Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1\chapter{Modular symbols}
2\label{chap:modsym}
3Modular symbols are central to most of the algorithms and computations
4of this thesis.  This chapter closely follows Merel~\cite{merel:1585}
5and Cremona~\cite{cremona:algs}.  However, we work with modular forms
6with character throughout; this makes the theory trickier, but is
7extremely import in computational applications.
8
9\section{The definition of modular symbols}
10\label{sec:defnofmodsyms}
11Fix a triple $(N,k,\eps)$ consisting of a level~$N$, a
12weight~$k$, and a mod~$N$ character
13$\eps:(\Z/N\Z)^*\ra\C^*$
14such that $\eps(-1)=(-1)^k$.
15
16Let $\sM$ be the torsion-free abelian group generated by symbols
17$\{\alp,\beta\}$ with $\alp, \beta\in\P^1(\Q)=\Q\union\{\infty\}$
18subject to the following relations:
19  $$\{\alp,\beta\}+\{\beta,\gamma\}+\{\gamma,\alp\} = 0, 20\qquad \text{ for all }\alp,\beta,\gamma\in\P^1(\Q).$$
21Because $\sM$ is torsion free, $\{\alp,\alp\}=0$ and
22$\{\alp,\beta\} = -\{\beta,\alp\}$.
23Let $V_{k-2}$\label{defn:vk} be the $\Z$-submodule of $\Z[X,Y]$ made up of
24homogeneous polynomials of degree $k-2$, and set
25   $\sM_k := V_{k-2}\tensor\sM.$
26\label{pg:higherweightmodsym}
27For $g=\abcd{a}{b}{c}{d}\in\GL_2(\Q)$ and $P\in V_{k-2}$, let
28\begin{align*}
29     gP(X,Y) &= P\left(\det(g)g^{-1}\vtwo{X}{Y}\right)
30             = P\left(\mtwo{\hfill d}{-b}{-c}{\hfill a}\vtwo{X}{Y}\right)\\
31             &= P(dX-bY,-cX+aY).
32\end{align*}
33This defines a left action of $\GL_2(\Q)$ on $V_{k-2}$;
34it is a left action because
35\begin{align*}
36 (gh)P(v) &= P(\det(gh)(gh)^{-1}v)
37          = P(\det(h)h^{-1}\det(g)g^{-1}v)\\
38          &= gP(\det(h)h^{-1}v) = g(hP(v)).
39\end{align*}
40Combining this action with the action of $\GL_2(\Q)$ on $\P^1(\Q)$
41by linear fractional transformations gives
42a left action of $\GL_2(\Q)$ on $\sM_k$:
43  $$g (P \tensor \{\alp,\beta\}) = g(P)\tensor\{g(\alp),g(\beta)\}.$$
44Finally, for $g=\abcd{a}{b}{c}{d}\in\Gamma_0(N)$, let
45$\eps(g) := \eps(\overline{a})$,
46where~$a$ reduces to $\overline{a}\in\Z/N\Z$.
47\begin{definition}[Modular symbols]\label{defn:modsym}
48The space $\sM_k(N,\eps)$ of \defn{modular symbols} is
49the torsion-free quotient of $\sM_k\tensor\Z[\eps]$ by the
50relations $gx=\eps(g)x$ for all $x\in\sM_k$
51and all $g\in\Gamma_0(N)$.
52\end{definition}
53Denote by $P\{\alp,\beta\}$ the image
54of $P\tensor\{\alp,\beta\}$ in $\sM_k(N,\eps)$.
55For any $\Z[\eps]$-algebra~$R$, let
56$\sM_k(N,\eps;R) := \sM_k(N,\eps)\tensor R$.
57
58
59
60\section{Cuspidal modular symbols}\label{cuspidalsymbols}
61Let~$\sB$ be the  free abelian group generated by the symbols
62$\{\alp\}$ for all $\alp\in\P^1(\Q)$.
63There is a left action of~$\GL_2(\Q)$ on~$\sB$ given by
64$g\{\alp\}=\{g(\alp)\}$.
65Let $\sB_k := V_{k-2}\tensor \sB$, and let $\GL_2(\Q)$ act
66on $\sB_k$ by $g(P\{\alp\}) = (gP)\{g(\alp)\}$.
67\begin{definition}[Boundary modular symbols]\label{def:boundarysymbols}
68The space of \defn{boundary modular symbols} is the
69torsion-free quotient
70of $\sB_k\tensor\Z[\eps]$ by the relations
71$gx = \eps(g) x$ for all
72$g\in \Gamma_0(N)$ and $x\in \sB_k$.
73\end{definition}
74Denote by $P\{\alp\}$ the image of $P\tensor\{\alp\}$.
75The \defn{boundary map} is
76  $$\delta: \sM_k(N,\eps) \ra \sB_k(N,\eps), 77 \qquad \delta(P\{\alp,\beta\}) = 78 P\{\beta\}-P\{\alp\}.$$
79\begin{definition}[Cuspidal modular symbols]\label{defn:cuspidalmodularsymbols}
80The space $\sS_k(N,\eps)$ of \defn{cuspidal modular symbols} is
81the kernel of~$\delta$:
82  $$0\ra \sS_k(N,\eps) \ra\sM_k(N,\eps)\xrightarrow{\,\delta\,} 83 \sB_k(N,\eps).$$
84\end{definition}
85
86
87\section{Duality between modular symbols and modular forms}
88For any integer~$k$, any $\C$-valued function~$f$ on
89the complex upper half plane $\h:=\{z \in \C : \im(z) > 0\}$,
90and any $\gamma\in\GL_2(\Q)$, define a function
91$f|[\gamma]$ on~$\h$ by
92 $$(f|[\gamma]_k)(z) = \det(\gamma)^{k-1}\frac{f(\gamma z)}{(cz+d)^{k}}.$$
93\begin{definition}[Cusp forms]
94Let $S_k(N,\eps)$ be the complex vector space of holomorphic
95functions $f(z)$ on~$\h$ that satisfy
96the equation
97  $$f|[\gamma]_k = \eps(\gamma)f$$
98for all $\gamma\in\Gamma_0(N)$, and such that~$f$
99is holomorphic at all cusps in the sense of
100\cite[pg.~42]{diamond-im}.
101\end{definition}
102
103\begin{definition}[Antiholomorphic cusp forms]
104Let $\Sbar_k(N,\eps)$ be the space of \defn{antiholomorphic
105cusp forms}; the definition is as above, except
106$$\frac{f(\gamma z)}{(c\overline{z}+d)^k} = \overline{\eps}(\gamma) f(z)$$
107for all $\gamma\in\Gamma_0(N)$.
