Sharedwww / job / Short.psOpen in CoCalc
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cleartomark

%%EndFont 
%%BeginFont: WNCYR10
%!PS-AdobeFont-1.1: WNCYR10 
%%CreationDate: 1992 Aug 30 15:56:44

% Cyrillic fonts were designed by the Humanities and Arts Computing Center at the University of Washington.
% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (WNCYR10) readonly def
/FamilyName (AMS) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /WNCYR10 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 88 /X put
readonly def
/FontBBox{-24 -264 1151 921}readonly def
/UniqueXX 5032002 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 65 /A 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: 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 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 182 /acute put
dup 35 /numbersign 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 54 /six put
dup 55 /seven 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
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dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
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dup 97 /a put
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dup 99 /c put
dup 100 /d put
dup 101 /e put
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dup 103 /g put
dup 104 /h put
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dup 108 /l put
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dup 110 /n put
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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{-34 -251 988 750}readonly def
/UniqueXX 5000794 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 173 /Omega put
dup 174 /alpha put
dup 175 /beta put
dup 176 /gamma put
dup 182 /iota put
dup 189 /rho put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 62 /greater put
dup 65 /A put
dup 66 /B put
dup 71 /G put
dup 74 /J put
dup 76 /L put
dup 78 /N put
dup 88 /X put
dup 96 /lscript put
dup 99 /c put
dup 110 /n put
dup 112 /p put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueXX 5087386 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 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 182 /acute put
dup 189 /ae put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 66 /B put
dup 67 /C put
dup 71 /G put
dup 73 /I put
dup 77 /M put
dup 79 /O put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j 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 121 /y put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 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 173 /circlemultiply put
dup 174 /circledivide put
dup 175 /circledot put
dup 176 /circlecopyrt put
dup 182 /reflexsuperset put
dup 189 /propersubset put
dup 95 /logicalor put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueXX 5000818 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 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 182 /acute put
dup 189 /ae put
dup 48 /zero put
dup 53 /five put
dup 111 /o put
dup 114 /r put
dup 116 /t put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueXX 5000791 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 173 /angbracketleftbig put
