Open in CoCalc
1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2
%% ribetofficial.tex --- Scribe notes for Ken Ribets Spring 1996 course. %%
3
%% %%
4
%% This file should compile with standard latex, the amssymb package, %%
5
%% and the diagrams package. The diagrams package is contained in the %%
6
%% file "diagrams.tex". Obtain it and put it in the same directory %%
7
%% as this file. %%
8
%% %%
9
%% This file is being maintained by William Stein ([email protected]).%%
10
%% This file was assembled by Lawren Smithline ([email protected]).%%
11
%% %%
12
%% Last modification date: 9/25/96. %%
13
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
14
15
\documentclass[12pt]{report}
16
17
%% Uncomment the following two lines if your system doesn't
18
%% define the \mathfrak font.
19
%\font\german=eufm10 scaled \magstep 1
20
%\def\mathfrak{\german}
21
22
\input diagrams
23
\usepackage{amssymb}
24
\marginparwidth 0pt
25
\oddsidemargin 0pt
26
\evensidemargin 0pt
27
\marginparsep 0pt
28
\topmargin 0pt
29
\textwidth 6.5in
30
\textheight 8.5 in
31
\def\nn{{\rm N}} % Roman N in math mode
32
\def\aa{{\cal A}}
33
\def\rr{{\cal R}}
34
\def\flt{F{\sc l}T} % Fermat's little theorem
35
\def\pf{{\sc Proof.\ }}
36
\def\st{\ \cdot\backepsilon\cdot\ } % ``such that'' symbol
37
\def\vs{\vspace{4pt}}
38
\def\A{{\bf A}} % Adele ring
39
\def\C{{\bf C}} % complex nos.
40
\def\Z{{\bf Z}} % integers
41
\def\F{{\bf F}} % field
42
\def\P{{\bf P}} % projective land
43
44
45
\def\N{{\mathfrak N}} % index of an ideal in a number ring (norm)
46
\def\H{{\mathfrak H}} % the complex upper halfplane
47
\def\R{{\bf R}} % reals
48
\def\Q{{\bf Q}} % rationals
49
\def\O{{\cal O}} % ring of integers
50
\def\T{{\bf T}} % Hecke algebra
51
\def\G{{\bf G}} % group scheme
52
\def\X{{X(N)}} % Modular curve
53
\def\E{{E_j}}
54
\def\K{{\overline{K}}}
55
\def\ff{{\cal F}} % Modular function field
56
57
\def\Qp{\Q_p} % p-adic numbers
58
\def\Zp{\Z_p} % p-adic integers
59
60
\def\0{{\bf 0}} % zero structure
61
\def\a{\alpha} % your basic Greeks
62
\def\b{\beta}
63
\def\c{\gamma}
64
\def\d{\delta}
65
\def\e{\epsilon}
66
\def\f{\zeta}
67
\def\G{\Gamma}
68
\def\ka{\kappa}
69
\def\la{\lambda}
70
\def\t{\tau}
71
\def\om{\omega}
72
\def\Om{\Omega}
73
\def\rh{{\overline % rho bar
74
{\rho_E}}}
75
\def\rhla{{\rho_\lambda}}
76
\def\th{{\theta}}
77
\def\g{\gamma}
78
79
\def\p{{\mathfrak p}} % Gothic p, a prime ideal
80
\def\q{{\mathfrak q}} % Gothic q
81
\def\M{{\mathfrak m}} % maximal ideal
82
\def\Tm{{\T_\M}} % T localized at m
83
\def\ia{{\mathfrak a}}
84
\def\ib{{\mathfrak b}}
85
\def\si{\sigma}
86
\def\cent{{\bf C}} % centralizer
87
\def\qed{\hfill $\blacksquare$\smallskip}
88
89
\def\tate{{\rm Ta}} % Tate module
90
\def\tatel{{{\rm Ta}_\l}} % l-adic Tate module
91
\def\tatem{{{\rm Ta}_\M}} % m-adic Tate module
92
\def\tatels{{{\rm Ta}_\l^*}} % contrav. l-adic T. m.
93
\def\tatems{{{\rm Ta}_\M^*}} % contrav. m-adic T. m.
94
95
\def\ve{\varepsilon} % epsilon the character
96
97
\def\lan{\langle} % angle brackets
98
\def\ran{\rangle}
99
\def\<{{\langle}} % < bracket
100
\def\>{{\rangle}} % > bracket
101
102
\def\l{\ell} % script l as in l-adic.
103
\def\Ql{{\Q_\l}} % l-adic numbers
104
\def\Zl{{\Z_\l}} % fewer l-adic numbers
105
\def\sigmaonf{
106
{}^\sigma\!f} % sigma acting on f, from upper left hand corner
107
\def\GL{{\rm GL}} % general linear group
108
\def\SL{{\rm SL}} % special linear group
109
\def\gal{{\cal G}al} % Galois group
110
\def\GQ{\gal(\overline\Q/\Q)}
111
% abs Galois gp of Q
112
\def\GQp{\gal(\overline\Qp/\Qp)}
113
% abs Galois gp of Qp
114
\def\modgp{\SL_2(\Z)} % modular group
115
\def\abcd{\left( % 2 x 2 matrix a b // c d
116
\begin{array}{cc}
117
a&b\\c&d\end{array}\right)}
118
\def\isom{\cong}
119
\def\tensor{\otimes}
120
121
\def\tr{{\rm tr}}
122
\def\frob{{\rm frob}}
123
\def\mod{\ {\rm mod}\,}
124
\def\im{{\rm im}}
125
\def\ord{{\rm ord}}
126
\def\rank{{\rm rank}}
127
\def\aut{{\rm Aut}}
128
\def\Hom{{\rm Hom}}
129
\def\plim{{\displaystyle\lim_{\longleftarrow}\,}} % Projective limit
130
\def\dlim{{\displaystyle\lim_{\longrightarrow}\,}} % Direct limit
131
\def\plimr{{\displaystyle\lim_{
132
\buildrel\longleftarrow\over r}\,}} % Projective limit over r
133
\def\Div{{\rm Div}}
134
\def\det{{\rm det }} % Determinant
135
\def\endo{{\rm End}}
136
\def\Cot{{\rm Cot}} % contangent space
137
\def\ver{{\rm ver}} % Vershibung endmorphism
138
139
\def\inj{\hookrightarrow}
140
\def\into{\rightarrow}
141
\def\onto{\twoheadrightarrow}
142
\def\isomap{{\buildrel \sim\over\rightarrow}}
143
\newarrow{To} ---->
144
\newarrow{Line} -----
145
146
\newtheorem{thm}{Theorem}
147
\newtheorem{dfn}{Definition}
148
\newtheorem{prop}{Proposition}
149
\newtheorem{lem}{Lemma}
150
\newtheorem{cor}{Corollary}
151
\newtheorem{res}{Result}
152
\newtheorem{eg}{Example}
153
\newtheorem{claim}{Claim}
154
\newtheorem{exercise}{Exercise}
155
\newtheorem{remark}{Remark}
156
\newtheorem{conj}{Conjecture}
157
\newtheorem{note}{Note}
158
159
\begin{document}
160
\title{Scribe notes for Ken Ribet's Math 274}
161
\author{}
162
\date{\null}
163
\maketitle
164
\section*{January 17, 1996}
165
\noindent{Scribe: Lawren Smithline, \tt <lawren@math>}
166
\bigskip
167
168
\noindent
169
Here are some topics to be discussed in this course:
170
171
\begin{tabular}{l}
172
Galois representations and modular forms, \\
173
Hecke algebras, \\
174
modular curves, and
175
Jacobians (Abelian varieties).
176
\end{tabular} \\
177
This lecture is a brief overview of some connection between these concepts,
178
and also an exercise in name-dropping.
179
180
We can describe an an elliptic curve, or the Jacobian of a higher genus
181
curve, or abelian variety using a lattice. For $E$, the lattice is $L =
182
H_1(E(\C),\Z) \hookrightarrow \C,$ by the map $\c \mapsto \int_\c
183
\omega.$
184
185
Weil considered curves over a finite field, $k$ of characteristic $p$.
186
There is an algebraic definition of $L/nL$ for $n \geq 1, \ \gcd(n, p) = 1.$
187
$$E[n] = \{ P \in E(\bar k) : nP = 0\} = \textstyle\frac1n L/L = L/nL.$$
188
For example, let $n = \l^\nu$ for $\nu \geq 1$, and $\l$ a prime different
189
from $p.$
190
191
Weil further considered the limit $$E[\l^\infty] = \bigcup_{\nu = 1}^\infty
192
E[\l^\nu].$$
193
Tate oberserved there is a map $E[\l^\nu] \stackrel{\l}\rightarrow
194
E[\l^{\nu-1}],$ and so of course this inverse limit, $$
195
\lim_\leftarrow E[\l^\nu] = T_\l(E)$$ is called the Tate module.
196
197
As $E[n]$ is free of rank 2 over $\Z / n\Z$, so is $T_\l(E)$ free of rank 2
198
over $\Z_\l$. Also, $V_l(e) = T_\l(E) \otimes \Q_\l$ is a 2 dimensional
199
vector space over $\Q_\l$. This is the first example of $\l$-adic \'etale
200
cohomology. For topological space $X$, $X \mapsto H_{\mathaccent 19
201
et}^i(X/\bar k, \Q_\l).$
202
203
Here are the names of some cool folks: Taniyama Shimura Mumford Tate.
204
205
Now, elliptic curve $E/\Q$ gets an action of $G = \gal(\bar \Q/\Q)$,
206
and so does $E[n]$. I.e. $\si(P+Q) = \si(P) + \si(Q).$ So we have a
207
homomorphism $\rho: G \rightarrow {\rm Aut}(E[n]) = GL_2(\Z/n\Z).$
208
Since we have an exact sequence $$1 \rightarrow \ker \rho \rightarrow G
209
\rightarrow {\rm im}\, \rho \rightarrow 0,$$
210
we get a tower of fields
211
$$\bar \Q \rightarrow K \rightarrow \Q,$$ and ${\cal G}al(K/\Q) =
212
{\rm im}\, \rho \subset GL_2( \Z/n\Z).$
213
\smallskip
214
215
And now for something completely different. We can also get to these
216
Galois representations via modular forms. Let $k$ be the weight, such as
217
2. Let $N$ be the level. The complex vector space $S_k(N)$ is the set of
218
cusp forms on $\Gamma_1(N)$, a finite dimensional vector space, namely, the
219
set of holomorphic functions $f$ on $\H$ such that
220
$$ f((az+b)/(cz+d)) = (cz+d)^kf(z)$$ for $$\left( \begin{array}{cc} a & b
221
\\
222
c & d \end{array} \right) \in \Gamma_1(N), \ \ a,d \equiv 1, c \equiv 0 (N).$$
223
Such an $f$ has a power series (or Fourier series) expansion in $q =
224
\exp(2\pi i z)$: $$f(z) =
225
\sum_1^\infty c_n q^n.$$
226
Here is a famous example observed by Ramanujan, and proved by Mordell using
227
(his) Hecke operators:
228
$$ q\prod_1^\infty (1 - q^n)^{24} = \sum_1^\infty \tau(n)q^n,$$
229
for Ramanujan's $\tau$ function. Now, $\tau(n)\tau(m) =\tau(nm)$ for
230
$\gcd(n,m) = 1$. Also, there is a recurrence for prime powers.
231
Amusingly, the normalized basis element of $S_{12}(1)$ is $$\Delta =
232
\sum_1^\infty \tau(n)\exp(2\pi i nz).$$ Even more amusingly, $$\tau(n)
233
\equiv \sum_{d \mid n} d^{11} \ \ (691).$$
234
\smallskip
235
236
Experience and Shimura have shown that there exist $f \in S_k(N)$ such that
237
$$T_n(f) =c_n \cdot f$$ for all $n \geq 1$ for some scalars $c_n$, and
238
$$f = \sum c_n q^n$$ for the same $c_n$, and that these $c_n$ are algebraic
239
integers in a finitely generated number field. That is, $[\Q(c_n: n\geq 1):
240
\Q]$ is finite.
241
242
How can we study and interpret this? We start with the Hecke ring,
243
$$\Q[T_n] \subseteq End(S_k(N)).$$
244
Serre in 1968 said there should be Galois representations attached to forms
245
of arbitrary weight. Deligne constructed them. In a broad stroke, one can
246
say that we get between Galois representations and modular forms via
247
Frobenius elements.
248
249
Next time, we continue with the semihistorical overview.
250
\section*{January 19, 1996}
251
\noindent{Scribe: William Stein, \tt <was@math>}
252
\bigskip
253
254
\noindent {\bf\large Modular Representations and Modular Curves} \smallskip
255
256
\subsection*{Arithmetic of Modular Forms}
257
Suppose $f=\sum_{n=1}^{\infty}a_n q^n$ is a cusp form in $S_k(N)$ which is
258
an eigenform for the Hecke operators. The Mellin transform associates
259
to $f$ the $L$-function $L(f,z)=\sum_{n=1}^{\infty} n^{-s}{a_n}$.
260
Let $K=\Q(a_1,a_2,\ldots)$. One can show that the $a_n$ are algebraic
261
integers and $K$ is a number field. When $k=2$, $f$ is associated
262
to $f$ an abelian variety
263
$A_f$ over $\Q$ of dimension $[K:\Q]$, and $A_f$ has a $K$ action. (See
264
Shimura,
265
{\em Introduction to the Arithmetic Theory of Automorphic Functions},
266
Theorem 7.14.)
267
268
\begin{eg}[Modular Elliptic Curves]
269
If $a_n\in\Q$ for all $n$, then $K=\Q$ and $[K:\Q]=1$. In this case, $A_f$
270
is a one dimensional abelian variety, which is an elliptic curve, since
271
it has nonzero genus.
272
An elliptic curve arising in this way is called modular.