108\end{definition}
109There is a canonical isomorphism of real vector spaces
110between $S_k(N,\eps)$ and $\Sbar_k(N,\eps)$ that associates
111to~$f$ the anitholomorphic cusp form $z\mapsto \overline{f(z)}$.
112
113\begin{theorem}[Duality]\label{thm:perfectpairing}
114There is a  pairing
115\begin{equation*}
116    \langle\,\, , \, \, \rangle:
117    (S_k(N,\eps)\oplus \Sbar_k(N,\eps)) \cross \sM_k(N,\eps;\C)
118   \ra \C
119\end{equation*}
120given by
121$$\langle f\oplus g, P\{\alp,\beta\}\rangle = 122 \int_{\alp}^{\beta} f(z)P(z,1) dz 123 + \int_{\alp}^{\beta} g(z)P(\zbar,1) d\zbar,$$
124where the path from~$\alp$ to~$\beta$ is,
125except for the endpoints, contained in~$\h$.
126The pairing is perfect when restricted to $\sS_k(N,\eps;\C)$.
127\end{theorem}
128\begin{proof}
129Take the~$\eps$ part of each side of~\cite[Thm.~3]{merel:1585}.
130\end{proof}
131
132
133\section{Linear operators}
134\label{sec:heckeops}
135\subsection{Hecke operators}\label{heckeops:modsym}
136The Hecke operators~$T_n$, for $n\geq 1$ act on
137modular symbols and modular forms.  The Hecke algebra~$\T$
138is the subring of the ring of endomorphism of modular
139symbols or modular forms generated by all~$T_n$.
140For the definition of Hecke operators on modular symbols,
141see~\cite[\S2]{merel:1585}.  For example, when $n=p$ is prime,
142$$T_p(x) = \left[ \mtwo{p}{0}{0}{1} + \sum_{r \md p} 143 \mtwo{1}{r}{0}{p}\right] x,$$
144where the first matrix is omitted if $p\mid N$.
145\begin{proposition}\label{prop:modsympairing}
146The pairing of Theorem~\ref{thm:perfectpairing} respects the
147action of the Hecke operators, in the sense that
148$\langle f T, x \rangle = \langle f , T x \rangle$
149for all $T\in \T$, $x\in\sM_k(\Gamma)$,
150 and $f\in S_k(\Gamma)\oplus \Sbar_k(\Gamma)$.
151\end{proposition}
152
153\subsection{The $*$-involution}\label{sec:starinvolution}
154The matrix $j=\abcd{-1}{0}{\hfill0}{1}$ defines
155an involution~$*$ of $\sM_k(N,\eps)$ given by
156$x\mapsto x^*=j(x)$.  Explicitly,
157\begin{equation*}
158(P(X,Y)\{\alp,\beta\})^* = P(X,-Y)\{-\alp,-\beta\}.
159\end{equation*}
160We are extremely fortunate to have the following proposition,
161because modular symbols correspond to both holomorphic and
162non-holomorphic cusp forms.
163\begin{proposition}
164The $*$-involution is well defined.
165\end{proposition}
166\begin{proof}
167$\sM_k(N,\eps)$ is the torsion-free quotient of the
168free $\Z[\eps]$-module generated by symbols
169$x=P\{\alp,\beta\}$ by the submodule generated by
170relations $\gamma x - \eps(\gamma)x$ for
171all $\gamma\in \Gamma_0(N)$.
172In order to check that~$*$ is well defined, it
173suffices to check that $*(\gamma x - \eps(\gamma)x)$ is of
174the form $\gamma' y - \eps(\gamma') y$.
175Note that if $\gamma=\abcd{a}{b}{c}{d}\in \Gamma_0(N)$, then
176$j\gamma j^{-1} = \abcd{\hfill a}{-b}{-c}{\hfill d}$ is also in $\Gamma_0(N)$
177and $\eps(j\gamma j^{-1}) = \eps(\gamma)$.  We have
178\begin{align*}
179    j(\gamma x - \eps(\gamma) x) &=
180           j \gamma x - j \eps(\gamma) x \\
181        &= j \gamma j^{-1} j x - \eps(\gamma) j x\\
182        &= (j\gamma j^{-1}) (j x) - \eps(j \gamma j^{-1}) (jx).
183\end{align*}
184\end{proof}
185
186If~$f$ is a modular form, let $f^*$ be the holomorphic
187function $\overline{f(-\overline{z})}$, where the bar
188denotes complex conjugation.
189   The Fourier coefficients
190of $f^*$ are the complex conjugates of those of~$f$; though $f^*$
191is again a holomorphic modular form, it's character
192is $\overline{\eps}$ instead of~$\eps$.
193The pairing of Theorem~\ref{thm:perfectpairing}
194is the restriction of a pairing on the full spaces without
195character, and we have the following proposition.
196\begin{proposition}\label{prop:starpairing}
197We have
198\begin{equation*}
199\langle f^*,  x^* \rangle = \overline{\langle f, x\rangle}.
200\end{equation*}
201\end{proposition}
202
203\begin{definition}[Plus one quotient]
204The \defn{plus one quotient} $\sM_k(N,\eps)_+$ is the torsion-free
205quotient of $\sM_k(N,\eps)$ by the relations
206$x^*-x=0$ for all $x\in \sM_k(N,\eps)$.
207The \defn{minus one quotient} is defined similarly.
208\end{definition}
209
210\begin{warning} The choice of~$*$ above
211agrees with~\cite[\S2.1.3]{cremona:algs},
212but not with~\cite[\S1.6]{merel:1585}.
213\end{warning}
214
215\subsection{The Atkin-Lehner involutions}\label{sec:atkin-lehner}
216In this section we assume
217that~$k$ is even and $\eps^2=1$.
218To each divisor~$d$ of $N$ such that $(d,N/d)=1$
219there is an \defn{Atkin-Lehner involution}~$W_d$ on $\sM_k(N,\eps)$,
220which is defined as follows.
221Using the Euclidean algorithm,
222choose integers $x,y,z,w$ such that $dxw - (N/d)yz = 1$;
223let $g=\abcd{dx}{y}{Nz}{dw}$ and define
224$W_d(x) \define g(x) / d^{\frac{k-2}{2}}.$
225For example, when $d=N$ we have $g=\abcd{0}{-1}{N}{\hfill 0}$.
226There is also an Atkin-Lehner involution, also denoted $W_d$,
227that acts on modular forms by $W_d(f) = f|[W_d]_k$.
228These involutions are compatible with the integration pairing:
229$\langle W_d(f), x\rangle = \langle f, W_d(x)\rangle$.
230
231\section{Degeneracy maps}
232\label{sec:degeneracymaps}
233\label{pg:degeneracymaps}
234In this section, we describe natural maps between spaces of
235modular symbols of different levels.  These are useful in
236investigating level lowering and raising questions,
237and in computing kernels of certain natural maps between
238Jacobians of modular curves.
239
240Fix a positive integer~$N$ and a Dirichlet character
241$\eps : (\Z/N\Z)^*\ra \C^*$.
242Let~$M$ be a positive divisor of~$N$ that
243is divisible by the conductor of~$\eps$, in the sense that~$\eps$
244factors through $(\Z/M\Z)^*$ via the natural
245map $(\Z/N\Z)^*\ra (\Z/M\Z)^*$ composed with
246some uniquely defined character $\eps':(\Z/M\Z)^*\ra\C^*$.
247For any positive divisor~$t$ of $N/M$, let
248$T=\abcd{1}{0}{0}{t}$ and
249fix a choice
250$D_t=\{T\gamma_i : i=1,\ldots, n\}$
251of coset representatives for $\Gamma_0(N)\backslash T\Gamma_0(M)$.
252
253{\bf Warning:} There is a mistake in \cite[\S2.6]{merel:1585}:
254 The quotient $\Gamma_1(N)\backslash\Gamma_1(M)T$'' should be replaced
255by $\Gamma_1(N)\backslash T\Gamma_1(M)$''.
256\begin{proposition}
257For each divisor $t$ of $N/M$ there are well-defined linear maps
258\begin{align*}
260      \alp_t(x) = tT^{-1}x = \mtwo{t}{0}{0}{1} x\\
262      \beta_t(x) = \sum_{T\gam_i\in D_t} \eps'(\gam_i)^{-1}T\gam_i{} x.
263\end{align*}
264Furthermore
265  $\alp_t\circ \beta_t$ is multiplication
266by $t\cdot [\Gamma_0(M) : \Gamma_0(N)].$
267\end{proposition}
268\begin{proof}
269To show that $\alp_t$ is well defined, we must show that for
270$x\in\sM_k(N,\eps)$ and $\gam=\abcdmat\in\Gamma_0(N)$, that
271$\alp_t(\gamma x -\eps(\gamma) x)=0\in\sM_k(M,\eps')$.
272We have
273$$\alp_t(\gam x) = \mtwo{t}{0}{0}{1}\gam x 274 = \mtwo{a}{tb}{c/t}{d}\mtwo{t}{0}{0}{1} x 275 = \eps'(a)\mtwo{t}{0}{0}{1} x,$$
276so
277$\alp_t(\gamma x -\eps(\gamma) x) 278 = \eps'(a)\alp_t(x) - \eps(\gamma)\alp_t(x) = 0$.
279
280We next verify that~$\beta_t$ is well defined.
281Suppose that $x\in\sM_k(M,\eps')$ and $\gamma\in\Gamma_0(M)$;
282then $\eps'(\gam)^{-1}\gam x = x$, so
283\begin{align*}
284\beta_t(x)
285    &= \sum_{T\gam_i\in D_t}
286        \eps'(\gam_i)^{-1}T\gam_i{}\eps'(\gam)^{-1}\gam{} x\\
287    &= \sum_{T\gam_i\gam\in D_t}
288        \eps'(\gam_i\gam)^{-1}T\gam_i{}\gam{} x.
289\end{align*}
290This computation shows that the definition of~$\beta_t$
291does not depend on the choice~$D_t$ of coset representatives.
292To show that~$\beta_t$ is well defined
293we must show that, for $\gam\in\Gamma_0(M)$, we have
294$\beta_t(\gam x) = \eps'(\gam)\beta_t(x)$ so that $\beta_t$
295respects the relations that define $\sM_k(M,\eps)$.
296Using that~$\beta_t$ does not depend on the choice of
297coset representative, we find that for $\gamma\in\Gamma_0(M)$,
298\begin{align*}
299  \beta_t(\gam x)
300     &= \sum_{T\gam_i\in D_t} \eps'(\gam_i)^{-1}T\gam_i{} \gam{} x\\
301     &= \sum_{T\gam_i\gam^{-1}\in D_t}
302         \eps'(\gam_i\gam^{-1})^{-1}T\gam_i{}\gam{}^{-1} \gam{} x\\
303     &= \eps'(\gam)\beta_t(x).\\
304\end{align*}
305To compute $\alp_t\circ\beta_t$, we use
306that $\#D_t = [\Gamma_0(N) : \Gamma_0(M)]$:
307\begin{align*}
308 \alp_t(\beta_t(x)) &=
309    \alp_t \left(\sum_{T\gamma_i}
310        \eps'(\gam_i)^{-1}T\gam_i x\right)\\
311  &= \sum_{T\gamma_i}
312        \eps'(\gam_i)^{-1}(tT^{-1})T\gam_i x\\
313  &= t\sum_{T\gamma_i}
314        \eps'(\gam_i)^{-1}\gam_i x\\
315  &= t\sum_{T\gamma_i} x \\
316  &= t \cdot [\Gamma_0(N) : \Gamma_0(M)] \cdot x.
317\end{align*}
318\end{proof}
319
320\begin{definition}[New and old modular symbols]\label{def:newandoldsymbols}
321The subspace $\sM_k(N,\eps)^{\new}$
322of \defn{new modular symbols} is the
323intersection of the kernels of the $\alp_t$ as~$t$
324runs through all positive divisors of $N/M$ and~$M$
325runs through positive divisors of~$M$ strictly less than~$N$
326and divisible by the conductor of~$\eps$.
327The subspace $\sM_k(N,\eps)^{\old}$
328of \defn{old modular symbols} is the
329subspace generated by the images of the $\beta_t$
330where~$t$ runs through all positive divisors of $N/M$ and~$M$
331runs through positive divisors of~$M$ strictly less than~$N$.
332\end{definition}
333
334\begin{remark}
335The map $\beta_t\circ\alp_t$ can not be multiplication by
336a scalar since $\sM_k(M,\eps')$
337usually has smaller dimension than $\sM_k(N,\eps)$.
338\end{remark}
339
340\begin{remark}
341The proposition implies that $\beta_t$ is injective.
342For each prime~$p$ there is also a map
343$\beta_{t,p}:\sM_k(M,\eps';\F_p[\eps]) \ra 344 \sM_k(N,\eps;\F_p[\eps])$.
345When~$p$ does not divide $t\cdot [\Gamma_0(M) : \Gamma_0(N)]$,
346the proposition shows that $\beta_{t,p}$ is injective.
347However, $\beta_t\tensor\F_p$ need not be injective for all~$p$.
348For example, suppose $M=14$, $N=28$, and $\eps=1$. Then there
349are basis with respect to which the matrix of $\beta_1$ is
350$$\left( 351\begin{matrix} 3521&0&0&1&0&0&0&0&0\\ 353 0&1&0&0&1&0&0&0&0\\ 354 0&0&1&0&0&1&0&0&0\\ 355 0&0&0&0&0&0&2&1&-1\\ 356 0&0&0&0&0&0&0&1&1 357\end{matrix} 358\right).$$
359The row vector $(0,0,0,1,1)$ is in the kernel of the mod~$2$
360reduction of this matrix.
361\end{remark}
362
363\begin{algorithm}\label{alg:degenreps}
364Let~$M$ be a positive divisor of~$N$ and~$t$
365a divisor of~$N/M$.  This algorithm
366computes a set~$D_t$ of representatives for
367the orbit space
368$\Gamma_0(M)\backslash T\Gamma_0(N).$
369      Let $\Gamma_0(N/t,t)$ denote the subgroup of $\SL_2(\Z)$
370consisting of matrices that are upper triangular modulo $N/t$ and lower
371triangular modulo~$t$.   Observe that two right cosets
372 of $\Gamma_0(N/t,t)$ in $\SL_2(\Z)$,  represented by
373$\abcd{a}{b}{c}{d}$ and $\abcd{a'}{b'}{c'}{d'}$,
374are equivalent if and only if
375$(a,b)=(a',b')$ as points of $\P^1(t)$, i.e.,
376$ab'\con ba'\pmod{t}$,
377and $(c,d)=(c',d')$ as points of $\P^1(N/t)$.
378Using the following algorithm, we compute right coset
379representatives for $\Gamma_0(N/t,t)$
380inside~$\Gamma_0(M)$.
381\begin{enumerate}
382       \item Compute the number $[\Gamma_0(M):\Gamma_0(N)]$ of cosets.
383       \item Compute a random element $x \in \Gamma_0(M)$.
384       \item If~$x$ is not equivalent to anything generated so
385              far, add it to the list.
386       \item Repeat steps (2) and (3) until the list is as long
387             as the bound of step (1).
388\end{enumerate}
389There is a natural bijection between
390       $\Gamma_0(N)\backslash T \Gamma_0(M)$
391and $\Gamma_0(N/t,t)\backslash \Gamma_0(M)$,
392under which~$T\gamma$ corresponds to~$\gamma$.
393Thus we obtain coset representatives for
394 $\Gamma_0(N)\backslash T\Gamma_0(M)$
395by left multiplying each
396coset representative of $\Gamma_0(N/t,t)\backslash\Gamma_0(M)$  by~$T$.
397\end{algorithm}
398
399\subsection{Compatibility}
400Suppose that the characteristic of the base field is zero.
401The degeneracy maps defined above
402are compatible with the corresponding degeneracy maps
403$\tilde{\alp}_t$ and $\tilde{\beta}_t$
404on modular forms.  This is because the degeneracy
405maps on modular forms are defined by summing over the
406same coset representatives $D_t$.
407Thus we have the following compatibilities.
408\begin{align*}
409  \langle \tilde{\alp}_t(f), x \rangle &= \langle f, \alp_t(x)\rangle,\\
410  \langle \tilde{\beta}_t(f), x\rangle &=  \langle f, \beta_t(x) \rangle .
411\end{align*}
412If~$p$ is prime to~$N$, then $T_p\alp_t = \alp_t T_p$
413   and $T_p\beta_t = \beta_t T_p$.
414
415
416\section{Manin symbols}
417\label{sec:maninsymbols}
418From the definition given in
419Section~\ref{sec:defnofmodsyms}, it is not even clear
420that $\sM_k(N,\eps)$ is of finite rank.  The Manin
421symbols provide a finite presentation of~$\sM_k(N,\eps)$
422that is much more useful from a computational point of view.
423
424\begin{definition}[Manin symbols]\label{defn:maninsymbols}
425The \defn{Manin symbols} are the set of pairs
426           $$[P(X,Y),(u,v)]$$
427where $P(X,Y)\in V_{k-2}$ and
428$0\leq u,v < N$ with $\gcd(u,v,N)=1$.
429\end{definition}
430Define a {\em right} action of $\GL_2(\Q)$ on
431the free $\Z[\eps]$-module~$M$ generated by the Manin
432symbols as follows. The element $g=\abcd{a}{b}{c}{d}$ acts by
433\begin{equation*}
434[P,(u,v)]g=[g^{-1}P(X,Y),(u,v) g]
435    = [P(aX+bY,cX+dY),(au+cv,bu+dv)].
436\end{equation*}
437Let $\sigma=\abcd{0}{-1}{1}{\hfill 0}$ and $\tau=\abcd{0}{-1}{1}{-1}$\label{defn:sigmatau}.
438Let $\sM_k(N,\eps)'$ be the torsion-free quotient
439of~$M$ by
440\begin{align*}
441\mbox{}x + x\sigma &= 0,\\
442\mbox{}x + x\tau+ x\tau^2 &= 0,\\
443   \eps(\lambda) [P,(u,v)]- [P,(\lambda u, \lambda v)] &=0.
444\end{align*}
445
446\begin{theorem}\label{thm:maninsymbols}
447There is a natural isomorphism
448$\vphi:\sM_k(N,\eps)'\lra\sM_k(N,\eps)$ given by
449$$[X^iY^{2-k-i},(u,v)] \mapsto g(X^iY^{k-2-i}\{ 0,\infty\})$$
450where $g=\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ is some matrix
451such that $(u,v)\con (c,d) \pmod{N}$.
452\end{theorem}
453\begin{proof}
454In~\cite[\S1.2, \S1.7]{merel:1585} it is proved that $\vphi\tensor\C$ is
455an isomorphism, so~$\vphi$ is injective and well defined.
456The discussion in Section~\ref{sec:modmanconv} below shows that
457every element in $\sM_k(N,\eps)$ is a $\Z$-linear combination
458of elements in the image, so~$\vphi$ is surjective as well.
459\end{proof}
460
461\subsection{Conversion between modular and Manin symbols}
462\label{sec:modmanconv}
463For some purposes it is better to work with modular symbols,
464and for others it is better to work with Manin symbols.
465For example, there are descriptions of the Atkin-Lehner involution
466in terms of both Manin and modular symbols; in practice it is
467more efficient to compute this involution using modular symbols.
468It is thus essential to convert between these two representations.
469The conversion from Manin  to modular symbols
470is straightforward, and follows immediately from
471Theorem~\ref{thm:maninsymbols}.
472The conversion back is accomplished using the
473method used to prove Theorem~\ref{thm:maninsymbols};
474it is known as Manin's trick'',
475and involves continued fractions.
476
477Given a Manin symbol $[X^iY^{k-2-i},(u,v)]$,
478we write down a corresponding modular symbol as follows.
479Choose $\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ such that
480$(c,d)\con (u,v)\pmod{N}$.  This is possible
481by Lemma~1.38 of~\cite[pg.~20]{shimura:intro}; in practice
482it is not completely trivial, but can be accomplished
483using the extended Euclidean algorithm.
484Then
485 \begin{eqnarray*}
486 [X^iY^{k-2-i},(u,v)] &\corrto&
487    \abcd{a}{b}{c}{d}(X^iY^{k-2-i}\{ 0,\infty\})\\
488    &&= (dX-bY)^i(-cX+aY)^{2-k-i}
489       \left\{\frac{b}{d},\,\frac{a}{c}\right\}.\\
490\end{eqnarray*}
491
492In the other direction, suppose that we are given a modular
493symbol $P(X,Y)\{\alp,\beta\}$ and wish to represent it as a
494sum of Manin symbols.
495Because
496$P\{a/b,c/d\} = P\{a/b,0\}+P\{0,c/d\}$
497it suffices to write $P\{0,a/b\}$ in
498terms of Manin symbols.
499Let
500$$0=\frac{p_{-2}}{q_{-2}} = \frac{0}{1},\,\, 501\frac{p_{-1}}{q_{-1}}=\frac{1}{0},\,\, 502\frac{p_0}{1}=\frac{p_0}{q_0},\,\, 503\frac{p_1}{q_1},\,\, 504\frac{p_2}{q_2},\,\ldots,\,\frac{p_r}{q_r}=\frac{a}{b}$$
505denote the continued fraction convergents of the
506rational number $a/b$.
507Then
508$$p_j q_{j-1} 509 - p_{j-1} q_j = (-1)^{j-1}\qquad \text{for }-1\leq j\leq r.$$
510If we let
511$g_j = \mtwo{(-1)^{j-1}p_j}{p_{j-1}}{(-1)^{j-1}q_j}{q_{j-1}}$,
512then $g_j\in\sltwoz$ and
513\begin{align*}
514  P(X,Y)\{0,a/b\}
515 &=P(X,Y)\sum_{j=-1}^{r}\left\{\frac{p_{j-1}}{q_{j-1}},\frac{p_j}{q_j}\right\}\\
516 &=\sum_{j=-1}^{r} g_j((g_j^{-1}P(X,Y))\{0,\infty\})\\
517 &=\sum_{j=-1}^{r} [g_j^{-1}P(X,Y),((-1)^{j-1}q_j,q_{j-1})].
518\end{align*}
519Note that in the $j$th summand, $g_j^{-1}P(X,Y)$, replaces $P(X,Y)$.
520Since $g_j\in\sltwoz$ and $P(X,Y)$ has integer coefficients,
521the polynomial $g_j^{-1}P(X,Y)$ also has integer coefficients,
522so no denominators are introduced.
523
524The continued fraction expansion $[c_1,c_2,\ldots,c_n]$
525of the rational number $a/b$ can be computed
526using the Euclidean algorithm.
527The first term, $c_1$, is the quotient'': $a = bc_1+r$,
528with $0\leq r < b$.
529Let $a'=b$, $b'=r$ and compute $c_2$ as
530$a'=b'c_2+r'$, etc., terminating when the
531remainder is $0$.  For example, the expansion
532of $5/13$ is $[0,2,1,1,2]$.
533The numbers $$d_i=c_1+\frac{1}{c_2+\frac{1}{c_3+\cdots}}$$
534will then be the (finite) convergents.
535For example if $a/b=5/13$, then the convergents are
536  $$0/1,\,\, 1/0,\,\, d_1=0,\,\, d_2=\frac{1}{2},\,\, d_3=\frac{1}{3},\,\, 537 d_4=\frac{2}{5},\,\, d_5=\frac{5}{13}.$$
538
539
540
541\subsection{Hecke operators on Manin symbols}
542\label{subsec:heckeonmanin}
543Thoerem~2 of \cite{merel:1585} gives a description of
544the Hecke operators~$T_n$ directly  on the space of Manin symbols.
545This avoids the expense of first converting a Manin
546symbol to a modular symbol, computing~$T_n$ on the modular symbol,
547and then converting back.  For the reader's convenience, we very
548briefly recall Merel's theorem here, along with an enhancement
549of Cremona.
550
551As in~\cite[\S2.4]{cremona:algs}, define~$R_p$ as follows.
552When $p=2$,
553$$R_2 := \left\{\mtwo{1}{0}{0}{2}, 554 \mtwo{2}{0}{0}{1}, \mtwo{2}{1}{0}{1}, 555 \mtwo{1}{0}{1}{2}\right\}.$$
556When~$p$ is odd,~$R_p$ is the set of $2\times 2$ integer
557matrices $\abcd{a}{b}{c}{d}$ with determinant~$p$, and either
558\begin{enumerate}
559\item $a>|b|>0$, $d>|c|>0$, and $bc<0$; or
560\item $b=0$, and $|c|<d/2$; or
561\item $c=0$, and $|b|<a/2$.
562\end{enumerate}
563\begin{proposition}
564For $[P(X,Y),(u,v)]\in\sM_k(N,\eps)$ and~$p$ a prime, we have
565\begin{align*}T_p([P(X,Y),(u,v)])
566  &= \sum_{g\in R_p} [P(X,Y),(u,v)].g \\
567  &= \sum_{\abcd{a}{b}{c}{d}\in R_p} [P(aX+bY,cX+dY),(au+cv,bu+dv)]
568\end{align*}
569where the sum is restricted to matrices $\abcd{a}{b}{c}{d}$
570such that $\gcd(au+cv,bu+dv,N)=1$.
571\end{proposition}
572\begin{proof}
573For the case $k=2$ and an algorithm to compute $R_p$,
574see \cite[\S2.4]{cremona:algs}.
575The general case follows from~\cite[Theorem 2]{merel:1585} applied
576to the set~$\sS$ of~\cite[\S3]{merel:1585} by observing that
577when~$p$ is an odd {\em prime} $\sS_p'$ is empty.
578\end{proof}
579
580\subsection{The cuspidal and boundary spaces in terms of Manin symbols}
581This section is a review  of Merel's explicit description
582of the boundary map in terms of Manin symbols for $\Gamma=\Gamma_1(N)$
583(see~\cite[\S1.4]{merel:1585}).  In the next section, we
584describe a very efficient way to compute the boundary map.
585
586Let~$\cR$ be the equivalence relation
587on $\Gamma\backslash\Q^2$ which identifies
588the element
589$[\Gamma\smallvtwo{\lambda u}{\lambda v}]$
590with $\sign(\lambda)^k[\Gamma\smallvtwo{u}{v}]$,
591for any $\lambda\in\Q^*$.  Denote by $B_k(\Gamma)$
592the finite dimensional $\Q$-vector space
593with basis the equivalence classes
594$(\Gamma\backslash\Q^2)/\cR$.
595The dimension of this space is $\#(\Gamma\backslash\P^1(\Q))$.
596\begin{proposition}
597The map
598$$\mu:\sB_k(\Gamma)\ra B_k(\Gamma), 599\qquad P\left\{\frac{u}{v}\right\}\mapsto 600 P(u,v)\left[\Gamma\vtwo{u}{v}\right]$$
601is well defined and injective.
602Here $u$ and $v$ are assumed coprime.
603\end{proposition}
604Thus the kernel of $\delta:\sS_k(\Gamma)\ra \sB_k(\Gamma)$
605is the same as the kernel of $\mu\circ \delta$.
606\begin{proposition}\label{prop:boundary}
607Let $P\in V_{k-2}$ and $g=\abcd{a}{b}{c}{d}\in\sltwoz$.  We have
608$$\mu\circ\delta([P,(c,d)]) 609 = P(1,0)[\Gamma\smallvtwo{a}{c}] 610 -P(0,1)[\Gamma\smallvtwo{b}{d}].$$
611\end{proposition}
612
613
614\subsection{Computing the boundary map}
615\label{sec:computeboundary}
616In this section we describe how to efficiently compute the
617map $\sM_k(N,\eps)\ra B_k(N,\eps)$
618given in the previous section.  Henceforth, we will denote
619this map by~$\delta$.
620Our algorithm generalizes~\cite[\S2.2]{cremona:algs}.
621To compute the image of $[P,(c,d)]$, with
622$g=\abcd{a}{b}{c}{d}\in\sltwoz$,
623we must compute the class of $[\smallvtwo{a}{c}]$ and of
624$[\smallvtwo{b}{d}]$.
625Instead of finding a canonical form for cusps, we
626use a quick test for equivalence modulo scalars.
627In the following algorithm, by the $i$th standard cusp we mean
628the $i$th basis vector for a basis of $B_k(N,\eps)$.  The
629basis is constructed as the algorithm is called successively.
630We present the algorithm first, and then prove the results
631used by the algorithm in testing equivalence.
632
633\begin{algorithm}\label{alg:cusplist}
634Given a cusp $[\smallvtwo{u}{v}]$ this algorithm computes
635an integer~$i$ and a scalar~$\alp$ such that
636$[\smallvtwo{u}{v}]$ is equivalent to~$\alp$ times
637the $i$th standard cusp.
638First, using Proposition~\ref{prop:cusp1} and
639Algorithm~\ref{alg:cusp1}, check
640whether or not $[\smallvtwo{u}{v}]$ is equivalent,
641modulo scalars, to any cusp found so far.  If so,
642 return the index of the representative and the scalar.
643If not, record $\smallvtwo{u}{v}$ in the representative list.
644Then, using Proposition~\ref{prop:cuspdies},
645check whether or not $\smallvtwo{u}{v}$
646is forced to equal zero by the relations.
647If it does not equal zero, return its position
648in the list and the scalar~$1$.
649If it equals zero, return the scalar~$0$ and the position~$1$;
650keep $\smallvtwo{u}{v}$ in the list, and record that it is zero.
651\end{algorithm}
652
653In the case considered in \cite{cremona:algs}, the relations
654between cusps involve only the trivial character, so they
655do not force any cusp classes to vanish. He gives the
656following two criteria for equivalence.
657\begin{proposition}[Cremona]\label{prop:cusp1}
658Let $\smallvtwo{u_i}{v_i}$, $i=1,2$ be written so that
659$\gcd(u_i,v_i)=1$.
660\begin{enumerate}
661\item There exists $g\in\Gamma_0(N)$ such that
662    $g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$ if and only if
663 $$s_1 v_2 \con s_2 v_1 \pmod{\gcd(v_1 v_2,N)},\, 664\text{ where s_j satisfies u_j s_j\con 1\pmod{v_j}}.$$
665\item There exists $g\in\Gamma_1(N)$ such that
666    $g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$ if and only if
667 $$v_2 \con v_1 \pmod{N}\text{ and } u_2 \con u_1 \pmod{\gcd(v_1,N)}.$$
668\end{enumerate}
669\end{proposition}
670\begin{proof}
671The first is Proposition 2.2.3 of \cite{cremona:algs}, and
672the second is Lemma 3.2 of \cite{cremona:gammaone}.
673\end{proof}
674
675\begin{algorithm}\label{alg:cusp1}
676Suppose $\smallvtwo{u_1}{v_1}$ and
677$\smallvtwo{u_2}{v_2}$
678are equivalent modulo $\Gamma_0(N)$.
679This algorithm computes a matrix $g\in\Gamma_0(N)$ such
680that $g\smallvtwo{u_1}{v_1}=\smallvtwo{u_2}{v_2}$.
681Let $s_1, s_2, r_1, r_2$ be solutions to
682$s_1 u_1 -r_1 v_1 =1$ and
683$s_2 u_2 -r_2 v_2 =1$.
684Find integers $x_0$ and $y_0$ such
685that $x_0v_1v_2+y_0N=1$.
686Let $x=-x_0(s_1v_2-s_2v_1)/(v_1v_2,N)$
687and $s_1' = s_1 + xv_1$.
688Then $g=\mtwo{u_2}{r_2}{v_2}{s_2} 689 \cdot \mtwo{u_1}{r_1}{v_1}{s_1'}^{-1}$
690sends $\smallvtwo{u_1}{v_1}$ to $\smallvtwo{u_2}{v_2}$.
691\end{algorithm}
692\begin{proof}
693This follows from the proof of Proposition~\ref{prop:cusp1} in
694\cite{cremona:algs}.
695\end{proof}
696
697
698To see how the~$\eps$ relations, for nontrivial~$\eps$,
699make the situation more complicated, observe that it is
700possible that $\eps(\alp)\neq \eps(\beta)$ but
701$$\eps(\alp)\left[\vtwo{u}{v}\right] =\left[\gamma_\alp \vtwo{u}{v}\right]= 702 \left[\gamma_\beta \vtwo{u}{v}\right]=\eps(\beta)\left[\vtwo{u}{v}\right];$$
703One way out of this difficulty  is to construct
704the cusp classes for $\Gamma_1(N)$, then quotient
705out by the additional~$\eps$ relations using
706Gauss elimination. This is far too
707inefficient to be useful in practice because the number of
708$\Gamma_1(N)$ cusp classes is huge.  Instead,
709we give a quick test to determine whether or not
710a cusp vanishes modulo the $\eps$-relations.
711
712\begin{lemma}\label{lem:canlift}
713Suppose $\alp$ and $\alp'$ are integers
714such that $\gcd(\alp,\alp',N)=1$.
715Then there exist integers $\beta$ and $\beta'$,
716congruent to $\alp$ and $\alp'$ modulo $N$, respectively,
717 such that $\gcd(\beta,\beta')=1$.
718\end{lemma}
719\begin{proof}
720By \cite[1.38]{shimura:intro} the map
721$\SL_2(\Z)\ra\SL_2(\Z/N\Z)$ is surjective.
722By the Euclidean algorithm, there exists
723integers $x$, $y$ and $z$ such that
724$x\alp + y\alp' + zN = 1$.
725Consider the matrix
726$\abcd{y}{-x}{\alp}{\hfill\alp'}\in \SL_2(\Z/N\Z)$
727and take $\beta$, $\beta'$ to be the bottom
728row of a lift of this matrix to $\SL_2(\Z)$.
729\end{proof}
730
731\begin{proposition}\label{prop:cuspdies}
732Let $N$ be a positive integer and $\eps$ a Dirichlet
733character modulo $N$.
734Suppose $\smallvtwo{u}{v}$ is a cusp with $u$ and $v$ coprime.
735Then $\smallvtwo{u}{v}$ vanishes modulo the relations
736$$\left[\gamma\smallvtwo{u}{v}\right]= 737\eps(\gamma)\left[\smallvtwo{u}{v}\right],\qquad 738\text{all \gamma\in\Gamma_0(N)}$$
739if and only if there exists $\alp\in(\Z/N\Z)^*$,
740with $\eps(\alp)\neq 1$, such that
741\begin{align*}
742 v &\con \alp v \pmod{N},\\
743 u &\con \alp u \pmod{\gcd(v,N)}.
744\end{align*}
745\end{proposition}
746\begin{proof}
747First suppose such an $\alp$ exists.
748By Lemma~\ref{lem:canlift}
749there exists $\beta, \beta'\in\Z$ lifting
750$\alp,\alp^{-1}$ such that $\gcd(\beta,\beta')=1$.
751The cusp $\smallvtwo{\beta u}{\beta' v}$
752has coprime coordinates so,
753by Proposition~\ref{prop:cusp1} and our
754congruence conditions on $\alp$, the cusps
755$\smallvtwo{\beta{}u}{\beta'{}v}$
756and $\smallvtwo{u}{v}$ are equivalent by
757an element of $\Gamma_1(N)$.
758This implies that $\left[\smallvtwo{\beta{}u}{\beta'{}v}\right] 759 =\left[\smallvtwo{u}{v}\right]$.
760Since $\left[\smallvtwo{\beta{}u}{\beta'{}v}\right] 761 = \eps(\alp)\left[\smallvtwo{u}{v}\right]$,
762our assumption that $\eps(\alp)\neq 1$
763forces $\left[\smallvtwo{u}{v}\right]=0$.
764
765Conversely, suppose $\left[\smallvtwo{u}{v}\right]=0$.
766Because all relations are two-term relations, and the
767$\Gamma_1(N)$-relations identify $\Gamma_1(N)$-orbits,
768there must exists $\alp$ and $\beta$ with
769  $$\left[\gamma_\alp \vtwo{u}{v}\right] 770 =\left[\gamma_\beta \vtwo{u}{v}\right] 771 \qquad\text{ and }\eps(\alp)\ne \eps(\beta).$$
772Indeed, if this did not occur,
773then we could mod out by the $\eps$ relations by writing
774each $\left[\gamma_\alp \smallvtwo{u}{v} \right]$
775in terms of  $\left[\smallvtwo{u}{v}\right]$, and there would
776be no further relations left to kill
777$\left[\smallvtwo{u}{v}\right]$.
778Next observe that
779$$780\left[\gamma_{\beta^{-1}\alp} 781 \vtwo{u}{v}\right] 782 = \left[\gamma_{\beta^{-1}}\gamma_\alp 783 \vtwo{u}{v}\right] 784 = \eps(\beta^{-1})\left[\gamma_\alp 785 \vtwo{u}{v}\right] 786 = \eps(\beta^{-1})\left[\gamma_\beta 787 \vtwo{u}{v}\right] 788 = \left[\vtwo{u}{v}\right].$$
789Applying Proposition~\ref{prop:cusp1} and
790noting that $\eps(\beta^{-1}\alp)\neq 1$ shows
791that $\beta^{-1}\alp$ satisfies the properties
792of the $\alp$'' in the statement of the
793proposition we are proving.
794\end{proof}
795
796The possible $\alp$ in Proposition~\ref{prop:cuspdies}
797can be enumerated as follows. Let $g=(v,N)$ and list the
798$\alp=v\cdot\frac{N}{g}\cdot{}a+1$, for $a=0,\ldots,g-1$,
799such that $\eps(\alp)\neq 0$.
800
801
802
803{\vspace{3ex}\em\par\noindent Working in the plus or minus quotient. }
804Let~$s$ be a sign, either~$+1$ or~$-1$.
805To compute $\sS_k(N,\eps)_s$ it is necessary
806to replace $B_k(N,\eps)$ by its quotient modulo the
808$\left[ \smallvtwo{-u}{\hfill v}\right] 809= s \left[\smallvtwo{u}{v}\right]$
810for all cusps $\smallvtwo{u}{v}$.
811Algorithm~\ref{alg:cusplist} can be modified to deal
812with this situation as follows.
813Given a cusp $x=\smallvtwo{u}{v}$, proceed as
814in Algorithm~\ref{alg:cusplist} and check if
815either $\smallvtwo{u}{v}$ or $\smallvtwo{-u}{\hfill v}$
816is equivalent (modulo scalars) to any cusp seen so far.  If not,
817use the following trick to determine whether
818the $\eps$ and $s$-relations
819kill the class of $\smallvtwo{u}{v}$:
820use the unmodified Algorithm~\ref{alg:cusplist}
821to compute the scalars $\alp_1, \alp_2$ and
822standard indices $i_1$, $i_2$ associated to
823$\smallvtwo{u}{v}$ and $\smallvtwo{-u}{\hfill v}$, respectively.
824The $s$-relation kills the class of  $\smallvtwo{u}{v}$
825if and only if $i_1=i_2$ but $\alp_1\neq s\alp_2$.
826
827
828\section{The complex torus attached to a modular form}
829\label{sec:tori}
830Fix integers $N\geq 1$, $k\geq 2$, and let~$\eps$ be a mod~$N$
831Dirichilet character.  {\bf For the rest of this section assume that
832$\eps^2=1$.} We construct a Hecke and complex
833conjugation invariant lattice in~$S$, hence a complex torus
834$J_k(N,\eps)$ equipped with an action of Hecke operators and an
835$\R$-structure.  The reader may wish to compare our construction to
836the one given by Shimura in~\cite{shimura:surles}, where in addition
837he observes that the Petterson pairing gives his torus the structure
838of an abelian variety.  When $k=2$, the torus comes from an abelian
839variety defined over~$\Q$; when $k>2$, the study of these complex tori
840is of interest in trying to understand the conjectures of Bloch and
841Kato (see \cite{bloch-kato}) on motifs attached to modular forms.
842
843Let $\sS=\sS_k(N,\eps)$ (resp., $S=S_k(N,\eps)$)
844be the associated space of cuspidal modular symbols (resp., forms).
845The Hecke algebra~$\T$ acts in a way compatible with the
846integration pairing
847$\langle\quad,\quad\rangle 848 : S \cross \sS \ra \C,$
849which we view as a $\T$-module homomorphism $\Phi:\sS\ra S^*=\Hom(S,\C)$
850called the \defn{period mapping}.
851Because $\eps^2=1$, the $*$-involution preserves~$S$.
852\begin{proposition}
853The period mapping~$\Phi$ is injective and the
854image of~$\Phi$ is a lattice in $S^*$.
855\end{proposition}
856\begin{proof}
857By Theorem~\ref{thm:perfectpairing},
858 $\sS\tensor_{\R}\C\isom 859 \Hom_\C(S\oplus \Sbar,\C).$
860Because $\eps^2=1$, we have $S = S_k(N,\eps;\R)\tensor_{\R}\C$.
861Set $S_\R := S_k(N,\eps;\R)$ and likewise defined $\Sbar_\R$.
862We have
863$$\Hom_\C(S\oplus \Sbar,\C) = 864 \Hom_\R(S_\R \oplus \Sbar_\R,\R)\tensor_\R \C.$$
865Let $\sS_{\R} = \sS_k(N,\eps;\R)$ and $\sS_{\R}^+$ be the
866subspace fixed under~$*$.  By Proposition~\ref{prop:starpairing}
867we have maps
868$$\sS_{\R}^+ \ra \Hom_{\R}(S_{\R}\oplus\Sbar_\R,\R) 869 \ra \Hom_{\R}(S_{\R},\R)$$
870and
871$$\sS_{\R}^- \ra \Hom_{\R}(S_{\R}\oplus\Sbar_\R,i\R) 872 \ra \Hom_{\R}(S_{\R},i\R).$$
873The map $\sS_{\R}^+\ra \Hom_{\R}(S_{\R},\R)$ is
874an isomorphism: the point is that if
875$\langle \bullet, x\rangle$, for $x\in \sS_{\R}^+$,
876vanishes on $S_\R$ then it  vanishes on  the
877whole of $S\oplus \Sbar$.  Likewise, the map
878$\sS_{\R}^-\ra \Hom_{\R}(S_{\R},i\R)$
879is an isomorphism.  Thus
880$$\sS_{\R} \isom \Hom_{\R}(S_{\R},\R) 881\oplus \Hom_{\R}(S_{\R},i\R) 882\isom \Hom_{\C}(S,\C).$$
883Finally, observing that $\sS$ is by definition
884torsion free completes the proof.
885\end{proof}
886
887Thus we have a torus
888$J_k(N,\eps)$
889that fits into an exact sequence
890$$0\lra \sS \xrightarrow{\quad\Phi\quad} 891 \Hom(S,\C) \lra J_k(N,\eps) \lra 0.$$
892Let $f\in S$ be a newform and $S_f$ the complex vector
893space spanned by the Galois conjugates of~$f$.
894The period map $\Phi_f$ associated to~$f$ is the map $\sS\ra \Hom(S_f,\C)$
895obtaind by composing~$\Phi$ with restriction to $S_f$.
896Set
897  $$A_f := \Hom(S_f,\C) / \Phi_f(\sS).$$
898
899Associate\label{pg:dual} an abelian subvariety of~$J$ to~$f$ as follows.
900Let $I_f = \Ann_{\T}(f)$ be the annihilator of $f$ in the Hecke algebra,
901and set
902$$\Adual_f := \Hom(S,\C)[I_f]/\Phi(\sS[I_f])$$
903where $\Hom(S,\C)[I_f] = \intersect_{t \in I_f} \ker(t)$.
904
905The following diagram summarizes the tori just defined;
906its columns are exact
907but its rows are {\em not}.
908\begin{equation}\label{dgm:uniformization}
909\[email protected]=.9pc{
910    0\ar[d]            &        0\ar[d]             &  0\ar[d]   \\
911  \sS[I_f]\ar[r]\ar[dd] &  \sS\ar[r]\ar[dd]&\Phi_f(\sS)\ar[dd] \\
912                      &                 &      \\
913\Hom(S,\C)[I_f]\ar[r]\ar[dd] &\Hom(S,\C)\ar[r]\ar[dd] &\Hom(S[I_f],\C)\ar[dd]\\
914                      &                 &      \\
916& J_k(N,\eps) \ar[r]\ar[d]& A_f \ar[d]\\
917    0   &   0    &  0   \\
918}\end{equation}
919
920
921\subsection{Weight~$2$}
922When $k=2$ and $\eps=1$ the above is just Shimura's classical
923association of an abelian variety to a modular form;
924see~\cite[Thm.~7.14]{shimura:intro} and~\cite{shimura:factors}.
925In this case $A_f$ and $\Adual_f$ are abelian varieties
926that are defined over~$\Q$ and are dual to each other.
927Furthermore $A_f$ is an \defn{optimal} quotient of $J$, in the
928sense that the kernel of $J\ra A_f$ is connected.
929
930
931
932
933
934
935
936
937
938
939
940
941