dup 174 /angbracketrightbig put
dup 175 /vextendsingle put
dup 176 /vextenddouble put
dup 182 /parenrightbigg put
dup 189 /braceleftbigg put
dup 81 /producttext put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 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 173 /Omega put
dup 174 /alpha put
dup 175 /beta put
dup 176 /gamma put
dup 182 /iota put
dup 189 /rho put
dup 65 /A put
dup 110 /n put
dup 112 /p put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueXX 5087383 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57

% 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 (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 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 162 /periodcentered put
dup 173 /circlemultiply put
dup 174 /circledivide put
dup 175 /circledot put
dup 176 /circlecopyrt put
dup 182 /reflexsuperset put
dup 189 /propersubset put
dup 33 /arrowright put
dup 54 /negationslash put
dup 92 /intersection put
dup 106 /bar put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueXX 5000820 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBXTI10
%!PS-AdobeFont-1.1: CMBXTI10 1.0
%%CreationDate: 1991 Aug 18 17:46:30

% 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 (CMBXTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMBXTI10 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 162 /Delta put
dup 173 /Omega put
dup 174 /ff put
dup 175 /fi put
dup 176 /fl put
dup 182 /acute put
dup 189 /ae put
dup 46 /period put
dup 65 /A put
dup 82 /R put
dup 83 /S put
dup 87 /W put
dup 97 /a put
dup 99 /c put
dup 101 /e put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 121 /y put
readonly def
/FontBBox{-29 -250 1274 754}readonly def
/UniqueXX 5000771 def
currentdict end
currentfile eexec
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b(h)434 b(are)f(higher)h(dimensional)g(analogues)g(of)g(elliptic)h
(curv)-36 b(es.)1951 36373 y(As)400 b(a)h(\257rst)f(step,)407
b(I)401 b(ha)-36 b(v)g(e)401 b(obtained)f(theorems)g(that)h(mak)-36
b(e)401 b(p)36 b(ossible)401 b(computation)f(of)i(some)e(of)i(the)0
37978 y(arithmetic)393 b(in)-36 b(v)-72 b(arian)-36 b(ts)392
b(of)i(mo)36 b(dular)393 b(ab)36 b(elian)393 b(v)-72
b(arieties.)566 b(My)393 b(ob)72 b(jectiv)-36 b(e)394
b(is)f(to)g(\257nd)f(w)-36 b(a)g(ys)394 b(to)f(explicitly)0
39583 y(compute)448 b(all)j(of)f(the)e(arithmetic)i(in)-36
b(v)-72 b(arian)-36 b(ts.)625 b(Cremona)450 b(has)f(en)-36
b(umerated)448 b(these)h(in)-36 b(v)-72 b(arian)-36 b(ts)449
b(for)h(the)0 41188 y(\257rst)537 b(few)h(thousand)e(elliptic)j(curv)
-36 b(es,)563 b(and)538 b(I)f(am)h(w)-36 b(orking)539
b(to)e(do)h(the)f(same)h(for)g(ab)36 b(elian)538 b(v)-72
b(arieties.)0 42793 y(It)639 b(is)h(hop)36 b(ed)639 b(that)g(this)g(w)
-36 b(ork)641 b(will)f(con)-36 b(tin)g(ue)639 b(to)h(yield)g
(theoretical)g(results.)1196 b(I)640 b(am)f(also)i(writing)0
44398 y(mo)36 b(dular)341 b(forms)g(soft)-36 b(w)g(are)342
b(that)e(I)h(hop)36 b(e)341 b(will)h(b)36 b(e)341 b(used)f(b)-36
b(y)341 b(man)-36 b(y)341 b(mathematicians)g(and)g(ha)-36
b(v)g(e)341 b(practical)0 46003 y(applications)434 b(in)f(the)g(dev)-36
b(elopmen)g(t)433 b(of)i(elliptic)f(curv)-36 b(e)433