273
\end{eg}
274
275
\begin{dfn}
276
Elliptic curves $E_1$ and $E_2$ are {\em isogenous} if there is
277
a morphism $E_1\into E_2$ of algebraic groups, which has a
278
finite kernel.
279
\end{dfn}
280
281
The following conjecture motivates much of the theory.
282
283
\begin{conj}
284
Every elliptic curve over $\Q$ is modular,
285
that is, isogenous to a curve constructed in the above way.
286
\end{conj}
287
288
For $k\geq 2$, Serre and Deligne found a way to associate to $f$ a family
289
of $\l$-adic representations. Let $\l$ be a prime number and $K$ be as
290
above. It is well known that $$K\otimes_{\Q} \Q_{\l}\isom
291
\prod_{\la|\l}K_{\la}.$$
292
One can associate to $f$ a family of representations
293
$$
294
\rho_{\l,f}:G=\gal(\overline{\Q}/\Q)
295
\rightarrow\GL(K\otimes_{\Q}\Q_{\l})
296
$$
297
unramified at all primes $p\not|\l N$.
298
By unramified we mean that for all primes $P$ lying over $p$,
299
the inertia group of the decomposition group at $P$ is contained
300
in the kernel of $\rho$. (The decomposition group $D_P$ at $P$ is the
301
set of those $g\in G$ which fix $P$ and the inertia group
302
is the kernel of the map $D_P\rightarrow
303
\gal(\O/P)$, where $\O$ is the ring of all algebraic integers.)
304
305
306
Now $I_P\subset D_P \subset \gal(\overline{\Q}/\Q)$ and
307
$D_P / I_P$ is cyclic, since it is isomorphic to a subgroup of the
308
galois group of a finite extension of finite fields.
309
So $D_P / I_P$
310
is generated by a Frobenious automorphism $\frob_p$ lying over $p$.
311
We have
312
$$
313
\tr(\rho_{\l,f}(\frob_p)) = a_p\in K \subset K\otimes \Q_{\l} $$
314
and
315
\begin{equation}\label{detrho}
316
\det(\rho_{\l}) = \chi_{\l}^{k-1}\ve,
317
\end{equation}
318
where $\chi_{\l}$ is the $\l$th cyclotomic character and
319
$\ve$ is a Dirichlet character.
320
321
\subsection*{Characters}
322
Let $f\in S_k(N)$. For all
323
$\abcd \in \modgp$ with $c\cong 0 \mod{N}$ we have
324
$$
325
f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k \ve(d) f(z),
326
$$
327
where $\ve:(\Z/n\Z)^*\rightarrow \C^*$
328
is a Dirichlet character mod $N$. If $f$ is an eigenform for
329
the diamond bracket operator $<d>$, (so that
330
$f|<d> = \ve(d) f$)
331
then $\ve$ actually takes values in $K$.
332
333
Let $\phi_n$ be the mod $n$ cyclotomic character.
334
The map $\phi_n: G \rightarrow (\Z/n\Z)^*$ takes $g\in G$ to
335
the automorphism induced by $g$ on the $n$th cyclotomic
336
extension $\Q(\mu_n)$ of $\Q$, where we identify
337
$\gal(\Q(\mu_n)/\Q)$ with $(\Z/n\Z)^*$.
338
The $\ve$ appearing in (\ref{detrho})
339
is really the composition
340
$$
341
G\stackrel{\phi_n}\longrightarrow(\Z/n\Z)^*
342
\stackrel{\ve}\longrightarrow \C^*.
343
$$
344
345
For each positive integer $\nu$ we consider the $\l^{\nu}$th
346
cyclotomic character on $G$,
347
$$
348
\phi_{\l^{\nu}}:G\rightarrow (\Z/\l^{\nu}\Z)^*.
349
$$
350
Putting these together give a map
351
$$
352
\phi_{\l^{\infty}}=\lim_{\stackrel\longleftarrow\nu}
353
\phi_{\l^{\nu}}:G\stackrel{\chi_{\l}}\longrightarrow\Z_{\l}^{*}.
354
$$
355
356
\subsection*{Parity Conditions}
357
358
Let $c\in\gal(\overline{\Q}/\Q)$ be complex conjugation.
359
We have $\phi_n(c)=-1$, so $\ve(c) = \ve(-1)$ and
360
$\chi_{\l}(c) = (-1)^{k-1}$. Let
361
$$\abcd
362
=\left(\begin{array}{cc} -1&0\\0&-1 \end{array}\right).$$
363
For $f\in S_k(N)$,
364
$$f(z) = (-1)^k\ve(-1)f(z),$$
365
so $(-1)^k\ve(-1) = 1$. Thus,
366
$$\det(\rho_{\l}(c)) = \epsilon(-1)(-1)^{k-1} = -1.$$
367
The $\det$ character is odd so the representation
368
$\rho_{\l}$ is odd.
369
370
\begin{remark} (Vague Question) How can one recognize representations
371
like $\rho_{\l,f}$ ``in nature''? Mazur and Fontaine have made
372
relevant conjectures. The Shimura-Taniyama conjecture can be reformulated
373
by saying that for any representation $\rho_{\l,E}$ comming
374
from an elliptic curve $E$ there is $f$ so that
375
$\rho_{\l,E}\isom \rho_{\l,f}$.
376
\end{remark}
377
378
\subsection*{Conjectures of Serre (mod $\l$ version)}
379
Suppose $f$ is a modular form, $\l$ a rational prime,
380
$\la$ a prime lying over $\l$, and the representation
381
$$\rho_{\la,f}:G\rightarrow \GL_2(K_{\la})$$
382
(constructed by Serre-Deligne) is irreducible.
383
Then $\rho_{\la,f}$ is conjugate to a representation
384
with image in $\GL_2(\O_{\la})$, where $\O_{\la}$
385
is the ring of integers of $K_{\la}$.
386
Reducing mod $\l$ gives a representation
387
$$\overline{\rho}_{\la,f}:G\rightarrow\GL_2(\F_{\la})$$
388
which has a well-defined trace and det, i.e., the det and trace
389
don't depend on the choice of conjugate used to reduce mod
390
$\la$. One knows from representation theory that if
391
such a representation is semisimple then it is completely determined
392
by its trace and det. Thus if $\overline{\rho}_{\la,f}$ is irreducible
393
it is unique in the sense that it doesn't depend on the choice
394
of conjugate.
395
396
We have the following conjecture of Serre which remains open.
397
\begin{conj}[Serre]
398
All irreducible representation of
399
$G$ over a finite field which are odd, i.e., $det(\sigma(c))=-1$, $c$
400
complex conjugation, are of the form $\overline{\rho}_{\la,f}$
401
for some representation $\rho_{\la,f}$ constructed as above.
402
\end{conj}
403
404
\begin{eg}
405
Let $E/\Q$ be an elliptic curve and let
406
$\sigma_{\l}:G\rightarrow\GL_2(\F_{\l})$ be
407
the representation induced by the action of $G$
408
on the $\l$-torsion of $E$. Then $\det \sigma_{\l} = \phi_{\l}$
409
is odd and $\sigma_{\l}$ is usually irreducible, so Serre's conjecture
410
would imply that $\sigma_{\l}$ is modular. From this one can, under Serre's
411
conjecture, prove that $E$ is modular.
412
\end{eg}
413
414
\begin{dfn}
415
Let $\sigma:G\rightarrow \GL_2(\F)$ ($\F$ is a finite field)
416
be a represenation of the galois group $G$. The we say that the
417
{\em representions $\sigma$ is
418
modular} if there is a modular form $f$, a prime $\la$, and an embedding
419
$\F\hookrightarrow \overline{\F}_{\la}$ such that
420
$\sigma\isom\overline{\rho}_{\la,f}$ over
421
$\overline{\F}_\la$.
422
\end{dfn}
423
424
\subsection*{Wile's Perspective}
425
426
Suppose $E/\Q$ is an elliptic curve and
427
$\rho_{\l,E}:G\rightarrow\GL_2(\Z_{\l})$
428
the associated $\l$-adic representation on the
429
Tate module $T_{\l}$. Then by reducing
430
we obtain a mod $\l$ representation
431
$$\overline{\rho}_{\l,E}=\sigma_{\l,E}:G
432
\rightarrow \GL_2(\F_{\l}).$$
433
If we can show this is modular for infinitely many $\l$
434
then we will know that $E$ is modular.
435
436
\begin{thm}[Langlands and Tunnel]
437
If $\sigma_{2,E}$ and $\sigma_{3,E}$ are irreducible, then they
438
are modular.
439
\end{thm}
440
441
This is proved by using the fact that $\GL_2(\F_2)$ and
442
$\GL_2(\F_3)$ are solvable so we may apply ``base-change''.
443
444
\begin{thm}[Wiles]
445
If $\rho$ is an $\l$-adic representation which is irreducible
446
and modular mod $\l$ with $\l>2$ and certain other reasonable
447
hypothesis are satisfied, then $\rho$ itself is modular.
448
\end{thm}
449
450
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
451
452
\section*{January 22, 1996}
453
\noindent{Scribe: Lawren Smithline, \tt <lawren@math>}
454
(Note. These were done about two months after the fact, since the assigned
455
person didn't. So they are terse. It was a pretty easy lecture.)
456
\bigskip
457
458
Today, we have a limited goal: to explain modular forms as functions on
459
lattices -- or elliptic curves. (See Serre's {\em Course in Arithmetic},
460
or Katz's paper in the Proceedings of the Antwerp Conference in the
461
Springer LNM series.)
462
463
Let the level $N = 1$. Consider a weight $k$ cusp form $f$. For $\t \in
464
\H$, we have $$f( a\t+b / c\t+d) = (c\t+d)^k f(\t).$$ So $f(\t+1) =
465
f(\t)$. By the map $\t \mapsto q= \exp(2\pi i \t)$, we map $\H$ to the
466
punctured disc $ \{ z : 0 < |z| < 1\}$. We abuse notation and think of $f$
467
as a function on this disc. Since $f$ is a cusp form, $f$ extends to 0,
468
and $f(0) = 0$. So we have a $q$-expansion $$ f = \sum _{n=1}^{\infty} a_n
469
q^n. $$
470
471
\subsection*{Lattices inside $\C$}
472
473
Let $L = \Z\om_1 \oplus Z\om_2$. We may assume that $\om_1 / \om_2 \in
474
\H$. Let ${\mathfrak R}$ be the set of lattices in $\C$. $\SL_2 \Z$ acts
475
on the left of $M = \{ (\om_1, \om_2) : \om_1, \om_2 \in \H\}$ by
476
multiplication of the column vector. This action fixes the lattice.
477
478
Here is the relation with elliptic curves. A lattice $L$ determines a
479
complex torus $\C / L$. There is a Weierstrass $\wp$ function on this
480
torus. Consider an elliptic curve $E$ over $\C$. There is a lattice given
481
by the inclusion $H_1(E(\C),\Z) \hookrightarrow \C$. Choose a nonzero $\om
482
\in H^0(E, \Om'_E).$ Then $\c \in H_1$ maps to $\int_\c \om \in \C$.
483
484
So maybe we should think of ${\mathfrak R}$ as the set of pairs $\{ (E,
485
\om) \}.$
486
487
We have a map $M / \C^\times \into \H$ sending $(\om_1, \om_2)$ to $\om_1 /
488
\om_2.$ Now take the quotient on the left by $\SL_2 \Z$:
489
$$ {\mathfrak R}/\C^{times} = \SL_2 \Z \backslash M / \C^\times
490
\longrightarrow \SL_2 \Z \backslash \H.$$
491
But this just is the space of elliptic curves over $\C$. So $f:\H
492
\rightarrow \C$ which is a modular form and $F:M \rightarrow \C$ satisfying
493
$F(\la L) = \la^{-k}F(L)$ amount to the same thing by a simple calculation.
494
495
\subsection*{Hecke Operators}
496
497
Let $F$ be a function on lattices. Define the Hecke operator $T_n$ as
498
$$T_n F (L) = \sum_{L' \subset L, (L:L') = n} F(L) n^{k-1}.$$
499
500
The essential case on elliptic curves is for $n = \l$ a prime. In this
501
case, the $L'$ correspond to the $\l+1$ subgroups of order $\l$ of $(\Z /
502
\l\Z)^ 2$.
503
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
504
\section*{January 24, 1996}
505
\noindent{Scribe: William Stein, \tt <was@math>}
506
\bigskip
507
508
\noindent {\bf \large More On Hecke Operators}\smallskip
509
510
We consider modular forms $f$ on $\Gamma_1(1)=\modgp$, that
511
is, holomorphic functions on $\H\cup\{\infty\}$ which satisfy
512
$$f(\tau)=f(\frac{a\tau+b}{c\tau+d})(c\tau+d)^{-k}$$
513
for all $\abcd\in\modgp$. Using a Fourier expansion we write
514
$$f(\tau)=\sum_{n=0}^{\infty} a_ne^{2\pi i\tau n},$$
515
and say $f$ is a cusp form if $a_0=0$.
516
There is a correspondence between modular forms $f$ and
517
lattice functions $F$ satisfying $F(\lambda L)=\lambda^{-k}F(L)$
518
given by $F(\Z\tau+\Z)=f(\tau)$.
519
520
\subsection*{Explicit Description of Sublattices}
521
The $n$th Hecke operator $T_n$ of weight $k$ is defined by
522
$$T_n(L)=n^{k-1}\sum_{{L'\subset L,\ (L:L')=n}} L'.$$
523
What are the $L'$ explicitly? Note that $L/L'$ is a group of order $n$ and
524
$$L'/nL\subset L/nL=(\Z/n\Z)^2.$$
525
Write $L=\Z\om_1+\Z\om_2$, let $Y_2$ be the cyclic subgroup
526
of $L/L'$ generated by $\om_2$ and let $d=\#Y_2$. Let
527
$Y_1=(L/L')/Y_2$. $Y_1$ is generated by the image
528
of $\om_1$ so it is a cyclic group of order $a=n/d$.
