is)470 b(holomorphic)g(since)f
(its)h(original)h(de\257nition)e(is)h(as)g(a)-3718 9697
y(\257nite)433 b(sum)g(of)h(holomorphic)g(functions.\))-3718
11856 y FC(R)-66 b(emark)463 b(3.6.3.)650 b FF(Setting)449
b Fz(\271)397 b FF(=)f(1)450 b(sho)-36 b(ws)450 b(that)g(the)f(co)36
b(e\261cien)-36 b(t)450 b(of)h Fz(q)497 b FF(in)450 b
Fz(T)32918 12055 y Fy(n)33545 11856 y Fz(f)591 b FF(is)36102
10859 y Fr(P)37504 12243 y Fy(a)p Fu(j)p Fx(1)39012 11856
y FF(1)39662 11374 y Fy(k)24 b Fu(\241)p Fx(1)41434 11856
y Fz(c)41994 12055 y Fy(n)43017 11856 y FF(=)396 b Fz(c)44985
12055 y Fy(n)45611 11856 y FF(.)627 b(As)-3718 13461
y(an)433 b(immediate)h(corollary)h(w)-36 b(e)434 b(ha)-36
b(v)g(e)434 b(the)f(follo)-36 b(wing)436 b(imp)36 b(ortan)-36
b(t)433 b(result.)-3718 16173 y FD(Corollary)500 b(3.6.4.)651
b FC(Supp)-66 b(ose)500 b Fz(f)642 b FC(is)499 b(a)h(cusp)g(form)f(for)
h(which)g Fz(T)28640 16372 y Fy(n)29266 16173 y Fz(f)642
b FC(has)500 b(0)g(as)g(c)-66 b(o)g(e\261cient)497 b(of)i
Fz(q)547 b FC(for)500 b(al)66 b(l)-3718 17778 y Fz(n)369
b Fw(\270)g FF(1)p FC(,)465 b(then)f Fz(f)511 b FF(=)369
b(0)p FC(.)-3718 20490 y(R)-66 b(emark)463 b(3.6.5.)650
b FF(When)489 b Fz(n)466 b FF(=)f Fz(p)490 b FF(is)g(prime)g(w)-36
b(e)490 b(get)g(an)g(in)-36 b(teresting)490 b(form)-36
b(ula)491 b(for)f(the)g(action)g(of)h Fz(T)45695 20689
y Fy(p)46715 20490 y FF(on)-3718 22095 y(the)433 b Fz(q)48
b FF(-expansion)433 b(of)h Fz(f)142 b FF(.)578 b(One)433
b(has)15157 25187 y Fz(T)15919 25386 y Fy(p)16448 25187
y Fz(f)511 b FF(=)18981 23925 y Fr(X)19056 26731 y Fy(\271)p
Fu(\270)p Fx(0)21121 23925 y Fr(X)21414 26803 y Fy(a)p
Fu(j)p Fy(n)21417 27809 y(a)p Fu(j)p Fy(\271)23261 25187
y Fz(a)23944 24639 y Fy(k)24 b Fu(\241)p Fx(1)25715 25187
y Fz(c)26408 25015 y Ft(n\271)p 26408 25224 1010 40 v
26458 25883 a(a)26906 25692 y Fn(2)27605 25187 y Fz(q)28230
24639 y Fy(\271)28851 25187 y Fz(:)-3718 30524 y FF(Since)454
b Fz(n)404 b FF(=)g Fz(p)454 b FF(is)g(prime)g(either)g
Fz(a)404 b FF(=)g(1)454 b(or)g Fz(a)404 b FF(=)g Fz(p)p
FF(.)640 b(When)454 b Fz(a)404 b FF(=)f(1,)460 b Fz(c)29863
30723 y Fy(p\271)31412 30524 y FF(o)36 b(ccurs)455 b(in)f(the)f(co)36
b(e\261cien)-36 b(t)455 b(of)g Fz(q)47467 30042 y Fy(\271)-3718
32129 y FF(and)433 b(when)g Fz(a)369 b FF(=)f Fz(p)p
FF(,)434 b(w)-36 b(e)434 b(can)g(write)f Fz(\271)369
b FF(=)g Fz(p\270)433 b FF(and)g(w)-36 b(e)434 b(get)f(terms)h
Fz(p)28973 31647 y Fy(k)24 b Fu(\241)p Fx(1)30744 32129
y Fz(c)31304 32328 y Fy(\270)32341 32129 y FF(in)434
b Fz(q)34484 31647 y Fy(\270p)35561 32129 y FF(.)578
b(Th)-36 b(us)12896 35221 y Fz(T)13658 35420 y Fy(n)14285
35221 y Fz(f)511 b FF(=)16817 33960 y Fr(X)16893 36766
y Fy(\271)p Fu(\270)p Fx(0)18958 35221 y Fz(c)19518 35420
y Fy(p\271)20612 35221 y Fz(q)21237 34673 y Fy(\271)22153
35221 y FF(+)295 b Fz(p)24113 34673 y Fy(k)24 b Fu(\241)p
Fx(1)26105 33960 y Fr(X)26189 36788 y Fy(\270)p Fu(\270)p
Fx(0)28246 35221 y Fz(c)28806 35420 y Fy(\270)29410 35221
y Fz(q)30035 34673 y Fy(p\270)31112 35221 y Fz(:)p eop
%%Page: 14 22
14 21 bop 1263 -6698 a FF(14)29447 b FA(CHAPTER)434 b(3.)1013
b(MODULAR)433 b(F)-36 b(ORMS)p eop
%%Page: 15 23
15 22 bop -3718 5686 a FE(Chapter)1033 b(4)-3718 11221
y(Em)-86 b(b)86 b(edding)1033 b(Hec)-86 b(k)g(e)1032
b(Op)86 b(erators)1034 b(in)f(the)-3718 14542 y(Dual)-3718
21405 y Fs(4.1)2151 b(The)717 b(Space)f(of)h(Mo)60 b(dular)716
b(F)-179 b(orms)-3718 24326 y FF(Let)433 b(\241)369 b(=)f(\241)1987
24525 y Fx(1)2513 24326 y FF(\(1\))h(=)f(SL)7460 24525
y Fx(2)7985 24326 y FF(\()p FD(Z)p FF(\))434 b(and)f(for)h
Fz(k)414 b Fw(\270)369 b FF(0)434 b(let)8244 27946 y
Fz(M)9502 28145 y Fy(k)10440 27946 y FF(=)368 b Fw(f)p
Fz(f)511 b FF(=)15506 26285 y Fu(1)15017 26684 y Fr(X)15090
29474 y Fy(n)p Fx(=0)17157 27946 y Fz(a)17840 28145 y
Fy(n)18466 27946 y Fz(q)19091 27397 y Fy(n)20086 27946
y FF(:)369 b Fz(f)575 b FF(is)434 b(a)g(mo)36 b(dular)434
b(form)g(for)g(\241)p Fw(g)10440 32224 y(\275)369 b Fz(S)12642
32423 y Fy(k)13580 32224 y FF(=)f Fw(f)p Fz(f)511 b FF(=)18646
30563 y Fu(1)18157 30962 y Fr(X)18230 33751 y Fy(n)p
Fx(=1)20297 32224 y Fz(a)20980 32423 y Fy(n)21606 32224
y Fz(q)22231 31675 y Fy(n)22857 32224 y Fw(g)-3718 35920
y FF(These)497 b(are)h(\257nite)e(dimensional)i FD(C)p
FF(-v)-36 b(ector)497 b(spaces)h(whose)f(dimensions)g(are)h(easily)h
(computed.)768 b(F)-108 b(ur-)-3718 37525 y(thermore,)489
b(they)478 b(are)g(generated)g(b)-36 b(y)477 b(familiar)j(elemen)-36
b(ts)478 b(\(see)g(Serre)f([24)q(])h(or)h(Lang)f([10)q(].\))711
b(The)478 b(main)-3718 39130 y(to)36 b(ol)434 b(is)g(the)f(form)-36
b(ula)15978 39800 y Fr(X)14734 42678 y Fy(p)p Fu(2)p
Fy(D)26 b Fu([f1g)20304 40164 y FF(1)p 19496 40757 2268
54 v 19496 41973 a Fz(e)p FF(\()p Fz(p)p FF(\))22117
41062 y(ord)23996 41261 y Fy(p)24525 41062 y FF(\()p
Fz(f)142 b FF(\))369 b(=)28492 40164 y Fz(k)p 28202 40757
1301 54 v 28202 41973 a FF(12)-3718 44646 y(where)433
b Fz(D)470 b FF(is)434 b(the)f(fundamen)-36 b(tal)433
b(domain)g(for)i(\241)e(and)15867 48981 y Fz(e)p FF(\()p
Fz(p)p FF(\))368 b(=)19884 45859 y Fr(8)19884 47055 y(>)19884
47453 y(<)19884 49844 y(>)19884 50243 y(:)21064 47108
y FF(1)1301 b(otherwise)21064 49034 y(2)g(if)435 b Fz(p)368
b FF(=)h Fz(i)21064 50960 y FF(3)1301 b(if)435 b Fz(p)368
b FF(=)h Fz(\275)-3718 53648 y FF(One)525 b(can)g(alternativ)-36
b(ely)527 b(de\257ne)e Fz(e)p FF(\()p Fz(p)p FF(\))g(as)h(follo)-36
b(ws.)856 b(If)526 b Fz(p)g FF(=)f Fz(\277)674 b FF(and)525
b Fz(E)603 b FF(=)525 b FD(C)p Fz(=)p FF(\()p FD(Z)p
Fz(\277)507 b FF(+)358 b FD(Z)p FF(\))525 b(then)g Fz(e)p
FF(\()p Fz(p)p FF(\))g(=)-3585 54730 y Fx(1)p -3585 54947
471 54 v -3585 55711 a(2)-2982 55253 y FF(#)221 b(Aut\()p
Fz(E)78 b FF(\).)-1767 56858 y(F)-108 b(or)433 b Fz(k)414
b Fw(\270)369 b FF(4)434 b(w)-36 b(e)434 b(de\257ne)e(the)h
FC(Eisenstein)464 b(series)433 b Fz(G)23010 57057 y Fy(k)24012
56858 y FF(b)-36 b(y)12073 60404 y Fz(G)13099 60603 y
Fy(k)13668 60404 y FF(\()p Fz(q)48 b FF(\))368 b(=)17186
59506 y(1)p 17186 60099 651 54 v 17186 61315 a(2)17969
60404 y Fz(\263)100 b FF(\(1)296 b Fw(\241)f Fz(k)45
b FF(\))295 b(+)24735 58744 y Fu(1)24246 59142 y Fr(X)24319
61932 y Fy(n)p Fx(=1)26386 60404 y Fz(\276)27125 60603
y Fy(k)24 b Fu(\241)p Fx(1)28897 60404 y FF(\()p Fz(n)p
FF(\))p Fz(q)31310 59856 y Fy(n)31935 60404 y Fz(;)-3718
64101 y FF(then)432 b(the)i(map)15115 66224 y Fz(\277)517
b Fw(7!)19467 64963 y Fr(X)17895 67841 y Fx(\()p Fy(m;n)p
Fx(\))p Fu(6)p Fx(=\(0)p Fy(;)p Fx(0\))18949 68789 y
Fy(m;n)p Fu(2)p Fv(Z)25891 65326 y FF(1)p 23311 65919
5810 54 v 23311 67136 a(\()p Fz(m\277)443 b FF(+)295
b Fz(n)p FF(\))28552 66752 y Fy(k)-3718 70781 y FF(di\256ers)421
b(from)h Fz(G)4228 70980 y Fy(k)5218 70781 y FF(b)-36
b(y)422 b(a)f(constan)-36 b(t)421 b(\(no)h(pro)36 b(of)93
b(\).)574 b(Also,)425 b Fz(\263)100 b FF(\(1)271 b Fw(\241)g
Fz(k)45 b FF(\))369 b Fw(2)f FD(Q)422 b FF(and)f(one)h(ma)-36
b(y)422 b(sa)-36 b(y)-108 b(,)424 b FC(symb)-66 b(olic)g(al)66
b(ly)-3718 73399 y FF(at)471 b(least,)482 b(\\)p Fz(\263)100
b FF(\(1)322 b Fw(\241)f Fz(k)45 b FF(\))433 b(=)9104
71738 y Fu(1)8615 72137 y Fr(X)8731 74966 y Fy(d)p Fx(=1)10755
73399 y Fz(d)11431 72850 y Fy(k)24 b Fu(\241)p Fx(1)13635
73399 y FF(=)433 b Fz(\276)15819 73598 y Fy(k)24 b Fu(\241)p
Fx(1)17591 73399 y FF(\(0\).")692 b(The)471 b Fz(n)p
FC(th)500 b(Bernoul)66 b(li)500 b(numb)-66 b(er)498 b
Fz(B)37475 73598 y Fy(n)38573 73399 y FF(is)472 b(de\257ned)e(b)-36
b(y)471 b(the)21534 77755 y(15)p eop
%%Page: 16 24
16 23 bop 1263 -6698 a FF(16)10060 b FA(CHAPTER)434 b(4.)1012
b(EMBEDDING)434 b(HECKE)g(OPERA)-108 b(TORS)433 b(IN)h(THE)g(DUAL)1263
-3169 y FF(equation)23048 -1319 y Fz(x)p 21687 -726 3462
54 v 21687 491 a(e)22290 107 y Fy(x)23170 491 y Fw(\241)295
b FF(1)25650 -420 y(=)27520 -2081 y Fu(1)27031 -1682
y Fr(X)27104 1108 y Fy(n)p Fx(=0)29304 -1319 y Fz(B)30292
-1120 y Fy(n)30918 -1319 y Fz(x)31657 -1801 y Fy(n)p
29304 -726 2980 54 v 30225 491 a Fz(n)p FF(!)32417 -420
y Fz(:)1263 3484 y FF(One)357 b(can)g(sho)-36 b(w)357
b(that)g Fz(\263)100 b FF(\(1)139 b Fw(\241)g Fz(k)45
b FF(\))369 b(=)g Fw(\241)19407 2916 y Fy(B)20117 3072
y Ft(k)p 19407 3178 1218 54 v 19759 3942 a Fy(k)21115
3484 y FF(so)357 b(the)g(constan)-36 b(t)356 b(co)36
b(e\261cien)-36 b(t)357 b(of)h Fz(G)38392 3683 y Fy(k)39318
3484 y FF(is)g Fw(\241)41716 2916 y Fy(B)42426 3072 y
Ft(k)p 41716 3178 V 41833 3942 a Fx(2)p Fy(k)43423 3484
y FF(whic)-36 b(h)357 b(is)h(rational.)1263 8344 y Fs(4.2)2152
b(Inner)716 b(Pro)60 b(duct)1263 11404 y FF(In)448 b(what)h(follo)-36
b(ws)450 b(w)-36 b(e)448 b(assume)h Fz(k)439 b Fw(\270)394
b FF(2)448 b(to)h(a)-36 b(v)g(oid)449 b(trivialities..)624
b(The)449 b(Hec)-36 b(k)g(e)448 b(op)36 b(erators)449
b Fz(T)45671 11603 y Fy(n)46746 11404 y FF(acts)f(on)g(the)1263
13009 y(space)500 b Fz(M)6063 13208 y Fy(k)6632 13009
y FF(.)775 b(Fix)500 b(a)g(subspace)e Fz(V)770 b Fw(\275)481
b Fz(M)21113 13208 y Fy(k)22182 13009 y FF(whic)-36 b(h)499
b(is)g(stable)h(under)e(the)g(action)i(of)g(the)f Fz(T)45812
13208 y Fy(n)46438 13009 y FF(.)776 b(Let)499 b FD(T)p
FF(\()p Fz(V)289 b FF(\))1263 14614 y(b)36 b(e)524 b(the)f
FD(C)p FF(-algebra)h(generated)g(b)-36 b(y)523 b(the)h(endomorphism)e
Fz(T)31454 14813 y Fy(n)32605 14614 y FF(acting)i(on)f
Fz(V)813 b FF(and)524 b(note)f(that)g FD(T)p FF(\()p
Fz(V)290 b FF(\))523 b(is)1263 16219 y(actually)398 b(a)f(\257nite)f
(dimensional)h FD(C)p FF(-v)-36 b(ector)397 b(space)f(since)h(it)g(is)g
(a)g(subspace)f(of)h Fz(E)78 b(nd)p FF(\()p Fz(V)289
b FF(\))396 b(and)h Fz(V)685 b FF(is)397 b(\257nite)1263
17824 y(dimensional.)579 b(Recall)435 b(that)e FD(T)h
FF(is)g(comm)-36 b(utativ)g(e.)3214 19498 y(There)434
b(is)g(a)f(bilinear)h(form)22120 22637 y FD(T)296 b Fw(\243)f
Fz(V)658 b Fw(!)370 b FD(C)22561 24574 y Fw(h)p Fz(T)108
b(;)221 b(f)142 b Fw(i)369 b(7!)h Fz(a)28580 24773 y
Fx(1)29105 24574 y FF(\()p Fz(f)142 b Fw(j)p Fz(T)181
b FF(\))1263 27714 y(where)434 b Fz(f)142 b Fw(j)p Fz(T)550
b FF(=)8865 26717 y Fr(P)10267 27068 y Fu(1)10267 28101
y Fy(n)p Fx(=0)12317 27714 y Fz(a)13000 27913 y Fy(n)13626
27714 y FF(\()p Fz(f)142 b Fw(j)p Fz(T)181 b FF(\))p
Fz(q)17358 27232 y Fy(n)17983 27714 y FF(.)579 b(W)-108
b(e)433 b(th)-36 b(us)433 b(get)h(maps)20817 30853 y
Fz(V)658 b Fw(!)370 b FF(Hom\()p FD(T)p Fz(;)221 b FD(C)p
FF(\))370 b(=)e FD(T)33141 30304 y Fu(\244)20664 32790
y FD(T)i Fw(!)f FF(Hom\()p Fz(V)72 b(;)221 b FD(C)p FF(\))370
b(=)f Fz(V)32780 32242 y Fu(\244)33306 32790 y Fz(:)1263
35998 y FD(Theorem)499 b(4.2.1.)652 b FC(The)464 b(ab)-66
b(ove)465 b(maps)g(ar)-66 b(e)464 b(isomorphisms.)1263
39053 y(Pr)-66 b(o)g(of.)649 b FF(It)409 b(just)g(remains)h(to)f(sho)
-36 b(w)409 b(eac)-36 b(h)409 b(map)g(is)h(injectiv)-36
b(e.)571 b(Then)409 b(since)g(a)g(\257nite)g(dimensional)g(v)-36
b(ector)1263 40658 y(space)583 b(and)f(its)h(dual)g(ha)-36
b(v)g(e)582 b(the)h(same)g(dimension)f(the)g(result)h(follo)-36
b(ws.)1027 b(First)583 b(supp)36 b(ose)582 b Fz(f)765
b Fw(7!)623 b FF(0)g Fw(2)1263 42263 y FF(Hom\()p FD(T)p
Fz(;)221 b FD(C)p FF(\),)573 b(then)543 b Fz(a)12374
42462 y Fx(1)12899 42263 y FF(\()p Fz(f)142 b Fw(j)p
Fz(T)181 b FF(\))557 b(=)f(0)544 b(for)h(all)f Fz(T)738
b Fw(2)556 b FD(T)544 b FF(so,)572 b(in)544 b(particular,)572
b Fz(a)38768 42462 y Fy(n)39950 42263 y FF(=)557 b Fz(a)42202
42462 y Fx(1)42727 42263 y FF(\()p Fz(f)142 b Fw(j)p
Fz(T)45147 42462 y Fy(n)45774 42263 y FF(\))556 b(=)g(0)544
b(for)h(all)1263 43868 y Fz(n)561 b Fw(\270)g FF(1.)917
b(Th)-36 b(us)546 b Fz(f)688 b FF(is)547 b(a)f(constan)-36
b(t,)575 b(but)545 b(since)h Fz(k)606 b Fw(\270)561 b
FF(2)547 b(this)f(implies)g Fz(f)703 b FF(=)561 b(0)546
b(\(otherwise)h Fz(f)688 b FF(w)-36 b(ouldn't)1263 45473
y(transform)434 b(correctly)g(with)g(resp)36 b(ect)433
b(to)h(the)f(action)h(of)g(the)f(mo)36 b(dular)433 b(group\).)3214
47147 y(Next)451 b(supp)36 b(ose)450 b Fz(T)580 b Fw(7!)398
b FF(0)h Fw(2)f FF(Hom\()p Fz(V)72 b(;)221 b FD(C)p FF(\),)457
b(then)449 b Fz(a)27412 47346 y Fx(1)27938 47147 y FF(\()p
Fz(f)142 b Fw(j)p Fz(T)181 b FF(\))398 b(=)g(0)451 b(for)g(all)h
Fz(f)540 b Fw(2)398 b Fz(V)289 b FF(.)630 b(Substiting)450
b Fz(f)142 b Fw(j)p Fz(T)50438 47346 y Fy(n)51515 47147
y FF(for)1263 48752 y Fz(f)576 b FF(and)433 b(using)g(the)g(comm)-36
b(utativit)g(y)435 b(of)f FD(T)g FF(w)-36 b(e)434 b(ha)-36
b(v)g(e)11379 51891 y Fz(a)12062 52090 y Fx(1)12588 51891
y FF(\(\()p Fz(f)142 b Fw(j)p Fz(T)15514 52090 y Fy(n)16140
51891 y FF(\))p Fw(j)p Fz(T)181 b FF(\))369 b(=)f(0)10117
b(for)434 b(all)h Fz(f)142 b FF(,)433 b Fz(n)370 b Fw(\270)f
FF(1)11379 53829 y Fz(a)12062 54028 y Fx(1)12588 53829
y FF(\(\()p Fz(f)142 b Fw(j)p Fz(T)181 b FF(\))p Fw(j)p
Fz(T)17332 54028 y Fy(n)17958 53829 y FF(\))369 b(=)f(0)10117
b(b)-36 b(y)433 b(comm)-36 b(utativit)g(y)14048 55766
y Fz(a)14731 55965 y Fy(n)15357 55766 y FF(\()p Fz(f)142
b Fw(j)p Fz(T)181 b FF(\))369 b(=)f(0)10117 b Fz(n)369
b Fw(\270)g FF(1)16369 57703 y Fz(f)142 b Fw(j)p Fz(T)550
b FF(=)368 b(0)10117 b(since)434 b Fz(k)414 b Fw(\270)369
b FF(2,)434 b(as)g(ab)36 b(o)-36 b(v)g(e)1263 60842 y(Th)g(us)433
b Fz(T)550 b FF(=)369 b(0)434 b(whic)-36 b(h)433 b(completes)h(the)f
(pro)36 b(of.)p 52128 60842 45 878 v 52173 60009 781
45 v 52173 60842 V 52953 60842 45 878 v 1263 64061 a
FC(R)-66 b(emark)464 b(4.2.2.)649 b FF(The)415 b(ab)36
b(o)-36 b(v)g(e)415 b(isomorphisms)g(are)g FD(T)p FC(-e)-66
b(quivariant)p FF(.)570 b(Hom\()p FD(T)p Fz(;)221 b FD(C)p
FF(\))416 b(is)f(a)g FD(T)p FF(-mo)36 b(dule)414 b(if)h(w)-36
b(e)1263 65666 y(let)489 b Fz(T)644 b Fw(2)462 b FD(T)489
b FF(act)g(on)g Fz(')463 b Fw(2)f FF(Hom)q(\()p FD(T)p
Fz(;)221 b FD(C)p FF(\))489 b(b)-36 b(y)489 b(\()p Fz(T)513
b Fw(\242)333 b Fz(')p FF(\)\()p Fz(T)28290 65184 y Fu(0)28600
65666 y FF(\))463 b(=)g Fz(')p FF(\()p Fz(T)181 b(T)34288
65184 y Fu(0)34598 65666 y FF(\).)744 b(If)489 b Fz(\256)472
b FF(:)463 b Fz(V)752 b Fw(!)463 b FF(Hom)q(\()p FD(T)p
Fz(;)221 b FD(C)p FF(\))489 b(is)g(the)1263 67271 y(ab)36
b(o)-36 b(v)g(e)462 b(isomorphism)g(\(so)g Fz(\256)425
b FF(:)418 b Fz(f)558 b Fw(7!)417 b Fz(')20564 67470
y Fy(f)21586 67271 y FF(:=)g(\()p Fz(T)24825 66789 y
Fu(0)25552 67271 y Fw(7!)g Fz(a)27980 67470 y Fx(1)28506
67271 y FF(\()p Fz(f)142 b Fw(j)p Fz(T)31107 66789 y
Fu(0)31417 67271 y FF(\)\)\))461 b(then)g(equiv)-72 b(ariance)462
b(is)g(the)f(statemen)-36 b(t)1263 68876 y(that)433 b
Fz(\256)8 b FF(\()p Fz(T)181 b(f)142 b FF(\))370 b(=)e
Fz(T)181 b(\256)8 b FF(\()p Fz(f)142 b FF(\))p Fz(:)434
b FF(This)g(follo)-36 b(ws)436 b(since)13853 72015 y
Fz(\256)8 b FF(\()p Fz(T)181 b(f)142 b FF(\)\()p Fz(T)18875
71467 y Fu(0)19186 72015 y FF(\))369 b(=)f Fz(')22293
72214 y Fy(T)131 b(f)23577 72015 y FF(\()p Fz(T)25026
71467 y Fu(0)25336 72015 y FF(\))369 b(=)g Fz(a)28275
72214 y Fx(1)28800 72015 y FF(\()p Fz(T)181 b(f)142 b
Fw(j)p Fz(T)32344 71467 y Fu(0)32655 72015 y FF(\))368
b(=)h Fz(a)35593 72214 y Fx(1)36119 72015 y FF(\()p Fz(f)142
b Fw(j)p Fz(T)38720 71467 y Fu(0)39030 72015 y Fz(T)181
b FF(\))20061 73953 y(=)368 b Fz(')22293 74152 y Fy(f)22898
73953 y FF(\()p Fz(T)24347 73404 y Fu(0)24658 73953 y
Fz(T)181 b FF(\))369 b(=)f Fz(T)181 b(')p FF(\()p Fz(T)31100
73404 y Fu(0)31411 73953 y FF(\))368 b(=)h Fz(T)181 b(\256)8
b FF(\()p Fz(f)142 b FF(\)\()p Fz(T)38688 73404 y Fu(0)38999
73953 y FF(\))p Fz(:)p eop
%%Page: 17 25
17 24 bop -3718 -6698 a FA(4.3.)1013 b(EIGENF)-36 b(ORMS)38538
b FF(17)-3718 -3169 y Fs(4.3)2151 b(Eigenforms)-3718
-249 y FF(W)-108 b(e)433 b(con)-36 b(tin)g(ue)433 b(to)h(assume)f(that)
g Fz(k)414 b Fw(\270)370 b FF(2.)-3718 2463 y FD(De\257nition)499
b(4.3.1.)652 b FF(A)552 b(mo)36 b(dular)552 b(form)h
Fz(f)713 b Fw(2)570 b Fz(M)21464 2662 y Fy(k)22586 2463
y FF(is)552 b(an)g FC(eigenform)572 b(for)i FD(T)552
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y Fz(f)694 b FF(for)553 b(all)-3718 4068 y Fz(n)369 b
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6780 y(Let)440 b Fz(f)582 b FF(b)36 b(e)440 b(an)g(eigenform,)k(then)
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6780 y FF(\))381 b(=)f Fz(\270)26853 6979 y Fy(n)27479
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b FF(\))440 b(so)h(if)g Fz(a)34408 6979 y Fx(1)34934
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b FF(\))380 b(=)g(0)-3718 8385 y(for)489 b(all)g Fz(n)462
b Fw(\270)g FF(1)488 b(so)h(since)f Fz(k)507 b Fw(\270)462
b FF(2)488 b(this)g(w)-36 b(ould)488 b(imply)h Fz(f)604
b FF(=)461 b(0.)743 b(Th)-36 b(us)487 b Fz(a)31412 8584
y Fx(1)31938 8385 y FF(\()p Fz(f)142 b FF(\))461 b Fw(6)p
FF(=)h(0)488 b(and)g(w)-36 b(e)488 b(ma)-36 b(y)489 b(as)g(w)-36
b(ell)-3718 9990 y(divide)506 b(through)e(b)-36 b(y)506
b Fz(a)7767 10189 y Fx(1)8293 9990 y FF(\()p Fz(f)142
b FF(\))505 b(to)g(obtain)h(the)f FC(normalize)-66 b(d)530
b(eigenform)32463 9467 y Fx(1)p 31577 9685 2243 54 v
31577 10449 a Fy(a)32077 10572 y Fn(1)32538 10449 y Fx(\()p
Fy(f)98 b Fx(\))33953 9990 y Fz(f)142 b FF(.)794 b(W)-108
b(e)506 b(th)-36 b(us)504 b(assume)i(that)-3718 11777
y Fz(a)-3035 11976 y Fx(1)-2510 11777 y FF(\()p Fz(f)142
b FF(\))369 b(=)g(1,)434 b(then)e(the)h(form)-36 b(ula)435
b(b)36 b(ecomes)433 b Fz(a)18308 11976 y Fy(n)18934 11777
y FF(\()p Fz(f)142 b FF(\))369 b(=)f Fz(\270)23237 11976
y Fy(n)24297 11777 y FF(and)433 b(so)h Fz(f)142 b Fw(j)p
Fz(T)30337 11976 y Fy(n)31333 11777 y FF(=)368 b Fz(a)33396
11976 y Fy(n)34022 11777 y FF(\()p Fz(f)142 b FF(\))p
Fz(f)g FF(,)434 b(for)g(all)g Fz(n)369 b Fw(\270)h FF(1.)-3718
14489 y FD(Theorem)499 b(4.3.2.)651 b FC(L)-66 b(et)637
b Fz(f)831 b Fw(2)689 b Fz(V)927 b FC(and)638 b(let)g
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b FC(in)637 b FF(Hom)q(\()p FD(T)p Fz(;)221 b FD(C)p
FF(\))p FC(,)681 b(thus)639 b Fz(\303)48 b FF(\()p Fz(T)181
b FF(\))688 b(=)-3718 16094 y Fz(a)-3035 16293 y Fx(1)-2510
16094 y FF(\()p Fz(f)142 b Fw(j)p Fz(T)181 b FF(\))p
FC(.)597 b(Then)465 b Fz(f)606 b FC(is)465 b(a)g(normalize)-66
b(d)464 b(eigenform)f(i\256)h Fz(\303)513 b FC(is)464
b(a)-1767 17699 y(ring)g(homomorphism.)-3718 20411 y(Pr)-66
b(o)g(of.)649 b FF(First)433 b(supp)36 b(ose)433 b Fz(f)576
b FF(is)434 b(a)g(normalized)f(eigenform)i(so)f Fz(f)142
b Fw(j)p Fz(T)28143 20610 y Fy(n)29138 20411 y FF(=)369
b Fz(a)31202 20610 y Fy(n)31828 20411 y FF(\()p Fz(f)142
b FF(\))p Fz(f)g FF(.)578 b(Then)11495 23345 y Fz(\303)48
b FF(\()p Fz(T)13657 23544 y Fy(n)14282 23345 y Fz(T)15044
23544 y Fy(m)15932 23345 y FF(\))369 b(=)g Fz(a)18871
23544 y Fx(1)19396 23345 y FF(\()p Fz(f)142 b Fw(j)p
Fz(T)21816 23544 y Fy(n)22443 23345 y Fz(T)23205 23544
y Fy(m)24093 23345 y FF(\))368 b(=)h Fz(a)27031 23544
y Fy(m)27918 23345 y FF(\()p Fz(f)142 b Fw(j)p Fz(T)30338
23544 y Fy(n)30965 23345 y FF(\))16807 25282 y(=)369
b Fz(a)18871 25481 y Fy(m)19758 25282 y FF(\()p Fz(a)20947
25481 y Fy(n)21573 25282 y FF(\()p Fz(f)142 b FF(\))p
Fz(f)g FF(\))368 b(=)h Fz(a)27089 25481 y Fy(m)27976
25282 y FF(\()p Fz(f)142 b FF(\))p Fz(a)30454 25481 y
Fy(n)31080 25282 y FF(\()p Fz(f)g FF(\))16807 27219 y(=)369
b Fz(\303)48 b FF(\()p Fz(T)20350 27418 y Fy(n)20975
27219 y FF(\))p Fz(\303)g FF(\()p Fz(T)23643 27418 y
Fy(m)24530 27219 y FF(\))p Fz(;)-3718 30152 y FF(so)434
b Fz(\303)481 b FF(is)433 b(a)h(homomorphism.)-1767 31758
y(Con)-36 b(v)g(ersely)-108 b(,)401 b(assume)391 b Fz(\303)439
b FF(is)392 b(a)g(homomorphism.)564 b(Then)391 b Fz(f)142
b Fw(j)p Fz(T)28024 31957 y Fy(n)29020 31758 y FF(=)30400
30761 y Fr(P)32024 31758 y Fz(a)32707 31957 y Fy(m)33594
31758 y FF(\()p Fz(f)g Fw(j)p Fz(T)36014 31957 y Fy(n)36641
31758 y FF(\))p Fz(q)37772 31275 y Fy(m)38658 31758 y
FF(,)401 b(so)391 b(to)h(sho)-36 b(w)392 b(that)-3718
33363 y Fz(f)142 b Fw(j)p Fz(T)-1804 33562 y Fy(n)-809
33363 y FF(=)369 b Fz(a)1255 33562 y Fy(n)1881 33363
y FF(\()p Fz(f)142 b FF(\))p Fz(f)550 b FF(w)-36 b(e)408
b(m)-36 b(ust)408 b(sho)-36 b(w)408 b(that)g Fz(a)16627
33562 y Fy(m)17514 33363 y FF(\()p Fz(f)142 b Fw(j)p
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Fz(a)23499 33562 y Fy(n)24125 33363 y FF(\()p Fz(f)142
b FF(\))p Fz(a)26603 33562 y Fy(m)27490 33363 y FF(\()p
Fz(f)g FF(\).)570 b(Recall)409 b(that)f Fz(\303)48 b
FF(\()p Fz(T)39065 33562 y Fy(n)39691 33363 y FF(\))369
b(=)f Fz(a)42629 33562 y Fx(1)43155 33363 y FF(\()p Fz(f)142
b Fw(j)p Fz(T)45575 33562 y Fy(n)46201 33363 y FF(\))369
b(=)-3718 34968 y Fz(a)-3035 35167 y Fy(n)-2409 34968
y FF(,)434 b(th)-36 b(us)8885 37901 y Fz(a)9568 38100
y Fy(n)10194 37901 y FF(\()p Fz(f)142 b FF(\))p Fz(a)12672
38100 y Fy(m)13559 37901 y FF(\()p Fz(f)g FF(\))369 b(=)f
Fz(a)17786 38100 y Fx(1)18312 37901 y FF(\()p Fz(f)142
b Fw(j)p Fz(T)20732 38100 y Fy(n)21358 37901 y FF(\))p
Fz(a)22547 38100 y Fx(1)23073 37901 y FF(\()p Fz(f)g
Fw(j)p Fz(T)25493 38100 y Fy(m)26380 37901 y FF(\))369
b(=)g Fz(\303)48 b FF(\()p Fz(T)30798 38100 y Fy(n)31423
37901 y FF(\))p Fz(\303)g FF(\()p Fz(T)34091 38100 y
Fy(m)34978 37901 y FF(\))15723 39838 y(=)368 b Fz(\303)48
b FF(\()p Fz(T)19265 40037 y Fy(n)19891 39838 y Fz(T)20653
40037 y Fy(m)21541 39838 y FF(\))369 b(=)f Fz(a)24479
40037 y Fx(1)25005 39838 y FF(\()p Fz(f)142 b Fw(j)p
Fz(T)27425 40037 y Fy(n)28051 39838 y Fw(j)p Fz(T)29182
40037 y Fy(m)30070 39838 y FF(\))15723 41776 y(=)368
b Fz(a)17786 41975 y Fy(m)18674 41776 y FF(\()p Fz(f)142
b Fw(j)p Fz(T)21094 41975 y Fy(n)21720 41776 y FF(\))-3718
44709 y(as)434 b(desired.)p 47147 46314 45 878 v 47192
45480 781 45 v 47192 46314 V 47972 46314 45 878 v eop
%%Page: 18 26
18 25 bop 1263 -6698 a FF(18)10060 b FA(CHAPTER)434 b(4.)1012
b(EMBEDDING)434 b(HECKE)g(OPERA)-108 b(TORS)433 b(IN)h(THE)g(DUAL)p
eop
%%Page: 19 27
19 26 bop -3718 5792 a FE(Chapter)1033 b(5)-3718 11433
y(Rationalit)-86 b(y)1033 b(and)g(In)-86 b(tegralit)g(y)1033
b(Questions)-3718 18403 y Fs(5.1)2151 b(Review)-3718
21539 y FF(In)488 b(the)h(previous)g(lecture)f(w)-36
b(e)489 b(lo)36 b(ok)-36 b(ed)490 b(at)f(subspaces)f
Fz(V)752 b Fw(\275)463 b Fz(M)27824 21738 y Fy(k)28856
21539 y Fw(\275)g FD(C)p FF([[)p Fz(q)48 b FF(]],)504
b(\()p Fz(k)k Fw(\270)463 b FF(4\),)503 b(and)488 b(considered)-3718
23144 y(the)504 b(space)h FD(T)491 b FF(=)f FD(T)p FF(\()p
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23144 y Fz(V)794 b FF(of)506 b(Hec)-36 b(k)g(e)505 b(op)36
b(erators)505 b(on)g Fz(V)290 b FF(.)792 b(W)-108 b(e)505
b(de\257ned)e(a)-3718 24749 y(pairing)398 b FD(T)222
b Fw(\243)g Fz(V)658 b Fw(!)369 b FD(C)398 b FF(b)-36
b(y)398 b(\()p Fz(T)108 b(;)221 b(f)142 b FF(\))370 b
Fw(7!)f Fz(a)15528 24948 y Fx(1)16054 24749 y FF(\()p
Fz(f)142 b Fw(j)p Fz(T)181 b FF(\))397 b(and)g(sho)-36
b(w)g(ed)398 b(this)f(pairing)h(is)g(nondegenerate)f(and)g(that)-3718
26355 y(it)434 b(induces)e(isomorphisms)i FD(T)11737
25986 y Fw(\273)11748 26410 y FF(=)13139 26355 y(Hom)q(\()p
Fz(V)72 b(;)221 b FD(C)p FF(\))434 b(and)f Fz(V)23732
25986 y Fw(\273)23743 26410 y FF(=)25134 26355 y(Hom\()p
FD(T)p Fz(;)221 b FD(C)p FF(\).)-3718 31446 y Fs(5.2)2151
b(In)-60 b(tegralit)g(y)-3718 34583 y FF(Fix)437 b Fz(k)419
b Fw(\270)374 b FF(4)437 b(and)f(let)h Fz(S)451 b FF(=)374
b Fz(S)10055 34782 y Fy(k)11061 34583 y FF(b)36 b(e)436
b(the)g(space)h(of)g(w)-36 b(eigh)g(t)437 b Fz(k)481
b FF(cusp)436 b(forms)h(with)g(resp)36 b(ect)436 b(to)h(the)f(action)h
(of)-3718 36188 y(SL)-2182 36387 y Fx(2)-1657 36188 y
FF(\()p FD(Z)p FF(\).)578 b(Let)16710 39440 y Fz(S)77
b FF(\()p FD(Q)p FF(\))369 b(=)g Fz(S)22271 39639 y Fy(k)23135
39440 y Fw(\\)294 b FD(Q)p FF([[)p Fz(q)48 b FF(]])16919
41377 y Fz(S)77 b FF(\()p FD(Z)p FF(\))369 b(=)g Fz(S)22271
41576 y Fy(k)23135 41377 y Fw(\\)294 b FD(Z)p FF([[)p
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b FC(Ther)-66 b(e)464 b(is)h(a)g FD(C)p FC(-b)-66 b(asis)465
b(of)g Fz(M)20465 44934 y Fy(k)21499 44735 y FC(c)-66
b(onsisting)463 b(of)h(forms)h(with)g(inte)-66 b(gr)g(al)462
b(c)-66 b(o)g(e\261cients.)-3718 47978 y(Pr)g(o)g(of.)649
b FF(This)434 b(is)g(seen)f(b)-36 b(y)434 b(exhibiting)f(a)h(basis.)579
b(Recall)435 b(that)e(for)h(all)h Fz(k)414 b Fw(\270)369
b FF(4)14387 52169 y Fz(G)15413 52368 y Fy(k)16351 52169
y FF(=)g Fw(\241)19023 51270 y Fz(b)19576 51469 y Fy(k)p
18898 51863 1372 54 v 18898 53080 a FF(2)p Fz(k)20697
52169 y FF(+)22493 50508 y Fu(1)22004 50907 y Fr(X)22105
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y Fy(d)p Fu(j)p Fy(k)26284 52169 y Fz(d)26960 51620 y
Fy(k)g Fu(\241)p Fx(1)28731 52169 y Fz(q)29356 51620
y Fy(n)-3718 56819 y FF(is)434 b(the)f Fz(k)45 b FF(th)433
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b(dular)434 b(form)g(of)g(w)-36 b(eigh)g(t)434 b Fz(k)478
b FF(and)15045 60656 y Fz(E)16008 60855 y Fy(k)16946
60656 y FF(=)369 b Fw(\241)19493 59758 y FF(2)p Fz(k)p
19493 60351 V 19618 61568 a(b)20171 61767 y Fy(k)21292
60656 y Fw(\242)295 b Fz(G)22982 60855 y Fy(k)23920 60656
y FF(=)369 b(1)295 b(+)g Fw(\242)221 b(\242)g(\242)-3718
64521 y FF(is)447 b(its)h(normalization.)621 b(Since)446
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b(b)29537 64720 y Fx(8)30510 64521 y FF(ha)-36 b(v)g(e)448
b(1)f(as)h(n)-36 b(umerator)447 b(\(this)f(isn't)-3718
66126 y(alw)-36 b(a)g(ys)473 b(the)d(case,)482 b Fz(b)6474
66325 y Fx(10)7903 66126 y FF(=)9716 65603 y Fx(5)p 9481
65820 941 54 v 9481 66584 a(66)10555 66126 y FF(\))471
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66126 y FF(and)h Fz(E)23970 66325 y Fx(6)24967 66126
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b(in)f FD(Z)g FF(and)g(constan)-36 b(t)471 b(term)-3718
67731 y(1.)579 b(F)-108 b(urthermore)431 b(one)j(sho)-36
b(ws)434 b(b)-36 b(y)433 b(dimension)g(and)g(indep)36
b(endence)432 b(argumen)-36 b(ts)433 b(that)g(the)g(forms)16484
70983 y Fw(f)p Fz(E)18189 70434 y Fy(a)18111 71311 y
Fx(4)18744 70983 y Fz(E)19785 70434 y Fy(b)19707 71311
y Fx(6)20243 70983 y Fw(j)p FF(4)p Fz(a)295 b FF(+)g(6)p
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b(a)g(basis)g(for)g Fz(M)6876 74434 y Fy(k)7445 74235
y FF(.)p 47147 74235 45 878 v 47192 73401 781 45 v 47192
74235 V 47972 74235 45 878 v 21534 77755 a(19)p eop
%%Page: 20 28
20 27 bop 1263 -6698 a FF(20)10729 b FA(CHAPTER)434 b(5.)1013
b(RA)-108 b(TIONALITY)434 b(AND)f(INTEGRALITY)i(QUESTIONS)1263
-3169 y Fs(5.3)2152 b(Victor)717 b(Miller's)f(Thesis)1263
-249 y FF(Let)418 b Fz(d)369 b FF(=)f(dim)8171 -50 y
Fv(C)9231 -249 y Fz(S)10031 -50 y Fy(k)10600 -249 y FF(,)422
b(then)417 b(Victor)h(Miller)h(sho)-36 b(w)g(ed)418 b(in)g(his)g
(thesis)g(\(see)g([11)q(],)k(c)-36 b(h.)573 b(X,)419
b(theorem)e(4.4\))i(that)1263 1356 y(there)433 b(exists)14841
2961 y Fz(f)15482 3160 y Fx(1)16008 2961 y Fz(;)221 b(:)g(:)g(:)445
b(;)221 b(f)19783 3160 y Fy(d)20692 2961 y Fw(2)369 b
Fz(S)22747 3160 y Fy(k)23316 2961 y FF(\()p FD(Z)p FF(\))1300
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2961 y FF(\()p Fz(f)35365 3160 y Fy(j)35852 2961 y FF(\))369
b(=)g Fz(\261)38684 3160 y Fy(ij)1263 5251 y FF(for)434
b(1)370 b Fw(\267)f Fz(i;)221 b(j)444 b Fw(\267)369 b
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5251 y FF(clearly)h(form)f(a)g(basis.)1263 7894 y FD(Prop)42
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b(T)23056 8093 y Fy(n)23683 7894 y Fz(;)g(:)g(:)g(:)j
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b FF(=)39522 6898 y Fr(L)40998 7248 y Fy(d)40998 8282
y(i)p Fx(=1)42798 7894 y FD(Z)p Fz(T)44473 8093 y Fy(i)44849
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b(o)683 b(see)f(that)g Fz(T)13679 10736 y Fx(1)14205
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b(T)18125 10736 y Fy(d)19457 10537 y Fw(2)792 b FD(T)h
FF(=)f FD(T)p FF(\()p Fz(S)27116 10736 y Fy(k)27685 10537
y FF(\))682 b(are)g(linearly)i(indep)36 b(enden)-36 b(t)680
b(o)-36 b(v)g(er)683 b FD(C)f FF(supp)36 b(ose)1263 11146
y Fr(P)2665 11496 y Fy(d)2665 12530 y(i)p Fx(=1)4465
12142 y Fz(c)5025 12341 y Fy(i)5400 12142 y Fz(T)6162
12341 y Fy(i)6907 12142 y FF(=)369 b(0,)434 b(then)13195
15283 y(0)370 b(=)e Fz(a)16278 15482 y Fx(1)16804 15283
y FF(\()p Fz(f)17951 15482 y Fy(j)18438 15283 y Fw(j)19028
14021 y Fr(X)21168 15283 y Fz(c)21728 15482 y Fy(i)22104
15283 y Fz(T)22866 15482 y Fy(i)23242 15283 y FF(\))g(=)25497
14021 y Fr(X)26296 16820 y Fy(i)27637 15283 y Fz(c)28197
15482 y Fy(i)28573 15283 y Fz(a)29256 15482 y Fy(i)29631
15283 y FF(\()p Fz(f)30778 15482 y Fy(j)31265 15283 y
FF(\))h(=)33520 14021 y Fr(X)34319 16820 y Fy(i)35660
15283 y Fz(c)36220 15482 y Fy(i)36596 15283 y Fz(\261)37172