b(cryptosystems.)1951 47608 y(My)609 b(long-range)g(goal)h(is)f(to)g
(giv)-36 b(e)610 b(a)g(general)f(h)-36 b(yp)36 b(othesis,)653
b(v)-72 b(alid)609 b(for)h(in\257nitely)f(man)-36 b(y)609
b(ab)36 b(elian)0 49213 y(v)-72 b(arieties,)678 b(under)626
b(whic)-36 b(h)628 b(the)g(full)g(BSD)g(conjecture)g(holds.)1161
b(My)629 b(approac)-36 b(h)627 b(in)-36 b(v)g(olv)g(es)629
b(com)-36 b(bining)0 50818 y(Euler)505 b(system)h(tec)-36
b(hniques)505 b(of)i(K.)f(Kato)g(and)f(K.)h(Rubin)f(with)h(visibilit)
-36 b(y)507 b(and)e(congruence)h(ideas)g(of)0 52424 y(Mazur)433
b(and)g(Rib)36 b(et.)0 56829 y Fq(3)2152 b(Mo)60 b(dular)716
b(ab)60 b(elian)715 b(v)-120 b(arieties)0 58892 y Fp(My)412
b(primary)h(ob)72 b(jectiv)-36 b(e)413 b(is)g(to)f(v)-36
b(erify)413 b(the)f(BSD)g(conjecture)g(for)h(sp)36 b(eci\257c)412
b(mo)36 b(dular)412 b(ab)36 b(elian)413 b(v)-72 b(arieties,)0
60497 y(b)-36 b(y)433 b(using)h(the)f(ric)-36 b(h)433
b(theory)g(of)i(their)e(arithmetic.)1951 62102 y(The)365
b(BSD)h(conjecture)g(asserts)f(that)h(if)g Fo(A)g Fp(is)g(a)g(mo)36
b(dular)366 b(ab)36 b(elian)366 b(v)-72 b(ariet)-36 b(y)366
b(with)g Fo(L)p Fp(\()p Fo(A;)221 b Fp(1\))370 b Fn(6)p
Fp(=)e(0,)380 b(then)16390 64656 y Fo(L)p Fp(\()p Fo(A;)221
b Fp(1\))p 16390 65249 4105 54 v 17592 66466 a(\255)18531
66665 y Fm(A)20996 65555 y Fp(=)24712 64656 y(#)p Fl(X)o
Fp(\()p Fo(A)p Fp(\))295 b Fn(\242)30287 63660 y Fk(Q)31763
64656 y Fo(c)32323 64855 y Fm(p)p 22510 65249 12545 54
v 22510 66466 a Fp(#)p Fo(A)p Fp(\()p Fj(Q)p Fp(\))26703
66665 y Fi(tor)28256 66466 y Fn(\242)g Fp(#)p Fo(A)30979
66082 y Fh(_)31662 66466 y Fp(\()p Fj(Q)p Fp(\))33796
66665 y Fi(tor)35187 65555 y Fo(:)0 69020 y Fp(Here)369
b Fo(A)p Fp(\()p Fj(Q)p Fp(\))6115 69219 y Fi(tor)7743
69020 y Fp(is)g(the)g(group)g(of)g(rational)h(torsion)g(p)36
b(oin)-36 b(ts)369 b(on)g Fo(A)p Fp(;)391 b(the)368 b(Shafarevic)-36
b(h-T)-108 b(ate)370 b(group)f Fl(X)p Fp(\()p Fo(A)p
Fp(\))0 70625 y(is)536 b(a)h(measure)e(of)i(the)f(failure)h(of)g(the)e
(lo)36 b(cal-to-global)538 b(principle;)587 b(the)536
b(T)-108 b(amaga)-36 b(w)g(a)538 b(n)-36 b(um)g(b)36
b(ers)534 b Fo(c)49006 70824 y Fm(p)50071 70625 y Fp(are)0
72230 y(the)446 b(orders)h(of)h(certain)f(comp)36 b(onen)-36
b(t)447 b(groups)g(asso)36 b(ciated)448 b(to)f Fo(A)p
Fp(;)455 b(the)447 b(real)g(n)-36 b(um)g(b)36 b(er)446
b(\255)42976 72429 y Fm(A)44184 72230 y Fp(is)h(the)g(v)-36
b(olume)0 73836 y(of)460 b Fo(A)p Fp(\()p Fj(R)p Fp(\))e(with)i(resp)36
b(ect)458 b(to)i(a)f(basis)h(of)g(di\256eren)-36 b(tials)459
b(ha)-36 b(ving)460 b(ev)-36 b(erywhere)460 b(nonzero)f(go)36
b(o)g(d)460 b(reduction;)0 75441 y(and)433 b Fo(A)3504
74959 y Fh(_)4620 75441 y Fp(is)h(the)f(dual)g(of)i Fo(A)p
Fp(.)p eop
%%Page: 2 2
2 1 bop -543 -2596 a Fs(Wil)135 b(liam)953 b(A.)f(Stein)10197
b(R)-135 b(ese)g(ar)g(ch)951 b(Summary)p 0 842 51806
45 v 0 3909 a Fg(3.1)1792 b(The)599 b(ratio)f Ff(L)p
Fe(\()p Ff(A;)266 b Fe(1\))p Ff(=)p Fe(\255)18871 4148
y Fd(A)0 5800 y Fp(Extending)523 b(Manin's)g(w)-36 b(ork)524
b(on)f(elliptic)g(curv)-36 b(es,)546 b(A.)524 b(Agash)-36
b(\266)-614 b(e)523 b(and)g(I)g(found)g(a)g(computable)g(form)-36
b(ula)0 7405 y(for)514 b(the)f(rational)i(n)-36 b(um)g(b)36
b(er)512 b Fo(L)p Fp(\()p Fo(A;)221 b Fp(1\))p Fo(=)p