529
We want to exhibit a basis of $L'$. Let
530
$\om_2'=d\om_2\in L'$ and use the fact that $Y_1$ is
531
generated by $\om_1$ to write $a\om_1=\om_1'+b\om_2$
532
for $b\in\Z$ and $\om_1'\in L'$. Since $b$ is only
533
well-defined modulo $d$ we may assume $0\leq b\leq d-1$.
534
Thus
535
$$
536
\left(\begin{array}{c}\om_1'\\ \om_2'\end{array}\right)
537
=
538
\left(\begin{array}{cc}a&b\\0&d\end{array}\right)
539
\left(\begin{array}{cc}\om_1\\ \om_2\end{array}\right)
540
$$
541
and the change of basis matrix has determinent $ad=n$.
542
Since
543
$$\Z\om_1'+\Z\om_2'\subset L' \subset L=\Z\om_1+\Z\om_2$$
544
and $(L:\Z\om_1'+\Z\om_2')=n$ (since the change of basis matrix has
545
determinent $n$) and $(L:L')=n$ we see that $L'=\Z\om_1'+\Z\om_2'$.
546
547
Thus there is a one-to-one correspondence between sublattices $L'\subset L$
548
of index $n$ and matrices
549
$\bigl(\begin{array}{cc}a&b\\0&d\end{array}\bigr)$
550
with $ad=n$ and $0\leq b\leq d-1$.
551
In particular, when $n=p$ is prime there $p+1$ of these. In general, the
552
number of such sublattices equals the sum of the positive divisors
553
of $n$.
554
555
\subsection*{Action of Hecke Operators on Modular Forms}
556
Now assume $f(\tau)=\sum_{m=0}^{\infty} c_m q^m$ is a modular
557
form with corresponding lattice function $F$. How can we describe the
558
action of the Hecke operator $T_n$ on $f(\tau)$? We have
559
$$\begin{array}{ll}
560
T_nF(\Z\tau+\Z) & = n^{k-1}\displaystyle\sum_{
561
\stackrel{\stackrel{\stackrel{a,b,d}{ab=n}}{0\leq b<d}}\null
562
}
563
F((a\tau+b)\Z + d\Z)\smallskip \\
564
& = n^{k-1}\displaystyle\sum d^{-k} F(\frac{a\tau+b}{d}\Z+\Z)\smallskip \\
565
& = n^{k-1}\displaystyle\sum d^{-k} f(\frac{a\tau+b}{d})\smallskip \\
566
& = n^{k-1}\displaystyle\sum_{a,d,b,m} d^{-k}c_m e^{2\pi i(\frac{a\tau+b}{d})m}\smallskip \\
567
& = n^{k-1}\displaystyle\sum_{a,d,m} d^{1-k}c_m e^{\frac{2\pi i a m \tau}{d}}
568
\frac{1}{d}\displaystyle\sum_{b=0}^{d-1} (e^{\frac{2\pi i m}{d}})^b\smallskip \\
569
& = n^{k-1}\displaystyle\sum_{\stackrel{ad=n}{m'\geq 0}\null}
570
d^{1-k} c_{dm'}e^{2\pi i a m'
571
\tau}\smallskip \\
572
& = \displaystyle\sum_{{ad=n, \ m'\geq 0}} a^{k-1} c_{dm'}q^{am'}.
573
\end{array}$$
574
In the second to the last expression we
575
let $m=dm'$, $m'\geq 0$, then used the fact that the
576
sum
577
$\frac{1}{d}\sum_{b=0}^{d-1} (e^{\frac{2\pi i m}{d}})^b$
578
is only nonzero if $d|m$.
579
580
Thus
581
$$T_nf(q)=\sum_{{ad=n, \ m\geq 0}} a^{k-1}c_{dm} q^{am}$$
582
and if $\mu\geq 0$ then the coefficient of $q^{\mu}$ is
583
$$\sum_{{a|n, \ a|\mu}}a^{k-1}c_{\frac{n\mu}{a^2}}.$$
584
585
\begin{remark}
586
When $k\geq 1$ the coefficients of $q^{\mu}$ for all $\mu$ belong
587
to the $\Z$-module generated by the $c_m$.
588
\end{remark}
589
590
\begin{remark}
591
Setting $\mu=0$ gives the constant coefficient of $T_n f$ which is
592
$$\sum_{a|n}a^{k-1}c_0 = \sigma_{k-1}(n)c_0.$$
593
Thus if $f$ is a cusp form so is $T_nf$. ($T_nf$ is holomorphic
594
since its original definition is as a finite sum of holomorphic
595
functions.)
596
\end{remark}
597
598
\begin{remark}
599
Setting $\mu=1$ shows that the coefficient of $q$ in $T_n f$ is
600
just $c_n$. As an immediate corollary we have the
601
following important result.
602
\end{remark}
603
604
\begin{cor}
605
Suppose $f$ is a cusp form for which $T_n f$ has 0 as coefficient
606
of $q$ for all $n\geq 1$, then $f=0$.
607
\end{cor}
608
609
\begin{remark}
610
When $n=p$ is prime we get an interesting formula for the
611
action of $T_p$ on the $q$-expansion of $f$.
612
One has
613
$$T_p f = \sum_{\mu\geq 0} \sum_{{a|n, \ a|\mu}}a^{k-1}
614
c_{\frac{n\mu}{a^2}} q^{\mu}. $$
615
Since $n=p$ is prime either $a=1$ or $a=p$. When
616
$a=1$, $c_{p\mu}$ occurs in the coefficient of $q^{\mu}$
617
and when $a=p$, we can write $\mu=p\lambda$ and we get
618
terms $p^{k-1}c_{\lambda}$ in $q^{\lambda p}$.
619
Thus
620
$$T_n f = \sum_{\mu\geq 0}c_{p\mu}q^{\mu}+
621
p^{k-1}\sum_{\lambda\geq 0} c_{\lambda}q^{p\lambda}.$$
622
\end{remark}
623
624
625
626
627
628
\section*{January 26, 1996}
629
\noindent{Scribe: Amod Agashe, \tt <amod@math>}
630
\bigskip
631
632
Following the notation of the last few lectures, let $M_k$ denote
633
the space of modular forms of weight $k$ for $SL_2(\Z)$ and $S_k$
634
denote the subspace of cusp forms.
635
636
Then we have:
637
638
\begin{prop}. $M_k$ is a finite dimensional $\C$-vector space and is generated
639
by modular forms having the coefficients of their Fourier expansion in $\Q$.
640
\end{prop}
641
{\sc Sketch of Proof}. (For details, refer Serre's ``A course in
642
Arithmetic'' or Lang's ``Introduction to modular forms''.) \smallskip
643
644
\noindent
645
The key ingredient that goes into proving finite dimensionality is
646
the following result, which can be obtained by contour integration:
647
648
Let $f\in M_k$ and let $D=\{z\in \C :Im(z)>0, \mid z\mid \geq 1,
649
\mid Re(z)\mid \leq 1/2\}$ be the fundamental domain for $SL_2(\Z)$. Then
650
$$\sum_{p \in D\cup\infty} \frac{1}{e_p} ord_p(f) = \frac{k}{12} $$
651
where
652
$$ e_p = \frac{1}{2} \# \{\gamma \in SL_2(\Z) : \gamma p =p \}
653
= \frac{1}{2} \# Aut(E_p)$$
654
Here, the latter equality follows from the observation that the
655
category of elliptic curves over $\C$ with isogenies is the same
656
as the category of lattices in $\C$ upto homothety with maps being
657
multiplication by elemets of $\C$. One can
658
show that the invertible maps that preserve the lattice $\Z \oplus
659
\Z p$ are in one-to-one correspondence with the set
660
$\{\gamma \in SL_2(\Z) : \gamma p =p \}$ and hence the latter equality.
661
662
In particular,
663
$$ e_p = \left\{ \begin{array}{ll}
664
2 & \mbox{if $p=i$} \\
665
3 & \mbox{if $p=\sqrt[3]{-1}$} \\
666
1 & \mbox{otherwise}
667
\end{array}
668
\right. $$
669
670
Using this formula and relating the dimensions of $M_k$ and
671
$S_k$, one can show that $M_k$ is finite dimensional and also
672
explicitly calculate its dimension.
673
674
To get a basis with Fourier coefficients in $\Q$, first observe
675
that $M_k$ is generated by the set of Eisenstein series $G_k$ for all $k$.
676
As a function on the complex upper half plane,
677
the Eisenstein series $G_k$ for $k \in \Z$ and $k>1$ is given by
678
$$G_k(\tau) = \sum_{(m,n) \in \Z^2 \backslash (0,0)
679
}
680
\frac{1}{(m\tau+n)^k}$$
681
One can then show that
682
$$G_k(\tau) = \frac{1}{2}\zeta(1-k) + \sum_{k=1}^\infty \sigma_{k-1}(n)q^n$$
683
where $q=e^{2\pi i \tau}$, $\zeta$ is the Riemann zeta function and
684
$$\sigma_k(n) = \sum_{d\mid n}d^k$$
685
There is a theorem due to Euler which states that $\zeta(1-k)= -\frac{b_k}{k}$
686
where $b_k$ are the Bernoulli numbers defined by the following power series
687
expansion:
688
$$\frac{x}{e^x-1} = \sum_{k=0}^\infty \frac{b_kx^k}{k!} $$
689
The constant term of $G_k$ is thus $-b_k/{2k}$, which is rational.
690
Thus the Fourier expansion of $G_k$ has rational coefficients and thus
691
we have found a basis with rational coefficients.
692
\bigskip
693
694
Next, let $V$ be a subspace of $M_k$ which is stable under the
695
action of all the Hecke operators $T_n$. For example, observing
696
that $T_n(G_k)=\sigma_{k-1}(n)G_k$, we see that $V=\C(G_k)$ is
697
one such subspace.
698
699
Let $\T=\T(V)$ = $\C$-algebra generated by the $T_n$'s inside $End(V)$
700
= $\C$-vector space generated by the $T_n$'s inside $End(V)$.
701
The latter equality of sets follows because the product of two Hecke
702
operators can be expressed as a linear combination
703
of finitely many Hecke operators.
704
705
For $k>0$, we define a bilinear map
706
$\T \times V \rightarrow \C$ by
707
$$(T,f) \mapsto a_1(f\mid T).$$
708
709
\begin{prop}. The induced maps $\T \rightarrow Hom(V,\C)$ and
710
$V \rightarrow Hom(\T, \C)$ are isomorphisms.
711
\end{prop}
712
\pf
713
We first show that the maps are injective.\\
714
Injectivity of the second map:
715
$$\begin{array}{l}
716
f \in V \mapsto 0 \\
717
\Rightarrow a_1(f \mid T) = 0 \ \forall T\in \T \\
718
\Rightarrow a_1(f \mid T_n) = 0 \ \forall n \\
719
\Rightarrow a_n(f) = 0 \ \forall n \geq 1 \\
720
\Rightarrow f\mbox{ is constant} \\
721
\Rightarrow f=0\mbox{ if }k>0
722
\end{array}$$
723
Injectivity of the first map:
724
$$\begin{array}{l}
725
T \in \T \mapsto 0 \\
726
\Rightarrow a_1(f \mid T)=0 \ \forall f\in V \\
727
\Rightarrow a_1((f\mid T_n)\mid T) = 0 \ \forall f\in V \\
728
\Rightarrow a_1((f\mid T)\mid T_n) = 0 \ \forall f \in V \\
729
\Rightarrow a_n(f\mid T) = 0 \ \forall n>0, \forall f\in V \\
730
\Rightarrow f\mid T = 0 \ \forall k>0, \forall f\in V \\
731
\Rightarrow T = 0
732
\end{array}$$
733
734
In the fourth line, $f$ is replaced by $f \mid T_n$.
735
Next observe that $V$, being a subspace of $M_k$, is finite dimensional.
736
Hence we have from the injectivities of both the above maps that each map
737
is actually an isomorphism. \smallskip
738
739
A map $\phi: M \rightarrow N$ of $\T$-modules is said to be $\T$-equivariant
740
if $\phi (Tm) = T\phi(m) \ \forall m\in M$.
741
742
\begin{prop}.
743
The isomorphisms $\T \cong Hom(V,\C)$
744
and $V \cong Hom(\T, \C)$
745
as defined above are $\T$-equivariant.
746
\end{prop}
747
\pf
748
Consider the first map.
749
Here is the $\T$-module structure on $Hom(V,\C)$.
750
Given $\psi \in Hom(V,\C)$, i.e. $\psi: V \rightarrow \C$,
751
define $T\psi: V \rightarrow \C$ by $(T\psi) (f)=\psi (f\mid T)$.
752
Let $\beta$ denote the map $\T \rightarrow Hom(V, \C)$.
753
Then given $T'\in \T$ and $T \in \T$,
754
we have to show that $\beta (T(T'))=T(\beta (T'))$. Let $f \in V$.
755
Now $(\beta (TT'))(f) = a_1(f\mid TT')$,
756
while $(T(\beta T'))(f) = \beta (T') (f\mid T) = a_1((f\mid T) \mid T')
757
= a_1(f\mid TT')$.
758
Thus $(\beta (TT'))(f)=(T(\beta (T')))(f) \ \forall f\in V$ and we
759
are done.
760
761
\noindent Next consider the second map.
762
We define the $\T$-module structure on $Hom(\T,\C)$.
763
Given $\phi \in Hom(\T,\C)$, i.e. $\phi: \T \rightarrow \C$,
764
define $T\phi: \T \rightarrow \C$ by $(T\phi) (T')=\phi (TT')$.
765
Let $\alpha$ denote the map $V \rightarrow Hom(\T, \C)$.
766
Then given $f\in V$ and $T \in \T$,
767
we have to show that $\alpha (T(f))=T(\alpha (f))$. Let $T' \in \T$.
768
Now $$(\alpha (Tf))(T') = a_1((f\mid T) \mid T') = a_1(f\mid TT'),$$
769
while $$(T(\alpha f))(T') = \alpha (f) (TT') = a_1(f\mid TT').$$
770
Thus $$(\alpha (Tf))(T')=(T(\alpha f))(T') \ \forall T'\in \T$$
771
and we are done.
772
773
\begin{dfn}.
774
An element $f$ of $M_k$ is said to be an eigenform if it is an eigenfunction
775
for all the Hecke operators.\\
776
i.e. $f\mid T_n = \lambda_n f$ for some $\lambda_n \in \C \ \forall n\geq 1$.
777
\end{dfn}
778
Let $f$ is an eigenform with eigenvalues $\lambda_n$. Then
779
$a_n(f) = a_1(f\mid T_n) = a_1(\lambda_n f) = \lambda_n a_1(f) \ \forall n\geq 1$. \\
780
Thus if $a_1(f) = 0$, then $a_n(f) = 0 \ \forall n\geq 1$. \\
781
If $k>0$ then this implies $f=0$.
782
Hence if $k>0$, then if $f\neq 0$, we can normalize $f$ to
783
$\frac{1}{a_1(f)}f$.
784
785
\begin{dfn}.
786
An eigenform $f$ is said to be normalized if $a_1(f) = 1$.
787
\end{dfn}
788
789
If $f$ is a normalized eigenform with eigenvalues $\lambda_n$,
790
then $a_n(f)=\lambda_n$ and $f\mid T_n = \lambda_n f = a_n(f)f$.
791
792
% Given $f \in M_k$ define $\phi_f: \T \rightarrow \C$ by
793
% $\phi_f(T) = a_1(f\mid T)$. Note that if $\alpha$ denotes the map
794
% $V \rightarrow Hom(\T, \C)$ induced by the bilinear pairing mentioned
795
% before, then $\phi_f$ is just $\alhpa(f)$.
796
% Then we have: \\
797
798
Again, let $\alpha$ denote the map $V \rightarrow Hom(\T, \C)$
799
induced by the bilinear pairing mentioned earlier. Then if $f \in V$,
800
we have the map $\alpha(f): \T \rightarrow \C$.
801
802
\begin{prop}.
803
Let $f$ be an eigenform.
804
Then $f$ is normalized $\Leftrightarrow$ $\alpha(f)$ is a ring homomorphism.
805
\end{prop}
806
\pf
807
If $f=0$ then the statement is trivial. So assume $f\neq 0$.
808
Then as discussed above, $a_1(f)\neq 0$. Also recall
809
from the same discussion that if $f\neq 0$ is an eigenform, then
810
$$f\mid T_n = \frac{a_n(f)}{a_1(f)}f.$$ For ease of notation, let
811
$\alpha_f$ denote the map $\alpha(f)$. So
812
$$\begin{array}{l}
813
\alpha_f(T_nT_m) \\ = a_1(f\mid T_nT_m) \\ = a_1((f\mid T_n)T_m)
814
\\ = a_m(f\mid T_n) \\ = a_m((a_n(f)/a_1(f))f) \\ = a_m(f)a_n(f)/a_1(f).
815
\end{array}$$
816
We have $$\alpha_f(T_n)\alpha_f(T_m) = a_1(f\mid T_n) a_1(f\mid T_m)
817
= a_n(f) a_m(f).$$
818
The following are equivalent:
819
820
\begin{tabular}{l}
821
$\alpha_f$ is a ring homomorphism, \\
822
$\alpha_f(T_nT_m) = \alpha_f(T_n)\alpha_f(T_m) \ \forall T_n,T_m$, \\
823
$a_1(f) = 1$, and \\
824
$f$ is normalized.
825
\end{tabular} \\
826
The first implication follows because $\T$ is generated by the $T_n$'s.
827
828
% We will only prove the ``$\Rightarrow$'' part and leave the reverse
829
% implication as an exercise. \\
830
% Assuming that $f$ is normalized, we have to show that
831
% $(\alpha(f))(T_nT_m) = (\alpha(f))(T_n)\phi_f(T_m) \ \forall n,m \geq 1$.
832
% Now $(\alpha(f))(T_nT_m) = a_1(f\mid T_nT_m) = a_1((f\mid T_n)T_m)
833
% = a_m(f\mid T_n) = a_m(a_n(f)f) = a_m(f)a_n(f)$. \\
834
% While $(\alpha(f))(T_n)\phi_f(T_m) = a_1(f\mid T_n) a_1(f\mid T_m)
835
% = a_n(f) a_m(f)$. \\
836
% Thus the two are equal and we are done.
837
838
839
\section*{January 29, 1996}
840
\noindent{Scribe: J\'anos Csirik, \tt <janos@math>}
841
\bigskip
842
843
Today we'll consider questions of rationality and integrality.
844
(References: Serre: {\em A Course in Arithmetic} and Lang: {\em
845
Introduction to Modular Forms}.) Let
846
$S=S_k$ be the space of cusp forms of weight $k$. Let
847
\[ S(\Q) = S_k \cap \Q [[ q ]] \]
848
and
849
\[ S(\Q) \supseteq S(\Z) = S_k \cap \Z [[ q ]]. \]
850
The following fact is easy to prove using explicit formul\ae: $S_k$
851
has a $\C$-basis consisting of forms with integral coefficients (see
852
Victor Miller's construction below).
853
854
Recall that for all even $k\geq4$, there is an Eisenstein series
855
\[ G_k = \frac{-b_k}{2k} +
856
\sum^\infty_{n=1} \left ( \sum_{d|n} d^{k-1} \right) q^n, \]
857
which is a modular form of weight $k$. Renormalize this to obtain
858
\[ E_k = \frac{2k}{-b_k}\cdot G_k = 1 + \cdots. \]
859
The first few Bernoulli numbers of even positive index are $b_2=1/6$,
860
$b_4=-1/30$, $b_6=1/42$, $b_8=-1/30$, $b_{10}=5/66$, $b_{12}=-691/2730$.
861
The fact that the first four of these have numerator 1 is closely
862
related to the arithmetic of cyclotomic fields.
863
864
The modular forms $E_4$ and $E_6$ have $q$-expansions with constant
865
terms equal to $1$, and all coefficients in $\Z$. The functions
866
$E_4^aE_6^b$ with $4a+6b=k$ form a basis for $M_k$.
867
It is easy to see that they are modular forms.
868
From the formula that the (weighted) number of zeros of any modular form
869
of weight $k$ is $k/12$, we deduce that $E_4$ has a simple zero at
870
$\rho$, and $E_6$ hasn't got a zero at $\rho$. Hence $E_4^aE_6^b$ has a
871
simple zero of order $a$ at $\rho$ so these expressions are linearly
872
independent over $\C$.
873
To
874
show that they span $M_k$, consider the following modular form of weight
875
$12$:
876
\[ \Delta = (E_4^3 - E_6^2)/1728. \]
877
Here the coefficient of $q$ is a simple number: $1$. $\Delta$ has a
878
simple zero at $\infty$. Since a cusp form of weight $k$ has $k/12$
879
zeros, it follows that (since
880
the weighting $e_\infty$ is $1$), that $\Delta$ does not vanish
881
anywhere on $\H$. Therefore $S_{k+12}=\Delta\cdot M_k$.
882
Since $E_k(i\infty)=1\neq0$, it follows that
883
$M_k=E_k\cdot\C\oplus S_k=E_k\cdot\C\oplus\Delta\cdot M_{k-12}$. Hence
884
${\rm dim}\,M_k={\rm dim}\,M_{k-12}+1$. Again using the fact that $f\in
885
M_k$ has $k/12$ zeros, we quickly deduce that the dimensions of $M_0$,
886
$M_2$, $M_4$, $M_6$, $M_8$, $M_{10}$ are $1$,$0$,$1$,$1$,$1$,$1$
887
respectively. (e.g., for $k=4$ any modular form must have just a simple
888
zero at $\rho$. So for any $f\in M_4$, it is the case that
889
$f(\tau)-f(i\infty)E_4(\tau)$ vanishes at
890
$\tau=i\infty$ and hence is identically zero. So $E_4$ spans $M_4$.) Thus
891
we have determined ${\rm dim}\,M_k$ for all $k\geq0$, and it is easy to
892
see that this number is equal to the number of solutions to $4a+6b=k$
893
for $a,b\geq0$. Hence the $E_4^aE_6^b$ span $M_k$ and therefore we have
894
proved that they form a basis.
895
896
The following construction comes from the first page of Victor Miller's
897
thesis. Let $d={\rm dim}_\C\, S_k$. Then there exist $f_1,\ldots,f_d\in
898
S(\Z)$ such that $a_i(f_j)=\delta_{ij}$ for $1\leq i,j\leq d$.
899
To show this, recall that $E_4\in M_4$ and $E_6\in M_6$ have
900
$q$-expansions with coefficients in $\Z$. $\Delta\in S_{12}$ has
901
constant coefficient $0$, and the coefficient of $q$ is $1$. Also
902
$\Delta\in S_{12}(\Z)$, as can be seen for example from the formula
903
\[ \Delta = q\prod_{n=1}^\infty (1-q^n)^{24}. \]
904
Now pick $a,b\geq0$ so that $14\geq 4a+6b\equiv k \pmod{12}$, with
905
$a=b=0$ when $k\equiv0\pmod{12}$. Note that then $12d+6a+4b=k$ by our
906
previous result on the dimension of $M_k$ (and the fact that the
907
dimension of $S_k$ is one less than that for $k\geq12$). Hence the
908
functions
909
\[ g_j = \Delta^j E_6^{2(d-j)+a}E_4^b \]
910
for $1\leq j\leq d$ will be cusp forms of weight $k$. By our previous
911
remarks on the coefficients of $\Delta$, $E_6$, $E_4$, we have $g_j\in
912
S_k(\Z)$ and
913
\[ a_i(g_j) = \delta_{ij} \]
914
for $i\leq j$. A straightforward elimination now yields the
915
$f_1,\ldots,f_d$ with the stated properties.
916
It is clear that these $f_1,\ldots,f_d$ are
917
linearly independent over $\C$, hence they form a basis of $S_k$.
918
919
If you take $T_1,\ldots,T_d\in\T=\T(S_k)$, they are also linearly
920
independent: for given any linear relation
921
\[ \sum_{i=1}^d c_iT_i=0, \]
922
apply this to $f_j$ and look at the first coefficient
923
\[ 0 = a_1\left(f_j\left|\sum_{i=1}^d c_iT_i\right.\right)
924
= \sum_{i=1}^d c_ia_i(f_j) = \sum_{i=1}^d c_i\delta_{ij} = c_j, \]
925
hence the linear relation given in the first place was trivial, so
926
$T_1,\ldots,T_d$ form a basis for $\T(S_k)$, since they are linearly
927
independent, and ${\rm dim}_\C\,\T={\rm dim}_\C\, V=d$.
928
929
Let ${\cal R}=\Z[\ldots T_n\ldots]\subseteq {\rm End}(S_k)$.
930
\begin{claim}
931
\[ {\cal R}=\bigoplus_{i=1}^d\Z T_i. \]
932
\end{claim}
933
934
\pf Since the $T_i$ form a basis of $\T$,
935
we have $T_n =\sum_{i=1}^d c_{n_i}T_i$ with $c_{n_i}\in\C$. We need
936
to check that $c_{n_i}\in\Z$. With the $f_j$ as above, consider
937
$$
938
\begin{array}{lll}
939
a_n(f_j) & = a_1(f_j|T_n)
940
\\ & = a_1\left(f_j\left|\sum_{i=1}^d c_{n_i}T_i\right.\right)
941
\\ & = \sum_{i=1}^d c_{n_i}a_1(f_j|T_i)
942
\\ & = \sum_{i=1}^d c_{n_i}a_i(f_j)
943
\\ & = \sum_{i=1}^d c_{n_i}\delta_{ij} & = c_{n_j}.
944
\end{array}$$
945
Hence $c_{n_i}\in\Z$. \qed
946
947
${\cal R}$ is called the {\em integral Hecke algebra}. It is a finite
948
$\Z$-module of rank $d$. We still have (from the last lecture) a
949
pairing
950
\begin{eqnarray*}
951
S(\Z)\times{\cal R} & \rightarrow & \Z \\
952
(f,T) & \mapsto & a_1(f|T).
953
\end{eqnarray*}
954
Now $S(\Z)\hookrightarrow{\rm Hom}({\cal R},\Z)\cong\Z^d$ by the
955
argument given before. Therefore $S(\Z)$ is a free $\Z$-module of
956
finite rank. But it also contains the $f_i$, so $S(\Z)\cong\Z^d$.
957
958
What is $S(\Z)$ as an ${\cal R}$-module?
959
960
\begin{exercise}
961
The map $S(\Z)\hookrightarrow{\rm Hom}({\cal R},\Z)$ is in fact an
962
isomorphism of $\T$-modules.
963
\end{exercise}
964
965
{\sc Hint.\ } The cokernel is a torsion (in fact finite) group.
966
So if we show it torsion free, we are done.
967
968
\begin{thm}
969
The $T_n$ are all diagonalizable on $S_k$.
970
\end{thm}
971
972
$S_n$ supports a Hermitian non-degenerate inner product, the Petersson
973
inner product
974
\[ (f,g)\mapsto \langle f,g\rangle\in\C. \]
975
We have $\langle f,f\rangle\geq0$, with equality iff $f=0$. Furthermore
976
\[ \langle f|T_n,g\rangle = \langle f,g|T_n\rangle, \]
977
i.e., $T_n$ is self-adjoint with respect to the given inner product.
978
979
An operator $T$ is {\em normal} if it commutes with $T^*$ (which denotes
980
its Hermitian transpose). Normal operators are diagonalizable (for a
981
proof, refer to Math H110). In our case, $T_n^*=T_n$, so this fact
982
applies. It is also true (same proof) that a commuting family of
983
semi-simple (i.e., diagonalizable) operators is simultaneously
984
diagonalizable.
985
986
\pf Put together the above facts. \qed
987
988
We can also prove that the eigenvalues are real. This depends on the
989
following trick. For $f\neq0$ consider
990
\[ a_n\langle f,f\rangle = \langle a_nf,f\rangle
991
= \langle f|T_n,f\rangle = \langle f,f|T_n\rangle
992
= \langle f,a_nf\rangle = \bar{a_n}\langle f,f\rangle. \]
993
$a_n\in\R$ now follows since $f\neq0$ implies
994
$\langle f,f\rangle\neq0$.
995
996
\begin{exercise}
997
The $a_n$ are totally real algebraic integers.
998
\end{exercise}
999
1000
{\sc Hint.\ } The space $S_k$ is stable under the action of
1001
${\rm Aut}(\C)$ ``on the coefficients''. Given a cusp form
1002
\[f=\sum_{n=1}^\infty c_nq^n\]
1003
and some $\sigma\in{\rm Aut}(\C)$, define
1004
$\sigmaonf=\sum_{n=1}^\infty \sigma(c_n)q^n$. This function is in
1005
$S_k$, since $S_k$ has a basis in $S(\Q)$, which is fixed by $\sigma$.
1006
Then $f$ is an eigenform iff $\sigmaonf$ is an eigenform.
1007
1008
We'll use a lame definition of the {\em Petersson inner product} for
1009
this section. Let $z=x+iy\in\H$. Then we have a volume form
1010
$y^{-2}{dx\,dy}$ which is invariant under $GL^+_2(\R)$ (the subgroup
1011
of the general linear group of matrices of positive determinant).
1012
To prove this, note that
1013
$dz\wedge d\bar z = (dx+i\,dy)\wedge(dx-i\,dy) = -2i(dx\wedge dy)$,
1014
and hence
1015
\[ dx\,dy = dx\wedge dy = \frac{-1}{2i} dz\wedge d\bar z \]
1016
Then for any
1017
$\alpha = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)
1018
\in GL^+_2(\R)$ we can consider the usual action
1019
\[ \alpha: z \mapsto \frac{az+b}{cz+d}
1020
= \frac{(az+b)(c\bar z+d)}{|cz+d|^2} \]
1021
where the imaginary part of the result is
1022
$|cz+d|^{-2}(ad-bc){\rm Im}\, z = y|cz+d|^{-2}\det(\alpha).$
1023
As for differentials,
1024
\[ d\left(\frac{az+b}{cz+d}\right)
1025
= \frac{a(cz+d)\,dz-(az+b)c\,dz}{(cz+d)^2}
1026
= \frac{ad-bc}{(cz+d)^2}\,dz \]
1027
hence under application of $\alpha$, $dz\wedge d\bar z$ takes on a
1028
factor of
1029
$$\frac{\det(\alpha)}{(cz+d)^2}\frac{\det(\alpha)}{(c\bar z+d)^2}
1030
= \left( \frac{\det(\alpha)}{|cz+d|^2} \right)^2.$$
1031
This finally proves that the differential $y^{-2}{dx\,dy}$
1032
is invariant under the action of $GL^+_2(\R)$.
1033
1034
The formula for the {\em Petersson inner product} is
1035
\[ \langle f,g\rangle=
1036
\int_{SL_2(\Z)\setminus\H}\left(f(z)\overline{g(z)}
1037
y^k\right)\,\frac{dx\,dy}{y^2}.\]
1038
1039
This could be considered either an integral over the fundamental region
1040
or, noting that the integrand is invariant under $SL_2(\Z)$, an integral
1041
over the quotient space. One then checks that for $f,g$ cusp forms, the
1042
integral will converge, since they go down exponentially as $z$ tends to
1043
infinity. It is then clear that this inner product is Hermitian.
1044
1045
It is not immediately clear that the Hecke operators are self-adjoint.
1046
\section*{January 31, 1996}
1047
\noindent{Scribe: William Stein, \tt <was@math>}
1048
\bigskip
1049
1050
{\bf \large Modular Curves}\smallskip
1051
1052
1053
1054
\subsection{Cusp Forms}
1055
Recall that if $N$ is a positive integer we define the congruence
1056
subgroups
1057
$\Gamma(N)\subset\Gamma_1(N)\subset\Gamma_0(N)$ by
1058
$$
1059
\begin{array}{cl}
1060
\Gamma_0(N) & = \{\abcd \in \modgp : c\equiv 0 \pmod{N}\}\\
1061
\Gamma_1(N) & = \{\abcd \in \modgp : a\equiv d\equiv 1, c\equiv 0 \pmod{N}\}\\
1062
\Gamma(N) & = \{\abcd \in \modgp : \abcd \equiv
1063
\bigl(\begin{array}{cc}1&0\\0&1\end{array}\bigr) \pmod{N}\}
1064
\end{array}
1065
$$
1066
1067
Let $\Gamma$ be one of the above subgroups.
1068
One can give a construction of the space $S_k(\Gamma)$ of cusp forms
1069
of weight $k$ for the action of $\Gamma$ using the language of
1070
algebraic geometry.
1071
Let $X_{\Gamma}=\Gamma\backslash\H^{*}$
1072
be the upper half plane (union the cusps)
1073
modulo the action of $\Gamma$. Then $X_{\Gamma}$ can be given the structure
1074
of Riemann surface. Furthermore,
1075
$S_2(\Gamma)=H^0(X_{\Gamma},\Omega^1)$ where
1076
$\Omega^1$ is the sheaf of differential 1-forms on $X_{\Gamma}$.
1077
This works since an element of $H^0(X_{\Gamma},\Omega^1)$
1078
is a differential form $f(z)dz$, holomorphic on $\H$ and
1079
the cusps, which is invariant with respect to the action
1080
of $\Gamma$. If $\gamma=\abcd\in\Gamma$ then
1081
$$d(\gamma(z))/dz=(cz+d)^{-2}$$
1082
so
1083
$$f(\gamma(z))d(\gamma(z))=f(z)dz$$
1084
iff $f$ satisfies the modular condition
1085
$$f(\gamma(z))=(cz+d)^{2}f(z).$$
1086
1087
There is a similiar construction when $k>2$.
1088
1089
\subsection{Modular Curves}
1090
$\modgp\backslash\H$ parametrizes isomorphism
1091
classes of elliptic curves. The other congruence subgroups also
1092
give rise to similiar parametrizations.
1093
$\Gamma_0(N)\backslash\H$ parametrizes pairs $(E,C)$ where
1094
$E$ is an elliptic curve and $C$ is a cyclic subgroup of order
1095
$N$, and $\Gamma_1(N)\backslash\H$ parametrizes pairs $(E,P)$ where
1096
$E$ is an elliptic curve and $P$ is a point of exact order $N$.
1097
Note that one can also give a point of exact order $N$ by giving
1098
an injection $\Z/N\Z\hookrightarrow E[N]$
1099
or equivalently an injection $\mu_N\hookrightarrow E[N]$
1100
where $\mu_N$ denotes the $N$th roots of unity.
1101
$\Gamma(N)\backslash\H$ parametrizes pairs $(E,\{\alpha,\beta\})$
1102
where $\{\alpha,\beta\}$ is a basis for
1103
$E[N]\isom(\Z/N\Z)^2$.
1104
1105
The above quotients spaces are {\em moduli spaces} for the
1106
{\em moduli problem} of determining equivalence classes of
1107
pairs ($E + $ extra structure).
1108
1109
\subsection{Classifying $\Gamma(N)$-structures}
1110
\begin{dfn}
1111
Let $S$ be an arbitrary scheme. An {\bfseries elliptic curve}
1112
$E/S$ is a proper smooth curve
1113
$$\begin{array}{c} E \\ \bigl| \\ S\end{array}$$
1114
with geometrically connected fibers all of genus one, together with a
1115
section ``0''.
1116
\end{dfn}
1117
1118
Loosely speaking, proper generalizes the notion of projective,
1119
and smooth generalizes nonsingularity. See Chapter III, section 10 of
1120
Hartshorne's {\em Algebraic Geometry} for the precise definitions.
1121
1122
\begin{dfn}
1123
Let $S$ be any scheme and $E/S$ an elliptic curve.
1124
A {\bfseries $\Gamma(N)$-structure} on $E/S$ is
1125
a group homomorphism
1126
$$\varphi:(\Z/N\Z)^2\into E[N](S)$$
1127
whose image ``generates'' $E[N](S)$.
1128
\end{dfn}
1129
1130
See Katz and Mazur, {\em Arithmetic Moduli of
1131
Elliptic Curves}, 1985, Princeton University Press, especially
1132
chapter 3.
1133
1134
Define a functor from the category of $\Q$-schemes to the
1135
category of sets by sending a scheme $S$ to the
1136
set of isomorphism classes of pairs
1137
$$(E, \Gamma(N)\mbox{\rm -structure})$$
1138
where $E$ is an elliptic curve defined over $S$ and
1139
isomorphisms (preserving the $\Gamma(N)$-structure) are taken
1140
over $S$. An isomorphism preserves the $\Gamma(N)$-structure
1141
if it takes the two distinguished generators to the two
1142
distinguished generators in the image (in the correct order).
1143
1144
\begin{thm}
1145
For $N\geq 4$ the functor defined above is representable and
1146
the object representing it is the modular curve $X(N)$ corresponding
1147
to $\Gamma(N)$.
1148
\end{thm}
1149
1150
What this means is that given a $\Q$-scheme $S$, the
1151
set $X(S)=Mor_{\Q\mbox{\rm -schemes}}(S,X)$ is isomorphic to
1152
the image of the functor's value on $S$.
1153
1154
There is a natural way to map a pair $(E,\Gamma(N)\mbox{\rm -structure})$
1155
to an $N$th root of unity.
1156
If $P,Q$ are the distinguished basis of $E[N]$ we send
1157
the pair $(E,\Gamma(N)\mbox{\rm -structure})$ to
1158
$$e_N(P,Q)\in\mu_N$$
1159
where $$e_N:E[N]\times E[N]\into \mu_N$$ is the Weil pairing. For
1160
the definition of this pairing see chapter III, section 8 of
1161
Silverman's {\em The Arithmetic of Elliptic Curves}. The Weil pairing
1162
is bilinear, alternating, non-degenerate, galois invariant, and
1163
maps surjectively onto $\mu_N$.
1164
\section*{February 2, 1996}
1165
\noindent{Scribe: Lawren Smithline, \tt <lawren@math>}
1166
(Note. These were done about a month after the fact, since the assigned
1167
person dropped the course. So they are terse.)
1168
\bigskip
1169
1170
Earlier, we looked at V. Miller's construction for eigenforms. (See, for
1171
instance, Lang X \S4 in the course references. This is a special miracle for
1172
$\SL_2(\Z)$.) Over $\Z$, $\T \cong \Z^d$ and $S(\Z)$ are dual, where $d =
1173
\dim S_k(\C)$.
1174
1175
Here is Shimura's explanation (Lang III \S5, VIII). The Hecke operator $\T_n$
1176
maps $S_k$ to itself. Let $A\subset \C$ be a subring and
1177
$$\T_A = A[\T_n : n > 0] \subseteq {\rm End}_\C S_k.$$
1178
Denote by $\T$, $\T_\Z$. There is a natural tensor product $\T_A \otimes_A
1179
\C \twoheadrightarrow \T_\C$.
1180
1181
There is also a complex conjugation automorphism of $S_k$ by
1182
$f \mapsto \overline{f(-\bar\tau)}$. This map is conjugate linear.
1183
The map $\tau \mapsto \exp(2\pi i \tau)$ becomes
1184
$\tau \mapsto \overline{\exp(-2\pi i \bar\tau)} = \exp(2 \pi i \tau)$.
1185
Say $f = \sum a_n q^n.$ Its conjugate
1186
$g = \sum \bar a_n q^n.$ If you know $S_k(\C) = \C \otimes_\Q S_k(\Q)$,
1187
then you know that modular forms can be conjugated in this sense.
1188
1189
There is an isomorphism $\T_\R \otimes_\R \C \stackrel\sim\longrightarrow
1190
\T_\C$ since the map is surjective and the complex dimensions on each side
1191
are equal.
1192
1193
Shimura (1959) exhibited the (Eichler-)Shimura isomorphism
1194
$$S_k(\C) \cong H^1(X_\Gamma,\R).$$
1195
We have $S_k(\C) = H^0(X_\Gamma, \Omega^1)$ and a map
1196
$H_1(X_\Gamma,\Z) \times S_k(\C) \rightarrow \C$.
1197
Now, $H_1(X,\Z) \cong \Z^{2d}$ embeds in $\Hom_\C(S_k(\C),\C) \cong
1198
\C^{2d}$ as a lattice.
1199
1200
So we have $S_k(\C) \rightarrow \Hom(H_1(X,\Z),\C))$ and
1201
$S_k(\C) \stackrel\sim\rightarrow \Hom(H_1(X,\Z),\R))$ as real vector
1202
spaces. By the Shimura isomorphism, this is isomorphic to $H^1(X,\R) \sim
1203
H^1_p(\Gamma,\R)$, the parabolic cohomology of $\Gamma$.
1204
1205
So $S_k(\C) \cong H^1_p(\Gamma, V_k)$, for a certain $d-1$ dimensional
1206
subspace $V_k$. (Let $W = \R \oplus \R$. $\Gamma$ acts by linear fractional
1207
transformation. Let $V_k = {\rm Sym}^{k-2} W$. There is a lattice in
1208
$S_k(\C)$
1209
corresponding to $H^1_p(\Gamma, {\rm Sym}^{k-2} \Z^2)$) We have an action
1210
of
1211
$\Gamma$ by $$f\cdot \c \mapsto \int_{\tau_0}^{\c(\tau_0)}
1212
f(\tau)\tau^{k-1}d\tau.$$
1213
1214
Recall $\T = \T_\Z$ is a set of endomorphisms of a lattice $L$, and $\T$
1215
has finite rank over $\Z$. We have the inclusion
1216
$S_k(\Z) \hookrightarrow S_k(\C)$, or equivalently, $$\Hom_\Z(\T,\Z)
1217
\hookrightarrow \Hom_\Z(\T,\C) = \Hom(\T_\C,\C) = S_k(\C) = S_k(\Z)
1218
\otimes_\Z \C.$$
1219
1220
Here is a nifty inner product (the Petersson innner product) on $S_k(\C)$.
1221
For $f,g \in S_k(\C)$, let
1222
$$\lan f,g \ran = \int_{\Gamma \backslash \H} f(\tau)g(\tau) y^k
1223
\frac{dx\,dy}{y^2}.$$
1224
The Hecke operators are self-adjoint for $(p, N) = 1$:
1225
$$\lan f | T_p, g \ran = \lan f, g | T_p \ran.$$
1226
Indeed, for $\a \in \GL_2^+ (\R)$,
1227
$$\lan f | \a , g | \a \ran = \lan f,g \ran.$$
1228
1229
\section*{February 5, 1996}
1230
\noindent{Scribe: Shuzo Takahashi, \tt <shuzo@math>}
1231
\bigskip
1232
1233
We have studied actions of $\T$ on
1234
$S_k(\C)$, $S_k(\Q)$ and $S_k(\Z)$ where
1235
$\Gamma = SL_2(\Z)$. What we know so far is
1236
$$S_k(\Z) \simeq {\rm Hom}_{\Z}(\T,\Z).$$
1237
Also we have studied the Peterson product. It is Hermitian, i.e.,
1238
$$\lan f|T_n, g\ran = \lan f,g|T_n\ran $$
1239
for $T_n \in \T$ and for $f,g \in S_k(\C)$.
1240
1241
\noindent
1242
Note: $T_n$ defined on $S_k(\C)$ preserves $S_k(\Z)$.
1243
1244
Today we study when $\Gamma = \G_1(N)$ or $\G_0(N)$ for
1245
$N \geq 1$.
1246
1247
\medbreak
1248
\noindent
1249
{\bf\large 1. The Diamond Operator and the Decomposition of
1250
$S_k(\G_1(N))$}
1251
1252
\begin{thm}
1253
$\G_1(N)$ is a normal subgroup of $\G_0(N)$ and we have
1254
$\G_0(N)/\G_1(N) \simeq (\Z/n\Z)^*$.
1255
\end{thm}
1256
1257
\begin{dfn}
1258
The diamond operator $< > $ is defined as follows: for
1259
$\pmatrix{a & b \cr c & d \cr} \in \G_0(N)$, the map on $S_k(\G_1(N))$
1260
$$f \rightarrow f | \pmatrix{a & b \cr c & d \cr} $$
1261
defines an endmorphism $< d> $ of $S_k(\G_1(N))$ which depends only
1262
on $d \ {\rm mod}\ N$. Thus we get an action $< > $ of
1263
$(\Z/n\Z)^*$ on $S_k(\G_1(N))$.
1264
\end{dfn}
1265
1266
Since $(\Z/n\Z)^*$ is a finite group, we have the following decomposition
1267
theorem:
1268
1269
\begin{thm}
1270
$$S_k(\G_1(N)) = \bigoplus_{\e} S_k(\G_0(N),\e)$$
1271
where $\e$ runs over the set of characters $(\Z/n\Z)^* \rightarrow \C^*$
1272
and $S_k(\G_0(N),\e)$ is defined as:
1273
$$S_k(\G_0(N),\e) = \{ f \in S_k(\G_1(N)) : f | < d> = \e(d) f\}.$$
1274
We have $S_k(\G_0(N),\e) = 0$ unless $\e(-1) = (-1)^k$.
1275
\end{thm}
1276
1277
\medbreak
1278
\noindent
1279
{\bf\large 2. The Hecke Operators on $S_k(\G_1(N))$}
1280
1281
For $n \geq 1$, we have the operation on $S_k(\G_1(N))$ of the $n$th Hecke
1282
operator $T_n$. The following are basic properties:
1283
1284
\begin{thm}
1285
1286
(1) $T_n$'s commute each other and with $< d> $.
1287
1288
(2) $T_n$'s preserve $S_k(\G_0(N),\e)$.
1289
1290
(3) if $(n,N) = 1$, then $\lan f | T_n,g\ran =
1291
\lan f,g|{< n>}^{-1}T_n\ran $.
1292
1293
(4) $\lan f | < d> ,g\ran =
1294
\lan f,g|{< d>}^{-1}\ran $.
1295
1296
(5) if $(n,N) = 1$, then $T_n$ is diagonalizable.
1297
1298
(6) if $(n,N) \neq 1$, then $T_n$ is not diagonalizable.
1299
1300
\end{thm}
1301
1302
The action of $T_n$ is described in the following theorem:
1303
1304
\begin{thm}
1305
Let $f = \sum_{n = 1}^\infty a_n q^n \in S_k(\G_0(N),\e)$. Then
1306
1307
%$$f | T_p =
1308
%\cases
1309
%\sum_{n=1}^{\infty} a_{pn} q^n + p^{k-1}\e(p) \sum_{n=1}^{\infty} a_n
1310
%q^{pn}
1311
%&\text{if $p \not| N$;} \\
1312
%\sum_{n=1}^{\infty} a_{pn} q^n
1313
%&\text{if $p | N$.}
1314
%\endcases$$
1315
1316
(1) $f | T_p = \sum_{n=1}^{\infty} a_{pn} q^n + p^{k-1}\e(p)
1317
\sum_{n=1}^{\infty} a_n q^{pn}$.
1318
(Note: when $p|N$, $\e(p)$ is considered to be $0$ and $U_p$ is used instead
1319
of $T_p$ which is called Atkin-Lehner operator.)
1320
1321
(2) if $(n,m)=1$, then $T_{nm} = T_n T_m$.
1322
1323
(3) if $p \not| N$, then $T_{p^l} = T_{p^{l-1}}T_p - p^{k-1}< p>
1324
T_{p^{l-2}}$.
1325
1326
(4) if $p | N$, then $T_{p^l} = (T_p)^l$.
1327
\end{thm}
1328
1329
The last formula in the theorem can be proved by comparing the coefficients
1330
of $q^{p^{l-1}}$ in both sides of the following formal identity:
1331
$$\left(\sum_{n=1}^{\infty} T_n q^n\right)|T_p =
1332
\sum_{n=1}^{\infty} T_{pn} q^n + p^{k-1}< p>
1333
\sum_{n=1}^{\infty} T_n q^{pn}.$$
1334
For example, the coefficient of $q^p$ in the LHS is $T_p T_p$. On the other
1335
hand, the coefficient of $q^p$ in the RHS is
1336
$T_{p^2} + p^{k-1}< p> T_1$. Thus, we have
1337
$$T_{p^2} = (T_p)^2 - p^{k-1}< p> Id$$
1338
where $< p> $ should be considered to be a null map if $p | N$.
1339
1340
\medbreak
1341
\noindent
1342
{\bf\large 3. The Old Forms}
1343
1344
Suppose $M|N$. Let $f \in S_k(\Gamma_1(M))$. Then for $d$ such that
1345
$d | \frac{N}{M}$, $f(d\tau) \in S_k(\G_1(N))$.
1346
Thus we have a map
1347
$$\phi_M : \bigoplus_{d | \frac{N}{M}} S_k(\Gamma_1(M)) \rightarrow
1348
S_k(\G_1(N)).$$
1349
The old part of $S_k(\G_1(N))$ is defined as the subspace generated by the
1350
images of $\phi_M$ for $M | N$, $M \neq N$.
1351
1352
\begin{eg} $\phi_M$ is not injective. Consider the case that $k = 12$,
1353
$M = p$ and $N = p^2$. $S_k(\Gamma_1(p))$ contains $\Delta(\tau)$ and
1354
$\Delta(p\tau)$. But $\phi_p$ maps both of them to $\Delta(p\tau)$ in
1355
$S_k(\Gamma_1(p^2))$.
1356
\end{eg}
1357
1358
\begin{thm}
1359
Suppose $p \nmid N$. Consider $f,g \in S_k(\G_1(N))$. Then $f$ and
1360
$g(p\tau)$ are both in $S_k(\Gamma_1(Np))$. Then we have
1361
$$f | U_p = (f | T_p) - p^{k-1}\e(p)(f(p\tau))$$
1362
and
1363
$$g(p\tau) | U_p = g(\tau)$$
1364
where $f | T_p$ is considered in $S_k(\G_1(N))$.
1365
\end{thm}
1366
\noindent
1367
{\bf Proof.} Let $f = \sum_{n=1}^{\infty} a_n q^n$. Then, considering in
1368
$S_k(\G_1(N))$, we have
1369
$$
1370
f | T_p =
1371
\sum_{n=1}^{\infty} a_{pn} q^n + p^{k-1}\e(p) \sum_{n=1}^{\infty} a_n
1372
q^{pn}.$$
1373
Also, considering in $S_k(\Gamma_1(Np))$, we have
1374
$$f | U_p = \sum_{n=1}^{\infty} a_{pn} q^n.$$
1375
Thus, we have
1376
$$f | U_p = (f | T_p) - p^{k-1}\e(p)(f(p\tau)).$$
1377
Now, let $g = \sum_{n=1}^{\infty} b_n q^n$. Then
1378
$$g(p\tau) | U_p =
1379
\left(\sum_{n=1}^{\infty} b_{n/p} q^n \right) | U_p = g(\tau)$$
1380
where $b_{n/p} = 0$ unless $p | n$.
1381
1382
1383
1384
\section*{February 7, 1996}
1385
\noindent{Scribe: Amod Agashe, \tt <amod@math>}
1386
\bigskip
1387
1388
We are in the process of showing that the Hecke operators $T_p$ acting
1389
on the space of cusp forms $S_k(\Gamma_1(N))$ are not necessarily
1390
semisimple if $p\mid N$.
1391
1392
Recall from last time that if $M \mid N$ then for every divisor $d$
1393
of $M/N$, we had a map $S_k(\Gamma_1(M)) \rightarrow S_k(\Gamma_1(N))$
1394
given by $f(\tau) \mapsto f(d\tau)$.
1395
1396
Note that the various $f(d\tau)$'s are linearly independent over $\C$,
1397
because the Fourier expansion of $f(d\tau)$ starts with $q^d$.
1398
1399
Let $f$ be an eigenfuntion for all the Hecke operators $T_n$ in
1400
$S_k(\Gamma_1(M))$. Let $p$ be a prime not dividing $M$.
1401
So $f\mid T_p = af$ where $a=a_p(f)$ and $f \mid <p>=\epsilon (p)f$
1402
where $\epsilon (p)$ is the character associated to the modular
1403
form $f$. Note that one can prove that if $f$ is an eigenfunction
1404
for the $T_n$'s then it is an eigenfunction for the diamond operators
1405
also (or alternatively, make it part of the definition of eigenform).
1406
Let $N=p^\alpha M$ with $\alpha \geq 1$. We will look at the action of
1407
the $p^{th}$ Hecke operator $U_p$ in $S_k(\Gamma_1(N))$ on the
1408
images of $f$ under the maps described above. Let $f_i(\tau) = f(p^i \tau)$
1409
for $0\leq i\leq \alpha$. As we showed earlier, \\
1410
$$f\mid T_p = \sum a_{np}q^n + \epsilon (p)p^{k-1}\sum a_nq^{pn}.$$
1411
So $$af = f_0\mid U_p + \epsilon (p)p^{k-1}f_1.$$
1412
Thus, $$f_0\mid U_p = af_0 -\epsilon(p)p^{k-1}f_1.$$
1413
From last time, we have $f_1\mid U_p = f_0.$ In fact, in general,
1414
one can see easily that $f_i\mid U_p=f_{i-1}$ for $i\geq 1$.
1415
1416
So $U_p$ preserves the 2-dimensional space spanned by $f_0$ and $f_1$.
1417
The matrix of $U_p$ (acting on the right) with this basis is given
1418
(from the equations above) by: \smallskip
1419
\[ \left( \begin{array}{cc}
1420
a & 1 \\
1421
-\epsilon(p)p^{k-1} & 0
1422
\end{array}
1423
\right)
1424
\]
1425
The characteristic polynomial of this matrix is $x^2-ax+p^{k-1}\epsilon(p)$.
1426
1427
There is the following striking coincidence:
1428
Let $E$ be the number field generated over $\Q$ by the coefficients of the
1429
Fourier series expansion of $f$ and let $\la$ be a prime ideal
1430
of $\O_E$ lying over some rational prime $l$.
1431
Then we have a Galois representation \smallskip
1432
$$\rho_\la: Gal(\overline{\Q}/\Q) \rightarrow \GL_2(E_\la)$$
1433
If $p\not|Nl$ then $\rho_\la$ is unramified and also
1434
$det\ \rho_\la(Frob_p)=\epsilon(p)p^{k-1}$ and
1435
$tr\ \rho_\la(Frob_p)=a_p(f)=a$. Thus the characteristic
1436
polynomial of $\rho_\la(Frob_p)$ is $x^2-ax+p^{k-1}\epsilon(p)$,
1437
the same as that of the matrix of $U_p$!
1438
1439
A question one can ask is: Is $U_p$ semisimple on the space spanned by
1440
$f_0$ and $f_1$? The answer is yes if the eigenvalues of $U_p$ are
1441
different.
1442
1443
Now, the eigenvalues are the same iff the
1444
discriminant of the characteristic polynomial is zero i.e.
1445
$a^2=4\epsilon(p)p^{k-1}$ i.e. $a=2p^{\frac{k-1}{2}}\zeta$ where
1446
$\zeta$ is some square root of $\epsilon(p)$.
