CoCalc -- Collaborative Calculation in the Cloud
Sharedwww / Tables / msri-intro.psOpen in CoCalc
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%
% Original Metafont design  Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation     Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility  Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997  Kristoffer H. Rose  <[email protected]>
%
% The Xy-pic macro package is free software; you can redistribute it 
% and/or modify it under the terms of the GNU General Public License 
% as published by the Free Software Foundation; either version 2  
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will 
% be useful, but WITHOUT ANY WARRANTY; without even the implied 
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License 
% along with this macro package; if not, write to the 
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.

11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def
/FullName (XYBTIP10) readonly def
/FamilyName (XYBTIP) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYBTIP10 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 37 /d37 put
dup 121 /d121 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueXX 5092839 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: XYATIP10
%!PS-AdobeFont-1.1: XYATIP10 001.104
%%CreationDate: 1997 Jul 20 21:19:17
%%RevisionDate: 1997 Sep 14 19:58:47
%
% XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style".
%
% Original Metafont design  Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation     Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility  Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997  Kristoffer H. Rose  <[email protected]>
%
% The Xy-pic macro package is free software; you can redistribute it 
% and/or modify it under the terms of the GNU General Public License 
% as published by the Free Software Foundation; either version 2  
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will 
% be useful, but WITHOUT ANY WARRANTY; without even the implied 
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License 
% along with this macro package; if not, write to the 
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.

11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def
/FullName (XYATIP10) readonly def
/FamilyName (XYATIP) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYATIP10 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 37 /d37 put
dup 121 /d121 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueXX 5092838 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
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$F2psEnd
rs

%%EndDocument
 @endspecial 1275 1309 a Fo(H)1344 1321 y Fp(1)1381 1309
y Fx(\()p Fo(X)1482 1321 y Fp(0)1519 1309 y Fx(\(39\))p
Fo(;)14 b Fn(Z)p Fx(\))1817 1287 y Fk(\030)1817 1313
y Fx(=)1905 1309 y Fn(Z)19 b Fk(\002)f Fn(Z)h Fk(\002)f
Fn(Z)g Fk(\002)g Fn(Z)h Fk(\002)f Fn(Z)h Fk(\002)f Fn(Z)1378
1575 y Fx(Figure)27 b(1:)36 b(The)28 b(homology)e(of)h
Fo(X)2451 1587 y Fp(0)2488 1575 y Fx(\(39\).)639 1844
y(The)c(\014rst)f(homology)e(group)h Fo(H)1640 1856 y
Fp(1)1678 1844 y Fx(\()p Fo(X)1779 1856 y Fp(0)1816 1844
y Fx(\()p Fo(N)9 b Fx(\))p Fo(;)14 b Fn(Z)p Fx(\))23
b(is)f(the)h(group)e(of)h(singular)g(1-cycles)e(mo)r(dulo)515
1944 y(homology)27 b(relations.)41 b(Recall)28 b(that)i(top)r
(ologically)d Fo(X)2259 1956 y Fp(0)2296 1944 y Fx(\()p
Fo(N)9 b Fx(\))30 b(is)f(a)g Fo(g)s Fx(-holed)f(torus,)h(where)g
Fo(g)i Fx(is)515 2044 y(the)d(gen)n(us)g(of)g Fo(X)1052
2056 y Fp(0)1089 2044 y Fx(\()p Fo(N)9 b Fx(\).)40 b(The)28
b(group)f Fo(H)1768 2056 y Fp(1)1806 2044 y Fx(\()p Fo(X)1907