Fp(\255)19815 7604 y Fm(A)20577 7405 y Fp(.)819 b(Using)514
b(similar)h(tec)-36 b(hniques,)534 b(I)514 b(hop)36 b(e)513
b(to)h(\257nd)f(computable)0 9010 y(form)-36 b(ulas)348
b(for)g(rational)h(parts)e(of)h(sp)36 b(ecial)348 b(v)-72
b(alues)349 b(of)f(t)-36 b(wists,)365 b(and)347 b(of)i
Fo(L)p Fp(-functions)e(attac)-36 b(hed)347 b(to)g(forms)i(of)0
10615 y(w)-36 b(eigh)g(t)427 b(greater)g(than)f(t)-36
b(w)g(o.)576 b(I)427 b(ha)-36 b(v)g(e)427 b(already)g(computed)f
Fo(L)p Fp(\()p Fo(A;)221 b Fp(1\))p Fo(=)p Fp(\255)34654
10814 y Fm(A)35842 10615 y Fp(for)427 b(sev)-36 b(eral)428
b(thousand)e(ab)36 b(elian)0 12220 y(v)-72 b(arieties,)435
b(and)e(hop)36 b(e)433 b(to)h(extend)f(these)g(computations.)0
16072 y Fg(3.2)1792 b(The)599 b(T)-149 b(amaga)-50 b(w)g(a)597
b(n)-50 b(um)g(b)50 b(ers)600 b Ff(c)24474 16311 y Fd(p)0
17963 y Fp(When)447 b Fo(A)g Fp(has)h(semistable)g(reduction)e(at)i
Fo(p)p Fp(,)j(I)d(ha)-36 b(v)g(e)447 b(found)g(a)h(w)-36
b(a)g(y)448 b(to)g(explicitly)h(compute)e(the)g(n)-36
b(um-)0 19568 y(b)36 b(er)433 b Fo(c)2836 19767 y Fm(p)3365
19568 y Fp(,)h(up)e(to)i(a)g(p)36 b(o)-36 b(w)g(er)433
b(of)i(2.)578 b(I)434 b(hop)36 b(e)434 b(to)f(\257nd)g(a)g(w)-36
b(a)g(y)435 b(to)f(compute)f Fo(c)34960 19767 y Fm(p)35922
19568 y Fp(in)g(the)g(remaining)h(cases.)0 23419 y Fg(3.3)1792
b(Bounding)598 b Fe(#)p Fc(X)0 25310 y Fp(V.)j(Kolyv)-72
b(agin)602 b(and)e(K.)h(Kato)h(obtained)e(upp)36 b(er)599
b(b)36 b(ounds)600 b(on)g(#)p Fl(X)p Fp(\()p Fo(A)p Fp(\).)1079
b(T)-108 b(o)601 b(v)-36 b(erify)602 b(the)e(full)h(BSD)0
26915 y(conjecture)g(for)h(certain)f(ab)36 b(elian)602
b(v)-72 b(arieties,)644 b(it)601 b(is)h(necessary)g(is)f(to)g(mak)-36
b(e)602 b(these)f(b)36 b(ounds)600 b(explicit.)0 28520
y(Kolyv)-72 b(agin's)460 b(b)36 b(ounds)458 b(in)-36
b(v)g(olv)g(e)459 b(computations)f(with)h(Heegner)f(p)36
b(oin)-36 b(ts,)464 b(and)458 b(Kato's)i(in)-36 b(v)g(olv)g(e)459
b(a)g(study)0 30126 y(of)341 b(the)g(Galois)h(represen)-36
b(tations)340 b(asso)36 b(ciated)342 b(to)f Fo(A)p Fp(.)547
b(I)342 b(plan)e(to)h(carry)h(out)f(suc)-36 b(h)340 b(computations)g
(in)h(man)-36 b(y)0 31731 y(sp)36 b(eci\257c)433 b(cases.)1951
33336 y(One)469 b(approac)-36 b(h)469 b(to)i(sho)-36
b(wing)470 b(that)g Fl(X)p Fp(\()p Fo(A)p Fp(\))f(is)i(as)f(large)h(as)
f(predicted)f(b)-36 b(y)470 b(the)f(BSD)h(conjecture)g(is)0
34941 y(suggested)533 b(b)-36 b(y)533 b(Mazur's)g(notion)g(of)h(the)f
(visible)h(part)f(of)h Fl(X)p Fp(\()p Fo(A)p Fp(\).)877
b(Consider)533 b(an)g(ab)36 b(elian)534 b(v)-72 b(ariet)-36
b(y)534 b Fo(A)0 36546 y Fp(that)468 b(sits)g(naturally)h(in)f(the)g
(Jacobian)h Fo(J)20867 36745 y Fi(0)21393 36546 y Fp(\()p
Fo(N)139 b Fp(\))467 b(of)i(the)f(mo)36 b(dular)468 b(curv)-36
b(e)468 b Fo(X)37627 36745 y Fi(0)38153 36546 y Fp(\()p
Fo(N)139 b Fp(\).)682 b(The)468 b Fb(visible)496 b(p)-66
b(art)468 b Fp(of)0 38151 y Fl(X)p Fp(\()p Fo(A)p Fp(\))498
b(is)h(the)g(collection)h(of)f(those)g(elemen)-36 b(ts)499
b(of)g Fl(X)p Fp(\()p Fo(A)p Fp(\))g(that)f(go)i(to)f(0)g(under)f(the)g
(natural)h(map)f(to)0 39756 y Fl(X)p Fp(\()p Fo(J)2771
39955 y Fi(0)3296 39756 y Fp(\()p Fo(N)139 b Fp(\)\).)581
b(Cremona)435 b(and)f(Mazur)g(observ)-36 b(ed)435 b(that)f(if)i(an)e