1447
Here is a curious fact: the Ramanujan-Petersson conjecture proved by Deligne
1448
says $|a|\leq 2p^{\frac{k-1}{2}}$; thus the above equality is allowed
1449
by it, so we do not get any conclusion about the semisimplicity of
1450
$U_p$.
1451
1452
Let us now specialize to $k=2$. Weil has shown that $\rho_\la(Frob_p)$
1453
is semisimple. Thus if the eigenvalues of $U_p$ are equal, then
1454
$\rho_\la(Frob_p)$ is a scalar. Edixhoven proved that it is not.
1455
So the eigenvalues of $U_p$ are different and hence $U_p$ is semisimple
1456
in this case. So this example (for k=2) does not give us an example
1457
of $U_p$ being not semisimple.
1458
1459
There is the following example given by Shimura which shows that the Hecke
1460
operator $U_p$ need not be semisimple. Let $W$ denote the space spanned
1461
by $f_0, f_1$ and $V$ denote the space spanned by $f_0,f_1,f_2,f_3$.
1462
$U_p$ preserves both spaces $W$ and $V$, so it acts on $V/W$. The action
1463
is given by $\overline{f_2}\mapsto \overline{f_1}=0$ and
1464
$\overline{f_3}\mapsto \overline{f_2}$ where the bar denotes the image
1465
in $V/W$. Thus the matrix of $U_p$ on the space $V/W$ is
1466
\[ \left( \begin{array}{cc}
1467
0 & 1 \\
1468
0 & 0
1469
\end{array}
1470
\right)
1471
\]
1472
which is nilpotent, and in particular not semisimple. If $U_p$ were
1473
semisimple on $V$ then it would be semisimple on $V/W$ also; but we
1474
have just shown that it is not. Thus $U_p$ is not semisimple
1475
on $V$, and hence not on $S_2(\Gamma_1(M))$ (because $V$ is invariant
1476
under $U_p$).
1477
1478
\bigskip
1479
1480
We next discuss the structure of the $\C$-algebra $\T =\T_\C$ generated
1481
by the Hecke and diamond operators and the structure of $S_k(\Gamma_1(N))$
1482
as a $\T$-module.
1483
1484
First we consider the case of level $1$ i.e. $N=1$.
1485
Then $\Gamma_1(1)=SL_2(\Z)$. All the $T_n$'s are diagonalizable.
1486
$S_k=S_k(\Gamma_1(N))$ has a basis of $f_1,....,f_d$ of normalized eigenforms
1487
where $d=dim(S_k)$. Thus $S_k\cong \C^d$ as a $\C$-vector space.
1488
Then we have the $\C$-algebra homomorphism $\T\rightarrow \C^d$ given by
1489
$T\mapsto (\la_1,....,\la_d$) where $f_i\mid T=\la_i f_i$.
1490
It is injective because if the image of $T$ is zero, then it kills all
1491
$f_i$ i.e. all of $S_k$ i.e. it is the zero operator. The map is
1492
surjective because $\T$ has dimension $d$.
1493
%and hence it is surjective
1494
%when we consider both sides as $\C$-vector spaces.
1495
Thus as a $\C$-algbebra, $\T\cong \C^d$.
1496
Next, we claim that the modular form $v=f_1+...+f_d$ generates
1497
$S_k$ as a $\T$-module. This follows because under the
1498
map $S_k\cong \C^d, v\mapsto (1,....,1)$ and our
1499
statement is just the trivial fact that $(1,....,1)$ generates
1500
$\C^d$ as a $\C^d$-module (acting component-wise).
1501
1502
Thus $S_k$ is free of rank $1$ as a $\T$-module.
1503
We already know that $S_k\cong Hom(\T,\C)$ as $\T$-modules.
1504
Thus $\T\cong Hom(\T,\C)$ as $\T$-modules. In fact the isomorphism
1505
is canonical since the $f_i$'s are normalized.
1506
We remark that $v$ in fact lies in $S_k(\Q)$.
1507
1508
Next, we deal with the general case where the level is not necessarily $1$.
1509
1510
First we need to talk about newforms. Recall the maps
1511
$S_k(\Gamma_1(M)) \rightarrow S_k(\Gamma_1(N))$
1512
for every divisor $d$ of $M/N$ mentioned at the beginning of this lecture.
1513
The old part of $S_k(\Gamma_1(N))$ is defined as the space generated by
1514
all the images of $S_k(\Gamma_1(M))$ for all $M\mid N, M\neq N$
1515
under these maps.
1516
The new part of $S_k(\Gamma_1(N))$ can be defined in two different ways.
1517
Firstly we can define it as the orthogonal complement of the old part
1518
with respect to the Petersson inner product.
1519
There is also an algbraic definition
1520
as follows. There are certain maps going the other way:
1521
$S_k(\Gamma_1(N)) \rightarrow S_k(\Gamma_1(M))$ for $M\mid N, M\neq N$.
1522
The new part is the space killed under all these maps. The space
1523
of newforms, denoted $S_k(\Gamma_1(N))_{new}$ is like $S_k(\Gamma(1))$ in the
1524
sense that all the $T_n$'s (including $U_p$) are semisimple and there
1525
is a basis consisting of newforms. A form of level $N$ is said to
1526
be new of level $N$ if it is in $S_k(\Gamma_1(N))_{new}$.
1527
1528
Next, one can show that the map
1529
$\bigoplus_{M\mid N,M\leq N} S_k(\Gamma_1(M))_{new} \rightarrow
1530
S_k(\Gamma_1(N))$ given by $f(\tau)\mapsto f(d\tau)$ for $d\mid \frac{N}{M}$
1531
is injective (See W.-C. W.Li, Newforms and functional equations,
1532
Math. Annalen, 212(1975), 285-315).
1533
Note that an eigenform in one of the subspaces of the source
1534
need not be an eigenfuntion for all the operators in the image.
1535
If $f$ is a newform, then let $M_f$ denote its level (i.e. $f$
1536
is new of level $M_f$). Let $S$ be the set of newforms of weight $k$
1537
and some level dividing $N$. Let
1538
$$v=\sum_{f\in S} f(\frac{N}{M_f}\tau).$$
1539
Then one can show that $S_k(\Gamma_1(N))$ is free of rank $1$ over
1540
$\T_\C$ with $v$ as the basis element. Also one can show that
1541
$v$ has rational coefficients.
1542
1543
\section*{February 9, 1996}
1544
\noindent{Scribe: J\'anos Csirik, \tt <janos@math>}
1545
\vspace{.2in}
1546
1547
\subsubsection*{Final comments about Hecke algebras}
1548
1549
Recall that for the case $\Gamma=SL(2,\Z)$, if we set $f_1,\ldots,f_d$
1550
to be the normalized eigenforms (newforms of level 1), then they have
1551
possibly complex coefficients but in any case $\{f_i\}$ is finite and
1552
stable under automorphisms of $\C$ and all the coefficients
1553
$a_n(f_i)$ lie in some number field. Furthermore this field is totally
1554
real: to show this we used that since the set is stable under
1555
conjugation, it suffices to show that all the $a_n(f_i)$ are real, which
1556
followed by remarking that they are eigenvalues of the operators $T_n$
1557
which are self-adjoint with respect to the Petersson inner product.
1558
1559
More generally, let $f\in S(\Gamma_1(N))$ be a normalized eigenform of
1560
character $\varepsilon$. Then $a_n=\varepsilon(n)\overline{a_n}$.
1561
1562
Note that the algebra $\T_\Q$ generated by the $T_i$ over $Q$ contains
1563
the diamond bracket operators: the formula relating $T_{p^2}$ and
1564
$(T_p)^2$ tells us that the difference is $p^{k-1}\varepsilon(p)$, so
1565
$\varepsilon(p)\in\T_\Q$. Using Dirichlet's Theorem on primes in
1566
arithmetic progressions, for any $d$ relatively prime to $N$ we can find
1567
a prime $p\equiv d\pmod{N}$, so $\varepsilon(d)=\varepsilon(p)\in\T_\Q$.
1568
1569
If the space of modular forms has dimension 1, then it is spanned by a
1570
(normalized) eigenform with rational coefficients, so the eigenvalues
1571
are all in $\Q$.
1572
1573
The next simplest example is $k=24$, which is the smallest weight such
1574
that the dimension is more than one. There are two eigenforms, which
1575
are conjugate to each other. If $f=\sum a_nq^n$ is an eigenform, then
1576
$\Q(\ldots a_n\ldots)=\Q(\sqrt{144169})$. In fact $S_{24}$ is spanned
1577
by $\Delta^2$ and $\Delta^2|T_2$. (Note that $\Delta^2$ is definitely
1578
not an eigenform since its $q$-coefficient is 0.) The action of $T_2$
1579
on $S_{24}$ with respect to this basis is described by a two by two
1580
matrix of trace $1080$ and determinant $-2^{10}3^2 2221$. For high $k$,
1581
eigenforms tend to form a single orbit; however, no proof
1582
is known for this.
1583
1584
For every newform $f$, let $E_f$ be the number field generated by its
1585
coefficients. Let $\Sigma$ be a set of representatives for $f$'s modulo
1586
$Gal(\bar\Q/\Q)$. Define
1587
\begin{eqnarray*}
1588
\T_\Q & \rightarrow & E_f \\
1589
T & \mapsto & \lambda_T,
1590
\end{eqnarray*}
1591
with $f|T=\lambda_Tf$. Thus $T_n\mapsto a_n$ so this map is surjective.
1592
1593
In fact the induced
1594
\[ \T_\Q \to \prod_{f\in\Sigma} E_f \]
1595
is an isomorphism of $\Q$-algebras: it is injective since if $T$ dies on
1596
the image then it acts as zero on $f\in\Sigma$ and so on all of f, by
1597
the rationality of Hecke operators: ${}^\sigma\!(g|T)={}^\sigma\!g|T$.
1598
(And if an operator acts as zero on everything, then it is zero.)
1599
1600
Here we used the fact that there were no oldforms around.
1601
1602
For example, consider $S_2(\Gamma(N))$ for $N$ prime. Then
1603
$S_2(\Gamma(1))$ is empty, hence we get an isomorphism
1604
\[
1605
\T_\Q\cong E_1\times\ldots\times E_t,
1606
\]
1607
with the right hand side a
1608
product of totally real number fields. $t>1$ is possible, e.g. for
1609
$N=37$, $\T_\Q=\Q\times\Q$.
1610
1611
In general, oldforms complicate the situation.
1612
1613
1614
1615
\subsubsection*{Final comments about Hecke algebras}
1616
1617
We'll only treat the case $k=2$. Then (for $\Gamma$ a congruence
1618
subgroup),
1619
\[ S_2(\Gamma)=H^0(X_\Gamma,\Omega^1) \]
1620
where $X_\Gamma=\Gamma\setminus\H\cup\Gamma\setminus\P^1(\Q)$, with
1621
$\Gamma\setminus\P^1(\Q)$ being the set of cusps that we need to adjoin
1622
to make it a compact Riemann surface.
1623
1624
Therefore
1625
\[ {\rm dim}\,S_2(\Gamma)= g(X_\Gamma). \]
1626
1627
{\sc Example.\ } $SL(2,\Z)\setminus\P^1(\Q)$ has just one point. The
1628
proof involves the Euclidean algorithm: any element of $\P^1(\Q)$ can be
1629
written as $\left( \begin{array}{cc} x\\y \end{array} \right)$ with $x$
1630
and $y$ relatively prime integers. By the Euclidean Algorithm, we can
1631
find $a$ and $b$ integers such that $ax+by=1$. Then
1632
\[
1633
\left(\begin{array}{cc} a & b \\ -y & x \end{array} \right)
1634
\left( \begin{array}{cc} x\\y \end{array} \right) =
1635
\left( \begin{array}{cc} 1\\0 \end{array} \right)
1636
\]
1637
1638
To calculate $g(X_\Gamma)$, use the following covering (recall that
1639
$X_\Gamma$ is the compactification of $\Gamma\setminus\H$ we obtain by
1640
adjoining the cusps)
1641
\[
1642
X_\Gamma \rightarrow X_{\Gamma(1)},
1643
\]
1644
keeping in mind the isomorphism
1645
\begin{eqnarray*}
1646
j:X_{\Gamma(1)} & \rightarrow & \P^1(\C) \\
1647
(i,\rho,\infty) & \mapsto & (1728,0,\infty)
1648
\end{eqnarray*}
1649
The only ramification in our covering
1650
occurs above the points $0,1728,\infty$.
1651
1652
{\sc Example.\ } Let $\Gamma=\Gamma_0(N)$. The degree of the covering
1653
is $(PSL(2,\Z):\Gamma_0(N)/\pm1)$ which is the number of cyclic subgroups
1654
of order $N$ in $SL(2,\Z/N\Z)$. We have a covering
1655
$Y_0(N)\to Y_{\Gamma(1)}$ where $Y_0(N)$ parametrizes elliptic curves
1656
$E$ with a cyclic subgroup of order $N$,
1657
$C\subseteq E[N]\cong\Z/N\Z\times\Z/n\Z$ up to isomorphism; and
1658
$Y_{\Gamma(1)}$ parametrises elliptic curves. The isomorphism
1659
$(E,C_1)\cong(E,C_2)$ is an automorphism $\alpha$ of $E$ with
1660
$\alpha:C_1\mapsto C_2$. Usually $\alpha=\pm1$, unless
1661
$j=1728,0,\infty$.