2056 y Fp(0)1944 2044 y Fx(\()p Fo(N)9 b Fx(\))p Fo(;)14
b Fn(Z)p Fx(\))29 b(is)f(th)n(us)h(a)f(free)g(ab)r(elian)g(group)f(of)h
(rank)515 2143 y(2)p Fo(g)38 b Fx(\(see,)h(e.g.,)f([8,)h(Ex.)d
(19.30]\),)h(with)g(t)n(w)n(o)f(generators)e(corresp)r(onding)g(to)j
(eac)n(h)f(hole,)i(as)515 2243 y(illustrated)27 b(in)h(the)g(case)f
Fo(N)k Fx(=)23 b(39)k(in)h(Diagram)e(1.)639 2343 y(The)i(Hec)n(k)n(e)f
(op)r(erators)f Fo(T)1467 2355 y Fi(p)1533 2343 y Fx(act)i(on)f
Fo(H)1856 2355 y Fp(1)1894 2343 y Fx(\()p Fo(X)1995 2355
y Fp(0)2032 2343 y Fx(\()p Fo(N)9 b Fx(\))p Fo(;)14 b
Fn(Z)p Fx(\),)29 b(and)f(in)n(tegration)e(de\014nes)i(a)g(nonde-)515
2442 y(generate)e(Hec)n(k)n(e-equiv)-5 b(arian)n(t)25
b(pairing)1388 2625 y Fk(h)14 b Fo(;)27 b Fk(i)d Fx(:)f
Fo(S)1637 2637 y Fp(2)1674 2625 y Fx(\()p Fo(N)9 b Fx(\))19
b Fk(\002)f Fo(H)1985 2637 y Fp(1)2022 2625 y Fx(\()p
Fo(X)2123 2637 y Fp(0)2160 2625 y Fx(\()p Fo(N)9 b Fx(\))p
Fo(;)14 b Fn(Z)p Fx(\))24 b Fk(!)f Fn(C)p Fo(:)515 2807
y Fx(Explicitly)-7 b(,)27 b(for)g(a)h(path)f Fo(x)p Fx(,)1616
2945 y Fk(h)p Fo(f)t(;)14 b(x)p Fk(i)24 b Fx(=)e(2)p
Fo(\031)s(i)2055 2832 y Fj(Z)2101 3021 y Fi(x)2156 2945
y Fo(f)9 b Fx(\()p Fo(z)t Fx(\))p Fo(dz)t(;)515 3136
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h(of)g(Manin)f(that)i(sho)n(ws)d(that)j(the)f(space)f(of)h(mo)r(dular)
515 1556 y(sym)n(b)r(ols)e(can)g(b)r(e)h(computed.)639
1656 y(By)20 b(reducing)f(mo)r(dulo)h Fo(N)9 b Fx(,)22
b(one)d(sees)h(that)g(the)g(group)f(\000)2393 1668 y
Fp(0)2430 1656 y Fx(\()p Fo(N)9 b Fx(\))21 b(has)e(\014nite)i(index)f
(in)g(SL)3339 1668 y Fp(2)3376 1656 y Fx(\()p Fn(Z)p
Fx(\).)515 1756 y(Let)g Fo(r)693 1768 y Fp(0)731 1756
y Fo(;)14 b(r)805 1768 y Fp(1)842 1756 y Fo(;)g(:)g(:)g(:)28
b(;)14 b(r)1078 1768 y Fi(m)1162 1756 y Fx(b)r(e)20 b(distinct)h(righ)n
(t)f(coset)f(represen)n(tativ)n(es)f(for)i(\000)2685
1768 y Fp(0)2722 1756 y Fx(\()p Fo(N)9 b Fx(\))21 b(in)f(SL)3070
1768 y Fp(2)3108 1756 y Fx(\()p Fn(Z)p Fx(\),)i(so)e(that)1125
1938 y(SL)1223 1950 y Fp(2)1260 1938 y Fx(\()p Fn(Z)p
Fx(\))k(=)f(\000)1546 1950 y Fp(0)1583 1938 y Fx(\()p
Fo(N)9 b Fx(\))p Fo(r)1760 1950 y Fi(o)1830 1938 y Fk([)33
b Fx(\000)1970 1950 y Fp(0)2007 1938 y Fx(\()p Fo(N)9
b Fx(\))p Fo(r)2184 1950 y Fp(1)2254 1938 y Fk([)33 b(\001)14
b(\001)g(\001)32 b([)h Fx(\000)2611 1950 y Fp(0)2648
1938 y Fx(\()p Fo(N)9 b Fx(\))p Fo(r)2825 1950 y Fi(m)2889
1938 y Fo(;)515 2121 y Fx(where)26 b(the)g(union)h(is)f(disjoin)n(t.)37
b(F)-7 b(or)26 b(example,)g(when)h Fo(N)35 b Fx(is)26
b(prime,)h(a)f(list)h(of)f(coset)g(represen-)515 2220
y(tativ)n(es)h(is)807 2315 y Fj(\022)868 2381 y Fx(1)82
b(0)868 2481 y(0)g(1)1034 2315 y Fj(\023)1109 2432 y
Fo(;)1146 2315 y Fj(\022)1207 2381 y Fx(1)g(0)1207 2481
y(1)g(1)1373 2315 y Fj(\023)1448 2432 y Fo(;)1485 2315
y Fj(\022)1546 2381 y Fx(1)g(0)1546 2481 y(2)g(1)1712
2315 y Fj(\023)1787 2432 y Fo(;)1824 2315 y Fj(\022)1885
2381 y Fx(1)g(0)1885 2481 y(3)g(1)2051 2315 y Fj(\023)2126
2432 y Fo(;)14 b(:)g(:)g(:)27 b(;)2324 2315 y Fj(\022)2563
2381 y Fx(1)82 b(0)2385 2481 y Fo(N)28 b Fk(\000)18 b
Fx(1)82 b(1)2729 2315 y Fj(\023)2804 2432 y Fo(;)2841
2315 y Fj(\022)2902 2381 y Fx(0)g Fk(\000)p Fx(1)2902
2481 y(1)147 b(0)3132 2315 y Fj(\023)3207 2432 y Fo(:)515
2665 y Fx(In)25 b(general,)f(the)i(righ)n(t)e(cosets)g(of)h(\000)1644
2677 y Fp(0)1681 2665 y Fx(\()p Fo(N)9 b Fx(\))26 b(in)f(SL)2039
2677 y Fp(2)2076 2665 y Fx(\()p Fn(Z)p Fx(\))i(are)d(in)h(bijection)g
(with)h(the)f(elemen)n(ts)g(of)515 2764 y Fn(P)580 2734
y Fp(1)617 2764 y Fx(\()p Fn(Z)p Fo(=)-5 b(N)9 b Fn(Z)p
Fx(\))29 b(\(see)e([6,)h Fk(x)p Fx(2.2])e(for)h(complete)h(details\).)