15482 y Fy(ij)38348 15283 y FF(=)g Fz(c)40289 15482 y
Fy(j)40775 15283 y Fz(:)1263 19308 y FF(F)-108 b(rom)423
b(the)g(isomorphism)h FD(T)15897 18939 y Fw(\273)15908
19363 y FF(=)17299 19308 y(Hom)q(\()p Fz(S)21315 19507
y Fy(k)21883 19308 y Fz(;)221 b FD(C)p FF(\))424 b(w)-36
b(e)424 b(kno)-36 b(w)424 b(that)g(dim)34741 19507 y
Fv(C)35802 19308 y FD(T)369 b FF(=)f Fz(d)p FF(,)426
b(so)e(w)-36 b(e)424 b(can)f(write)h(an)-36 b(y)424 b
Fz(T)52442 19507 y Fy(n)1263 20913 y FF(as)434 b(a)g
FD(C)p FF(-linear)g(com)-36 b(bination)19947 23432 y
Fz(T)20709 23631 y Fy(n)21704 23432 y FF(=)23802 21771
y Fy(d)23085 22170 y Fr(X)23283 24969 y Fy(i)p Fx(=1)25225
23432 y Fz(c)25785 23631 y Fy(n)26356 23766 y Ft(i)26762
23432 y Fz(T)27524 23631 y Fy(i)27900 23432 y Fz(;)1523
b(c)30344 23631 y Fy(n)30915 23766 y Ft(i)31690 23432
y Fw(2)369 b FD(C)p Fz(:)1263 26805 y FF(But)9607 29324
y FD(Z)g Fw(3)g Fz(a)12827 29523 y Fy(n)13453 29324 y
FF(\()p Fz(f)14600 29523 y Fy(j)15087 29324 y FF(\))g(=)f
Fz(a)18025 29523 y Fx(1)18551 29324 y FF(\()p Fz(f)19698
29523 y Fy(j)20185 29324 y Fw(j)p Fz(T)21316 29523 y
Fy(n)21942 29324 y FF(\))h(=)24915 27664 y Fy(d)24198
28062 y Fr(X)24396 30862 y Fy(i)p Fx(=1)26338 29324 y
Fz(c)26898 29523 y Fy(n)27469 29658 y Ft(i)27875 29324
y Fz(a)28558 29523 y Fx(1)29084 29324 y FF(\()p Fz(f)30231
29523 y Fy(j)30718 29324 y Fw(j)p Fz(T)31849 29523 y
Fy(i)32225 29324 y FF(\))f(=)35197 27664 y Fy(d)34480
28062 y Fr(X)34678 30862 y Fy(i)p Fx(=1)36620 29324 y
Fz(c)37180 29523 y Fy(n)37751 29658 y Ft(i)38157 29324
y Fz(a)38840 29523 y Fy(i)39216 29324 y FF(\()p Fz(f)40363
29523 y Fy(j)40850 29324 y FF(\))g(=)h Fz(c)43665 29523
y Fy(n)44236 29658 y Ft(j)1263 32713 y FF(so)434 b(the)f
Fz(c)5660 32912 y Fy(n)6231 33047 y Ft(i)7071 32713 y
FF(all)i(lie)f(in)f FD(Z)h FF(whic)-36 b(h)433 b(completes)h(the)f(pro)
36 b(of.)p 52128 32713 45 878 v 52173 31879 781 45 v
52173 32713 V 52953 32713 45 878 v 3214 35291 a(Th)-36
b(us)433 b Fz(R)445 b FF(is)434 b(an)f(in)-36 b(tegral)434
b(Hec)-36 b(k)g(e)434 b(algebra)g(of)h(\257nite)e(rank)g
Fz(d)h FF(o)-36 b(v)g(er)434 b FD(Z)p FF(.)578 b(W)-108
b(e)434 b(ha)-36 b(v)g(e)434 b(a)g(map)21262 38144 y
Fz(S)77 b FF(\()p FD(Z)p FF(\))295 b Fw(\243)h Fz(R)380
b Fw(!)369 b FD(Z)23441 40081 y FF(\()p Fz(f)70 b(;)221
b(T)181 b FF(\))369 b Fw(7!)g Fz(a)29438 40280 y Fx(1)29963
40081 y FF(\()p Fz(f)142 b Fw(j)p Fz(T)181 b FF(\))1263
42934 y(whic)-36 b(h)434 b(induces)e(an)i(em)-36 b(b)36
b(edding)19760 45786 y Fz(S)77 b FF(\()p FD(Z)p FF(\))369
b Fz(,)-221 b Fw(!)370 b FF(Hom\()p Fz(R)11 b(;)221 b
FD(Z)p FF(\))31355 45418 y Fw(\273)31366 45842 y FF(=)32758
45786 y FD(Z)33671 45238 y Fy(d)34210 45786 y Fz(:)1263
48639 y FC(Exer)-66 b(cise)464 b(5.3.2.)649 b FF(Pro)-36
b(v)g(e)558 b(that)g(the)f(map)h Fz(S)77 b FF(\()p FD(Z)p
FF(\))581 b Fz(,)-221 b Fw(!)580 b FF(Hom)q(\()p Fz(R)11
b(;)221 b FD(Z)p FF(\))558 b(is)g(in)g(fact)h(an)e(isomorphism)h(of)h
FD(T)p FF(-)1263 50244 y(mo)36 b(dules.)578 b([Hin)-36
b(t:)579 b(Sho)-36 b(w)433 b(the)g(cok)-36 b(ernel)434
b(is)g(torsion)g(free.])1263 54669 y Fs(5.4)2152 b(P)-60
b(etersson)716 b(Inner)h(Pro)60 b(duct)1263 57590 y FF(The)434
b(main)f(theorem)h(is)1263 60233 y FD(Theorem)499 b(5.4.1.)652
b FC(The)464 b Fz(T)15153 60432 y Fy(n)16149 60233 y
Fw(2)369 b FD(T)p FF(\()p Fz(S)19749 60432 y Fy(k)20317
60233 y FF(\))465 b FC(ar)-66 b(e)465 b(al)66 b(l)465
b(diagonalizable)g(over)f FD(C)p FC(.)3214 62876 y FF(T)-108
b(o)570 b(pro)-36 b(v)g(e)570 b(this)f(w)-36 b(e)570
b(note)g(that)f Fz(S)20410 63075 y Fy(k)21548 62876 y
FF(supp)36 b(orts)569 b(a)h(non-degenerate)f(p)36 b(ositiv)-36
b(e)570 b(de\257nite)f(Hermitean)1263 64481 y(inner)433
b(pro)36 b(duct)433 b(\(the)f(P)-36 b(etersson)434 b(inner)f(pro)36
b(duct\))21795 67333 y(\()p Fz(f)70 b(;)221 b(g)48 b
FF(\))369 b Fw(7!)g(h)p Fz(f)70 b(;)221 b(g)48 b Fw(i)368
b(2)g FD(C)1263 70186 y FF(suc)-36 b(h)433 b(that)g Fw(h)p
Fz(f)142 b Fw(j)p Fz(T)9446 70385 y Fy(n)10072 70186
y Fz(;)221 b(g)48 b Fw(i)369 b FF(=)f Fw(h)p Fz(f)70
b(;)221 b(g)48 b Fw(j)p Fz(T)17203 70385 y Fy(n)17829
70186 y Fw(i)p FF(.)578 b(W)-108 b(e)434 b(need)f(some)h(bac)-36
b(kground)433 b(facts.)1263 72829 y FD(De\257nition)500
b(5.4.2.)651 b FF(An)533 b(op)36 b(erator)535 b Fz(T)714
b FF(is)534 b FC(normal)g FF(if)g(it)g(comm)-36 b(utes)534
b(with)g(its)g(adjoin)-36 b(t,)559 b(th)-36 b(us)533
b Fz(T)181 b(T)50992 72347 y Fu(\244)52057 72829 y FF(=)1263
74434 y Fz(T)2206 73952 y Fu(\244)2732 74434 y Fz(T)g
FF(.)p eop
%%Page: 21 29
21 28 bop -3718 -6698 a FA(5.4.)1013 b(PETERSSON)433
b(INNER)h(PR)-36 b(ODUCT)27694 b FF(21)-1767 -3169 y
Fz(T)-1005 -2970 y Fy(n)55 -3169 y FF(is)434 b(clearly)h(normal)f
(since)f Fz(T)14055 -3651 y Fu(\244)13874 -2841 y Fy(n)14950
-3169 y FF(=)369 b Fz(T)17093 -2970 y Fy(n)17719 -3169
y FF(,)-3718 -1007 y FD(Theorem)499 b(5.4.3.)651 b FC(A)465
b(normal)f(op)-66 b(er)g(ator)465 b(is)f(diagonalizable.)-1767
1156 y FF(Th)-36 b(us)433 b(eac)-36 b(h)433 b Fz(T)5216
1355 y Fy(n)6277 1156 y FF(is)g(diagonalizable.)-3718
3318 y FD(Theorem)499 b(5.4.4.)651 b FC(A)437 b(c)-66
b(ommuting)436 b(family)g(of)i(semisimple)e(\(=diagonalizable\))i(op)
-66 b(er)g(ators)437 b(c)-66 b(an)437 b(b)-66 b(e)437
b(si-)-3718 4923 y(multane)-66 b(ously)465 b(diagonalize)-66
b(d.)-1767 7086 y FF(Since)411 b(the)f Fz(T)4586 7285
y Fy(n)5624 7086 y FF(comm)-36 b(ute)410 b(this)h(implies)h
Fz(S)18908 7285 y Fy(k)19888 7086 y FF(has)f(a)g(basis)h(consisting)f
(of)h(normalized)f(eigenforms)h Fz(f)142 b FF(.)-3718
8691 y(Their)434 b(eigen)-36 b(v)-72 b(alues)434 b(are)g(real)g(since)
10935 10984 y Fz(a)11618 11183 y Fy(n)12244 10984 y FF(\()p
Fz(f)142 b FF(\))p Fw(h)p Fz(f)70 b(;)221 b(f)142 b Fw(i)368
b FF(=)g Fw(h)p Fz(a)20097 11183 y Fy(n)20723 10984 y
FF(\()p Fz(f)142 b FF(\))p Fz(f)70 b(;)221 b(f)142 b
Fw(i)369 b FF(=)f Fw(h)p Fz(f)142 b Fw(j)p Fz(T)29291
11183 y Fy(n)29917 10984 y Fz(;)221 b(f)142 b Fw(i)17517
13170 y FF(=)368 b Fw(h)p Fz(f)70 b(;)221 b(a)21390 13369
y Fy(n)22016 13170 y FF(\()p Fz(f)142 b FF(\))p Fz(f)g
Fw(i)369 b FF(=)p 26860 12014 3104 54 v 368 w Fz(a)27543
13369 y Fy(n)28169 13170 y FF(\()p Fz(f)142 b FF(\))p
Fw(h)p Fz(f)70 b(;)221 b(f)142 b Fw(i)p Fz(:)-3718 15462
y FC(Exer)-66 b(cise)464 b(5.4.5.)649 b FF(The)309 b(co)36
b(e\261cien)-36 b(ts)309 b Fz(a)14959 15661 y Fy(n)15894
15462 y FF(of)g(the)f(eigenforms)i(are)f(totally)h(real)g(algebraic)g
(in)-36 b(tegers.)536 b([Hin)-36 b(t:)-3718 17067 y(The)338
b(space)h Fz(S)3041 17266 y Fy(k)3948 17067 y FF(is)g(stable)f(under)f
(the)h(action)h(of)g(Aut)o(\()p FD(C)p FF(\))g(on)f(co)36
b(e\261cien)-36 b(ts:)531 b(if)339 b Fz(f)511 b FF(=)36773
16071 y Fr(P)38175 16422 y Fu(1)38175 17455 y Fy(n)p
Fx(=1)40225 17067 y Fz(c)40785 17266 y Fy(n)41411 17067
y Fz(q)42036 16585 y Fy(n)43030 17067 y Fw(2)369 b Fz(S)45085
17266 y Fy(k)45992 17067 y FF(and)-3718 18672 y Fz(\276)417
b Fw(2)368 b FF(Aut\()p FD(C)p FF(\))422 b(then)g Fz(\276)48
b FF(\()p Fz(f)142 b FF(\))369 b(=)10692 17676 y Fr(P)12094
18027 y Fu(1)12094 19060 y Fy(n)p Fx(=1)14144 18672 y
Fz(\276)48 b FF(\()p Fz(c)15997 18871 y Fy(n)16623 18672
y FF(\))p Fz(q)17754 18190 y Fy(n)18802 18672 y FF(is)423
b(again)h(in)f Fz(S)25864 18871 y Fy(k)26855 18672 y
FF(\(c)-36 b(hec)g(k)423 b(this)f(b)-36 b(y)423 b(writing)g
Fz(f)565 b FF(in)422 b(terms)h(of)g(a)-3718 20278 y(basis)433
b Fz(f)116 20477 y Fx(1)641 20278 y Fz(;)221 b(:)g(:)g(:)446
b(;)221 b(f)4417 20477 y Fy(d)5326 20278 y Fw(2)368 b
Fz(S)77 b FF(\()p FD(Z)p FF(\)\).)578 b(Next)433 b(use)f(the)g(fact)h
(that)f Fz(f)574 b FF(is)433 b(an)f(eigenform)i(i\256)e
Fz(\276)48 b FF(\()p Fz(f)142 b FF(\))432 b(is)h(an)f(eigenform.])-1767
22161 y(Let)12111 23766 y Fw(H)382 b FF(=)368 b Fw(f)p
Fz(x)296 b FF(+)e Fz(iy)417 b FF(:)369 b Fz(x;)221 b(y)418
b Fw(2)368 b FD(R)p Fz(;)655 b FF(and)433 b Fz(y)417
b(>)369 b FF(0)p Fw(g)-3718 25816 y FF(b)36 b(e)433 b(the)g(upp)36
b(er)433 b(half)h(plane.)578 b(Then)433 b(the)g(v)-36
b(olume)434 b(form)23873 25232 y Fy(dx)p Fu(^)p Fy(dy)p
23873 25511 2623 54 v 24705 26274 a(y)25202 26022 y Fn(2)27062
25816 y FF(is)g(in)-36 b(v)-72 b(arian)-36 b(t)433 b(under)f(the)h
(action)h(of)10736 28337 y(GL)12569 27780 y Fx(+)12569
28665 y(2)13356 28337 y FF(\()p FD(R)p FF(\))369 b(=)f
Fw(f)p Fz(M)509 b Fw(2)368 b FF(GL)22750 28536 y Fx(2)23276
28337 y FF(\()p FD(R)p FF(\))p Fw(j)221 b FF(det)o(\()p
Fz(M)139 b FF(\))369 b Fz(>)g FF(0)p Fw(g)p Fz(:)-3718
30629 y FF(If)434 b Fz(\256)378 b FF(=)167 29553 y Fr(\241)998
30205 y Fy(a)401 b(b)1044 31005 y(c)407 b(d)2564 29553
y Fr(\242)3542 30629 y Fw(2)368 b FF(GL)6630 30072 y
Fx(+)6630 30958 y(2)7417 30629 y FF(\()p FD(R)p FF(\))433
b(then)12939 29553 y Fr(\241)13770 30205 y Fy(a)401 b(b)13816
31005 y(c)407 b(d)15336 29553 y Fr(\242)16378 30629 y
FF(acts)434 b(on)g Fw(H)446 b FF(b)-36 b(y)15710 32355
y Fr(\263)16504 33016 y Fz(a)1168 b(b)16565 34621 y(c)g(d)18969
32355 y Fr(\264)20132 33829 y FF(:)1669 b Fz(z)429 b
Fw(7!)25025 32931 y Fz(az)354 b FF(+)295 b Fz(b)p 25025
33524 3502 54 v 25025 34741 a(cz)354 b FF(+)295 b Fz(d)-3718
36823 y FF(and)433 b(one)g(has)15107 38829 y(Im\()17299
37930 y Fz(az)354 b FF(+)295 b Fz(b)p 17299 38523 V 17299
39740 a(cz)354 b FF(+)295 b Fz(d)20933 38829 y FF(\))369
b(=)23877 37930 y(det)o(\()p Fz(\256)8 b FF(\))p 23321
38523 4765 54 v 23321 39740 a Fw(j)p Fz(cz)355 b FF(+)294
b Fz(d)p Fw(j)27560 39356 y Fx(2)28219 38829 y Fz(y)48
b(:)-3718 41765 y FF(Di\256eren)-36 b(tiating)5067 41242
y Fy(az)37 b Fx(+)p Fy(b)p 5067 41460 2110 54 v 5072
42223 a(cz)g Fx(+)p Fy(d)7743 41765 y FF(giv)-36 b(es)11165
45009 y Fz(d)p FF(\()12480 44110 y Fz(az)354 b FF(+)295
b Fz(b)p 12480 44703 3502 54 v 12480 45920 a(cz)354 b
FF(+)295 b Fz(d)16113 45009 y FF(\))369 b(=)18502 44110
y Fz(a)p FF(\()p Fz(cz)354 b FF(+)294 b Fz(d)p FF(\))p
Fz(dz)354 b Fw(\241)296 b Fz(c)p FF(\()p Fz(az)354 b
FF(+)295 b Fz(b)p FF(\))p Fz(dz)p 18502 44703 14571 54
v 23267 45920 a FF(\()p Fz(cz)355 b FF(+)294 b Fz(d)p
FF(\))27780 45536 y Fx(2)16988 48590 y FF(=)18502 47691
y(\()p Fz(ad)g Fw(\241)i Fz(bc)p FF(\))p Fz(dz)p 18502
48284 6446 54 v 19205 49501 a FF(\()p Fz(cz)354 b FF(+)295
b Fz(d)p FF(\))23718 49117 y Fx(2)16988 52170 y FF(=)19223
51272 y Fz(det)p FF(\()p Fz(\256)8 b FF(\))p 18502 51865
5039 54 v 18502 53082 a(\()p Fz(cz)354 b FF(+)295 b Fz(d)p
FF(\))23015 52698 y Fx(2)23673 52170 y Fz(dz)-3718 55134
y FF(Th)-36 b(us,)433 b(under)f(the)h(action)h(of)h Fz(\256)8
b FF(,)434 b Fz(dz)355 b Fw(^)295 b Fz(d)p 16369 54403
664 54 v(z)492 b FF(tak)-36 b(es)434 b(on)g(a)g(factor)g(of)15468
57243 y(det)o(\()p Fz(\256)8 b FF(\))19121 56761 y Fx(2)p
12519 57836 10077 54 v 12519 59053 a FF(\()p Fz(cz)354
b FF(+)295 b Fz(d)p FF(\))17032 58669 y Fx(2)17558 59053
y FF(\()p Fz(c)p 18624 58322 664 54 v(z)354 b FF(+)295
b Fz(d)p FF(\))22071 58669 y Fx(2)23098 58142 y FF(=)24479
56667 y Fr(\263)25960 57243 y FF(det\()p Fz(\256)8 b
FF(\))p 25405 57836 4765 54 v 25405 59053 a Fw(j)p Fz(cz)354
b FF(+)295 b Fz(d)p Fw(j)29644 58669 y Fx(2)30302 56667
y Fr(\264)31096 56965 y Fx(2)31622 58142 y Fz(:)-3718
61106 y FD(De\257nition)499 b(5.4.6.)652 b FF(The)433
b FC(Petersson)465 b(inner)f(pr)-66 b(o)g(duct)433 b
FF(of)h(forms)g Fz(f)70 b(;)221 b(g)417 b Fw(2)368 b
Fz(S)33537 61305 y Fy(k)34540 61106 y FF(is)434 b(de\257ned)e(b)-36
b(y)11835 63983 y Fz(<)368 b(f)70 b(;)221 b(g)417 b(>)p
FF(=)17940 62175 y Fr(Z)18678 65182 y Fx(\241)p Fu(nH)20593
63983 y FF(\()p Fz(f)142 b FF(\()p Fz(z)59 b FF(\))p
23557 62827 2346 54 v Fz(g)48 b FF(\()p Fz(z)59 b FF(\))p
Fz(y)26585 63435 y Fy(k)27153 63983 y FF(\))27792 63084
y Fz(dx)295 b Fw(^)g Fz(dy)p 27792 63678 4249 54 v 29313
64894 a(y)29995 64511 y Fx(2)32173 63983 y Fz(;)-3718
67198 y FF(where)433 b(\241)369 b(=)g(SL)4138 67397 y
Fx(2)4663 67198 y FF(\()p FD(Z)p FF(\).)-1767 69361 y(In)-36
b(tegrating)463 b(o)-36 b(v)g(er)463 b(\241)p Fw(nH)476
b FF(can)463 b(b)36 b(e)463 b(tak)-36 b(en)463 b(to)g(mean)g(in)-36
b(tegrating)463 b(o)-36 b(v)g(er)464 b(a)f(fundamen)-36
b(tal)463 b(domain)g(for)-3718 70966 y(the)349 b(action)g(of)h
Fw(H)13 b FF(.)550 b(Sho)-36 b(wing)349 b(that)g(the)g(op)36
b(erators)349 b Fz(T)22099 71165 y Fy(n)23075 70966 y
FF(are)h(self-adjoin)-36 b(t)350 b(with)f(resp)36 b(ect)349
b(to)g(the)g(P)-36 b(etersson)-3718 72571 y(inner)434
b(pro)36 b(duct)433 b(is)i(a)g(harder)f(computation)h(than)f(Serre)g
([24)q(])g(migh)-36 b(t)435 b(lead)g(one)g(to)f(b)36
b(eliev)-36 b(e)436 b(|)e(it)h(tak)-36 b(es)-3718 74176
y(a)434 b(bit)f(of)h(though)-36 b(t.)p eop
%%Page: 22 30
22 29 bop 1263 -6698 a FF(22)10729 b FA(CHAPTER)434 b(5.)1013
b(RA)-108 b(TIONALITY)434 b(AND)f(INTEGRALITY)i(QUESTIONS)p
eop
%%Page: 23 31
23 30 bop -3718 5686 a FE(Chapter)1033 b(6)-3718 11221
y(Mo)86 b(dular)1034 b(Curv)-86 b(es)-3718 18084 y Fs(6.1)2151
b(Cusp)717 b(F)-179 b(orms)-3718 21005 y FF(Recall)449
b(that)f(if)g Fz(N)586 b FF(is)449 b(a)f(p)36 b(ositiv)-36
b(e)449 b(in)-36 b(teger)448 b(w)-36 b(e)448 b(de\257ne)f(the)g
(congruence)h(subgroups)f(\241\()p Fz(N)139 b FF(\))392
b Fw(\275)h FF(\241)43943 21204 y Fx(1)44469 21005 y
FF(\()p Fz(N)139 b FF(\))392 b Fw(\275)-3718 22610 y
FF(\241)-2905 22809 y Fx(0)-2380 22610 y FF(\()p Fz(N)139
b FF(\))433 b(b)-36 b(y)6238 24990 y(\241)7051 25189
y Fx(0)7576 24990 y FF(\()p Fz(N)139 b FF(\))368 b(=)h
Fw(f)12182 23914 y Fr(\241)13012 24565 y Fy(a)402 b(b)13058
25365 y(c)407 b(d)14579 23914 y Fr(\242)15556 24990 y
Fw(2)369 b FF(SL)18347 25189 y Fx(2)18872 24990 y FF(\()p
FD(Z)p FF(\))g(:)g Fz(c)g Fw(\264)g FF(0)1329 b(\(mo)36
b(d)442 b Fz(N)139 b FF(\))p Fw(g)6238 27043 y FF(\241)7051
27242 y Fx(1)7576 27043 y FF(\()p Fz(N)g FF(\))368 b(=)h
Fw(f)12182 25967 y Fr(\241)13012 26618 y Fy(a)402 b(b)13058
27418 y(c)407 b(d)14579 25967 y Fr(\242)15556 27043 y
Fw(2)369 b FF(SL)18347 27242 y Fx(2)18872 27043 y FF(\()p
FD(Z)p FF(\))g(:)g Fz(a)g Fw(\264)g Fz(d)g Fw(\264)g
FF(1)p Fz(;)221 b(c)370 b Fw(\264)f FF(0)1329 b(\(mo)36
b(d)442 b Fz(N)139 b FF(\))p Fw(g)6764 29096 y FF(\241\()p
Fz(N)g FF(\))367 b(=)i Fw(f)12182 28020 y Fr(\241)13012
28672 y Fy(a)402 b(b)13058 29472 y(c)407 b(d)14579 28020
y Fr(\242)15556 29096 y Fw(2)369 b FF(SL)18347 29295
y Fx(2)18872 29096 y FF(\()p FD(Z)p FF(\))g(:)21896 28020
y Fr(\241)22726 28672 y Fy(a)402 b(b)22773 29472 y(c)k(d)24293
28020 y Fr(\242)25271 29096 y Fw(\264)26673 28020 y Fr(\241)27503
28672 y Fx(1)362 b(0)27503 29428 y(0)g(1)29027 28020
y Fr(\242)30964 29096 y FF(\(mo)36 b(d)442 b Fz(N)139
b FF(\))p Fw(g)p Fz(:)-1767 31476 y FF(Let)424 b(\241)f(b)36
b(e)424 b(one)g(of)h(the)f(ab)36 b(o)-36 b(v)g(e)424
b(subgroups.)574 b(One)424 b(can)g(giv)-36 b(e)425 b(a)f(construction)g
(of)h(the)e(space)i Fz(S)44223 31675 y Fy(k)44791 31476
y FF(\(\241\))f(of)-3718 33081 y(cusp)316 b(forms)i(of)g(w)-36
b(eigh)g(t)318 b Fz(k)362 b FF(for)318 b(the)f(action)h(of)g(\241)f
(using)g(the)g(language)h(of)g(algebraic)g(geometry)-108
b(.)541 b(Let)316 b Fz(X)46063 33280 y Fx(\241)47076
33081 y FF(=)p -3718 33531 3138 54 v -3718 34686 a(\241)p
Fw(nH)-1106 34302 y Fu(\244)-206 34686 y FF(b)36 b(e)375
b(the)f(compactifaction)i(of)g(the)e(upp)36 b(er)374
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%%Page: 24 32
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%%Page: 25 33
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Fw(\\)g FD(R)p FF([[)p Fz(q)48 b FF(]])524 b(and)g(since)h(theorem)f
(5.1)i(assures)e(us)g(that)g(there)g(is)h(a)g FD(C)p
FF(-)-3718 36730 y(basis)434 b(of)h Fz(S)1758 36929 y
Fy(k)2327 36730 y FF(\()p FD(C)p FF(\))e(consisting)i(of)g(forms)f
(with)g(in)-36 b(tegral)434 b(co)36 b(e\261cien)-36 b(ts,)435
b(w)-36 b(e)434 b(see)g(that)g Fz(S)38208 36929 y Fy(k)38777
36730 y FF(\()p FD(R)p FF(\))41273 36361 y Fw(\273)41283
36785 y FF(=)42675 36730 y FD(R)43790 36248 y Fy(d)44764
36730 y FF(where)-3718 38335 y Fz(d)540 b FF(=)g(dim)1217
38534 y Fv(C)2278 38335 y Fz(S)3078 38534 y Fy(k)3646
38335 y FF(\()p FD(C)p FF(\).)881 b(\(An)-36 b(y)533
b(elemen)-36 b(t)534 b(of)h Fz(S)17654 38534 y Fy(k)18223
38335 y FF(\()p FD(R)p FF(\))e(is)i(a)f FD(C)p FF(-linear)h(com)-36
b(bination)534 b(of)h(the)e(in)-36 b(tegral)535 b(basis,)-3718
39940 y(hence)397 b(equating)h(real)g(and)f(imaginary)i(parts,)404
b(an)398 b FD(R)p FF(-linear)f(com)-36 b(bination)397
b(of)i(the)e(in)-36 b(tegral)398 b(basis,)405 b(and)-3718
41545 y(the)528 b(in)-36 b(tegral)529 b(basis)g(sta)-36
b(ys)530 b(indep)36 b(enden)-36 b(t)526 b(o)-36 b(v)g(er)530
b FD(R)p FF(.\))863 b(By)530 b(considering)e(the)h(explicit)g(form)-36
b(ula)530 b(for)f(the)-3718 43150 y(action)345 b(of)g(the)f(Hec)-36
b(k)g(e)345 b(op)36 b(erators)345 b Fz(T)13731 43349
y Fy(n)14702 43150 y FF(on)f Fz(S)17219 43349 y Fy(k)18133
43150 y FF(\(see)g(section)h(3\))g(w)-36 b(e)345 b(see)f(that)g
FD(T)34014 43349 y Fv(R)35225 43150 y FF(lea)-36 b(v)g(es)346
b Fz(S)39665 43349 y Fy(k)40233 43150 y FF(\()p FD(R)p
FF(\))e(in)-36 b(v)-72 b(arian)-36 b(t,)-3718 44756 y(th)g(us)11824
46432 y FD(T)12863 46631 y Fv(R)14098 46432 y FF(=)369
b FD(R)p FF([)p Fz(:)221 b(:)g(:)444 b(;)221 b(T)20268
46631 y Fy(d)20809 46432 y Fz(;)g(:)g(:)g(:)i FF(])370
b Fw(\275)f FF(End)27602 46631 y Fv(R)28689 46432 y Fz(S)29489
46631 y Fy(k)30058 46432 y FF(\()p FD(R)p FF(\))p Fz(:)-3718
48839 y FF(In)433 b(section)h(4)g(w)-36 b(e)434 b(de\257ned)e(a)i(\\p)
36 b(erfect")434 b(pairing)17131 51844 y FD(T)18170 52043
y Fv(C)19304 51844 y Fw(\243)296 b Fz(S)21433 52043 y
Fy(k)22002 51844 y FF(\()p FD(C)p FF(\))369 b Fw(!)g
FD(C)-3718 54848 y FF(whic)-36 b(h)559 b(allo)-36 b(w)g(ed)561
b(us)e(to)h(sho)-36 b(w)559 b(that)g FD(T)15722 55047
y Fv(C)17145 54480 y Fw(\273)17156 54904 y FF(=)18762
54848 y Fz(S)19562 55047 y Fy(k)20130 54848 y FF(\()p
FD(C)p FF(\))p Fz(:)h FF(By)g(restricting)g(to)f FD(R)h
FF(w)-36 b(e)559 b(again)i(get)f(a)f(p)36 b(erfect)-3718
56453 y(pairing)434 b(so)g(w)-36 b(e)433 b(see)h(that)f
FD(T)10161 56652 y Fv(R)11396 56085 y Fw(\273)11407 56509
y FF(=)12799 56453 y Fz(S)13599 56652 y Fy(k)14167 56453
y FF(\()p FD(R)p FF(\))16663 56085 y Fw(\273)16673 56509
y FF(=)18065 56453 y FD(R)19180 55971 y Fy(d)20153 56453
y FF(whic)-36 b(h)433 b(implies)h(that)17249 59458 y
FD(T)18288 59657 y Fy(R)19353 59458 y Fw(\255)20386 59657
y Fv(R)21547 59458 y FD(C)23388 58705 y Fu(\273)22996
59458 y Fw(\241)-846 b(!)370 b FD(T)25920 59657 y Fv(C)26759
59458 y Fz(:)p 47147 62463 45 878 v 47192 61629 781 45
v 47192 62463 V 47972 62463 45 878 v -1767 65277 a FF(This)434
b(also)g(sho)-36 b(ws)434 b(that)f Fz(S)11165 65476 y
Fy(k)11734 65277 y FF(\()p FD(C)p FF(\))14194 64908 y
Fw(\273)14205 65333 y FF(=)15596 65277 y FD(C)295 b Fw(\255)18003
65476 y Fv(R)19165 65277 y Fz(S)19965 65476 y Fy(k)20533
65277 y FF(\()p FD(R)p FF(\))433 b(so)h(w)-36 b(e)434
b(ha)-36 b(v)g(e)434 b(complex)g(conjugation)g(o)-36
b(v)g(er)434 b FD(R)p FF(.)-3718 69860 y Fs(6.7)2151
b(The)717 b(Eic)-60 b(hler-Shim)g(ura)716 b(Isomorphism)-3718
72829 y FF(Our)570 b(goal)h(in)g(this)f(section)h(is)g(to)g(outline)g
(a)g(homological)h(in)-36 b(terpretation)570 b(of)i Fz(S)37769
73028 y Fy(k)38337 72829 y FF(.)990 b(F)-108 b(or)570
b(details)h(see)-3718 74434 y(c)-36 b(hapter)433 b(6)g(of)i(Lang)e([11)
q(],)h(the)f(original)i(pap)36 b(er)433 b(of)i(Shim)-36
b(ura)432 b([32)q(],)i(or)g(c)-36 b(hapter)433 b(VI)36
b(I)g(I)434 b(of)h(Shim)-36 b(ura)432 b([31)q(].)p eop
%%Page: 26 34
26 33 bop 1263 -6698 a FF(26)28833 b FA(CHAPTER)435 b(6.)1012
b(MODULAR)433 b(CUR)-145 b(VES)3214 -3169 y FF(Ho)-36
b(w)524 b(is)f Fz(S)8463 -2970 y Fy(k)9032 -3169 y FF(\()p
FD(C)p FF(\))g(sort)g(of)h(isomorphic)f(to)h Fz(H)25448
-3651 y Fx(1)25973 -3169 y FF(\()p Fz(X)27558 -2970 y
Fx(\241)28201 -3169 y Fz(;)221 b FD(R)p FF(\)?)847 b(Supp)36
b(ose)522 b Fz(k)566 b FF(=)521 b(2)j(and)e(\241)f Fw(\275)h
FF(SL)48048 -2970 y Fx(2)48573 -3169 y FF(\()p FD(Z)p
FF(\))h(is)g(a)1263 -1564 y(congruence)382 b(subgroup,)391
b(let)383 b Fz(X)16802 -1365 y Fx(\241)17814 -1564 y
FF(=)p 19195 -2720 2612 54 v 369 w(\241)p Fw(nH)394 b
FF(b)36 b(e)382 b(the)g(Riemann)g(surface)g(obtained)g(b)-36
b(y)382 b(compactifying)h(the)1263 41 y(upp)36 b(er)433
b(half)h(plane)f(mo)36 b(dulo)434 b(the)f(action)h(of)g(\241.)578
b(Then)433 b Fz(S)29250 240 y Fy(k)29819 41 y FF(\()p
FD(C)p FF(\))369 b(=)g Fz(H)34841 -441 y Fx(0)35366 41
y FF(\()p Fz(X)36951 240 y Fx(\241)37594 41 y Fz(;)221
b FF(\255)39115 -441 y Fx(1)39642 41 y FF(\))433 b(so)h(w)-36
b(e)434 b(ha)-36 b(v)g(e)434 b(a)g(pairing)20135 2683
y Fz(H)21213 2882 y Fx(1)21738 2683 y FF(\()p Fz(X)23323
2882 y Fx(\241)23967 2683 y Fz(;)221 b FD(Z)p FF(\))295
b Fw(\243)h Fz(S)28392 2882 y Fy(k)28961 2683 y FF(\()p
FD(C)p FF(\))369 b Fw(!)g FD(C)1263 5325 y FF(giv)-36
b(en)434 b(b)-36 b(y)434 b(in)-36 b(tegration)23148 7502
y(\()p Fz(\260)72 b(;)221 b(!)48 b FF(\))369 b Fw(7!)28413
5694 y Fr(Z)29151 8701 y Fy(\260)29965 7502 y Fz(!)48
b(:)1263 10501 y FF(This)434 b(giv)-36 b(es)435 b(an)e(em)-36
b(b)36 b(edding)14763 13143 y FD(Z)15676 12595 y Fx(2)p
Fy(d)17055 12775 y Fw(\273)17066 13199 y FF(=)18458 13143
y Fz(H)19536 13342 y Fx(1)20061 13143 y FF(\()p Fz(X)21646
13342 y Fx(\241)22289 13143 y Fz(;)221 b FD(Z)p FF(\))370
b Fz(,)-221 b Fw(!)369 b FF(Hom)29207 13342 y Fv(C)30045
13143 y FF(\()p Fz(S)31351 13342 y Fy(k)31920 13143 y
FF(\()p FD(C)p FF(\))p Fz(;)221 b FD(C)p FF(\))36548
12775 y Fw(\273)36558 13199 y FF(=)37950 13143 y FD(C)39029
12595 y Fy(d)1263 15785 y FF(of)464 b(a)f(\\lattice")h(in)f
FD(C)11817 15303 y Fy(d)12356 15785 y FF(.)667 b(\(W)-108
b(e)462 b(sa)-36 b(y)464 b(\\lattice")g(since)f(there)f(w)-36
b(ere)463 b(some)g(commen)-36 b(ts)462 b(b)-36 b(y)463
b(Rib)36 b(et)463 b(that)f Fz(Z)52058 15303 y Fx(2)p
Fy(d)1263 17391 y FF(isn't)453 b(a)g(lattice)h(b)36 b(ecause)453
b(the)f(rank)h(migh)-36 b(t)453 b(b)36 b(e)452 b(to)36
b(o)454 b(small)g(since)f(a)g(subring)f(of)i FD(C)42175
16908 y Fy(d)43167 17391 y FF(ha)-36 b(ving)454 b FD(Z)p
FF(-rank)f(2)p Fz(d)1263 18996 y FF(migh)-36 b(t)455
b(not)h(spans)f FD(C)11997 18514 y Fy(d)12992 18996 y
FF(o)-36 b(v)g(er)456 b FD(C)p FF(\).)644 b(P)-36 b(assing)456
b(to)g(the)f(quotien)-36 b(t)455 b(\(and)g(compactifying\))i(giv)-36
b(es)456 b(a)g(complex)1263 20601 y(torus)433 b(called)i(the)e
(Jacobian)h(of)g Fz(X)18517 20800 y Fx(\241)19161 20601
y Fz(:)f FF(Again)i(using)e(the)g(ab)36 b(o)-36 b(v)g(e)434
b(pairing)g(w)-36 b(e)434 b(get)g(an)f(em)-36 b(b)36
b(edding)15100 23243 y FD(C)16179 22694 y Fy(d)17087
22874 y Fw(\273)17098 23298 y FF(=)18490 23243 y Fz(S)19290
23442 y Fy(k)19858 23243 y FF(\()p FD(C)p FF(\))369 b
Fz(,)-221 b Fw(!)370 b FF(Hom\()p Fz(H)28449 23442 y
Fx(1)28974 23243 y FF(\()p Fz(X)30559 23442 y Fx(\241)31202
23243 y Fz(;)221 b FD(Z)p FF(\))p Fz(;)g FD(C)p FF(\))35741
22874 y Fw(\273)35752 23298 y FF(=)37143 23243 y FD(C)38222
22694 y Fx(2)p Fy(d)1263 25885 y FF(whic)-36 b(h,)434
b(up)36 b(on)433 b(taking)h(the)f(real)h(part,)f(giv)-36
b(es)11323 28527 y Fz(S)12123 28726 y Fy(k)12692 28527
y FF(\()p FD(C)p FF(\))369 b Fw(!)g FF(Hom)q(\()p Fz(H)21143
28726 y Fx(1)21668 28527 y FF(\()p Fz(X)23253 28726 y
Fx(\241)23896 28527 y Fz(;)221 b FD(Z)p FF(\))p Fz(;)g
FD(R)p FF(\))28470 28158 y Fw(\273)28481 28582 y FF(=)29872
28527 y Fz(H)31053 27979 y Fx(1)31578 28527 y FF(\()p
Fz(X)33163 28726 y Fx(\241)33807 28527 y Fz(;)g FD(R)p
FF(\))36379 28158 y Fw(\273)36389 28582 y FF(=)37781
28527 y Fz(H)38962 27979 y Fx(1)38859 28855 y Fy(p)39487
28527 y FF(\(\241)p Fz(;)g FD(R)p FF(\))1263 31169 y(where)578
b Fz(H)6346 30687 y Fx(1)6243 31497 y Fy(p)6871 31169
y FF(\(\241)p Fz(;)221 b FD(R)p FF(\))577 b(denotes)g(the)g
FC(p)-66 b(ar)g(ab)g(olic)576 b FF(group)h(cohomology)j(of)e(\241)f
(with)h(resp)36 b(ect)577 b(to)g(the)g(trivial)1263 32774
y(action.)694 b(It)473 b(is)f(this)g(result,)481 b(that)472
b(w)-36 b(e)472 b(ma)-36 b(y)473 b(view)g Fz(S)26688
32973 y Fy(k)27257 32774 y FF(\()p FD(C)p FF(\))f(as)g(the)g
(cohomology)i(group)e Fz(H)45795 32292 y Fx(1)45692 33103
y Fy(p)46320 32774 y FF(\(\241)p Fz(;)221 b FD(R)p FF(\),)481
b(that)1263 34379 y(w)-36 b(as)434 b(alluded)g(to)f(ab)36
b(o)-36 b(v)g(e.)3214 35984 y(Shim)g(ura)433 b(generalized)h(this)f
(for)h(arbitrary)g Fz(k)414 b Fw(\270)369 b FF(2)434
b(so)g(that)21830 38626 y Fz(S)22630 38825 y Fy(k)23199
38626 y FF(\()p FD(C)p FF(\))25659 38258 y Fw(\273)25670
38682 y FF(=)27061 38626 y Fz(H)28242 38078 y Fx(1)28139
38955 y Fy(p)28767 38626 y FF(\(\241)p Fz(;)221 b(V)31426
38825 y Fy(k)31996 38626 y FF(\))1263 41269 y(where)493
b Fz(V)5838 41468 y Fy(k)6900 41269 y FF(is)g(a)g Fz(k)381
b Fw(\241)336 b FF(1)493 b(dimensional)g FD(R)p FF(-v)-36
b(ector)492 b(space.)757 b(The)493 b(isomorphism)f(is)i(\(appro)-36
b(ximately\))493 b(the)1263 42874 y(follo)-36 b(wing:)581
b Fz(f)511 b Fw(2)368 b Fz(S)10467 43073 y Fy(k)11036
42874 y FF(\()p FD(C)p FF(\))433 b(is)h(sen)-36 b(t)433
b(to)h(the)f(map)15305 46192 y Fz(\260)441 b Fw(7!)369
b FF(Re)19875 44384 y Fr(Z)21203 44735 y Fy(\260)52 b(\277)22158
44858 y Fn(0)20613 47391 y Fy(\277)21031 47514 y Fn(0)22896
46192 y Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))p Fz(\277)26117
45644 y Fy(i)26492 46192 y Fz(d\277)76 b(;)1522 b(i)369
b FF(=)g(0)p Fz(;)221 b(:)g(:)g(:)445 b(;)221 b(k)341
b Fw(\241)295 b FF(2)p Fz(:)1263 49585 y FF(Let)433 b
Fz(W)550 b FF(=)368 b FD(R)295 b Fw(\251)h FD(R)p FF(,)433
b(then)g(\241)g(acts)h(on)f Fz(W)614 b FF(b)-36 b(y)18918
51055 y Fr(\265)19896 52114 y Fz(a)1168 b(b)19957 53719
y(c)h(d)22361 51055 y Fr(\266)23708 52928 y FF(:)24438
51055 y Fr(\265)25416 52114 y Fz(x)25445 53719 y(y)26155
51055 y Fr(\266)27502 52928 y Fw(7!)29200 51055 y Fr(\265)30177
52114 y Fz(ax)295 b FF(+)g Fz(by)30177 53719 y(cx)h FF(+)e
Fz(dy)34436 51055 y Fr(\266)1263 56256 y FF(so)434 b(\241)f(acts)h(on)
18239 57861 y Fz(V)18997 58060 y Fy(k)19935 57861 y FF(=)369
b(Sym)23809 57304 y Fy(k)24 b Fu(\241)p Fx(2)25801 57861
y Fz(W)550 b FF(=)369 b Fz(W)30369 57313 y Fu(\255)p
Fy(k)24 b Fu(\241)p Fx(2)32871 57861 y Fz(=S)34321 58060
y Fy(k)g Fu(\241)p Fx(2)1263 60061 y FF(where)434 b Fz(S)5821
60260 y Fy(k)24 b Fu(\241)p Fx(2)8025 60061 y FF(is)434
b(the)f(symmetric)h(group)f(on)h Fz(k)340 b Fw(\241)295
b FF(2)434 b(sym)-36 b(b)36 b(ols)434 b(\(note)g(that)f(dim)221
b Fz(V)41149 60260 y Fy(k)42087 60061 y FF(=)369 b Fz(k)340
b Fw(\241)295 b FF(1\).)579 b(Let)19429 62703 y Fz(L)369
b FF(=)g Fz(H)23245 62155 y Fx(1)23142 63031 y Fy(p)23770
62703 y FF(\(\241)p Fz(;)221 b FF(Sym)28164 62146 y Fy(k)24
b Fu(\241)p Fx(2)29935 62703 y FF(\()p FD(Z)295 b Fw(\251)h
FD(Z)p FF(\)\))1263 65345 y(then)433 b(under)f(the)h(isomorphism)21937
66950 y Fz(S)22737 67149 y Fy(k)23305 66950 y FF(\()p
FD(C)p FF(\))25765 66582 y Fw(\273)25776 67006 y FF(=)27168
66950 y Fz(H)28349 66402 y Fx(1)28246 67279 y Fy(p)28874
66950 y FF(\(\241)p Fz(;)221 b FD(R)p FF(\))1263 69150
y Fz(L)302 b FF(is)h(a)g(sublattice)f(of)h Fz(S)12531
69349 y Fy(k)13100 69150 y FF(\()p FD(C)p FF(\))f(of)i
FD(Z)p FF(-rank)e(2)h(whic)-36 b(h)302 b(is)g Fz(T)27539
69349 y Fy(n)28166 69150 y FF(-stable)g(for)h(all)g Fz(n)p
FF(.)535 b(Th)-36 b(us)302 b(w)-36 b(e)303 b(ha)-36 b(v)g(e)302
b(an)h(em)-36 b(b)36 b(edding)22071 71792 y FD(T)23110
71991 y Fv(Z)24198 71792 y FF(=)369 b FD(T)g Fz(,)-221
b Fw(!)369 b FF(End)221 b Fz(L)1263 74434 y FF(and)433
b(so)h FD(T)6428 74633 y Fv(R)7663 74434 y Fw(\275)370
b FF(End)11396 74633 y Fv(R)12262 74434 y FF(\()p Fz(L)294
b Fw(\255)i FD(R)p FF(\))433 b(and)g FD(T)20898 74633
y Fv(Z)21912 74434 y Fw(\255)22945 74633 y Fv(Z)23958
74434 y FD(R)25442 74066 y Fw(\273)25453 74490 y FF(=)26844
74434 y FD(T)27883 74633 y Fv(R)29183 74434 y FF(whic)-36
b(h)433 b(has)h(rank)g Fz(d)p FF(.)p eop
%%Page: 27 35
27 34 bop -3718 -6698 a FA(6.8.)1013 b(THE)434 b(PETTERSON)f(INNER)h
(PR)-36 b(ODUCT)435 b(IS)e(HECKE)h(COMP)-108 b(A)g(TIBLE)8242
b FF(27)-3718 -3169 y Fs(6.8)2151 b(The)586 b(P)-60 b(etterson)586
b(Inner)g(Pro)60 b(duct)586 b(is)g(Hec)-60 b(k)g(e)587
b(Compatible)-3718 -249 y FD(Theorem)499 b(6.8.1.)651
b FC(L)-66 b(et)464 b FF(\241)369 b(=)g(SL)13095 -50
y Fx(2)13620 -249 y FF(\()p FD(Z)p FF(\))o FC(,)465 b(let)f
Fz(f)70 b(;)221 b(g)417 b Fw(2)369 b Fz(S)22622 -50 y
Fy(k)23191 -249 y FF(\()p FD(C)p FF(\))p FC(,)464 b(and)h(let)13597
3343 y Fw(h)p Fz(f)70 b(;)221 b(g)48 b Fw(i)368 b FF(=)18344
1535 y Fr(Z)19082 4543 y Fx(\241)p Fu(nH)21219 3343 y
Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))p 23727 2188 2396
54 v Fz(g)48 b FF(\()p Fz(\277)148 b FF(\))o Fz(y)26804
2795 y Fy(k)27505 2445 y Fz(dxdy)p 27505 3038 2773 54
v 28288 4255 a(y)28970 3871 y Fx(2)30411 3343 y Fz(:)-3718
7199 y FC(Then)472 b(this)g(inte)-66 b(gr)g(al)470 b(is)j(wel)66
b(l-de\257ne)-66 b(d)472 b(and)g(He)-66 b(cke)471 b(c)-66
b(omp)g(atible,)472 b(that)g(is,)i Fw(h)p Fz(f)142 b
Fw(j)p Fz(T)35209 7398 y Fy(n)35835 7199 y Fz(;)221 b(g)48
b Fw(i)383 b FF(=)f Fw(h)p Fz(f)70 b(;)221 b(g)48 b Fw(j)p
Fz(T)42994 7398 y Fy(n)43620 7199 y Fw(i)473 b FC(for)f(al)66
b(l)-3718 8804 y Fz(n)p FC(.)-3718 11516 y(Pr)-66 b(o)g(of.)649
b FF(See)433 b(Chapter)g(3)h(of)h(Lang)e([11)q(].)p 47147
11516 45 878 v 47192 10683 781 45 v 47192 11516 V 47972
11516 45 878 v eop
%%Page: 28 36
28 35 bop 1263 -6698 a FF(28)28833 b FA(CHAPTER)435 b(6.)1012
b(MODULAR)433 b(CUR)-145 b(VES)p eop
%%Page: 29 37
29 36 bop -3718 5686 a FE(Chapter)1033 b(7)-3718 11221
y(Higher)g(W)-258 b(eigh)-86 b(t)1032 b(Mo)86 b(dular)1034
b(F)-258 b(orms)-3718 17254 y FF(W)-108 b(e)381 b(are)g(considering)g
(the)g(spaces)g Fz(S)14255 17453 y Fy(k)14824 17254 y
FF(\()p FD(C)p FF(\),)392 b Fz(S)18468 17453 y Fy(k)19037
17254 y FF(\()p FD(Q)p FF(\))381 b(and)f Fz(S)24828 17453
y Fy(k)25397 17254 y FF(\()p FD(Z)p FF(\))h(whic)-36
b(h)381 b(all)h(ha)-36 b(v)g(e)381 b(rank)g Fz(d)p FF(.)561
b(Eac)-36 b(h)381 b(is)h(acted)-3718 18859 y(up)36 b(on)404
b(b)-36 b(y)405 b(the)f(Hec)-36 b(k)g(e)406 b(algebra)g
FD(T)p FF(.)569 b(W)-108 b(e)405 b(de\257ned)e(a)j(Hec)-36
b(k)g(e)405 b(compatible)g(inner)g(pro)36 b(duct)404
b(\(the)g(P)-36 b(eterrson)-3718 20464 y(pro)36 b(duct\))432
b(and)h(used)g(it)h(to)f(sho)-36 b(w)434 b(that)15984
23094 y Fz(S)16784 23293 y Fy(k)17353 23094 y FF(\()p
FD(Z)p FF(\))19647 22725 y Fw(\273)19658 23149 y FF(=)21049
23094 y(Hom)23758 23293 y Fv(Z)24477 23094 y FF(\()p
FD(T)p Fz(;)221 b FD(Z)p FF(\))p Fz(:)-3718 27486 y Fs(7.1)2151
b(De\257nitions)716 b(of)h(T)-466 30407 y FF(\\I)425
b(ma)-36 b(y)425 b(b)36 b(e)425 b(asking)h(y)-36 b(ou)425
b(to)g(explain)g(something)g(w)-36 b(e)425 b(ha)-36 b(v)g(e)425
b(already)h(discussed,)g(but)e(ha)-36 b(v)g(e)-466 32012
y(w)g(e)433 b(in)-36 b(trinsically)435 b(de\257ned)d(the)h(Hec)-36
b(k)g(e)434 b(op)36 b(erators)434 b(y)-36 b(et?")-466
34127 y({)433 b(Saul)h(Sc)-36 b(hliemer)-1767 36824 y(The)385
b Fz(T)1620 37023 y Fy(n)2633 36824 y FF(are)h(de\257ned)e(as)i(op)36
b(erators)385 b(on)h Fz(S)18996 37023 y Fy(k)19565 36824
y FF(\()p FD(C)p FF(\))f(b)-36 b(y)386 b(de\257ning)f(their)g(action)h
(on)f(mo)36 b(dular)386 b(forms)g(and)-3718 38429 y(noting)359
b(from)h(explicit)h(form)-36 b(ulas)360 b(that)f Fz(S)16518
38628 y Fy(k)17087 38429 y FF(\()p FD(C)p FF(\))g(is)h(preserv)-36
b(ed.)553 b(But)359 b(the)g Fz(T)32477 38628 y Fy(n)33464
38429 y FF(can)g(b)36 b(e)360 b(though)-36 b(t)358 b(of)i(in)g(other)
-3718 40034 y(w)-36 b(a)g(ys,)435 b(for)f(example,)g(since)12948
41639 y Fz(S)13748 41838 y Fy(k)14317 41639 y FF(\()p
FD(C)p FF(\))16777 41270 y Fw(\273)16788 41695 y FF(=)18179
41639 y Fz(H)19360 41091 y Fx(1)19257 41968 y Fy(p)19885
41639 y FF(\(\241)p Fz(;)221 b FF(Sym)24279 41082 y Fy(k)24
b Fu(\241)p Fx(2)26051 41639 y FF(\()p FD(R)294 b Fw(\251)i
FD(R)p FF(\)\))-3718 43834 y(one)433 b(ma)-36 b(y)435
b(giv)-36 b(e)434 b(an)g FC(explicit)d FF(description)j(of)g(the)f
(action)h(of)g Fz(T)26938 44033 y Fy(n)27998 43834 y
FF(on)g Fz(H)30986 43352 y Fx(1)30883 44162 y Fy(p)31511
43834 y FF(.)-3718 48226 y Fs(7.2)2151 b(Double)717 b(Cosets)-3718
51146 y FF(Let)433 b Fz(p)g FF(b)36 b(e)434 b(a)g(prime)f(and)12741
52751 y Fw(M)14336 52950 y Fy(p)15234 52751 y FF(=)369
b Fw(f)p Fz(\256)378 b Fw(2)368 b Fz(M)20996 52950 y
Fx(2)21522 52751 y FF(\()p FD(Z)p FF(\))h(:)h(det)o(\()p
Fz(\256)8 b FF(\))369 b(=)g Fz(p)p Fw(g)p Fz(:)-3718
54946 y FF(Let)410 b Fz(F)549 b FF(:)370 b Fw(L)f(!)g
FD(C)411 b FF(b)36 b(e)410 b(a)h(function)f(on)g(the)g(free)h(ab)36
b(elian)411 b(group)f(of)h(lattices)g(and)f(recall)h(that)f
Fz(T)43117 55145 y Fy(p)44057 54946 y FF(acts)h(on)-3718
56551 y Fz(F)614 b FF(b)-36 b(y)13582 58315 y(\()p Fz(T)14850
58514 y Fy(p)15379 58315 y Fz(F)181 b FF(\)\()p Fz(L)p
FF(\))368 b(=)g Fz(p)21206 57766 y Fy(k)24 b Fu(\241)p
Fx(1)24128 57053 y Fr(X)23932 59948 y Fy(L)24572 59636
y Fo(0)24870 59948 y Fu(\275)p Fy(L)23199 60805 y Fx(\()p
Fy(L)p Fx(:)p Fy(L)25106 60493 y Fo(0)25404 60805 y Fx(\)=)p
Fy(p)27197 58315 y Fz(F)181 b FF(\()p Fz(L)29611 57766
y Fu(0)29921 58315 y FF(\))p Fz(:)-3718 62782 y FF(One)433
b(can)g(write)h Fw(M)6331 62981 y Fy(p)7294 62782 y FF(as)g(a)g
(disjoin)-36 b(t)434 b(union)f(of)h(left)g(cosets,)6856
66226 y Fw(M)8451 66425 y Fy(p)9349 66226 y FF(=)369
b(SL)12266 66425 y Fx(2)12791 66226 y FF(\()p FD(Z)p
FF(\))14937 64353 y Fr(\265)15915 65412 y Fz(p)1107 b
FF(0)15916 67017 y(0)i(1)18325 64353 y Fr(\266)19524
66226 y FF(SL)21060 66425 y Fx(2)21585 66226 y FF(\()p
FD(Z)p FF(\))369 b(=)25932 64964 y Fr([)25576 67793 y
Fy(ad)p Fx(=)p Fy(p)25260 68700 y Fx(0)p Fu(\267)p Fy(b<d)28302
64353 y Fr(\265)29280 65412 y Fz(a)1168 b(b)29296 67017
y FF(0)1124 b Fz(d)31745 64353 y Fr(\266)32945 66226
y FF(SL)34481 66425 y Fx(2)35006 66226 y FF(\()p FD(Z)p
FF(\))221 b Fz(:)-3718 71113 y FF(Then)-197 70117 y Fr(P)1205
71501 y Fx(\()p Fy(L)p Fx(:)p Fy(L)3112 71249 y Fo(0)3410
71501 y Fx(\)=)p Fy(p)5258 71113 y Fz(L)6143 70631 y
Fu(0)7012 71113 y FF(ma)-36 b(y)559 b(b)36 b(e)558 b(though)-36
b(t)557 b(of)i(as)g(the)f(sum)g(of)h(the)f(images)h(of)g
Fz(L)f FF(under)f(the)h(action)h(of)-3718 72829 y(the)533
b(left)i(cosets)f(of)g Fw(M)8047 73028 y Fy(p)8577 72829
y FF(.)879 b(F)-108 b(or)533 b(a)h(complete)g(exp)36
b(osition,)561 b(in)533 b(greater)h(generalit)-36 b(y)-108
b(,)560 b(see)534 b(Shim)-36 b(ura)533 b([31)q(],)-3718
74434 y(esp)36 b(ecially)435 b(c)-36 b(hapter)433 b(3.)21534
77755 y(29)p eop
%%Page: 30 38
30 37 bop 1263 -6698 a FF(30)17672 b FA(CHAPTER)435 b(7.)1012
b(HIGHER)434 b(WEIGHT)g(MODULAR)f(F)-36 b(ORMS)1263 -3169
y Fs(7.3)2152 b(More)717 b(General)g(Congruence)f(Subgroups)1263
-249 y FD(De\257nition)500 b(7.3.1.)651 b FF(A)434 b
FD(Diric)-42 b(hlet)500 b(c)-42 b(haracter)436 b FF(mo)36
b(d)434 b Fz(N)571 b FF(is)434 b(a)g(homomorphism)22081
2431 y Fz(")369 b FF(:)g(\()p FD(Z)p Fz(=n)p FD(Z)p FF(\))28053
1882 y Fu(\244)28949 2431 y Fw(!)g FD(C)31725 1882 y
Fu(\244)1263 5111 y FF(extended)433 b(to)h FD(Z)p Fz(=)-72
b(N)139 b FD(Z)433 b FF(b)-36 b(y)434 b(putting)e Fz(")p
FF(\()p Fz(m)p FF(\))368 b(=)h(0)434 b(if)g(\()p Fz(m;)221
b(N)139 b FF(\))368 b Fw(6)p FF(=)h(1.)3214 7605 y(Fix)434
b(in)-36 b(tegers)434 b Fz(k)414 b Fw(\270)369 b FF(0)434
b(and)f Fz(N)507 b Fw(\270)369 b FF(1.)579 b(In)433 b(this)h(section)f
(w)-36 b(e)434 b(consider)g(the)f(spaces)23657 10285
y Fz(S)24457 10484 y Fy(k)25025 10285 y FF(\(\241)26344
10484 y Fx(1)26870 10285 y FF(\()p Fz(N)139 b FF(\))p
Fz(;)221 b(\262)p FF(\))1263 12965 y(for)434 b(Diric)-36
b(hlet)434 b(c)-36 b(haracters)434 b Fz(")f FF(mo)36
b(d)433 b Fz(N)572 b FF(and)433 b(explicitly)j(describ)36
b(e)433 b(the)g(action)h(of)g(the)f(Hec)-36 b(k)g(e)434
b(op)36 b(erators)21168 14462 y Fr(\()22238 15823 y Fz(T)23000
16022 y Fy(n)23627 15823 y Fz(;)1621 b(n)370 b Fw(\270)f
FF(1)22238 17749 y Fw(h)p Fz(d)p Fw(i)p Fz(;)1300 b(d)369
b Fw(2)g FF(\()p FD(Z)p Fz(=)-72 b(N)139 b FD(Z)p FF(\))32506
17267 y Fu(\244)1263 20502 y FF(on)434 b(these)f(spaces.)1263
22551 y FC(R)-66 b(emark)464 b(7.3.2.)649 b FF(Let)487
b Fz(n)g FF(b)36 b(e)487 b(a)g(p)36 b(ositiv)-36 b(e)488
b(in)-36 b(teger.)738 b(If)487 b(\()p Fz(n;)221 b(N)139
b FF(\))460 b(=)g(1,)500 b(then)486 b(the)h Fz(T)40776
22750 y Fy(n)41889 22551 y FF(b)36 b(eha)-36 b(v)g(e)487
b(lik)-36 b(e)488 b(they)f(do)1263 24157 y(for)434 b(SL)4786
24356 y Fx(2)5311 24157 y FF(\()p FD(Z)p FF(\).)579 b(In)433
b(fact,)h(the)g Fz(T)15730 24356 y Fy(n)16790 24157 y
FF(and)f Fw(h)p Fz(d)p Fw(i)f FF(comm)-36 b(ute)433 b(and)19988
26836 y(\()p Fz(f)142 b Fw(j)p Fz(T)22408 27035 y Fy(n)23035
26836 y Fz(;)221 b(g)48 b FF(\))369 b(=)f(\()p Fz(f)70
b(;)221 b(g)48 b Fw(jh)p Fz(n)p Fw(i)31192 26288 y Fu(\241)p
Fx(1)32449 26836 y Fz(T)33211 27035 y Fy(n)33838 26836
y FF(\))20573 29516 y(\()p Fz(f)142 b Fw(jh)p Fz(d)p
Fw(i)p Fz(;)221 b(g)48 b FF(\))367 b(=)i(\()p Fz(f)70
b(;)221 b(g)48 b Fw(jh)p Fz(d)p Fw(i)31997 28968 y Fu(\241)p
Fx(1)33253 29516 y FF(\))1263 31732 y(so)468 b(the)f
Fz(T)5930 31931 y Fy(n)7024 31732 y FF(\(for)h Fz(n)g
FF(prime)f(to)h Fz(N)139 b FF(\))467 b(and)g Fw(h)p Fz(d)p
Fw(i)f FF(are)i(sim)-36 b(ultaneously)468 b(diagonalizable.)682
b(But)467 b(if)h(\()p Fz(n;)221 b(N)139 b FF(\))427 b
Fw(6)p FF(=)f(1)1263 33337 y(then)433 b Fz(T)4988 33536
y Fy(n)6048 33337 y FF(ma)-36 b(y)434 b(not)f(b)36 b(e)434
b(diagonalizable.)1263 35832 y FD(De\257nition)500 b(7.3.3.)651
b FF(Let)11338 38512 y Fz(S)12138 38711 y Fy(k)12707
38512 y FF(\(\241)14026 38711 y Fx(1)14551 38512 y FF(\()p
Fz(N)139 b FF(\)\))368 b(=)h Fw(f)p Fz(f)511 b FF(:)369
b Fz(f)142 b FF(\()p Fz(\260)72 b(\277)148 b FF(\))369
b(=)f(\()p Fz(c\277)443 b FF(+)295 b Fz(d)p FF(\))31112
37963 y Fy(k)31681 38512 y Fz(f)142 b FF(\()p Fz(\277)148
b FF(\))433 b(all)i Fz(\260)440 b Fw(2)369 b FF(\241)39612
38711 y Fx(1)40138 38512 y FF(\()p Fz(N)139 b FF(\))p
Fw(g)1263 41191 y FF(where)446 b(the)f Fz(f)588 b FF(are)446
b(assumed)f(holomorphic)h(on)g Fw(H)316 b([)303 b(f)p
FF(cusps)p Fw(g)p FF(.)615 b(F)-108 b(or)446 b(eac)-36
b(h)445 b(Diric)-36 b(hlet)447 b(c)-36 b(haracter)445
b Fz(")h FF(mo)36 b(d)1263 42797 y Fz(N)572 b FF(let)7533
44402 y Fz(S)8333 44601 y Fy(k)8902 44402 y FF(\(\241)10221
44601 y Fx(1)10746 44402 y FF(\()p Fz(N)139 b FF(\))p
Fz(;)221 b(")p FF(\))368 b(=)h Fw(f)p Fz(f)511 b FF(:)369
b Fz(f)142 b FF(\()p Fz(\260)72 b(\277)148 b FF(\)\()p
Fz(c\277)443 b FF(+)295 b Fz(d)p FF(\))26749 43853 y
Fu(\241)p Fy(k)28418 44402 y FF(=)368 b Fz(")p FF(\()p
Fz(d)p FF(\))p Fz(f)575 b FF(all)435 b Fz(\260)440 b
FF(=)37614 43326 y Fr(\241)38444 43977 y Fy(a)402 b(b)38490
44777 y(c)407 b(d)40011 43326 y Fr(\242)40989 44402 y
Fw(2)368 b FF(\241)43056 44601 y Fx(0)43582 44402 y FF(\()p
Fz(N)139 b FF(\))p Fw(g)p Fz(:)3214 46896 y FF(When)433
b Fz(")369 b Fw(6)p FF(=)g(0)433 b(and)h Fz(f)510 b Fw(2)369
b Fz(S)16185 47095 y Fy(k)16754 46896 y FF(\(\241)18073
47095 y Fx(1)18598 46896 y FF(\()p Fz(N)139 b FF(\))p
Fz(;)221 b(")p FF(\))433 b(one)g(calls)i Fz(")e FF(the)g
FD(neb)42 b(en)-42 b(t)g(ypus)435 b FF(of)g Fz(f)142
b FF(.)3214 48501 y(Let)515 b Fz(d)509 b Fw(2)f FF(\()p
FD(Z)p Fz(=)-72 b(N)139 b FD(Z)p FF(\))12802 48019 y
Fu(\244)13843 48501 y FF(and)515 b(let)g Fz(f)651 b Fw(2)508
b Fz(S)21900 48700 y Fy(k)22469 48501 y FF(\(\241)23788
48700 y Fx(1)24313 48501 y FF(\()p Fz(N)139 b FF(\)\).)823
b(Let)515 b Fz(\260)581 b FF(=)33384 47425 y Fr(\241)34214
48077 y Fy(a)402 b(b)34229 48877 y Fx(0)376 b Fy(d)35781
47425 y Fr(\242)36898 48501 y Fw(2)508 b FF(\241)39105
48700 y Fx(1)39631 48501 y FF(\()p Fz(N)139 b FF(\))514
b(b)36 b(e)515 b(a)h(matrix)g(whose)1263 50106 y(lo)-36
b(w)g(er)434 b(righ)-36 b(t)434 b(en)-36 b(try)433 b(is)h(congruen)-36
b(t)432 b(to)i Fz(d)f FF(mo)36 b(d)434 b Fz(N)139 b FF(.)578
b(Then)433 b(w)-36 b(e)434 b(de\257ne)19259 52786 y Fz(f)142
b FF(\()p Fz(\277)148 b FF(\))p Fw(jh)p Fz(d)p Fw(i)367
b FF(=)i Fz(f)142 b FF(\()p Fz(\260)72 b(\277)148 b FF(\)\()p
Fz(c\277)443 b FF(+)295 b Fz(d)p FF(\))33412 52238 y
Fu(\241)p Fy(k)34712 52786 y Fz(:)3214 55466 y FF(Since)444
b Fz(f)142 b Fw(jh)p Fz(d)p Fw(i)387 b FF(=)g Fz(")p
FF(\()p Fz(d)p FF(\))p Fz(f)142 b FF(,)446 b Fz(S)15956
55665 y Fy(k)16525 55466 y FF(\(\241)17844 55665 y Fx(1)18369
55466 y FF(\()p Fz(N)139 b FF(\))p Fz(;)221 b(")p FF(\))444
b(is)h(the)f Fz(")p FF(\()p Fz(d)p FF(\))f(eigenspace)i(of)g
Fw(h)p Fz(d)p Fw(i)f FF(and)g Fw(h)p Fz(d)p Fw(i)f FF(is)i
(diagonalizable)1263 57071 y(so)434 b(one)g(has)f(a)h(direct)f(sum)g
(decomp)36 b(osition)16145 59910 y Fz(S)16945 60109 y
Fy(k)17514 59910 y FF(\(\241)18833 60109 y Fx(1)19358
59910 y FF(\()p Fz(N)139 b FF(\)\))368 b(=)26150 58648
y Fr(M)23805 61526 y Fy(")p Fx(:\()p Fv(Z)p Fy(=)-52
b(N)94 b Fv(Z)p Fx(\))27817 61274 y Fo(\244)28298 61526
y Fu(!)p Fv(C)30022 61274 y Fo(\244)30723 59910 y Fz(S)31523
60109 y Fy(k)32092 59910 y FF(\(\241)33411 60109 y Fx(1)33937
59910 y FF(\()p Fz(N)139 b FF(\))p Fz(;)221 b(")p FF(\))p
Fz(:)1263 64062 y FF(If)434 b Fz(f)511 b Fw(2)369 b Fz(S)5770
64261 y Fy(k)6339 64062 y FF(\(\241)7658 64261 y Fx(1)8183
64062 y FF(\()p Fz(N)139 b FF(\))p Fz(;)221 b(")p FF(\))433
b(then)21374 64569 y Fr(\265)22352 65628 y Fw(\241)p
FF(1)1624 b(0)22869 67233 y(0)f Fw(\241)p FF(1)26826
64569 y Fr(\266)28173 66442 y Fw(2)368 b FF(\241)30240
66641 y Fx(0)30766 66442 y FF(\()p Fz(N)139 b FF(\))1263
69008 y(so)19617 70613 y Fz(f)j FF(\()p Fw(\241)p Fz(\277)148
b FF(\)\()p Fw(\241)p FF(1\))25853 70065 y Fu(\241)p
Fy(k)27523 70613 y FF(=)368 b Fz(")p FF(\()p Fw(\241)p
FF(1\))p Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))1263
72829 y(so)502 b(that)g Fz(S)6615 73028 y Fy(k)7184 72829
y FF(\(\241)8503 73028 y Fx(1)9028 72829 y FF(\()p Fz(N)139
b FF(\))p Fz(;)221 b(")p FF(\))484 b(=)h(0)503 b(unless)e
Fz(")p FF(\()p Fw(\241)p FF(1\))485 b(=)g(\()p Fw(\241)p
FF(1\))27945 72347 y Fy(k)28514 72829 y FF(.)784 b(Th)-36
b(us)501 b(ab)36 b(out)502 b(half)g(of)h(the)e(direct)h(summands)1263
74434 y(v)-72 b(anish.)p eop
%%Page: 31 39
31 38 bop -3718 -6698 a FA(7.4.)1013 b(EXPLICIT)434 b(F)-36
b(ORMULAS)33202 b FF(31)-3718 -3169 y Fs(7.4)2151 b(Explicit)717
b(F)-179 b(orm)-60 b(ulas)-3718 -249 y FF(Let)14206 2036
y Fz(f)511 b FF(=)17227 376 y Fu(1)16738 774 y Fr(X)16811
3564 y Fy(n)p Fx(=1)18878 2036 y Fz(a)19561 2235 y Fy(n)20187
2036 y Fz(q)20812 1488 y Fy(n)21807 2036 y Fw(2)368 b
Fz(S)23861 2235 y Fy(k)24430 2036 y FF(\(\241)25749 2235
y Fx(1)26275 2036 y FF(\()p Fz(N)139 b FF(\))p Fz(;)221
b(")p FF(\))-3718 5450 y(and)433 b(let)h Fz(p)f FF(b)36
b(e)433 b(a)h(prime,)g(then)8918 11314 y Fz(f)142 b Fw(j)p
Fz(T)10832 11513 y Fy(p)11730 11314 y FF(=)13111 7395
y Fr(8)13111 8591 y(>)13111 8989 y(>)13111 9388 y(>)13111
9786 y(<)13111 12177 y(>)13111 12576 y(>)13111 12974
y(>)13111 13373 y(:)14781 7661 y Fu(1)14292 8059 y Fr(X)14365
10849 y Fy(n)p Fx(=1)16432 9321 y Fz(a)17115 9520 y Fy(np)18215
9321 y Fz(q)18840 8773 y Fy(n)19760 9321 y FF(+)295 b
Fz(p)21720 8773 y Fy(k)24 b Fu(\241)p Fx(1)23491 9321
y Fz(")p FF(\()p Fz(p)p FF(\))26475 7661 y Fu(1)25986
8059 y Fr(X)26058 10849 y Fy(n)p Fx(=1)28126 9321 y Fz(a)28809
9520 y Fy(n)29435 9321 y Fz(q)30060 8773 y Fy(pn)31159
9321 y Fz(;)1301 b(p)-74 b Fw(6)369 b(j)p Fz(N)14781
11496 y Fu(1)14292 11894 y Fr(X)14365 14684 y Fy(n)p
Fx(=1)16432 13156 y Fz(a)17115 13355 y Fy(np)18215 13156
y Fz(q)18840 12608 y Fy(n)19760 13156 y FF(+)295 b(0)p
Fz(;)10743 b(p)p Fw(j)p Fz(N)-3718 17252 y FF(When)433
b Fz(p)p Fw(j)p Fz(N)139 b FF(,)433 b Fz(T)3834 17451
y Fy(p)4797 17252 y FF(is)h(often)f(denoted)g Fz(U)15189
17451 y Fy(p)16152 17252 y FF(and)g(called)h(an)g(A)-36
b(tkin-Lehner)432 b(op)36 b(erator.)-1767 18857 y(W)-108
b(e)433 b(ha)-36 b(v)g(e)434 b(the)f(relations)14784
21791 y Fz(T)15546 21990 y Fy(m)16434 21791 y Fz(T)17196
21990 y Fy(n)18192 21791 y FF(=)368 b Fz(T)20334 21990
y Fy(mn)21793 21791 y Fz(;)1523 b FF(\()p Fz(m;)221 b(n)p
FF(\))369 b(=)g(1)16023 25040 y Fz(T)16785 25296 y Fy(p)17259
25043 y Ft(k)18192 25040 y FF(=)19572 22768 y Fr(\()20643
24130 y FF(\()p Fz(T)21911 24329 y Fy(p)22440 24130 y
FF(\))22946 23648 y Fy(k)23514 24130 y Fz(;)1301 b(p)p
Fw(j)p Fz(N)20643 26056 y FF(?)p Fz(;)3558 b(p)-74 b
Fw(6)369 b(j)p Fz(N)-3718 30141 y Fs(7.5)2151 b(Old)717
b(and)g(New)g(F)-179 b(orms)-3718 33061 y FD(W)-125 b(arning:)561
b Fz(T)3597 33260 y Fy(p)4522 33061 y FF(is)395 b(not)h(necessarily)g
(diagonalizable)i(if)e Fz(p)p Fw(j)p Fz(N)139 b FF(.)565
b(There)395 b(is)h(an)f(example)i(due)e(to)g(Shim)-36
b(ura,)-3718 34667 y(to)434 b(presen)-36 b(t)432 b(it)i(w)-36
b(e)434 b(m)-36 b(ust)432 b(\257rst)h(in)-36 b(tro)36
b(duce)433 b(old)h(and)f(new)g(forms.)-1767 36272 y(Let)353
b Fz(M)492 b FF(and)352 b Fz(N)492 b FF(b)36 b(e)352
b(p)36 b(ositiv)-36 b(e)354 b(in)-36 b(tegers)353 b(suc)-36
b(h)353 b(that)f Fz(M)139 b Fw(j)p Fz(N)492 b FF(and)352
b(let)i Fz(d)p Fw(j)31762 35749 y Fy(N)p 31684 35966
998 54 v 31684 36730 a(M)32814 36272 y FF(.)552 b(If)353
b Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))369 b Fw(2)g
Fz(S)39878 36471 y Fy(k)40446 36272 y FF(\(\241)41765
36471 y Fx(1)42291 36272 y FF(\()p Fz(M)139 b FF(\)\))352
b(then)-3718 37877 y Fz(f)142 b FF(\()p Fz(d\277)148
b FF(\))368 b Fw(2)h Fz(S)1889 38076 y Fy(k)2458 37877
y FF(\(\241)3777 38076 y Fx(1)4302 37877 y FF(\()p Fz(N)139
b FF(\)\).)546 b(W)-108 b(e)341 b(th)-36 b(us)340 b(ha)-36
b(v)g(e)341 b(a)g(map)g Fz(S)20319 38076 y Fy(k)20887
37877 y FF(\(\241)22206 38076 y Fx(1)22732 37877 y FF(\()p
Fz(M)139 b FF(\)\))368 b Fw(!)i Fz(S)28513 38076 y Fy(k)29081
37877 y FF(\(\241)30400 38076 y Fx(1)30926 37877 y FF(\()p
Fz(N)139 b FF(\)\))339 b(for)j(eac)-36 b(h)340 b Fz(d)p
Fw(j)39948 37354 y Fy(N)p 39870 37571 V 39870 38335 a(M)41001
37877 y FF(.)547 b(Com)-36 b(bining)-3718 39482 y(these)433
b(giv)-36 b(es)435 b(a)f(map)12696 41246 y Fz(')13548
41445 y Fy(M)14970 41246 y FF(:)15700 39984 y Fr(M)15777
43013 y Fy(d)p Fu(j)16720 42651 y Ft(N)p 16655 42812
844 40 v 16655 43361 a(M)17929 41246 y Fz(S)18729 41445
y Fy(k)19298 41246 y FF(\(\241)20617 41445 y Fx(1)21142
41246 y FF(\()p Fz(d)p FF(\)\))368 b Fw(!)i Fz(S)26202
41445 y Fy(k)26770 41246 y FF(\(\241)28089 41445 y Fx(1)28615
41246 y FF(\()p Fz(N)139 b FF(\)\))p Fz(:)-3718 45708
y FD(De\257nition)499 b(7.5.1.)652 b FF(The)454 b FD(old)522
b(part)456 b FF(of)e Fz(S)18106 45907 y Fy(k)18675 45708
y FF(\(\241)19994 45907 y Fx(1)20520 45708 y FF(\()p
Fz(N)139 b FF(\)\))452 b(is)j(the)f(subspace)f(generated)h(b)-36
b(y)454 b(the)g(images)h(of)-3718 47313 y(the)433 b Fz(')-626
47512 y Fy(M)861 47313 y FF(for)h Fz(M)139 b Fw(j)p Fz(N)g
FF(,)433 b Fz(M)508 b Fw(6)p FF(=)369 b Fz(N)139 b FF(.)-1767
50025 y(W)-108 b(e)522 b(remark)g(that)f(the)g FD(new)600
b(part)523 b FF(of)f Fz(S)19157 50224 y Fy(k)19726 50025
y FF(\(\241)21045 50224 y Fx(1)21570 50025 y FF(\()p
Fz(N)139 b FF(\)\))521 b(is)h(the)f(orthogonal)h(complemen)-36
b(t)522 b(of)g(the)f(old)-3718 51630 y(part)433 b(with)h(resp)36
b(ect)433 b(to)g(the)g(P)-36 b(etersson)434 b(inner)f(pro)36
b(duct.)p eop
%%Page: 32 40
32 39 bop 1263 -6698 a FF(32)17672 b FA(CHAPTER)435 b(7.)1012
b(HIGHER)434 b(WEIGHT)g(MODULAR)f(F)-36 b(ORMS)p eop
%%Page: 33 41
33 40 bop -3718 5718 a FE(Chapter)1033 b(8)-3718 11284
y(New)f(F)-258 b(orms)-3718 17348 y FF(T)-108 b(o)36
b(da)-36 b(y)491 b(w)-36 b(e)491 b(discuss)f(ho)-36 b(w)491
b(the)f(Hec)-36 b(k)g(e)491 b(op)36 b(erators)490 b Fz(T)22147
17547 y Fy(n)23264 17348 y FF(on)h Fz(S)25928 17547 y
Fy(k)26497 17348 y FF(\(\241)27816 17547 y Fx(1)28341
17348 y FF(\()p Fz(N)139 b FF(\)\))489 b(can)i(fail)g(to)g(b)36
b(e)490 b(diagonalizable.)-3718 18953 y(Let)433 b Fz(N)572
b FF(b)36 b(e)433 b(a)h(p)36 b(ositiv)-36 b(e)435 b(in)-36
b(teger)433 b(and)g Fz(M)573 b FF(a)434 b(divisor)g(of)g
Fz(N)139 b FF(.)578 b(F)-108 b(or)433 b(eac)-36 b(h)434
b Fz(d)p Fw(j)31984 18430 y Fy(N)p 31906 18647 998 54
v 31906 19411 a(M)33470 18953 y FF(w)-36 b(e)433 b(de\257ne)g(a)h(map)
8640 21980 y Fz(\256)9467 22179 y Fy(d)10376 21980 y
FF(:)369 b Fz(S)11906 22179 y Fy(k)12475 21980 y FF(\(\241)13794
22179 y Fx(1)14319 21980 y FF(\()p Fz(M)139 b FF(\)\))368
b Fw(!)i Fz(S)20100 22179 y Fy(k)20669 21980 y FF(\(\241)21988
22179 y Fx(1)22513 21980 y FF(\()p Fz(N)139 b FF(\)\))368
b(:)1670 b Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))368
b Fw(7!)i Fz(f)142 b FF(\()p Fz(d\277)148 b FF(\))p Fz(:)-3718
25007 y FF(Note)502 b(that)g(when)g Fz(T)6607 25206 y
Fy(p)7638 25007 y FF(acts)h(on)f(the)f(image)i(space)g
Fz(S)22742 25206 y Fy(k)23310 25007 y FF(\(\241)24629
25206 y Fx(1)25155 25007 y FF(\()p Fz(N)139 b FF(\)\))501
b(w)-36 b(e)502 b(will)i(often)e(denote)g(it)g(b)-36
b(y)502 b Fz(U)44607 25206 y Fy(p)45136 25007 y FF(.)784
b(W)-108 b(e)-3718 26613 y(m)-36 b(ust)433 b(c)-36 b(hec)g(k)433
b(that)g Fz(f)142 b FF(\()p Fz(d\277)148 b FF(\))369
b Fw(2)f Fz(S)11434 26812 y Fy(k)12003 26613 y FF(\(\241)13322
26812 y Fx(1)13848 26613 y FF(\()p Fz(N)139 b FF(\)\).)577
b(De\257ne)433 b(for)h Fz(\260)441 b FF(=)25997 25537
y Fr(\241)26827 26188 y Fy(a)402 b(b)26873 26988 y(c)407
b(d)28393 25537 y Fr(\242)29002 26613 y FF(,)10458 29774
y(\()p Fz(f)142 b Fw(j)p FF([)p Fz(\260)72 b FF(])13585
29973 y Fy(k)14154 29774 y FF(\)\()p Fz(\277)148 b FF(\))369
b(=)f(det\()p Fz(\260)72 b FF(\))21700 29225 y Fy(k)24
b Fu(\241)p Fx(1)23470 29774 y FF(\()p Fz(cz)355 b FF(+)294
b Fz(d)p FF(\))27983 29225 y Fu(\241)p Fy(k)29284 29774
y Fz(f)142 b FF(\()p Fz(\260)72 b FF(\()p Fz(\277)148
b FF(\)\))p Fz(:)-3718 32801 y FF(Th)-36 b(us)369 b Fz(f)511
b Fw(2)368 b Fz(S)2719 33000 y Fy(k)3288 32801 y FF(\(\241)4607
33000 y Fx(1)5132 32801 y FF(\()p Fz(N)139 b FF(\)\))369
b(i\256)g Fz(f)142 b Fw(j)p FF([)p Fz(\260)72 b FF(])12310
33000 y Fy(k)12879 32801 y FF(\()p Fz(\277)148 b FF(\))369
b(=)g Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))369 b(\(and)g
Fz(f)511 b FF(is)370 b(holomorphic\).)557 b(No)-36 b(w)370
b(let)g Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))369 b
Fw(2)f FF(\241)42688 33000 y Fx(1)43214 32801 y FF(\()p
Fz(M)139 b FF(\))369 b(and)-3718 34406 y(let)476 b Fz(\266)-1337
34605 y Fy(d)-357 34406 y FF(=)1095 33330 y Fr(\241)1925
34004 y Fy(d)362 b Fx(0)1932 34759 y(0)369 b(1)3462 33330
y Fr(\242)4071 34406 y FF(.)705 b(Then)475 b Fz(f)142
b Fw(j)p FF([)p Fz(\266)10548 34605 y Fy(d)11088 34406
y FF(])11449 34605 y Fy(k)12018 34406 y FF(\()p Fz(\277)148
b FF(\))441 b(=)f Fz(d)16312 33924 y Fy(k)24 b Fu(\241)p
Fx(1)18083 34406 y Fz(f)142 b FF(\()p Fz(d\277)148 b
FF(\))475 b(is)h(a)g(mo)36 b(dular)476 b(form)g(on)g(\241)35202
34605 y Fx(1)35727 34406 y FF(\()p Fz(N)139 b FF(\))475
b(since)h Fz(\266)42084 33856 y Fu(\241)p Fx(1)42084
34776 y Fy(d)43341 34406 y FF(\241)44154 34605 y Fx(1)44680
34406 y FF(\()p Fz(M)139 b FF(\))p Fz(\266)47549 34605
y Fy(d)-3718 36011 y FF(con)-36 b(tains)502 b(\241)2265
36210 y Fx(1)2790 36011 y FF(\()p Fz(N)139 b FF(\))501
b(\(c)-36 b(hec)g(k)502 b(this)g(directly)g(b)-36 b(y)502
b(conjugating)h(an)f(elemen)-36 b(t)501 b(of)i(\241)35069
36210 y Fx(1)35595 36011 y FF(\()p Fz(N)139 b FF(\))500
b(b)-36 b(y)502 b Fz(\266)40623 36210 y Fy(d)41163 36011
y FF(\).)783 b(Moreo)-36 b(v)g(er)-3718 37616 y(if)530
b Fz(f)671 b FF(is)529 b(a)h(cusp)f(form)g(then)f(so)i(is)g
Fz(f)142 b Fw(j)p FF([)p Fz(\266)15825 37815 y Fy(d)16364
37616 y FF(])16725 37815 y Fy(k)17294 37616 y FF(.)866
b(If)529 b Fz(f)674 b Fw(2)532 b Fz(S)23449 37815 y Fy(k)24017
37616 y FF(\(\241)25336 37815 y Fx(1)25862 37616 y FF(\()p
Fz(M)139 b FF(\)\))528 b(is)i(nonzero,)553 b(then)528
b(as)i Fz(d)f FF(v)-72 b(aries)530 b(o)-36 b(v)g(er)-3718
39221 y(divisors)434 b(of)2722 38698 y Fy(N)p 2643 38916
V 2643 39679 a(M)3774 39221 y FF(,)g(the)f(v)-72 b(arious)434
b Fz(f)142 b FF(\()p Fz(d\277)148 b FF(\))433 b(are)h(linearly)h(indep)
36 b(enden)-36 b(t.)-1767 40858 y(Supp)36 b(ose)420 b
Fz(f)511 b Fw(2)369 b Fz(S)6529 41057 y Fy(k)7098 40858
y FF(\(\241)8417 41057 y Fx(1)8942 40858 y FF(\()p Fz(M)139
b FF(\)\))421 b(is)h(a)f(normalized)h(eigenform)h(for)f(all)g(of)g(the)
f(Hec)-36 b(k)g(e)422 b(op)36 b(erators)422 b Fz(T)44944
41057 y Fy(n)45992 40858 y FF(and)-3718 42463 y Fw(h)p
Fz(n)p Fw(i)p FF(,)433 b(and)g Fz(p)h FF(is)g(a)f(prime)h(not)f
(dividiing)h Fz(M)139 b FF(.)579 b(Then)13041 45490 y
Fz(f)142 b Fw(j)p Fz(T)14955 45689 y Fy(p)15853 45490
y FF(=)369 b Fz(af)1442 b FF(and)1300 b Fz(f)142 b Fw(jh)p
Fz(p)p Fw(i)368 b FF(=)h Fz(")p FF(\()p Fz(p)p FF(\))p
Fz(f)70 b(:)-3718 48517 y FF(Assume)433 b Fz(N)507 b
FF(=)369 b Fz(p)4684 48035 y Fy(r)5190 48517 y Fz(M)572
b FF(where)434 b Fz(r)470 b FF(is)434 b(an)f(in)-36 b(teger)434
b Fw(\270)369 b FF(1.)578 b(Let)17990 51545 y Fz(f)18631
51744 y Fy(i)19007 51545 y FF(\()p Fz(\277)148 b FF(\))369
b(=)f Fz(f)142 b FF(\()p Fz(p)24423 50996 y Fy(i)24799
51545 y Fz(\277)148 b FF(\))p Fz(;)-3718 54572 y FF(so)485
b Fz(f)-1429 54771 y Fx(0)-903 54572 y Fz(;)221 b(:)g(:)g(:)446
b(;)221 b(f)2873 54771 y Fy(r)3865 54572 y FF(are)485
b(the)g(images)h(of)g Fz(f)627 b FF(under)484 b(the)h(maps)g(de\257ned)
f(ab)36 b(o)-36 b(v)g(e)486 b(and)f Fz(f)599 b FF(=)456
b Fz(f)39182 54771 y Fx(0)39708 54572 y FF(.)734 b(Consider)485
b(the)-3718 56177 y(action)434 b(of)g Fz(U)2548 56376
y Fy(p)3511 56177 y FF(on)g(the)f Fz(f)8199 56376 y Fy(i)8574
56177 y FF(.)579 b(F)-108 b(rom)433 b(previous)h(w)-36
b(ork)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)11655 59476
y Fz(f)142 b Fw(j)p Fz(T)13569 59675 y Fy(p)14467 59476
y FF(=)15847 58214 y Fr(X)15920 61020 y Fy(n)p Fu(\270)p
Fx(1)17988 59476 y Fz(a)18671 59675 y Fy(np)19770 59476
y Fz(q)20395 58928 y Fy(n)21316 59476 y FF(+)295 b Fz(")p
FF(\()p Fz(p)p FF(\))p Fz(p)25550 58928 y Fy(k)24 b Fu(\241)p
Fx(1)27541 58214 y Fr(X)29681 59476 y Fz(a)30364 59675
y Fy(n)30990 59476 y Fz(q)31615 58928 y Fy(pn)14467 62909
y FF(=)368 b Fz(f)16488 63108 y Fx(0)17014 62909 y Fw(j)p
Fz(U)18266 63108 y Fy(p)19091 62909 y FF(+)295 b Fz(")p
FF(\()p Fz(p)p FF(\))p Fz(p)23325 62360 y Fy(k)24 b Fu(\241)p
Fx(1)25095 62909 y Fz(f)25736 63108 y Fx(1)-3718 65936
y FF(so)9382 67635 y Fz(f)10023 67834 y Fx(0)10548 67635
y Fw(j)p Fz(U)11800 67834 y Fy(p)12699 67635 y FF(=)368
b Fz(f)142 b Fw(j)p Fz(T)15993 67834 y Fy(p)16818 67635
y Fw(\241)295 b Fz(")p FF(\()p Fz(p)p FF(\))p Fz(p)21073
67086 y Fy(k)24 b Fu(\241)22373 67635 y Fz(f)23014 67834
y Fx(1)23909 67635 y FF(=)368 b Fz(af)26613 67834 y Fx(0)27434
67635 y Fw(\241)296 b Fz(")p FF(\()p Fz(p)p FF(\))p Fz(p)31690
67086 y Fy(k)24 b Fu(\241)p Fx(1)33460 67635 y Fz(f)34101
67834 y Fx(1)34626 67635 y Fz(:)-3718 70069 y FF(Also)10989
71768 y Fz(f)11630 71967 y Fx(1)12156 71768 y Fw(j)p
Fz(U)13408 71967 y Fy(p)14306 71768 y FF(=)369 b(\()16193
70506 y Fr(X)18333 71768 y Fz(a)19016 71967 y Fy(n)19642
71768 y Fz(q)20267 71219 y Fy(pn)21366 71768 y FF(\))p
Fw(j)p Fz(U)23124 71967 y Fy(p)24022 71768 y FF(=)25403
70506 y Fr(X)27543 71768 y Fz(a)28226 71967 y Fy(n)28852
71768 y Fz(q)29477 71219 y Fy(n)30472 71768 y FF(=)f
Fz(f)32493 71967 y Fx(0)33019 71768 y Fz(:)-3718 74434
y FF(More)433 b(generally)i(one)f(can)f(sho)-36 b(w)434
b(that)f Fz(f)16620 74633 y Fy(i)16996 74434 y Fw(j)p
Fz(U)18248 74633 y Fy(p)19146 74434 y FF(=)369 b Fz(f)21168
74633 y Fy(i)p Fu(\241)p Fx(1)22746 74434 y FF(.)21534
77755 y(33)p eop
%%Page: 34 42
34 41 bop 1263 -6698 a FF(34)33167 b FA(CHAPTER)435 b(8.)1012
b(NEW)434 b(F)-36 b(ORMS)3214 -3169 y Fz(U)4097 -2970
y Fy(p)5077 -3169 y FF(preserv)g(es)451 b(the)f(t)-36
b(w)g(o)451 b(dimensional)g(v)-36 b(ector)452 b(space)e(spanned)g(b)-36
b(y)451 b Fz(f)37542 -2970 y Fx(0)38518 -3169 y FF(and)g
Fz(f)41706 -2970 y Fx(1)42232 -3169 y FF(.)630 b(The)450
b(matrix)i(of)f Fz(U)52539 -2970 y Fy(p)1263 -1564 y
FF(is)21267 545 y Fz(A)369 b FF(=)23991 -929 y Fr(\263)27308
-269 y Fz(a)3631 b FF(1)24785 1337 y Fw(\241)p Fz(")p
FF(\()p Fz(p)p FF(\))p Fz(p)28745 855 y Fy(k)24 b Fu(\241)p
Fx(1)31622 1337 y FF(0)32272 -929 y Fr(\264)1263 3643
y FF(whic)-36 b(h)434 b(has)f(c)-36 b(haracteristic)434
b(p)36 b(olynomial)18476 6638 y Fz(\302)19291 6837 y
Fy(A)20051 6638 y FF(\()p Fz(X)104 b FF(\))369 b(=)g
Fz(X)25179 6089 y Fx(2)26001 6638 y Fw(\241)295 b Fz(aX)400
b FF(+)295 b Fz(p)31451 6089 y Fy(k)24 b Fu(\241)p Fx(1)33222
6638 y Fz(")p FF(\()p Fz(p)p FF(\))p Fz(:)1263 11201
y Fs(8.1)2152 b(Connection)716 b(With)g(Galois)h(Represen)-60
b(tations)1263 14163 y FF(This)496 b(leads)f(to)g(a)h(striking)g
(connection)f(with)g(Galois)h(represen)-36 b(tations.)762
b(Let)495 b Fz(f)637 b FF(b)36 b(e)495 b(a)g(mo)36 b(dular)495
b(form)1263 15768 y(and)383 b Fz(E)460 b FF(b)36 b(e)383
b(the)g(\257eld)f(generated)h(o)-36 b(v)g(er)383 b FD(Q)h
FF(b)-36 b(y)383 b(the)f(co)36 b(e\261cien)-36 b(ts)383
b(of)h Fz(f)142 b FF(.)561 b(Let)383 b Fz(`)g FF(b)36
b(e)383 b(a)g(prime)g(and)f Fz(\270)h FF(a)g(prime)1263
17374 y(lying)435 b(o)-36 b(v)g(er)434 b Fz(`)p FF(.)578
b(Then)433 b(one)h(constructs)e(a)i(represen)-36 b(tation)19005
20369 y Fz(\275)19676 20568 y Fy(\270)20649 20369 y FF(:)369
b Fw(G)79 b Fz(al)29 b FF(\()p 23853 19298 1123 54 v
FD(Q)q Fz(=)p FD(Q)p FF(\))369 b Fw(!)h FF(GL)o(\(2)p
Fz(;)221 b(E)33855 20568 y Fy(\270)34460 20369 y FF(\))p
Fz(:)1263 23364 y FF(If)515 b Fz(p)64 b Fw(6)508 b(j)p
Fz(N)139 b(`)p FF(,)534 b(then)514 b Fz(\275)10575 23563
y Fy(\270)11693 23364 y FF(is)h(unrami\257ed)f(at)g Fz(p)p
FF(,)536 b(so)515 b(there)f(is)h(a)g(F)-108 b(rob)36
b(enious)513 b(elemen)-36 b(t)515 b(frob)44404 23563
y Fy(p)45440 23364 y Fw(2)506 b(G)79 b Fz(al)29 b FF(\()p
49306 22293 V FD(Q)q Fz(=)p FD(Q)p FF(\).)1263 24969
y(One)433 b(can)h(sho)-36 b(w)433 b(that)19988 27964
y(det)o(\()p Fz(\275)22971 28163 y Fy(\270)23575 27964
y FF(\(frob)26357 28163 y Fy(p)26886 27964 y FF(\)\))369
b(=)f Fz(p)30300 27415 y Fy(k)24 b Fu(\241)p Fx(1)32071
27964 y Fz(")p FF(\()p Fz(p)p FF(\))20457 29901 y(T)-108
b(r\()p Fz(\275)22971 30100 y Fy(\270)23575 29901 y FF(\(frob)26357
30100 y Fy(p)26886 29901 y FF(\)\))369 b(=)f Fz(a)30330
30100 y Fy(p)31228 29901 y FF(=)h Fz(a;)1263 32896 y
FF(so)434 b(the)f(c)-36 b(haracteristic)434 b(p)36 b(olynomial)435
b(of)f Fz(\275)21964 33095 y Fy(\270)22568 32896 y FF(\(frob)25350
33095 y Fy(p)25879 32896 y FF(\))369 b Fw(2)g FF(GL)29842
33095 y Fx(2)30368 32896 y FF(\()p Fz(E)31837 33095 y
Fy(\270)32441 32896 y FF(\))433 b(is)20971 35891 y Fz(X)22154
35342 y Fx(2)22976 35891 y Fw(\241)296 b Fz(a)24988 36090
y Fy(p)25516 35891 y Fz(X)400 b FF(+)295 b Fz(p)28955
35342 y Fy(k)24 b Fu(\241)p Fx(1)30726 35891 y Fz(")p
FF(\()p Fz(p)p FF(\))p Fz(:)1263 40454 y Fs(8.2)2152
b(Semisimplicit)-60 b(y)715 b(of)i Fp(U)24271 40741 y
Fz(p)1263 43417 y FD(Question)500 b(8.2.1.)651 b FC(Is)465
b Fz(U)14076 43616 y Fy(p)15070 43417 y FC(semisimple)f(on)h(the)f(sp)
-66 b(an)465 b(of)g Fz(f)30888 43616 y Fx(0)31879 43417
y FC(and)g Fz(f)35044 43616 y Fx(1)35569 43417 y FC(?)3214
46231 y FF(If)444 b(the)f(eigen)-36 b(v)-72 b(alues)443
b(are)h(distinct)e(the)h(answ)-36 b(er)443 b(is)h(clearly)g(y)-36
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b(are)g(the)e(same,)1263 47836 y(then)433 b Fz(X)5409
47354 y Fx(2)6231 47836 y Fw(\241)295 b Fz(aX)400 b FF(+)295
b Fz(p)11681 47354 y Fy(k)24 b Fu(\241)p Fx(1)13452 47836
y Fz(")p FF(\()p Fz(p)p FF(\))432 b(has)i(discriminan)-36
b(t)433 b(zero,)h(that)f(is,)h(4)p Fz(")p FF(\()p Fz(p)p
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47836 y FF(=)368 b Fz(a)41338 47354 y Fx(2)42297 47836
y FF(so)22407 51052 y Fz(a)h FF(=)g(2)p Fz(p)26276 50130
y Ft(k)14 b Fo(\241)p Fn(1)p 26276 50302 1499 40 v 26822
50851 a(2)27962 49852 y Fr(p)p 29290 49852 2274 54 v
1200 x Fz(")p FF(\()p Fz(p)p FF(\))o Fz(:)1263 54047
y FF(Is)290 b(this)g(p)36 b(ossible?)531 b(The)290 b(answ)-36
b(er)289 b(is)i(still)f FC(unknown)p FF(,)319 b(although)289
b(it)h(is)h(a)f(curious)f(fact)i(that)e(the)h(Raman)-36
b(ujan)1263 55652 y(conjectures)346 b(\(pro)-36 b(v)g(ed)346
b(b)-36 b(y)346 b(Delign)h(in)f(1973\))h(imply)g(that)e
Fw(j)p Fz(a)p Fw(j)369 b(\267)g FF(2)p Fz(p)34120 54789
y Ft(k)14 b Fo(\241)p Fn(1)p 34120 54961 1499 40 v 34666
55509 a(2)35807 55652 y FF(,)364 b(so)346 b(the)g(ab)36
b(o)-36 b(v)g(e)346 b(equalit)-36 b(y)348 b(remains)1263
57257 y(taun)-36 b(ting.)3214 58883 y(When)548 b Fz(k)610
b FF(=)564 b(2)548 b(W)-108 b(eil)549 b(sho)-36 b(w)g(ed)548
b(that)g Fz(\275)22465 59082 y Fy(\270)23069 58883 y
FF(\(frob)25851 59082 y Fy(p)26380 58883 y FF(\))g(is)h(semisimple)g
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Fz(U)50257 59082 y Fy(p)51335 58883 y FF(are)1263 60488
y(equal)390 b(then)e Fz(\275)8240 60687 y Fy(\270)8845
60488 y FF(\(frob)11627 60687 y Fy(p)12156 60488 y FF(\))h(is)g(a)h
(scalar.)564 b(But)389 b(Edixho)-36 b(v)g(en)389 b(and)g(Coleman)h([5)q
(])f(sho)-36 b(w)390 b(that)e(it)i(is)f(not)g(a)h(scalar)1263
62093 y(b)-36 b(y)434 b(lo)36 b(oking)435 b(at)f(the)f(ab)36
b(elian)434 b(v)-72 b(ariet)-36 b(y)434 b(attac)-36 b(hed)433
b(to)h Fz(f)142 b FF(.)1263 66657 y Fs(8.3)2152 b(Shim)-60
b(ura's)715 b(Example)j(of)e(Nonsemisimple)f Fp(U)43682
66944 y Fz(p)1263 69619 y FF(Let)481 b Fz(W)662 b FF(b)36
b(e)482 b(the)f(space)g(spanned)g(b)-36 b(y)481 b Fz(f)20770
69818 y Fx(0)21296 69619 y Fz(;)221 b(f)22519 69818 y
Fx(1)23527 69619 y FF(and)481 b(let)h Fz(V)771 b FF(b)36
b(e)481 b(the)g(space)h(spanned)e(b)-36 b(y)481 b Fz(f)44798
69818 y Fx(0)45324 69619 y Fz(;)221 b(f)46547 69818 y
Fx(1)47074 69619 y Fz(;)g(f)48297 69818 y Fx(2)48824
69619 y Fz(;)g(f)50047 69818 y Fx(3)50573 69619 y FF(.)722
b Fz(U)52539 69818 y Fy(p)1263 71224 y FF(acts)419 b(on)g
Fz(V)145 b(=W)600 b FF(b)-36 b(y)p 10894 70142 1167 54
v 419 w Fz(f)11535 71423 y Fx(2)12429 71224 y Fw(7!)370
b FF(0)419 b(and)p 17710 70142 V 418 w Fz(f)18351 71423
y Fx(3)19246 71224 y Fw(7!)p 20944 70142 V 370 w Fz(f)21585
71423 y Fx(2)22111 71224 y FF(.)573 b(Th)-36 b(us)418
b(the)h(matrix)g(of)h(the)e(action)i(of)f Fz(U)42692
71423 y Fy(p)43641 71224 y FF(on)f Fz(V)145 b(=W)600
b FF(is)50106 70148 y Fr(\241)50936 70799 y Fx(0)362
b(1)50936 71555 y(0)g(0)52460 70148 y Fr(\242)1263 72829
y FF(whic)-36 b(h)508 b(is)g(nonzero)f(and)g(nilp)36
b(oten)-36 b(t)507 b(hence)g(not)h(semisimple.)801 b(Since)507
b Fz(W)688 b FF(is)508 b(in)-36 b(v)-72 b(arian)-36 b(t)508
b(under)e Fz(U)49929 73028 y Fy(p)50966 72829 y FF(this)1263
74434 y(sho)-36 b(ws)434 b(that)f Fz(U)8700 74633 y Fy(p)9663
74434 y FF(is)h(not)f(semisimple)h(on)g Fz(V)289 b FF(.)p
eop
%%Page: 35 43
35 42 bop -3718 -6698 a FA(8.4.)1013 b(AN)433 b(INTERESTING)h(DUALITY)
29277 b FF(35)-3718 -3169 y Fs(8.4)2151 b(An)717 b(In)-60
b(teresting)716 b(Dualit)-60 b(y)-3718 -249 y FF(No)-36
b(w)421 b(supp)36 b(ose)421 b Fz(N)507 b FF(=)369 b(1)421
b(th)-36 b(us)420 b(\241)11774 -50 y Fx(1)12300 -249
y FF(\()p Fz(N)139 b FF(\))367 b(=)i(SL)17777 -50 y Fx(2)18302
-249 y FF(\()p FD(Z)p FF(\).)574 b(Because)421 b(of)h(the)f(P)-36
b(etersson)420 b(pro)36 b(duct)420 b(all)i(the)f Fz(T)45306
-50 y Fy(n)46353 -249 y FF(are)-3718 1356 y(diagonalizable,)435
b(so)f Fz(S)7494 1555 y Fy(k)8432 1356 y FF(=)369 b Fz(S)10613
1555 y Fy(k)11182 1356 y FF(\(\241)12501 1555 y Fx(1)13026
1356 y FF(\(1\)\))433 b(has)h(a)g(basis)19443 4153 y
Fz(f)20084 4352 y Fx(1)20610 4153 y Fz(;)221 b(:)g(:)g(:)446
b(;)221 b(f)24386 4352 y Fy(d)-3718 6950 y FF(of)434
b(normalized)g(eigenforms)g(where)g Fz(d)369 b FF(=)f(dim)222
b Fz(S)20258 7149 y Fy(k)20826 6950 y FF(.)579 b(Let)433
b FD(T)369 b FF(=)g FD(T)27924 7149 y Fv(C)28763 6950
y FF(,)434 b(then)e(there)h(is)h(a)g FC(c)-66 b(anonic)g(al)432
b FF(map)13264 9747 y FD(T)14303 9946 y Fv(C)15511 9747
y Fz(,)-221 b Fw(!)369 b FD(C)18427 9199 y Fy(d)19336
9747 y FF(:)1670 b Fz(T)550 b Fw(7!)369 b FF(\()p Fz(\270)25641
9946 y Fx(1)26167 9747 y Fz(;)221 b(:)g(:)g(:)445 b(;)221
b(\270)30060 9946 y Fy(d)30600 9747 y FF(\))-3718 12544
y(where)480 b Fz(f)727 12743 y Fy(i)1103 12544 y Fw(j)p
Fz(T)629 b FF(=)447 b Fz(\270)5081 12743 y Fy(i)5457
12544 y Fz(f)6098 12743 y Fy(i)6473 12544 y FF(.)718
b(This)480 b(map)g(is)h(clearly)g(injectiv)-36 b(e)481
b(and)e(w)-36 b(e)481 b(kno)-36 b(w)480 b(b)-36 b(y)480
b(previous)h(argumen)-36 b(ts)479 b(that)-3718 14149
y(dim)221 b FD(T)-290 14348 y Fv(C)918 14149 y FF(=)369
b Fz(d)433 b FF(so)h(the)f(map)g(is)h(an)g(isomorphism)g(of)g
FD(C)p FF(-v)-36 b(ector)434 b(spaces.)-1767 15754 y(The)f(form)17378
17359 y Fz(v)416 b FF(=)369 b Fz(f)20445 17558 y Fx(1)21266
17359 y FF(+)295 b Fw(\242)221 b(\242)g(\242)296 b FF(+)e
Fz(f)26365 17558 y Fy(n)-3718 19625 y FF(generates)441
b Fz(S)2805 19824 y Fy(k)3815 19625 y FF(as)h(a)g FD(T)p
FF(-mo)36 b(dule.)601 b(Since)441 b Fz(v)489 b FF(corresp)36
b(onds)440 b(to)i(the)e(v)-36 b(ector)442 b(\(1)p Fz(;)221
b(:)g(:)g(:)445 b(;)221 b FF(1\))443 b(and)d FD(T)42365
19257 y Fw(\273)42376 19681 y FF(=)43780 19625 y FD(C)44859
19143 y Fy(d)45840 19625 y FF(acts)-3718 21231 y(on)i
Fz(S)-1103 21430 y Fy(k)-150 20862 y Fw(\273)-140 21286
y FF(=)1267 21231 y FD(C)2346 20749 y Fy(d)3328 21231
y FF(comp)36 b(onen)-36 b(t)g(wise)442 b(this)g(is)h(just)f(the)f
(statemen)-36 b(t)442 b(that)g FD(C)30987 20749 y Fy(d)31969
21231 y FF(is)h(generated)f(b)-36 b(y)442 b(\(1)p Fz(;)221
b(:)g(:)g(:)445 b(;)221 b FF(1\))443 b(as)-3718 22836
y(a)434 b FD(C)-1555 22354 y Fy(d)-1015 22836 y FF(-mo)36
b(dule)433 b({)g(whic)-36 b(h)434 b(is)g(clear.)579 b(Th)-36
b(us)433 b(w)-36 b(e)433 b(ha)-36 b(v)g(e)434 b(sim)-36
b(ultaneously:)-1767 24441 y(1\))434 b Fz(S)623 24640
y Fy(k)1625 24441 y FF(is)g(free)g(of)g(rank)g(1)g(o)-36
b(v)g(er)434 b FD(T)p FF(,)g(and)-1767 26046 y(2\))g
Fz(S)623 26245 y Fy(k)1560 26046 y FF(=)369 b(Hom)5650
26245 y Fv(C)6489 26046 y FF(\()p FD(T)p Fz(;)221 b FD(C)p
FF(\))434 b(as)g FD(T)p FF(-mo)36 b(dules,)434 b(th)-36
b(us)16968 29256 y FD(T)18377 28887 y Fw(\273)18387 29311
y FF(=)19779 29256 y(Hom)22488 29455 y Fv(C)23327 29256
y FF(\()p FD(T)p Fz(;)221 b FD(C)p FF(\))p Fz(:)-3718
31522 y FF(The)547 b(isomorphism)g(sends)f(an)g(elemen)-36
b(t)547 b(of)g Fz(T)743 b Fw(2)561 b FD(T)547 b FF(to)g
Fz(T)181 b(v)609 b Fw(2)562 b Fz(S)29482 31721 y Fy(k)30050
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b Fz(S)45945 31721 y Fy(k)47076 31522 y FF(=)-3718 33127
y(Hom)-1009 33326 y Fv(C)-170 33127 y FF(\()p FD(T)p
Fz(;)221 b FD(C)p FF(\))633 b(w)-36 b(as)633 b(constructed)e(using)h
(the)g(P)-36 b(etersson)632 b(pro)36 b(duct)631 b(it)i(is)g(canonical)g
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y Fx(1)16822 34732 y Fz(;)221 b(:)g(:)g(:)445 b(;)221
b(f)20597 34931 y Fy(d)21620 34732 y FF(is)483 b(canonical)h(w)-36
b(e)483 b(see)g(that)f(the)h(isomorphism)g Fz(T)47054
34364 y Fw(\273)47065 34788 y FF(=)-3718 36337 y(Hom)-1009
36536 y Fv(C)-170 36337 y FF(\()p FD(T)p Fz(;)221 b FD(C)p
FF(\))434 b(is)g(canonical.)-3718 38933 y FD(Prop)42
b(osition)500 b(8.4.1.)651 b Fz(v)416 b Fw(2)369 b Fz(S)11593
39132 y Fy(k)12162 38933 y FF(\()p FD(Q)p FF(\))-3718
41528 y FC(Pr)-66 b(o)g(of.)649 b FF(Let)403 b Fz(\276)417
b Fw(2)369 b(G)79 b Fz(al)29 b FF(\()p 7539 40457 1123
54 v FD(Q)p Fz(=)p FD(Q)p FF(\),)411 b(then)403 b(if)i
Fz(f)16448 41727 y Fy(i)17228 41528 y FF(is)f(a)g(normalized)g
(eigenform)h(so)f(is)h Fz(\276)48 b FF(\()p Fz(f)36890
41727 y Fy(i)37265 41528 y FF(\))404 b(\(from)g(the)g(explicit)-3718
43133 y(form)-36 b(ula\).)579 b(Th)-36 b(us)433 b Fz(\276)48
b FF(\()p Fz(f)7292 43332 y Fx(1)8112 43133 y FF(+)295
b Fw(\242)221 b(\242)g(\242)296 b FF(+)f Fz(f)13212 43332
y Fy(n)13838 43133 y FF(\))369 b(=)g Fz(f)16735 43332
y Fx(1)17556 43133 y FF(+)295 b Fw(\242)221 b(\242)g(\242)296
b FF(+)e Fz(f)22655 43332 y Fy(n)23715 43133 y FF(for)434
b(all)h Fz(\276)481 b FF(as)434 b(desired.)p 47147 43133
45 878 v 47192 42299 781 45 v 47192 43133 V 47972 43133
45 878 v -1767 45695 a(No)-36 b(w)434 b(w)-36 b(e)434
b(consider)f(the)g(case)h(for)g(general)h Fz(N)139 b
FF(.)577 b(Recall)435 b(that)e(w)-36 b(e)434 b(ha)-36
b(v)g(e)434 b(de\257ned)e(maps)15132 48492 y Fz(S)15932
48691 y Fy(k)16501 48492 y FF(\(\241)17820 48691 y Fx(1)18345
48492 y FF(\()p Fz(M)139 b FF(\)\))369 b Fw(!)g Fz(S)24126
48691 y Fy(k)24695 48492 y FF(\(\241)26014 48691 y Fx(1)26539
48492 y FF(\()p Fz(N)139 b FF(\)\))-3718 51289 y(for)434
b(all)g Fz(M)573 b FF(dividing)434 b Fz(N)572 b FF(and)433
b(all)h(divisors)h Fz(d)e FF(of)20427 50766 y Fy(N)p
20348 50983 998 54 v 20348 51747 a(M)21479 51289 y FF(.)-3718
53884 y FD(De\257nition)499 b(8.4.2.)652 b FF(The)381
b FD(old)438 b(part)383 b FF(of)f Fz(S)17804 54083 y
Fy(k)18372 53884 y FF(\(\241)19691 54083 y Fx(1)20217
53884 y FF(\()p Fz(N)139 b FF(\)\))380 b(is)h(the)g(space)g(generated)g
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y(maps)425 b(with)g Fz(M)139 b Fw(j)p Fz(N)563 b FF(but)424
b Fz(M)508 b Fw(6)p FF(=)369 b Fz(N)139 b FF(.)575 b(The)425
b FD(new)489 b(part)426 b FF(is)f(the)g(orthogonal)h(complemen)-36
b(t)424 b(of)i(the)f(old)g(part)-3718 57094 y(with)433
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b(pro)36 b(duct.)-1767 59689 y(There)433 b(is)h(an)g(algebraic)g
(de\257nition)f(of)h(the)f(new)h(part.)578 b(One)433
b(de\257nes)f(certain)i(trace)g(maps)15132 62486 y Fz(S)15932
62685 y Fy(k)16501 62486 y FF(\(\241)17820 62685 y Fx(1)18345
62486 y FF(\()p Fz(N)139 b FF(\)\))368 b Fw(!)h Fz(S)23909
62685 y Fy(k)24478 62486 y FF(\(\241)25797 62685 y Fx(1)26323
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b(all)g Fz(M)582 b(<)442 b(N)139 b FF(,)488 b Fz(M)139
b Fw(j)p Fz(N)615 b FF(whic)-36 b(h)477 b(are)h(the)e(adjoin)-36
b(ts)478 b(to)f(the)f(ab)36 b(o)-36 b(v)g(e)478 b(maps)f(\(w.r.t)h(P)
-36 b(etersson)476 b(pro)36 b(duct\).)-3718 66888 y(Then)433
b Fz(f)576 b FF(is)433 b(in)h(the)f(new)g(part)h(of)g
Fz(S)13733 67087 y Fy(k)14302 66888 y FF(\(\241)15621
67087 y Fx(1)16146 66888 y FF(\()p Fz(N)139 b FF(\)\))432
b(i\256)i Fz(f)575 b FF(is)434 b(killed)h(b)-36 b(y)433
b(all)i(of)f(these)f(maps.)-1767 68493 y(It)595 b(can)h(b)36
b(e)595 b(sho)-36 b(wn)596 b(that)f(the)g Fz(T)14534
68692 y Fy(n)15756 68493 y FF(act)g(semisimply)i(on)e
Fz(S)27714 68692 y Fy(k)28283 68493 y FF(\(\241)29602
68692 y Fx(1)30127 68493 y FF(\()p Fz(M)139 b FF(\)\))33042
68692 y Fx(new)35313 68493 y FF(for)596 b(all)h Fz(M)784
b Fw(\270)644 b FF(1.)1065 b(Th)-36 b(us)-3718 70099
y Fz(S)-2918 70298 y Fy(k)-2349 70099 y FF(\(\241)-1030
70298 y Fx(1)-505 70099 y FF(\()p Fz(M)139 b FF(\)\))2410
70298 y Fx(new)4519 70099 y FF(has)433 b(a)h(basis)g(of)g(eigenforms.)
580 b(W)-108 b(e)433 b(ha)-36 b(v)g(e)434 b(a)g(natural)f(map)12929
71866 y Fr(M)12883 74744 y Fy(M)94 b Fu(j)p Fy(N)15205
73128 y Fz(S)16005 73327 y Fy(k)16573 73128 y FF(\(\241)17892
73327 y Fx(1)18418 73128 y FF(\()p Fz(M)139 b FF(\)\))21333
73327 y Fx(new)23377 73128 y Fz(,)-221 b Fw(!)369 b Fz(S)26014
73327 y Fy(k)26583 73128 y FF(\(\241)27902 73327 y Fx(1)28427
73128 y FF(\()p Fz(N)139 b FF(\)\))p Fz(:)p eop
%%Page: 36 44
36 43 bop 1263 -6698 a FF(36)33167 b FA(CHAPTER)435 b(8.)1012
b(NEW)434 b(F)-36 b(ORMS)1263 -3169 y FF(The)457 b(image)i(in)e
Fz(S)10083 -2970 y Fy(k)10651 -3169 y FF(\(\241)11970
-2970 y Fx(1)12496 -3169 y FF(\()p Fz(N)139 b FF(\)\))456
b(of)i(an)f(eigenform)h Fz(f)599 b FF(for)458 b(some)f
Fz(S)32304 -2970 y Fy(k)32873 -3169 y FF(\(\241)34192
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-2970 y Fx(new)39765 -3169 y FF(is)458 b(called)f(a)h
FD(newform)g FF(of)1263 -1564 y(lev)-36 b(el)387 b Fz(M)5436
-1365 y Fy(f)6411 -1564 y FF(=)368 b Fz(M)139 b FF(.)563
b(Note)386 b(that)f(a)i(newform)f(is)h(not)e(necessarily)i(an)f
(eigenform)h(for)g(the)e(Hec)-36 b(k)g(e)387 b(op)36
b(erators)1263 41 y(acting)434 b(on)g Fz(S)7772 240 y
Fy(k)8340 41 y FF(\(\241)9659 240 y Fx(1)10185 41 y FF(\()p
Fz(N)139 b FF(\)\).)577 b(Let)19095 2151 y Fz(v)416 b
FF(=)21521 889 y Fr(X)22205 3718 y Fy(f)23661 2151 y
Fz(f)142 b FF(\()p Fz(q)26001 937 y Ft(N)p 25707 1098
1302 40 v 25707 1647 a(M)26478 1879 y(f)27197 2151 y
FF(\))369 b Fw(2)f Fz(S)30126 2350 y Fy(k)30695 2151
y FF(\(\241)32014 2350 y Fx(1)32539 2151 y FF(\()p Fz(N)139
b FF(\)\))1263 5562 y(where)320 b(the)f(sum)g(is)h(tak)-36
b(en)320 b(o)-36 b(v)g(er)320 b(all)g(newforms)h Fz(f)461
b FF(of)321 b(w)-36 b(eigh)g(t)320 b Fz(k)364 b FF(and)319
b(some)h(lev)-36 b(el)321 b Fz(M)139 b Fw(j)p Fz(N)g
FF(.)540 b(This)320 b(generalizes)1263 7167 y(the)602
b Fz(v)650 b FF(constructed)602 b(ab)36 b(o)-36 b(v)g(e)603
b(when)f Fz(N)795 b FF(=)656 b(1)603 b(and)f(has)h(man)-36
b(y)603 b(of)g(the)f(same)h(go)36 b(o)g(d)603 b(prop)36
b(erties.)1085 b(F)-108 b(or)1263 8772 y(example,)599
b Fz(S)7683 8971 y Fy(k)8252 8772 y FF(\(\241)9571 8971
y Fx(1)10096 8772 y FF(\()p Fz(N)139 b FF(\)\))563 b(is)i(free)h(of)f
(rank)g(1)g(o)-36 b(v)g(er)565 b FD(T)g FF(with)g(basis)g(elemen)-36
b(t)565 b Fz(v)48 b FF(.)971 b(The)565 b(co)36 b(e\261cien)-36
b(ts)565 b(of)g Fz(v)1263 10377 y FF(lie)478 b(in)g FD(Q)p
FF(,)489 b(but)476 b(to)i(sho)-36 b(w)478 b(this)f(w)-36
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13587 y(the)402 b(P)-36 b(etersson)402 b(pro)36 b(duct)401
b(whic)-36 b(h)402 b(is)h(an)f(analytic)h(construction.)568
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b(ations)716 b(on)g Fp(T)23107 19817 y Fz(n)1263 22451
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b FD(Q)p FF([)p Fw(\242)221 b(\242)g(\242)445 b Fz(;)221
b(T)12145 22650 y Fy(n)12772 22451 y Fz(;)g Fw(\242)g(\242)g(\242)i
FF(])450 b(and)f(\241)397 b(=)f(\241\(1\))h(=)f(SL)26916
22650 y Fx(2)27441 22451 y FF(\()p FD(Z)p FF(\).)627
b(Let)450 b Fz(f)33342 22650 y Fx(1)33868 22451 y Fz(;)221
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24056 y(of)434 b(normalized)g(eigenforms.)1263 26192
y FD(Prop)42 b(osition)500 b(8.5.1.)651 b FC(The)465
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b(acting)h(on)f(the)g(co)36 b(e\261cien)-36 b(ts)639
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31539 y(of)580 b Fw(G)79 b Fz(al)29 b FF(\()p FD(C)p
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b(y)579 b Fz(i)p FF(,)615 b Fz(a)17278 31738 y Fy(n)17904
31539 y FF(\()p Fz(f)19051 31738 y Fy(i)19427 31539 y
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31738 y Fy(i)38220 31539 y Fw(j)p Fz(T)39351 31738 y
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31539 y FF(\()p Fz(f)44678 31738 y Fy(i)45053 31539 y
FF(\))p Fz(f)46200 31738 y Fy(i)46576 31539 y FF(,)f(and)579
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FF(\))g(m)-36 b(ust)421 b(b)36 b(e)422 b(real.)575 b(Th)-36
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33144 y FF(\))f(are)g(real)h(and)e(there)h(are)g(only)1263
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Fz(a)21499 34948 y Fy(n)22126 34749 y FF(\()p Fz(f)23273
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b(e)434 b Fz(a)30264 34948 y Fy(n)30890 34749 y FF(\()p
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b Fz(d)p FF(.)p 52128 34749 45 878 v 52173 33915 781
45 v 52173 34749 V 52953 34749 45 878 v 1263 36885 a
FD(Prop)42 b(osition)500 b(8.5.2.)651 b FC(The)465 b(op)-66
b(er)g(ators)464 b Fw(h)p Fz(d)p Fw(i)g FC(on)h Fz(S)26610
37084 y Fy(k)27179 36885 y FF(\(\241)28498 37084 y Fx(1)29023
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37084 y Fy(n)40788 36885 y Fz(;)g(:)g(:)g(:)j FF(])p
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41283 y Fz(p)22112 40735 y Fy(k)24 b Fu(\241)p Fx(1)23883
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y Fy(p)28610 40726 y Fx(2)29431 41283 y Fw(\241)295 b
Fz(T)31521 41506 y Fy(p)31995 41253 y Fn(2)32512 41283
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b(theorem)g(on)g(prime's)g(in)g(arithmetic)f(progression,)496
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b Fz(N)139 b FF(.)778 b(Since)500 b Fz(p)30690 44668
y Fy(k)24 b Fu(\241)p Fx(1)32961 45150 y FF(and)500 b
Fz(q)36182 44668 y Fy(k)24 b Fu(\241)p Fx(1)38453 45150
y FF(are)500 b(relativ)-36 b(ely)502 b(prime)e(there)1263
46755 y(exist)434 b(in)-36 b(tegers)434 b Fz(a)f FF(and)g
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Fx(1)21603 46755 y FF(+)295 b Fz(bq)24088 46273 y Fy(k)24
b Fu(\241)p Fx(1)26227 46755 y FF(=)369 b(1.)578 b(Then)11613
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Fx(1)20643 49017 y FF(+)294 b Fz(bq)23127 48468 y Fy(k)24
b Fu(\241)p Fx(1)24898 49017 y FF(\))369 b(=)f Fz(a)p
FF(\()p Fz(T)29104 49216 y Fy(p)29633 48459 y Fx(2)30454
49017 y Fw(\241)296 b Fz(T)32545 49239 y Fy(p)33019 48987
y Fn(2)33535 49017 y FF(\))f(+)g Fz(b)p FF(\()p Fz(T)37464
49216 y Fy(q)37971 48459 y Fx(2)38792 49017 y Fw(\241)h
Fz(T)40883 49239 y Fy(q)41335 48987 y Fn(2)41851 49017
y FF(\))p Fz(:)p 52128 51278 V 52173 50445 781 45 v 52173
51278 V 52953 51278 45 878 v 3214 53688 a FF(Let)485
b(\247)g(b)36 b(e)484 b(a)i(set)e(of)i(represen)-36 b(tativ)g(es)484
b(of)i Fw(f)p Fz(f)25346 53887 y Fx(1)25872 53688 y Fz(;)221
b(:)g(:)g(:)445 b(;)221 b(f)29647 53887 y Fy(d)30188
53688 y Fw(gnG)79 b Fz(al)29 b FF(\()p FD(C)p Fz(=)p
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b(e)426 b(larger)g(than)e(one,)k(that)d(is,)i(whether)e(the)g
(eigenforms)h(are)g(all)g(conjugate)g(under)e(the)1263
56898 y(action)434 b(of)g(Galois.)580 b(Let)433 b Fz(K)14580
57097 y Fy(f)15553 56898 y FF(=)369 b FD(Q)p FF(\()p
Fz(:)221 b(:)g(:)445 b(;)221 b(a)21797 57097 y Fy(n)22423
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Fv(Q)18979 59159 y Fw(!)369 b Fz(K)21783 59358 y Fy(f)22757
59159 y FF(:)g Fz(T)24249 59358 y Fy(n)25245 59159 y
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b(a)g(set)g(of)g(represen)-36 b(tativ)g(es)433 b(of)h(the)f
Fz(f)33322 61620 y Fy(i)34132 61421 y FF(yields)h(a)g(map)23190
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63841 y Fw(\241)-845 b(!)27433 62580 y Fr(Y)27353 65408
y Fy(f)98 b Fu(2)p Fx(\247)29431 63841 y Fz(K)30538 64040
y Fy(f)1263 67403 y FF(whic)-36 b(h)434 b(one)f(can)h(sho)-36
b(w)433 b(is)h(an)g(isomorphism)f(of)i FD(Q)p FF(-algebras.)1263
69274 y FC(Example)465 b(8.5.3.)649 b FF(Consider)434
b Fz(S)16593 69473 y Fx(2)17119 69274 y FF(\(\241)18438
69473 y Fx(0)18963 69274 y FF(\()p Fz(N)139 b FF(\)\))432
b(with)i Fz(N)572 b FF(prime,)433 b(then)22205 71536
y FD(T)23244 71735 y Fv(Q)24484 71167 y Fw(\273)24494
71591 y FF(=)25886 71536 y Fz(E)26849 71735 y Fx(1)27669
71536 y Fw(\243)296 b(\242)221 b(\242)g(\242)h Fz(E)31732
71735 y Fy(t)1263 73797 y FF(with)434 b(the)f Fz(E)7429
73996 y Fy(i)8238 73797 y FF(totally)i(real)f(\257elds.)578
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73996 y Fv(Q)32674 73428 y Fw(\273)32685 73853 y FF(=)34076
73797 y FD(Q)296 b Fw(\243)f FD(Q)p FF(.)p eop
%%Page: 37 45
37 44 bop -3718 5729 a FE(Chapter)1033 b(9)-3718 11306
y(Some)g(Explicit)f(Gen)-86 b(us)1033 b(Computations)-3718
18212 y Fs(9.1)2151 b(Computing)716 b(the)h(Dimension)f(of)h
Fp(S)30409 18499 y Fz(k)31185 18212 y Fm(\(\241\))-3718
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21418 y Fx(2)24826 21219 y FF(\()p FD(Z)p FF(\))433 b(b)36
b(e)433 b(a)h(congruence)f(subgroup.)578 b(Then)16491
24280 y Fz(S)17291 24479 y Fx(2)17817 24280 y FF(\(\241\))368
b(=)g Fz(H)22571 23732 y Fx(0)23097 24280 y FF(\()p Fz(X)24682
24479 y Fx(\241)25325 24280 y Fz(;)221 b FF(\255)26846
23732 y Fx(1)27373 24280 y FF(\))-3718 27341 y(where)14634
29074 y Fz(X)15713 29273 y Fx(\241)16725 29074 y FF(=)369
b(\(\241)p Fw(nH)13 b FF(\))294 b Fw([)h FF(\(\241)p
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b Fz(H)7674 31065 y Fx(0)8199 31547 y FF(\()p Fz(X)9784
31746 y Fx(\241)10427 31547 y Fz(;)g FF(\255)11948 31065
y Fx(1)12475 31547 y FF(\))433 b(is)h(the)f(gen)-36 b(us)433
b(of)i Fz(X)23107 31746 y Fx(\241)23750 31547 y FF(.)-3718
33834 y FC(Exer)-66 b(cise)464 b(9.1.1.)649 b FF(Pro)-36
b(v)g(e)434 b(that)f(when)g(\241)369 b(=)f(SL)19248 34033
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FD(Q)p FF(\))h(is)f(a)h(p)36 b(oin)-36 b(t.)-1767 36120
y(Since)433 b(\241)369 b Fw(\275)g FF(\241\(1\))433 b(there)g(is)h(a)g
(co)-36 b(v)g(ering)12976 39015 y(\241)p Fw(nH)1494 b(\241)-406
b(\241)-295 b(\241)-405 b(!)1251 b Fz(X)22720 39214 y
Fx(\241)13839 39955 y Fr(?)13839 40752 y(?)13839 41549
y(y)22060 39955 y(?)22060 40752 y(?)22060 41549 y(y)12145
44242 y FF(\241\(1\))p Fw(nH)663 b(\241)-406 b(\241)-295
b(\241)-405 b(!)650 b Fz(X)22119 44448 y Fx(\241\(1\))26060
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b(is)f(the)g(index)g(\(SL)42973 51323 y Fx(2)43498 51124
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Fz(Y)45370 52928 y Fx(0)45896 52729 y FF(\()p Fz(N)139
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y Fs(9.2)2151 b(Application)716 b(of)h(Riemann-Hurwitz)-3718
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b(us)514 b(of)i Fz(X)15307 62240 y Fx(\241)16465 62041
y FF(b)-36 b(y)515 b(applying)h(the)e(Riemann-Hurwitz)h(form)-36
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b(Euler)h(c)-36 b(harcteristic)364 b(should)g(b)36 b(e)364
b(totally)h(additiv)-36 b(e,)379 b(that)363 b(is,)379
b(if)365 b Fz(A)f FF(and)f Fz(B)432 b FF(are)364 b(disjoin)-36
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Fz(A)295 b Fw([)g Fz(B)67 b FF(\))369 b(=)g Fz(\302)p
FF(\()p Fz(A)p FF(\))294 b(+)h Fz(\302)p FF(\()p Fz(B)67
b FF(\))p Fz(:)-3718 69768 y FF(Let)384 b Fz(X)490 b
FF(b)36 b(e)385 b(a)g(compact)g(Riemann)g(surface)h(of)f(gen)-36
b(us)385 b Fz(g)48 b FF(,)394 b(then)384 b Fz(\302)p
FF(\()p Fz(X)104 b FF(\))370 b(=)e(2)196 b Fw(\241)g
FF(2)p Fz(g)48 b FF(.)563 b(Since)384 b Fz(\302)p FF(\()p
Fw(f)p FF(p)36 b(oin)-36 b(t)p Fw(g)p FF(\))369 b(=)f(1)-3718
71373 y(w)-36 b(e)434 b(should)f(ha)-36 b(v)g(e)433 b(that)7030
74434 y Fz(\302)p FF(\()p Fz(X)400 b Fw(\241)295 b(f)p
Fz(p)12475 74633 y Fx(1)13001 74434 y Fz(;)221 b(:)g(:)g(:)445
b(;)221 b(p)16788 74633 y Fy(n)17415 74434 y Fw(g)p FF(\))369
b(=)g Fz(\302)p FF(\()p Fz(X)104 b FF(\))295 b Fw(\241)h
Fz(n\302)p FF(\(1\))369 b(=)f(\(2)296 b Fw(\241)f FF(2)p
Fz(g)48 b FF(\))295 b Fw(\241)g Fz(n:)21534 77755 y FF(37)p
eop
%%Page: 38 46
38 45 bop 1263 -6698 a FF(38)14610 b FA(CHAPTER)435 b(9.)1012
b(SOME)432 b(EXPLICIT)j(GENUS)e(COMPUT)-108 b(A)g(TIONS)1263
-3169 y FF(If)437 b(w)-36 b(e)437 b(ha)-36 b(v)g(e)436
b(an)g(umrami\257ed)f(co)-36 b(v)g(ering)437 b Fz(X)479
b Fw(!)373 b Fz(Y)725 b FF(of)437 b(degree)f Fz(d)g FF(then)g
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b Fw(\242)g Fz(\302)p FF(\()p Fz(Y)288 b FF(\).)586 b(Consider)436
b(the)1263 -1564 y(co)-36 b(v)g(ering)18611 -156 y Fz(X)19690
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b(ts)434 b(o)-36 b(v)g(er)434 b(0)p Fz(;)221 b FF(1728)p
Fz(;)g Fw(1)s(g)26723 784 y Fr(?)26723 1581 y(?)26723
2378 y(y)21355 4779 y Fz(X)22434 4986 y Fx(\241\(1\))24575
4779 y Fw(\241)295 b(f)p FF(0)p Fz(;)221 b FF(1728)p
Fz(;)g Fw(1g)1263 7084 y FF(Since)525 b Fz(X)5830 7290
y Fx(\241\(1\))8200 7084 y FF(has)g(gen)-36 b(us)525
b(0,)548 b Fz(X)16925 7290 y Fx(\241\(1\))19128 7084
y Fw(\241)358 b(f)p FF(0)p Fz(;)221 b FF(1728)p Fz(;)g
Fw(1g)529 b FF(has)c(Euler)f(c)-36 b(haracteristic)525
b(2)358 b Fw(\241)g FF(3)525 b(=)f Fw(\241)p FF(1.)853
b(If)525 b(w)-36 b(e)1263 8689 y(let)448 b Fz(g)441 b
FF(=)393 b Fz(\302)p FF(\()p Fz(X)8025 8888 y Fx(\241)8668
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%%Page: 39 47
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71864 y FF(if)434 b Fz(N)507 b Fw(\264)370 b FF(2)1328
b(\(mo)36 b(d)443 b(3\))16202 73267 y Fy(N)94 b Fu(\241)p
Fx(1)p 16202 73484 V 16988 74248 a(3)18673 73790 y FF(+)295
b(2)1301 b(if)434 b Fz(N)507 b Fw(\264)370 b FF(1)1328
b(\(mo)36 b(d)443 b(3\))p eop
%%Page: 40 48
40 47 bop 1263 -6698 a FF(40)14610 b FA(CHAPTER)435 b(9.)1012
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-3169 y FC(Exer)-66 b(cise)464 b(9.5.2.)649 b FF(It)501
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593 y(By)434 b(similar)h(reasoning)f(one)f(sho)-36 b(ws)434
b(that)15861 4612 y Fz(m)16999 4811 y Fx(1728)19305 4612
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21889 3396 2044 54 v 22675 4160 a(2)27618 3702 y FF(if)434
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b(4\))21889 5105 y Fy(N)94 b Fu(\241)p Fx(1)p 21889 5322
V 22675 6086 a(2)24360 5628 y FF(+)295 b(2)1301 b(if)434
b Fz(N)507 b Fw(\264)370 b FF(1)1328 b(\(mo)36 b(d)443
b(4\))1263 8704 y(W)-108 b(e)491 b(can)g(no)-36 b(w)491
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b(2.)656 b(Similarly)-108 b(,)466 b Fz(X)33890 10508
y Fx(0)34416 10309 y FF(\(13\))460 b(has)f(gen)-36 b(us)458
b(0)i(and)e Fz(X)47885 10508 y Fx(0)48411 10309 y FF(\(11\))i(has)1263
11914 y(gen)-36 b(us)433 b(1.)579 b(In)433 b(general,)i
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b Fz(N)30 b(=)p FF(12.)3214 13520 y(Serre)521 b(constructed)e(a)j(nice)
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b(gen)-36 b(us.)841 b(Supp)36 b(ose)520 b Fz(N)656 b(>)517
b FF(3)522 b(is)f(a)h(prime)e(and)1263 15125 y(write)434
b Fz(N)507 b FF(=)369 b(12)p Fz(a)296 b FF(+)e Fz(b)434
b FF(with)g(0)369 b Fw(\267)g Fz(b)g Fw(\267)g FF(11.)579
b(Then)433 b(Serre's)h(form)-36 b(ula)434 b(is)p 19881
16565 14570 45 v 19859 18170 45 1606 v 20604 17688 a
Fz(b)p 21858 18170 V 2540 w FF(1)2498 b(5)1362 b(7)2162
b(11)p 34429 18170 V 19881 18214 14570 45 v 19859 19819
45 1606 v 20545 19337 a Fz(g)p 21858 19819 V 1376 w(a)295
b Fw(\241)h FF(1)1328 b Fz(a)g(a)g(a)295 b FF(+)g(1)p
34429 19819 V 19881 19863 14570 45 v 1263 23304 a Fs(9.6)2152
b(Mo)60 b(dular)716 b(F)-179 b(orms)717 b(mo)60 b(d)716
b Fp(p)1263 26224 y FF(Let)433 b Fz(N)572 b FF(b)36 b(e)433
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b(9.6.1.)651 b FF(Let)433 b Fz(M)15963 29132 y Fy(k)16532
28933 y FF(\(\241)p Fz(;)221 b FD(Z)p FF(\))370 b(=)e
Fz(M)22860 29132 y Fy(k)23429 28933 y FF(\(\241)p Fz(;)221
b FD(C)p FF(\))296 b Fw(\\)f FD(Z)p FF([[)p Fz(q)48 b
FF(]],)434 b(then)18961 31864 y Fz(M)20219 32063 y Fy(k)20788
31864 y FF(\(\241)p Fz(;)221 b FD(F)23628 32063 y Fy(p)24158
31864 y FF(\))369 b(=)f Fz(M)27671 32063 y Fy(k)28240
31864 y FF(\(\241)p Fz(;)221 b FD(Z)p FF(\))296 b Fw(\255)32889
32063 y Fv(Z)33902 31864 y FD(F)34841 32063 y Fy(p)1263
34794 y FF(is)434 b(the)f(space)h(of)g FD(mo)42 b(dular)500
b(forms)f(mo)42 b(d)498 b Fz(p)433 b FF(of)i(w)-36 b(eigh)g(t)434
b Fz(k)45 b FF(.)3214 37503 y(Supp)36 b(ose)433 b Fz(p)368
b FF(=)h Fz(N)139 b FF(,)433 b(then)g(one)g(has)h FD(Serre's)499
b(Equalit)-42 b(y)p FF(:)17347 40434 y Fz(M)18605 40633
y Fy(p)p Fx(+1)20337 40434 y FF(\(SL)22379 40633 y Fx(2)22904
40434 y FF(\()p FD(Z)p FF(\))p Fz(;)221 b FD(F)26350
40633 y Fy(p)26880 40434 y FF(\))368 b(=)h Fz(M)30393
40633 y Fx(2)30919 40434 y FF(\(\241)32238 40633 y Fx(0)32763
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40633 y Fy(p)36479 40434 y FF(\))3214 43364 y(The)536
b(map)f(from)h(the)e(righ)-36 b(t)535 b(hand)g(side)g(to)h(the)f(left)h
(hand)e(side)h(is)h(accomplished)g(via)g(a)g(certain)1263
44969 y(normalized)434 b(Eisenstein)f(series.)579 b(Recall)435
b(that)e(in)g(SL)27865 45168 y Fx(2)28391 44969 y FF(\()p
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48912 y FF(=)368 b Fw(\241)23391 48013 y Fz(B)24379 48212
y Fy(k)p 23391 48606 1557 54 v 23484 49823 a FF(2)p Fz(k)25376
48912 y FF(+)27172 47251 y Fu(1)26683 47650 y Fr(X)26756
50439 y Fy(n)p Fx(=1)28602 48912 y FF(\()29108 47650
y Fr(X)29409 50528 y Fy(d)p Fu(j)p Fy(n)31248 48912 y
Fz(d)31924 48363 y Fy(k)24 b Fu(\241)p Fx(1)33695 48912
y FF(\))p Fz(q)34826 48363 y Fy(n)1263 53240 y FF(and)18802
55526 y Fz(E)19765 55725 y Fy(k)20702 55526 y FF(=)369
b(1)295 b Fw(\241)24582 54627 y FF(2)p Fz(k)p 24490 55220
V 24490 56437 a(B)25478 56636 y Fy(k)26890 53865 y Fu(1)26401
54264 y Fr(X)26474 57053 y Fy(n)p Fx(=1)28320 55526 y
FF(\()28826 54264 y Fr(X)29127 57142 y Fy(d)p Fu(j)p
Fy(n)30966 55526 y Fz(d)31642 54977 y Fy(k)24 b Fu(\241)p
Fx(1)33413 55526 y FF(\))p Fz(q)34544 54977 y Fy(n)35169
55526 y Fz(:)1263 59498 y FF(One)320 b(\257nds)g(ord)8777
59697 y Fy(p)9306 59498 y FF(\()p Fw(\241)10978 58930
y Fy(B)11688 59086 y Ft(k)p 10977 59192 1218 54 v 11094
59956 a Fx(2)p Fy(k)12328 59498 y FF(\))h(using)f(Kummer)h
(congruences.)540 b(In)321 b(particular,)343 b(ord)39113
59697 y Fy(p)39642 59498 y FF(\()p Fz(B)41136 59697 y
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Fx(1)52036 59498 y Fw(\264)1263 61103 y FF(1)591 b(\(mo)36
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b(ultiplication)480 b(b)-36 b(y)479 b Fz(E)22727 61302
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b(but)f(do)36 b(es)480 b(not)e(c)-36 b(hange)479 b(the)1263
62708 y Fz(q)48 b FF(-expansion)433 b(mo)36 b(d)433 b
Fz(p)p FF(.)579 b(W)-108 b(e)433 b(th)-36 b(us)433 b(get)g(a)h(map)
17500 65638 y Fz(M)18758 65837 y Fx(2)19284 65638 y FF(\(\241)20603
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b FD(F)24314 65837 y Fy(p)24844 65638 y FF(\))368 b Fw(!)i
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y Fy(p)p Fx(+1)19576 70174 y FF(\(\241)20895 70373 y
Fx(0)21420 70174 y FF(\()p Fz(p)p FF(\))p Fz(;)221 b
FD(F)24606 70373 y Fy(p)25135 70174 y FF(\))369 b Fw(!)h
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70373 y Fx(2)33264 70174 y FF(\()p FD(Z)p FF(\))p Fz(;)221
b FD(F)36710 70373 y Fy(p)37240 70174 y FF(\))1263 72497
y(is)424 b(the)e(trace)h(map)g(\(whic)-36 b(h)423 b(is)g(dual)g(to)h
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(form)g(in)-36 b(v)-72 b(arian)-36 b(t)433 b(under)f(SL)36804
74301 y Fx(2)37329 74102 y FF(\()p FD(Z)p FF(\).)p eop
%%Page: 41 49
41 48 bop -3718 5686 a FE(Chapter)1033 b(10)-3718 11221
y(The)g(Field)e(of)i(Mo)86 b(duli)-3718 17254 y FF(In)368
b(this)g(c)-36 b(hapter)367 b(w)-36 b(e)368 b(will)i(study)e(the)f
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FD(P)15827 19982 y Fx(1)15827 20802 y Fv(Q)17108 20464
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b Fz(E)489 b FF(is)412 b(an)f(elliptic)h(curv)-36 b(e)411
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24680 24800 5317 54 v 24680 26017 a Fz(g)25351 25559
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Fz(g)29471 25559 y Fx(2)29423 26342 y(3)30130 25106 y
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b Fz(E)78 b(=)p FD(Q)p FF(\()p Fz(t)p FF(\))-3718 29770
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32684 y Fz(y)13296 32135 y Fx(2)14191 32684 y FF(=)369
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b Fz(E)78 b(=k)535 b FF(b)36 b(e)490 b(an)g(arbitrary)h(elliptic)g
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y Fx(2)7852 37133 y FF(.)571 b(Let)413 b Fz(k)45 b FF(\()p
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b(ordinates)413 b(of)h(the)-3718 38738 y Fz(N)139 b FF(-torsion)523
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V 523 w Fz(k)568 b Fw(\276)523 b Fz(k)45 b FF(\()p Fz(E)78
b FF([)p Fz(N)139 b FF(]\))522 b Fw(\276)h Fz(k)45 b
FF(.)850 b(There)524 b(is)g(a)h(Galois)-3718 40344 y(represen)-36
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42092 y Ft(E)36 b(;N)15818 42855 y Fw(\241)-237 b(\241)g(!)369
b FF(Aut\()p Fz(E)78 b FF([)p Fz(N)139 b FF(]\))25635
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b FF(\()p 1312 44095 V Fz(k)45 b(=k)g FF(\()p Fz(E)78
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14062 47316 1123 54 v 371 w FD(Q)168 b Fw(\\)g FD(Q)p
FF(\()p Fz(t)p FF(\)\()p Fz(E)78 b FF([)p Fz(N)139 b
FF(]\))370 b(is)h(con)-36 b(tained)371 b(in)g Fz(Q)p
FF(\()-59 b Fz(\271)33369 48350 y(\271)33399 48387 y(\271)34181
48586 y Fy(N)35078 48387 y FF(\),)384 b Fz(X)104 b FF(\()p
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51597 y FF(with)g(the)f(natural)g(map)h(GL)21374 51796
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b(ctive.)-3718 58426 y(Pr)g(o)g(of.)649 b FF(If)p 1786
57694 V 566 w Fz(\275)2457 58739 y Fy(E)3817 58426 y
FF(is)566 b(surjectiv)-36 b(e)566 b(then)f(either)18125
57350 y Fr(\241)18956 57955 y Fx(0)361 b Fu(\241)p Fx(1)18956
58803 y(1)727 b(0)21211 57350 y Fr(\242)22385 58426 y
FF(or)566 b(its)g(negativ)-36 b(e)567 b(lies)f(in)f(the)h(image)g(of)g
Fz(\275)p FF(.)975 b(Th)-36 b(us)-3718 58955 y Fr(\241)-2888
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Fx(1)99 58955 y Fr(\242)1210 60031 y FF(lie)502 b(in)f(the)g(image)h
(of)h Fz(\275)p FF(.)782 b(Since)p 17558 59299 V 501
w Fz(\275)18229 60344 y Fy(E)19525 60031 y FF(is)501
b(surjectiv)-36 b(e)502 b(this)g(implies)g(that)f Fz(\275)g
FF(is)h(surjectiv)-36 b(e.)782 b(The)-3718 61636 y(con)-36
b(v)g(erse)434 b(is)g(trivial.)p 47147 61636 45 878 v
47192 60802 781 45 v 47192 61636 V 47972 61636 45 878
v -3718 65983 a Fs(10.1)2151 b(Digression)717 b(on)f(Mo)60
b(duli)-3718 68903 y Fz(X)-2639 69102 y Fx(0)-2113 68903
y FF(\()p Fz(N)139 b FF(\)\()p 586 67836 1202 54 v Fz(K)94
b FF(\))358 b(is)h(the)f(set)g(of)p 9411 67836 V 360
w Fz(K)95 b FF(-isomorpisms)358 b(classes)i(of)f(pairs)g(\()p
Fz(E)78 b(;)221 b(C)95 b FF(\))358 b(where)h Fz(E)78
b(=)p 36340 67836 V(K)453 b FF(is)359 b(an)f(elliptic)i(curv)-36
b(e)-3718 70508 y(and)416 b Fz(C)512 b FF(is)417 b(a)g(cyclic)i
(subgroup)c(of)j(order)e Fz(N)139 b FF(.)572 b Fz(X)19819
70707 y Fx(0)20346 70508 y FF(\()p Fz(N)139 b FF(\)\()p
Fz(K)95 b FF(\))415 b(is)i(the)g(set)f(of)i(isomorphism)f(classes)h(of)
f(pairs)-3718 72113 y(\()p Fz(E)78 b(;)221 b(C)95 b FF(\))433
b(suc)-36 b(h)433 b(that)g(for)h(all)h Fz(\276)417 b
Fw(2)368 b(G)79 b Fz(al)29 b FF(\()p 14806 71046 V Fz(K)95
b(=K)g FF(\),)434 b Fz(\276)48 b FF(\()p Fz(E)78 b(;)221
b(C)95 b FF(\))369 b(=)g(\()p Fz(E)78 b(;)221 b(C)95
b FF(\).)578 b(There)434 b(is)g(a)f(map)5943 74434 y
Fw(f)p Fz(k)45 b FF(-isomorphism)433 b(classes)i(of)f(pairs)g(\()p
Fz(E)78 b(;)221 b(C)95 b FF(\))p Fz(=K)g Fw(g)369 b(!)h
Fz(X)33495 74633 y Fx(0)34021 74434 y FF(\()p Fz(N)139
b FF(\)\()p Fz(K)95 b FF(\))21534 77755 y(41)p eop
%%Page: 42 50
42 49 bop 1263 -6698 a FF(42)25420 b FA(CHAPTER)434 b(10.)1013
b(THE)433 b(FIELD)h(OF)f(MODULI)1263 -3169 y FF(whic)-36
b(h)373 b(is)h(\\notoriously")h(non-injectiv)-36 b(e.)558
b(Deligne)374 b(and)f(Rapap)36 b(ort)374 b([13)q(])f(pro)-36
b(v)g(e)373 b(the)g(map)g(is)h(surjectiv)-36 b(e.)1263
-1564 y(When)377 b Fz(N)507 b FF(=)369 b(1)377 b(they)g(observ)-36
b(e)378 b(that)f(the)f(map)h(is)h(surjectiv)-36 b(e,)389
b(then)376 b(for)i Fz(N)507 b(>)369 b FF(1)378 b(they)f(sho)-36
b(w)377 b(that)g(certain)1263 41 y(obstructions)433 b(v)-72
b(anish.)578 b(A)434 b(related)g(question)f(is)1263 2373
y FD(Question)500 b(10.1.1.)651 b FC(If)493 b Fz(E)78
b(=)p 15527 1306 1202 54 v(K)588 b FC(is)493 b(isomorphic)f(to)h(al)66
b(l)494 b(its)f(Galois)g(c)-66 b(onjugates,)499 b(is)493
b(ther)-66 b(e)492 b Fz(E)47225 1891 y Fu(0)47535 2373
y Fz(=K)589 b FC(which)1263 3978 y(is)465 b(isomorphic)f(to)h
Fz(E)543 b FC(over)p 15255 2911 V 464 w Fz(K)95 b FC(?)1263
8350 y Fs(10.2)2152 b(When)716 b(is)h Fp(\275)16684 8637
y Fz(E)18497 8350 y Fs(Surjectiv)-60 b(e?)1263 11271
y FD(Prop)42 b(osition)500 b(10.2.1.)651 b FC(L)-66 b(et)331
b Fz(E)17294 11470 y Fx(1)18152 11271 y FC(and)h Fz(E)21506
11470 y Fx(2)22364 11271 y FC(b)-66 b(e)331 b(el)66 b(liptic)331
b(curves)h(de\257ne)-66 b(d)331 b(over)h Fz(K)427 b FC(with)332
b(e)-66 b(qual)332 b Fz(j)75 b FC(-invariants,)1263 12876
y(thus)412 b Fz(E)4962 13075 y Fx(1)5857 12507 y Fw(\273)5867
12931 y FF(=)7259 12876 y Fz(E)8222 13075 y Fx(2)9159
12876 y FC(over)p 11977 11809 V 410 w Fz(K)95 b FC(.)580
b(Assume)411 b Fz(E)19918 13075 y Fx(1)20855 12876 y
FC(and)g Fz(E)24288 13075 y Fx(2)25225 12876 y FC(do)g(not)g(have)g(c)
-66 b(omplex)410 b(multiplic)-66 b(ation)409 b(over)p
47966 11809 V 411 w Fz(K)95 b FC(.)579 b(Then)1263 14481
y Fz(\275)1934 14680 y Fy(E)2624 14803 y Fn(1)3605 14481
y FC(is)465 b(surje)-66 b(ctive)463 b(i\256)h Fz(\275)13126
14680 y Fy(E)13816 14803 y Fn(2)14797 14481 y FC(is)h(surje)-66
b(ctive.)1263 16813 y(Pr)g(o)g(of.)649 b FF(Assume)434
b Fz(\275)10826 17012 y Fy(E)11516 17135 y Fn(1)12466
16813 y FF(is)h(surjectiv)-36 b(e.)580 b(Since)434 b
Fz(E)24466 17012 y Fx(1)25425 16813 y FF(has)h(no)f(complex)g(m)-36
b(ultiplication)435 b(o)-36 b(v)g(er)p 45776 15746 V
434 w Fz(K)95 b FF(,)435 b(Aut)221 b Fz(E)51162 17012
y Fx(1)52057 16813 y FF(=)1263 18418 y Fw(f\247)p FF(1)p
Fw(g)p FF(.)625 b(Cho)36 b(ose)450 b(an)f(isomorphism)g
Fz(')395 b FF(:)g Fz(E)22204 18617 y Fx(1)23517 17665
y Fu(\273)23125 18418 y Fw(\241)-845 b(!)395 b Fz(E)25999
18617 y Fx(2)26974 18418 y FF(o)-36 b(v)g(er)p 29771
17351 V 449 w Fz(K)95 b FF(.)624 b(Then)449 b(for)g Fz(\276)443
b Fw(2)395 b(G)79 b Fz(al)29 b FF(\()p 42309 17351 V
Fz(K)95 b(=K)g FF(\))449 b(w)-36 b(e)449 b(ha)-36 b(v)g(e)449
b(the)1263 20023 y(diagram)23165 21714 y Fz(E)24128 21913
y Fx(1)26859 20907 y Fy(')25506 21714 y Fw(\241)-407
b(\241)-295 b(\241)-405 b(!)852 b Fz(E)30641 21913 y
Fx(2)22735 24017 y(=)23466 22654 y Fr(?)23466 23451 y(?)23466
24248 y(y)29248 24017 y Fx(=)29980 22654 y Fr(?)29980
23451 y(?)29980 24248 y(y)22963 26254 y Fy(\276)23367
26736 y Fz(E)24330 26935 y Fx(1)26575 25929 y Fy(\276)32
b(')25506 26736 y Fw(\241)-407 b(\241)-295 b(\241)-405
b(!)29477 26254 y Fy(\276)29880 26736 y Fz(E)30843 26935
y Fx(2)1263 28749 y FF(Th)-36 b(us)627 b Fz(\276)48 b(')699
b FF(=)f Fw(\247)p Fz(')628 b FF(for)g(all)g Fz(\276)747
b Fw(2)698 b(G)79 b Fz(al)29 b FF(\()p 21038 27682 V
Fz(K)95 b(=K)g FF(\),)676 b(so)628 b Fz(')699 b FF(:)g
Fz(E)31000 28948 y Fx(1)31526 28749 y FF([)p Fz(N)139
b FF(])698 b Fw(!)h Fz(E)37117 28948 y Fx(2)37643 28749
y FF([)p Fz(N)139 b FF(])627 b(de\257nes)f(an)i(equiv)-72
b(alence)p 1263 29623 671 54 v 1263 30355 a Fz(\275)1934
30668 y Fy(E)2624 30791 y Fn(1)3805 29986 y Fw(\273)3815
30410 y FF(=)p 5502 29623 V 5502 30355 a Fz(\275)6173
30668 y Fy(E)6863 30791 y Fn(2)7379 30355 y FF(.)1100
b(Since)607 b Fz(\275)13081 30554 y Fy(E)13771 30677
y Fn(1)14894 30355 y FF(is)g(surjectiv)-36 b(e)608 b(this)f(implies)p
29672 29623 V 608 w Fz(\275)30342 30668 y Fy(E)31032
30791 y Fn(2)32156 30355 y FF(is)g(surjectiv)-36 b(e)608
b(whic)-36 b(h,)651 b(b)-36 b(y)607 b(the)g(previous)1263
31960 y(prop)36 b(osition,)434 b(implies)g Fz(\275)13546
32159 y Fy(E)14236 32282 y Fn(2)15186 31960 y FF(is)g(surjectiv)-36
b(e.)p 52128 31960 45 878 v 52173 31126 781 45 v 52173
31960 V 52953 31960 45 878 v 3214 34435 a(Let)564 b Fz(K)687
b FF(=)592 b FD(C)p FF(\()p Fz(j)75 b FF(\),)598 b(with)564
b Fz(j)640 b FF(transcenden)-36 b(tal)563 b(o)-36 b(v)g(er)565
b FD(C)p FF(.)972 b(Let)564 b Fz(E)78 b(=K)660 b FF(b)36
b(e)564 b(an)h(elliptic)g(curv)-36 b(e)565 b(with)f Fz(j)75
b FF(-)1263 36040 y(in)-36 b(v)-72 b(arian)-36 b(t)434
b Fz(j)75 b FF(.)578 b(Fix)434 b(a)g(p)36 b(ositiv)-36
b(e)434 b(in)-36 b(teger)434 b Fz(N)572 b FF(and)433
b(let)18356 38530 y Fz(\275)19027 38729 y Fy(E)20190
38530 y FF(:)369 b Fw(G)79 b Fz(al)29 b FF(\()p 23394
37463 1202 54 v Fz(K)95 b(=K)g FF(\))370 b Fw(!)f FF(GL)30854
38729 y Fx(2)31380 38530 y FF(\()p FD(Z)p Fz(=)-72 b(N)139
b FD(Z)p FF(\))1263 41021 y(b)36 b(e)611 b(the)g(asso)36
b(ciated)612 b(Galois)g(represen)-36 b(tation.)1111 b(Then)611
b(det)221 b Fz(\275)31977 41220 y Fy(E)33383 41021 y
FF(is)611 b(the)g(cyclotomic)i(c)-36 b(haracter)611 b(whic)-36
b(h)1263 42626 y(is)558 b(trivial)i(since)e FD(C)g FF(con)-36
b(tains)558 b(the)f Fz(N)139 b FF(th)556 b(ro)36 b(ots)559
b(of)f(unit)-36 b(y)-108 b(.)951 b(Th)-36 b(us)557 b(the)h(image)h(of)f
Fz(\275)43324 42825 y Fy(E)44677 42626 y FF(lands)f(inside)h(of)1263
44231 y(SL)2798 44430 y Fx(2)3324 44231 y FF(\()p FD(Z)p
Fz(=)-72 b(N)139 b FD(Z)p FF(\).)947 b(Our)555 b(next)i(theorem)f
(states)g(that)g(a)h(generic)g(elliptic)g(curv)-36 b(e)556
b(has)g(maximal)i(p)36 b(ossible)1263 45836 y(Galois)435
b(action)f(on)f(its)h(division)g(p)36 b(oin)-36 b(ts.)1263
48169 y FD(Theorem)499 b(10.2.2.)652 b Fz(\275)13153
48368 y Fy(E)14316 48169 y FF(:)370 b Fw(G)79 b Fz(al)29
b FF(\()p 17521 47101 V Fz(K)95 b(=K)g FF(\))369 b Fw(!)g
FF(SL)24682 48368 y Fx(2)25208 48169 y FF(\()p FD(Z)p
Fz(=)-72 b(N)139 b FD(Z)p FF(\))465 b FC(is)f(surje)-66
b(ctive.)3214 50501 y FF(Igusa)410 b([8)q(])g(found)f(an)h(algebraic)h
(pro)36 b(of)410 b(of)h(this)e(theorem)h(W)-108 b(e)409
b(will)i(no)-36 b(w)410 b(mak)-36 b(e)410 b(some)h(commen)-36
b(ts)409 b(on)1263 52106 y(ho)-36 b(w)434 b(an)f(analytic)i(pro)36
b(of)434 b(go)36 b(es.)1263 54581 y FC(Pr)-66 b(o)g(of.)649
b FF(Let)332 b FD(C)p FF(\()p Fz(j)75 b FF(\))369 b(=)g
Fz(K)464 b FF(=)368 b Fw(F)15924 54780 y Fx(1)16783 54581
y FF(b)36 b(e)332 b(the)g(\257eld)g(of)i(mo)36 b(dular)332
b(functions)g(for)i(SL)38683 54780 y Fx(2)39208 54581
y FF(\()p FD(Z)p FF(\).)544 b(Supp)36 b(ose)332 b Fz(N)507
b Fw(\270)369 b FF(3)333 b(and)1263 56186 y(let)422 b
Fw(F)4085 56385 y Fy(N)5403 56186 y FF(b)36 b(e)421 b(the)g(\257eld)g
(of)h(mereomorphic)f(functions)h(for)g(\241\()p Fz(N)139
b FF(\).)573 b(Then)421 b Fw(F)37956 56385 y Fy(N)38852
56186 y Fz(=)p Fw(F)40457 56385 y Fx(1)41404 56186 y
FF(is)h(a)g(Galois)g(extension)1263 57791 y(with)434
b(Galois)g(group)f(SL)13437 57990 y Fx(2)13963 57791
y FF(\()p FD(Z)p Fz(=)-72 b(N)139 b FD(Z)p FF(\))p Fz(=)p
Fw(f\247)p FF(1)p Fw(g)p FF(.)3214 59396 y(Let)445 b
Fz(E)524 b FF(b)36 b(e)445 b(an)h(elliptic)g(curv)-36
b(e)446 b(o)-36 b(v)g(er)446 b Fz(K)541 b FF(with)446
b Fz(j)75 b FF(-in)-36 b(v)-72 b(arian)-36 b(t)445 b
Fz(j)75 b FF(.)614 b(W)-108 b(e)446 b(will)h(sho)-36
b(w)446 b(that)f Fw(G)79 b Fz(al)29 b FF(\()p Fw(F)48134
59595 y Fy(N)49031 59396 y Fz(=)p Fw(F)50636 59595 y
Fx(1)51162 59396 y FF(\))389 b(=)1263 61001 y(SL)2798
61200 y Fx(2)3324 61001 y FF(\()p FD(Z)p Fz(=)-72 b(N)139
b FD(Z)p FF(\))p Fz(=)p Fw(f\247)p FF(1)p Fw(g)420 b
FF(acts)g(transitiv)-36 b(ely)420 b(on)f(the)g Fz(x)p
FF(-co)36 b(ordinates)419 b(of)h(the)f Fz(N)558 b FF(torsion)419
b(p)36 b(oin)-36 b(ts)419 b(of)h Fz(E)78 b FF(.)573 b(This)1263
62606 y(will)374 b(sho)-36 b(w)374 b(that)p 9579 61875
671 54 v 372 w Fz(\275)10250 62920 y Fy(E)11417 62606
y FF(maps)f(surjectiv)-36 b(ely)375 b(on)-36 b(to)372
b(SL)25973 62805 y Fx(2)26499 62606 y FF(\()p FD(Z)p
Fz(=)-72 b(N)139 b FD(Z)p FF(\))p Fz(=)p Fw(f\247)p FF(1)p
Fw(g)p FF(.)558 b(Then)373 b(b)-36 b(y)373 b(prop)36
b(osition)373 b(10.0.2,)387 b Fz(\275)52274 62805 y Fy(E)1263
64212 y FF(maps)434 b(surjectiv)-36 b(ely)434 b(on)-36
b(to)434 b(SL)16000 64411 y Fx(2)16526 64212 y FF(\()p
FD(Z)p Fz(=)-72 b(N)139 b FD(Z)p FF(\),)433 b(as)h(claimed.)3214
65817 y(W)-108 b(e)438 b(will)i(no)-36 b(w)438 b(construct)f(the)h
Fz(x)p FF(-co)36 b(ordinates)438 b(of)h Fz(E)78 b FF([)p
Fz(N)139 b FF(])437 b(as)i(functions)f(on)g Fw(H)451
b FF(whic)-36 b(h)437 b(are)i(in)-36 b(v)-72 b(arian)-36
b(t)1263 67422 y(under)432 b(\241\()p Fz(N)139 b FF(\).)577
b(\(Th)-36 b(us)433 b Fz(K)95 b FF(\()p Fz(E)78 b FF([)p
Fz(N)139 b FF(])p Fz(=)p Fw(f\247)p FF(1)p Fw(g)p FF(\))370
b Fw(\275)f(F)24239 67621 y Fy(N)25135 67422 y FF(.\))3214
69027 y(Let)380 b Fz(\277)517 b Fw(2)369 b(H)393 b FF(and)380
b(let)h Fz(L)14530 69226 y Fy(\277)15474 69027 y FF(=)368
b FD(Z)p Fz(\277)336 b FF(+)187 b FD(Z)p FF(.)560 b(Consider)380
b Fz(})p FF(\()p Fz(z)59 b(;)221 b(L)30538 69226 y Fy(\277)31114
69027 y FF(\))381 b(whic)-36 b(h)380 b(giv)-36 b(es)381
b(the)f Fz(x)h FF(co)36 b(ordinate)381 b(of)g FD(C)p
Fz(=L)52493 69226 y Fy(\277)1263 70632 y FF(in)434 b(it)f(standard)g
(form.)579 b(De\257ne,)433 b(for)h(eac)-36 b(h)434 b(nonzero)f(\()p
Fz(r)-36 b(;)221 b(s)p FF(\))369 b Fw(2)g FF(\(\()p FD(Z)p
Fz(=)-72 b(N)139 b FD(Z)p FF(\))p Fz(=)p Fw(f\247)p FF(1)p
Fw(g)p FF(\))40880 70150 y Fx(2)41406 70632 y FF(,)434
b(a)g(function)15026 73855 y Fz(f)15667 74062 y Fx(\()p
Fy(r)-26 b(;s)p Fx(\))17918 73855 y FF(:)369 b Fw(H)382
b(!)369 b FD(C)g FF(:)1670 b Fz(\277)517 b Fw(7!)28241
72956 y Fz(g)28864 73155 y Fx(2)29389 72956 y FF(\()p
Fz(\277)148 b FF(\))p 28241 73550 2874 54 v 28241 74766
a Fz(g)28864 74965 y Fx(3)29389 74766 y FF(\()p Fz(\277)g
FF(\))31247 73855 y Fz(})p FF(\()32714 72956 y Fz(r)36
b(\277)443 b FF(+)295 b Fz(s)p 32714 73550 3551 54 v
33898 74766 a(N)36397 73855 y(;)221 b(L)37864 74054 y
Fy(\277)38439 73855 y FF(\))p Fz(:)p eop
%%Page: 43 51
43 50 bop -3718 -6698 a FA(10.3.)1013 b(OBSER)-145 b(V)g(A)-108
b(TIONS)36652 b FF(43)-3718 -3169 y(First)433 b(notice)h(that)f(for)h
(an)-36 b(y)434 b Fz(\256)378 b FF(=)13127 -4245 y Fr(\241)13958
-3594 y Fy(a)401 b(b)14004 -2794 y(c)406 b(d)15524 -4245
y Fr(\242)16502 -3169 y Fw(2)368 b FF(SL)19292 -2970
y Fx(2)19817 -3169 y FF(\()p FD(Z)p FF(\),)15025 -585
y Fz(f)15666 -378 y Fx(\()p Fy(r)-26 b(;s)p Fx(\))17548
-585 y FF(\()p Fz(\256)8 b(\277)148 b FF(\))370 b(=)e
Fz(f)22499 87 y Fx(\()p Fy(r)-26 b(;s)p Fx(\))24326 -989
y Fr(\241)25092 -227 y Fy(a)402 b(b)25138 573 y(c)407
b(d)26594 -989 y Fr(\242)27258 -585 y FF(\()p Fz(\277)148
b FF(\))p Fz(:)-3718 2585 y FF(Indeed,)377 b Fz(})364
b FF(is)g(homogenous)g(of)g(degree)g Fw(\241)p FF(2,)378
b Fz(g)18985 2784 y Fx(2)19875 2585 y FF(is)364 b(mo)36
b(dular)364 b(of)g(w)-36 b(eigh)g(t)364 b(4)g(and)g Fz(g)35767
2784 y Fx(3)36656 2585 y FF(is)g(mo)36 b(dular)364 b(of)g(w)-36
b(eigh)g(t)-3718 4190 y(6,)434 b(so)7109 7434 y Fz(f)7750
7640 y Fx(\()p Fy(r)-26 b(;s)p Fx(\))9632 7434 y FF(\()p
Fz(\256)8 b(\277)148 b FF(\))369 b(=)14075 6535 y Fz(g)14698
6734 y Fx(2)15224 6535 y FF(\()p Fz(\256)8 b(\277)148
b FF(\))p 14075 7128 3710 54 v 14075 8345 a Fz(g)14698
8544 y Fx(3)15224 8345 y FF(\()p Fz(\256)8 b(\277)148
b FF(\))17917 7434 y Fz(})p FF(\()19384 6535 y Fz(r)36
b(\256)8 b(\277)444 b FF(+)295 b Fz(s)p 19384 7128 4386
54 v 20986 8345 a(N)23902 7434 y FF(\))12561 11014 y(=)369
b(\()p Fz(c\277)443 b FF(+)295 b Fz(d)p FF(\))18505 10466
y Fu(\241)p Fx(2)19895 10116 y Fz(g)20518 10315 y Fx(2)21044
10116 y FF(\()p Fz(\277)148 b FF(\))p 19895 10709 2874
54 v 19895 11926 a Fz(g)20518 12125 y Fx(3)21044 11926
y FF(\()p Fz(\277)g FF(\))22901 11014 y Fz(})p FF(\()24368
10116 y Fz(r)36 b(a\277)443 b FF(+)295 b Fz(r)36 b(b)296
b FF(+)f Fz(cs\277)442 b FF(+)295 b Fz(sd)p 24368 10709
11174 54 v 27083 11926 a(N)139 b FF(\()p Fz(c\277)443
b FF(+)295 b Fz(d)p FF(\))35675 11014 y(\))12561 14595
y(=)14075 13697 y Fz(g)14698 13896 y Fx(2)15224 13697
y FF(\()p Fz(\277)148 b FF(\))p 14075 14290 2874 54 v
14075 15507 a Fz(g)14698 15706 y Fx(3)15224 15507 y FF(\()p
Fz(\277)g FF(\))17081 14595 y Fz(})p FF(\()18548 13697
y(\()p Fz(r)36 b(a)295 b FF(+)g Fz(sc)p FF(\))p Fz(\277)442
b FF(+)295 b Fz(r)36 b(b)296 b FF(+)f Fz(sd)p 18548 14290
11473 54 v 23694 15507 a(N)30153 14595 y FF(\))369 b(=)f
Fz(f)33049 14802 y Fx(\()p Fy(r)-26 b(;s)p Fx(\))p Fy(\256)35536
14595 y FF(\()p Fz(\277)148 b FF(\))-1767 17925 y(If)542
b Fz(\277)700 b Fw(2)551 b(H)j FF(with)541 b Fz(g)7712
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b(Conclude)574 b(that)f(there)h(is)g(an)g(elliptic)h(curv)-36
b(e)574 b Fz(E)36383 57489 y Fx(0)36908 57290 y Fz(=K)670
b FF(suc)-36 b(h)573 b(that)h Fz(E)46331 57489 y Fx(0)47464
56922 y Fw(\273)47475 57346 y FF(=)49106 57290 y Fz(E)651
b FF(o)-36 b(v)g(er)1263 58896 y Fz(L)p FF(.)719 b(The)481
b(conclusion)g(ma)-36 b(y)481 b(fail)h(to)e(hold)h(if)g
Fz(L)f FF(is)h(a)g(\257nite)f(extension)h(of)g Fz(K)95
b FF(,)493 b(but)479 b(the)h(exercise)i(is)e(true)1263
60501 y(when)466 b Fz(L)424 b FF(=)p 7437 59433 V 424
w Fz(K)95 b FF(.)676 b(First)466 b(w)-36 b(e)466 b(giv)-36
b(e)467 b(a)g(descen)-36 b(t)465 b(argumen)-36 b(t)465
b(whic)-36 b(h)466 b(holds)g(when)f Fz(L)425 b FF(=)p
42488 59433 V 424 w Fz(K)561 b FF(and)465 b(then)g(giv)-36
b(e)467 b(a)1263 62106 y(coun)-36 b(terexample)434 b(to)f(the)g(more)h
(general)g(statemen)-36 b(t.)3214 63717 y(F)-108 b(or)453
b Fz(g)48 b(;)221 b(h)401 b Fw(2)g Fz(G)h FF(=)f Fw(G)79
b Fz(al)29 b FF(\()p Fz(L=K)95 b FF(\))453 b(w)-36 b(e)454
b(de\257ne)d(an)i(automorphism)f Fz(c)p FF(\()p Fz(g)48
b(;)221 b(h)p FF(\))401 b Fw(2)g FF(Aut)221 b Fz(E)480
b FF(=)401 b Fw(f\247)p FF(1)p Fw(g)p FF(.)637 b(Cho)36
b(ose)1263 65322 y(for)434 b(ev)-36 b(ery)435 b Fz(g)416
b Fw(2)369 b(G)79 b Fz(al)29 b FF(\()p Fz(L=K)95 b FF(\))433
b(some)h(isomorphism)23465 68273 y Fz(\270)24224 68472
y Fy(g)25127 68273 y FF(:)25857 67724 y Fy(g)26171 68273
y Fz(E)27973 67520 y Fu(\273)27581 68273 y Fw(\241)-846
b(!)369 b Fz(E)78 b(:)1263 71224 y FF(If)403 b(the)g
Fz(\270)5501 71423 y Fy(g)6438 71224 y FF(w)-36 b(ere)403
b(to)f(all)i(satisfy)g(the)e(compatibilit)-36 b(y)403
b(criterion)g Fz(\270)32827 71423 y Fy(g)32 b(h)34273
71224 y FF(=)369 b Fz(\270)36413 71423 y Fy(g)37179 71224
y Fw(\261)38075 70742 y Fy(g)38389 71224 y Fz(\270)39148
71423 y Fy(h)40148 71224 y FF(then)401 b(b)-36 b(y)403
b(descen)-36 b(t)402 b(theory)1263 72829 y(w)-36 b(e)537
b(could)f(\257nd)g(a)g Fz(K)95 b FF(-structure)535 b(on)h
Fz(E)78 b FF(,)563 b(that)536 b(is)g(a)h(mo)36 b(del)537
b(for)g Fz(E)614 b FF(de\257ned)535 b(o)-36 b(v)g(er)537
b Fz(K)632 b FF(and)536 b(isomorphic)1263 74434 y(to)510
b Fz(E)588 b FF(o)-36 b(v)g(er)511 b Fz(L)p FF(.)808
b(De\257ne)509 b Fz(c)p FF(\()p Fz(g)48 b(;)221 b(h)p
FF(\))510 b(b)-36 b(y)510 b Fz(c)p FF(\()p Fz(g)48 b(;)221
b(h)p FF(\))p Fz(\270)23797 74633 y Fy(g)32 b(h)25372
74434 y FF(=)499 b Fz(\270)27642 74633 y Fy(g)28524 74434
y Fw(\261)29536 73952 y Fy(g)29849 74434 y Fz(\270)30608
74633 y Fy(h)31715 74434 y FF(so)511 b Fz(c)p FF(\()p
Fz(g)48 b(;)221 b(h)p FF(\))509 b(measures)h(ho)-36 b(w)511
b(m)-36 b(uc)g(h)509 b(the)g Fz(\270)52534 74633 y Fy(g)p
eop
%%Page: 45 53
45 52 bop -3718 -6698 a FA(10.6.)1013 b(A)-36 b(CTION)434
b(OF)f FF(GL)9801 -6499 y Fx(2)46787 -6698 y FF(45)-3718
-3169 y(fail)530 b(to)e(satisfy)i(the)e(compatibilit)-36
b(y)529 b(criterion.)863 b(Since)528 b Fz(c)p FF(\()p
Fz(g)48 b(;)221 b(h)p FF(\))527 b(is)i(a)g(co)36 b(cycle)530
b(it)e(de\257nes)f(an)i(elemen)-36 b(ts)-3718 -1564 y(of)558
b Fz(H)-932 -2046 y Fx(2)-407 -1564 y FF(\()p Fz(G;)221
b Fw(f\247)p FF(1)p Fw(g)p FF(\).)951 b(W)-108 b(e)557
b(w)-36 b(an)g(t)558 b(to)f(kno)-36 b(w)558 b(that)f(this)g(elemen)-36
b(t)557 b(is)h(trivial.)951 b(When)557 b Fz(L)580 b FF(=)p
41115 -2631 1202 54 v 579 w Fz(K)95 b FF(,)589 b(the)557
b(map)-3718 41 y Fz(H)-2537 -441 y Fx(2)-2012 41 y FF(\()p
Fz(G;)221 b Fw(f\247)p FF(1)p Fw(g)p FF(\))370 b Fw(!)f
Fz(H)6867 -441 y Fx(2)7393 41 y FF(\()p Fz(G;)221 b(L)10392
-441 y Fu(\244)10918 41 y FF(\))433 b(is)h(injectiv)-36
b(e.)579 b(T)-108 b(o)434 b(see)g(this)f(\257rst)g(consider)g(the)g
(exact)i(sequence)14205 2965 y(0)369 b Fw(!)h(f\247)p
FF(1)p Fw(g)f(!)p 22000 1898 V 370 w Fz(K)23202 2142
y Fu(\244)24489 2212 y Fx(2)24096 2965 y Fw(\241)-885
b(!)p 25720 1898 V 148 w Fz(K)26922 2142 y Fu(\244)27817
2965 y Fw(!)369 b FF(0)-3718 5595 y(where)433 b(2)370
b(:)p 1789 4528 V 369 w Fz(K)2991 4772 y Fu(\244)3886
5595 y Fw(!)p 5583 4528 V 369 w Fz(K)6785 4772 y Fu(\244)7744
5595 y FF(is)434 b(the)f(squaring)h(map.)578 b(T)-108
b(aking)435 b(cohomology)h(yields)e(an)f(exact)i(sequence)10214
8225 y Fz(H)11395 7677 y Fx(1)11921 8225 y FF(\()p Fz(G;)p
14035 7158 V 221 w(K)15237 7402 y Fu(\244)15763 8225
y FF(\))369 b Fw(!)g Fz(H)19516 7677 y Fx(2)20041 8225
y FF(\()p Fz(G;)221 b Fw(f\247)p FF(1)p Fw(g)p FF(\))370
b Fw(!)f Fz(H)28920 7677 y Fx(2)29446 8225 y FF(\()p
Fz(G;)p 31560 7158 V 221 w(K)32762 7402 y Fu(\244)33288
8225 y FF(\))p Fz(:)-3718 10855 y FF(By)474 b(Hilb)36
b(ert's)473 b(theorem)g(90)h(\([25)q(])f(Ch.)698 b(X,)473
b(Prop.)697 b(2\),)484 b Fz(H)25164 10373 y Fx(1)25689
10855 y FF(\()p Fz(G;)221 b Fw(f\247)p FF(1)p Fw(g)p
FF(\))438 b(=)e(0.)698 b(Th)-36 b(us)473 b(w)-36 b(e)473
b(ha)-36 b(v)g(e)474 b(an)f(exact)-3718 12460 y(sequence)10872
14065 y(0)369 b Fw(!)g Fz(H)14769 13517 y Fx(2)15294
14065 y FF(\()p Fz(G;)221 b Fw(f\247)p FF(1)p Fw(g)p
FF(\))370 b Fw(!)g Fz(H)24174 13517 y Fx(2)24699 14065
y FF(\()p Fz(G;)p 26813 12998 V 221 w(K)28015 13242 y
Fu(\244)28541 14065 y FF(\)[2])g Fw(!)f FF(0)p Fz(:)-3718
16260 y FF(Th)-36 b(us)433 b Fz(H)758 15778 y Fx(2)1283
16260 y FF(\()p Fz(G;)221 b Fw(f\247)p FF(1)p Fw(g)p
FF(\))435 b(naturally)f(sits)f(inside)h Fz(H)20149 15778
y Fx(2)20674 16260 y FF(\()p Fz(G;)221 b(L)23673 15778
y Fu(\244)24199 16260 y FF(\).)-1767 17865 y(T)-108 b(o)513
b(\257nish)e(Rib)36 b(et)513 b(do)36 b(es)513 b(something)f(with)h
(di\256eren)-36 b(tials)512 b(and)g Fz(H)30694 17383
y Fx(0)31219 17865 y FF(\()31725 17383 y Fy(g)32038 17865
y Fz(E)78 b(;)221 b FF(\255)34600 17383 y Fx(1)35127
17865 y FF(\))512 b(whic)-36 b(h)512 b(I)h(don't)f(under-)-3718
19470 y(stand.)-1767 21075 y(The)589 b(coun)-36 b(terexample)589
b(in)h(the)e(case)i(when)f Fz(L=K)685 b FF(is)589 b(\257nite)g(w)-36
b(as)590 b(pro)-36 b(vided)589 b(b)-36 b(y)589 b(Kevin)h(Buzzard)-3718
22680 y(\(who)476 b(said)f(Coates)i(ga)-36 b(v)g(e)477
b(it)e(to)h(him\).)704 b(Let)475 b Fz(L)441 b FF(=)f
FD(Q)p FF(\()p Fz(i)p FF(\),)486 b Fz(K)536 b FF(=)440
b FD(Q)476 b FF(and)f Fz(E)553 b FF(b)36 b(e)475 b(the)h(elliptic)g
(curv)-36 b(e)475 b(with)-3718 24285 y(W)-108 b(eirstrass)409
b(equation)h Fz(iy)8978 23803 y Fx(2)9872 24285 y FF(=)369
b Fz(x)11992 23803 y Fx(3)12763 24285 y FF(+)245 b Fz(x)g
FF(+)g(1.)570 b(Then)409 b Fz(E)487 b FF(is)409 b(isomorphic)h(to)f
(its)g(conjugate)h(o)-36 b(v)g(er)409 b Fz(L)g FF(but)f(one)-3718
25890 y(can)433 b(sho)-36 b(w)434 b(directly)g(that)f
Fz(E)511 b FF(has)434 b(no)f(mo)36 b(del)434 b(o)-36
b(v)g(er)434 b FD(Q)p FF(.)-3718 30283 y Fs(10.6)2151
b(Action)717 b(of)g Fm(GL)14206 30570 y FF(2)-3718 33203
y(Let)536 b Fz(N)683 b(>)545 b FF(3)537 b(b)36 b(e)537
b(an)f(in)-36 b(teger)537 b(and)f Fz(E)78 b(=)p FD(Q)p
FF(\()p Fz(j)d FF(\))537 b(an)g(elliptic)g(curv)-36 b(e)537
b(with)g Fz(j)75 b FF(-in)-36 b(v)-72 b(arian)-36 b(t)536
b Fz(j)75 b FF(\()p Fz(E)j FF(\))544 b(=)h Fz(j)75 b
FF(.)888 b(Then)-3718 34808 y(there)433 b(is)h(a)g(Galois)g(extension)
15202 36042 y Fw(F)16157 36241 y Fy(N)17422 36042 y FF(=)369
b FD(Q)p FF(\()p Fz(j)75 b FF(\)\()p Fz(E)j FF([)p Fz(N)139
b FF(])p Fz(=)p Fw(f\247)p FF(1)p Fw(g)p FF(\))22000
37648 y Fw(j)19196 39253 y(F)20151 39452 y Fx(1)21046
39253 y FF(=)368 b FD(Q)p FF(\()p Fz(j)75 b FF(\))-3718
41431 y(with)369 b(Galois)g(group)g(GL)8559 41630 y Fx(2)9085
41431 y FF(\()p FD(Z)p Fz(=)-72 b(N)139 b FD(Z)p FF(\))p
Fz(=)p Fw(f\247)p FF(1)p Fw(g)p FF(.)558 b(Think)369
b(of)g FD(Q)p FF(\()p Fz(j)75 b FF(\)\()p Fz(E)j FF([)p
Fz(N)139 b FF(])p Fz(=)p Fw(f\247)p FF(1)p Fw(g)370 b
FF(as)f(the)f(\257eld)h(obtained)f(from)-3718 43036 y
FD(Q)p FF(\()p Fz(j)75 b FF(\))451 b(b)-36 b(y)452 b(adjoining)g(the)f
Fz(x)p FF(-co)36 b(ordinates)451 b(of)i(the)e Fz(N)139
b FF(-torsion)450 b(p)36 b(oin)-36 b(ts)451 b(of)i Fz(E)78
b FF(.)631 b(Note)452 b(that)f(this)g(situation)-3718
44641 y(di\256ers)433 b(from)h(the)f(previous)h(situation)f(in)h(that)f
(the)g(base)h(\257eld)f FD(C)h FF(has)f(b)36 b(een)433
b(replaced)g(b)-36 b(y)434 b FD(Q)p FF(.)-1767 46246
y(Consider)18992 47851 y Fw(F)501 b FF(=)21828 46589
y Fr([)22146 49408 y Fy(N)23526 47851 y Fw(F)24481 48050
y Fy(N)-3718 51164 y FF(whic)-36 b(h)478 b(corresp)36
b(onds)478 b(to)h(a)g(pro)72 b(jectiv)-36 b(e)479 b(system)g(of)g(mo)36
b(dular)479 b(curv)-36 b(es)478 b(.)714 b(Let)478 b FD(A)35768
51363 y Fy(f)36852 51164 y FF(b)36 b(e)478 b(the)g(ring)h(of)g
(\257nite)-3718 52769 y(ad)-36 b(\265)-614 b(eles,)434
b(th)-36 b(us)14326 54532 y FD(A)15454 54731 y Fy(f)16429
54532 y FF(=)18045 54193 y(^)17809 54532 y FD(Q)370 b
FF(=)20813 54193 y(^)20681 54532 y FD(Z)296 b Fw(\255)f
FD(Q)369 b Fw(\275)26112 53271 y Fr(Y)26723 56060 y Fy(p)28030
54532 y FD(Q)29152 54731 y Fy(p)29682 54532 y Fz(:)-3718
57988 y FD(A)-2590 58187 y Fy(f)-1551 57988 y FF(can)433
b(b)36 b(e)434 b(though)-36 b(t)432 b(of)j(as)12944 59593
y Fw(f)p FF(\()p Fz(x)14853 59792 y Fy(p)15382 59593
y FF(\))369 b(:)g Fz(x)17726 59792 y Fy(p)18625 59593
y Fw(2)f FD(Z)20792 59792 y Fy(p)21755 59593 y FF(for)434
b(almost)g(all)h Fz(p)p Fw(g)p Fz(:)-3718 61787 y FF(The)564
b(group)f(GL)4735 61986 y Fx(2)5260 61787 y FF(\()p FD(A)6894
61986 y Fy(f)7500 61787 y FF(\))g(acts)i(on)e Fw(F)132
b FF(.)969 b(T)-108 b(o)565 b(understand)d(what)i(this)f(action)i(is)f
(w)-36 b(e)564 b(\257rst)f(consider)h(the)-3718 63531
y(subgroup)432 b(GL)3759 63730 y Fx(2)4284 63531 y FF(\()4921
63192 y(^)4790 63531 y FD(Z)p FF(\))i(of)g(GL)9957 63730
y Fx(2)10483 63531 y FF(\()p FD(A)12117 63730 y Fy(f)12722
63531 y FF(\).)-1767 65137 y(It)f(can)h(b)36 b(e)433
b(sho)-36 b(wn)434 b(that)7596 67767 y Fw(F)501 b FF(=)368
b FD(Q)p FF(\()p Fz(f)12701 67973 y Fy(N)42 b(;)p Fx(\()p
Fy(r)-26 b(;s)p Fx(\))16003 67767 y FF(:)369 b(\()p Fz(r)-36
b(;)221 b(s)p FF(\))369 b Fw(2)g FF(\()p FD(Z)p Fz(=)-72
b(N)139 b FD(Z)p FF(\))25711 67218 y Fx(2)26531 67767
y Fw(\241)296 b(f)p FF(\(0)p Fz(;)221 b FF(0\))p Fw(g)p
Fz(;)g(N)509 b Fw(\270)369 b FF(1\))-3718 70397 y(where)433
b Fz(f)680 70603 y Fy(N)42 b(;)p Fx(\()p Fy(r)-26 b(;s)p
Fx(\))4046 70397 y FF(is)434 b(a)g(function)9519 73855
y Fz(f)10160 74062 y Fy(N)42 b(;)p Fx(\()p Fy(r)-26 b(;s)p
Fx(\))13462 73855 y FF(:)369 b Fw(H)381 b(!)370 b FD(C)f
FF(:)1670 b Fz(\277)517 b Fw(7!)23784 72956 y Fz(g)24407
73155 y Fx(2)24933 72956 y FF(\()p Fz(\277)148 b FF(\))p
23784 73550 2874 54 v 23784 74766 a Fz(g)24407 74965
y Fx(3)24933 74766 y FF(\()p Fz(\277)g FF(\))26791 73855
y Fz(})p FF(\()28258 72956 y Fz(r)36 b(\277)443 b FF(+)295
b Fz(s)p 28258 73550 3551 54 v 29442 74766 a(N)31940
73855 y(;)221 b(L)33407 74054 y Fy(\277)33983 73855 y
FF(\))p Fz(:)p eop
%%Page: 46 54
46 53 bop 1263 -6698 a FF(46)25420 b FA(CHAPTER)434 b(10.)1013
b(THE)433 b(FIELD)h(OF)f(MODULI)1263 -3169 y FF(W)-108
b(e)486 b(de\257ne)f(the)g(action)h(of)h(GL)16978 -2970
y Fx(2)17504 -3169 y FF(\()18141 -3509 y(^)18010 -3169
y FD(Z)p FF(\))e(on)h Fw(F)617 b FF(as)487 b(follo)-36
b(ws.)736 b(Let)486 b Fz(g)505 b Fw(2)458 b FF(GL)36615
-2970 y Fx(2)37141 -3169 y FF(\()37778 -3509 y(^)37647
-3169 y FD(Z)p FF(\),)499 b(then)484 b(to)i(giv)-36 b(e)487
b(the)e(action)1263 -1564 y(of)466 b Fz(g)512 b FF(on)465
b Fz(f)6390 -1358 y Fy(N)42 b(;)p Fx(\()p Fy(r)-26 b(;s)p
Fx(\))9788 -1564 y FF(\257rst)464 b(map)h Fz(g)512 b
FF(in)-36 b(to)465 b(GL)21060 -1365 y Fx(2)21585 -1564
y FF(\()p FD(Z)p Fz(=)-72 b(N)139 b FD(Z)p FF(\))465
b(via)h(the)e(natural)h(reduction)f(map,)473 b(then)464
b(note)h(that)1263 41 y(GL)3096 240 y Fx(2)3622 41 y
FF(\()p FD(Z)p Fz(=)-72 b(N)139 b FD(Z)p FF(\))433 b(acts)h(on)g
Fz(f)13781 247 y Fy(N)42 b(;)p Fx(\()p Fy(r)-26 b(;s)p
Fx(\))17147 41 y FF(b)-36 b(y)13834 2606 y Fr(\241)14442
2868 y Fz(a)1169 b(b)14504 4473 y(c)f(d)16908 2606 y
Fr(\242)17812 3682 y Fw(\242)295 b Fz(f)19117 3888 y
Fy(N)42 b(;)p Fx(\()p Fy(r)-26 b(;s)p Fx(\))22419 3682
y FF(=)368 b Fz(f)24440 4752 y Fy(N)42 b(;)p Fx(\()p
Fy(r)-26 b(;s)p Fx(\))27317 3277 y Fr(\263)28268 4438
y Fy(a)401 b(b)28314 5238 y(c)407 b(d)29770 3277 y Fr(\264)30987
3682 y FF(=)369 b Fz(f)33009 3888 y Fy(N)42 b(;)p Fx(\()p
Fy(r)26 b(a)p Fx(+)p Fy(sc;r)g(b)p Fx(+)p Fy(sd)p Fx(\))40137
3682 y Fz(:)3214 7814 y FF(Let)433 b Fz(E)512 b FF(b)36
b(e)433 b(an)g(elliptic)i(curv)-36 b(e.)578 b(Then)433
b(the)g(univ)-36 b(ersal)434 b(T)-108 b(ate)434 b(mo)36
b(dule)433 b(is)18621 10743 y Fz(T)181 b FF(\()p Fz(E)78
b FF(\))368 b(=)h(lim)23366 11353 y Fw(\241)-555 b(!)25394
10743 y Fz(E)78 b FF([)p Fz(N)139 b FF(])368 b(=)30087
9481 y Fr(Y)30699 12270 y Fy(p)32006 10743 y Fz(T)32768
10942 y Fy(p)33297 10743 y FF(\()p Fz(E)78 b FF(\))p
Fz(:)1263 15113 y FF(There)632 b(is)g(an)g(isomorphism)f
Fz(\256)715 b FF(:)19270 14774 y(^)19139 15113 y FD(Z)20052
14631 y Fx(2)21676 14360 y Fu(\273)21284 15113 y Fw(\241)-845
b(!)707 b Fz(T)181 b FF(\()p Fz(E)78 b FF(\).)1172 b(Via)632
b(righ)-36 b(t)632 b(comp)36 b(osition)632 b(GL)43296
15312 y Fx(2)43822 15113 y FF(\()44459 14774 y(^)44328
15113 y FD(Z)p FF(\))g(acts)g(on)f(the)1263 16718 y(collection)593
b(of)g(all)g(suc)-36 b(h)591 b(isomorphism)h Fz(\256)8
b FF(.)1054 b(So)592 b(GL)27707 16917 y Fx(2)28233 16718
y FF(\()28870 16379 y(^)28739 16718 y FD(Z)p FF(\))f(acts)i(naturally)f
(on)g(pairs)g(\()p Fz(E)78 b(;)221 b(\256)8 b FF(\))592
b(but)f(the)1263 18323 y(action)485 b(do)36 b(es)485
b(nothing)g(to)f Fz(E)78 b FF(.)732 b(One)484 b(of)h(the)g(\257rst)f
(imp)36 b(ortan)-36 b(t)484 b(things)g(w)-36 b(e)485
b(m)-36 b(ust)484 b(do)h(in)g(understanding)1263 19928
y(the)383 b(construction)g(of)h(things)e(lik)-36 b(e)385
b(Shim)-36 b(ura)382 b(v)-72 b(arieties)384 b(is)g(to)f(free)h(ourselv)
-36 b(es)384 b(and)e(allo)-36 b(w)385 b(GL)46961 20127
y Fx(2)47487 19928 y FF(\()48124 19589 y(^)47993 19928
y FD(Z)p FF(\))e(to)h(act)1263 21533 y(on)434 b(the)f
Fz(E)78 b FF('s)433 b(as)h(w)-36 b(ell.)3214 23138 y(Let)20741
25186 y Fz(g)416 b FF(=)23161 23712 y Fr(\263)23954 24373
y Fz(a)1168 b(b)24016 25978 y(c)g(d)26420 23712 y Fr(\264)27582
25186 y Fw(2)369 b FF(GL)30670 24629 y Fx(+)30670 25515
y(2)31457 25186 y FF(\()p FD(Q)p FF(\))1263 28216 y(\(th)-36
b(us)314 b Fz(g)363 b FF(has)315 b(p)36 b(ositiv)-36
b(e)316 b(determinen)-36 b(t\).)538 b(Let)314 b Fz(\277)518
b Fw(2)368 b(H)328 b FF(and)315 b(let)g Fz(E)447 b FF(=)369
b Fz(E)34551 28415 y Fy(\277)35441 28216 y FF(b)36 b(e)315
b(the)g(elliptic)h(curv)-36 b(e)315 b(determined)1263
29821 y(b)-36 b(y)434 b(the)f(lattice)h Fz(L)10169 30020
y Fy(\277)11113 29821 y FF(=)369 b FD(Z)295 b FF(+)g
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32790 y Fy(\277)22164 32591 y FF(:)370 b Fz(L)23780 32790
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b(!)369 b FD(Z)33413 32043 y Fx(2)1263 35361 y FF(b)36
b(e)434 b(the)f(isomorphism)g(de\257ned)f(b)-36 b(y)434
b Fz(\277)517 b Fw(7!)369 b FF(\(1)p Fz(;)221 b FF(0\))435
b(and)e(1)369 b Fw(7!)h FF(\(0)p Fz(;)221 b FF(1\).)579
b(No)-36 b(w)434 b(view)h Fz(\256)378 b FF(=)368 b Fz(\256)43723
35560 y Fy(\277)44732 35361 y FF(as)434 b(a)g(map)20550
38132 y Fz(\256)378 b FF(:)369 b FD(Z)23398 37583 y Fx(2)24685
37379 y Fu(\273)24293 38132 y Fw(\241)-845 b(!)369 b
Fz(H)27256 38331 y Fx(1)27781 38132 y FF(\()p Fz(E)78
b FF(\()p FD(C)p FF(\))p Fz(;)221 b FD(Z)p FF(\))p Fz(:)1263
40902 y FF(T)-108 b(ensoring)434 b(with)f FD(Q)h FF(then)f(giv)-36
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b(!)461 b Fz(L)23489 51501 y Fu(0)24260 51983 y Fw(\275)h
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b(;)221 b FD(Q)p FF(\))488 b(where)f Fz(L)36299 51501
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b Fz(E)10036 53106 y Fu(0)10346 53588 y Fz(=)p FD(C)g
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53106 y Fu(0)25961 53588 y Fz(;)221 b(E)78 b FF(\))296
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FF(\))26296 55605 y Fu(\273)25904 56358 y Fw(\241)-845
b(!)369 b Fz(L)28674 55810 y Fu(0)29354 56358 y Fw(\275)g
Fz(H)31834 56557 y Fx(1)32359 56358 y FF(\()p Fz(E)78
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60734 y(No)-36 b(w)399 b(w)-36 b(e)399 b(can)g(de\257ne)e(an)i(action)g
(on)g(pairs)f(\()p Fz(E)78 b(;)221 b(\256)8 b FF(\))400
b(b)-36 b(y)398 b(sending)g(\()p Fz(E)78 b(;)221 b(\256)8
b FF(\))400 b(to)e(\()p Fz(E)41936 60252 y Fu(0)42246
60734 y Fz(;)221 b(\256)43663 60252 y Fu(0)43975 60734
y FF(\).)567 b(Here)398 b Fz(\256)49279 60252 y Fu(0)49989
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61857 y Fu(0)5669 62339 y FF(:)369 b FD(Z)7312 61857
y Fx(2)8207 62339 y Fw(!)g Fz(H)10982 62538 y Fx(1)11508
62339 y FF(\()p Fz(E)13055 61857 y Fx(1)13580 62339 y
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b(y)433 b(the)g(comp)36 b(osition)19940 65573 y FD(Z)20853
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b(!)369 b Fz(L)24870 65025 y Fu(0)25942 64820 y Fy(\270)26491
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b Fz(H)29431 65772 y Fx(1)29956 65573 y FF(\()p Fz(E)31503
65025 y Fx(1)32028 65573 y Fz(;)221 b FD(Z)p FF(\))p
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(is)20713 71114 y Fz(g)417 b FF(:)369 b(\()p Fz(E)23952
71313 y Fy(\277)24527 71114 y Fz(;)221 b(\256)25936 71313
y Fy(\277)26512 71114 y FF(\))369 b Fw(7!)g FF(\()p Fz(E)30631
70565 y Fu(0)30553 71442 y Fy(\277)31128 71114 y Fz(;)221
b(\256)32545 70565 y Fu(0)32537 71442 y Fy(\277)33113
71114 y FF(\))1263 73884 y(where)434 b Fz(\277)5734 73402
y Fu(0)6413 73884 y FF(=)369 b Fz(g)48 b(\277)517 b FF(=)11060
73361 y Fy(a\277)102 b Fx(+)p Fy(b)p 11060 73578 2154
54 v 11065 74342 a(c\277)g Fx(+)p Fy(d)13347 73884 y
Fz(:)434 b FF([[)g(CHECK)g(THIS)g(SOMETIME)f(SOON!!]])p
eop
%%Page: 47 55
47 54 bop -3718 5686 a FE(Chapter)1033 b(11)-3718 11221
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b FF(of)360 b FD(C)p FF(.)554 b(The)360 b(Hec)-36 b(k)g(e)-3718
22610 y(algebras)505 b(o)36 b(ccur)505 b(naturally)g(as)g(op)36
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b(dular)504 b(forms.)792 b(W)-108 b(e)505 b(are)g(aiming)-3718
24215 y(for)417 b(an)g(arithmetic)g(p)36 b(ersp)g(ectiv)-36
b(e.)573 b(One)416 b(w)-36 b(a)g(y)417 b(is)g(to)g(study)g(the)f
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b(2)-3718 25820 y(for)383 b(congruence)f(subgroups)g(lik)-36
b(e)384 b(\241)14140 26019 y Fx(0)14665 25820 y FF(\()p
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25820 y FF(\()p Fz(N)139 b FF(\).)561 b(These)382 b(cusp)g(forms)i
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b(v)g(e)508 b(constructed)f(mo)36 b(dels)508 b(for)g(eac)-36
b(h)508 b(of)h(these)-3718 29030 y(o)-36 b(v)g(er)434
b FD(Q)p FF(.)-1767 30635 y(When)419 b Fz(N)3193 30153
y Fu(0)3503 30635 y Fw(j)p Fz(N)558 b FF(there)419 b(is)h(a)g(natural)f
(map)h Fz(X)104 b FF(\()p Fz(N)139 b FF(\))369 b Fw(!)g
Fz(X)104 b FF(\()p Fz(N)26873 30153 y Fu(0)27184 30635
y FF(\).)573 b(Th)-36 b(us)419 b(w)-36 b(e)420 b(get)g(a)g(to)-36
b(w)g(er)420 b(of)g(curv)-36 b(es)420 b(and)-3718 32240
y(a)434 b(corresp)36 b(onding)433 b(to)-36 b(w)g(er)434
b(of)g(n)-36 b(um)g(b)36 b(er)432 b(\257elds.)19267 34957
y Fw(\242)221 b(\242)g(\242)3544 b(\242)221 b(\242)g(\242)19820
36563 y(#)4208 b([)18464 38168 y Fz(X)104 b FF(\()p Fz(N)139
b FF(\))2369 b Fw(F)25164 38367 y Fy(N)19820 39773 y
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40896 y Fu(0)21489 41378 y FF(\))2214 b Fw(F)25296 40896
y Fu(0)25164 41713 y Fy(N)19820 42983 y Fw(#)4208 b([)19267
44588 y(\242)221 b(\242)g(\242)3544 b(\242)221 b(\242)g(\242)-3718
47401 y FF(T)-108 b(aking)434 b(limits)h(giv)-36 b(es)434
b(a)g(curv)-36 b(e)434 b Fz(X)473 b FF(=)369 b(lim)14910
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b(!)40396 47401 y(F)41351 47600 y Fy(N)42247 47401 y
FF(.)-1767 49006 y(There)408 b(is)g(an)g(action)h(of)g(GL)12194
49205 y Fx(2)12720 49006 y FF(\()p FD(A)14354 49205 y
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y Fy(p)30611 50611 y FF(.)887 b(The)537 b(subscript)e
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b(is)g FD(A)633 b FF(=)g FD(A)42262 52415 y Fy(f)43268
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y FD(Q)369 b FF(=)21961 61356 y Fr(Y)23879 62618 y FD(Z)24792
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62618 y FD(Q)h FF(=)30771 62279 y(^)30535 62618 y FD(Q)q
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b FF(\))433 b(is)h(free)g(of)g(rank)g(2)g(o)-36 b(v)g(er)14518
65057 y(^)14387 65396 y FD(Z)p FF(.)578 b(Let)13026 68458
y Fz(V)290 b FF(\()p Fz(E)78 b FF(\))368 b(=)h Fz(T)181
b FF(\()p Fz(E)78 b FF(\))294 b Fw(\255)22199 68657 y
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72821 y Fy(p)19071 72622 y FF(\()p Fz(E)78 b FF(\))653
b(=)g Fz(T)24204 72821 y Fy(p)24733 72622 y FF(\()p Fz(E)78
b FF(\))408 b Fw(\255)28227 72821 y Fv(Z)28890 72932
y Ft(p)29837 72622 y FD(Q)30959 72821 y Fy(p)31488 72622
y FF(.)1080 b(View)601 b Fz(g)701 b Fw(2)653 b FF(GL)41079
72821 y Fx(2)41605 72622 y FF(\()p FD(A)43239 72821 y
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74434 y FF(.)928 b(Then)549 b Fz(\256)383 b Fw(\261)375
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b(a)h(lattice)f Fz(T)33864 73952 y Fu(0)34742 74434 y
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b(As)550 b(a)h(lemma,)21534 77755 y(47)p eop
%%Page: 48 56
48 55 bop 1263 -6698 a FF(48)11168 b FA(CHAPTER)434 b(11.)1013
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b Fz(V)289 b FF(\()p Fz(E)13664 -2046 y Fu(0)13974 -1564
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38339 y Fx(0)26546 38140 y FF(\()p Fz(pN)139 b FF(\))21633
39423 y Fy(\256)21271 40612 y Fw(.)32110 39369 y Fy(\257)31732
40612 y Fw(&)20037 42217 y Fz(X)21116 42416 y Fx(0)21642
42217 y FF(\()p Fz(N)g FF(\))6663 b Fz(X)31577 42416
y Fx(0)32103 42217 y FF(\()p Fz(N)139 b FF(\))1263 44583
y(The)441 b(maps)g Fz(\256)450 b FF(and)440 b Fz(\257)515
b FF(are)441 b(degeneracy)h(maps)e(whic)-36 b(h)441 b(forget)h(data.)
601 b(T)-108 b(o)441 b(de\257ne)f(them)g(view)i Fz(X)48746
44782 y Fx(0)49273 44583 y FF(\()p Fz(N)139 b FF(\))439
b(as)1263 46188 y(classifying)448 b(pairs)d(\()p Fz(E)78
b(;)221 b(C)95 b FF(\))445 b(where)g Fz(E)523 b FF(is)445
b(an)g(elliptic)h(curv)-36 b(e)445 b(and)g Fz(C)34911
45820 y Fw(\273)34921 46244 y FF(=)36333 46188 y FD(Z)p
Fz(=)-72 b(N)139 b FD(Z)445 b FF(is)g(a)h(cyclic)g(subgroup)e(of)1263
47794 y(order)455 b Fz(N)139 b FF(.)641 b(Similarly)456
b Fz(X)13492 47993 y Fx(0)14018 47794 y FF(\()p Fz(pN)139
b FF(\))454 b(classi\257es)h(pairs)g(\()p Fz(E)78 b(;)221
b(G)p FF(\))455 b(where)g Fz(G)405 b FF(=)g Fz(C)f Fw(\251)310
b Fz(D)40713 47425 y Fw(\273)40724 47849 y FF(=)42152
47794 y FD(Z)p Fz(=)-72 b(N)139 b FD(Z)309 b Fw(\251)h
FD(Z)p Fz(=p)p FD(Z)455 b FF(and)1263 49399 y Fz(D)470
b FF(is)434 b(cyclic)h(of)f(order)f Fz(p)p FF(.)578 b(Then)17873
52382 y Fz(\256)378 b FF(:)369 b(\()p Fz(E)78 b(;)221
b(G)p FF(\))369 b Fw(7!)g FF(\()p Fz(E)78 b(;)221 b(C)95
b FF(\))17901 54320 y Fz(\257)443 b FF(:)369 b(\()p Fz(E)78
b(;)221 b(G)p FF(\))369 b Fw(7!)g FF(\()p Fz(E)78 b(=D)36
b(;)221 b FF(\()p Fz(C)391 b FF(+)295 b Fz(D)36 b FF(\))p
Fz(=D)g FF(\))1263 57303 y(No)-36 b(w)506 b(w)-36 b(e)506
b(no)-36 b(w)505 b(translate)h(this)f(in)-36 b(to)505
b(the)g(language)h(of)g(complex)g(analysis.)795 b(The)505
b(\257rst)g(map)g Fz(\256)514 b FF(corre-)1263 58908
y(sp)36 b(onds)433 b(to)h(the)f(map)20477 60564 y(\241)21290
60763 y Fx(0)21816 60564 y FF(\()p Fz(pN)139 b FF(\))p
Fw(nH)380 b(!)370 b FF(\241)29339 60763 y Fx(0)29864
60564 y FF(\()p Fz(N)139 b FF(\))p Fw(nH)1263 62947 y
FF(induced)391 b(b)-36 b(y)392 b(the)f(inclusion)h(\241)16224
63146 y Fx(0)16750 62947 y FF(\()p Fz(pN)139 b FF(\))368
b Fz(,)-221 b Fw(!)369 b FF(\241)22614 63146 y Fx(0)23139
62947 y FF(\()p Fz(N)139 b FF(\).)564 b(The)392 b(second)f(map)h
Fz(\257)466 b FF(is)392 b(constructed)f(b)-36 b(y)392
b(comp)36 b(osing)1263 64552 y(the)433 b(map)16854 66909
y(\241)17667 67108 y Fx(0)18192 66909 y Fw(nH)20752 66156
y Fu(\273)20360 66909 y Fw(\241)-845 b(!)22245 65434
y Fr(\263)23038 66095 y Fz(p)1107 b FF(0)23040 67700
y(0)h(1)25448 65434 y Fr(\264)26242 66909 y FF(\241)27055
67108 y Fx(0)27580 66909 y FF(\()p Fz(pN)139 b FF(\))30426
65434 y Fr(\263)31218 66095 y Fz(p)1107 b FF(0)31220
67700 y(0)h(1)33628 65434 y Fr(\264)34422 65732 y Fu(\241)p
Fx(1)35679 66909 y Fw(nH)1263 70067 y FF(with)434 b(the)f(map)g(to)h
(\241)11759 70266 y Fx(0)12284 70067 y FF(\()p Fz(N)139
b FF(\))p Fw(nH)446 b FF(induced)432 b(b)-36 b(y)434
b(the)f(inclusion)17798 73825 y(\241)18611 74024 y Fx(0)19137
73825 y FF(\()p Fz(N)139 b FF(\))367 b Fw(\275)23100
72351 y Fr(\263)23893 73012 y Fz(p)1107 b FF(0)23894
74617 y(0)i(1)26303 72351 y Fr(\264)27096 73825 y FF(\241)27909
74024 y Fx(0)28435 73825 y FF(\()p Fz(pN)139 b FF(\))31281
72351 y Fr(\263)32073 73012 y Fz(p)1107 b FF(0)32074
74617 y(0)i(1)34483 72351 y Fr(\264)35276 72648 y Fu(\241)p
Fx(1)p eop
%%Page: 49 57
49 56 bop -3718 -6698 a FA(11.3.)1013 b(GENERALITIES)433
b(ON)h(CORRESPONDENCES)20225 b FF(49)-1767 -3169 y(The)433
b(maps)h Fz(\256)442 b FF(and)433 b Fz(\257)508 b FF(induce)432
b(maps)9893 -312 y Fz(\256)10728 -860 y Fu(\244)11255
-312 y Fz(;)221 b(\257)12645 -860 y Fu(\244)13540 -312
y FF(:)369 b Fz(H)15451 -860 y Fx(0)15976 -312 y FF(\()p
Fz(X)17561 -113 y Fx(0)18087 -312 y FF(\()p Fz(N)139
b FF(\))p Fz(;)221 b FF(\255)21801 -860 y Fx(1)22327
-312 y FF(\))369 b Fw(!)g Fz(H)26080 -860 y Fx(0)26605
-312 y FF(\()p Fz(X)28190 -113 y Fx(0)28717 -312 y FF(\()p
Fz(pN)139 b FF(\))p Fz(;)221 b FF(\255)33084 -860 y Fx(1)33609
-312 y FF(\))p Fz(:)-3718 2546 y FF(W)-108 b(e)456 b(can)g(iden)-36
b(tify)456 b Fz(S)6506 2745 y Fx(2)7032 2546 y FF(\(\241)8351
2745 y Fx(0)8876 2546 y FF(\()p Fz(N)139 b FF(\)\))455
b(with)h Fz(H)16196 2064 y Fx(0)16721 2546 y FF(\()p
Fz(X)18306 2745 y Fx(0)18832 2546 y FF(\()p Fz(N)139
b FF(\))p Fz(;)221 b FF(\255)22546 2064 y Fx(1)23528
2546 y FF(b)-36 b(y)456 b(sending)f(the)g(cusp)h(form)g
Fz(f)142 b FF(\()p Fz(\277)148 b FF(\))456 b(to)g(the)f(holo-)-3718
4151 y(morphic)433 b(di\256eren)-36 b(tial)433 b Fz(f)142
b FF(\()p Fz(\277)148 b FF(\))p Fz(d\277)g FF(.)579 b(No)-36
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y Fu(\244)16235 7009 y FF(:)369 b Fz(S)17765 7208 y Fx(2)18291
7009 y FF(\(\241)19610 7208 y Fx(0)20135 7009 y FF(\()p
Fz(N)139 b FF(\)\))368 b Fw(!)h Fz(S)25699 7208 y Fx(2)26225
7009 y FF(\(\241)27544 7208 y Fx(0)28069 7009 y FF(\()p
Fz(pN)139 b FF(\)\))p Fz(:)-3718 9867 y FC(Exer)-66 b(cise)464
b(11.2.2.)649 b FF(Sho)-36 b(w)388 b(that)f Fz(\256)12927
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b(=)g Fz(f)529 b FF(though)-36 b(t)387 b(of)h(as)g(a)g(cusp)f(form)i
(with)e(resp)36 b(ect)388 b(to)g(the)f(smaller)-3718
11472 y(group)433 b(\241)780 11671 y Fx(0)1306 11472
y FF(\()p Fz(pN)139 b FF(\).)577 b(Then)433 b(sho)-36
b(w)434 b(that)16406 15342 y Fz(\257)17214 14794 y Fu(\244)17740
15342 y FF(\()p Fz(f)142 b FF(\))368 b(=)h Fz(p)22647
13682 y Fu(1)22158 14080 y Fr(X)22231 16870 y Fy(n)p
Fx(=1)24298 15342 y Fz(a)24981 15541 y Fy(n)25607 15342
y Fz(q)26232 14794 y Fy(pN)27602 15342 y Fz(:)-3718 19362
y FF(\(Rib)36 b(et)433 b(w)-36 b(as)434 b(unsure)f(whether)g(the)g
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23788 y Fs(11.3)2151 b(Generalities)716 b(on)h(Corresp)60
b(ondences)-3718 26709 y FF(Let)466 b Fz(X)104 b FF(,)477
b Fz(Y)288 b FF(,)476 b(and)466 b Fz(C)562 b FF(b)36
b(e)467 b(curv)-36 b(es)466 b(and)h(let)g Fz(\256)475
b FF(and)467 b Fz(\257)540 b FF(b)36 b(e)467 b(nonconstan)-36
b(t)466 b(holomorphic)h(maps)g(so)g(that)f(w)-36 b(e)-3718
28314 y(ha)g(v)g(e)434 b(the)f(corresp)36 b(ondence)21672
29807 y Fz(C)19598 31090 y Fy(\256)19236 32278 y Fw(.)24182
31036 y Fy(\257)23805 32278 y Fw(&)19309 33883 y Fz(X)3559
b(Y)-3718 36085 y FF(Then)433 b(w)-36 b(e)434 b(ha)-36
b(v)g(e)434 b(maps)10842 38009 y Fz(H)12023 37460 y Fx(0)12548
38009 y FF(\()p Fz(X)32 b(;)221 b FF(\255)15686 37460
y Fx(1)16213 38009 y FF(\))17480 37256 y Fy(\256)18084
36943 y Fo(\244)17088 38009 y Fw(\241)-493 b(!)369 b
Fz(H)20506 37460 y Fx(0)21031 38009 y FF(\()p Fz(C)23
b(;)221 b FF(\255)24012 37460 y Fx(1)24538 38009 y FF(\))25805
37202 y Fy(\257)26333 37313 y Fo(\244)25413 38009 y Fw(\241)-569
b(!)369 b Fz(H)28755 37460 y Fx(0)29280 38009 y FF(\()p
Fz(Y)72 b(;)221 b FF(\255)32134 37460 y Fx(1)32661 38009
y FF(\))p Fz(:)-3718 40301 y FF(The)433 b(comp)36 b(osition)435
b Fz(\257)6959 40500 y Fu(\244)7780 40301 y Fw(\261)295
b Fz(\256)9574 39819 y Fu(\244)10534 40301 y FF(is)434
b(a)g(map)15236 43159 y Fz(H)16417 42611 y Fx(0)16942
43159 y FF(\()p Fz(X)32 b(;)221 b FF(\255)20080 42611
y Fx(1)20607 43159 y FF(\))369 b Fw(!)h Fz(H)24361 42611
y Fx(0)24886 43159 y FF(\()p Fz(Y)72 b(;)221 b FF(\255)27740
42611 y Fx(1)28266 43159 y FF(\))p Fz(:)-3718 46017 y
FF(Switc)-36 b(hing)433 b(the)g(roles)h(of)h Fz(X)538
b FF(and)433 b Fz(Y)722 b FF(giv)-36 b(es)435 b(a)f(map)15236
48875 y Fz(H)16417 48326 y Fx(0)16942 48875 y FF(\()p
Fz(Y)72 b(;)221 b FF(\255)19796 48326 y Fx(1)20323 48875
y FF(\))368 b Fw(!)i Fz(H)24076 48326 y Fx(0)24601 48875
y FF(\()p Fz(X)32 b(;)221 b FF(\255)27739 48326 y Fx(1)28266
48875 y FF(\))p Fz(:)-1767 51733 y FF(In)433 b(this)h(con)-36
b(text)433 b(w)-36 b(e)434 b(can)g(iden)-36 b(tify)433
b Fz(T)16813 51932 y Fy(p)17776 51733 y FF(b)-36 b(y)434
b(viewing)h(it)e(as)h(the)f(map)8324 54590 y Fz(\257)9058
54789 y Fu(\244)9879 54590 y Fw(\261)296 b Fz(\256)11674
54042 y Fu(\244)12569 54590 y FF(:)370 b Fz(H)14481 54042
y Fx(0)15006 54590 y FF(\()p Fz(X)16591 54789 y Fx(0)17117
54590 y FF(\()p Fz(N)139 b FF(\))p Fz(;)221 b FF(\255)20831
54042 y Fx(1)21357 54590 y FF(\))368 b Fw(!)i Fz(H)25110
54042 y Fx(0)25635 54590 y FF(\()p Fz(X)27220 54789 y
Fx(0)27746 54590 y FF(\()p Fz(N)139 b FF(\))p Fz(;)221
b Fw(j)p Fz(O)36 b(meg)48 b(a)35015 54042 y Fx(1)35539
54590 y FF(\))-3718 57448 y(and)400 b(using)h(the)f(fact)i(that)e
Fz(S)10474 57647 y Fx(2)11000 57448 y FF(\(\241)12319
57647 y Fx(0)12844 57448 y FF(\()p Fz(N)139 b FF(\)\))368
b(=)h Fz(H)18473 56966 y Fx(0)18998 57448 y FF(\()p Fz(X)20583
57647 y Fx(0)21109 57448 y FF(\()p Fz(N)139 b FF(\))p
Fz(;)221 b FF(\255)24823 56966 y Fx(1)25349 57448 y FF(\).)567
b(One)400 b(should)g(reco)-36 b(v)g(er)401 b(the)g(explicit)h(repre-)
-3718 59053 y(sen)-36 b(tation)11014 60658 y Fz(T)11776
60857 y Fy(p)12674 60658 y FF(:)13405 59397 y Fr(X)15545
60658 y Fz(a)16228 60857 y Fy(n)16854 60658 y Fz(q)17479
60110 y Fy(n)18474 60658 y Fw(7!)20171 59397 y Fr(X)22311
60658 y Fz(a)22994 60857 y Fy(np)24094 60658 y Fz(q)24719
60110 y Fy(n)25639 60658 y FF(+)295 b Fz(p)27820 59397
y Fr(X)29961 60658 y Fz(a)30644 60857 y Fy(n)31270 60658
y Fz(q)31895 60110 y Fy(np)32994 60658 y Fz(:)-1767 63109
y FF(No)-36 b(w)451 b(lets)h(think)e(more)h(generally)i(ab)36
b(out)451 b(corresp)36 b(ondences.)629 b(Supp)36 b(ose)450
b Fz(')399 b FF(:)g Fz(X)504 b Fw(!)399 b Fz(Y)739 b
FF(is)452 b(a)f(map)g(of)-3718 64714 y(curv)-36 b(es.)578
b(Let)433 b(\241)369 b Fw(\275)g Fz(X)400 b Fw(\243)296
b Fz(Y)722 b FF(b)36 b(e)433 b(the)g(graph)g(of)i Fz(')p
FF(.)578 b(This)434 b(giv)-36 b(es)435 b(a)e(stupid)g(corresp)36
b(ondence)21778 67460 y(\241)19705 68743 y Fy(\256)19343
69931 y Fw(.)24075 68689 y Fy(\257)23698 69931 y Fw(&)19415
71536 y Fz(X)3346 b(Y)-3718 74434 y FF(W)-108 b(e)433
b(can)h(reconstruct)f Fz(')g FF(since)h Fz(')p FF(\()p
Fz(x)p FF(\))369 b(=)f Fz(\257)74 b FF(\()p Fz(\256)18679
73952 y Fu(\241)p Fx(1)19937 74434 y FF(\()p Fz(x)p FF(\)\).)p
eop
%%Page: 50 58
50 57 bop 1263 -6698 a FF(50)11168 b FA(CHAPTER)434 b(11.)1013
b(HECKE)434 b(OPERA)-108 b(TORS)433 b(AS)g(CORRESPONDENCES)3214
-3169 y FF(More)504 b(generally)i(supp)36 b(ose)503 b
Fz(\256)498 b FF(:)490 b(\241)f Fw(!)g Fz(X)609 b FF(has)504
b(degree)g Fz(d)490 b Fw(\270)f FF(1.)790 b(View)505
b Fz(\256)39377 -3651 y Fu(\241)p Fx(1)40636 -3169 y
FF(\()p Fz(x)p FF(\))f(as)g(a)h(divisor)g(on)f(\241.)1263
-1564 y(Then)433 b Fz(\257)74 b FF(\()p Fz(\256)6808
-2046 y Fu(\241)p Fx(1)8066 -1564 y FF(\))434 b(is)g(a)g(divisor)g(on)f
Fz(Y)289 b FF(.)579 b(W)-108 b(e)433 b(th)-36 b(us)433
b(get)g(a)h(map)20695 1275 y(Div)22755 718 y Fy(n)23602
1275 y Fz(X)25547 468 y Fy(\257)46 b Fu(\261)p Fy(\256)27195
155 y Fo(\241)p Fn(1)25155 1275 y Fw(\241)-300 b(\241)-295
b(\241)c(!)369 b FF(Div)31116 718 y Fy(dn)32448 1275
y Fz(Y)72 b(:)1263 3596 y FF(In)538 b(particular,)564
b(when)537 b Fz(d)546 b FF(=)g(0,)564 b(there)537 b(w)-36
b(e)538 b(get)g(a)g(map)f(Div)31297 3039 y Fx(0)32044
3596 y Fz(X)651 b Fw(!)547 b FF(Div)37708 3039 y Fx(0)38455
3596 y Fz(Y)72 b(:)538 b FF(No)-36 b(w)538 b(apply)g(this)g(to)f(the)
1263 5201 y(sp)36 b(ecial)435 b(case)f(of)g Fz(T)10494
5400 y Fy(p)11023 5201 y FF(.)579 b Fz(T)12725 5400 y
Fy(p)13688 5201 y FF(is)433 b(the)g(corresp)36 b(ondence)24941
7523 y Fz(X)26020 7722 y Fx(0)26546 7523 y FF(\()p Fz(pN)139
b FF(\))21633 8807 y Fy(\256)21271 9995 y Fw(.)32110
8752 y Fy(\257)31732 9995 y Fw(&)20037 11600 y Fz(X)21116
11799 y Fx(0)21642 11600 y FF(\()p Fz(N)g FF(\))6663
b Fz(X)31577 11799 y Fx(0)32103 11600 y FF(\()p Fz(N)139
b FF(\))1263 13831 y(and)433 b(the)g(induced)g(map)g(is)11446
16471 y(\()p Fz(E)78 b(;)221 b(C)95 b FF(\))15598 15718
y Fy(\256)16202 15405 y Fo(\244)15476 16471 y Fw(7!)17795
15209 y Fr(X)17174 18087 y Fy(D)26 b Fu(2)p Fy(E)50 b
Fx([)p Fy(p)p Fx(])20334 16471 y FF(\()p Fz(E)78 b(;)221
b(C)391 b Fw(\251)295 b Fz(D)36 b FF(\))27481 15663 y
Fy(\257)27104 16471 y Fw(7!)29422 15209 y Fr(X)28801
18087 y Fy(D)26 b Fu(2)p Fy(E)50 b Fx([)p Fy(p)p Fx(])31962
16471 y FF(\()p Fz(E)78 b(=D)36 b(;)221 b FF(\()p Fz(C)391
b FF(+)295 b Fz(D)36 b FF(\))p Fz(=D)g FF(\))1263 20190
y(Th)-36 b(us)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434
b(a)g(map)13171 21328 y Fr(n)14490 22803 y FF(divisors)h(on)e
Fz(X)22123 23002 y Fx(0)22649 22803 y FF(\()p Fz(N)139
b FF(\))25275 21328 y Fr(o)26529 22803 y Fw(\241)-221
b(!)29039 21328 y Fr(n)29924 22803 y FF(divisors)435
b(on)e Fz(X)37557 23002 y Fx(0)38083 22803 y FF(\()p
Fz(N)139 b FF(\))40275 21328 y Fr(o)1263 25416 y FF(This)545
b(strongly)h(resem)-36 b(bles)545 b(the)f(original)i(de\257nition)f(of)
g Fz(T)30834 25615 y Fy(p)31909 25416 y FF(as)g(a)g(corresp)36
b(ondence)544 b(of)i(lattices.)913 b(But)1263 27021 y(there)425
b(w)-36 b(as)426 b(a)g(factor)g(of)g Fz(p)13986 26539
y Fy(k)24 b Fu(\241)p Fx(1)16183 27021 y FF(then)424
b(whic)-36 b(h)425 b(anno)-36 b(y)g(ed)426 b(Ogus.)575
b(It)425 b(disapp)36 b(ears)426 b(here)f(though,)h(and)f(this)g(is)1263
28626 y(related)434 b(to)f(the)g(`p')h(in)g(the)f(exercise.)1263
32973 y Fs(11.4)2152 b(Jacobians)717 b(of)f(Curv)-60
b(es)1263 35893 y FF(Let)433 b Fz(X)539 b FF(b)36 b(e)433
b(a)h(curv)-36 b(e)433 b(of)i(gen)-36 b(us)433 b Fz(g)481
b FF(o)-36 b(v)g(er)434 b(a)g(\257eld)f Fz(k)45 b FF(.)578
b(There)434 b(is)g(an)f(imp)36 b(ortan)-36 b(t)433 b(asso)36
b(ciation)9634 37031 y Fr(n)10954 38506 y FF(curv)-36
b(es)433 b Fz(X)l(=k)17815 37031 y Fr(o)19070 38506 y
Fw(\241)-221 b(!)21579 37031 y Fr(n)22898 38506 y FF(Jacobians)435
b Fz(J)123 b(ac)p FF(\()p Fz(X)104 b FF(\))369 b(=)f
Fz(J)123 b FF(\()p Fz(X)104 b FF(\))434 b(of)g(curv)-36
b(es)43812 37031 y Fr(o)1263 41118 y FF(b)36 b(et)-36
b(w)g(een)433 b(curv)-36 b(es)434 b(and)f(their)g(asso)36
b(ciated)434 b(Jacobians.)1263 43306 y FD(De\257nition)500
b(11.4.1.)651 b FF(Let)566 b Fz(X)671 b FF(b)36 b(e)566
b(a)h(curv)-36 b(e,)599 b(then)566 b(the)f FD(Jacobian)j
FF(of)f Fz(X)671 b FF(is)567 b(an)f(ab)36 b(elian)567
b(v)-72 b(ariet)-36 b(y)567 b(of)1263 44911 y(dimension)391
b Fz(g)439 b FF(whose)392 b(underlying)f(group)g(is)h(isomorphic)g(to)f
(the)g(group)h(of)g(divisors)g(of)g(degree)g(0)f(on)h
Fz(X)1263 46516 y FF(mo)36 b(dulo)434 b(linear)g(equiv)-72
b(alence.)3214 48703 y(There)389 b(are)g(man)-36 b(y)388
b(constructions)g(of)i(the)e(Jacobian)h(of)g(a)g(curv)-36
b(e.)564 b(W)-108 b(e)388 b(\257rst)g(consider)h(the)f(Albanese)1263
50308 y(construction.)682 b(Recall)470 b(that)e(o)-36
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%%Page: 51 59
51 58 bop -3718 -6698 a FA(11.5.)1013 b(MORE)433 b(ON)g(HECKE)h(OPERA)
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%%Page: 52 60
52 59 bop 1263 -6698 a FF(52)11168 b FA(CHAPTER)434 b(11.)1013
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%%Page: 55 63
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53188 y Fx(0)17174 52989 y FF(\()p Fz(N)139 b FF(\))12261
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55734 y FF(\()p Fz(J)19497 55933 y Fx(0)20022 55734 y
FF(\()p Fz(N)139 b FF(\)\))368 b Fw(\275)h FF(End)26821
55933 y Fv(C)27660 55734 y FF(\()p Fz(J)28885 55933 y
Fx(0)29411 55734 y FF(\()p Fz(N)139 b FF(\)\))13669 57672
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FF(.)1068 b(If)597 b Fz(g)645 b FF(is)597 b(the)f(gen)-36
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FF(acts)g(on)g Fz(X)104 b FF(.)890 b(Similarly)539 b(if)-3718
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b(t)486 b(then)e(there)h(is)h(an)g(asso)36 b(ciated)486
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b(haracteristic)434 b(0)g(to)g(giv)-36 b(e)434 b([[ho)-36
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72793 y Fz(T)18373 72992 y Fy(p)19271 72793 y FF(=)369
b Fz(')21504 72992 y Fy(p)22328 72793 y FF(+)295 b Fz(p')25140
72245 y Fu(\241)p Fx(1)25140 73122 y Fy(p)26397 72793
y Fz(:)p eop
%%Page: 58 66
58 65 bop 1263 -6698 a FF(58)11168 b FA(CHAPTER)434 b(11.)1013
b(HECKE)434 b(OPERA)-108 b(TORS)433 b(AS)g(CORRESPONDENCES)p
eop
%%Page: 59 67
59 66 bop -3718 5686 a FE(Chapter)1033 b(12)-3718 11221
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b Fz(\274)29368 37166 y Fy(f)30341 36967 y FF(=)369 b(\(0)p
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b FD(T)21008 38771 y Fx(0)21939 38572 y Fw(\275)406 b
FD(T)455 b FF(and)g FD(T)g FF(is)h(free)f(o)-36 b(v)g(er)456
b FD(Z)p FF(,)461 b(these)454 b(pro)72 b(jectors)456
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b FF(is)h(comm)-36 b(utativ)g(e)507 b(and)f(the)g Fz(\274)25650
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b(w)514 b(that)f(dim)14512 49154 y Fy(E)15202 49310 y
Ft(f)16008 48955 y Fl(A)17074 49154 y Fy(f)18193 48955
y FF(is)h(the)f(n)-36 b(um)g(b)36 b(er)512 b(of)j(divisors)f(of)35438
48432 y Fy(N)p 34798 48650 2123 54 v 34798 49413 a(N)94
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b FF(\()p Fz(f)j FF(\))513 b(is)h(the)-3718 50560 y(lev)-36
b(el)434 b(of)h(the)e(newform)h Fz(f)142 b FF(.)-1767
52486 y(Let's)354 b(examine)h(a)f(particular)g(case)h(of)g(this)f
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b(a)g(prime,)370 b(supp)36 b(ose)354 b Fz(f)496 b FF(is)354
b(a)h(newform)-3718 54092 y(of)434 b(lev)-36 b(el)435
b Fz(N)139 b FF(\()p Fz(f)j FF(\),)432 b(and)i Fz(N)507
b FF(=)368 b Fz(N)139 b FF(\()p Fz(f)j FF(\))p Fz(q)480
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FF(\()p Fz(f)j FF(\).)578 b(W)-108 b(e)433 b(sho)-36
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y FF([)p Fz(U)139 b FF(])p Fz(=)p FF(\()p Fz(U)23093
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y Fy(f)8106 58874 y FF(=)368 b(2)423 b(whic)-36 b(h,)425
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y FF(.)578 b(Recall)434 b(that)e Fz(U)24955 64293 y Fy(q)25895
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y Fx(2)38298 66449 y Fy(q)39258 66121 y Fw(\241)295 b
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y Fy(f)11650 68047 y FF(in)g(general)g(as)17239 70438
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74434 y(4.4)434 b(of)h([4])f(for)g(the)f(answ)-36 b(er.])21534
77755 y(59)p eop
%%Page: 60 68
60 67 bop 1263 -6698 a FF(60)10376 b FA(CHAPTER)435 b(12.)1012
b(ABELIAN)434 b(V)-145 b(ARIETIES)435 b(FR)-36 b(OM)433
b(MODULAR)g(F)-36 b(ORMS)3214 -3169 y FF(One)536 b(can)g(de\257ne)f(an)
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b Fz(A)26236 -2970 y Fy(f)27377 -3169 y FF(of)h Fz(J)29680
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21404 y Fx(0)8489 21205 y FF(\()p Fz(N)139 b FF(\))439
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52128 74434 45 878 v 52173 73601 781 45 v 52173 74434
V 52953 74434 45 878 v eop
%%Page: 61 69
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44952 y Fy(`)26784 44623 y Fw(\275)f Fz(E)29227 44075
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b(densit)-36 b(y)433 b(implies)h(something.]])p 47147
49108 45 878 v 47192 48275 781 45 v 47192 49108 V 47972
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%%Page: 62 70
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%%Page: 63 71
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(alternating)g(and)f(bilinear)h(form)g(so)g(it)f(de\257nes)g(a)h(map)
14841 74434 y Fw(h\242)p Fz(;)221 b Fw(\242i)369 b FF(:)g(\244)19197
73886 y Fx(2)19197 74763 y Fy(E)19887 74919 y Ft(\270)20707
74434 y FF(T)-108 b(ate)23273 74633 y Fy(\270)24246 74434
y Fw(!)369 b FD(Q)27065 74633 y Fy(`)27505 74434 y FF(\(1\))p
Fz(:)p eop
%%Page: 64 72
64 71 bop 1263 -6698 a FF(64)10376 b FA(CHAPTER)435 b(12.)1012
b(ABELIAN)434 b(V)-145 b(ARIETIES)435 b(FR)-36 b(OM)433
b(MODULAR)g(F)-36 b(ORMS)1263 -3169 y FF(It)504 b(is)h(not)f
FC(a)530 b(priori)503 b FF(true)h(that)g(w)-36 b(e)504
b(can)g(tak)-36 b(e)505 b(the)f(w)-36 b(edge)504 b(pro)36
b(duct)503 b(o)-36 b(v)g(er)505 b Fz(E)39469 -2970 y
Fy(\270)40577 -3169 y FF(instead)f(of)h FD(Q)47809 -2970
y Fy(`)48249 -3169 y FF(,)522 b(but)503 b(w)-36 b(e)1263
-1564 y(can)434 b(b)36 b(ecause)433 b Fw(h)p Fz(tx;)221
b(y)48 b Fw(i)369 b FF(=)f Fw(h)p Fz(x;)221 b(ty)48 b
Fw(i)433 b FF(for)h(an)-36 b(y)434 b Fz(t)369 b Fw(2)f
Fz(E)25156 -1365 y Fy(\270)25761 -1564 y FF(.)578 b(Let)433
b Fz(D)405 b FF(=)369 b(\244)32799 -2046 y Fx(2)33546
-1564 y FF(T)-108 b(ate)36111 -1365 y Fy(\270)37149 -1564
y FF(and)433 b(note)g(that)g(dim)47554 -1365 y Fy(E)48244
-1209 y Ft(\270)49064 -1564 y Fz(D)406 b FF(=)368 b(1.)3214
41 y(There)434 b(is)g(a)f(trace)h(map)f(whic)-36 b(h)434
b(giv)-36 b(es)434 b(an)g(iden)-36 b(ti\257cation)16355
2882 y(Hom)19064 3081 y Fv(Q)19879 3237 y Ft(`)20334
2882 y FF(\()p Fz(D)36 b(;)221 b FD(Q)23660 3081 y Fy(`)24101
2882 y FF(\(1\)\))26637 2514 y Fw(\273)26648 2938 y FF(=)28039
2882 y(Hom)30749 3081 y Fy(E)31439 3237 y Ft(\270)32038
2882 y FF(\()p Fz(D)36 b(;)221 b(E)35205 3081 y Fy(\270)35809
2882 y FF(\(1\)\))1263 5724 y(Th)-36 b(us)467 b Fw(h\242)p
Fz(;)221 b Fw(\242i)468 b FF(is)g(an)g(isomorphism)g
Fz(D)19776 5355 y Fw(\273)19787 5779 y FF(=)21236 5724
y Fz(E)22199 5923 y Fy(\270)22803 5724 y FF(\(1\).)681
b([[I)469 b(don't)e(see)h(this)g(implication)g(righ)-36
b(t)468 b(no)-36 b(w.]])682 b(It)468 b(no)-36 b(w)1263
7329 y(follo)g(ws)436 b(that)d(det)221 b Fz(\275)11051
7528 y Fy(\270)12024 7329 y FF(=)368 b Fz(\302)14219
7528 y Fy(`)14659 7329 y FF(.)1263 11753 y Fs(12.5)2152
b(The)716 b(Concrete)h(Pro)60 b(of)1263 14673 y FF(No)-36
b(w)584 b(w)-36 b(e)583 b(consider)g(the)f(concrete)h(pro)36
b(of.)1027 b(Supp)36 b(ose)582 b Fz(\276)671 b Fw(2)623
b FF(Gal\()p 34381 13602 1123 54 v FD(Q)q Fz(=)p FD(Q)p
FF(\),)e(then)582 b(w)-36 b(e)583 b(m)-36 b(ust)582 b(sho)-36
b(w)583 b(that)1263 16278 y(det\()p Fz(\276)48 b FF(\))465
b(=)h Fz(\302)7627 16477 y Fy(`)8066 16278 y FF(\()p
Fz(\276)48 b FF(\).)749 b(Cho)36 b(ose)491 b(v)-36 b(ectors)491
b Fz(x;)221 b(y)515 b Fw(2)465 b FF(T)-108 b(ate)26413
16477 y Fy(\270)27508 16278 y FF(whic)-36 b(h)490 b(are)h(linearly)h
(indep)36 b(enden)-36 b(t)488 b(o)-36 b(v)g(er)491 b
Fz(E)49458 16477 y Fy(\270)50062 16278 y FF(.)749 b(Let)1263
17883 y Fz(a;)221 b(b;)g(c;)g(d)437 b Fw(2)e Fz(E)8202
18082 y Fy(\270)9278 17883 y FF(b)36 b(e)472 b(de\257ned)f(b)-36
b(y)473 b Fz(\276)48 b FF(\()p Fz(x)p FF(\))434 b(=)h
Fz(ax)322 b FF(+)f Fz(cy)520 b FF(and)472 b Fz(\276)48
b FF(\()p Fz(y)g FF(\))435 b(=)f Fz(bx)322 b FF(+)f Fz(dy)48
b FF(.)695 b(Let)472 b Fz(\261)485 b FF(=)435 b Fz(ad)321
b Fw(\241)h Fz(bc)g Fw(\241)f Fz(\302)50469 18082 y Fy(`)50909
17883 y FF(\()p Fz(\276)48 b FF(\),)1263 19488 y(th)-36
b(us)433 b(w)-36 b(e)434 b(w)-36 b(an)g(t)433 b(to)h(sho)-36
b(w)433 b(that)g Fz(\261)419 b FF(=)369 b(0.)579 b(W)-108
b(e)433 b(ha)-36 b(v)g(e)12298 22330 y Fz(\302)13113
22529 y Fy(`)13553 22330 y FF(\()p Fz(\276)48 b FF(\))p
Fw(h)p Fz(x;)221 b(y)48 b Fw(i)368 b FF(=)h Fw(h)p Fz(\276)48
b(x;)221 b(\276)48 b(y)g Fw(i)18757 24267 y FF(=)369
b Fw(h)p Fz(ax)294 b FF(+)h Fz(cy)48 b(;)221 b(bx)296
b FF(+)f Fz(dy)48 b Fw(i)18757 26204 y FF(=)369 b Fw(h)p
Fz(ax;)221 b(bx)p Fw(i)295 b FF(+)g Fw(h)p Fz(ax;)221
b(dy)48 b Fw(i)294 b FF(+)h Fw(h)p Fz(cy)48 b(;)221 b(bx)p
Fw(i)295 b FF(+)g Fw(h)p Fz(cy)48 b(;)221 b(dy)48 b Fw(i)18757
28142 y FF(=)369 b Fw(h)p Fz(ax;)221 b(dy)48 b Fw(i)294
b FF(+)h Fw(h)p Fz(cy)48 b(;)221 b(bx)p Fw(i)18757 30079
y FF(=)369 b Fw(h)p Fz(adx;)221 b(y)48 b Fw(i)294 b(\241)i(h)p
Fz(bcx;)221 b(y)48 b Fw(i)368 b FF(=)h Fw(h)p FF(\()p
Fz(ad)294 b Fw(\241)h Fz(bc)p FF(\))p Fz(x;)221 b(y)48
b Fw(i)1263 32920 y FF(T)-108 b(o)434 b(see)g(that)f
Fw(h)p Fz(ax;)221 b(bx)p Fw(i)369 b FF(=)f(0)434 b(w)-36
b(e)434 b(noted)f(that)14668 35762 y Fw(h)p Fz(ax;)221
b(bx)p Fw(i)369 b FF(=)f Fw(h)p Fz(abx;)221 b(x)p Fw(i)369
b FF(=)g Fw(\241h)p Fz(x;)221 b(abx)p Fw(i)369 b FF(=)g
Fw(\241h)p Fz(ax;)221 b(bx)p Fw(i)p Fz(:)1263 38604 y
FF([[Of)526 b(course)e(it)h(lo)36 b(oks)527 b(lik)-36
b(e)525 b(the)g(next)f(step)h(is)g(to)36 b(o)525 b(use)f(nondegeneracy)
h(to)g(conclude)f(that)h Fz(\302)49295 38803 y Fy(`)49734
38604 y FF(\()p Fz(\276)48 b FF(\))524 b(=)1263 40209
y Fz(ad)310 b Fw(\241)h Fz(bc)p FF(,)461 b(but)455 b(it)h(is)g(not)g
(clear)g(to)g(me)g(ho)-36 b(w)456 b(to)g(safely)i(do)e(this.)645
b(My)456 b(problem)f(is)i(that)e(the)g(expression)1263
41814 y(\()p Fz(ad)322 b Fw(\241)h Fz(bc)p FF(\))p Fw(h)p
Fz(x;)221 b(y)48 b Fw(i)473 b FF(do)36 b(es)474 b(not)f(really)i(mak)
-36 b(e)475 b(sense)e(unless)g(w)-36 b(e)474 b(kno)-36
b(w)475 b(that)e Fz(ad)322 b Fw(\241)h Fz(bc)437 b Fw(2)g
FD(Q)44859 42013 y Fy(\270)45463 41814 y FF(\(1\))474
b(and,)484 b(ev)-36 b(en)1263 43419 y(then,)474 b(wh)-36
b(y)467 b(is)g(this)g(equal)g(to)g(the)f(last)h(expression?)678
b(I)466 b(clearly)i(lac)-36 b(k)468 b(some)f(crucial)g(bit)f(of)i(kno)
-36 b(wledge)1263 45024 y(ab)36 b(out)434 b FD(Q)6107
45223 y Fy(\270)6711 45024 y FF(\(1\)]])1263 49447 y
Fs(12.6)2152 b(The)716 b(Construction)g(for)g Fp(X)29481
49734 y FF(1)30188 49447 y Fm(\()p Fp(N)200 b Fm(\))1263
52368 y FF(Let)419 b Fz(f)511 b FF(=)6112 51372 y Fr(P)7735
52368 y Fz(a)8418 52567 y Fy(n)9044 52368 y Fz(q)9669
51886 y Fy(n)10714 52368 y FF(b)36 b(e)420 b(a)f(newform)h(for)g(\241)
21623 52567 y Fx(1)22149 52368 y FF(\()p Fz(N)139 b FF(\),)421
b(so)f Fz(f)142 b Fw(j)p Fz(T)28621 52567 y Fy(n)29617
52368 y FF(=)368 b Fz(a)31680 52567 y Fy(n)32306 52368
y Fz(f)562 b FF(for)420 b(all)g Fz(n)369 b Fw(\270)g
FF(1)420 b(and)f Fz(f)142 b Fw(jh)p Fz(d)p Fw(i)368 b
FF(=)h Fz(")p FF(\()p Fz(d)p FF(\))p Fz(f)560 b FF(for)1263
53973 y(all)385 b Fz(d)369 b Fw(2)f FF(\()p FD(Z)p Fz(=)-72
b(N)139 b FD(Z)p FF(\))9916 53491 y Fu(\244)10442 53973
y FF(.)561 b(In)384 b(the)g(case)g(of)h Fz(X)20345 54172
y Fx(1)20871 53973 y FF(\()p Fz(N)139 b FF(\),)393 b
Fz(E)24781 54172 y Fy(f)25770 53973 y FF(is)384 b(either)f(totally)j
(real)e(or)g(it)g(can)g(b)36 b(e)384 b(a)g(CM)g(\257eld.)561
b(If)385 b(it)1263 55578 y(is)396 b(a)f(CM)g(\257eld)g(then)f(it)h(has)
g(a)g(canonical)h(complex)g(conjugation)p 34129 54828
369 54 v 396 w Fw(\242)369 b FF(:)g Fz(a)36280 55777
y Fy(n)37275 55578 y Fw(7!)h Fz(")p FF(\()p Fz(n)p FF(\))41370
55096 y Fu(\241)p Fx(1)42627 55578 y Fz(a)43310 55777
y Fy(n)43936 55578 y FF(.)565 b(The)396 b(analogous)1263
57183 y(form)-36 b(ulas)434 b(are)21254 60025 y(T)-108
b(r)o(\()p Fz(\275)23767 60224 y Fy(`)24207 60025 y FF(\(F)g(rob)27332
60224 y Fy(p)27861 60025 y FF(\)\))368 b(=)h Fz(a)31305
60224 y Fy(p)20784 61962 y FF(det)o(\()p Fz(\275)23767
62161 y Fy(`)24207 61962 y FF(\(F)-108 b(rob)27332 62161
y Fy(p)27861 61962 y FF(\)\))368 b(=)h Fz(")p FF(\()p
Fz(p)p FF(\))p Fz(p)3214 64804 y FF(There)489 b(are)g(some)h(details)f
(concerning)g(signs)g(whic)-36 b(h)489 b(w)-36 b(e)489
b(m)-36 b(ust)488 b(consider.)745 b(First,)502 b(for)490
b(ev)-36 b(ery)490 b Fz(p)e FF(w)-36 b(e)1263 66409 y(can)339
b(de\257ne)f Fz(T)7978 66608 y Fy(p)8847 66409 y FF(as)h(a)h(corresp)36
b(ondence)338 b(on)h Fz(X)23111 66608 y Fx(1)23637 66409
y FF(\()p Fz(N)139 b FF(\))338 b(with)i Fz(\256)347 b
FF(and)339 b Fz(\257)413 b FF(de\257ned)338 b(in)h(an)g(analogous)h(w)
-36 b(a)g(y)340 b(to)f(the)1263 68014 y(w)-36 b(a)g(y)442
b(they)e(w)-36 b(ere)441 b(de\257ned)e(in)h(the)g(case)i(of)f
Fz(X)23433 68213 y Fx(0)23959 68014 y FF(\()p Fz(N)139
b FF(\).)598 b(But)440 b(no)-36 b(w)441 b(w)-36 b(e)441
b(m)-36 b(ust)440 b(c)-36 b(ho)36 b(ose)441 b(if)g(the)f(corresp)36
b(onding)1263 69619 y(endomorphism)372 b(of)h Fz(J)12092
69818 y Fx(1)12618 69619 y FF(\()p Fz(N)139 b FF(\))371
b(should)h(b)36 b(e)373 b(\()p Fz(T)22225 69818 y Fy(p)22754
69619 y FF(\))23260 69818 y Fu(\244)24158 69619 y FF(or)g(\()p
Fz(T)26955 69818 y Fy(p)27484 69619 y FF(\))27990 69137
y Fu(\244)28515 69619 y FF(.)558 b(The)373 b(second)f(c)-36
b(hoice)373 b(concerns)f(what)h(happ)36 b(ens)1263 71224
y(in)581 b(the)e(Eic)-36 b(hler-Shim)g(ura)579 b(relation.)1020
b(Recall)582 b(that)e Fz(X)28844 71423 y Fx(1)29370 71224
y FF(\()p Fz(N)139 b FF(\))579 b(classi\257es)j(pairs)e(\()p
Fz(E)78 b(;)221 b(p)p FF(\))581 b(where)f Fz(E)658 b
FF(is)581 b(an)1263 72829 y(elliptic)537 b(curv)-36 b(e)536
b(and)g Fz(p)g FF(is)g(a)h(p)36 b(oin)-36 b(t)536 b(of)h(order)e
Fz(N)139 b FF(.)886 b(Equiv)-72 b(alen)-36 b(tly)-108
b(,)563 b Fz(X)35620 73028 y Fx(1)36146 72829 y FF(\()p
Fz(N)139 b FF(\))535 b(classi\257es)i(pairs)f(consisting)1263
74434 y(of)478 b(an)e(elliptic)h(curv)-36 b(e)477 b(and)f(an)h(em)-36
b(b)36 b(edding)475 b FD(Z)p Fz(=)-72 b(N)139 b FD(Z)442
b Fz(,)-221 b Fw(!)443 b Fz(E)78 b FF(.)707 b(But)476
b(this)h(is)f(a)h(bad)g(w)-36 b(a)g(y)477 b(to)g(lo)36
b(ok)478 b(at)f(this.)p eop
%%Page: 65 73
65 72 bop -3718 -6698 a FA(12.6.)1013 b(THE)434 b(CONSTR)-36
b(UCTION)434 b(F)-36 b(OR)433 b Fz(X)18746 -6499 y Fx(1)19273
-6698 y FF(\()p Fz(N)139 b FF(\))25321 b(65)-3718 -3169
y(A)547 b(b)36 b(etter)547 b(w)-36 b(a)g(y)548 b(is)g(to)f(view)i
Fz(X)11852 -2970 y Fx(1)12378 -3169 y FF(\()p Fz(N)139
b FF(\))546 b(as)i(classifying)i(pairs)d(consisting)h(of)g(an)f
(elliptic)h(curv)-36 b(e)548 b(and)f(an)-3718 -1564 y(em)-36
b(b)36 b(edding)330 b Fz(\271)2784 -1601 y(\271)2814
-1564 y(\271)3596 -1365 y Fy(N)4862 -1564 y Fz(,)-221
b Fw(!)369 b Fz(E)468 b FF(b)36 b(ecause)389 b(the)h(asso)36
b(ciated)391 b(T)-108 b(ate)390 b(curv)-36 b(e)389 b(has)h(a)g(cop)-36
b(y)391 b(of)331 b Fz(\271)35339 -1601 y(\271)35368 -1564
y(\271)36151 -1365 y Fy(N)37437 -1564 y FF(naturally)391
b(sitting)f(in)-3718 41 y(it.)572 b(When)412 b(passing)i(to)f
Fz(J)8689 240 y Fx(1)9215 41 y FF(\()p Fz(N)139 b FF(\))412
b(w)-36 b(e)414 b(tak)-36 b(e)414 b(the)f(Picard)g(functoralit)-36
b(y)414 b(suc)-36 b(h)412 b(that)h(if)h Fz(T)37519 240
y Fy(p)38418 41 y FF(:)369 b Fz(J)39867 240 y Fx(1)40393
41 y FF(\()p Fz(N)139 b FF(\))367 b Fw(!)j Fz(J)45370
240 y Fx(1)45896 41 y FF(\()p Fz(N)139 b FF(\))-3718
1646 y(then)432 b(\()p Fz(T)693 1164 y Fu(_)512 1974
y Fy(p)1376 1646 y FF(\))1882 1164 y Fu(\244)2841 1646
y FF(on)i Fz(H)5829 1164 y Fx(0)6354 1646 y FF(\()p Fz(X)7939
1845 y Fx(1)8465 1646 y FF(\()p Fz(N)139 b FF(\))p Fz(;)221
b FF(\255)12179 1164 y Fx(1)12705 1646 y FF(\))368 b(=)h
Fz(S)15760 1845 y Fx(2)16286 1646 y FF(\(\241)17605 1845
y Fx(1)18130 1646 y FF(\()p Fz(N)139 b FF(\)\))432 b(is)i(the)f
(classical)j Fz(T)30572 1845 y Fy(p)31101 1646 y FF(.)p
eop
%%Page: 66 74
66 73 bop 1263 -6698 a FF(66)10376 b FA(CHAPTER)435 b(12.)1012
b(ABELIAN)434 b(V)-145 b(ARIETIES)435 b(FR)-36 b(OM)433
b(MODULAR)g(F)-36 b(ORMS)p eop
%%Page: 67 75
67 74 bop -3718 5699 a FE(Chapter)1033 b(13)-3718 11247
y(The)g(Gorenstein)g(Prop)86 b(ert)-86 b(y)-3718 17293
y FF(Supp)36 b(ose)331 b Fz(f)510 b FF(=)3814 16297 y
Fr(P)5438 17293 y Fz(a)6121 17492 y Fy(n)6747 17293 y
Fz(q)7372 16811 y Fy(n)8329 17293 y FF(is)332 b(a)g(newform)g(of)h
(exact)f(lev)-36 b(el)333 b Fz(N)470 b FF(and)332 b(w)-36
b(eigh)g(t)332 b(2)g(for)g(the)f(congruence)h(subgroup)-3718
18898 y(\241)-2905 19097 y Fx(0)-2380 18898 y FF(\()p
Fz(N)139 b FF(\).)573 b(Let)421 b Fz(E)446 b FF(=)369
b FD(Q)p FF(\()p Fz(:)221 b(:)g(:)444 b(;)221 b(a)10717
19097 y Fy(n)11344 18898 y Fz(;)g(:)g(:)g(:)j FF(\))420
b(and)g(let)h Fz(\270)f FF(b)36 b(e)421 b(a)g(place)g(of)g
Fz(E)498 b FF(lieing)422 b(o)-36 b(v)g(er)421 b(the)f(prime)g
Fz(`)h FF(of)g FD(Q)p FF(.)574 b(The)-3718 20503 y(action)539
b(of)g(Gal\()p 4413 19432 1123 54 v FD(Q)q Fz(=)p FD(Q)p
FF(\))g(on)f(the)g Fz(`)p FF(-adic)g(T)-108 b(ate)539
b(mo)36 b(dule)539 b(of)g(the)f(ab)36 b(elian)539 b(v)-72
b(ariet)-36 b(y)540 b Fz(A)38169 20702 y Fy(f)39312 20503
y FF(asso)36 b(ciated)540 b(to)f Fz(f)-3718 22108 y FF(giv)-36
b(es)434 b(rise)g(to)g(a)g(represen)-36 b(tation)14526
25081 y Fz(\275)15197 25280 y Fy(\270)16170 25081 y FF(:)369
b(Gal)q(\()p 19439 24010 V FD(Q)p Fz(=)p FD(Q)p FF(\))g
Fw(!)h FF(GL)26739 25280 y Fx(2)27265 25081 y FF(\()p
Fz(E)28734 25280 y Fy(\270)29337 25081 y FF(\))-3718
28053 y(with)642 b(det)221 b Fz(\275)2152 28252 y Fy(\270)3480
28053 y FF(=)723 b Fz(\302)6030 28252 y Fy(`)7112 28053
y FF(and)642 b(T)-108 b(r)221 b Fz(\275)12079 28252 y
Fy(\270)12683 28053 y FF(\(F)-108 b(rob)15808 28252 y
Fy(p)16337 28053 y FF(\))724 b(=)f Fz(a)19985 28252 y
Fy(p)21156 28053 y FF(for)643 b Fz(p)280 b Fw(6)724 b(j)p
Fz(`N)139 b FF(.)1203 b(W)-108 b(e)642 b(ask,)696 b(ho)-36
b(w)642 b(can)g(w)-36 b(e)642 b(reduce)g(this)-3718 29658
y(represen)-36 b(tation)433 b(mo)36 b(dule)433 b Fz(\270)p
FF(?)-3718 32435 y FD(Lemma)499 b(13.0.1.)651 b FC(L)-66
b(et)492 b Fw(O)530 b FC(b)-66 b(e)492 b(the)g(ring)g(of)h(inte)-66
b(gers)491 b(of)i Fz(E)25831 32634 y Fy(\270)26435 32435
y FC(.)681 b(Then)493 b Fz(\275)31602 32634 y Fy(\270)32699
32435 y FC(is)g(e)-66 b(quivalent)491 b(to)i(a)g(r)-66
b(epr)g(esen-)-3718 34040 y(tation)464 b(which)h(takes)f(values)i(in)e
FF(GL)14321 34239 y Fx(2)14847 34040 y FF(\()p Fw(O)37
b FF(\))p FC(.)-3718 36817 y(Pr)-66 b(o)g(of.)649 b FF(View)385
b(GL)5425 37016 y Fx(2)5951 36817 y FF(\()p Fz(E)7420
37016 y Fy(\270)8024 36817 y FF(\))e(as)i(the)e(group)h(of)h
(automorphisms)e(of)i(a)f(2)g(dimensional)h Fz(E)38363
37016 y Fy(\270)38967 36817 y FF(-v)-36 b(ector)384 b(space)g
Fz(V)289 b FF(.)-3718 38422 y(A)403 b(lattice)h(is)f(a)g(free)g
Fw(O)37 b FF(-mo)f(dule)402 b(of)i(rank)f(2)h(suc)-36
b(h)402 b(that)g Fz(L)233 b Fw(\255)g Fz(E)26992 38621
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29893 38956 V FD(Q)p Fz(=)p FD(Q)p FF(\).)606 b(F)-108
b(or)443 b(then)e(the)i(matrices)g(of)-3718 41632 y(Gal\()p
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45614 781 45 v 47192 46447 V 47972 46447 45 878 v -3718
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b(is)g(expressed)f(b)-36 b(y)434 b(the)f(diagram)14266
52278 y(Gal\()p 16804 51207 1123 54 v FD(Q)p Fz(=)p FD(Q)q
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14151 64072 671 54 v 352 w Fz(\275)14822 65117 y Fy(\270)15778
64804 y FF(b)36 b(e)351 b(the)g(semisimpli\257cation)i(of)f(the)f
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b(If)492 b Fz(g)516 b Fw(2)468 b FF(Gal\()p 7181 68548
V FD(Q)p Fz(=)p FD(Q)q FF(\))491 b(then)g(the)g(c)-36
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b(of)e Fz(g)475 b FF(in)428 b(the)f(semisimpli\257cation.)-3718
72829 y(The)376 b(represen)-36 b(tation)p 7338 72098
671 54 v 375 w Fz(\275)8009 73143 y Fy(\270)8990 72829
y FF(is)376 b(semisimple)h(\(since)f(it)h(is)f(a)h
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%%Page: 68 76
68 75 bop 1263 -6698 a FF(68)20985 b FA(CHAPTER)434 b(13.)1013
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24581 -691 V 433 w Fz(\275)25252 354 y Fy(\270)26290
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Fx(2)32012 2992 y FF(:)557 b Fz(G)h Fw(!)f FF(GL\()p
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6202 y(that)459 b(for)f(every)g Fz(g)416 b Fw(2)369 b
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b(of)h Fz(\275)10004 8006 y Fx(2)10530 7807 y FF(\()p
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y(The)289 b(Hec)-36 b(k)g(e)290 b(op)36 b(erators)289
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Fw(\275)28388 51176 y Fr(Y)30307 52438 y Fw(O)31365 52637
y Fy(E)32055 52772 y Ft(i)32461 52438 y Fz(:)1263 55748
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y Fy(\270)17720 63199 y FF(=)584 b Fz(T)20078 63398 y
Fy(p)21492 63199 y FF(mo)36 b(d)443 b FD(m)560 b FF(\(b)36
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b(ha)-36 b(v)g(e)560 b(set)f(things)h(up)e Fz(T)52539
63398 y Fy(p)1263 64804 y FF(pla)-36 b(ys)590 b(the)e(role)i(of)g
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b(of)e(ev)-36 b(ery)p 48005 65677 V 554 w Fz(\275)48676
66722 y Fy(\270)49280 66409 y FF(\()p Fz(g)48 b FF(\))552
b(for)1263 68014 y Fz(g)594 b Fw(2)546 b FF(Gal\()p 6450
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52128 74434 45 878 v 52173 73601 781 45 v 52173 74434
V 52953 74434 45 878 v eop
%%Page: 69 77
69 76 bop -3718 -6698 a FA(13.1.)1013 b(THE)434 b(GORENSTEIN)f(PR)-36
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%%Page: 71 79
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b(t)11927 67957 y Fu(_)12610 68439 y Fz(y)48 b Fw(i)p
Fz(:)470 b FF(It)h(is)f(more)h(con)-36 b(v)g(enien)g(t)469
b(to)i(use)f(an)g(adapted)g(pairing)g(de\257ned)f(as)-3718
70044 y(follo)-36 b(ws.)574 b(Let)413 b Fz(w)404 b FF(=)369
b Fz(w)7007 70243 y Fy(V)8188 70044 y Fw(2)f FF(End)221
b Fz(J)12713 70243 y Fx(0)13239 70044 y FF(\()p Fz(N)139
b FF(\))412 b(b)36 b(e)413 b(the)h(A)-36 b(tkin-Lehner)412
b(in)-36 b(v)g(olution)413 b(so)h(that)f Fz(t)38727 69562
y Fu(_)39779 70044 y FF(=)368 b Fz(w)36 b(tw)g FF(.)571
b(De\257ne)-3718 71650 y(a)434 b(new)f FD(T)p FF(-compatible)h(pairing)
g(b)-36 b(y)433 b([)p Fz(x;)221 b(y)48 b FF(])371 b(:=)d
Fw(h)p Fz(x;)221 b(w)36 b(y)48 b Fw(i)p Fz(:)434 b FF(Then)8260
74434 y([)p Fz(tx;)221 b(y)48 b FF(])369 b(=)g Fw(h)p
Fz(tx;)221 b(w)36 b(y)48 b Fw(i)368 b FF(=)h Fw(h)p Fz(x;)221
b(t)21735 73886 y Fu(_)22417 74434 y Fz(w)36 b(y)48 b
Fw(i)368 b FF(=)h Fw(h)p Fz(x;)221 b(w)36 b(ty)48 b Fw(i)368
b FF(=)h([)p Fz(x;)221 b(ty)48 b FF(])p Fz(:)p eop
%%Page: 72 80
72 79 bop 1263 -6698 a FF(72)20985 b FA(CHAPTER)434 b(13.)1013
b(THE)434 b(GORENSTEIN)f(PR)-36 b(OPER)-108 b(TY)3214
-3169 y FF(The)434 b(pairing)g([)p Fw(\242)p Fz(;)221
b Fw(\242)p FF(])434 b(de\257nes)f(an)g(isomorphism)h(of)g
FD(T)296 b Fw(\255)f FD(Z)31508 -2970 y Fy(`)31948 -3169
y FF(-mo)36 b(dules)17644 -249 y(T)-108 b(ate)20209 -50
y Fy(`)20870 -249 y Fz(J)22473 -1002 y Fu(\273)22081
-249 y Fw(\241)-845 b(!)369 b FF(Hom)26676 -50 y Fv(Z)27339
106 y Ft(`)27794 -249 y FF(\(T)-108 b(ate)30865 -50 y
Fy(`)31526 -249 y Fz(J)-22 b(;)221 b FD(Z)33718 -50 y
Fy(`)34159 -249 y FF(\(1\)\))p Fz(:)1263 2671 y FF(Since)362
b FD(Z)5501 2870 y Fy(`)5941 2671 y FF(\(1\))h(is)f(a)h(free)g(mo)36
b(dule)363 b(of)g(rank)f(1)h(o)-36 b(v)g(er)363 b FD(Z)26129
2870 y Fy(`)26932 2671 y FF(a)f(suitable)h(c)-36 b(hoice)363
b(of)g(basis)g(giv)-36 b(es)363 b(an)g(isomorphism)1263
4276 y(of)434 b FD(T)296 b Fw(\255)f FD(Z)6320 4475 y
Fy(`)6760 4276 y FF(-mo)36 b(dules)7391 7196 y(\(T)-108
b(ate)10462 6639 y Fu(\244)10462 7525 y Fy(`)11210 7196
y Fz(J)123 b FF(\)\(1\))14588 6827 y Fw(\273)14599 7251
y FF(=)15990 7196 y(Hom)18700 7395 y Fv(Z)19363 7551
y Ft(`)19818 7196 y FF(\(T)-108 b(ate)22889 7395 y Fy(`)23550
7196 y Fz(J)-22 b(;)221 b FD(Z)25742 7395 y Fy(`)26183
7196 y FF(\(1\)\))28720 6827 y Fw(\273)28730 7251 y FF(=)30122
7196 y(Hom)32831 7395 y Fv(Z)33494 7551 y Ft(`)33949
7196 y FF(\(T)-108 b(ate)37020 7395 y Fy(`)37681 7196
y Fz(J)-22 b(;)221 b FD(Z)39873 7395 y Fy(`)40314 7196
y FF(\))369 b(=)g(T)-108 b(ate)45135 6639 y Fu(\244)45135
7525 y Fy(`)45882 7196 y Fz(J)-22 b(:)1263 10116 y FF(Th)-36
b(us)433 b(T)-108 b(ate)7123 10315 y Fy(`)7784 10116
y Fz(J)9387 9363 y Fu(\273)8995 10116 y Fw(\241)-845
b(!)369 b FF(T)-108 b(ate)13446 9559 y Fu(\244)13446
10445 y Fy(`)14193 10116 y Fz(J)123 b FF(.)1263 12714
y FC(Pr)-66 b(o)g(of.)649 b FF(\(That)455 b Fz(J)123
b FF([)p FD(m)p FF(])405 b Fw(6)p FF(=)f(0.\))641 b(The)455
b(p)36 b(oin)-36 b(t)454 b(is)h(that)f(the)g(con)-36
b(tra)g(v)-72 b(arian)-36 b(t)454 b(T)-108 b(ate)455
b(mo)36 b(dule)454 b(Hom)q(\()p Fz(J)47976 12913 y Fy(`)48415
12714 y Fz(;)221 b FD(Q)50119 12913 y Fy(`)50560 12714
y Fz(=)p FD(Z)52123 12913 y Fy(`)52563 12714 y FF(\))1263
14319 y(is)493 b(the)g(P)-36 b(on)g(trjagin)493 b(dual)g(of)g
Fz(T)16654 14518 y Fy(`)17095 14319 y FF(.)756 b(Ho)-36
b(w)494 b(do)36 b(es)493 b(this)f(relate)i(to)f(T)-108
b(ate)34709 14518 y Fv(m)35890 14319 y Fz(J)123 b FF(?)756
b(Since)493 b FD(T)336 b Fw(\255)f FD(Z)45214 14518 y
Fy(`)46124 13950 y Fw(\273)46135 14374 y FF(=)47628 13323
y Fr(Q)48882 14706 y Fv(m)p Fu(j)p Fy(`)50709 14319 y
FD(T)51748 14518 y Fv(m)52707 14319 y FF(,)1263 16035
y(T)-108 b(ate)3828 16234 y Fy(`)4489 16035 y Fz(J)5700
15666 y Fw(\273)5711 16090 y FF(=)7103 15038 y Fr(Q)8357
16422 y Fv(m)p Fu(j)p Fy(`)10184 16035 y FF(T)g(ate)12749
16234 y Fv(m)13930 16035 y Fz(J)526 b FF(so)404 b(w)-36
b(e)404 b(can)f(de\257ne)g(T)-108 b(ate)27274 15478 y
Fu(\244)27274 16363 y Fv(m)28455 16035 y Fz(J)492 b FF(:=)369
b(Hom)34117 16234 y Fv(Z)34780 16390 y Ft(`)35235 16035
y FF(\(T)-108 b(ate)38306 16234 y Fv(m)39487 16035 y
Fz(J)-22 b(;)221 b FD(Z)41679 16234 y Fy(`)42120 16035
y FF(\).)568 b(W)-108 b(eil)404 b(pro)-36 b(v)g(ed)403
b(that)1263 17640 y(T)-108 b(ate)3828 17839 y Fv(m)5009
17640 y Fz(J)6220 17271 y Fw(\273)6231 17695 y FF(=)7622
17640 y Fz(T)181 b(ate)10321 17158 y Fu(\244)10321 17968
y Fv(m)11280 17640 y Fz(J)504 b FF(is)381 b(nonzero.)561
b(View)382 b(T)-108 b(ate)24888 17083 y Fu(\244)24888
17968 y Fv(m)26069 17640 y Fz(J)504 b FF(as)381 b(b)36
b(eing)381 b(dual)g(to)g Fz(J)37382 17839 y Fv(m)38722
17640 y FF(in)g(the)g(sense)g(of)h(P)-36 b(on)g(trjagin)1263
19245 y(dualit)g(y)518 b(and)e(so)i(\(T)-108 b(ate)13119
18688 y Fu(\244)13119 19573 y Fv(m)14299 19245 y Fz(J)123
b FF(\))p Fz(=)p FF(\()p FD(m)221 b FF(T)-108 b(ate)20835
18688 y Fu(\244)20835 19573 y Fv(m)22016 19245 y Fz(J)123
b FF(\))517 b(is)g(dual)g(to)g Fz(J)123 b FF([)p FD(m)p
FF(].)830 b(If)518 b Fz(J)123 b FF([)p FD(m)p FF(])512
b(=)f(0)517 b(then)g(this)f(quotien)-36 b(t)517 b(is)1263
20850 y(0,)609 b(so)574 b(Nak)-72 b(a)-36 b(y)g(ama's)576
b(lemma)e(w)-36 b(ould)573 b(imply)h(that)f(T)-108 b(ate)29533
20293 y Fu(\244)29533 21178 y Fv(m)30714 20850 y Fz(J)730
b FF(=)607 b(0.)998 b(This)574 b(w)-36 b(ould)574 b(con)-36
b(tradicts)573 b(W)-108 b(eil's)1263 22455 y(assertion.)579
b(Therefore)434 b Fz(J)123 b FF([)p FD(m)p FF(])370 b
Fw(6)p FF(=)e(0.)p 52128 22455 45 878 v 52173 21621 781
45 v 52173 22455 V 52953 22455 45 878 v 1263 26890 a
Fs(13.2)2152 b(Pro)60 b(of)716 b(the)h(Gorenstein)f(Prop)60
b(ert)-60 b(y)1263 29811 y FF(W)-108 b(e)464 b(are)g(considering)g(the)
f(situation)h(with)f(resp)36 b(ect)464 b(to)g Fz(J)30101
30010 y Fx(0)30626 29811 y FF(\()p Fz(N)139 b FF(\))463
b(although)h(w)-36 b(e)464 b(could)f(consider)h Fz(J)49990
30010 y Fx(1)50516 29811 y FF(\()p Fz(N)139 b FF(\).)1263
31416 y(Let)543 b FD(T)555 b Fw(\275)g FF(End)221 b Fz(J)10156
31615 y Fx(0)10681 31416 y FF(\()p Fz(N)139 b FF(\))542
b(b)36 b(e)543 b(the)f(Hec)-36 b(k)g(e)543 b(algebra)h(and)e(let)h
FD(m)556 b Fw(\275)f FD(T)543 b FF(b)36 b(e)543 b(a)g(maximal)h(ideal.)