1662
1663
If we understand ramification, we can use the {\em Riemann-Hurwitz
1664
formula}. The following mnemonic way of thinking about it is due to
1665
N. Katz. The Euler characteristic (alternating sum of the dimensions of
1666
cohomology groups) should be thought of as totally additive:
1667
\[ \chi(A\coprod B) = \chi(A) + \chi(B). \]
1668
1669
If $X$ is a Riemann surface of genus $G$, $\chi(X)=2-2G$. A single
1670
point has Euler characteristic $\chi(P)=1$. Hence
1671
\[ \chi(X\setminus\{P_1,\ldots,P_n\})=2-2G-n. \]
1672
Therefore
1673
\[ \chi(X_{\Gamma(1)}\setminus\{0,1728,\infty\})=-1 \]
1674
and
1675
\[ \chi(X_\Gamma\setminus\{\mbox{points over $1728,0,\infty$}\})=2-2g-n \]
1676
if $n$ points lie over ${0,1728,\infty}$.
1677
If the covering map $X_\Gamma\to X_{\Gamma(1)}$ has degree $d$, then we
1678
can think of the top space as $d$ copies of the bottom space, so
1679
\[ d \cdot (-1) = 2-2g-n \]
1680
and therefore
1681
\[ 2g-2 = d-n = d-n_0 - n_{1728} -n_\infty.\]
1682
1683
{\sc Example.\ } Let $\Gamma=\Gamma(N)$. What happens over $j=0$?
1684
This corresponds to an elliptic curve $E$ with an automorphism $\alpha$
1685
of order three. Let $N>3$.
1686
1687
For any $(E,P,Q)$, we have $(E,\alpha P,\alpha Q)$ and
1688
$(E,\alpha^2 P,\alpha^2 Q)$ which are isomorphic to it and hence they
1689
are the same point on $X_\Gamma$. So $E$ only has one-third the usual
1690
number of points lying over it, $n_0=d/3$. (Except if the above three
1691
points are equal, i.e., $\alpha$ fixes $E[N]$. This can't happen since
1692
$N>3$.)
1693
1694
Similarly we get $n_{1728}=d/2$.
1695
1696
To determine the degree $d$, fix $E$ and count the points lying over it:
1697
these are all of the form $(\C/\Z\oplus\tau\Z,1/n,\tau/N)$ with Weil
1698
pairing $e^{2\pi i/N}$ and all such occur, so we need to
1699
count the number of $P,Q\in E[N]$ which form a basis of
1700
$E[N]$ and $e_N\langle P,Q\rangle=e^{2\pi i/N}$. This gives us the
1701
order of $SL(2,\Z/n\Z)$. However, (since $N\neq2$) we have to take into
1702
account that $(E,P,Q)\cong(E,-P,-Q)$ but they are not equal
1703
so the degree of the covering is
1704
$\#(SL(2,\Z/N\Z))/2$.
1705
1706
We also have $d=(PSL(2,\Z):\Gamma(N)/(\Gamma(N)\cap\pm1))$ since
1707
$\Gamma(N)\setminus SL(2,\Z)\cong SL(2,\Z/N\Z)$.
1708
1709
So we have established that
1710
\[ 2g-2 = d-n_0 - n_{1728} -n_\infty = d/6 - n_\infty. \]
1711
To determine $n_\infty$, note that $SL(2,\Z)$ acts on
1712
$\P^1(\Q)\ni\infty=
1713
\left( \begin{array}{cc} 1 \\ 0 \end{array} \right)$.
1714
The stabilizer $\Gamma(N)_\infty$
1715
of $\left( \begin{array}{cc} 1 \\ 0 \end{array} \right)$
1716
is $U=\pm\left( \begin{array}{cc} 1 & * \\ 0 & 1 \end{array} \right)$
1717
So the index
1718
$((\modgp/\pm1)_\infty:\Gamma(N)_\infty)=N$ is the ramification degree
1719
of a point over $\infty$, so $n_\infty=N/d$.
1720
1721
Hence $2g(X(N))-2=d/6-d/N$.
1722
\section*{February 12, 1996}
1723
\noindent{Scribe: J\'anos Csirik, \tt <janos@math>}
1724
\vspace{.2in}
1725
1726
\def\T{{\bf T}} % Hecke algebra
1727
\def\qed{\hfill $\blacksquare$}
1728
1729
Plug: A useful reference for the next lecture is Andrew Ogg: {\em Rational
1730
points on certain elliptic modular curves} (1972).
1731
1732
From last lecture's results on easily deduces that
1733
\[ g(X(N)) = 1 + \frac{d}{12N}(N-6). \]
1734
1735
{\sc Example.} Let $N=5,7$. If $N$ is prime then the degree of the
1736
covering is $\frac{(N^2-1)(N^2-N)}{N-1}$. Therefore $N=5$ gives $d=60$
1737
and $g=0$ (the Galois group of the covering in this case is $A_5$.)
1738
Similarly, $N=7$ yields $g=3$ and $d=168$. (Remark: For $p\geq5$ prime, the
1739
group $SL(2,\Z/p\Z)/\pm1=L_2(p)$ is simple.)
1740
1741
{\sc Example.} What is the genus of $X_0(N)$, for $N$ a prime?
1742
1743
It is an exercise to show that there are two cusps:
1744
$\left( \begin{array}{cc} 1 \\ 0 \end{array} \right)=\infty$
1745
and
1746
$\left( \begin{array}{cc} 0 \\ 1 \end{array} \right)=0$.
1747
The covering $X_0(N)\to X(1)$ is easily seen to have degree $N+1$, since
1748
an elliptic curve has $N+1$ cyclic subgroups of order $N$. Here
1749
$\infty$ is unramified and $0$ has ramification index $N$.
1750
Hence
1751
\[ 2g-2 = N+1 -n_\infty-n_{1728}-n_0 \]
1752
with $n_\infty=2$, $n_{1728}$ approximately $d/2$ and
1753
$n_0$ approximately $d/3$.
1754
1755
To calculate $n_0$, we need the number of isomorphism classes $(E,C)$
1756
with $E$ fixed with $End(E)=\Z[\mu_6]$. $\mu_6$ acts on the
1757
set of $C$'s. Since $\pm1$ is acting trivially, we really have an
1758
action of $\mu_3$ with $(E,C)\cong(E,\zeta C)\cong(E,\zeta^2C)$ with
1759
$\zeta$ some third root of unity.
1760
1761
If we consider $E=\C/\O$ with $\O=\Z[(-1+i\sqrt3)/2]$, then
1762
$E[N]=\O/N\O$ as an $\O$-module. In fact
1763
\[ \O/N\O =
1764
\left\{ \begin{array}{ll}
1765
\F_N\oplus\F_N & \mbox{if N splits in $\O$} \\
1766
\F_{N^2} & \mbox{if N does not split in $\O$.}
1767
\end{array}
1768
\right.
1769
\]
1770
The above covers all possibilities, because $N>3$ can't ramify in $\O$.
1771
By Kummer's theorem, $N$ splits in $\O$ iff
1772
$\left(\frac{-3}{N}\right)=1$, and
1773
there are exactly two $\O$-stable submodules of $\O/N\O$. In the second
1774
case, which happens iff $\left(\frac{-3}{N}\right)=-1$, $\O/N\O$ has no
1775
$\O$-stable submodules. Therefore
1776
\[ n_0 =
1777
\left\{ \begin{array}{ll}
1778
\frac{N-1}{3} + 2 &
1779
\mbox{if $N$ splits in $\O$, i.e., $N\equiv1\pmod3$.}\\
1780
\frac{N+1}{3} &
1781
\mbox{if $N$ does not split in $\O$, i.e., $N\equiv2\pmod3$}
1782
\end{array}
1783
\right.
1784
\]
1785
In the first case, note that the two $\O$-stable submodules have to be
1786
counted separately.
1787
1788
Similarly we obtain
1789
\[ n_{1728} =
1790
\left\{ \begin{array}{ll}
1791
\frac{N-1}{2}+2 & \mbox{if $N\equiv1\pmod4$} \\
1792
\frac{N+1}{2} & \mbox{if $N\equiv3\pmod4$}
1793
\end{array}
1794
\right.
1795
\]
1796
depending on whether$N$ splits in $\Q(i)$.
1797
1798
{\sc Examples.} For $N=37$ the genus is $2$. Using the formula, we get
1799
$2g-2=36-(2+18)-14=2$ (so it works).
1800
1801
Using the formula we can also obtain $g(X_0(13))=0$ and $g(X_0(11))=1$.
1802
1803
It is therefore clear that the genus of $X_0(N)$ is approximately
1804
$N/12$. In the article by Serre in the Lecture Notes in Mathematics 349
1805
(Antwerp), we find the following table. Write $N=12a+b$ with $0\leq
1806
b\leq11$. Then
1807
\begin{center}
1808
\begin{tabular}{c|cccc}
1809
$b$ & $1$ & $5$ & $7$ & $11$ \\ \hline
1810
$g$ & $a-1$ & $a$ & $a$ & $a+1$. \\
1811
\end{tabular}
1812
\end{center}
1813
Hence
1814
\[
1815
1 + g(X_0(p)) = {\rm dim}\,M_{p+1}(\modgp).
1816
\]
1817
1818
It was Serre's idea to think of ``modular forms mod $p$'', for some
1819
congruence subgroup
1820
$\Gamma\ni\left(\begin{array}{cc}1&1\\0&1\end{array}\right)$, like
1821
$\Gamma_0(N)$ or $\Gamma_1(N)$. We could use our moduli theoretic
1822
interpretations, but instead we'll define
1823
\[
1824
M_k(\Gamma,\F_p) \subseteq \F_p[[q]].
1825
\]
1826
1827
By Shimura's cohomology trick, we know that $M_k(\Gamma,\Z)$ is a
1828
lattice in $M_k(\Gamma,\C)$. Hence we can set
1829
\[
1830
M_k(\Gamma,\F_p) = M_k(\Gamma,\Z) \otimes_{\Z} \F_p.
1831
\]
1832
Then {\em Serre's equality} states that for a prime $p$,
1833
\[
1834
M_{p+1}(\modgp,\F_p) = M_2(\Gamma_0(p),\F_p)
1835
\]
1836
in $\F_p[[q]]$. The philosophy is {\em mod $p$ forms with $p$ in the
1837
level can be taken to mod $p$ formswith no $p$ in the level, but of a
1838
higher weight}. So for example
1839
$M_k(\Gamma_1(p^\alpha N),\F_p)$ is a subset of
1840
$M_?(\Gamma_1(N),\F_p)$ of forms of some higher level.
1841
1842
Finally, consider the map from the right hand side to the left hand
1843
side in Serre's equality. Recall that
1844
\[ G_k = \frac{-B_k}{2k} + \sum^\infty_{n=1}\sigma_k(n) q^n
1845
\in M_k(\modgp). \]
1846
By Kummer, ${\rm ord}_p(B_{p-1}) = -1$, so
1847
\[
1848
E_{p-1}=1+\frac{-2(p-1)}{B_{p-1}}\sum\sigma_{p-1}(n)q^n \equiv 1 \pmod{p}.
1849
\]
1850
Hence we got the map from the right to the left:
1851
multiply by $E_{p-1}$ to get to $M_{p+1}(\Gamma_0(p),\F_p)$. Then take
1852
the trace to get to $M_{p+1}(\modgp,\F_p)$. The trace map is dual to
1853
the inclusion and is expressed by
1854
\[
1855
\tr(f) = \sum_{i=1}^{p+1} f | \gamma_i
1856
\qquad
1857
\gamma_i\in\Gamma_0(p)\setminus\modgp.
1858
\]
1859
\section*{February 14 and 16, 1996}
1860
\noindent{Scribe: Jessica Polito, \tt <polito@math>}
1861
\bigskip
1862
1863
1864
Our goal, for these two days, is to define the modular curves $\X$ over
1865
$\Q(\mu_N)$, and
1866
$X_1(N)$, and $X_0(N)$ over $\Q$.
1867
These notes will spell out the construction of $\X$, with some
1868
discussion of the construction of the other two types of curves.
1869
The idea comes from Shimura.
1870
1871
Let $\Q(t)$ be the function field of $\P^1/\Q$, and pick an elliptic
1872
curve $\E/\Q(t)$ with $j$-invariant $t$:
1873
$$\E: y^2 = 4x^3 - \frac{27t}{t-1728}x - \frac{27t}{t-1728}$$
1874
(Note that the general formula for the $j$-invariant of a curve $y^2 =
1875
4x^3 - g_2x -g_3$ is $j = \frac{1728g_2^3}{g_2^3 - 27g_3^2}$, and
1876
that $j(E)$ determines the isomorphism class of the given curve $E$
1877
over the algebraic closure of the field of definition.)
1878
1879
By substituting in a given value $j$ for $t$, we would get a formula
1880
for an elliptic curve over $\Q(j)$ with $j$-invariant $j$; for $j= 0 $
1881
or 1728, we could pick a diferent formula for $\E$ which would give an
1882
isomorphic curve over $\Q(t)$, for which that substitution would make
1883
sense.
1884
1885
Notice that, in general, if we have an elliptic curve $E/K$, with $K$
1886
some field of charictaristic prime to $N$, then we can consider
1887
$E[N](\K)$, the set of all $N$-torsion point of $E$ defined over $\K$,
1888
which is isomorphic to $(\Z/N\Z)^2$. Then we let $K(E[N])$ be the
1889
smallest extension of $K$ over which all the points of $E[N]$ are
1890
defined. Notice also that $E/K$ and $N$ together define a
1891
representation of the Galois groups $\gal(\K/K)$ into the
1892
automorphisms of the $N$-torsion points of $E$, as they are defined by
1893
polymial equations with coefficients in $E$. We get
1894
$$\rho_{E,N}: \gal(\K/K) \into \aut(E[N]) \isom \GL_2(\Z/N\Z)$$
1895
with $\gal(\K/K(E[N])) = \ker(\rho_{E,N})$. Unsurprisingly, we will
1896
frequently leave out the $N$, writing simply $\rho_E$.
1897
1898
Now, to return to our construction, where $K=\Q(t)$. We will show
1899
that $\Q(t)(E[N])$ is the function field of a curve defined over
1900
$\Q(\mu_N)$ (so $\Q(t)(E[N]) \cap \overline{\Q} =