639 2864 y(The)34 b(follo)n(wing)f(tric)n(k)g(of)g(Manin)h(\(see)g([12)
o(,)h Fk(x)p Fx(1.5])e(and)g([6,)i Fk(x)p Fx(2.1.6]\))e(allo)n(ws)f(us)
i(to)f(write)515 2963 y(ev)n(ery)c(mo)r(dular)h(sym)n(b)r(ol)g(as)g(a)g
Fn(Z)p Fx(-linear)g(com)n(bination)g(of)g(sym)n(b)r(ols)g(of)g(the)h
(form)g Fo(r)3227 2975 y Fi(i)3255 2963 y Fk(f)p Fx(0)p
Fo(;)14 b Fk(1g)p Fx(.)515 3063 y(In)28 b(particular,)e(the)i
(\014nitely)g(man)n(y)f(sym)n(b)r(ols)g Fo(r)2027 3075
y Fi(i)2055 3063 y Fk(f)p Fx(0)p Fo(;)14 b Fk(1g)26 b
Fx(generate)d Fk(M)2657 3061 y(M)2659 3063 y(M)2759 3075
y Fp(2)2796 3063 y Fx(\()p Fo(N)9 b Fx(\).)639 3163 y(Because)28
b(of)h(the)g(relation)e Fk(f)p Fo(\013;)14 b(\014)t Fk(g)25
b Fx(=)f Fk(f)p Fx(0)p Fo(;)14 b(\014)t Fk(g)k(\000)h(f)p
Fx(0)p Fo(;)14 b(\013)p Fk(g)p Fx(,)28 b(it)h(su\016ces)g(to)f
(consider)g(mo)r(dular)515 3262 y(sym)n(b)r(ols)36 b(of)h(the)g(form)g
Fk(f)p Fx(0)p Fo(;)14 b(b=a)p Fk(g)p Fx(,)37 b(where)f(the)h(rational)f
(n)n(um)n(b)r(er)h Fo(b=a)f Fx(is)g(in)i(lo)n(w)n(est)e(terms.)515
3362 y(Expand)20 b Fo(b=a)f Fx(as)h(a)g(con)n(tin)n(ued)g(fraction)g
(and)h(consider)e(the)i(successiv)n(e)e(con)n(v)n(ergen)n(ts)g(in)h(lo)
n(w)n(est)515 3462 y(terms:)989 3528 y Fo(b)1025 3540
y Fl(\000)p Fp(2)p 985 3565 134 4 v 985 3641 a Fo(a)1029
3653 y Fl(\000)p Fp(2)1151 3584 y Fx(=)1249 3528 y(0)p
1249 3565 42 4 v 1249 3641 a(1)1300 3584 y Fo(;)1434
3528 y(b)1470 3540 y Fl(\000)p Fp(1)p 1430 3565 134 4
v 1430 3641 a Fo(a)1474 3653 y Fl(\000)p Fp(1)1596 3584
y Fx(=)1694 3528 y(1)p 1694 3565 42 4 v 1694 3641 a(0)1745
3584 y Fo(;)1879 3528 y(b)1915 3540 y Fp(0)p 1875 3565
82 4 v 1875 3641 a Fo(a)1919 3653 y Fp(0)1989 3584 y
Fx(=)2087 3528 y Fo(b)2123 3540 y Fp(0)p 2087 3565 73
4 v 2102 3641 a Fx(1)2169 3584 y Fo(;)28 b(:)14 b(:)g(:)28
b(;)2479 3528 y(b)2515 3540 y Fi(n)p Fl(\000)p Fp(1)p
2475 3565 175 4 v 2475 3641 a Fo(a)2519 3653 y Fi(n)p
Fl(\000)p Fp(1)2659 3584 y Fo(;)2793 3528 y(b)2829 3540
y Fi(n)p 2789 3565 90 4 v 2789 3641 a Fo(a)2833 3653
y Fi(n)2911 3584 y Fx(=)3012 3528 y Fo(b)p 3008 3565
44 4 v 3008 3641 a(a)515 3776 y Fx(where)f(the)h(\014rst)f(t)n(w)n(o)g
(are)g(added)g(formally)-7 b(.)36 b(Then)1506 3958 y
Fo(b)1542 3970 y Fi(k)1583 3958 y Fo(a)1627 3970 y Fi(k)q
Fl(\000)p Fp(1)1771 3958 y Fk(\000)18 b Fo(b)1890 3970
y Fi(k)q Fl(\000)p Fp(1)2016 3958 y Fo(a)2060 3970 y
Fi(k)2123 3958 y Fx(=)23 b(\()p Fk(\000)p Fx(1\))2382
3924 y Fi(k)q Fl(\000)p Fp(1)2507 3958 y Fo(;)515 4141
y Fx(so)k(that)1353 4271 y Fo(g)1393 4283 y Fi(k)1457
4271 y Fx(=)1545 4153 y Fj(\022)1610 4220 y Fo(b)1646
4232 y Fi(k)1782 4220 y Fx(\()p Fk(\000)p Fx(1\))1953
4190 y Fi(k)q Fl(\000)p Fp(1)2078 4220 y Fo(b)2114 4232