907 b(Let)542 b Fz(`)h FF(b)36 b(e)1263 33021 y(the)470
b(c)-36 b(haracteristic)471 b(of)g(the)e(residue)h(class)h(\257eld)f
FD(T)p Fz(=)p FD(m)p FF(.)690 b(Let)470 b FD(T)33147
33220 y Fv(m)34538 33021 y FF(=)431 b(lim)35981 33632
y Fw(\303)-555 b(\241)38009 33021 y FD(T)p Fz(=)p FD(m)40943
32539 y Fy(i)41320 33021 y FD(T)p FF(.)689 b(Then)469
b FD(T)321 b Fw(\255)49234 33220 y Fv(Z)50273 33021 y
FD(Z)51186 33220 y Fy(`)52057 33021 y FF(=)1263 33630
y Fr(Q)2518 35013 y Fv(m)p Fu(j)p Fy(`)4344 34626 y FD(T)5383
34825 y Fv(m)6343 34626 y FF(.)582 b([[I)436 b(w)-36
b(an)g(t)434 b(to)h(put)f(a)h(go)36 b(o)g(d)436 b(reference)f(for)g
(this)g(A)-36 b(tiy)g(ah-Macdonald)434 b(lik)-36 b(e)436
b(fact)g(here.]])582 b(Eac)-36 b(h)1263 36342 y FD(T)2302
36541 y Fv(m)3695 36342 y FF(acts)434 b(on)g(T)-108 b(ate)10748
36541 y Fy(`)11409 36342 y Fz(J)12128 36541 y Fx(0)12654
36342 y FF(\()p Fz(N)139 b FF(\))432 b(so)i(w)-36 b(e)434
b(obtain)g(a)f(pro)36 b(duct)433 b(decomp)36 b(osition)18228
39494 y(T)-108 b(ate)20794 39693 y Fy(`)21455 39494 y
Fz(J)22174 39693 y Fx(0)22699 39494 y FF(\()p Fz(N)139
b FF(\))368 b(=)26641 38233 y Fr(Y)26715 41111 y Fv(m)p
Fu(j)p Fy(`)28560 39494 y FF(T)-108 b(ate)31125 39693
y Fv(m)32306 39494 y Fz(J)33025 39693 y Fx(0)33550 39494
y FF(\()p Fz(N)139 b FF(\))p Fz(:)1263 43813 y FF(W)-108
b(e)434 b(ha)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing)435
b(t)-36 b(w)g(o)434 b(facts:)2853 46514 y(1.)651 b(T)-108
b(ate)7080 46713 y Fv(m)8261 46514 y Fz(J)8980 46713
y Fx(0)9506 46514 y FF(\()p Fz(N)139 b FF(\))367 b Fw(6)p
FF(=)i(0)2853 49222 y(2.)651 b(T)-108 b(ate)7080 49421
y Fy(`)7741 49222 y Fz(J)8460 49421 y Fx(0)8986 49222
y FF(\()p Fz(N)139 b FF(\))432 b(is)i FD(T)296 b Fw(\255)f
FD(Z)16495 49421 y Fy(`)16935 49222 y FF(-auto)36 b(dual)433
b(and)g(eac)-36 b(h)434 b(T)-108 b(ate)30845 49421 y
Fv(m)32026 49222 y Fz(J)32745 49421 y Fx(0)33270 49222
y FF(\()p Fz(N)139 b FF(\))433 b(is)h FD(Z)38117 49421
y Fy(`)38557 49222 y FF(-auto)36 b(dual.)1263 51923 y([[auto)g(dualit)
-36 b(y)435 b(for)f(whic)-36 b(h)433 b(dual?)578 b(I)434
b(think)f(it)h(is)g(the)f(linear)h(dual)f(since)h(this)f(is)h(used.]])
3214 53528 y(Let)d Fz(W)549 b FF(=)369 b Fz(J)123 b FF([)p
FD(m)p FF(],)432 b(then)e(the)h(action)g(of)h(Gal\()p
25416 52457 1123 54 v FD(Q)p Fz(=)p FD(Q)q FF(\))f(on)f
Fz(W)612 b FF(giv)-36 b(es)432 b(a)f(represen)-36 b(tation)430
b(of)i(Gal\()p 49668 52457 V FD(Q)p Fz(=)p FD(Q)q FF(\))1263
55133 y(o)-36 b(v)g(er)508 b(the)e(\257eld)h FD(T)p Fz(=)p
FD(m)p FF(.)799 b(W)-108 b(e)507 b(compared)g Fz(W)687
b FF(with)507 b(a)h(certain)f(t)-36 b(w)g(o)507 b(dimensional)g
(represen)-36 b(tation)506 b Fz(\275)51254 55332 y Fv(m)52707
55133 y FF(:)1263 56738 y(Gal\()p 3801 55667 V FD(Q)p
Fz(=)p FD(Q)q FF(\))664 b Fw(!)h Fz(V)896 b FF(o)-36
b(v)g(er)607 b FD(T)p Fz(=)p FD(m)p FF(.)1100 b(Assume)606
b(unless)h(otherwise)g(stated)g(that)f Fz(V)896 b FF(is)608
b(irreducible)e(as)h(a)1263 58343 y(Gal\()p 3801 57272
V FD(Q)p Fz(=)p FD(Q)q FF(\)-mo)36 b(dule.)540 b(Let)320
b(T)-108 b(ate)17474 58542 y Fy(`)18283 58343 y FF(=)368
b(T)-108 b(ate)22229 58542 y Fy(`)22890 58343 y Fz(J)23609
58542 y Fx(0)24134 58343 y FF(\()p Fz(N)139 b FF(\))320
b(and)f(T)-108 b(ate)31628 58542 y Fv(m)32956 58343 y
FF(=)369 b(T)-108 b(ate)36902 58542 y Fv(m)38083 58343
y Fz(J)38802 58542 y Fx(0)39327 58343 y FF(\()p Fz(N)139
b FF(\).)540 b(A)320 b(formal)h(argumen)-36 b(t)1263
59948 y(due)433 b(to)h(Mazur)f(sho)-36 b(w)g(ed)433 b(that)19328
61553 y Fz(W)20737 61005 y Fx(s.s.)22426 61185 y Fw(\273)22436
61609 y FF(=)23828 61553 y Fz(V)584 b Fw(\243)296 b(\242)221
b(\242)g(\242)296 b(\243)f Fz(V)658 b FF(=)369 b Fz(V)33516
61005 y Fu(\251)p Fy(t)34643 61553 y Fz(:)1263 63872
y FF(W)-108 b(e)434 b(ha)-36 b(v)g(e)434 b(not)f(y)-36
b(et)434 b(determined)e Fz(t)h FF(but)g(w)-36 b(e)434
b(w)-36 b(ould)433 b(lik)-36 b(e)435 b(to)e(sho)-36 b(w)434
b(that)f Fz(t)369 b FF(=)g(1.)1263 66573 y FD(De\257nition)500
b(13.2.1.)651 b FF(The)289 b FD(P)-42 b(on)g(trjagin)335
b(dual)290 b FF(of)g(a)f(mo)36 b(dule)289 b Fz(M)428
b FF(is)289 b(the)g(mo)36 b(dule)289 b Fz(M)43463 66091
y Fu(^)44515 66573 y FF(:=)368 b(Hom)q(\()p Fz(M)66 b(;)221
b FD(Q)p Fz(=)p FD(Z)p FF(\))1263 68178 y(where)359 b
Fz(M)498 b FF(is)359 b(view)-36 b(ed)359 b(as)g(an)g(ab)36
b(elian)359 b(group)g(\(if)g Fz(M)498 b FF(is)359 b(top)36
b(ological,)376 b(only)359 b(tak)-36 b(e)360 b(those)e(homomorphisms)
1263 69783 y(whose)532 b(k)-36 b(ernel)531 b(is)h(compact\).)871
b(The)531 b FD(linear)611 b(dual)533 b FF(of)e(a)h(mo)36
b(dule)531 b Fz(M)670 b FF(o)-36 b(v)g(er)532 b(a)f(ring)h
Fz(R)542 b FF(is)531 b(the)g(mo)36 b(dule)1263 71388
y Fz(M)2660 70906 y Fu(\244)3555 71388 y FF(=)369 b(Hom)7645
71587 y Fy(R)8414 71388 y FF(\()p Fz(M)66 b(;)221 b(R)11
b FF(\).)1263 74089 y FC(Exer)-66 b(cise)464 b(13.2.2.)649
b FF(Note)434 b(that)f(\()p FD(Q)18503 74288 y Fy(`)18943
74089 y Fz(=)p FD(Z)20506 74288 y Fy(`)20946 74089 y
FF(\))21452 73607 y Fu(^)22504 74089 y FF(=)369 b FD(Z)24798
74288 y Fy(`)25671 74089 y FF(and)433 b FD(Z)29113 73607
y Fu(^)29113 74434 y Fy(`)30165 74089 y FF(=)369 b FD(Q)32668
74288 y Fy(`)33108 74089 y Fz(=)p FD(Z)34671 74288 y
Fy(`)35111 74089 y FF(.)p eop
%%Page: 73 81
73 80 bop -3718 -6698 a FA(13.2.)1013 b(PR)-36 b(OOF)433
b(THE)h(GORENSTEIN)f(PR)-36 b(OPER)-108 b(TY)21850 b
FF(73)-3718 -3169 y FC(Solution.)649 b FF(W)-108 b(e)434
b(can)f(think)g(of)i FD(Q)12640 -2970 y Fy(`)13080 -3169
y Fz(=)p FD(Z)14643 -2970 y Fy(`)15517 -3169 y FF(as)12235
887 y Fw(f)13573 -773 y Fu(\241)p Fx(1)13214 -375 y Fr(X)12899
2454 y Fy(n)p Fx(=)p Fu(\241)p Fy(k)15669 887 y Fz(a)16352
1086 y Fy(n)16978 887 y Fz(`)17524 339 y Fy(n)18519 887
y FF(:)369 b Fz(k)414 b(>)369 b FF(0)434 b(and)f(0)369
b Fw(\267)h Fz(a)28438 1086 y Fy(i)29182 887 y Fz(<)f(`)p
Fw(g)p Fz(:)-3718 5107 y FF(Let)599 b(\()p Fz(b)-163
5306 y Fy(i)213 5107 y FF(\))652 b Fw(2)f FD(Z)3821 5306
y Fy(`)4861 5107 y FF(so)600 b Fz(b)7177 5306 y Fy(i)8205
5107 y Fw(2)651 b FD(Z)p Fz(=`)11851 4625 y Fy(i)12227
5107 y FD(Z)600 b FF(and)g Fz(b)16989 5306 y Fy(i)p Fx(+1)19219
5107 y Fw(\264)652 b Fz(b)21457 5306 y Fy(i)22423 5107
y FF(\(mo)36 b(d)442 b Fz(`)26410 4625 y Fy(i)26786 5107
y FF(\).)1077 b(De\257ne)599 b(a)i(map)e FD(Q)38353 5306
y Fy(`)38793 5107 y Fz(=)p FD(Z)40356 5306 y Fy(`)41448
5107 y Fw(!)653 b FD(Q)p Fz(=)p FD(Z)601 b FF(b)-36 b(y)-3718
6712 y(1)p Fz(=`)-1872 6230 y Fy(i)-978 6712 y Fw(7!)519
b Fz(b)1422 6911 y Fy(i)1797 6712 y Fz(=`)2993 6230 y
Fy(i)3369 6712 y FF(.)841 b(T)-108 b(o)522 b(c)-36 b(hec)g(k)521
b(that)g(this)f(is)i(w)-36 b(ell-de\257ned)520 b(it)h(su\261ces)g(to)g
(c)-36 b(hec)g(k)522 b(that)e(1)p Fz(=`)40215 6230 y
Fy(i)41113 6712 y FF(maps)h(to)g(the)-3718 8318 y(same)434
b(place)g(as)g Fz(`)295 b Fw(\242)g FF(1)p Fz(=`)7813
7836 y Fy(i)p Fx(+1)9391 8318 y FF(.)579 b(No)-36 b(w)434
b(1)p Fz(=`)15139 7836 y Fy(i)15884 8318 y Fw(7!)369
b Fz(b)18134 8517 y Fy(i)18510 8318 y Fz(=`)19706 7836
y Fy(i)20515 8318 y FF(and)12575 11173 y Fz(`)295 b Fw(\242)g
FF(1)p Fz(=`)15926 10624 y Fy(i)p Fx(+1)17873 11173 y
Fw(7!)370 b Fz(`)295 b Fw(\242)g Fz(b)21629 11372 y Fy(i)p
Fx(+1)23207 11173 y Fz(=`)24403 10624 y Fy(i)p Fx(+1)26350
11173 y FF(=)368 b Fz(b)28283 11372 y Fy(i)p Fx(+1)29861
11173 y Fz(=`)31057 10624 y Fy(i)31433 11173 y Fz(:)-3718
14028 y FF(So)433 b(w)-36 b(e)434 b(just)g(need)f(to)g(c)-36
b(hec)g(k)434 b(that)14922 16883 y Fz(b)15475 17082 y
Fy(i)p Fx(+1)17053 16883 y Fz(=`)18249 16334 y Fy(i)18994
16883 y Fw(\264)369 b Fz(b)20949 17082 y Fy(i)21325 16883
y Fz(=`)22521 16334 y Fy(i)24225 16883 y FF(\(mo)36 b(d)443
b FD(Z)p FF(\))p Fz(:)-3718 19738 y FF(This)434 b(is)g(just)f(the)g
(assertion)h(that)f(\()p Fz(b)14683 19937 y Fy(i)p Fx(+1)16556
19738 y Fw(\241)295 b Fz(b)18437 19937 y Fy(i)18813 19738
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54317 y Fv(m)32486 54118 y Fw(\255)33519 54317 y Fv(Z)34182
54473 y Ft(`)34637 54118 y FD(Q)35759 54317 y Fy(`)1263
57365 y FF(w)-36 b(e)434 b(can)g(tensor)f(with)g FD(C)h
FF(to)g(see)g(that)18435 59524 y(T)-108 b(ate)21000 59723
y Fy(`)21661 59524 y Fw(\255)22694 59723 y Fv(Z)23357
59879 y Ft(`)23812 59524 y FD(C)369 b FF(=)26641 58262
y Fr(Y)27038 61052 y Fv(m)28560 59524 y FF(T)-108 b(ate)31125
59723 y Fv(m)32306 59524 y Fw(\255)33339 59723 y Fv(Z)34002
59879 y Ft(`)34457 59524 y FD(C)p Fz(:)1263 62846 y FF(But)289
b(T)-108 b(ate)6267 63045 y Fy(`)6928 62846 y Fw(\255)7961
63045 y Fv(Z)8624 63201 y Ft(`)9079 62846 y FD(C)369
b FF(=)g Fz(L)p Fw(\255)13826 63045 y Fv(Z)14544 62846
y FD(C)290 b FF(is)f(free)h(of)f(rank)h(2)f(o)-36 b(v)g(er)289
b FD(T)p Fw(\255)p FD(C)p FF(.)531 b(Therefore)290 b(the)f(pro)36
b(duct)43809 61850 y Fr(Q)45063 63233 y Fv(m)46244 62846
y FF(T)-108 b(ate)48809 63045 y Fv(m)49990 62846 y Fw(\255)51023
63045 y Fv(Z)51686 63201 y Ft(`)52141 62846 y FD(C)1263
64451 y FF(is)434 b(free)g(of)g(rank)g(2)g(o)-36 b(v)g(er)434
b FD(T)295 b Fw(\255)h FD(C)p FF(.)578 b(Since)14498
66610 y FD(T)295 b Fw(\255)h FD(C)369 b FF(=)f(\()p FD(T)296
b Fw(\255)22863 66809 y Fv(Z)23876 66610 y FD(Z)24789
66809 y Fy(`)25229 66610 y FF(\))f Fw(\255)27063 66809
y Fv(Z)27726 66965 y Ft(`)28477 66610 y FD(C)369 b FF(=)31306
65348 y Fr(Y)31702 68138 y Fv(m)33003 66610 y FF(\()p
FD(T)34548 66809 y Fv(m)35803 66610 y Fw(\255)36836 66809
y Fv(Z)37499 66965 y Ft(`)38249 66610 y FD(C)p FF(\))1263
69858 y(w)-36 b(e)434 b(conclude)f(that)g(for)h(eac)-36
b(h)434 b FD(m)p FF(,)g(T)-108 b(ate)20862 70057 y Fv(m)22043
69858 y Fw(\255)23076 70057 y Fv(Z)23739 70213 y Ft(`)24194
69858 y FD(C)434 b FF(is)g(free)g(of)g(rank)g(2)g(o)-36
b(v)g(er)434 b FD(T)38893 70057 y Fv(m)40148 69858 y
Fw(\255)41181 70057 y Fv(Z)41844 70213 y Ft(`)42594 69858
y FD(C)p FF(.)579 b(This)434 b(implies)16229 72017 y(dim)18397
72216 y Fv(C)19457 72017 y FF(T)-108 b(ate)22023 72216
y Fv(m)23203 72017 y Fw(\255)24236 72216 y Fv(Z)24899
72372 y Ft(`)25354 72017 y FD(C)370 b FF(=)e(2)221 b(dim)31223
72216 y Fv(C)32283 72017 y FD(T)33322 72216 y Fv(m)34577
72017 y Fw(\255)35610 72216 y Fv(Z)36273 72372 y Ft(`)37023
72017 y FD(C)1263 74176 y FF(whic)-36 b(h)434 b(completes)f(the)g(pro)
36 b(of.)p 52128 74176 45 878 v 52173 73342 781 45 v
52173 74176 V 52953 74176 45 878 v eop
%%Page: 75 83
75 82 bop -3718 -6698 a FA(13.3.)1013 b(FINITE)434 b(FLA)-108
b(T)433 b(GR)-36 b(OUP)433 b(SCHEMES)26330 b FF(75)-3718
-3169 y Fq(13.2.1)1792 b(V)-149 b(ague)597 b(Commen)-50
b(ts)-3718 -639 y FF(Ogus)405 b(commen)-36 b(ted)405
b(that)g(this)h(same)g(pro)36 b(of)406 b(sho)-36 b(ws)406
b(that)f FD(T)239 b Fw(\255)27165 -440 y Fv(Z)28122 -639
y FD(C)406 b FF(is)g(Gorenstein.)568 b(Then)406 b(he)f(said)h(that)
-3718 966 y(something)433 b(called)i(\\faithfully)g(\260at)e(descen)-36
b(t")433 b(could)h(then)e(sho)-36 b(w)434 b(that)f FD(T)296
b Fw(\255)34837 1165 y Fv(Z)35850 966 y FD(Q)434 b FF(is)g(Gorenstein.)
-1767 2605 y(W)-108 b(e)305 b(ha)-36 b(v)g(e)305 b(giv)-36
b(en)306 b(the)e(classical)j(argumen)-36 b(t)304 b(of)i(Mazur)f(that)f
FD(T)28188 2804 y Fv(m)29453 2605 y FF(is)h(Gorenstein,)330
b(but)304 b(w)-36 b(e)306 b(still)g(ha)-36 b(v)g(en't)-3718
4210 y(sho)g(wn)516 b(that)g Fz(J)123 b FF([)p FD(m)p
FF(])517 b(has)g(dimension)f(2.)827 b(This)517 b(will)h(b)36
b(e)516 b(accomplished)h(next)f(time)h(using)f(Dieudonn)-36
b(\266)-614 b(e)-3718 5815 y(mo)36 b(dules.)-3718 10457
y Fs(13.3)2151 b(Finite)716 b(Flat)h(Group)g(Sc)-60 b(hemes)-3718
13445 y FD(De\257nition)499 b(13.3.1.)652 b FF(Let)461
b Fz(S)540 b FF(b)36 b(e)461 b(a)h(sc)-36 b(heme.)663
b(Then)462 b(a)g FD(group)532 b(sc)-42 b(heme)532 b(o)-42
b(v)g(er)532 b Fz(S)539 b FF(is)462 b(a)g(group)g(ob)72
b(ject)-3718 15050 y(in)433 b(the)g(category)i(of)f Fz(S)77
b FF(-sc)-36 b(hemes.)-1767 17928 y(Th)g(us)294 b(a)h(group)f(sc)-36
b(heme)294 b(o)-36 b(v)g(er)295 b Fz(S)371 b FF(is)295
b(a)g(sc)-36 b(heme)294 b Fz(G=S)372 b FF(equipp)36 b(ed)294
b(with)g Fz(S)77 b FF(-morphisms)294 b Fz(m)368 b FF(:)i
Fz(G)11 b Fw(\243)g Fz(G)369 b Fw(!)g Fz(G)p FF(,)-3718
19533 y(in)-36 b(v)369 b(:)g Fz(G)g Fw(!)h Fz(G)433 b
FF(and)g(a)h(section)g(1)12273 19732 y Fy(G)13431 19533
y FF(:)369 b Fz(S)446 b Fw(!)370 b Fz(G)433 b FF(satisfying)i(the)e
(usual)h(group)f(axioms.)-1767 21171 y(Supp)36 b(ose)405
b Fz(G)h FF(is)g(a)h(group)f(sc)-36 b(heme)405 b(o)-36
b(v)g(er)407 b(the)f(\257nite)f(\257eld)h FD(F)27151
21370 y Fy(q)27658 21171 y FF(.)570 b(If)406 b Fz(R)418
b FF(is)406 b(an)g FD(F)35267 21370 y Fy(q)35775 21171
y FF(-algebra)g(then)f Fz(G)p FF(\()p Fz(R)11 b FF(\))369
b(=)-3718 22776 y(Mor\(sp)36 b(ec)221 b Fz(R)11 b(;)221
b(G)p FF(\))434 b(is)g(a)g(group.)578 b(It)433 b(is)h(the)f(group)g(of)
i Fz(R)11 b FF(-v)-72 b(alued)433 b(p)36 b(oin)-36 b(ts)433
b(of)h Fz(G)p FF(.)-1767 24415 y(W)-108 b(e)433 b(consider)h(sev)-36
b(eral)434 b(standard)f(examples)h(of)h(group)e(sc)-36
b(hemes.)-3718 26673 y FC(Example)465 b(13.3.2.)649 b
FF(The)358 b(m)-36 b(ultiplicativ)g(e)358 b(group)g(sc)-36
b(heme)357 b Fz(G)25783 26872 y Fy(m)27028 26673 y FF(is)h
Fz(G)29286 26872 y Fy(m)30543 26673 y FF(=)368 b(sp)36
b(ec)221 b FD(Z)p FF([)p Fz(x;)37331 26150 y Fx(1)p 37302
26367 530 54 v 37302 27131 a Fy(x)37964 26673 y FF(])358
b(with)g(morphisms.)-3718 28278 y([[giv)-36 b(e)435 b(maps,)f(etc.]])
579 b(The)433 b(additiv)-36 b(e)434 b(group)f(sc)-36
b(heme)434 b(is)f(sp)36 b(ec)222 b FD(Z)p FF([)p Fz(x)p
FF(]...)-3718 30536 y FC(Example)465 b(13.3.3.)649 b
FF(The)543 b(group)f(sc)-36 b(heme)484 b Fz(\271)17157
30499 y(\271)17187 30536 y(\271)17969 30735 y Fy(p)19041
30536 y FF(is)543 b(the)f(k)-36 b(ernel)543 b(of)h(the)e(morphism)g
Fz(G)37898 30735 y Fy(m)39340 30536 y Fw(!)555 b Fz(G)42249
30735 y Fy(m)43680 30536 y FF(induced)-3718 32141 y(b)-36
b(y)624 b Fz(x)694 b Fw(7!)h Fz(x)2474 31659 y Fy(p)3003
32141 y FF(.)1151 b(Th)-36 b(us)565 b Fz(\271)7971 32104
y(\271)8001 32141 y(\271)8784 32340 y Fy(p)10007 32141
y FF(=)693 b(sp)36 b(ec)221 b FD(Z)p FF([)p Fz(x)p FF(])p
Fz(=)p FF(\()p Fz(x)18630 31659 y Fy(p)19586 32141 y
Fw(\241)425 b FF(1\))625 b(and)f(so)h(for)g(an)-36 b(y)625
b FD(F)33098 32340 y Fy(q)33605 32141 y FF(-algebra)g
Fz(R)635 b FF(w)-36 b(e)625 b(ha)-36 b(v)g(e)625 b(that)-3777
33746 y Fz(\271)-3748 33709 y(\271)-3718 33746 y(\271)-2936
33945 y Fy(p)-2407 33746 y FF(\()p Fz(R)11 b FF(\))458
b(=)f Fw(f)p Fz(r)494 b Fw(2)457 b Fz(R)468 b FF(:)458
b Fz(r)7519 33264 y Fy(p)8506 33746 y FF(=)f(1)p Fw(g)p
FF(.)735 b(The)486 b(group)f(sc)-36 b(heme)486 b Fz(\256)24179
33945 y Fy(p)25193 33746 y FF(is)g(the)f(k)-36 b(ernel)486
b(of)h(the)e(morphism)g Fz(G)43708 33945 y Fy(a)44720
33746 y Fw(!)458 b Fz(G)47532 33945 y Fy(a)-3718 35352
y FF(induced)516 b(b)-36 b(y)518 b([[what!!)832 b(what)518
b(is)g(alphap??)831 b(it)518 b(should)f(b)36 b(e)518
b(the)f(additiv)-36 b(e)518 b(group)f(sc)-36 b(heme)518
b(of)g(order)g(p,)-3718 36957 y(no?]].)-1767 39215 y(Let)381
b Fz(A)g FF(b)36 b(e)382 b(a)g(\257nite)e(algebra)j(o)-36
b(v)g(er)382 b FD(F)16059 39414 y Fy(p)16969 39215 y
FF(and)f(supp)36 b(ose)381 b Fz(G)369 b FF(=)g(sp)36
b(ec)221 b Fz(A)382 b FF(a\261ne)f(group)g(sc)-36 b(heme)381
b(\(o)-36 b(v)g(er)382 b FD(F)46691 39414 y Fy(p)47220
39215 y FF(\).)-3718 40820 y(Then)433 b(the)g FD(order)i
FF(of)f Fz(G)f FF(is)h(de\257ned)e(to)i(b)36 b(e)433
b(the)g(dimension)h(of)g Fz(A)g FF(as)g(an)f FD(F)33542
41019 y Fy(p)34505 40820 y FF(v)-36 b(ector)434 b(space.)-3718
43078 y FC(Example)465 b(13.3.4.)649 b FF(Let)450 b Fz(E)78
b(=)p FD(F)11026 43277 y Fy(p)12006 43078 y FF(b)36 b(e)450
b(an)g(elliptic)h(curv)-36 b(e.)628 b(Then)450 b Fz(G)398
b FF(=)f Fz(E)78 b FF([)p Fz(p)p FF(])450 b(is)g(a)h(group)f(sc)-36
b(heme)450 b(of)h(order)-3718 44683 y Fz(p)-3065 44201
y Fx(2)-2115 44683 y FF([[wh)-36 b(y)425 b(is)f(this)g(true?]].)576
b(This)424 b(is)h(w)-36 b(onderful)424 b(b)36 b(ecause)424
b(this)g(is)h(the)e(order)h(that)g Fz(E)78 b FF([)p Fz(p)p
FF(])424 b(should)g(ha)-36 b(v)g(e)425 b(in)-3718 46288
y(analogy)435 b(with)f(the)f(c)-36 b(haracteristic)434
b(0)f(situation.)579 b(When)433 b(w)-36 b(e)434 b(just)f(lo)36
b(ok)436 b(at)d(p)36 b(oin)-36 b(ts)433 b(w)-36 b(e)434
b(ha)-36 b(v)g(e)13146 50410 y(#)p Fz(G)p FF(\()p 15762
49339 1469 54 v FD(F)16701 50609 y Fy(p)17230 50410 y
FF(\))369 b(=)19485 48139 y Fr(\()20555 49500 y FF(1)1737
b(sup)36 b(ersingular)20555 51426 y Fz(p)1734 b FF(ordinary)30862
50410 y Fz(:)-3718 55716 y Fs(13.4)2151 b(Reform)-60
b(ulation)716 b(of)h Fp(V)947 b Fm(=)532 b Fp(W)977 b
Fs(problem)-3718 58704 y FF(Let)379 b Fz(J)492 b FF(=)369
b Fz(J)1869 58903 y Fx(0)2394 58704 y FF(\()p Fz(N)139
b FF(\))379 b(b)36 b(e)380 b(the)f(Jacobian)i(of)f Fz(X)16759
58903 y Fx(0)17285 58704 y FF(\()p Fz(N)139 b FF(\).)560
b(Then)379 b Fz(J)503 b FF(is)380 b(de\257ned)e(o)-36
b(v)g(er)380 b FD(Q)h FF(and)e(has)h(go)36 b(o)g(d)380
b(reduction)-3718 60309 y(at)460 b(all)h(primes)e(not)h(dividing)g
Fz(N)139 b FF(.)657 b(Assume)460 b Fz(`)f FF(is)i(a)f(prime)f(not)h
(dividing)g Fz(N)139 b FF(.)657 b Fz(J)123 b FF([)p Fz(`)p
FF(])460 b(extends)g(to)g(a)g(\257nite)-3718 61914 y(\260at)553
b(group)h(sc)-36 b(heme)553 b(o)-36 b(v)g(er)554 b FD(Z)p
FF([)11586 61391 y Fx(1)p 11400 61609 842 54 v 11400
62372 a Fy(N)12375 61914 y FF(].)939 b(This)554 b(is)g(a)g(non)-36
b(trivial)555 b(result)e(of)i(Grothendiec)-36 b(k)553
b(\(SGA)f(7I,)j(LNM)-3718 63519 y(288\).)579 b(Since)433
b Fz(`)-74 b Fw(6)369 b(j)p Fz(N)139 b FF(,)433 b Fz(J)123
b FF([)p Fz(`)p FF(])434 b(giv)-36 b(es)435 b(rise)f(to)f(a)h(group)f
(sc)-36 b(heme)434 b(o)-36 b(v)g(er)434 b FD(F)28913
63718 y Fy(`)29353 63519 y FF(.)-1767 65158 y(W)-108
b(e)595 b(ha)-36 b(v)g(e)595 b(\\forcefully")j(constructed)c(a)i
(Galois)g(represen)-36 b(tation)594 b Fz(\275)32835 65357
y Fv(m)34438 65158 y FF(:)645 b Fz(G)f Fw(!)g Fz(V)885
b FF(of)596 b(dimension)-3718 66763 y(2)535 b(o)-36 b(v)g(er)534
b FD(T)p Fz(=)p FD(m)p FF(.)882 b(Our)533 b(goal)j(is)f(to)f(sho)-36
b(w)535 b(that)e(this)h(is)h(isomorphic)g(to)f(the)g(naturally)h
(de\257ned)e(Galois)-3718 68368 y(represen)-36 b(tation)433
b Fz(W)549 b FF(=)369 b Fz(J)123 b FF([)p FD(m)p FF(].)579
b(So)433 b(far)i(w)-36 b(e)433 b(kno)-36 b(w)435 b(that)16739
71401 y(0)369 b Fw(\275)h Fz(V)658 b Fw(\275)369 b Fz(W)550
b Fw(\275)369 b Fz(J)123 b FF([)p Fz(`)p FF(])p Fz(:)-3718
74434 y FF(Our)432 b(assumptions)i(are)f(that)g Fz(`)-74
b Fw(6)369 b(j)p Fz(N)139 b FF(,)434 b Fz(V)722 b FF(is)434
b(irreducible,)g(and)f Fz(`)368 b Fw(6)p FF(=)h(2.)p
eop
%%Page: 76 84
76 83 bop 1263 -6698 a FF(76)20985 b FA(CHAPTER)434 b(13.)1013
b(THE)434 b(GORENSTEIN)f(PR)-36 b(OPER)-108 b(TY)3214
-3169 y FF(Let)386 b Fz(J)p 5497 -2957 842 54 v 510 w
FF(b)36 b(e)386 b Fz(J)510 b FF(though)-36 b(t)385 b(of)j(as)f(a)g(sc)
-36 b(heme)386 b(o)-36 b(v)g(er)387 b FD(Z)26582 -2970
y Fy(`)27022 -3169 y FF(.)563 b(Grothendiec)-36 b(k)386
b(sho)-36 b(w)g(ed)386 b(that)g Fz(J)p 43140 -2957 V
510 w FF(is)h(the)f(sp)36 b(ectrum)1263 -1564 y(of)431
b(a)f(free)h(\257nite)e FD(Z)10544 -1365 y Fy(`)10984
-1564 y FF(-mo)36 b(dule.)577 b(Ra)-36 b(ynaud)430 b(\(1974\))h(sho)-36
b(w)g(ed)430 b(that)f(if)i Fz(`)369 b Fw(6)p FF(=)g(2)430
b(then)f(essen)-36 b(tially)431 b(ev)-36 b(erything)1263
41 y(ab)36 b(out)472 b Fz(J)p 5023 253 V 123 w FF([)p
Fz(`)p FF(])h(can)f(b)36 b(e)472 b(seen)h(in)f(terms)g(of)h
Fz(J)123 b FF([)p Fz(`)p FF(]\()p 24054 -1026 1031 54
v Fz(Q)25084 354 y Fy(`)25524 41 y FF(\).)695 b(He)472
b(go)36 b(es)473 b(on)g(to)f(construct)g(group)g(sc)-36
b(heme)472 b Fz(V)p 49454 253 1048 54 v 761 w FF(and)1263
1646 y Fz(W)p 1263 1858 1409 54 v 614 w FF(o)-36 b(v)g(er)434
b FD(Z)6800 1845 y Fy(`)7674 1646 y FF(suc)-36 b(h)433
b(that)22931 3251 y Fz(V)p 22931 3463 1048 54 v 658 w
Fw(\275)370 b Fz(W)p 25750 3463 1409 54 v 549 w Fw(\275)f
Fz(J)p 28929 3463 842 54 v 123 w FF([)p Fz(`)p FF(])p
Fz(:)3214 5276 y FF(Our)504 b(goal)i(is)e(to)h(pro)-36
b(v)g(e)505 b(that)f(the)g(inclusion)h Fz(V)779 b(,)-221
b Fw(!)490 b Fz(W)685 b FF(of)505 b(Galois)h(mo)36 b(dules)504
b(is)h(an)f(isomorphism.)1263 6881 y(Ra)-36 b(ynaud)458
b(sho)-36 b(w)g(ed)458 b(noted)f(that)h(the)f(category)i(of)g(\257nite)
f(\260at)f(group)h(sc)-36 b(hemes)458 b(o)-36 b(v)g(er)458
b FD(Z)44925 7080 y Fy(`)45823 6881 y FF(is)g(an)g(ab)36
b(elian)1263 8486 y(category)517 b(so)f(the)g(cok)-36
b(ernel)516 b Fz(Q)p 15726 8957 1031 54 v 509 w FF(=)509
b Fz(W)p 18786 8698 1409 54 v 180 w(=V)p 20844 8698 1048
54 v 806 w FF(is)516 b(de\257ned.)823 b(F)-108 b(urthermore,)535
b Fz(V)798 b FF(=)509 b Fz(W)696 b FF(i\256)516 b Fz(Q)p
43664 8957 1031 54 v 509 w FF(=)509 b(0.)825 b(Since)516
b Fz(Q)p 52039 8957 V 1263 10091 a FF(is)448 b(\260at)g
Fz(Q)p 4912 10562 V 579 x Fv(F)6623 10826 y Ft(`)7526
10091 y FF(has)f(the)g(same)i(dimension)e(o)-36 b(v)g(er)448
b FD(F)25285 10290 y Fy(`)26173 10091 y FF(as)g Fz(Q)p
27784 10562 V 579 x Fv(Q)29629 10826 y Ft(`)30532 10091
y FF(has)g(o)-36 b(v)g(er)448 b FD(Q)36784 10290 y Fy(`)37224
10091 y FF(.)621 b(P)-36 b(assing)448 b(to)g(c)-36 b(haracteristic)448
b Fz(`)1263 11876 y FF(yields)434 b(an)g(exact)g(sequence)18756
13481 y(0)369 b Fw(!)g Fz(V)p 21472 13693 1048 54 v 22519
13802 a Fv(F)23200 13958 y Ft(`)24024 13481 y Fw(!)h
Fz(W)p 25722 13693 1409 54 v 27130 13802 a Fv(F)27811
13958 y Ft(`)28635 13481 y Fw(!)f Fz(Q)p 30332 13951
1031 54 v 31363 14060 a Fv(F)32044 14216 y Ft(`)32867
13481 y Fw(!)h FF(0)p Fz(:)1263 15669 y FF(Th)-36 b(us)500
b Fz(V)772 b(,)-221 b Fw(!)483 b Fz(W)680 b FF(is)501
b(an)f(isomorphism)h(i\256)f Fz(V)p 22551 15881 1048
54 v 23598 15990 a Fv(F)24279 16146 y Ft(`)25217 15669
y Fz(,)-221 b Fw(!)482 b Fz(W)p 27167 15881 1409 54 v
28576 15990 a Fv(F)29257 16146 y Ft(`)30212 15669 y FF(is)501
b(an)f(isomorphism.)779 b(Since)499 b Fz(V)p 45230 15881
1048 54 v 290 w FF(,)517 b Fz(W)p 47156 15881 1409 54
v 181 w FF(,)g(and)500 b Fz(Q)p 52039 16140 1031 54 v
1263 17274 a FF(ha)-36 b(v)g(e)498 b(an)g(action)h(of)g
Fz(k)523 b FF(=)479 b FD(T)p Fz(=)p FD(m)499 b FF(that)e(are)h
Fz(k)45 b FF(-v)-36 b(ector)498 b(space)g(sc)-36 b(hemes.)772
b(This)498 b(leads)h(us)e(to)h(Dieudonn)-36 b(\266)-614
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