(elemen)-36 b(t)435 b(of)g(order)f Fo(p)h Fp(in)g Fl(X)p
Fp(\()p Fo(A)p Fp(\))f(is)h(visible,)0 41361 y(then)372
b(it)h(is)g(explained)g(b)-36 b(y)373 b(a)g(jump)g(in)f(the)h(rank)g
(of)g(Mordell-W)-108 b(eil,)386 b(in)373 b(the)f(sense)h(that)f(there)h
(is)g(another)0 42966 y(ab)36 b(elian)376 b(sub)-36 b(v)-72
b(ariet)-36 b(y)376 b Fo(B)436 b Fn(\275)369 b Fo(J)14167
43165 y Fi(0)14693 42966 y Fp(\()p Fo(N)139 b Fp(\))375
b(suc)-36 b(h)375 b(that)g Fo(p)369 b Fn(j)g Fp(#\()p
Fo(A)177 b Fn(\\)g Fo(B)67 b Fp(\))376 b(and)g Fo(B)443
b Fp(has)375 b(man)-36 b(y)377 b(rational)f(p)36 b(oin)-36
b(ts.)559 b(I)376 b(am)0 44571 y(trying)502 b(to)g(\257nd)f(the)g
(precise)h(degree)g(to)g(whic)-36 b(h)502 b(this)f(observ)-72
b(ation)503 b(can)f(b)36 b(e)501 b(turned)g(around:)714
b(if)503 b(there)0 46176 y(is)435 b(another)f(ab)36 b(elian)435
b(v)-72 b(ariet)-36 b(y)436 b Fo(B)501 b Fp(with)435
b(man)-36 b(y)435 b(rational)g(p)36 b(oin)-36 b(ts)435
b(and)f Fo(p)370 b Fn(j)h Fp(#\()p Fo(A)296 b Fn(\\)f
Fo(B)67 b Fp(\),)435 b(then)f(under)f(what)0 47782 y(h)-36
b(yp)36 b(otheses)433 b(is)h(there)f(an)g(elemen)-36
b(t)433 b(of)i Fl(X)p Fp(\()p Fo(A)p Fp(\))e(of)h(order)f
Fo(p)p Fp(?)0 52219 y Fq(4)2152 b(Icosahedral)716 b(Galois)h(represen)
-60 b(tations)0 54281 y Fp(E.)301 b(Artin)g(conjectured)g(that)f(the)h
Fo(L)p Fp(-series)g(asso)36 b(ciated)302 b(to)g(an)-36
b(y)301 b(con)-36 b(tin)g(uous)300 b(irreducible)h(represen)-36
b(tation)0 55886 y Fo(\275)369 b Fp(:)g Fo(G)2796 56085
y Fa(Q)4035 55886 y Fn(!)g Fp(GL)7566 56085 y Fm(n)8192
55886 y Fp(\()p Fj(C)p Fp(\),)383 b(with)370 b Fo(n)g(>)e
Fp(1,)384 b(is)371 b(en)-36 b(tire.)556 b(Recen)-36 b(t)370
b(exciting)i(w)-36 b(ork)371 b(of)g(T)-108 b(a)-36 b(ylor)371
b(and)f(others)g(suggests)0 57491 y(that)444 b(a)h(complete)f(pro)36
b(of)445 b(of)g(Artin's)g(conjecture,)i(in)e(the)e(case)i(when)g
Fo(n)387 b Fp(=)h(2)444 b(and)g Fo(\275)h Fp(is)g(o)36
b(dd,)447 b(is)d(on)h(the)0 59097 y(horizon.)1951 60702
y(By)307 b(com)-36 b(bining)306 b(the)g(main)g(result)h(of)g(a)g(recen)
-36 b(t)305 b(pap)36 b(er)306 b(of)h(K.)g(Buzzard)f(and)g(T)-108
b(a)-36 b(ylor)308 b(with)e(a)h(computer)0 62307 y(computation,)525
b(Buzzard)506 b(and)h(I)g(recen)-36 b(tly)506 b(pro)-36
b(v)g(ed)507 b(that)f(the)h(icosahedral)g(Artin)g(represen)-36
b(tations)506 b(of)0 63912 y(conductor)411 b(1376)370
b(=)f(2)11049 63430 y Fi(5)11826 63912 y Fn(\242)251
b Fp(43)412 b(are)h(mo)36 b(dular.)571 b(If)412 b(I)g(can)g(extend)g(a)
g(congruence)f(result)h(of)h(J.)f(Sturm,)j(then)0 65517
y(our)381 b(metho)36 b(d)380 b(will)i(yield)g(sev)-36
b(eral)382 b(more)f(examples.)562 b(These)381 b(ongoing)h(computations)
f(are)g(la)-36 b(ying)382 b(a)g(part)0 67122 y(of)325
b(the)f(foundation)g(necessary)h(for)g(a)g(full)g(pro)36
b(of)326 b(of)f(the)f(Artin)g(conjecture)g(for)h(o)36
b(dd)325 b(t)-36 b(w)g(o-dimensional)324 b Fo(\275)p
Fp(,)0 68727 y(as)361 b(w)-36 b(ell)362 b(as)f(stim)-36
b(ulating)361 b(the)g(dev)-36 b(elopmen)g(t)360 b(of)i(new)f
(algorithms)h(for)f(computing)g(with)g(mo)36 b(dular)361
b(forms)0 70332 y(in)433 b(c)-36 b(haracteristic)434
b Fo(`)p Fp(.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF