CoCalc Shared Fileswww / papers / thesis-old / compgroup.tex
Author: William A. Stein
1% compgroup.tex
2\chapter{Component groups of optimal quotients}
3
4Let~$A$ be an abelian variety over a finite extension~$K$ of the
5$p$-adic numbers~$\Qp$.  Let~$\O$ be the ring of integers of~$K$,~$\m$
6its maximal ideal and $k=\O/\m$ the residue class field.
7The N\'{e}ron model of~$A$ is a smooth commutative group scheme~$\A$
8over~$\O$ such that~$A$ is its generic fiber and satisfying the
9property: the restriction map
10$$\Hom_\O(S,\A)\lra \Hom_K(S_K,A)$$
11is bijective for all smooth schemes~$S$ over~$\O$.
12The closed fiber~$\A_k$ is a group scheme over~$k$,
13which need not be connected; denote by~$\A_k^0$ the
14connected component containing the identity.
15There is an exact sequence
16     $$0\lra\A_k^0\lra\A_k\lra\Phi_A\lra0$$
17with~$\Phi_A$ a finite \'{e}tale group scheme over~$k$, i.e.,
18a finite abelian group
19equipped with an action of $\Gal(\kbar/k)$.
20\label{defn:componentgroup}
21
22In this chapter we study the groups~$\Phi_A$
23attached to quotients~$A$ of Jacobians of modular
24curves~$X_0(N)$.  When~$A$ has semistable reduction,
25Grothendieck described the component group in terms
26of a monodromy pairing on certain free abelian groups.
27When $A=J$ is the Jacobian of $X_0(N)$, this pairing
28can be explicitly computed, hence so can~$\Phi_J$;
29this has been done in many cases in
30\cite{mazur:eisenstein} and \cite{edixhoven:eisen}.
31Suppose now that $A=A_f$ is a quotient of~$J$ corresponding
32to a newform~$f$, so the kernel of
33the map $\pi:J\ra A$ is connected.
34There is a natural map $\pi_*:\Phi_J\ra \Phi_A$.
35We describe how to compute
36the image and the order of the cokernel of $\pi_*$.
37
38We now state our main result more precisely.
39Suppose $\pi:J\ra A$ is an optimal quotient,
40with~$J$ a semistable Jacobian and~$A$ purely toric.
41We express the component group of~$A$ in terms of
42the monodromy pairing associated to~$J$.
43Let $m_A=\sqrt{\deg(\theta_A)}$ where
44$\theta_A:A'\ra A$ is induced by
45the canonical principal polarization of $J$.
46Let $X_J$ be the character group of the toric part
47of the special fiber of $J$.  Let $\cL$ be the saturation
48of the image of $X_A$ in $X_J$.  The monodromy pairing
49defines a map $\alp:X_J\ra \Hom(\cL,\Z)$.
50Let $\Phi_X$ be the cokernel of $\alp$ and
51$m_X=[\alp(X_J):\alp(\cL)]$ be the order of the finite
52group $\alp(X_J)/\alp(\cL)$.   We prove that
53   $$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_X|}{m_X}.$$
54More precisely, $\Phi_X$ is the image of $\Phi_J \ra \Phi_A$
55and the cokernel of $\Phi_J \ra \Phi_A$ has order $m_A/m_X$.
56If the optimal quotient $J\ra A$ arises from a modular
57form on $\Gamma_0(N)$,
58then the quantities $m_A$, $m_X$ and $\Phi_X$ can
59be explicitly computed, hence so can $|\Phi_A|$.
60Having done this, we present some tables and discuss the
61conjectures that they suggest.
62
63\section{Optimal quotients of Jacobians}
64Let~$J$ be a Jacobian equipped with its canonical principal
65polarization~$\theta_J$.
66An \defn{optimal quotient} of $J$ is an
67abelian variety $A$ and a surjective
68map $\pi: J \ra A$ whose
69kernel is an abelian subvariety $B$ of $J$.
70Denote by $J'$ and $A'$ the abelian varieties dual to $J$ and $A$,
71respectively.
72Dualizing $\pi$ and composing with $\theta_J'=\theta_J$
73we obtain a map
74    $A'\xrightarrow{\pi'} J'\xrightarrow{\theta_J} J$.
75\begin{proposition}
76$A'\ra J$ is injective.
77\end{proposition}
78\begin{proof}
79Since $\theta_J$ is an isomorphism it suffices to prove
80that $\pi'$ is injective.
81Since the dual of $\pi'$ is $(\pi')'=\pi$ and $\pi$ is surjective,
82$\pi'$ must have finite kernel.
83Thus $A' \ra C=\im(\pi')$ is
84an isogeny.  Let $G$ denote the kernel,
85and dualize. By \cite[\S11]{milne:abvars} we have have
86$$\xymatrix{ 87 G\ar[r] & A'\[email protected]{->>}[r] \ar[dr]_{\pi'} 88 & C\ar[d]\\ 89 && J' 90}\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad 91\xymatrix{ 92 A & C\ar[l] & G'\ar[l] \\ 93 & J\ar[u]_{\vphi}\ar[ul]^{\pi} 94}$$
95with $G'$ the Cartier dual of $G$.
96Since $G'$ is finite, $\ker(\vphi)\subset\ker(\pi)$
97is of finite index.
98Since $\ker(\pi)$ is an abelian  variety it is divisible.
99Thus $\ker(\vphi)=\ker(\pi)$ so $G'=0$ and hence $G=0$.
100\end{proof}
101We denote the map $A'\ra J$ by $\pi'$.
102The kernel of $\theta_A$ measures the intersection of
103$A'$ and $B=\ker(\theta_A)$ inside of $J$
104as shown in the following diagram.
105$$\xymatrix{ 106A'\intersect B\ar[r]\ar[d] & B\ar[d] \\ 107 A'\[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\ 108 & A 109}$$
110Because $\theta_A$ is a polarization, $|\ker(\theta_A)|$ is
111a square \cite[Theorem 13.3]{milne:abvars}.   The
112\defn{congruence modulus} is the integer
113  $$m_A=\sqrt{|\ker(\theta_A)|}.$$
114
115\section{The special fiber of the N\'{e}ron model}
116Let~$K$ be a finite extension of $\Qp$ with ring of integers~$\O$
117and residue class field~$k$.
118Let $A/K$ be an abelian variety and denote its N\'{e}ron model
119by~$\cA$.
120Let $\Phi_A$ be the group of connected components of
121the special fiber $\cA_k$. This group
122is a finite \'{e}tale group scheme over~$k$, i.e.,
123a finite abelian group equipped with an action of
124$\Gal(\kbar/k)$. There is an exact sequence of group schemes
125    $$0 \ra \cA_k^0 \ra \cA_k \ra \Phi_A \ra 0.$$
126The group scheme $\cA_k^0$ is an extension of an abelian variety~$\cB$
127of dimension~$a$ by a group scheme~$\cC$; we have a diagram
128$$\[email protected]=.3cm{ 129&0\ar[d]\\ 130&{\cT}\ar[d]\\ 1310\ar[r]&{\cC}\ar[r]\ar[d]&{\cA_k^0}\ar[r]&{\cB}\ar[r]&0\\ 132&{\cU}\ar[d]\\ 133&0}$$
134with~$\cT$ a torus of dimension~$t$
135and of a~$\cU$ a unipotent group of dimension~$u$.
136The abelian variety~$A$ is said to have \defn{purely toric} reduction
137if $t=\dim A$, and is \defn{semistable} if $u=0$.
138The character group $X_A = \Hom(\cT,\Gm)$\label{defn:chargroup} is a
139free abelian group of rank~$t$ contravariantly associated to~$A$.
140If~$A$ is semistable there is a monodromy pairing
141$X_A\cross X_{A'}\ra \Z$ and an exact sequence
142  $$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
143
144\section{Rigid uniformization}
145In this section we review the rigid analytic uniformization of
146a semistable abelian variety over a finite extension~$K$ of the
147maximal unramified extension $\Qp^{\ur}$ of $\Qp$.   We use this to prove
148that if~$A$ is purely toric and $\phi:A'\ra A$ is a
149symmetric isogeny, then
150  $$\deg(\phi) = (\# \coker(X_A\ra X_{A'}))^2.$$
151We also prove a some lemmas about character groups.
152
153\subsection{Raynaud and van der Put uniformization}\label{subsec:raynaud}
154
155\begin{theorem}[Raynaud]\label{raynaud}
156If~$A$ is a semistable Abelian variety, its universal
157covering is isomorphic to an extension~$G$ of an abelian
158variety~$B$ with good reduction by a torus~$T$, the
159covering map from~$G$ to~$A$ is a homomorphism and
160its kernel is a twisted free Abelian group~$\Gamma$ of finite rank.
161\end{theorem}
162This may be summarized by the following diagram,
163$$\xymatrix{ 164 &\Gamma\ar[d] \\ 165 T\ar[r] & G\ar[r]\ar[d] & B\\ 166 & A 167}$$
168which we call the \defn{uniformization cross} of~$A$.
169The group~$\Gamma$ can be identified with the character
170group $X_A$ of the previous section.
171The uniformization cross of the dual abelian variety
172$A'$ is
173$$\xymatrix{ 174 &\Gamma'\ar[d] \\ 175 T'\ar[r] & G'\ar[r]\ar[d] & B'\\ 176 & A' 177}$$
178where $\Gamma'=\Hom(T,\Gm)$ and $T'=\Hom(\Gamma,\Gm)$
179and the morphisms $\Gamma'\ra G'$ and $T'\ra G'$ are the
180one-motive duals of the morphisms $T\ra G$ and $\Gamma\ra G$,
181respectively.
182For more details see, e.g., \cite{coleman:monodromy}.
183
184\begin{example}[Tate curve]
185If $E/\Qp$ is an elliptic curve with multiplicative reduction
186then the uniformization is $E=\Gm/q^\Z$ where $q=q(j)$ is
187obtained by inverting the expression for~$j$ as a function of
188$q(z)=e^{2\pi iz}$.
189\end{example}
190
191
192\subsection{Some lemmas}
193Let $\pi:J\ra A$ be an optimal quotient,
194with~$J$ semistable and~$A$ purely toric.
195\begin{lemma}\label{lem:surj}
196The map $\Gamma_J\ra \Gamma_A$ induced by~$\pi$ is surjective.
197\end{lemma}
198\begin{proof}
199Because $G_J$ is simply connected,~$\pi$ induces a map
200$G_J\ra T_A$ and a map $\Gamma_J\ra \Gamma_A$.
201Because~$\pi$ is surjective and $T_A$ is a
202torus, the map $G_J\ra T_A$ is surjective.
203The snake lemma applied to the following diagram gives
204a surjective map from $B=\ker(\pi)$ to
205$M=\coker(\Gamma_J\ra\Gamma_A)$.
206$$\xymatrix{ 207 & \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\ 208 & G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\ 209B\ar[r] & J\ar[r]^{\pi}& A 210}$$
211Because~$\pi$ is optimal,~$B$ is connected so~$M$ must also be connected.
212Since~$M$ is discrete it follows that $M=0$.
213\end{proof}
214
215\subsubsection{Purely toric abelian varieties}
216Assume that~$A$ is purely toric. Then
217$B=0$, and  the uniformization cross becomes
218     $$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A}$$
219Let $\vphi:A'\ra A$ be a \defn{symmetric isogeny}, i.e.,
220$\vphi':A'\ra (A')'=A$ is equal to~$\vphi$.
221Denote by $\vphi_t:T'\ra T$ and $\vphi_a:\Gamma'\ra\Gamma$ the
222induced maps.
223\begin{proposition}\label{prop:kerphi}
224There is an exact sequence
225$$0\ra\ker(\vphi_t) \ra \ker(\vphi) \ra \coker(\vphi_a)\ra 0,$$
226and $\ker(\vphi_t)$ is dual to $\coker(\vphi_a)$.
227\end{proposition}
228\begin{proof}
229Since~$\vphi$ is an isogeny we have the following diagram:
230$$\xymatrix{ 231 0\ar[r]\ar[d] & \Gamma'\ar[r]\ar[d] & \Gamma\ar[r]\ar[d] 232 & \coker(\vphi_a)\ar[d]\\ 233 \ker(\vphi_t)\ar[r]\ar[d]& T'\ar[d]\ar[r] & T\ar[r]\ar[d] & 0\\ 234 \ker(\vphi)\ar[r] & A'\ar[r]^{\vphi} & A}$$
235The snake lemma then gives the claimed exact sequence.
236For the second assertion observe that the one-motive dual of the diagram
237$$\xymatrix{ 238 & \Gamma'\ar[r]\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\ 239 \ker(\vphi_t)\ar[r] & T'\ar[r]^{\vphi} & T}$$
240is the diagram
241$$\xymatrix{ 242 & T & T'\ar[l]_{\vphi'} & \coker(\vphi_a)'\ar[l]\\ 243 \ker(\vphi_t)'& \Gamma\ar[l]\ar[u] &\Gamma'\ar[l]\ar[u] 244}$$
245Since $\vphi$ is symmetric, $\vphi'=\vphi$ and so
246     $$\ker(\vphi_t) = \coker(\vphi_a)'.$$
247\end{proof}
248
249\begin{lemma}\label{lem:isogcoker}
250$|\ker(\vphi)|=|\coker(\vphi_a)|^2$
251\end{lemma}
252\begin{proof}
253The order of a finite group scheme equals the order of its
254dual.
255\end{proof}
256
257\section{The main theorem}
258Let $\pi:J\ra A$ be an optimal quotient,
259with~$J$ a semistable Jacobian and~$A$ purely toric.
260Let $X_A$, $X_{A'}$, and $X_J$ denote the
261character groups of the toric parts of the
262special fibers.
263
264\subsection{Monodromy description of the component group}
265There is a pairing
266$X_A\cross X_{A'}\ra \Z$ called
267the monodromy pairing.  We have an exact sequence
268  $$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
269If~$J$ is a Jacobian then~$J$ is canonically self-dual so
270the monodromy pairing on~$J$
271can be viewed as a pairing $X_J\cross X_J \ra \Z$ and
272there is an exact sequence
273  $$0 \ra X_{J} \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0.$$
274
275\begin{example}[Tate curve]
276Suppose $E=\Gm/q^{\Z}$ is a Tate curve.
277The monodromy pairing on $X_E=q^{\Z}$ is
278$$\langle q, q\rangle = \ord_p(q)=-\ord_p(j).$$
279Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
280\end{example}
281
282
283  \subsubsection{Proof of the main theorem}
284We now prove the main theorem.
285The key diagrams are
286$$\[email protected]=3pc{A' \[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta} 287 & J \[email protected]{->>}[d]^{\pi}\\ 288 &A} 289\qquad\qquad\qquad 290 \[email protected]=3pc{X_A \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*} 291 & X_J \[email protected]{->>}[d]^{\pi_*} \\ 292 & X_{A'}\[email protected]/^1.5pc/[ul]^{\theta_*}} 293$$
294The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
295The injectivity of $\pi^*$ follows because
296$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
297and multiplication by $\deg(\theta)$ on a free abelian
298group is injective.
299
300Let
301 $$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
302be the map defined by the monodromy pairing restricted
303to $X_J\cross \pi^* X_A$.
304\begin{lemma}\label{lem:twokers}
305$\ker(\pi_*) = \ker(\alp)$
306\end{lemma}
307\begin{proof}
308Suppose $x\in \ker(\pi_*)$ and let $y=\pi^* z$ with
309$z\in X_A$.  Then
310$$\langle x, y \rangle = \langle x, \pi^* z \rangle 311 = \langle \pi_* x, z \rangle = 0$$
312so $x\in\ker(\alp)$.
313Next let $x\in\ker(\alp)$.
314Then for all $z\in X_A$,
315$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle$$
316so $\pi_* x$ is in the kernel of the
317monodromy map
318$$X_{A'} \ra \Hom(X_A,\Z).$$
319Since $X_{A'}$ and $\Hom(X_A,\Z)$ are free of the same rank
320and the cokernel is torsion, the monodromy map is injective.
321Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
322\end{proof}
323
324\begin{lemma}\label{lem:compphi}
325There is an exact sequence
326$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
327\end{lemma}
328\begin{proof}
329Lemma~\ref{lem:twokers} gives the following
330commutative diagram with exact rows
331$$\xymatrix{0\ar[r] 332 & X_J/\ker(\alp)\ar[d]^{\isom} \ar[r] 333 & {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\ 334 0\ar[r] & X_{A'}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0}$$
335By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
336The second is an isomorphism because it is induced by the
337isomorphism $\pi^*:X_A\ra \pi^* X_A$.  It follows that
338$\coker(\alp)\isom \Phi_A$, as claimed.
339\end{proof}
340
341Let $\cL$ be the \defn{saturation} of $\pi^* X_A$ in $X_J$, i.e.,
342$[\cL:\pi^*X_A]$ is finite and $X_J/\cL$ is torsion free.
343Suppose $L$ is of finite index in $\cL$.
344Define the \defn{congruence modulus} of $L$
345   $$m_L = [\alp(X_J):\alp(L)]$$
346and the \defn{component group} by
347  $$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
348When $L=\cL$ we often set $m_X=m_\cL$ and $\Phi_X=\Phi_\cL$
349and think of $m_X$ and $\Phi_X$ as the character group
350congruence modulus and component group.''
351
352\begin{lemma}\label{lem:homog}
353The rational number $\ds \frac{|\Phi_L|}{m_L}$ does not
354depend on the choice of~$L$.
355\end{lemma}
356\begin{proof}
357If $L'$ is another choice let $n=[L:L']\in\Q$.
358Then since $\alp$ is injective when restricted to $\cL$,
359 $$m_{L'} = [\alp(X_J):\alp(L')] 360 = [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n$$
361and similarly $|\Phi_{L'}| = |\Phi_L|\cdot n$.
362\end{proof}
363
364Recall that we defined
365\begin{eqnarray*}
366  m_A &=& \sqrt{\deg(\theta)}\\
367  \Phi_A &=& \coker(X_{A'}\ra \Hom(X_A,\Z))
368\end{eqnarray*}
369
370\begin{theorem}\label{formula}
371For any $L$ of finite index in $\cL$
372the following relation holds:
373$$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_L|}{m_L}.$$
374\end{theorem}
375\begin{proof}
376By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
377With this choice of $L$, Lemma~\ref{lem:compphi} says that
378$\Phi_L \isom \Phi_A$.
379By Lemma~\ref{lem:twokers}, properties of the index,
380and Lemma~\ref{lem:isogcoker} we have
381\begin{eqnarray*}
382m_L&=&[\alp(X_J):\alp(L)] \\
383   &=& [\pi_*(X_J):\pi_*(L)]\\
384   &=& [X_{A'}:\pi_*(\pi^*X_A)]\\
385   &=& [X_{A'}:\theta^* X_A]\\
386   &=& \#\coker(\theta^*) \\
387   &=& \sqrt{\deg(\theta)} = m_A.
388\end{eqnarray*}
389\end{proof}
390
391\begin{proposition}\label{prop:compim}
392$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\cL.$$
393\end{proposition}
394\begin{proof}
395Because $\pi^*X_A\subset \cL \subset X_J$, by
396Lemma~\ref{lem:compphi} we obtain a commutative diagram
397with exact rows
398$$\xymatrix{ 399 X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\ 400 X_J\ar[r]\[email protected]{=}[d]& \Hom(\cL,\Z)\ar[r]\ar[d]& \Phi_\cL \ar[r]\ar[d] & 0\\ 401 X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0 402}$$
403The map $\Hom(\cL,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism
404so $\Phi_\cL\ra\Phi_A$ is injective, hence
405 $$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\cL).$$
406The cokernel of $\Hom(X_J,\Z)\ra\Hom(\cL,\Z)$
407surjects onto the cokernel of $\Phi_J\ra \Phi_\cL$.
408Using the exact sequence
409$$0\ra \cL \ra X_J \ra X_J/\cL \ra 0,$$
410we find that
411$$\coker(\Hom(X_J,\Z)\ra\Hom(\cL,\Z)) \subset \Ext^1(X_J/\cL,\Z)=0,$$
412where $\Ext^1$ vanishes because $\cL$ is saturated
413so that $X_J/\cL$ is torsion free.  Thus the cokernel of
414$\Phi_J\ra\Phi_\cL$ is $0$, from which the proposition follows.
415\end{proof}
416
417The following corollary
418follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
419\begin{corollary}\label{cor:div}
420$$|\coker(X_J\ra X_A)| = \frac{m_A}{m_\cL}.$$
421As a consequence, $m_\cL | m_A.$
422\end{corollary}
423
424\section{Optimal quotients of $J_0(N)$}
425Let $X_0(N)/\Q$ be the modular curve associated to the congruence
426subgroup $\Gamma_0(N)\subset\sltwoz$ of matrices which are upper
427triangular modulo $N$.  Let $p$ be a prime divisor of $N$ which is
428coprime to $M=N/p$.  The Jacobian $J=J_0(N)$ of $X_0(N)$ has semistable
429reduction at $p$.   The Hecke algebra
430$$\T=\Z[\ldots T_n\ldots]\subset\End(J)$$
431is a commutative ring of endomorphisms of $J$ of $\Z$-rank $=\dim J$.
432The character group $X_J$ is equipped with a
433functorial action of $\T$.
434The Hecke algebra $\T$ also act on the cusp
435forms $$S = S_2(\Gamma_0(N),\C).$$
436A newform $f$ is an eigenform normalized so that the coefficient
437of $q$ in the expansion of $f$ at the cusp $\infty$ is $1$, and
438such that $f$ does not occur at any level $N'\mid N$ with $N'\neq N$.
439If $f$ is a newform, let $I_f$ be the ideal in $\T$ of
440elements which annihilate $f$.  Then $\O_f=\T/I_f$ is an
441order in the ring of integers of the totally real number field
442$K_f$ obtained by adjoining the Fourier coefficients of $f$ to $\Q$.
443The quotient
444  $$A_f = J_0(N)/ I_f J_0(N)$$
445 is a purely toric optimal quotient of dimension $[K_f:\Q]$.
446
447Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
448complex algebraic curve $X_0(N)$.  Integration defines a
449$\T$-equivariant nondegenerate
450pairing  $S \cross H \ra \C$ which we view as a map
451$\alp: H \ra \Hom_\C(S,\C)$.
452
453\begin{theorem}\label{Af}
454We have the following commutative diagram of $\T$-modules:
455$$\xymatrix{ 456 H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d] 457 & \alp(H)\ar[d]\[email protected]{^(->}[d]\\ 458\Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d] 459 &\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\ 460 A_f'(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\ 461}$$
462\end{theorem}
463\begin{proof}
464This can be deduced from \cite{shimura:factors}.
465\end{proof}
466
467\begin{corollary}\label{moduluscomp}
468$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
469\end{corollary}
470\begin{proof}
471Recall that $m_A$ is defined to be $\sqrt{\deg(\theta_A)}$.
472The kernel of an isogeny of complex tori is
473isomorphic to the cokernel of the induced map
474on lattices.  The corollary now follows from
475the diagram of Theorem~\ref{Af}
476which indicates that the index $[\alp(H):\alp(H[I_f])]$
477is the cokernel of the map $H[I_f]\ra \alp(H).$
478\end{proof}
479
480Let $\Frob_p:X_J\ra X_J$ denote the map induced by Frobenius.
481One has $\Frob_p=-W_p$, where $W_p$ is the map induced
482by the Atkin-Lehner involution on $J_0(p)$.
483Let $f$ be a newform, $A=A_f$ the corresponding optimal
484quotient, and $w_p$ the sign of the eigenvalue of
485$W_p$ on $f$.
486\begin{proposition}
487$$\Phi_A(\Fp) 488 = \begin{cases} 489 \Phi_A(\Fpbar) & \text{if w_p=-1},\\ 490 \Phi_A(\Fpbar)[2] & \text{if w_p=1.} 491 \end{cases}$$
492\end{proposition}
493\begin{proof}
494If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
495of $\Phi_A(\Fpbar)$ is trivial.  Thus in this case,
496$\Phi(\Fp)=\Phi(\Fpbar)$.
497Next suppose $w_p=1$.  We have an exact sequence
498   $$0\ra X_{A'} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
499Since $W_p$ acts as $+1$ on $f$ it acts as $+1$ on
500$A$, on $X_A$, on $\Hom(X_A,\Z)$, and on $\Phi_A$.
501Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$.  The $2$-torsion
502in a finite abelian group equals the fixed points under $-1$.
503\end{proof}
504
505{\bf WARNING:} When extending this result to the whole
506of $J_0(N)$ be careful!  The action of $\Frob_p=T_p$ is
507not by $\pm 1$, even though it must be by an involution
508of order $2$.  For example, the component group of
509$J_0(65)/\F_5$ is cyclic of order $42$.  The action
510of $\Frob_5$ is by multiplication by $-13$.  Note that
511$(-13)^2 = 1 \pmod{42}$.   The fixed points of
512multipliction by $-13$ is the order $14$ subgroup
513generated by $3$.
514
515\subsubsection{Computation}
516Using the algorithms of Chapter~\ref{chap:computing},
517we can enumerate the optimal
518quotients $A_f$ and compute $m_A$.
519The method of graphs \cite{mestre:graphs} and \cite{kohel:hecke}
520can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
521and the monodromy pairing.  We can then compute
522   $$\cL=\bigcap_{t\in I_f} \ker(t|_X),$$
523$m_X:=m_\cL$, and $\Phi_X:=\Phi_\cL$.
524By Theorem~\ref{formula} we can now compute
525  $$|\Phi_A| = |\Phi_X| \cdot \frac{m_A}{m_X}.$$
526We have computed $\Phi_A$ in a number of cases.  In the
527next subsection we discuss two conjectures suggested by
528our numerical computations.
529
530\subsection{Conjectures}
531Our numerical computations suggest the following conjectures.
532Suppose that $N=pM$ with $(p,M)=1$.
533Let
534$$H_{\new} = 535 \ker\,\Bigl( H_1(X_0(N),\Z)\lra 536 (H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
537where the map is induced by the two natural
538degeneracy maps $X_0(N)\ra X_0(M)$.
539The Hecke algebra $\T$ acts on $H_{\new}$,
540and on the submodule $H_{\new}[I_f]$ of elements annihilated
541by $I_f$. Integration defines a map
542$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
543Define the homology congruence modulus $m_H$ by
544$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
545We expect that there is a very close relationship
546between $m_X$ and $m_H$.
547\begin{conjecture}\label{conj:deg}
548  $m_X = m_H.$
549\end{conjecture}
550
551When $N=p$ is prime we make the following conjecture.
552\begin{conjecture}\label{conj:iso}
553Let~$p$ be a prime and let $f_1,\ldots,f_n$ be a set of
554representatives for the Galois conjugacy classes of newforms
555in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the corresponding
556optimal quotients.  Then
557$\#A_i(\Q)=\#\Phi_{A_i}$ for each~$i$ and
558$\#\Phi_{J_0(p)}= \prod_{i=1}^d \#\Phi_{A_i}$.
559\end{conjecture}
560Note, the natural map
561$\Phi_{J_0(113)}\ra \prod_{i=1}^4 \Phi_{A_i}$
562is not an isomorphism because two of the $\Phi_{A_i}$
563have order two, so the product is not cyclic.
564
565\section{Tables}
566We computed several component groups of optimal quotients
567$A_f$ of $J_0(N)$ associated to newforms $f$.
568We denote such an optimal quotient by
569\begin{center}
570{\bf  N\, isogeny-class\, dimension}
571\end{center}
572The dimension frequently determines the factor, so it
573is included in the notation.
574
575
576\subsection{Table 1: Some large component groups predicted by
577the Birch and Swinnerton-Dyer conjecture}
578Using the algorithm described in \cite{stein:vissha} we computed
579the special value $L(A,1)/\Omega$ (up to a Manin constant)
580for every optimal quotient $A=A_f$ of level $\leq 1500$.
581We found exactly five for which the numerator of
582$L(A,1)/\Omega$ is nonzero and divisible by a
583prime number $>10^9$.
584These are given below.
585$$\begin{array}{|lcc|}\hline 586 A & N & \text{\qquad L(A,1)/\Omega\cdot \text{Manin constant}\qquad }\\\hline 587\text{\bf 1154E20}&2\cdot 577 & 2^?\cdot 85495047371/17^2\\ 588\text{\bf 1238G19}& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\ 589\text{\bf 1322E21}& 2\cdot 661 & 2^?\cdot 57851840099/331\\ 590\text{\bf 1382D20}& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\ 591\text{\bf 1478J20}& 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\\hline 592\end{array}$$
593The Birch and Swinnerton-Dyer conjecture predicts that these large
594prime divisors must divide either $|\Phi_A|$ or
595the Shafarevich-Tate group of $A$.  We computed $\Phi_A$ and
596found that this was the case.
597$$\begin{array}{|lccccc|}\hline 598 A & p & w & |\Phi_X| & m_X & |\Phi_A| \\\hline 599\text{\bf 1154E20}&2 & - & 17^2 & 2^{24} 600 & 2^?\cdot 17^2 \cdot 85495047371 \\ 601 &577& + & 1 & 2^{26}\cdot85495047371 602 & 2^? \\ 603\vspace{-1ex}&&&&&\\ 604\text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\ 605 & 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\ 606\vspace{-1ex}&&&&&\\ 607\text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\ 608 & 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\ 609\vspace{-1ex}&&&&&\\ 610\text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\ 611 & 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\ 612\vspace{-1ex}&&&&&\\ 613\text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31} 614 & 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\ 615 & 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463 616 & 2^? \\ 617\hline 618\end{array}$$
619\vfill
620
621\subsection{Table 2: Some quotients of $J_0(N)$}
622In this table we give the invariants defined above for
623the optimal quotients of levels $65$, $66$, $68$, and $69$.
624$$\begin{array}{|lccccccc|}\hline 625 A & p & w & |\Phi_X| & m_X & m_H & m_A & |\Phi_A| \\\hline 626\text{\bf 65A1} & 5 & +& 1 &2 & ? & 2 & 1\\ 627 & 13 &+& 1 & 2& ? & & 1\\ 628 629\text{\bf 65B2} & 5 &+& 3 & 2^2&? & 2^2 & 3\\ 630 & 13 &- & 3 & 2^2&? & & 3\\ 631 632\text{\bf 65C2} & 5 &-& 7 & 2^2&? & 2^2 &7 \\ 633 & 13 &+ & 1 & 2^2&? & & 1\\ 634 635\vspace{-1ex} & & & & & & & \\ 636\text{\bf 66A1}& 2 &+ & 1 &2 & ?& 2^2&2 \\ 637 & 3 &- & 3 &2^2 & ?& & 3\\ 638 & 11 &+ & 1 &2^2 & ?& &1 \\ 639 640\text{\bf 66B1}& 2 &- & 2 &2 & ?& 2^2& 2^2\\ 641 & 3 &+ & 1 &2^2& ?& & 1\\ 642 & 11 &+ & 1 &2^2 &? & & 1\\ 643 644\text{\bf 66C1}& 2 & -& 1 & 2&? & 2^2\cdot 5& 2\cdot5\\ 645 & 3 &- & 1 & 2^2&? & & 5\\ 646 & 11 &- & 1 & 2^2\cdot5&? & &1 \\ 647 648\vspace{-1ex} & & & & & & & \\ 649\text{\bf 68A2} &17&+&2 &2\cdot3 &? &2\cdot3 & 2 \\ 650 651\vspace{-1ex} & & & & & & & \\ 652\text{\bf 69A1} &3 &-&2 &2 & ?&2& 2\\ 653 &23 &+& 1&2 &? & & 1\\ 654 655\text{\bf 69B2} &3 &+&2 &2 &? &2\cdot11& 2\cdot11 \\ 656 &23 &-&2 &2\cdot11 &? && 2 \\ 657 658\hline 659\end{array}$$
660
661
662\subsection{Table 3: Some quotients of $J_0(p)$}
663Using the method of graphs and modular symbols we computed
664the quantities $m_A$, $m_L$ and $\Phi_L$ for each abelian
665variety $A=A_f$ associated to a newform of prime level
666$p\leq 757$.  The results were as follows:
667\begin{enumerate}
668\item In all cases $m_A=m_L$, so the map $\Phi_J\ra \Phi_A$
669is surjective.
670\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
671$w_p$ on $A$ is $1$.
672\item $\prod |\Phi_A| = |\Phi_J|$
673\end{enumerate}
674Table 1 lists those $A$ of level $\leq 631$ for which $w_p=-1$, along with
675the order of the component group.
676
677\newpage
678Table 3: Some quotients of $J_0(p)$~%
679$$680\begin{array}{|lc}\hline 681\vspace{-2ex}\\ 682A & |\Phi_A| \\ 683\vspace{-2ex}\\\hline 68411\text{A}1&5\\ 68517\text{A}1&2^2\\ 68619\text{A}1&3\\ 68723\text{A}2&11\\ 688\vspace{-2ex} &\\ 68929\text{A}2&7\\ 69031\text{A}2&5\\ 69137\text{B}1&3\\ 69241\text{A}3&2\cdot5\\ 693\vspace{-2ex} &\\ 69443\text{B}2&7\\ 69547\text{A}4&23\\ 69653\text{B}3&13\\ 69759\text{A}5&29\\ 698\vspace{-2ex} &\\ 69961\text{B}3&5\\ 70067\text{A}1&1\\ 70167\text{C}2&11\\ 70271\text{A}3&5\\ 703\vspace{-2ex} &\\ 70471\text{B}3&7\\ 70573\text{A}1&2\\ 70673\text{C}2&3\\ 70779\text{B}5&13\\ 708\vspace{-2ex} &\\ 70983\text{B}6&41\\ 71089\text{B}1&2\\ 71189\text{C}5&11\\ 71297\text{B}4&2^3\\ 713\vspace{-2ex} &\\ 714101\text{B}7&5^2\\ 715103\text{B}6&17\\ 716107\text{B}7&53\\ 717109\text{A}1&1\\ 718\vspace{-2ex} &\\ 719109\text{C}4&3^2\\ 720113\text{A}1&2\\ 721113\text{B}2&2\\ 722113\text{D}3&7\\ 723\vspace{-2ex} &\\ 724127\text{B}7&3\cdot7\\ 725131\text{B}10&5\cdot13\\ 726137\text{B}7&2\cdot17\\ 727139\text{A}1&1\\ 728\vspace{-2ex} &\\ 729139\text{C}7&23\\ 730149\text{B}9&37\\ 731151\text{B}3&1\\ 732151\text{C}6&5^2\\ 733\hline\end{array} 734\begin{array}{lc}\hline 735\vspace{-2ex}\\ 736A & |\Phi_A| \\ 737\vspace{-2ex}\\\hline 738157\text{B}7&13\\ 739163\text{C}7&3^3\\ 740167\text{B}12&83\\ 741173\text{B}10&43\\ 742\vspace{-2ex} &\\ 743179\text{A}1&1\\ 744179\text{C}11&89\\ 745181\text{B}9&3\cdot5\\ 746191\text{B}14&5\cdot19\\ 747\vspace{-2ex} &\\ 748193\text{C}8&2^4\\ 749197\text{C}10&7^2\\ 750199\text{A}2&1\\ 751199\text{C}10&3\cdot11\\ 752\vspace{-2ex} &\\ 753211\text{A}2&5\\ 754211\text{D}9&7\\ 755223\text{C}12&37\\ 756227\text{B}2&1\\ 757\vspace{-2ex} &\\ 758227\text{C}2&1\\ 759227\text{E}10&113\\ 760229\text{C}11&19\\ 761233\text{A}1&2\\ 762\vspace{-2ex} &\\ 763233\text{C}11&29\\ 764239\text{B}17&7\cdot17\\ 765241\text{B}12&2^2\cdot5\\ 766251\text{B}17&5^3\\ 767\vspace{-2ex} &\\ 768257\text{B}14&2^6\\ 769263\text{B}17&131\\ 770269\text{C}16&67\\ 771271\text{B}16&3^2\cdot5\\ 772\vspace{-2ex} &\\ 773277\text{B}3&1\\ 774277\text{D}9&23\\ 775281\text{B}16&2\cdot5\cdot7\\ 776283\text{B}14&47\\ 777\vspace{-2ex} &\\ 778293\text{B}16&73\\ 779307\text{A}1&1\\ 780307\text{B}1&1\\ 781307\text{C}1&1\\ 782\vspace{-2ex} &\\ 783307\text{D}1&1\\ 784307\text{E}2&3\\ 785307\text{F}9&17\\ 786311\text{B}22&5\cdot31\\ 787\hline\end{array} 788\begin{array}{lc}\hline 789\vspace{-2ex}\\ 790A & |\Phi_A| \\ 791\vspace{-2ex}\\\hline 792313\text{A}2&1\\ 793313\text{C}12&2\cdot13\\ 794317\text{B}15&79\\ 795331\text{D}16&5\cdot11\\ 796\vspace{-2ex} &\\ 797337\text{B}15&2^2\cdot7\\ 798347\text{D}19&173\\ 799349\text{B}17&29\\ 800353\text{A}1&2\\ 801\vspace{-2ex} &\\ 802353\text{B}3&2\\ 803353\text{D}14&2\cdot11\\ 804359\text{D}24&179\\ 805367\text{B}19&61\\ 806\vspace{-2ex} &\\ 807373\text{C}17&31\\ 808379\text{B}18&3^2\cdot7\\ 809383\text{C}24&191\\ 810389\text{A}1&1\\ 811\vspace{-2ex} &\\ 812389\text{E}20&97\\ 813397\text{B}2&1\\ 814397\text{C}5&11\\ 815397\text{D}10&3\\ 816\vspace{-2ex} &\\ 817401\text{B}21&2^2\cdot5^2\\ 818409\text{B}20&2\cdot17\\ 819419\text{B}26&11\cdot19\\ 820421\text{B}19&5\cdot7\\ 821\vspace{-2ex} &\\ 822431\text{B}1&1\\ 823431\text{D}3&1\\ 824431\text{F}24&5\cdot43\\ 825433\text{A}1&1\\ 826\vspace{-2ex} &\\ 827433\text{B}3&1\\ 828433\text{D}16&2^2\cdot3^2\\ 829439\text{C}25&73\\ 830443\text{C}1&1\\ 831\vspace{-2ex} &\\ 832443\text{E}22&13\cdot17\\ 833449\text{B}23&2^4\cdot7\\ 834457\text{C}20&2\cdot19\\ 835461\text{D}26&5\cdot23\\ 836\vspace{-2ex} &\\ 837463\text{B}22&7\cdot11\\ 838467\text{C}26&233\\ 839479\text{B}32&239\\ 840487\text{A}2&1\\ 841\hline\end{array} 842\begin{array}{lc|}\hline 843\vspace{-2ex}&\\ 844A & |\Phi_A| \\ 845\vspace{-2ex}&\\\hline 846487\text{B}2&3\\ 847487\text{C}3&1\\ 848487\text{D}16&3^3\\ 849491\text{C}29&5\cdot7^2\\ 850\vspace{-2ex} &\\ 851499\text{C}23&83\\ 852503\text{B}1&1\\ 853503\text{C}1&1\\ 854503\text{D}3&1\\ 855\vspace{-2ex} &\\ 856503\text{F}26&251\\ 857509\text{B}28&127\\ 858521\text{B}29&2\cdot5\cdot13\\ 859523\text{C}26&3\cdot29\\ 860\vspace{-2ex} &\\ 861541\text{B}24&3^2\cdot5\\ 862547\text{C}25&7\cdot13\\ 863557\text{B}1&1\\ 864557\text{D}26&139\\ 865\vspace{-2ex} &\\ 866563\text{A}1&1\\ 867563\text{E}31&281\\ 868569\text{B}31&2\cdot71\\ 869571\text{A}1&1\\ 870\vspace{-2ex} &\\ 871571\text{B}1&1\\ 872571\text{C}2&1\\ 873571\text{D}2&1\\ 874571\text{F}4&1\\ 875\vspace{-2ex} &\\ 876571\text{I}18&5\cdot19\\ 877577\text{A}2&3\\ 878577\text{B}2&1\\ 879577\text{C}3&1\\ 880\vspace{-2ex} &\\ 881577\text{D}18&2^4\\ 882587\text{C}31&293\\ 883593\text{B}1&2\\ 884593\text{C}2&1\\ 885\vspace{-2ex} &\\ 886593\text{E}27&2\cdot37\\ 887599\text{C}37&13\cdot23\\ 888601\text{B}29&2\cdot5^2\\ 889607\text{D}31&101\\ 890\vspace{-2ex} &\\ 891613\text{C}27&3\cdot17\\ 892617\text{B}28&2\cdot7\cdot11\\ 893619\text{B}30&103\\ 894631\text{B}32&3\cdot5\cdot7\\ 895\hline\end{array} 896$$
897
898