Author: William A. Stein
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44\chapter{Component groups of optimal quotients}%
45\index{Component group}%
46\index{Optimal quotient!component groups of}%
47\label{chap:compgroups}%
48Let~$A$ be an abelian variety over the rational numbers~$\Q$. The
49Birch and Swinnerton-Dyer
50conjecture\index{BSD conjecture!and component groups}
51supplies a formula for the order
52of the Shafarevich-Tate group of~$A$.  A key step in computing this
53order is to find each of the Tamagawa numbers~$c_p$ of~$A$.  The
54Tamagawa numbers are defined as follows, where the definition of
55N\'eron model and component group is given below.
56\begin{definition}[Tamagawa number]
57\index{Tamagawa numbers|textit}%
58\index{Optimal quotient!Tamagawa numbers of}%
59\index{Component group!rational points of}%
60Let~$p$ be a prime,
61let $\cA$ be a N\'eron model\index{N\'eron model}
62of~$A$ over the $p$-adic integers~$\Z_p$, and let $\Phi_{A,p}$
63be the component group of~$\cA$ at~$p$.  Then the
64\defn{Tamagawa number} $c_p$ of~$A$ is the order of the group
65$\Phi_{A,p}(\F_p)$ of $\F_p$-rational points
66of $\Phi_{A,p}(\Fbar_p)$.
67\end{definition}
68\begin{remark}
69We warn the reader that the Tamagawa number is defined in a different
70way in some other papers.  The definitions are equivalent.
71\end{remark}
72In this chapter we present a method for computing the Tamagawa numbers
73$c_p$, up to a power of~$2$, under the hypothesis that~$A$ has purely
74toric reduction at~$p$.  Such~$A$ are plentiful among the modular abelian
75varieties; for example, if~$A$ is a new optimal quotient of $J_0(N)$
76and~$p$ exactly divides~$N$, then~$A$ is purely toric at~$p$.
77
78In Sections~\ref{sec:compgrpintro}--\ref{sec:compgrpmaintheorem} we
79state and prove an explicit formula involving component groups of
80fairly general abelian varieties.  Then in Section~\ref{sec:compj0n}
81we turn to quotients of modular Jacobians $J_0(N)$.
82\index{Jacobian!of $X_0(N)$}
83We give several tables and issue a conjecture and a question.
84
85The results of this chapter were inspired by a letter that Ribet\index{Ribet}
86wrote to Mestre\index{Mestre},
87in which he treats the case when~$A$ is an elliptic curve.
88
89\section{Main results}\label{sec:compgrpintro}
90\subsection{N\'eron models and component groups}
91Let~$A$ be an abelian variety over a finite extension~$K$ of the
92$p$-adic numbers~$\Qp$.  Let~$\O$ be the ring of integers of~$K$,
93let~$\m$ be its maximal ideal, and let $k=\O/\m$ be the residue class field.
94\begin{definition}[N\'eron model]\index{N\'eron model|textit}
95A \defn{N\'{e}ron model} of~$A$ is a smooth commutative group
96scheme~$\A$ over~$\O$ such that~$A$ is its generic fiber and~$\A$
97satisfies the N\'eron mapping property: the restriction map
98       $$\Hom_\O(S,\A)\lra \Hom_K(S_K,A)$$
99is bijective for all {\em smooth} schemes~$S$ over~$\O$.
100\end{definition}
101The N\'eron mapping property implies that~$\A$ is unique up to a
102unique isomorphism, so we will refer without hesitation to the''
103N\'eron model of~$A$.
104
105The closed fiber~$\A_k$ of~$\A$ is a group scheme over~$k$,
106which need not be connected; denote by~$\A_k^0$ the
107connected component containing the identity.
108There is an exact sequence
109     $$0\lra\A_k^0\lra\A_k\lra\Phi_A\lra0,$$
110where~$\Phi_A$ a finite \'{e}tale group scheme
111over~$k$.  Equivalently,~$\Phi_A$ may be viewed as
112a finite abelian group equipped
113with an action of $\Gal(\kbar/k)$.
114\begin{definition}[Component group]%
115\index{Component group|textit}%
116\label{defn:componentgroup}%
117The \defn{component group} of an abelian variety~$\A$ over a
118local field~$K$ is the group scheme $\Phi_A=A_k/A_k^0$
119defined above.
120\end{definition}
121
122\subsection{Motivating problem}
123This chapter is motivated by the problem of
124computing the groups~$\Phi_{A,p}$ attached to quotients~$A$
125of Jacobians of modular curves~$X_0(N)$.\index{Jacobian!of $X_0(N)$}
126When~$A$ has semistable
127reduction, Grothendieck\index{Grothendieck} and Mumford\index{Mumford}
128described the component group in terms of a monodromy pairing on
129certain free abelian groups.  When $A=J=J_0(N)$ is the Jacobian of
130$X_0(N)$, this pairing can be explicitly computed, hence the component
131group~$\Phi_J$ can also be computed; this has been done in many cases
132in \cite{mazur:eisenstein} and \cite{edixhoven:eisen}.
133
134Suppose now that $A=A_f$ is an optimal quotient of $J_0(N)$ that is
135attached to a newform~$f$, so that the kernel of the map $\pi:J\ra A$
136is connected.  There is a natural map $\pi_*:\Phi_J\ra \Phi_A$.  We
137wish to compute the image and the order of the cokernel of
138$\pi_*$.
139
140\subsection{The main result}
141We now state our main result more precisely, necessarily supressing
142some of the definitions of the terms used until later.  Suppose
143$\pi:J\ra A$ is an optimal quotient, with~$J$
144a Jacobian\index{Jacobian!semistable}
145with semistable reduction and~$A$ having purely toric
146reduction.  We express the component group\index{Component group}
147of~$A$ in terms of the monodromy pairing\index{Monodromy pairing}
148associated to~$J$.
149
150Let $m_A=\sqrt{\deg(\theta_A)}$, where $\theta_A:A^{\vee}\ra A$ is
151induced by the canonical principal polarization\index{Canonical
152polarization} of~$J$ arising from the $\theta$-divisor.  Let $X_J$ be
153the character group\index{Character group of torus} of the toric
154part\index{Toric part} of the closed fiber of the N\'eron model
155of~$J$.  Let~$\cL$ be the saturation of the image of $X_A$ in $X_J$.
156The monodromy pairing\index{Monodromy pairing} induces a map
157$\alp:X_J\ra \Hom(\cL,\Z)$.  Let $\Phi_X$ be the cokernel of~$\alp$
158and $m_X=[\alp(X_J):\alp(\cL)]$ be the order of the finite group
159$\alp(X_J)/\alp(\cL)$.  We obtain the equality
160   $$\frac{\#\Phi_A}{m_A} = \frac{\#\Phi_X}{m_X}.$$
161Using the snake lemma\index{Snake lemma}, one see that
162$\Phi_X$ is isomorphic to the image of the natural map
163$\Phi_J \ra \Phi_A$, and the above formula implies that the cokernel of
164the map $\Phi_J \ra \Phi_A$ has order $m_A/m_X$.
165
166If the optimal quotient $J\ra A$ arises from a
167modular form on $\Gamma_0(N)$, then the quantities $m_A$, $m_X$ and
168$\Phi_X$ can be explicitly computed, hence we can compute $\#\Phi_A$.
169
170\section{Optimal quotients of Jacobians}
171Let~$J$ be a Jacobian\index{Jacobian}, and let
172$\theta_J$ be the canonical principal polarization
173arising from the $\theta$-divisor.
174Recall that an \defn{optimal quotient}\index{Optimal quotient|textit}
175of~$J$ is an abelian variety~$A$ and a surjective
176map $\pi: J \ra A$ whose
177kernel is an abelian subvariety~$B$ of~$J$.
178Denote by $J^{\vee}$ and $A^{\vee}$ the abelian varieties dual
179to~$J$ and~$A$, respectively.
180Upon composing the dual of~$\pi$ with $\theta_J^{\vee}=\theta_J$,
181we obtain a map
182    $$A^{\vee}\xrightarrow{\pi^{\vee}} J^{\vee}\xrightarrow{\theta_J} J.$$
183\begin{proposition}
184\index{Optimal quotient!dual map is injective}
185The map $A^{\vee}\ra J$ is injective.
186\end{proposition}
187\begin{proof}
188Since $\theta_J$ is an isomorphism it suffices to prove
189that $\pi^{\vee}$ is injective.
190Since the dual of $\pi^{\vee}$ is
191$(\pi^{\vee})^{\vee}=\pi$ and $\pi$ is surjective,
192the map $\pi^{\vee}$ must have finite kernel.
193Thus $A^{\vee} \ra C=\im(\pi^{\vee})$ is
194an isogeny.  Let~$G$ denote the kernel of this isogeny,
195and dualize.  By \cite[\S11]{milne:abvars} we have
196the following two commutative diagrams:
197$$\xymatrix{ 198 G\ar[r] & A^{\vee}\[email protected]{->>}[r] \ar[dr]_{\pi^{\vee}} 199 & C\ar[d]\\ 200 && J^{\vee} 201}\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad 202\xymatrix{ 203 A & C^{\vee}\ar[l] & G^{\vee}\ar[l] \\ 204 & J,\ar[u]_{\vphi}\ar[ul]^{\pi} 205}$$
206where $G^{\vee}$ is the Cartier dual\index{Cartier dual} of~$G$.
207Since $G^{\vee}$ is finite, $\ker(\vphi)$ is of
208finite index in $\ker(\pi)$.
209Since $\ker(\pi)$ is an abelian variety, as a group it is divisible.
210But a divisible group has no nontrivial finite-index subgroups
211(divisibility is a property inherited by quotients, and nonzero
212finite groups are not divisible).
213Thus $\ker(\vphi)=\ker(\pi)$, so $G^{\vee}=0$.  It follows that $G=0$.
214\end{proof}
215Henceforth we will abuse notation and
216denote the injection $A^{\vee}\ra J$ by $\pi^{\vee}$.
217The kernel of $\theta_A$ equals the intersection of
218$A^{\vee}$ and $B=\ker(\pi)$,
219as depicted in the following diagram:
220$$\xymatrix{ 221A^{\vee}\intersect B\ar[r]\ar[d] & B\ar[d] \\ 222 A^{\vee}\[email protected]{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\ 223 & A. 224}$$
225
226Since $\theta_A$ is a polarization\index{Polarization},
227the degree $\#\ker(\theta_A)$ of $\theta_A$ is
228a perfect square (see \cite[Thm.~13.3]{milne:abvars}).
229Recall that
230the \defn{modular degree}\index{Modular degree|textit} is the integer
231  $$m_A=\sqrt{\#\ker(\theta_A)}.$$
232For an algorithm to compute $m_A$,
233see Section~\ref{sec:moddeg} and Corollary~\ref{moduluscomp}.
234
235\section{The closed fiber of the N\'{e}ron model}%
236\index{N\'eron model!closed fiber of}%
237\index{Closed fiber of N\'eron model|textit}%
238Let~$K$ be a finite extension of $\Qp$ with ring of integers~$\O$
239and residue class field~$k$.
240Let~$A$ be an abelian variety over~$K$ and denote its
241N\'{e}ron model\index{N\'eron model} by~$\cA$.
242Let $\Phi_A$ be the group of connected
243components of the closed fiber $\cA_k$. This group
244is a finite \'{e}tale group scheme\index{\'Etale group scheme|textit}
245over~$k$; equivalently, it is
246a finite abelian group equipped with an action of
247$\Gal(\kbar/k)$. There is an exact sequence of group schemes
248    $$0 \ra \cA_k^0 \ra \cA_k \ra \Phi_A \ra 0.$$
249The group scheme $\cA_k^0$ is an extension of an abelian variety~$\cB$
250of some dimension~$a$ by a group scheme~$\cC$; we have a diagram
251$$\[email protected]=.3cm{ 252&0\ar[d]\\ 253&{\cT}\ar[d]\\ 2540\ar[r]&{\cC}\ar[r]\ar[d]&{\cA_k^0}\ar[r]&{\cB}\ar[r]&0\\ 255&{\cU}\ar[d]\\ 256&0}$$
257with~$\cT$ a torus of dimension~$t$
258and~$\cU$ a unipotent group of dimension~$u$.
259The abelian variety~$A$ is said to have \defn{purely toric reduction}%
260\index{Purely toric reduction}
261if $t=\dim A$, and have \defn{semistable reduction} if $u=0$.%
262\index{Semistable reduction}
263\begin{definition}[Character group of torus]
264The \defn{character group}\index{Character group of torus|textit}
265  $$X_A = \Hom_{\kbar}(\cT_{/{\kbar}},{\Gm}_{/{\kbar}})\label{defn:chargroup}$$
266is a free abelian group of rank~$t$ contravariantly associated to~$A$.
267\end{definition}
268As discussed in, e.g., \cite{ribet:modreps}, if~$A$ is semistable
269there is a \defn{monodromy pairing}
270$X_A\cross X_{A^{\vee}}\ra \Z$ and an exact sequence
271  $$0 \ra X_{A^{\vee}} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
272
273\section{Rigid uniformization}
274\index{Rigid uniformization|textit}
275In this section we review the
276rigid analytic uniformization
277of a semistable\index{Semistable reduction!and uniformization}
278abelian variety over a finite extension~$K$ of the
279maximal unramified extension $\Qp^{\ur}$ of $\Qp$.   We use this
280uniformization to prove
281that if~$A$ has purely toric reduction\index{Purely toric reduction},
282and $\phi:A^{\vee}\ra A$ is a
283symmetric isogeny\index{Symmetric isogeny} (as defined below), then
284  $$\deg(\phi) = (\# \coker(X_A\ra X_{A^{\vee}}))^2.$$
285We also prove some lemmas about character groups\index{Character group of torus}.
286
287It is possible to prove the assertions we will need without recourse
288to rigid uniformization, as Ahmed Abbes has pointed out to the author.
289
290\subsection{Raynaud's uniformization}%
291\label{subsec:raynaud}%
292\begin{theorem}[Raynaud\index{Raynaud}]\label{raynaud}
293If~$A$ is a semistable abelian variety, its universal
294covering (as defined in \cite{coleman:monodromy})
295is isomorphic to an extension~$G$ of an abelian
296variety~$B$ with good reduction by a torus~$T$.  The
297covering map from~$G$ to~$A$ is a homomorphism, and
298its kernel is a twisted free abelian group~$\Gamma$ of finite rank.
299\end{theorem}
300This may be summarized by the diagram
301$$\xymatrix{ 302 &\Gamma\ar[d] \\ 303 T\ar[r] & G\ar[r]\ar[d] & B\\ 304 & A, 305}$$
306which we call the
307\defn{uniformization cross}\index{Uniformization cross}
308of~$A$.
309
310\begin{remark}
311The group~$\Gamma$ can be identified with the character
312group $X_{A^{\vee}}$ of the previous and latter sections.
313\end{remark}
314
315The uniformization cross
316of the dual abelian variety $A^{\vee}$ is
317$$\xymatrix{ 318 &\Gamma^{\vee}\ar[d] \\ 319 T^{\vee}\ar[r] & G^{\vee}\ar[r]\ar[d] & B^{\vee}\\ 320 & A^{\vee}, 321}$$
322where $\Gamma^{\vee}=\Hom(T,\Gm)$, where $T^{\vee}=\Hom(\Gamma,\Gm)$,
323and the morphisms $\Gamma^{\vee}\ra G^{\vee}$ and $T^{\vee}\ra G^{\vee}$
324are the one-motif duals\index{One-motif dual}
325of the morphisms $T\ra G$ and $\Gamma\ra G$, respectively.
326For more details see, e.g., \cite{coleman:monodromy}.
327
328To avoid confusion when considering the uniformization of more than
329one abelian variety, we will often denote the objects $T$, $G$,
330$\Gamma$, and~$B$ connected with $A$ by $T_A$, $G_A$, $\Gamma_A$, and
331$B_A$, respectively.
332
333\begin{example}[Tate curve]\index{Tate curve!uniformization of}
334If $E/\Qp$ is an elliptic curve with split multiplicative reduction,
335then the uniformization is $E=\Gm/q^\Z$ where $q=q(j)$ is
336obtained by inverting the expression for~$j$ as a function of
337$q(z)=e^{2\pi iz}$.
338\end{example}
339
340
341\subsection{Some lemmas}
342Let $\pi:J\ra A$ be an optimal quotient%
343\index{Optimal quotient!and semistable reduction},
344assume that~$J$ has semistable\index{Semistable reduction} reduction,
345and that~$A$ has purely
346toric\index{Purely toric reduction} reduction.
347\begin{lemma}\label{lem:surj}
348The map $\Gamma_J\ra \Gamma_A$ induced by~$\pi$ is surjective.
349\end{lemma}
350\begin{proof}
351Since $G_J$ is simply connected,~$\pi$ induces a map
352$G_J\ra T_A$ and a map $\Gamma_J\ra \Gamma_A$.
353Because~$\pi$ is surjective and $T_A$ is a
354torus, the map $G_J\ra T_A$ is surjective.
355Upon applying the snake lemma\index{Snake lemma} to the
356following diagram, we obtain a surjective map from $B=\ker(\pi)$ to
357$M=\coker(\Gamma_J\ra\Gamma_A)$:
358$$\xymatrix{ 359 & \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\ 360 & G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\ 361B\ar[r] & J\ar[r]^{\pi}& A. 362}$$
363Since~$\pi:J\ra A$ is an optimal quotient, the kernel~$B$ is connected.
364Thus~$M$ must also be connected.
365Since~$M$ is discrete it follows that $M=0$.
366\end{proof}
367
368\subsubsection{Abelian varieties with purely toric reduction}
369Assume that~$A$ has purely toric reduction\index{Purely toric reduction}.
370Then $B=0$, and the uniformization
371cross\index{Uniformization cross} is simply
372     $$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A.}$$
373\begin{definition}[Symmetric isogeny]\index{Symmetric isogeny|textit}
374A \defn{symmetric isogeny} $\vphi:A^{\vee}\ra A$
375is an isogeny such that the map
376 $\vphi^{\vee}:A^{\vee}\ra (A^{\vee})^{\vee}=A$
377is equal to~$\vphi$.
378\end{definition}
379Let $\vphi:A^{\vee}\ra A$ be a symmetric isogeny.
380Denote by $\vphi_t:T^{\vee}\ra T$ and $\vphi_a:\Gamma^{\vee}\ra\Gamma$ the
381maps induced by~$\vphi$.
382\begin{proposition}\label{prop:kerphi}
383There is an exact sequence
384$$0\ra\ker(\vphi_t) \ra \ker(\vphi) \ra \coker(\vphi_a)\ra 0,$$
385and $\ker(\vphi_t)$ is the Cartier dual of $\coker(\vphi_a)$.
386\end{proposition}
387\begin{proof}
388Since~$\vphi$ is an isogeny we obtain the following diagram:
389$$\xymatrix{ 390 0\ar[r]\ar[d] & \Gamma^{\vee}\ar[r]^{\vphi_a}\ar[d] & \Gamma\ar[r]\ar[d] 391 & \coker(\vphi_a)\ar[d]\\ 392 \ker(\vphi_t)\ar[r]\ar[d]& T^{\vee}\ar[d]\ar[r]^{\vphi_t} & T\ar[r]\ar[d] & 0\\ 393 \ker(\vphi)\ar[r] & A^{\vee}\ar[r]^{\vphi} & A.}$$
394The snake lemma\index{Snake lemma} then gives the claimed exact sequence.
395
396For the second assertion, observe that if we take one-motif
397duals of every object in the diagram
398$$\xymatrix{ 399 & \Gamma^{\vee}\ar[r]^{\vphi_a}\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\ 400 \ker(\vphi_t)\ar[r] & T^{\vee}\ar[r]^{\vphi_t} & T}$$
401we obtain the following diagram:
402$$\xymatrix{ 403 & T & T^{\vee}\ar[l]_{\vphi_a^{\vee}} & \coker(\vphi_a)^{\vee}\ar[l]\\ 404 \ker(\vphi_t)^{\vee}& \Gamma\ar[l]\ar[u] &\Gamma^{\vee}.\ar[l]^{\vphi_t^{\vee}}\ar[u] 405}$$
406Since $\vphi$ is symmetric, $\vphi_a^{\vee}=\vphi_t$, so
407     $$\ker(\vphi_t) = \coker(\vphi_a)^{\vee}.$$
408\end{proof}
409
410\begin{lemma}\label{lem:isogcoker}
411$\#\ker(\vphi)=\#\coker(\vphi_a)^2$
412\end{lemma}
413\begin{proof}
414Use the exact sequence of Proposition~\ref{prop:kerphi} together
415with the observation that the
416order of a finite group scheme equals the order of its Cartier dual.
417\end{proof}
418
419\section{The main theorem}
420\label{sec:compgrpmaintheorem}
421Let $\pi:J\ra A$ be an optimal quotient\index{Optimal quotient},
422with~$J$ a Jacobian\index{Jacobian!semistable}
423having semistable reduction and~$A$
424an abelian variety having purely toric reduction.
425Let $X_A$, $X_{A^{\vee}}$, and $X_J$ denote the
426character groups\index{Character group of torus} of the toric parts
427of the closed fibers of the abelian varieties~$A$,
428$A^{\vee}$, and~$J$, respectively.
429
430\subsection{Description of the component group in terms of the monodromy pairing}
431Recall that there is a pairing
432$X_A\cross X_{A^{\vee}}\ra \Z$ called
433the monodromy pairing\index{Monodromy pairing}.
434We have an exact sequence
435  $$0 \ra X_{A^{\vee}} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
436If~$J$ is a Jacobian\index{Jacobian!is principally polarized}
437then~$J$ is canonically self-dual via the
438$\theta$-polarization, so
439the monodromy pairing on~$J$
440can be viewed as a pairing $X_J\cross X_J \ra \Z$, and
441there is an exact sequence
442  $$0 \ra X_{J} \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0.$$
443
444\begin{example}[Tate curve]\index{Tate curve!and monodromy pairing}%
445\index{Tate curve!component group of}%
446Suppose $E=\Gm/q^{\Z}$ is a Tate curve over $\Qp^{\ur}$.
447The monodromy pairing on $X_E=q^{\Z}$ is
448$$\langle q, q\rangle = \ord_p(q)=-\ord_p(j).$$
449Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
450\end{example}
451
452
453\subsubsection{Proof of the main theorem}
454We now prove the main theorem.
455Let $\pi : J\ra A$ be an optimal quotient, and let
456 $\theta:A^{\vee}\ra A$ denote the induced polarization.
457Let $\pi_*$, $\pi^*$, $\theta_*$, and $\theta^*$ be the
458maps induced on character groups by the various functorialities,
459as indicated in the following two key diagrams:
460$$\[email protected]=3pc{A^{\vee} \[email protected]{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\theta} 461 & J \[email protected]{->>}[d]^{\pi}\\ 462 &A} 463\qquad\qquad\qquad 464 \[email protected]=3pc{X_A \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*} 465 & X_J \[email protected]{->>}[d]^{\pi_*} \\ 466 & X_{A^{\vee}}.\[email protected]/^1.5pc/[ul]^{\theta_*}} 467$$
468The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
469The injectivity of $\pi^*$ follows because
470$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
471and multiplication by a nonzero integer on a free abelian
472group is injective.
473
474Let
475 $$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
476be the map defined by the monodromy pairing\index{Monodromy pairing}
477restricted to $X_J\cross \pi^* X_A$.
478\begin{lemma}\label{lem:twokers}
479$\ker(\pi_*) = \ker(\alp)$
480\end{lemma}
481\begin{proof}
482Suppose $x\in \ker(\pi_*)$, and let $y=\pi^* z$ with
483$z\in X_A$.  Then
484$$\langle x, y \rangle = \langle x, \pi^* z \rangle 485 = \langle \pi_* x, z \rangle = 0,$$
486so $x\in\ker(\alp)$.
487Next let $x\in\ker(\alp)$.
488Then for all $z\in X_A$,
489$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle,$$
490so $\pi_* x$ is in the kernel of the
491monodromy map
492$$X_{A^{\vee}} \ra \Hom(X_A,\Z).$$
493Since $X_{A^{\vee}}$ and $\Hom(X_A,\Z)$ are free of the same finite
494rank and the cokernel is torsion, the monodromy map is injective.
495Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
496\end{proof}
497
498\begin{lemma}\label{lem:compphi}
499There is an exact sequence
500$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
501\end{lemma}
502\begin{proof}
503Lemma~\ref{lem:twokers} gives the following
504commutative diagram with exact rows
505$$\xymatrix{0\ar[r] 506 & X_J/\ker(\alp)\ar[d]^{\isom} \ar[r] 507 & {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\ 508 0\ar[r] & X_{A^{\vee}}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0.}$$
509By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
510The second is an isomorphism because it is induced by the
511isomorphism $\pi^*:X_A\ra \pi^* X_A$.  It follows that
512$\coker(\alp)\isom \Phi_A$, as claimed.
513\end{proof}
514
515Let $\cL$ be the \defn{saturation} of $\pi^* X_A$ in $X_J$;
516thus $\pi^*X_A$ is a finite-index subgroup of~$\cL$
517and the quotient $X_J/\cL$ is torsion free.
518For~$L$ of finite index in $\cL$,
519define the
520\defn{modular degree}\index{Modular degree!and character group|textit}
521of~$L$ to be
522   $$m_L = [\alp(X_J):\alp(L)],$$
523and the
524\defn{component group}\index{Component group!and character group|textit}
525of~$L$ to be
526  $$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
527When $L=\cL$ and $A$ is fixed, we often slightly abuse notation and
528write $m_X=m_\cL$ and $\Phi_X=\Phi_\cL$.
529We think of $m_X$ and $\Phi_X$ as the character group
530modular degree and component group''
531of~$A$.
532
533\begin{lemma}\label{lem:homog}
534Choose a subgroup~$L$ of finite index in~$\cL$.
535The rational number
536$\ds \frac{\#\Phi_L}{m_L}$
537is independent of the choice of~$L$.
538\end{lemma}
539\begin{proof}
540Suppose $L'$ is another finite index subgroup of~$\cL$,
541and let $n=[L:L']$.  Here~$n$ is a rational number, the lattice
542index of~$L'$ in~$L$.
543Since~$\alp$ is injective when restricted to $\cL$, it follows that
544 $$m_{L'} = [\alp(X_J):\alp(L')] 545 = [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n.$$
546Similarly, $\#\Phi_{L'} = \#\Phi_L\cdot n$.
547\end{proof}
548
549Recall that $m_A = \sqrt{\deg(\theta)}$ and
550$$\Phi_A \isom \coker(X_{A^{\vee}}\ra \Hom(X_A,\Z)),$$
551where $m_A$ is the modular degree of~$A$ and $\Phi_A$ is the
552component group of~$A$.
553
554\begin{theorem}\label{formula}
555For any subgroup~$L$ of finite index in $\cL$,
556the following relation holds:
557$$\frac{\#\Phi_A}{m_A} = \frac{\#\Phi_L}{m_L}.$$
558\end{theorem}
559\begin{proof}
560By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
561With this choice of $L$, Lemma~\ref{lem:compphi} asserts that
562$\Phi_L \isom \Phi_A$.
563By Lemma~\ref{lem:twokers}, properties of the index,
564and Lemma~\ref{lem:isogcoker} we have
565\begin{eqnarray*}
566m_L&=&[\alp(X_J):\alp(L)] \\
567   &=& [\pi_*(X_J):\pi_*(L)]\\
568   &=& [X_{A^{\vee}}:\pi_*(\pi^*X_A)]\\
569   &=& [X_{A^{\vee}}:\theta^* X_A]\\
570   &=& \#\coker(\theta^*) \\
571   &=& \sqrt{\deg(\theta)} = m_A.
572\end{eqnarray*}
573\end{proof}
574
575\begin{proposition}\label{prop:compim}
576$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\cL.$$
577\end{proposition}
578\begin{proof}
579Since $\pi^*X_A\subset \cL \subset X_J$, an application
580of Lemma~\ref{lem:compphi} gives the following commutative diagram
581with exact rows:
582$$\xymatrix{ 583 X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\ 584 X_J\ar[r]\[email protected]{=}[d]& \Hom(\cL,\Z)\ar[r]\ar[d]& \Phi_\cL \ar[r]\ar[d] & 0\\ 585 X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0. 586}$$
587The map $\Hom(\cL,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism,
588so the map $\Phi_\cL\ra\Phi_A$ is injective.  Thus
589 $$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\cL).$$
590The cokernel of $\Hom(X_J,\Z)\ra\Hom(\cL,\Z)$
591surjects onto the cokernel of $\Phi_J\ra \Phi_\cL$.
592Using the exact sequence
593$$0\ra \cL \ra X_J \ra X_J/\cL \ra 0,$$
594we find that
595$$\coker(\Hom(X_J,\Z)\ra\Hom(\cL,\Z)) \subset \Ext^1(X_J/\cL,\Z).$$
596Because~$\cL$ is saturated, the quotient $X_J/\cL$ is torsion free,
597so the indicated $\Ext^1$ group vanishes.
598Thus the map $\Phi_J\ra\Phi_\cL$ is surjective,
599from which the proposition follows.
600\end{proof}
601
602The following corollary
603follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
604\begin{corollary}\label{cor:div}
605$$\#\coker(X_J\ra X_A) = \frac{m_A}{m_\cL}.$$
606\end{corollary}
607\begin{remark}
608A non-obvious consequence of this corollary is that
609   $$m_\cL \mid m_A.$$
610\end{remark}
611
612
613\section{Optimal quotients of $J_0(N)$}\label{sec:optquoj0n}
614\label{sec:compj0n}\index{Optimal quotient!of $J_0(N)$}
615We now summarize some facts about $J_0(N)$ that will be used in
616our numerical computations.  Some of these facts were discussed in
617greater generality in the previous chapters of this thesis.
618
619\subsection{Modular curves and semistability}
620Let $X_0(N)$ be the modular curve\index{Modular curve}
621associated to the subgroup $\Gamma_0(N)$ of $\sltwoz$
622that consists of those matrices which are upper
623triangular modulo~$N$.  Initially, $X_0(N)$ is constructed as
624a Riemann surface as the quotient
625$$\Gamma_0(N)\backslash (\{z : z \in \C,\,\Im(z)>0\}\union\P^1(\Q)).$$
626With some work, we find that $X_0(N)$ has a canonical
627structure of algebraic curve over~$\Q$.
628
629Suppose that~$p$ is a prime divisor of~$N$ such that $N/p$ is
630coprime to~$p$.  We write $p\mid\mid N$.  In this situation,
631it is well-known that the Jacobian\index{Jacobian!of $X_0(N)$}
632$J_0(N)$ of $X_0(N)$ has semistable
633reduction at~$p$.
634
635\subsection{Newforms and optimal quotients}
636The Hecke algebra\index{Hecke algebra|textit}
637         $$\T=\Z[\ldots T_n\ldots]\subset\End(J_0(N))$$
638is a commutative ring of endomorphisms of~$J_0(N)$ of $\Z$-rank
639equal to the dimension $J_0(N)$.
640The character group $X_{J_0(N)}$ of $J_0(N)$ at~$p$
641is equipped with a functorial action of~$\T$.
642The Hecke algebra~$\T$ also acts on the complex vector space
643$S = S_2(\Gamma_0(N),\C)$
644of cusp forms.
645
646
647A newform~$f$ is an eigenform normalized so that the coefficient
648of~$q$ in the Fourier expansion of~$f$ at the cusp~$\infty$ is~$1$, and
649such that~$f$ is not a modular form of any level $N'\mid N$, with $N'$ a
650proper divisor of~$N$.
651
652Let~$f$ be a newform, and associate to~$f$ the ideal $I_f$ of
653the Hecke algebra~$\T$ of elements which
654annihilate~$f$.  Then $\O_f=\T/I_f$ is an
655order in the ring of integers of the totally real number field
656$K_f$ obtained by adjoining the Fourier coefficients of~$f$ to~$\Q$.
657The quotient
658  $$A_f = J_0(N)/ I_f J_0(N)$$
659is an optimal quotient\index{Optimal quotient} of $J_0(N)$
660of dimension equal to $[K_f:\Q]$.  It is purely
661toric\index{Purely toric reduction} at~$p$, since $p\mid \mid N$.
662
663\subsection{Homology and the modular degree}
664Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
665complex algebraic curve $X_0(N)$.  Integration defines a
666$\T$-equivariant nondegenerate\index{Integration pairing}
667pairing
668   $S \cross H \ra \C$.
669This pairing induces a map
670   $\alp: H \ra \Hom_\C(S,\C)$.
671
672\begin{theorem}\label{Af}
673We have the following commutative diagram of $\T$-modules:
674$$\xymatrix{ 675 H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d] 676 & \alp(H)\ar[d]\[email protected]{^(->}[d]\\ 677\Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d] 678 &\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\ 679 A_f^{\vee}(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\ 680}$$
681\end{theorem}
682\begin{proof}
683This can be deduced from \cite{shimura:factors}.
685\end{proof}
686
687\begin{corollary}\label{moduluscomp}
688$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
689\end{corollary}
690\begin{proof}
691Recall that $m_A$ is by definition equal to $\sqrt{\deg(\theta_A)}$.
692The kernel of an isogeny between complex tori is
693isomorphic to the cokernel of the induced map
694on lattices.  The corollary now follows from
695the diagram of Theorem~\ref{Af},
696which indicates that the index $[\alp(H):\alp(H[I_f])]$
697is the cokernel of the map $H[I_f]\ra \alp(H).$
698
699For more details, see Section~\ref{sec:moddeg}.
700\end{proof}
701
702\subsection{Rational points of the component group (Tamagawa numbers)}
703Let $\Frob_p:X_J\ra X_J$ denote the map induced by the
704Frobenius automorphism.
705We have $\Frob_p=-W_p$, where $W_p$ is the map induced
706by the Atkin-Lehner\index{Atkin-Lehner involution} involution on $J_0(p)$.
707Let~$f$ be a newform, $A=A_f$ the corresponding optimal
708quotient, and $w_p$ the sign of the eigenvalue of
709$W_p$ on $f$.
710\begin{proposition}
711$$\Phi_A(\Fp) 712 = \begin{cases} 713 \Phi_A(\Fpbar) & \text{if w_p=-1},\\ 714 \Phi_A(\Fpbar) & \text{if w_p=1.} 715 \end{cases}$$
716\end{proposition}
717\begin{proof}
718If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
719of $\Phi_A(\Fpbar)$ is trivial.  In this case
720$\Phi(\Fp)=\Phi(\Fpbar)$.
721Next suppose $w_p=1$.  Recall that we have an exact sequence
722   $$0\ra X_{A^{\vee}} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
723Since $W_p$ acts as $+1$ on~$f$, it also acts as $+1$ on
724each of the four modules~$A$,~$X_A$,~$\Hom(X_A,\Z)$, and~$\Phi_A$.
725Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$.  Since the subgroup
726of $2$-torsion elements of a finite abelian group equals the subgroup
727of elements fixed under $-1$, it follows that
728   $\Phi_A(\Fp) = \Phi_A(\Fpbar)$.
729\end{proof}
730
731{\bf WARNING:} When we extend this result to the whole
732of $J_0(N)$, it is necessary to be exceedingly careful!
733The action of $\Frob_p=T_p$ need
734not be by $\pm 1$, even though it must be by an involution
735of order~$2$.  For example, the component group of
736$J_0(65)$ at~$5$ is cyclic of order~$42$.  The action
737of $\Frob_5$ is by multiplication by $-13$.  Note that
738$(-13)^2 = 1 \pmod{42}$.   The fixed points of
739multiplication by~$-13$ is the order~$14$ subgroup
740of $\Z/42\Z$.
741
742\section{Computations}
743Using the algorithms of Chapter~\ref{chap:computing},
744we can enumerate the optimal
745quotients $A_f$ of $J_0(N)$ and compute the modular degree $m_A$.
746The method of graphs\index{Method of graphs}
747(see \cite{mestre:graphs}) and
748quaternion algebras (see \cite{kohel:hecke})\index{Quaternion algebras}
749can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
750and the monodromy pairing\index{Monodromy pairing}.  We can then
751compute the following three modules:
752the saturated submodule
753   $\cL=\bigcap_{t\in I_f} \ker(t)$
754of~$X$,
755the character group modular degree $m_X=m_\cL$,
756and $\Phi_X=\Phi_\cL$.
757By Theorem~\ref{formula} we obtain
758  $$\#\Phi_A = \#\Phi_X \cdot \frac{m_A}{m_X}.$$
759
760Using this method, we have computed $\#\Phi_A$ in a number of cases.
761We give tables that report on some of these computations in
762Secton~\ref{sec:compgrptables}.
763In the next section we discuss a conjecture and a question,
764which were both suggested by our numerical computations.
765
766\subsection{Conjectures and questions}%
767\label{sec:compgroupconjectures}%
769Suppose that $N=pM$ with $(p,M)=1$.
770Let
771$$H_{\new} = 772 \ker\,\Bigl( H_1(X_0(N),\Z)\lra 773 H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
774where the map is induced by the two natural
775degeneracy maps $X_0(N)\ra X_0(M)$.
776
777The Hecke algebra\index{Hecke algebra}~$\T$ acts on $H_{\new}$,
778and also on the submodule $H_{\new}[I_f]$ of those elements that
779are annihilated by $I_f$.
780Integration defines a map
781$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
782Define the $p$-new homology modular degree $m_H$ by
783$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
784We expect that there is a very close relationship
785between $m_X$ and $m_H$.
786\begin{question}
787  Is $m_X$ equal to $m_H$?
788\end{question}
789
790The following conjecture offers
791a refinement of some of the results of \cite{mazur:eisenstein}.
792\index{Conjecture!refined Eisenstein}
793\begin{conjecture}[Refined Eisenstein conjecture]\label{conj:iso}
794Let~$p$ be a prime and let $f_1,\ldots,f_n$ be a set of
795representatives for the Galois-conjugacy classes of newforms
796in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the optimal
797quotients associated to $f_1,\ldots,f_n$, respectively.
798Then for each~$i$, $i=1,\ldots,n$, we have
799$$\#A_i(\Q)_{\tor}=\#\Phi_{A_i}(\Fpbar)=\#\Phi_{A_i}(\Fp).$$
800Furthermore,
801$$\#\Phi_{J_0(p)}(\Fpbar)= \prod_{i=1}^d \#\Phi_{A_i}(\Fpbar).$$
802\end{conjecture}
803
804We have verified Conjecture~\ref{conj:iso} for all $p\leq 757$,
805and, up to a power of~$2$, for all $p< 2000$.
806
807\begin{remark}
808It is tempting to guess that, e.g., the natural map
809$$\Phi_{J_0(113)}(\Fpbar)\ra \prod_{i=1}^4 \Phi_{A_i}(\Fpbar)$$
810is an isomorphism.  Two of the $\Phi_{A_i}(\Fpbar)$ have
811order~$2$, so the product $\prod \Phi_{A_i}(\Fpbar)$ can not
812be a cyclic group.  However, the groups
813$\Phi_{J_0(p)}(\Fpbar)$ are known to be
814cyclic for all primes~$p$.
815\end{remark}
816
817\subsection{Tables}\label{sec:compgrptables}%
818We have computed component groups of many optimal quotients
819$A_f$ of $J_0(N)$.
820In this section we provide tables, which hint at the data
821we have gathered.  Our notation for optimal quotients
822is described in Section~\ref{sec:optquo-notation}.
824
825\subsubsection{Table~\ref{tbl:lowlevel}: Component groups at low level}
826Table~\ref{tbl:lowlevel} gives the component groups of the
827quotients $A_f$ of $J_0(N)$ for $N\leq 106$.
828The column labeled $d$ contains the
829dimensions of the $A_f$,
830and the column labeled $\#\Phi_{A,p}$ contains a list
831of the orders of the component groups of $A_f$,
832one for each divisor~$p$ of~$N$, ordered by increasing~$p$.
833An entry of ?'' indicates that $p^2\mid N$, so our algorithm
834does not apply.
835A component group order is starred if the $\Gal(\Fpbar/\Fp)$-action is
836nontrivial.
837
838
839\subsubsection{Table~\ref{table:big1}--\ref{table:big2}: Big component groups}
840Using the algorithms described in Section~\ref{sec:ratpartformula},
841we computed the rational numbers $L(A,1)/\Omega_A$
842for every optimal quotient~$A$ that is attached to a newform
843of level $\leq 1500$.
844There are exactly~$5$ optimal quotients~$A$ such that the numerator of
845$L(A,1)/\Omega_A$ is nonzero and divisible by a prime $>10^9$.
846The Birch and Swinnerton-Dyer conjecture%
847\index{BSD conjecture!predicts large component groups}
848predicts that these large
849prime divisors must divide either $\#\Phi_A$ or
850the Shafarevich-Tate\index{Shafarevich-Tate group}
851group of~$A$.   This is the case, as Table~\ref{table:big2} shows.
852
853
854\subsubsection{Table~\ref{table:compj0n}: Quotients of $J_0(N)$}
855Table~\ref{table:compj0n} contains all of the invariants involved
856in the computation of component groups for
857each of the newform optimal quotients of levels $65$, $66$, $68$, and $69$.
858
859\subsubsection{Table~\ref{table:compprime}:
860Quotients of $J_0(p)^-$}
861We computed
862the quantities $m_A$, $m_X$ and $\Phi_X$ for each abelian
863variety $A=A_f$ associated to a newform of prime level~$p$ with
864$p\leq 757$.  The results are as follows:
865\begin{enumerate}
866\item In all cases $m_A=m_X$, so the map $\Phi_J\ra \Phi_A$
867is surjective.
868\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
869$w_p$ on~$A$ is~$1$.
870\item $\prod \#\Phi_A(\Fpbar) = \#\Phi_J(\Fpbar)$
871\end{enumerate}
872Table~\ref{table:compprime}
873lists those~$A$ of level $\leq 631$ for which $w_p=-1$, along with
874the order of the corresponding component group.
875
876\begin{table}
877\ssp
878\begin{center}
879\caption{Component groups at low level\label{tbl:lowlevel}}
880\end{center}%
881\vspace{-.3in}%
882\index{Table of!component groups at low level}%
883\index{Component group!table of}%
884$$885\begin{array}{lcl} 886 A & \, d \, & \, \#\Phi_{A,p}\, \\ 887\vspace{-2ex}\\ 888{\bf 11A} & 1 & 5\\ 889{\bf 14A} & 1 & 6^*,3\\ 890{\bf 15A} & 1 & 4^*,4\\ 891{\bf 17A} & 1 & 4\\ 892\vspace{-2ex}\\ 893{\bf 19A} & 1 & 3\\ 894{\bf 20A} & 1 & ?,2^*\\ 895{\bf 21A} & 1 & 4,2^*\\ 896{\bf 23A} & 2 & 11\\ 897\vspace{-2ex}\\ 898{\bf 24A} & 1 & ?,2^*\\ 899{\bf 26A} & 1 & 3^*,3\\ 900{\bf 26B} & 1 & 7,1^*\\ 901{\bf 27A} & 1 & ?\\ 902\vspace{-2ex}\\ 903{\bf 29A} & 2 & 7\\ 904{\bf 30A} & 1 & 4^*,3,1^*\\ 905{\bf 31A} & 2 & 5\\ 906{\bf 32A} & 1 & ?\\ 907\vspace{-2ex}\\ 908{\bf 33A} & 1 & 6^*,2\\ 909{\bf 34A} & 1 & 6,1^*\\ 910{\bf 35A} & 1 & 3^*,3\\ 911{\bf 35B} & 2 & 8,4^*\\ 912\vspace{-2ex}\\ 913{\bf 36A} & 1 & ?,?\\ 914{\bf 37A} & 1 & 1^*\\ 915{\bf 37B} & 1 & 3\\ 916{\bf 38A} & 1 & 9^*,3\\ 917\vspace{-2ex}\\ 918{\bf 38B} & 1 & 5,1^*\\ 919{\bf 39A} & 1 & 2^*,2\\ 920{\bf 39B} & 2 & 14,2^*\\ 921{\bf 40A} & 1 & ?,2\\ 922\vspace{-2ex}\\ 923{\bf 41A} & 3 & 10\\ 924{\bf 42A} & 1 & 8,2^*,1^*\\ 925{\bf 43A} & 1 & 1^*\\ 926{\bf 43B} & 2 & 7\\ 927\vspace{-2ex}\\ 928{\bf 44A} & 1 & ?,1^*\\ 929{\bf 45A} & 1 & ?,1^*\\ 930{\bf 46A} & 1 & 10^*,1\\ 931{\bf 47A} & 4 & 23\\ 932\vspace{-2ex}\\ 933{\bf 48A} & 1 & ?,2\\ 934{\bf 49A} & 1 & ?\\ 935{\bf 50A} & 1 & 1^*,?\\ 936{\bf 50B} & 1 & 5,?\\ 937\end{array}\quad 938\begin{array}{lcl} 939 A & \, d \, & \, \#\Phi_{A,p}\, \\ 940\vspace{-2ex}\\ 941{\bf 51A} & 1 & 3,1^*\\ 942{\bf 51B} & 2 & 16^*,4\\ 943{\bf 52A} & 1 & ?,2^*\\ 944{\bf 53A} & 1 & 1^*\\ 945\vspace{-2ex}\\ 946{\bf 53A} & 1 & 1^*\\ 947{\bf 53B} & 3 & 13\\ 948{\bf 54A} & 1 & 3^*,?\\ 949{\bf 54B} & 1 & 3,?\\ 950\vspace{-2ex}\\ 951{\bf 55A} & 1 & 2,2^*\\ 952{\bf 55B} & 2 & 14^*,2\\ 953{\bf 56A} & 1 & ?,1\\ 954{\bf 56B} & 1 & ?,1^*\\ 955\vspace{-2ex}\\ 956{\bf 57A} & 1 & 2^*,1^*\\ 957{\bf 57B} & 1 & 2,2^*\\ 958{\bf 57C} & 1 & 10,1^*\\ 959{\bf 58A} & 1 & 2^*,1^*\\ 960\vspace{-2ex}\\ 961{\bf 58B} & 1 & 10,1^*\\ 962{\bf 59A} & 5 & 29\\ 963{\bf 61A} & 1 & 1^*\\ 964{\bf 61B} & 3 & 5\\ 965\vspace{-2ex}\\ 966{\bf 62A} & 1 & 4,1^*\\ 967{\bf 62B} & 2 & 66^*,3\\ 968{\bf 63A} & 1 & ?,1^*\\ 969{\bf 63B} & 2 & ?,3\\ 970\vspace{-2ex}\\ 971{\bf 64A} & 1 & ?\\ 972{\bf 65A} & 1 & 1^*,1^*\\ 973{\bf 65B} & 2 & 3^*,3\\ 974{\bf 65C} & 2 & 7,1^*\\ 975\vspace{-2ex}\\ 976{\bf 66A} & 1 & 2^*,3,1^*\\ 977{\bf 66B} & 1 & 4,1^*,1^*\\ 978{\bf 66C} & 1 & 10,5,1\\ 979{\bf 67A} & 1 & 1\\ 980\vspace{-2ex}\\ 981{\bf 67B} & 2 & 1^*\\ 982{\bf 67C} & 2 & 11\\ 983{\bf 68A} & 2 & ?,2^*\\ 984{\bf 69A} & 1 & 2,1^*\\ 985\vspace{-2ex}\\ 986{\bf 69B} & 2 & 22^*,2\\ 987{\bf 70A} & 1 & 4,2^*,1^*\\ 988{\bf 71A} & 3 & 5\\ 989{\bf 71B} & 3 & 7\\ 990\end{array}\quad 991\begin{array}{lcl} 992 A & \, d \, & \, \#\Phi_{A,p}\, \\ 993\vspace{-2ex}\\ 994{\bf 72A} & 1 & ?,?\\ 995{\bf 73A} & 1 & 2\\ 996{\bf 73B} & 2 & 1^*\\ 997{\bf 73C} & 2 & 3\\ 998\vspace{-2ex}\\ 999{\bf 74A} & 2 & 9^*,3\\ 1000{\bf 74B} & 2 & 95,1^*\\ 1001{\bf 75A} & 1 & 1^*,?\\ 1002{\bf 75B} & 1 & 1,?\\ 1003\vspace{-2ex}\\ 1004{\bf 75C} & 1 & 5,?\\ 1005{\bf 76A} & 1 & ?,1^*\\ 1006{\bf 77A} & 1 & 2^*,1^*\\ 1007{\bf 77B} & 1 & 3^*,2\\ 1008\vspace{-2ex}\\ 1009{\bf 77C} & 1 & 6,3^*\\ 1010{\bf 77D} & 2 & 2,2^*\\ 1011{\bf 78A} & 1 & 16^*,5^*,1\\ 1012{\bf 79A} & 1 & 1^*\\ 1013\vspace{-2ex}\\ 1014{\bf 79B} & 5 & 13\\ 1015{\bf 80A} & 1 & ?,2\\ 1016{\bf 80B} & 1 & ?,2^*\\ 1017{\bf 81A} & 2 & ?\\ 1018\vspace{-2ex}\\ 1019{\bf 82A} & 1 & 2^*,1^*\\ 1020{\bf 82B} & 2 & 28,1^*\\ 1021{\bf 83A} & 1 & 1^*\\ 1022{\bf 83B} & 6 & 41\\ 1023\vspace{-2ex}\\ 1024{\bf 84A} & 1 & ?,1^*,2^*\\ 1025{\bf 84B} & 1 & ?,3,2\\ 1026{\bf 85A} & 1 & 2^*,1\\ 1027{\bf 85B} & 2 & 2^*,1^*\\ 1028\vspace{-2ex}\\ 1029{\bf 85C} & 2 & 6,1^*\\ 1030{\bf 86A} & 2 & 21^*,3\\ 1031{\bf 86B} & 2 & 55,1^*\\ 1032{\bf 87A} & 2 & 5,1^*\\ 1033\vspace{-2ex}\\ 1034{\bf 87B} & 3 & 92^*,4\\ 1035{\bf 88A} & 1 & ?,1^*\\ 1036{\bf 88B} & 2 & ?,2^*\\ 1037{\bf 89A} & 1 & 1^*\\ 1038\vspace{-2ex}\\ 1039{\bf 89B} & 1 & 2\\ 1040{\bf 89C} & 5 & 11\\ 1041{\bf 90A} & 1 & 2^*,?,3\\ 1042{\bf 90B} & 1 & 6,?,1^*\\ 1043\end{array}\quad 1044\begin{array}{lcl} 1045 A & \, d \, & \, \#\Phi_{A,p}\, \\ 1046\vspace{-2ex}\\ 1047{\bf 90C} & 1 & 4,?,1\\ 1048{\bf 91A} & 1 & 1^*,1^*\\ 1049{\bf 91B} & 1 & 1,1\\ 1050{\bf 91C} & 2 & 7,1^*\\ 1051\vspace{-2ex}\\ 1052{\bf 91D} & 3 & 4^*,8\\ 1053{\bf 92A} & 1 & ?,1^*\\ 1054{\bf 92B} & 1 & ?,1\\ 1055{\bf 93A} & 2 & 4^*,1^*\\ 1056\vspace{-2ex}\\ 1057{\bf 93B} & 3 & 64,2^*\\ 1058{\bf 94A} & 1 & 2,1^*\\ 1059{\bf 94B} & 2 & 94^*,1\\ 1060{\bf 95A} & 3 & 10,2^*\\ 1061\vspace{-2ex}\\ 1062{\bf 95B} & 4 & 54^*,6\\ 1063{\bf 96A} & 1 & ?,2\\ 1064{\bf 96B} & 1 & ?,2^*\\ 1065{\bf 97A} & 3 & 1^*\\ 1066\vspace{-2ex}\\ 1067{\bf 97B} & 4 & 8\\ 1068{\bf 98A} & 1 & 2^*,?\\ 1069{\bf 98B} & 2 & 14,?\\ 1070{\bf 99A} & 1 & ?,1^*\\ 1071\vspace{-2ex}\\ 1072{\bf 99B} & 1 & ?,1\\ 1073{\bf 99C} & 1 & ?,1^*\\ 1074{\bf 99D} & 1 & ?,1^*\\ 1075{\bf 100A} & 1 & ?,?\\ 1076\vspace{-2ex}\\ 1077{\bf 101A} & 1 & 1^*\\ 1078{\bf 101B} & 7 & 25\\ 1079{\bf 102A} & 1 & 2^*,2^*,1^*\\ 1080{\bf 102B} & 1 & 6^*,6,1^*\\ 1081\vspace{-2ex}\\ 1082{\bf 102C} & 1 & 8,4,1\\ 1083{\bf 103A} & 2 & 1^*\\ 1084{\bf 103B} & 6 & 17\\ 1085{\bf 104A} & 1 & ?,1^*\\ 1086\vspace{-2ex}\\ 1087{\bf 104B} & 2 & ?,2\\ 1088{\bf 105A} & 1 & 1,1,1\\ 1089{\bf 105B} & 2 & 10^*,2^*,2\\ 1090{\bf 106A} & 1 & 4^*,1^*\\ 1091\vspace{-2ex}\\ 1092{\bf 106B} & 1 & 5^*,1\\ 1093{\bf 106C} & 1 & 24,1^*\\ 1094{\bf 106D} & 1 & 3,1^*\\ 1095&&\\ 1096\end{array}$$
1097\end{table}
1098
1099
1100\begin{table}
1101\ssp
1102\caption{Big $L(A,1)/\Omega_A$\label{table:big1}}
1103\index{Table of!big $L(A,1)/\Omega_A$}
1104\index{Component group!table of}
1105$$\begin{array}{lccc} 1106 A & \text{dim} & 1107 N & \text{\qquad L(A,1)/\Omega_A\cdot \text{Manin constant}\qquad }\\ 1108\vspace{-2ex} && & \\ 1109\text{\bf 1154E} & 20&2\cdot 577 & 2^?\cdot 85495047371/17^2\\ 1110\text{\bf 1238G} & 19& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\ 1111\text{\bf 1322E} & 21& 2\cdot 661 & 2^?\cdot 57851840099/331\\ 1112\text{\bf 1382D} & 20& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\ 1113\text{\bf 1478J} & 20 1114 & 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\ 1115\end{array}$$
1116\end{table}
1117
1118\begin{table}
1119\ssp
1120\caption{Big component groups\label{table:big2}}
1121\index{Table of!big component groups}
1122\index{Component group!table of}
1123$$\begin{array}{lcccccc} 1124 A &p & w & \#\Phi_X & m_X & \#\Phi_A(\Fpbar) \\ 1125\vspace{-1ex} && & & & & \\ 1126\text{\bf 1154E} &2 & - & 17^2 & 2^{24} 1127 & 2^?\cdot 17^2 \cdot 85495047371 \\ 1128 &577& + & 1 & 2^{26}\cdot85495047371 1129 & 2^? \\ 1130\vspace{-1ex}&&&&&\\ 1131\text{\bf 1238G} & 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\ 1132 & 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\ 1133\vspace{-1ex}&&&&&\\ 1134\text{\bf 1322E} & 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\ 1135 & 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\ 1136\vspace{-1ex}&&&&&\\ 1137\text{\bf 1382D} & 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\ 1138 & 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\ 1139\vspace{-1ex}&&&&&\\ 1140\text{\bf 1478J} & 2 & - & 5\cdot37 &2^{31} 1141 & 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\ 1142 & 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463 1143 & 2^? \\ 1144\end{array}$$
1145\end{table}
1146
1147
1148\begin{table}
1149\ssp
1150\caption{Component groups of quotients of $J_0(N)$\label{table:compj0n}}%
1151\index{Table of!component groups of quotients}%
1152\index{Component group!table of}%
1153$$\begin{array}{lccccccc} 1154 A & \text{dim} & p & w_p & \#\Phi_X & m_X & m_A & \#\Phi_A \\ 1155\vspace{-1ex} && & & & & & \\ 1156\text{\bf 65A} & 1& 5 & +& 1 &2 & 2 & 1\\ 1157 && 13 &+& 1 & 2 & & 1\\ 1158 1159\text{\bf 65B} &2& 5 &+& 3 & 2^2 & 2^2 & 3\\ 1160 && 13 &- & 3 & 2^2 & & 3\\ 1161 1162\text{\bf 65C}&2 & 5 &-& 7 & 2^2& 2^2 &7 \\ 1163 && 13 &+ & 1 & 2^2& & 1\\ 1164 1165\vspace{-1ex} && & & & & & \\ 1166\text{\bf 66A}&1& 2 &+ & 1 &2 & 2^2&2 \\ 1167 && 3 &- & 3 &2^2 & & 3\\ 1168 && 11 &+ & 1 &2^2 & &1 \\ 1169 1170\text{\bf 66B}&1& 2 &- & 2 &2 & 2^2& 2^2\\ 1171 && 3 &+ & 1 &2^2& & 1\\ 1172 && 11 &+ & 1 &2^2 & & 1\\ 1173 1174\text{\bf 66C}&1& 2 & -& 1 & 2& 2^2\cdot 5& 2\cdot5\\ 1175 && 3 &- & 1 & 2^2 & & 5\\ 1176 && 11 &- & 1 & 2^2\cdot5 & &1 \\ 1177 1178\vspace{-1ex} && & & & & & \\ 1179\text{\bf 68A}&2&17&+&2 &2\cdot3 &2\cdot3 & 2 \\ 1180 1181\vspace{-1ex} && & & & & & \\ 1182\text{\bf 69A} &1&3 &-&2 &2 & 2& 2\\ 1183 &&23 &+& 1&2 & & 1\\ 1184 1185\text{\bf 69B} &2&3 &+&2 &2 &2\cdot11& 2\cdot11 \\ 1186 &&23 &-&2 &2\cdot11 && 2 \\ 1187 1188\end{array}$$
1189\end{table}
1190
1191
1192\begin{table}
1193\ssp
1194\caption{Component groups of quotients of $J_0(p)^{-}$
1195\index{Table of!component groups at prime level}
1196\index{Component group!table of}
1197\label{table:compprime}}%
1198\vspace{-.3in}%
1199$$1200\begin{array}{lcc} 1201\vspace{-2ex}\\ 1202A & d & \#\Phi_A \\ 1203\vspace{-2ex}\\ 1204{\bf 11A}&1&5\\ 1205{\bf 17A}&1&2^2\\ 1206{\bf 19A}&1&3\\ 1207{\bf 23A}&2&11\\ 1208\vspace{-2ex}& &\\ 1209{\bf 29A}&2&7\\ 1210{\bf 31A}&2&5\\ 1211{\bf 37B}&1&3\\ 1212{\bf 41A}&3&2\cdot5\\ 1213\vspace{-2ex}& &\\ 1214{\bf 43B}&2&7\\ 1215{\bf 47A}&4&23\\ 1216{\bf 53B}&3&13\\ 1217{\bf 59A}&5&29\\ 1218\vspace{-2ex}& &\\ 1219{\bf 61B}&3&5\\ 1220{\bf 67A}&1&1\\ 1221{\bf 67C}&2&11\\ 1222{\bf 71A}&3&5\\ 1223\vspace{-2ex}& &\\ 1224{\bf 71B}&3&7\\ 1225{\bf 73A}&1&2\\ 1226{\bf 73C}&2&3\\ 1227{\bf 79B}&5&13\\ 1228\vspace{-2ex}& &\\ 1229{\bf 83B}&6&41\\ 1230{\bf 89B}&1&2\\ 1231{\bf 89C}&5&11\\ 1232{\bf 97B}&4&2^3\\ 1233\vspace{-2ex}& &\\ 1234{\bf 101B}&7&5^2\\ 1235{\bf 103B}&6&17\\ 1236{\bf 107B}&7&53\\ 1237{\bf 109A}&1&1\\ 1238\vspace{-2ex}& &\\ 1239{\bf 109C}&4&3^2\\ 1240{\bf 113A}&1&2\\ 1241{\bf 113B}&2&2\\ 1242{\bf 113D}&3&7\\ 1243\vspace{-2ex}& &\\ 1244{\bf 127B}&7&3\cdot7\\ 1245{\bf 131B}&10&5\cdot13\\ 1246{\bf 137B}&7&2\cdot17\\ 1247{\bf 139A}&1&1\\ 1248\vspace{-2ex}& &\\ 1249{\bf 139C}&7&23\\ 1250{\bf 149B}&9&37\\ 1251{\bf 151B}&3&1\\ 1252{\bf 151C}&6&5^2\\ 1253\end{array}\,\, 1254\begin{array}{lcc} 1255\vspace{-2ex}\\ 1256A & d& \#\Phi_A \\ 1257\vspace{-2ex}\\ 1258{\bf 157B}&7&13\\ 1259{\bf 163C}&7&3^3\\ 1260{\bf 167B}&12&83\\ 1261{\bf 173B}&10&43\\ 1262\vspace{-2ex} &\\ 1263{\bf 179A}&1&1\\ 1264{\bf 179C}&11&89\\ 1265{\bf 181B}&9&3\cdot5\\ 1266{\bf 191B}&14&5\cdot19\\ 1267\vspace{-2ex} &\\ 1268{\bf 193C}&8&2^4\\ 1269{\bf 197C}&10&7^2\\ 1270{\bf 199A}&2&1\\ 1271{\bf 199C}&10&3\cdot11\\ 1272\vspace{-2ex} &\\ 1273{\bf 211A}&2&5\\ 1274{\bf 211D}&9&7\\ 1275{\bf 223C}&12&37\\ 1276{\bf 227B}&2&1\\ 1277\vspace{-2ex} &\\ 1278{\bf 227C}&2&1\\ 1279{\bf 227E}&10&113\\ 1280{\bf 229C}&11&19\\ 1281{\bf 233A}&1&2\\ 1282\vspace{-2ex} &\\ 1283{\bf 233C}&11&29\\ 1284{\bf 239B}&17&7\cdot17\\ 1285{\bf 241B}&12&2^2\cdot5\\ 1286{\bf 251B}&17&5^3\\ 1287\vspace{-2ex} &\\ 1288{\bf 257B}&14&2^6\\ 1289{\bf 263B}&17&131\\ 1290{\bf 269C}&16&67\\ 1291{\bf 271B}&16&3^2\cdot5\\ 1292\vspace{-2ex} &\\ 1293{\bf 277B}&3&1\\ 1294{\bf 277D}&9&23\\ 1295{\bf 281B}&16&2\cdot5\cdot7\\ 1296{\bf 283B}&14&47\\ 1297\vspace{-2ex} &\\ 1298{\bf 293B}&16&73\\ 1299{\bf 307A}&1&1\\ 1300{\bf 307B}&1&1\\ 1301{\bf 307C}&1&1\\ 1302\vspace{-2ex} &\\ 1303{\bf 307D}&1&1\\ 1304{\bf 307E}&2&3\\ 1305{\bf 307F}&9&17\\ 1306{\bf 311B}&22&5\cdot31\\ 1307\end{array}\,\, 1308\begin{array}{lcc} 1309\vspace{-2ex}\\ 1310A & d &\#\Phi_A \\ 1311\vspace{-2ex}\\ 1312{\bf 313A}&2&1\\ 1313{\bf 313C}&12&2\cdot13\\ 1314{\bf 317B}&15&79\\ 1315{\bf 331D}&16&5\cdot11\\ 1316\vspace{-2ex} &\\ 1317{\bf 337B}&15&2^2\cdot7\\ 1318{\bf 347D}&19&173\\ 1319{\bf 349B}&17&29\\ 1320{\bf 353A}&1&2\\ 1321\vspace{-2ex} &\\ 1322{\bf 353B}&3&2\\ 1323{\bf 353D}&14&2\cdot11\\ 1324{\bf 359D}&24&179\\ 1325{\bf 367B}&19&61\\ 1326\vspace{-2ex} &\\ 1327{\bf 373C}&17&31\\ 1328{\bf 379B}&18&3^2\cdot7\\ 1329{\bf 383C}&24&191\\ 1330{\bf 389A}&1&1\\ 1331\vspace{-2ex} &\\ 1332{\bf 389E}&20&97\\ 1333{\bf 397B}&2&1\\ 1334{\bf 397C}&5&11\\ 1335{\bf 397D}&10&3\\ 1336\vspace{-2ex} &\\ 1337{\bf 401B}&21&2^2\cdot5^2\\ 1338{\bf 409B}&20&2\cdot17\\ 1339{\bf 419B}&26&11\cdot19\\ 1340{\bf 421B}&19&5\cdot7\\ 1341\vspace{-2ex} &\\ 1342{\bf 431B}&1&1\\ 1343{\bf 431D}&3&1\\ 1344{\bf 431F}&24&5\cdot43\\ 1345{\bf 433A}&1&1\\ 1346\vspace{-2ex} &\\ 1347{\bf 433B}&3&1\\ 1348{\bf 433D}&16&2^2\cdot3^2\\ 1349{\bf 439C}&25&73\\ 1350{\bf 443C}&1&1\\ 1351\vspace{-2ex} &\\ 1352{\bf 443E}&22&13\cdot17\\ 1353{\bf 449B}&23&2^4\cdot7\\ 1354{\bf 457C}&20&2\cdot19\\ 1355{\bf 461D}&26&5\cdot23\\ 1356\vspace{-2ex} &\\ 1357{\bf 463B}&22&7\cdot11\\ 1358{\bf 467C}&26&233\\ 1359{\bf 479B}&32&239\\ 1360{\bf 487A}&2&1\\ 1361\end{array}\,\, 1362\begin{array}{lcc} 1363\vspace{-2ex}&\\ 1364A & d & \#\Phi_A \\ 1365\vspace{-2ex}&\\ 1366{\bf 487B}&2&3\\ 1367{\bf 487C}&3&1\\ 1368{\bf 487D}&16&3^3\\ 1369{\bf 491C}&29&5\cdot7^2\\ 1370\vspace{-2ex} &\\ 1371{\bf 499C}&23&83\\ 1372{\bf 503B}&1&1\\ 1373{\bf 503C}&1&1\\ 1374{\bf 503D}&3&1\\ 1375\vspace{-2ex} &\\ 1376{\bf 503F}&26&251\\ 1377{\bf 509B}&28&127\\ 1378{\bf 521B}&29&2\cdot5\cdot13\\ 1379{\bf 523C}&26&3\cdot29\\ 1380\vspace{-2ex} &\\ 1381{\bf 541B}&24&3^2\cdot5\\ 1382{\bf 547C}&25&7\cdot13\\ 1383{\bf 557B}&1&1\\ 1384{\bf 557D}&26&139\\ 1385\vspace{-2ex} &\\ 1386{\bf 563A}&1&1\\ 1387{\bf 563E}&31&281\\ 1388{\bf 569B}&31&2\cdot71\\ 1389{\bf 571A}&1&1\\ 1390\vspace{-2ex} &\\ 1391{\bf 571B}&1&1\\ 1392{\bf 571C}&2&1\\ 1393{\bf 571D}&2&1\\ 1394{\bf 571F}&4&1\\ 1395\vspace{-2ex} &\\ 1396{\bf 571I}&18&5\cdot19\\ 1397{\bf 577A}&2&3\\ 1398{\bf 577B}&2&1\\ 1399{\bf 577C}&3&1\\ 1400\vspace{-2ex} &\\ 1401{\bf 577D}&18&2^4\\ 1402{\bf 587C}&31&293\\ 1403{\bf 593B}&1&2\\ 1404{\bf 593C}&2&1\\ 1405\vspace{-2ex} &\\ 1406{\bf 593E}&27&2\cdot37\\ 1407{\bf 599C}&37&13\cdot23\\ 1408{\bf 601B}&29&2\cdot5^2\\ 1409{\bf 607D}&31&101\\ 1410\vspace{-2ex} &\\ 1411{\bf 613C}&27&3\cdot17\\ 1412{\bf 617B}&28&2\cdot7\cdot11\\ 1413{\bf 619B}&30&103\\ 1414{\bf 631B}&32&3\cdot5\cdot7\\ 1415\end{array}$$
1416\end{table}
1417
1418
1419