y Fi(k)q Fl(\000)p Fp(1)1606 4320 y Fo(a)1650 4332 y
Fi(k)1773 4320 y Fx(\()p Fk(\000)p Fx(1\))1944 4290 y
Fi(k)q Fl(\000)p Fp(1)2070 4320 y Fo(a)2114 4332 y Fi(k)q
Fl(\000)p Fp(1)2240 4153 y Fj(\023)2324 4271 y Fk(2)c
Fx(SL)2500 4283 y Fp(2)2538 4271 y Fx(\()p Fn(Z)p Fx(\))p
Fo(:)515 4465 y Fx(Hence)1351 4477 y Fj(\032)1428 4538
y Fo(b)1464 4550 y Fi(k)q Fl(\000)p Fp(1)p 1423 4575
170 4 v 1423 4651 a Fo(a)1467 4663 y Fi(k)q Fl(\000)p
Fp(1)1603 4594 y Fo(;)1654 4538 y(b)1690 4550 y Fi(k)p
1650 4575 85 4 v 1650 4651 a Fo(a)1694 4663 y Fi(k)1745
4477 y Fj(\033)1830 4594 y Fx(=)g Fo(g)1958 4606 y Fi(k)1998
4594 y Fk(f)p Fx(0)p Fo(;)14 b Fk(1g)22 b Fx(=)g Fo(r)2390
4606 y Fi(i)2418 4594 y Fk(f)p Fx(0)p Fo(;)14 b Fk(1g)p
Fo(;)515 4788 y Fx(for)27 b(some)g Fo(i)p Fx(,)g(is)h(of)f(the)h
(required)f(sp)r(ecial)g(form.)1998 5255 y(6)p eop
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7 6 bop 515 523 a Fh(Example)30 b(2.2.)44 b Fx(Let)27
b Fo(N)32 b Fx(=)23 b(11,)k(and)g(consider)g(the)h(mo)r(dular)f(sym)n
(b)r(ol)g Fk(f)p Fx(0)p Fo(;)14 b Fx(4)p Fo(=)p Fx(7)p
Fk(g)p Fx(.)34 b(W)-7 b(e)28 b(ha)n(v)n(e)1702 689 y(4)p
1702 726 42 4 v 1702 802 a(7)1776 745 y(=)23 b(0)18 b(+)2144
689 y(1)p 2017 726 296 4 v 2017 806 a(1)g(+)2219 773
y Fp(1)p 2170 787 133 4 v 2170 839 a(1+)2264 817 y Ff(1)p
2264 826 29 3 v 2264 859 a(3)2322 745 y Fo(;)515 1008
y Fx(so)27 b(the)h(partial)e(con)n(v)n(ergen)n(ts)f(are)850
1173 y Fo(b)886 1185 y Fl(\000)p Fp(2)p 846 1210 134
4 v 846 1286 a Fo(a)890 1298 y Fl(\000)p Fp(2)1012 1229
y Fx(=)1109 1173 y(0)p 1109 1210 42 4 v 1109 1286 a(1)1161
1229 y Fo(;)1295 1173 y(b)1331 1185 y Fl(\000)p Fp(1)p
1291 1210 134 4 v 1291 1286 a Fo(a)1335 1298 y Fl(\000)p
Fp(1)1457 1229 y Fx(=)1554 1173 y(1)p 1554 1210 42 4
v 1554 1286 a(0)1606 1229 y Fo(;)1740 1173 y(b)1776 1185
y Fp(0)p 1736 1210 82 4 v 1736 1286 a Fo(a)1780 1298
y Fp(0)1850 1229 y Fx(=)1947 1173 y(0)p 1947 1210 42
4 v 1947 1286 a(1)1999 1229 y Fo(;)2133 1173 y(b)2169
1185 y Fp(1)p 2129 1210 82 4 v 2129 1286 a Fo(a)2173
1298 y Fp(1)2243 1229 y Fx(=)2341 1173 y(1)p 2341 1210
42 4 v 2341 1286 a(1)2392 1229 y Fo(;)2526 1173 y(b)2562
1185 y Fp(2)p 2522 1210 82 4 v 2522 1286 a Fo(a)2566
1298 y Fp(2)2636 1229 y Fx(=)2734 1173 y(1)p 2734 1210
42 4 v 2734 1286 a(2)2785 1229 y Fo(;)2919 1173 y(b)2955
1185 y Fp(3)p 2915 1210 82 4 v 2915 1286 a Fo(a)2959
1298 y Fp(3)3029 1229 y Fx(=)3127 1173 y(4)p 3127 1210
42 4 v 3127 1286 a(7)3178 1229 y Fo(:)639 1455 y Fx(Th)n(us)884
1637 y Fk(f)p Fx(0)p Fo(;)14 b Fx(4)p Fo(=)p Fx(7)p Fk(g)80
b Fx(=)i Fk(f)p Fx(0)p Fo(;)14 b Fk(1g)j Fx(+)h Fk(f1)p
Fo(;)c Fx(0)p Fk(g)j Fx(+)h Fk(f)p Fx(0)p Fo(;)c Fx(1)p
Fk(g)j Fx(+)h Fk(f)p Fx(1)p Fo(;)c Fx(1)p Fo(=)p Fx(2)p
Fk(g)h Fx(+)j Fk(f)p Fx(1)p Fo(=)p Fx(2)p Fo(;)c Fx(4)p
Fo(=)p Fx(7)p Fk(g)1253 1812 y Fx(=)1400 1695 y Fj(\022)1462
1761 y Fx(1)82 b Fk(\000)p Fx(1)1462 1861 y(2)g Fk(\000)p
Fx(1)1692 1695 y Fj(\023)1767 1812 y Fk(f)p Fx(0)p Fo(;)14
b Fk(1g)j Fx(+)2113 1695 y Fj(\022)2174 1761 y Fx(4)83
b(1)2174 1861 y(7)g(2)2340 1695 y Fj(\023)2415 1812 y
Fk(f)p Fx(0)p Fo(;)14 b Fk(1g)1253 2044 y Fx(=)82 b(2)18
b Fk(\001)1502 1927 y Fj(\024\022)1607 1993 y Fx(1)82
b(4)1607 2093 y(1)g(5)1773 1927 y Fj(\023)1848 2044 y
Fk(f)p Fx(0)p Fo(;)14 b Fk(1g)2094 1927 y Fj(\025)515
2317 y Fy(2.2)112 b(Manin)38 b(sym)m(b)s(ols)515 2470
y Fx(As)20 b(ab)r(o)n(v)n(e,)h(\014x)f(coset)g(represen)n(tativ)n(es)e
Fo(r)1790 2482 y Fp(0)1828 2470 y Fo(;)c(:)g(:)g(:)27
b(;)14 b(r)2063 2482 y Fi(m)2147 2470 y Fx(for)20 b(\000)2319
2482 y Fp(0)2356 2470 y Fx(\()p Fo(N)9 b Fx(\))21 b(in)f(SL)2704
2482 y Fp(2)2742 2470 y Fx(\()p Fn(Z)p Fx(\).)35 b(Denote)21
b(the)f(mo)r(d-)515 2570 y(ular)28 b(sym)n(b)r(ol)g Fo(r)1009
2582 y Fi(i)1037 2570 y Fk(f)p Fx(0)p Fo(;)14 b Fk(1g)27
b Fx(b)n(y)h([)p Fo(r)1486 2582 y Fi(i)1514 2570 y Fx(].)40
b(The)28 b(sym)n(b)r(ols)g([)p Fo(r)2149 2582 y Fp(0)2187
2570 y Fx(])p Fo(;)14 b(:)g(:)g(:)27 b(;)14 b Fx([)p
Fo(r)2468 2582 y Fi(m)2532 2570 y Fx(])29 b(are)e(called)h
Fh(Manin)j(symb)l(ols)p Fx(,)515 2669 y(and)c(they)g(are)f(equipp)r(ed)
i(with)g(a)f(righ)n(t)f(action)h(of)g(SL)2254 2681 y
Fp(2)2291 2669 y Fx(\()p Fn(Z)p Fx(\),)i(whic)n(h)e(is)g(giv)n(en)f(b)n
(y)h([)p Fo(r)3176 2681 y Fi(i)3204 2669 y Fx(])p Fo(g)f
Fx(=)d([)p Fo(r)3441 2681 y Fi(j)3476 2669 y Fx(],)515
2769 y(where)31 b(\000)811 2781 y Fp(0)848 2769 y Fx(\()p
Fo(N)9 b Fx(\))p Fo(r)1025 2781 y Fi(j)1090 2769 y Fx(=)29
b(\000)1236 2781 y Fp(0)1273 2769 y Fx(\()p Fo(N)9 b
Fx(\))p Fo(r)1450 2781 y Fi(i)1479 2769 y Fo(g)s Fx(.)48
b(Recall)31 b(that)h(SL)2129 2781 y Fp(2)2166 2769 y
Fx(\()p Fn(Z)p Fx(\))h(is)e(generated)f(b)n(y)i(the)f(t)n(w)n(o)g
(matrices)515 2868 y Fo(\033)26 b Fx(=)676 2801 y Fj(\000)728
2838 y Fp(0)d Fl(\000)p Fp(1)728 2895 y(1)75 b(0)883
2801 y Fj(\001)948 2868 y Fx(and)28 b Fo(\034)33 b Fx(=)1266
2801 y Fj(\000)1318 2838 y Fp(1)23 b Fl(\000)p Fp(1)1318
2895 y(1)75 b(0)1473 2801 y Fj(\001)1539 2868 y Fx(\(see)27
b(Theorem)g(2)g(of)h([18)o(,)g(VI)r(I.1.2]\).)515 3034
y Fn(Theorem)i(2.3)h(\(Manin\).)41 b Fh(The)24 b(Manin)g(symb)l(ols)g
Fx([)p Fo(r)2265 3046 y Fp(0)2303 3034 y Fx(])p Fo(;)14
b(:)g(:)g(:)27 b(;)14 b Fx([)p Fo(r)2584 3046 y Fi(m)2648
3034 y Fx(])23 b Fh(satisfy)h(the)g(fol)t(lowing)i(r)l(e-)515
3134 y(lations:)1889 3317 y Fx([)p Fo(r)1949 3329 y Fi(i)1977
3317 y Fx(])19 b(+)f([)p Fo(r)2162 3329 y Fi(i)2190 3317
y Fx(])p Fo(\033)27 b Fx(=)22 b(0)1598 3452 y([)p Fo(r)1658
3464 y Fi(i)1686 3452 y Fx(])d(+)f([)p Fo(r)1871 3464
y Fi(i)1899 3452 y Fx(])p Fo(\034)28 b Fx(+)18 b([)p
Fo(r)2129 3464 y Fi(i)2157 3452 y Fx(])p Fo(\034)2225
3417 y Fp(2)2287 3452 y Fx(=)k(0)p Fo(:)515 3634 y Fh(F)-6
b(urthermor)l(e,)30 b(these)g(r)l(elations)g(gener)l(ate)g(al)t(l)h(r)l
(elations)f(\(mo)l(dulo)h(torsion)f(r)l(elations\).)639
3800 y Fx(This)25 b(theorem,)f(whic)n(h)h(is)f(pro)n(v)n(ed)f(in)h([12)
o(,)i Fk(x)p Fx(1.7],)e(pro)n(vides)f(a)h(\014nite)h(presen)n(tation)e
(for)h(the)515 3900 y(space)j(of)g(mo)r(dular)g(sym)n(b)r(ols.)515
4132 y Fy(2.3)112 b(Hec)m(k)m(e)37 b(op)s(erators)h(on)g(mo)s(dular)e
(sym)m(b)s(ols)515 4286 y Fx(When)28 b Fo(p)f Fx(is)h(a)f(prime)g(not)h
(dividing)g Fo(N)9 b Fx(,)27 b(de\014ne)1094 4513 y Fo(T)1143
4525 y Fi(p)1181 4513 y Fk(f)p Fo(\013;)14 b(\014)t Fk(g)23
b Fx(=)1516 4396 y Fj(\022)1578 4463 y Fo(p)82 b Fx(0)1578
4562 y(0)g(1)1744 4396 y Fj(\023)1819 4513 y Fk(f)p Fo(\013;)14
b(\014)t Fk(g)k Fx(+)2202 4434 y Fj(X)2145 4613 y Fi(r)i
Fp(mo)r(d)i Fi(p)2393 4396 y Fj(\022)2454 4463 y Fx(1)83
b Fo(r)2454 4562 y Fx(0)f Fo(p)2620 4396 y Fj(\023)2695
4513 y Fk(f)p Fo(\013;)14 b(\014)t Fk(g)p Fo(:)515 4786
y Fx(As)28 b(men)n(tioned)g(b)r(efore,)g(this)g(de\014nition)h(is)f
(compatible)f(with)i(the)f(in)n(tegration)f(pairing)g
Fk(h)14 b Fo(;)28 b Fk(i)515 4886 y Fx(of)d(Section)g(1,)g(in)g(the)h
(sense)f(that)g Fk(h)p Fo(f)9 b(T)1739 4898 y Fi(p)1777
4886 y Fo(;)14 b(x)p Fk(i)23 b Fx(=)g Fk(h)p Fo(f)t(;)14
b(T)2167 4898 y Fi(p)2205 4886 y Fo(x)p Fk(i)p Fx(.)37
b(When)26 b Fo(p)c Fk(j)h Fo(N)9 b Fx(,)26 b(the)f(de\014nition)h(is)f
(the)515 4986 y(same,)i(except)g(that)h(the)g(matrix)1599
4918 y Fj(\000)1651 4953 y Fi(p)23 b Fp(0)1652 5014 y(0)g(1)1755
4918 y Fj(\001)1821 4986 y Fx(is)28 b(dropp)r(ed.)1998
5255 y(7)p eop
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8 7 bop 639 523 a Fx(F)-7 b(or)27 b(example,)g(when)h
Fo(N)k Fx(=)23 b(11)j(w)n(e)i(ha)n(v)n(e)1125 706 y Fo(T)1174
718 y Fp(2)1210 706 y Fk(f)p Fx(0)p Fo(;)14 b Fx(1)p
Fo(=)p Fx(5)p Fk(g)80 b Fx(=)j Fk(f)p Fx(0)p Fo(;)14
b Fx(2)p Fo(=)p Fx(5)p Fk(g)h Fx(+)j Fk(f)p Fx(0)p Fo(;)c
Fx(1)p Fo(=)p Fx(10)p Fk(g)h Fx(+)j Fk(f)p Fx(1)p Fo(=)p
Fx(2)p Fo(;)c Fx(3)p Fo(=)p Fx(5)p Fk(g)1579 830 y Fx(=)83
b Fk(\000)p Fx(2)p Fk(f)p Fx(0)p Fo(;)14 b Fx(1)p Fo(=)p
Fx(5)p Fk(g)p Fo(:)639 1013 y Fx(In)21 b([14],)h(L.)f(Merel)g(giv)n(es)
f(a)g(description)h(of)g(the)g(action)g(of)g Fo(T)2516
1025 y Fi(p)2574 1013 y Fx(directly)g(on)g(Manin)g(sym)n(b)r(ols)515
