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2\documentclass[11pt]{article}
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7
8\title{Manin symbols and modular forms\\{\large (first draft)}}
9\author{W. Stein\footnote{UC Berkeley, Department of Mathematics, Berkeley, CA  94720, USA.}}
10\include{macros}
11\begin{document}
12\maketitle
13%\begin{abstract}
14%\end{abstract}
15\tableofcontents
16\section{Introduction}
17\begin{quote}
18The object of numerical computation is theoretical advance.''
19
20-- Atkin
21\end{quote}
22
23The definition of the
24spaces $S_k(\Gamma)$ of modular forms as functions on the upper half
25plane $\h$ satisfying a certain equation is very abstract.
26The definition of the Hecke operators even more so.
27
28We are fortunate that we now have methods available which allow us
29to transform the vector space of cusp forms of given weight $\geq 2$
30and level $N$ into a concrete object, which can be
31explicitely computed.  We have the work
32of Atkin-Lehner, Birch-Swinnerton-Dyer,
33Cremona, Manin, Mazur, Merel, and many others
34to thank for this.
35
36The Eichler-Selberg trace formulas, as developed in \cite{hijikata} and \cite{wada},
37can be used to compute characteristic polynomials of Hecke operators and hence
38gain some information about spaces of modular forms.
39It is also sometimes possible to write down explicit basis in terms of
40$\Theta$-series and to compute the action of Hecke operators on their $q$-expansions.
41Other methods include computing the Hecke operators and $q$-expansions
42using Brandt matrices and quaternion algebras as in \cite{pizer}
43or \cite{kohel}, or the module of supersingular points in
44characteristic $N$'' as  exploited by Mestre and Oesterle in \cite{mestre}.
45
46Though the above methods are each beautiful and well suited to certain applications,
47we will not discuss them further here.  Instead we focus
48on the modular symbols method, as it also has many advantages.  We will only
49discusss the theory in this summary paper, leaving an explicit description
50of the objects involved for later.  Nonetheless there is a definite
51gap between the {\em theory} on the one hand, and an efficient running
52machine implimentation on the other.  To impliment the algorithms
53hinted at below requires making absolutely everything completely explnicit
54and then finding intelligent and efficient ways of performing the
55necessary manipulations.  This is a nontrivial and tedious task, with
56room for error at every step.
57
58Our exposition follows very closely that of \cite{merel}.
59
60\subsection{Notation}
61Let $\Gamma$ be a finite index subgroup of $\sltwoz$ and
62$k\geq 2$ an integer. If $k$ is odd, assume $-1\not\in\Gamma$,
63so that the modular forms theory is nonempty.
64Let $\P^1(\Q) = \Q\union\{\infty\}$.
65
66\section{Modular symbols and modular forms}
67\subsection{Modular symbols}
68Let $\M$ be the $\Z$-module generated by formal symbols
69$\{\alpha,\beta\}$, $\alpha, \beta\in\P^1(\Q)$, subject
70to the relations
71$$\{\alpha,\beta\}+\{\beta,\gamma\}+\{\gamma,\alpha\}=0.$$
72Thus $\{\alpha,\beta\}=-\{\beta,\alpha\}$
73and $\{\alpha,\alpha\}=0$. There is a left action
74of $g\in\gltwoq$ given by
75$g.\{\alpha,\beta\}=\{g\alpha,g\beta\}$.
76
77
78\comment{% This map has no good properties, so far as I can see!
79There is a natural  homomorphism
80$$\cD_0\into\M:\quad \alp\mapsto \{0,\alp\}$$
81By Manin's trick Lemma~\ref{maninstrick} this map is surjective,
82[and I think the relations defining $\M$ exactly make
83it injective].
84}
85
86Let $$V_k=\Sym^{k-2}_\Z(\Z\cross\Z)=\Z_{k-2}[X,Y]$$ be the
87free $\Z$-module of homogeneous polynomimals in two variables
88of degree $k-2$.
89There is a {\em left} action of $g=\abcd{a}{b}{c}{d}\in\mtwoz$
90given by $$g.P(X,Y) = P(\det(g)g^{-1}(X,Y)) = P(dX-bY,-cX+aY).$$
91%Note that $g$ induces an automorphism of the algebra
92%$\Z[X,Y]$ which restricts to a linear endomorphism
93%of each homogeneous piece, and that
94%$$g.(h.P(X,Y))=g.P(\det(h)h^{-1}(X,Y))=P(((X,Y)g)h)=(gh).P(X,Y).$$
95The space
96$$\M_k := V_k\tensor_\Z \M.$$
97is equipped with a left action of $\mtwoz$ given by
98$$g.(P\tensor x) 99 =g.P\tensor g.x.$$
100%Note that $\M_k$ is {\em not} $V_k\tensor_{\Z[\mtwoz]}\M_k$
101%with its induced action.
102Let $$\M_k(\Gamma):=H_0(\Gamma,\M_k)$$ be the zeroth homology group.
103Thus $\M_k(\Gamma)$ is the quotient of $\M_k$ by the relations
104$g.x=x$ for all $x\in\M_k$ and $g\in\Gamma$.
105The elements of $\M_k(\Gamma)$ are called
106{\em modular symbols of weight $k$ for $\Gamma$}.
107As we will see later, using Shapiro's lemma and an explicit computation,
108$$\M_k(\Gamma)\tensor\C\isom H^1(\Gamma,V_k\tensor\C).$$
109The theory of Eichler and Shimura embeds modular forms
110in $H^1(\Gamma,V_k\tensor\C)$.
111
112\subsection{Manin symbols}
113Let $e=\{0,\infty\}\in\M$.
114
115\comment{Note furthermore that $\psltwoz=\sltwoz/\{\pm 1\}$
116is the free product of the
117cyclic group of order $2$ generated by $S$ and the cyclic group of order
118$3$ generated by $\tau$.}
119
120\begin{proposition}[Manin's trick]~\label{maninstrick}
121The elements $g.e$ for $g\in\sltwoz$ generate $\M$.
122\end{proposition}
123\begin{proof}
124(From \cite{cremona1}.) Writing $\{\alpha,\beta\}=\{0,\beta\}-\{0,\alp\}$,
125it suffices to show that every symbol of the form
126$\{0,\alp\}$ is in the group generated by
127the $g.e$.   Let
128$$\frac{p_{-2}}{q_{-2}} = \frac{0}{1},\, 129\frac{p_{-1}}{q_{-1}}=\frac{1}{0},\, 130\frac{p_0}{1}=\frac{p_0}{q_0},\, 131\frac{p_1}{q_1},\, 132\frac{p_2}{q_2},\ldots,\frac{p_r}{q_r}=\alp$$
133denote the continued fraction convergents of the rational number $\alp$.
134Then
135$$p_j q_{j-1} - p_{j-1} q_j = (-1)^{j-1}\qquad \text{for }-1\leq j\leq r.$$
136Hence
137$$\{0,\alp\} 138=\sum_{j=-1}^{r}\left\{\frac{p_{j-1}}{q_{j-1}},\frac{p_j}{q_j}\right\} 139 = \sum_{j=-1}^{r}g_j.e$$
140where
141$g_j = \mtwo{(-1)^{j-1}p_j}{p_{j-1}}{(-1)^{j-1}q_j}{q_{j-1}}$.
142\end{proof}
143
144To a pair $g\in\sltwoz$ and $P\in V_k$ define
145the {\em Manin symbol}
146$$[P,g]=g.(P\tensor e)\in\M_k(\Gamma).$$
147
148The matrices
149$$\sigma=\mtwo{0}{-1}{1}{0}\quad \text{and} \quad \tau=\mtwo{0}{-1}{1}{-1}$$
150satisfy
151$$\sigma^4=1,\,\,\tau^3=1$$
152and generate $\sltwoz$.
153
154\begin{proposition}~\label{maninsymbols}
155Let $g\in\sltwoz$ and $P\in V_k$. The symbol
156$[P,g]$ depends only on $P$ and the class $\Gamma g$.  When
157$g$ runs through $\sltwoz$ and $P$ runs through $V_k$,
158the Manin symbols generate $\M_k(\Gamma)$ ({\em maybe}, at
159least this should be true after tensoring with $\Q$).
160Furthermore, they satisfy
161\begin{eqnarray*}
162\mbox{}[P,g]+[\sigma^{-1}P,g\sigma]&=&0,\\
163\mbox{}[P,g]+[\tau^{-1}P,g\tau]+[\tau^{-2}P,g\tau^2]&=&0, \\
164\end{eqnarray*}
165\end{proposition}
166\begin{proof}
167The first assertion follows from the construction of $\M_k(\Gamma)$.
168The correct version of the second assertion
170For the third assertion note the following relations:
171\begin{eqnarray*}
172 e + \sigma(e) &=& \{0,\infty\}
173        + \{\sigma(0),\sigma(\infty)\}\\
174       &=&\{0,\infty\}+\{\infty,0\}=0,\\
175\\
176e+\tau(e)+\tau^2(e) &=&
177           \{0,\infty\} + \{\tau(0),\tau(\infty)\}
178                      + \{\tau^2(0),\tau^2(\infty)\}\\
179          &=& \{0,\infty\} + \{1,0\} + \{\infty,1\} = 0.
180\end{eqnarray*}
181Thus
182\begin{eqnarray*}
183\mbox{}[P,g]+[\sigma^{-1}P,g\sigma] &=&
184            g(P\tensor e) + g\sigma(\sigma^{-1}P\tensor e)\\
185    &=& gP\tensor e + gP\tensor g\sigma e\\
186    &=& gP\tensor g(e+\sigma(e)) = 0,\\
187\\
188\mbox{}[P,g]+[\tau^{-1}P,g\tau]+[\tau^{-2}P,g\tau^2]&=&
189        g(P\tensor e) + g\tau(\tau^{-1}P\tensor e) + g\tau^2(\tau^{-2}P\tensor e) \\
190&=& gP\tensor g(e+\tau(e)+\tau^2(e)) = 0.
191\end{eqnarray*}
192\end{proof}
193
194\begin{theorem}
195The above relations generate all relations satisfied by the Manin symbols.
196(Maybe one must tensor with $\Q$.)
197\end{theorem}
198\begin{remark}
199Not only do not know whether or not this is true before tensoring with $\Q$,
200I do not know how to prove this.
201\end{remark}
202
203%Note that the relation $[P,g]=[j^{-1}P,gj]$ is imposed
204%by repeated application of the $\sigma$ relation because $\sigma^2=j$.
205
206The Manin symbol $[P,g]$ can be written as $\Z$-linear combinations of
207Manin symbols $$[X^q Y^{k-2-q},g],\quad \text{with } 0 \leq q \leq k-2.$$  Since
208$[\sltwoz:\Gamma]$ is finite, $\M_k(\Gamma)$ is a finitely
209generated abelian group.   In particular, we can write down an
210explicit basis which is then readily amenable to machine computation.
211We write $[X^q Y^{k-2-q},g]=[q,g]$ to simplify notation.
212Let $g_1,\ldots,g_n$ be a set of coset representatives for
213$\Gamma$ in $\sltwoz$.  Then $\M_k(\Gamma)$ is generated by
214$$\{[q,a_i] : 0\leq q\leq k-2, \, 1\leq i\leq n\}$$
215subject to the relations given by Proposition~\ref{maninsymbols}.
216
217Note that $\M_k(\Gamma)$ may contain nontrivial torsion, which is
218not well understood (by me!).
219
220
221% what is the torsion?
222
223\subsection{Cuspidal modular symbols}
224
225\comment{
226%About my paper "Universal...", there are a few mistakes there
227%(one reason other than my own fault, is the fact that that I never
228%saw the proofs). Some of them are insignificant. One you have to
229%be careful about is the fact that  the parameterization of the cusps
230%of X_1(N) is wrong. It is easy to fix it anyway. One should replace
231%the set I called P_N by the disjoint union of the (Z/gcd(d,N)Z)^* where
232%d runs through Z/NZ. Then everything works mutatis mutandis.
233}
234In this section we assume $k$ is even.  There is a similar definition
235when $k$ is odd.
236
237Let $\cD_0=\Div^0(\P^1(\Q))$ be the group of divisors
238of degree zero supported on $\P^1(\Q)$ and note that
239$\gltwoq$ acts on $\cD_0$ on the left by linear
240fractional transformations.
241
242Let $\cC(\Gamma)=H_0(\Gamma)=H_0(\Gamma,\cD_0)$ be the zeroth
243homology of $\Gamma\subset\sltwoz$ acting on $\cD_0=\Div^0(\P^1(\Q))$,
244so $H_0(\Gamma)$ is the free abelian group generated
245by the set of orbits $\Gamma\backslash P^1(\Q)$.  It is the free abelian group on the
246cusps of the modular curve $X({\Gamma})$.  There is a map
247$\M_k(\Gamma)\into\cC(\Gamma)$ which on Manin symbols is
248$$[P,g]\mapsto P(1,0)[g(\infty)]-P(0,1)[g(0)].$$
249Let $\cS_k(\Gamma)$ be the kernel of this map.
250
251\subsection{Duality between modular symbols and modular forms}
252\subsubsection{Weight 2}
253The first homology group $H_1(X_{\Gamma},\Z)$ of the modular curve
254$X_{\Gamma}$, viewed as a real $2$-manifold, is a free abelian group
255of rank $2g$, where $g$ is the genus of $X_{\Gamma}$.
256The global differentials $\Omega(X) = H^0(X,\Omega)$
257on $X_{\Gamma}$, now viewed as a Riemann surface, form a
258$g$ dimensional complex vector space.  It is equal to the
259complex vector space $S_2(\Gamma)$ of cusp forms.
260There is a nondegenerate pairing
261\begin{eqnarray*}
262H_1(X_{\Gamma},\Z)\tensor \Omega(X)&\ra& \C\\
263\langle \gamma, \omega \rangle &\mapsto& \int_{\gamma} \omega.
264\end{eqnarray*}
265Taking coefficients in $\R$ we have
266$$H_1(X_{\Gamma},\R)=H_1(X_{\Gamma},\Z)\tensor\R.$$
267Extending the above pairing gives
268a natural injection of $H_1(X_{\Gamma},\R)$
269into the dual space of $\Omega(X)$.  Since the
270two spaces have the same real dimension, this injection
271must be an isomorphism.
272
273Suppose now that $k=2$.  We can identify modular symbols
274$\{\alp,\beta\}$ for $\Gamma$ as elements of
275$H_1(X_{\Gamma},\R)$, and we have the formula,
276$$\langle \{\alp,\beta\}, \omega\rangle 277 = \int_{\alp}^{\beta} \omega.$$
278In fact, modular symbols were first introduced in this way
279by Birch in \cite{birch} in his work
280with Swinnerton-Dyer on the special value at $s=1$ of
281the $L$-function associated to a (modular) elliptic curve.
282
283\subsubsection{Higher weight}
284The duality generalizes to higher weight.