1112 y([)p Fo(r)575 1124 y Fi(i)603 1112 y Fx(])28 b(\(see)f(also,)g
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b Fh(Mo)l(dular)c(functions)e(of)h(one)f(variable.)i(IV)p
Fx(,)685 3771 y(Springer-V)-7 b(erlag,)26 b(Berlin,)h(1975,)f(Lecture)h
(Notes)g(in)h(Mathematics,)f(V)-7 b(ol.)28 b(476.)556
3930 y([4])41 b(W.)20 b(Bosma,)f(J.)f(Cannon,)i(and)f(C.)f(Pla)n(y)n
(oust,)h Fh(The)k(Magma)f(algebr)l(a)h(system.)f(I.)g(The)h(user)685
4030 y(language)p Fx(,)40 b(J.)c(Sym)n(b)r(olic)g(Comput.)h
Fn(24)f Fx(\(1997\),)h(no.)f(3-4,)i(235{265,)d(Computational)685
4129 y(algebra)26 b(and)i(n)n(um)n(b)r(er)f(theory)g(\(London,)g
(1993\).)556 4289 y([5])41 b(C.)24 b(Breuil,)g(B.)f(Conrad,)g(F.)g
(Diamond,)i(and)e(R.)g(T)-7 b(a)n(ylor,)23 b Fh(On)i(the)h(mo)l
(dularity)h(of)f(el)t(liptic)685 4388 y(curves)k(over)h
Fn(Q)p Fh(,)f(or)g(Wild)g Fx(3)p Fh(-adic)h(exer)l(cises)p
Fx(,)d(\(2000\),)685 4488 y Fw(http://www.math.ha)n(rva)n
(rd.edu/HTML/Individuals/Richa)m(rd)p 2728 4488 25 4
v 25 w(T)-7 b(a)n(ylo)n(r.html)p Fx(.)556 4648 y([6])41
b(J.)14 b(E.)30 b(Cremona,)f Fh(A)n(lgorithms)k(for)f(mo)l(dular)h(el)t
(liptic)h(curves)p Fx(,)d(second)e(ed.,)i(Cam)n(bridge)685
4747 y(Univ)n(ersit)n(y)c(Press,)f(Cam)n(bridge,)h(1997.)556
4907 y([7])41 b(F.)33 b(Diamond)g(and)g(J.)f(Im,)i Fh(Mo)l(dular)i
(forms)f(and)g(mo)l(dular)g(curves)p Fx(,)f(Seminar)e(on)h(F)-7
b(er-)685 5006 y(mat's)28 b(Last)f(Theorem,)g(Pro)n(vidence,)f(RI,)h
(1995,)f(pp.)i(39{133.)1998 5255 y(9)p eop
%%Page: 10 10
10 9 bop 556 523 a Fx([8])41 b(M.)14 b(J.)40 b(Green)n(b)r(erg)f(and)h
(J.)14 b(R.)40 b(Harp)r(er,)i Fh(A)n(lgebr)l(aic)h(top)l(olo)l(gy)p
Fx(,)i(Benjamin/Cummings)685 623 y(Publishing)36 b(Co.)f(Inc.)h(Adv)-5
b(anced)36 b(Bo)r(ok)f(Program,)g(Reading,)i(Mass.,)g(1981,)f(A)h
(\014rst)685 722 y(course.)556 888 y([9])k(H.)e(Hijik)-5
b(ata,)41 b Fh(Explicit)f(formula)g(of)h(the)e(tr)l(ac)l(es)g(of)h(He)l
(cke)g(op)l(er)l(ators)g(for)g Fx(\000)3215 900 y Fp(0)3252
888 y Fx(\()p Fo(N)9 b Fx(\),)41 b(J.)685 988 y(Math.)28
b(So)r(c.)g(Japan)e Fn(26)h Fx(\(1974\),)g(no.)g(1,)g(56{82.)515
1154 y([10])40 b(D.)15 b(R.)28 b(Kohel,)e Fh(He)l(cke)k(mo)l(dule)h
(structur)l(e)d(of)i(quaternions)p Fx(,)e(preprin)n(t)f(\(1998\).)515
1320 y([11])40 b(S.)35 b(Lang,)h Fh(Intr)l(o)l(duction)g(to)g(mo)l
(dular)h(forms)p Fx(,)g(Springer-V)-7 b(erlag,)34 b(Berlin,)i(1995,)e
(With)685 1420 y(app)r(endixes)28 b(b)n(y)f(D.)h(Zagier)e(and)i(W.)g(F)
-7 b(eit,)28 b(Corrected)e(reprin)n(t)h(of)h(the)g(1976)d(original.)515
1586 y([12])40 b(J.)14 b(I.)30 b(Manin,)g Fh(Par)l(ab)l(olic)j(p)l
(oints)f(and)g(zeta)g(functions)f(of)h(mo)l(dular)h(curves)p