285Let $f:\h\into\C$ be a map.  For $g=\abcd{a}{b}{c}{d}\in\gltwoq$ and $z\in\h$,
286define
287\begin{eqnarray*}
288f|[g]_k(z)&=&(cz+d)^{-k} f(gz)(\det g)^{k-1}\\
289f|[\overline{g}]_k(z)&=&(c\overline{z}+d)^{-k} f(gz)(\det g)^{k-1}
290\end{eqnarray*}
291We denote by $S_k(\Gamma)$ (resp. $\overline{S_k(\Gamma)}$) the complex
292vector space of holomorphic (resp. antiholomorphic) cusp forms of
293weight $k$ for $\Gamma$.   There is a canonical isomorphism of
294real vector spaces between $S_k(\Gamma)$ and $\overline{S_k(\Gamma)}$
295which associates to $f$ the antiholomorphic modular form
296$z\mapsto \overline{f(z)}$.
297
298There is a pairing
299$$(S_k(\Gamma)\oplus \overline{S_k(\Gamma)})\cross \M_k(\Gamma)\into\C$$
300given by the rule
301$$\langle f_1+f_2,P\tensor\{\alpha,\beta\}\rangle 302 = \int_{\alp}^{\beta} f_1(z)P(z,1)dz 303 + \int_{\alp}^{\beta} f_2(z)P(\overline{z},1)dz$$
304where $f_1\in S_k(\Gamma)$ and $f_2\in \overline{S_k(\Gamma)}$.
305
306\begin{theorem}~\label{pairing}
307The following pairing, obtained from the above one,
308is nondegenerate:
309$$(S_k(\Gamma)\oplus\overline{S_k(\Gamma)})\cross \cS_k(\Gamma) 310\into \C.$$
311\end{theorem}
312
313\subsection{Complex conjugation}
314Let $\eta=\abcd{-1}{0}{0}{1}$ and $\tilde{\eta}=\abcd{1}{0}{0}{-1}$.
315Assume in this section that  $\eta^{-1}\Gamma\eta=\Gamma$.
316\begin{proposition} The map $\iota$ which associates to
317$f\in S_k(\Gamma)\oplus\overline{ S_k(\Gamma)}$ the
318function $z\mapsto f(-\overline{z})$ is a complex
319linear involution of $S_k(\Gamma)\oplus\overline{ S_k(\Gamma)}$
320which exchanges $S_k(\Gamma)$ and $\overline{ S_k(\Gamma)}$.
321\end{proposition}
322Define an involution $\iota^{\star}$ on $\M_k(\Gamma)$ by
323$$\iota^{\star}(P\tensor x)= -\tilde{\eta}P\tensor\eta x.$$
324This involution is adjoint to $\iota$ with respect to
325the pairing of Theorem~\ref{pairing}.  Moreover $\iota^{\star}$
326acts as follows on Manin symbols
327$$\iota^{\star}([P,g]) = -[\tilde{\eta}P,\eta g \eta^{-1}].$$
328Let $\cS_k(\Gamma)^{+}$ denote the subspace of elements
329of $\cS_k(\Gamma)$ fixed by $\iota^{\star}$.
330\begin{proposition}
331The bilinear pairing induced by the pairing $\langle . , . \rangle$
332$$S_k(\Gamma)\cross \cS_k(\Gamma)^{+}\into\C$$
333is nondegenerate.
334\end{proposition}
335
336\subsection{Eichler-Shimura}
337Eichler and Shimura found a way to embed
338modular forms into a cohomology group.
339There is also a way to embed modular symbols into the same cohomology
340group.
341
342The complex
343vector space $V_k\tensor_{\Gamma} \sltwoz$ is endowed
344with a {\em right} action of $\sltwoz$ given by the formula
345$$(P\tensor g).\gamma = (\gamma^{-1}P)\tensor (g\gamma).$$
346\begin{proposition}
347We have an isomorphism of complex vector spaces
348$$H^1(\sltwoz,V_k\tensor_{\Gamma} \sltwoz\tensor\C)\isom\M_k(\Gamma)\tensor\C.$$
349\end{proposition}
350\begin{proof}
351This is Proposition 9 of \cite{merel}.
352The proof involves explicit computations with cocycles using the fact that
353$\sltwoz$ is generated by $\sigma$ and $\tau$.
354\end{proof}
355
356\begin{remark}
357It might be possible to replace tensoring with $\C$ by something
358less severe.
359\end{remark}
360
361\begin{lemma}[Shapiro]
362Let $H$ be a subgroup of a group $G$ and let $A$ be a $\Z[H]$-module.
363Then
364$$H^q(G,\Hom_H(\Z[G],A))=H^q(H,A)\qquad\text{for all q\geq 0}.$$
365\end{lemma}
366
367
368\begin{corollary}
369There is an isomorphism
370$$\M_k(\Gamma)\tensor\C \isom H^1(\Gamma, V_k\tensor\C).$$
371\end{corollary}
372\begin{proof}
373Since $\Gamma$ has finite index in $\sltwoz$ there is an isomorphism
374$$\Hom_{\Gamma}(\Z[\sltwoz],V_k\tensor\C) 375 \isom V_k\tensor_{\Gamma} \sltwoz\tensor\C.$$
376Now apply Shapiro's lemma.
377\end{proof}
378
379
380Define the {\em parabolic} cohomology group $H^1_P$ by the exactness
381of the following sequence
382$$0\into H^1_P(\Gamma,V_k\tensor \C) \into 383 H^1(\Gamma,V_k\tensor\C) \into 384 \bigoplus_{\text{cusps \alp}} H^1(\Gamma_{\alp},V_k\tensor\C)$$
385where $\Gamma_{\alp}$ is the stabilizer in $\Gamma$ of the cusp $\alp$
386of $X_{\Gamma}$.
387
388For $f\in M_k(\Gamma)$ define a class in $H^1(\Gamma,V_k\tensor\C)$
389by the cocycle
390$$\gamma\mapsto \int_{z_0}^{\gamma(z_0)} f(z) 391\binom{z}{1}^{k-2}dz.$$
392Here $z_0$ is a basepoint, $v^{k-2}$ denotes the image
393of $v\tensor\cdots\tensor v$ in $\Sym^{k-2}(\C\cross\C)$
394and the integral is that of a vector-valued differential.
395There is a similiar construction for holomorphic
396differentials.
397
398\begin{theorem}[Eichler-Shimura]
399The map above gives rise to isomorphisms
400\begin{eqnarray*}
401M_k(\Gamma)\oplus\overline{S_k(\Gamma)}&\into &H^1(\Gamma,V_k\tensor \C)\\
402S_k(\Gamma)\oplus\overline{S_k(\Gamma)}&\into &H^1_P(\Gamma,V_k\tensor \C).
403\end{eqnarray*}
404\end{theorem}
405
406
407%\subsection{Characters}
408% put Hijikata trace formula for the case $N=1$ here.
409\section{Linear maps}
410
411\subsection{Linear operators}
412Let $\Delta\subset\mtwoz$ such that
413$\Gamma\Delta\Gamma=\Delta$ and such that
414$\Gamma\backslash\Delta$ is finite. Note
415that $\Delta$ is a union of double cosets
416of $\Gamma\backslash\mtwoz/\Gamma$.  Let
417$R$ be a set of representatives of $\Gamma\backslash\Delta$.
418
419\comment{
420Two examples are $\Gamma=\Gamma_1(N)$ with
421$$\Delta = \Delta_1(N) = \{\abcd{a}{b}{c}{d}\in M_2(\Z) 422 : \det > 0, \,\,c\con a-1\con 0\pmod {N}\},$$
423and $\Gamma=\Gamma_0(N)$ with
424$$\Delta = \Delta_0(N) = \{\abcd{a}{b}{c}{d}\in M_2(\Z) 425 : \det > 0,\,\, c\con 0\pmod {N}, \,(a,N)=1\}.$$
426
427Define the {\em Hecke ring} $R(\Gamma,\Delta)$ as follows.
428It is the free $\Z$-module generated by the double cosets
429$\Gamma\alp\Gamma$, $\alp\in\Delta$.  Define multiplication
430between two double cosets $u=\Gamma\alp\Gamma$ and
431$v=\Gamma\beta\Gamma$ as follows.  Consider their coset decompositions
432$\Gamma\alp\Gamma = \coprod_{i}\Gamma\alpha_i$
433and $\Gamma\beta\Gamma = \coprod_{i}\Gamma\beta_i$.
434Then $\Gamma\alp\Gamma\beta\Gamma=\union_{i,j}\Gamma\alp_i\beta_j$
435(not necessarily disjoint), and so $\Gamma\alp\Gamma\beta\Gamma$
436is a finite union of double cosets of the form $\Gamma\gamma\Gamma$.
437Define
438$$u\cdot v = \sum_{w} m(u,v;w)w$$
439where the sum is extended over all double cosets
440$w=\Gamma\gamma\Gamma\subset\Gamma\alp\Gamma\beta\Gamma$, and
441$$m(u,v;w)=\#\{(i,j) : \Gamma\alp_i\beta_j = \Gamma\gamma\}$$
442for $w=\Gamma\gamma\Gamma$.  Thus equipped,
443$R(\Gamma,\Delta)$ becomes an associate, and in fact commutative,
444ring with $\Gamma=\Gamma\cdot 1\cdot\Gamma$ as the unit element.
445}
446\subsubsection{Action on modular forms}
447Let $M_k(\Gamma)$ be the space of modular forms of weight $k$ for $\Gamma$.
448For $f\in M_k(\Gamma)$, define an operator $T_{\Delta}$ by
449$$T_\Delta(f) = \sum_{\alp\in R} f|[\alp]_k$$
450This is a well-defined linear action
451on $M_k(\Gamma)$ which preserves the subspace $S_k(\Gamma)$.
452
453\subsubsection{Action on modular symbols}
454Similiarly, define an operator $T_{\Delta}$ on the
455space $\M_k(\Gamma)$ of modular symbols by
456$$T_\Delta(x) = \sum_{\alp\in R} \alp.x.$$
457
458\subsubsection{Hecke operators}
459Suppose now that $\Gamma=\Gamma_1(N)$.  Let $n\geq 1$ be an
460integer and set
461$$\Delta_n=\{\abcd{a}{b}{c}{d}\in M_2(\Z) : 462 \det = n,\,N|c,\, N|(a-1)\}.$$
463Then the $n$th {\em Hecke operator} is $T_{\Delta_n}$.
464If $\Gamma=\Gamma_0(N)$ the condition that $N|(a-1)$ is relaxed to
465$(N,a)=1$.
466
467\subsection{Action on Manin symbols}
468We now describe how to explicitely compute the action of the
469Hecke operators on $\M_k(\Gamma)$.
470Recall, we have an explicit generators and relations''
471description of $\M_k(\Gamma)$ in terms of Manin symbols.
472The action of the Hecke operators (and other linear operators)
473described in the previous section is given in terms of
474modular symbols.  We {\em could} describe the action of
475an operator on a Manin symbol by taking the Manin symbol,
476finding the corresponding modular symbol, acting by the
477operator, and then converting back to a sum of Manin symbols.
478This process is painfully inefficient as it involves repeated
479application of Proposition~\ref{maninstrick}.  This was
480how computations were originally done until Mazur and Merel
481described the action of the Hecke operators directly in terms
482of Manin symbols.
483
484Let $n>0$ be an integer.  We denote by $\mtwoz_n$ the set of
485matrices of $\mtwoz$ of determinant $n$.
486
487\begin{definition}[Condition (M)]
488We say that an element $\sum_{g} a_g g\in\C[\mtwoz_n]$
489satisfies condition (M) if for all cosets
490$\cC\in \mtwoz_n/\sltwoz$, we have in $\C[\P^1(\Q)]$,
491$$\sum_{g\in \cC} a_g ([g(\infty)] - [g(0)]) = [\infty]-.$$
492\end{definition}
493Note that the condition (M) depends neither on the level
494or the weight.
495
496Suppose now that $\Gamma=\Gamma_1(N)$ (or $\Gamma_0(N)$).
497There is a bijection between
498cosets $\Gamma\backslash\sltwoz$ and pairs of integers
499$(u,v)$ satisfying a certain equivalence.  The bijection associates
500to a $2\times 2$ matrix its bottom two entries.  We may thus view
501the Manin symbols as pairs $[P,(u,v)]$.
502
503\begin{theorem}[Merel]
504Let $[P,(u,v)]$ be a Manin symbol. Suppose $\sum_{g} a_g g\in\C[\mtwoz_n]$
505satisfies condition (M).  Then we have
506$$T_n([P(X,Y),(u,v)]) = \sum_{g=\abcd{a}{b}{c}{d}\in\mtwoz_n} 507 a_g [P(aX+bY,cX+dY),(au+cv,bu+dv)]$$
508where the sum is restricted to the matrices $g$ such that
509$\gcd(au+cv,bu+dv)=1$ (if $(n,N)=1$ this restriction is
510unnecessary).
511\end{theorem}
512\begin{proof}
513See section 2 of \cite{merel}.
514\end{proof}
515
516The element $\sum_{g \in\mtwoz_n} g \in \C[\mtwoz_n]$
517satisfies condition (M).  In Merel's paper one can find other families of simpler
518(more sparse) elements satisfying condition (M).
519
520\comment{
521For one of the families $S_n$ there is an
522asymptotic formula for the number $|S_n|$ of nonzero summands:
523$$|S_n| \sim \frac{12\log 2}{\pi^2} \sigma(n)\log n, \quad\text{as n\ra\infty}$$
524where $\sigma(n)$ is the sum of the positive divisors of $n$.
525Let $s(n)$ be the right hand side, rounded down to the nearest integer.  Then
526$$s(10)=34,\quad s(100)=842,\quad s(250)=2177,\quad 527 s(500)=5719,\quad s(1000)=13622.$$
528Finding a family minimizing these numbers is extremely important in
529computing many Hecke eigenvalues.
530
531\begin{remark}
532Cremona has improved on this slightly to give even simpler
533elements $\mathcal{X}_n\in\C[\mtwoz_n]$ which can be used to compute
534the Hecke action, but which I don't think satisfies condition (M).
535He proves this in his book for weight $2$, but I think his elements
536work for any positive weight.
537\end{remark}
538}
539%\subsection{Atkin-Lehner operators}
540%\subsection{Newforms}
541\comment{
542\subsection{Characters}
543Let $N$ be a positive integer and let
544$\chi:(\Z/N\Z)^{\star}\into\C^{\star}$ be
545a character. Let $\Z[\chi]=\Z[\chi(\Z/N\Z)^{\star}]$.
546Define $\M_k(N,\chi)$ to be the quotient of
547$\M_k(\Gamma_1(N))\tensor\Z[\chi]$ by the equivalence relation
548which identifies the Manin symbol
549$[P,(\lambda u,\lambda v)]$ with $\chi(\lambda)[P,(u,v)]$.
550Define $\S_k(N,\chi)\subset \M_k(N,\chi)$.
551\begin{proposition}
552The pairing
553$$(S_k(N,\chi)\sum\overline{ S_k(N,\chi)} \cross 554 \cS_k(N,\chi)\into \C$$
555is nondegenerate.
556\end{proposition}
557Thus the space $\S_k(N,\chi)\tensor\C$ obtained from modular symbols
558is related to the cusp forms $S_k(N,\chi)$.
559}
560
561\comment{
562\section{Computation}
563\subsection{Coset representatives~\label{cosetrep}}
564\begin{proposition}
565For $j=1,2$, let $g_j=\abcd{a_j}{b_j}{c_j}{d_j}\in\sltwoz$.
566The following are equivalent.
567\begin{enumerate}
568\item The right cosets $\Gamma_0(N)g_1$ and
569$\Gamma_0(N) g_2$ are equal,
570\item $c_1d_2\con c_2 d_1\pmod{N}$,
571\item There exists $u\in(\Z/N\Z)^{\star}$ such that
572$c_1\con u c_2$ and $d_1\con u d_2$  $\pmod{N}$.
573\end{enumerate}
574\end{proposition}
575\begin{proof}
576Proposition 2.2.1 of \cite{cremona1}.
577\end{proof}
578
579\begin{proposition}
580For $j=1,2$, let $\alp_j=p_j/q_j$ be cusps written in lowest terms.
581The following are equivalent:
582\begin{enumerate}
583\item $\alp_2=g\alp_1$ for some $g\in\Gamma_0(N)$,
584\item $q_2\con u q_1\pmod{N}$ and $up_2\con p_1\pmod{\gcd(q_1,N)}$,
585with $\gcd(u,N)=1$,
586\item $s_1 q_2\con s_2 q_1\pmod{gcd(q_1 q_2,N)}$, where
587$s_j$ satisfies $p_j s_j \con 1\pmod {q_j}$.
588\end{enumerate}
589\end{proposition}
590\begin{proof}
591Proposition 2.2.3 of \cite{cremona1}.
592\end{proof}
593}
594
595%\subsection{Cusp equivalence}
596%\subsection{Newforms}
597%\subsection{Computing Hecke eigenvalues}
598
599%\subsection{Computing $L^{(r)}(f,1)$}
600%\section{Computing in characteristic $p$}
601%\section{Equations for modular curves}
602%\section{Congruences between newforms}
603%\section{$\Spec(\T)$}
604%\section{Implimentation}
605%\subsection{Coset representatives}
606%\subsection{Linear algebra}
607%\subsection{Complexity}
608
609%\section{Examples}
610%\subsection{Level $37$}
611%\subsection{$X_0(389)$}
612
613
614\section{Generating $H_1(\Gamma,\Z)$}
615How can we generate $H_1(\Gamma,\Z)$ using modular symbols?
616\begin{theorem}
617Choose any $\alp\in\Q\union\{\infty\}$, it doesn't
618matter which.  Then the map
619$$\Gamma\ra H_1(\Gamma,\Z): \quad \gamma\mapsto\{\alp,\gamma(\alp)\}$$
620is a surjective group homomorphism.
621\end{theorem}
622So, knowing generators for $\Gamma$ would be enough.
623
624Now specialize to the case $\Gamma=\Gamma_0(N)$.
625Here is one guess for what {\em might} be true.
626\begin{question}
627Do the Manin symbols $(c,d)$ with $(c,N)=(d,N)=1$
628generate $H_1(X_0(N),\Z)$?
629\end{question}
630When $N$ is prime those Manin symbols lie in
631$H_1(X_0(N),\Z)$ because they correspond
632to paths from the non-$\infty$ cusp to itself.
633Let $(c,d)$ be such a Manin symbol and choose
634$a,b$ so that $M=\abcd{a}{b}{c}{d}\in\sltwoz$.
635Then the Modular symbol corresponding to $(c,d)$
636is $M.\e=M.\{0,\infty\}=\{M(0),M(\infty)\}$.
637Since $c$ and $d$ are both coprime to $N$, the
638cusps $[\frac{a}{c}]$ and $[\frac{b}{d}]$ are
639the same (remember, we are assuming $N$ is prime
640so that there are only two cusps).   The only
641other way to force the cusps to be the same would
642be to force $b$ and $d$ to both be divisible by $N$,
643but then $(c,d)$ would not be a Manin symbol.
644
645I do not know if these Manin symbols are enough
646to generate all integral modular symbols.  But,
647we can set up some computer computations to get an
648idea of whether or not we should expect this.
649
650{\bf Computation 1.} Let $H=H_1(X_0(N),\Z)$.
651Let $V$ be the submodule of $H$ generated by
652the modular symbols $\{\infty,\gamma(\infty)\}$
653where $\gamma=\abcd{a}{b}{N}{d}$ and $0<a<N$.
654Let $W$ be the submodule of $H_1(X_0(N),\Q)$ generated
655by the Manin symbols $(c,d)$ for which both
656$c$ and $d$ are coprime to $N$.
657If things were as easy as imaginable then
658both $V$ and $W$ would be equal to $H$ (and
659to each other).  If $W$ is properly contained
660in $V$ then we learn that the answer to the question
661is {\em NO}.  In the prime case, if $V$ is properly
662contained in $W$ we learn that $V$ does not
663generate $H$, which is also interesting.
664
665We compute $V$ and $W$ for $11\leq N\leq 100$, and the
666module index $[V:W]$.
667Whenever there is a - for the index, this means that
668that {\em neither} $V$ nor $W$ span $H_1(X_0(N),\Q)$
669over $\Q$.  In these cases we didn't compute the actual index.
670
671\begin{center}
672\begin{tabular}{|c|c|}\hline
673$N$&$[W:V]$\\ \hline
67411 & 1\\
67514 & 1\\
67615 & 1\\
67717 & 1\\
67819 & 1\\
67920 & 1\\
68021 & 1\\
68122 & 1\\
68223 & 1\\
68324 & 1\\
68426 & 1\\
68527 & 1\\
68628 & 1\\
68729 & 1\\
68830 & -\\
68931 & 1\\
69032 & 1\\
69133 & 1\\
69234 & 1\\
69335 & 1\\
69437 & 1\\
69538 & 1\\
69639 & 1\\
69741 & 1\\
69842 & -\\
69943 & 1\\
70044 & 1\\
70145 & 1\\
70246 & 1\\
70347 & 1\\
70448 & -\\
70549 & 1\\
70650 & 1\\
70751 & 1\\
70852 & 1\\
70953 & 1\\
71054 & -\\
71155 & 1\\
71256 & -\\
71357 & 1\\
71458 & 1\\
71559 & 1\\
71660 & -\\
71761 & 1\\
71862 & 1\\
719\hline
720\end{tabular}
721\begin{tabular}{|c|c|}\hline
722$N$&$[W:V]$\\ \hline
72363 & 1\\
72464 & 1\\
72565 & 1\\
72666 & -\\
72767 & 1\\
72868 & 1\\
72969 & 1\\
73070 & -\\
73171 & 1\\
73272 & -\\
73373 & 1\\
73474 & 1\\
73575 & 1\\
73676 & 1\\
73777 & 1\\
73878 & -\\
73979 & 1\\
74080 & -\\
74181 & 1\\
74282 & 1\\
74383 & 1\\
74484 & -\\
74585 & 1\\
74686 & 1\\
74787 & 1\\
74888 & -\\
74989 & 1\\
75090 & -\\
75191 & 1\\
75292 & 1\\
75393 & 1\\
75494 & 1\\
75595 & 1\\
75696 & -\\
75797 & 1\\
75898 & 1\\
75999 & 1\\
760100 & -\\
761101 & 1\\
762102 & -\\
763103 & 1\\
764104 & -\\
765105 & -\\
766106 & 1\\
767107 & 1\\
768\hline
769\end{tabular}
770\begin{tabular}{|c|c|}\hline
771$N$&$[W:V]$\\ \hline
772108 & -\\
773109 & 1\\
774110 & -\\
775111 & 1\\
776112 & -\\
777113 & 1\\
778114 & -\\
779115 & 1\\
780116 & 1\\
781117 & 1\\
782118 & 1\\
783119 & 1\\
784120 & -\\
785121 & 1\\
786122 & 1\\
787123 & 1\\
788124 & 1\\
789125 & 1\\
790126 & -\\
791127 & 1\\
792128 & 1\\
793129 & 1\\
794130 & -\\
795131 & 1\\
796132 & -\\
797133 & 1\\
798134 & 1\\
799135 & -\\
800136 & -\\
801137 & 1\\
802138 & -\\
803139 & 1\\
804140 & -\\
805141 & 1\\
806142 & 1\\
807143 & 1\\
808144 & -\\
809145 & 1\\
810146 & 1\\
811147 & 1\\
812148 & 1\\
813149 & 1\\
814150 & -\\
815151 & 1\\
816152 & -\\
817\hline
818\end{tabular}
819\begin{tabular}{|c|c|}\hline
820$N$&$[W:V]$\\ \hline
821153 & 1\\
822154 & -\\
823155 & 1\\
824156 & -\\
825157 & 1\\
826158 & 1\\
827159 & 1\\
828160 & -\\
829161 & 1\\
830162 & -\\
831163 & 1\\
832164 & 1\\
833165 & -\\
834166 & 1\\
835167 & 1\\
836168 & -\\
837169 & 1\\
838170 & -\\
839171 & 1\\
840172 & 1\\
841173 & 1\\
842174 & -\\
843175 & 1\\
844176 & -\\
845177 & 1\\
846178 & 1\\
847179 & 1\\
848180 & -\\
849181 & 1\\
850182 & -\\
851183 & 1\\
852184 & -\\
853185 & 1\\
854186 & -\\
855187 & 1\\
856188 & 1\\
857189 & -\\
858190 & -\\
859191 & 1\\
860192 & -\\
861193 & 1\\
862194 & 1\\
863195 & -\\
864196 & -\\
865197 & 1\\
866
867\hline
868\end{tabular}
869\end{center}
870
871{\bf Conclusion:} Neither obvious set of elements
872of $H_1(X_0(N),\Z)$ will, in general, generate.
873It might be necessary to look at
874\begin{verbatim}
875R. S. Kulkarni, An arithmetic-geometry method of the study
876of the subgroups of the modular group, American Journal of mathematics
877113, 1991, 1053-1133
878\end{verbatim}
879and find explicit generators.
880
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937\end{document}
938