Fx(,)d(Izv.)f(Ak)-5 b(ad.)685 1685 y(Nauk)28 b(SSSR)g(Ser.)f(Mat.)h
Fn(36)f Fx(\(1972\),)f(19{66.)515 1851 y([13])40 b(B.)j(Mazur,)j
Fh(Courb)l(es)e(el)t(liptiques)h(et)e(symb)l(oles)i(mo)l(dulair)l(es)p
Fx(,)j(S)n(\023)-39 b(eminaire)41 b(Bourbaki,)685 1951
y(24)n(\022)-39 b(eme)29 b(ann)n(\023)-39 b(ee)29 b(\(1971/1972\),)e
(Exp.)j(No.)h(414,)f(Springer,)g(Berlin,)g(1973,)g(pp.)h(277{294.)685
2051 y(Lecture)d(Notes)f(in)h(Math.,)g(V)-7 b(ol.)27
b(317.)515 2217 y([14])40 b(L.)19 b(Merel,)i Fh(Universal)h(Fourier)h
(exp)l(ansions)f(of)h(mo)l(dular)f(forms)p Fx(,)g(On)d(Artin's)g
(conjecture)685 2316 y(for)27 b(o)r(dd)h(2-dimensional)e(represen)n
(tations)g(\(Berlin\),)i(Springer,)e(1994,)g(pp.)i(59{94.)515
2482 y([15])40 b(J.-F.)34 b(Mestre,)h Fh(L)l(a)g(m)n(\023)-40
b(etho)l(de)37 b(des)f(gr)l(aphes.)h(Exemples)f(et)f(applic)l(ations)p
Fx(,)j(Pro)r(ceedings)685 2582 y(of)30 b(the)g(in)n(ternational)f
(conference)g(on)g(class)g(n)n(um)n(b)r(ers)g(and)g(fundamen)n(tal)h
(units)g(of)g(al-)685 2682 y(gebraic)d(n)n(um)n(b)r(er)g(\014elds)h
(\(Katata\))e(\(1986\),)h(217{242.)515 2848 y([16])40
b(A.)28 b(Pizer,)e Fh(A)n(n)i(algorithm)j(for)f(c)l(omputing)f(mo)l
(dular)h(forms)f(on)h Fx(\000)2796 2860 y Fp(0)2833 2848
y Fx(\()p Fo(N)9 b Fx(\),)27 b(J.)g(Algebra)f Fn(64)685
2947 y Fx(\(1980\),)h(no.)g(2,)g(340{390.)515 3113 y([17])40
b(K.)14 b(A.)33 b(Rib)r(et)h(and)f(W.)15 b(A.)33 b(Stein,)i
Fh(L)l(e)l(ctur)l(es)f(on)h(Serr)l(e's)g(c)l(onje)l(ctur)l(es)p
Fx(,)f(IAS/P)n(ark)d(Cit)n(y)685 3213 y(Mathematics)d(Institute)g
(1999.)515 3379 y([18])40 b(J-P)-7 b(.)31 b(Serre,)g
Fh(A)i(Course)g(in)h(A)n(rithmetic)p Fx(,)e(Springer-V)-7
b(erlag,)30 b(New)i(Y)-7 b(ork,)31 b(1973,)g(T)-7 b(rans-)685
3479 y(lated)28 b(from)f(the)h(F)-7 b(renc)n(h,)27 b(Graduate)g(T)-7
b(exts)28 b(in)g(Mathematics,)f(No.)g(7.)515 3645 y([19])40
b(G.)d(Shim)n(ura,)h Fh(Intr)l(o)l(duction)f(to)g(the)h(arithmetic)h
(the)l(ory)f(of)h(automorphic)g(functions)p Fx(,)685
3744 y(Princeton)28 b(Univ)n(ersit)n(y)g(Press,)f(Princeton,)h(NJ,)g
(1994,)f(Reprin)n(t)i(of)f(the)h(1971)e(original,)685
3844 y(Kan)g(Memorial)g(Lectures,)g(1.)515 4010 y([20])40
b(W.)15 b(A.)35 b(Stein,)i Fh(Explicit)g(appr)l(o)l(aches)h(to)f(mo)l
(dular)g(ab)l(elian)g(varieties)p Fx(,)h(U.)14 b(C.)35
b(Berk)n(eley)685 4110 y(Ph.D.)28 b(thesis)g(\(2000\).)515
4276 y([21])40 b(H.)26 b(W)-7 b(ada,)26 b Fh(T)-6 b(ables)28
b(of)h(He)l(cke)e(op)l(er)l(ations.)j(I)p Fx(,)25 b(Seminar)g(on)g(Mo)r
(dern)g(Metho)r(ds)g(in)h(Num-)685 4375 y(b)r(er)i(Theory)e(\(T)-7
b(oky)n(o\),)27 b(Inst.)h(Statist.)g(Math.,)g(1971,)e(p.)h(10.)1977
5255 y(10)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF