CoCalc
Sharedwww / ants2.texOpen in CoCalc
Author: William A. Stein
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\documentclass[draft,11pt]{llncs}
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\newcommand{\capt}{\caption}
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\newcommand{\cA}{\mathcal{A}}
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\newcommand{\cT}{\mathcal{T}}
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\newcommand{\cX}{\mathcal{X}}
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\newcommand{\cO}{\mathcal{O}}
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\newcommand{\coker}{\mbox{\rm coker}}
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\newcommand{\F}{\mathbf{F}}
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\newcommand{\G}{\mathbf{G}}
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\newcommand{\Fbar}{\overline{\F}}
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\newcommand{\Fp}{\F_p}
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\newcommand{\Fpbar}{\Fbar_p}
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\newcommand{\Frob}{\mbox{\rm Frob}}
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\newcommand{\Gal}{\mbox{\rm Gal}}
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\newcommand{\Hom}{\mbox{\rm Hom}}
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\newcommand{\Q}{\mathbf{Q}}
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\newcommand{\SL}{\mbox{\rm SL}}
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\newcommand{\T}{\mathbf{T}}
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\newcommand{\Z}{\mathbf{Z}}
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\newcommand{\isom}{\cong}
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\newcommand{\tdot}{\!\cdot\!}
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\newcommand{\rta}{\rightarrow}
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\begin{document}
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\title{Component Groups of Quotients of $J_0(N)$}
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\titlerunning{Component Groups}
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\author{David Kohel\inst{1} \and William A. Stein\inst{2}}
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\authorrunning{Kohel \and Stein}
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\tocauthor{David Kohel (University of Sydney),
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William A. Stein (University of California at Berkeley)}
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%
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\institute{University of Sydney\\
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\email{kohel@maths.usyd.edu.au}\\
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\texttt{http://www.maths.usyd.edu.au:8000/u/kohel/}
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\and
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University of California at Berkeley,\\
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\email{was@math.berkeley.edu}\\
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\texttt{http://shimura.math.berkeley.edu/\~{}was}
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}
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\maketitle
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\begin{abstract}
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Let~$f$ be a newform of weight~$2$ on $\Gamma_0(N)$, and let~$A_f$ be
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the corresponding optimal abelian variety quotient of $J_0(N)$. We
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describe an algorithm to compute the order of the component group of
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$A_f$ at primes~$p$ that exactly divide~$N$. We give a table of
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orders of component groups for all~$f$ of level $N\leq 127$ and four
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examples in which the component group is very large, as predicted by
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the Birch and Swinnerton-Dyer conjecture.
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\end{abstract}
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\section{Introduction}
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Let $X_0(N)$ be the Riemann surface obtained by compactifying the
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quotient of the upper half-plane by the action of $\Gamma_0(N)$.
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Then $X_0(N)$ has a canonical structure of algebraic curve
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over~$\Q$; denote its Jacobian by $J_0(N)$. It is equipped with
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an action of a commutative ring $\T=\Z[\ldots T_n \ldots]$ of
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Hecke operators. For more details on modular curves, Hecke operators,
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and modular forms see, e.g.,~\cite{diamond-im}.
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Now suppose that~$f=\sum_{n=1}^{\infty} a_n q^n$ is a modular newform
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of weight~$2$ for the congruence subgroup $\Gamma_0(N)$.
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The Hecke operators also act on~$f$ by $T_n(f) = a_n f$.
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The eigenvalues $a_n$ generate an order $R_f = \Z[\ldots a_n \ldots]$
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in a number field $K_f$. Let $I_f$ be the kernel ideal of the map
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$\T \rta R_f$ sending $T_n$ to~$a_n$.
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Following Shimura~\cite{shimura:factors}, we associate to~$f$ the
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quotient $A_f = J_0(N)/I_f J_0(N)$ of
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$J_0(N)$. Then $A_f$ is an abelian variety over~$\Q$ of dimension
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$[K_f:\Q]$, with bad reduction exactly at the primes dividing~$N$.
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One dimensional quotients of $J_0(N)$ have been intensely studied
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in recent years, both computationally and theoretically.
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The original conjectures of Birch and
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Swinnerton-Dyer~\cite{birch-swd-I,birch-swd-II},
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which concern elliptic curves,
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were greatly influenced by computations.
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The scale of these computations was extended and systematized by
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Cremona in~\cite{cremona:algs}.
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In another direction, Wiles~\cite{wiles:fermat} and
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Taylor-Wiles~\cite{taylor-wiles} proved a special case of the
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conjecture of Shimura-Taniyama, which asserts that every elliptic
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curve over~$\Q$ is a quotient of some $J_0(N)$; this allowed them to establish
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Fermat's Last Theorem. The full Shimura-Taniyama
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conjecture was later proved by Breuil, Conrad, Diamond,
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and Taylor (see~\cite{breuil-conrad-diamond-taylor}).
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This illustrates the central roll played by quotients of $J_0(N)$.
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\section{Component Groups of $A_f$}
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The N\'eron model $\cA/\Z$ of an abelian variety $A/\Q$ is by definition
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a smooth commutative group scheme over~$\Z$ with generic fiber~$A$
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such that for any smooth scheme~$S$ over~$\Z$, the restriction map
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$$\Hom_\Z(S,\cA) \rta \Hom_\Q(S_\Q,A)$$ is a bijection.
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For more details, including a proof of existence, see,
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e.g.,~\cite{neronmodels}.
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Suppose that $A_f$ is a quotient of $J_0(N)$ corresponding to a newform~$f$,
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and let $\cA_f$ be the N\'eron model of~$A_f$. For any prime divisor~$p$
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of~$N$, the closed fiber~${\cA_f}_{/\Fp}$ is a group scheme
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over~$\Fp$, which need not be connected. Denote the connected
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component of the identity by~${\cA^{\circ}_f}_{/\Fp}$.
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There is an exact sequence
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$$
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0 \rightarrow {\cA^{\circ}_f}_{/\Fp}
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\rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p}
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\rightarrow 0
111
$$
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with $\Phi_{A_f,p}$ a finite \'{e}tale group scheme over~$\Fp$
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called the {\em component group} of $A_f$ at~$p$.
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We note that a finite \'etale group scheme is equivalent to a
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finite abelian group together with an action of $\Gal(\Fpbar/\Fp)$.
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In this paper we describe an algorithm for computing the order of
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this group.
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\section{The Algorithm}
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Let~$J = J_0(N)$. As the Jacobian of a curve, $J$ is canonically
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self-dual, so the projection $J \rta A_f$ induces a polarization
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$A_f^{\vee} \rta A_f$, where $A_f^{\vee}$ denotes the abelian variety
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dual of $A_f$. The {\em modular degree} $\delta_{A_f}$
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is the positive square root of the degree of this polarization.
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128
Let $T_{J,p}$ be the toric part of the closed fiber of the
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N\'eron model of $J$ at~$p$, and define $\cX_{J,p}$ to
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be the group $\Hom_{\Fpbar}(T_{J,p},\G_m)$ of characters of $T_{J,p}$.
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Then $\cX_{J,p}$ is a free abelian group equipped with an action
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of both~$\T$ and $\Gal(\Fpbar/\Fp)$
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(see, e.g., Ribet~\cite{ribet:modreps}); there is a bilinear
134
pairing
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$$
136
\langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z
137
$$
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called the {\em monodromy pairing} such that
139
$$
140
\Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)).
141
$$
142
Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$ for~$t$
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in $I_f$, and let
144
$$
145
\alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z)
146
$$
147
be the map induced by the monodromy pairing.
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The following theorem of Stein~\cite{stein:phd}, provides the
149
basis for the computation of orders of component groups.
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151
\begin{theorem}\label{thm:main}
152
$$
153
\#\Phi_{A_f,p}
154
= \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}}
155
{\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,.
156
$$
157
\end{theorem}
158
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\subsection{Computing the modular degree $\delta_{A,f}$}
160
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To compute the modular degree we require
162
$H_1(X_0(N),\Z;\mbox{\rm cusps})$ together with the action
163
of the Hecke ring $\T$.
164
Using modular symbols (see, e.g., Cremona~\cite{cremona:algs}), we
165
first compute $H_1(X_0(N),\Q;\mbox{\rm cusps})$.
166
Using lattice reduction, we compute the $\Z$-submodule generated
167
by all Manin symbols $(c,d)$; then $H_1(X_0(N),\Z)$ is the
168
{\em integer} kernel of the boundary map. We have a natural restriction
169
map
170
$$
171
\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow
172
\Hom(H_1(X_0(N),\Z)[I_f],\Z)
173
$$
174
where $t \in \T$ acts on $\varphi \in \Hom(H_1(X_0(N),\Z),\Z)$ by
175
$(t.\varphi)(x) = \varphi(tx)$.
176
By considering $J_0(N)$, $A_f$, and $A_f^{\vee}$
177
as complex lattices, and applying the snake lemma,
178
we see that the order of the cokernel of this map is the square of the
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modular degree $\delta_f$.
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181
\subsection{Computing the character group $\cX_{J,p}$}
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When $N=Mp$, with~$M$ small, the algorithm of Mestre and
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Oesterl\'e~\cite{mestre:graphs} can be used. This algorithm
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constructs the graph of isogenies between $\Fpbar$-isomorphism
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classes of pairs consisting of a supersingular elliptic curve
187
and an $M$-torsion subgroup. In particular, the
188
method is elementary to apply when $X_0(M)$ is of genus~$0$.
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In general, the above category of ``enhanced'' supersingular
191
elliptic curves can be replaced
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by one of left (or right) ideals of a quaternion order~$\cO$ of
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level~$M$ in the quaternion algebra over~$\Q$ ramified at~$p$.
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This gives an equivalent category, in which the computation of
195
homomorphisms is efficient. The character group $\cX_{J,p}$ is
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known by Deligne-Rapoport~\cite{deligne-rapoport} to be canonically
197
equivalent to the degree zero subgroup $\cX(\cO)$ of the free
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abelian ``divisor group'' on the isomorphism classes of enhanced
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supersingular elliptic curves and of quaternion ideals. Moreover,
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this isomorphism is compatible with the operation of Hecke operators,
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which are effectively computable in $\cX(\cO)$ in terms of ideal
202
homomorphisms.
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204
The inner product of two classes in this setting is defined
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to be the number of isomorphisms between any two representatives.
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The linear extension to $\cX(\cO)$ gives an inner product which
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agrees, under the isomophism, with the monodromy pairing on
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$\cX_{J,p}$. This gives, in particular, an isomorphism $\Phi_{J,p}
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\isom \coker(\cX(\cO) \rightarrow \Hom(\cX(\cO),\Z))$, and an
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effective means of computing $\#\coker(\alpha_f)$ and
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$\#(\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))$.
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The ideal arithmetic of quaternions has been implemented in
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{\sc Magma} \cite{magma} by the first author. Additional details and
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the application to Shimura curves, generalizing $X_0(N)$, can
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be found in Kohel~\cite{kohel}.
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\subsection{The Galois action on $\Phi_{A_f,p}$}
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To determine the Galois action on $\Phi_{A_f,p}$, we need only
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know the action of the Frobenius automorphism $\Frob_p$. However
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$\Frob_p$ is known to equal $-W_p$ where $W_p$ is the $p$th
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Atkin-Lehner involution, which can be computed using modular symbols.
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Since~$f$ is an eigenform, the involution $W_p$ acts as either
225
$+1$ or $-1$ on $\Phi_{A_f,p}$.
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Moreover, the operator $W_p$ is determined by an involution on
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the set of quaternion ideals, so can be determined explicitly
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on the character group.
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\section{Tables}
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\subsection{Component groups at low level}
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Table~\ref{tbl:lowlevel} gives the component groups of the quotients
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$A_f$ of $J_0(N)$ for $N\leq 127$; it was
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computed using a {\sc Magma} implementation of an
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algorithm based on Theorem~\ref{thm:main}. The column~$d$ is
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the dimension, and the column $\#\Phi_{A_f,p}$ contains a list
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of orders of component groups, one for each divisor~$p$ of~$N$.
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An order is starred if the $\Gal(\Fpbar/\Fp)$-action is nontrivial.
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More data along these lines can be obtained from the second author.
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\begin{table}
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\begin{center}
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\caption{Component groups at low level}
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\end{center}
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\label{tbl:lowlevel}
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$$
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\begin{array}{lcl}
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N & \, d \, & \, \#\Phi_{A_f,p}\, \\
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11 & 1 & 5\\
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14 & 1 & 6^*,3\\
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15 & 1 & 4^*,4\\
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17 & 1 & 4\\
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19 & 1 & 3\\
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20 & 1 & ?,2^*\\
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21 & 1 & 4,2^*\\
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23 & 2 & 11\\
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24 & 1 & ?,2^*\\
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26 & 1 & 3^*,3\\
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& 1 & 7,1^*\\
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27 & 1 & ?\\
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29 & 2 & 7\\
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30 & 1 & 4^*,3,1^*\\
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31 & 2 & 5\\
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32 & 1 & ?\\
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33 & 1 & 6^*,2\\
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34 & 1 & 6,1^*\\
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35 & 1 & 3^*,3\\
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& 2 & 8,4^*\\
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36 & 1 & ?,?\\
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37 & 1 & 1^*\\
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& 1 & 3\\
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38 & 1 & 9^*,3\\
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& 1 & 5,1^*\\
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39 & 1 & 2^*,2\\
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& 2 & 14,2^*\\
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40 & 1 & ?,2\\
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41 & 3 & 10\\
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42 & 1 & 8,2^*,1^*\\
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43 & 1 & 1^*\\
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& 2 & 7\\
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44 & 1 & ?,1^*\\
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45 & 1 & ?,1^*\\
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46 & 1 & 10^*,1\\
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47 & 4 & 23\\
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48 & 1 & ?,2\\
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49 & 1 & ?\\
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50 & 1 & 1^*,?\\
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& 1 & 5,?\\
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51 & 1 & 3,1^*\\
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& 2 & 16^*,4\\
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52 & 1 & ?,2^*\\
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53 & 1 & 1^*\\
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\end{array}\quad
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\begin{array}{lcl}
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N & \, d \, & \, \#\Phi_{A_f,p}\, \\
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& 3 & 13\\
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54 & 1 & 3^*,?\\
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& 1 & 3,?\\
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55 & 1 & 2,2^*\\
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& 2 & 14^*,2\\
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56 & 1 & ?,1\\
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& 1 & ?,1^*\\
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57 & 1 & 2^*,1^*\\
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& 1 & 2,2^*\\
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& 1 & 10,1^*\\
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58 & 1 & 2^*,1^*\\
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& 1 & 10,1^*\\
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59 & 5 & 29\\
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61 & 1 & 1^*\\
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& 3 & 5\\
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62 & 1 & 4,1^*\\
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& 2 & 66^*,3\\
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63 & 1 & ?,1^*\\
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& 2 & ?,3\\
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64 & 1 & ?\\
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65 & 1 & 1^*,1^*\\
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& 2 & 3^*,3\\
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& 2 & 7,1^*\\
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66 & 1 & 2^*,3,1^*\\
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& 1 & 4,1^*,1^*\\
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& 1 & 10,5,1\\
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67 & 1 & 1\\
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& 2 & 1^*\\
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& 2 & 11\\
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68 & 2 & ?,2^*\\
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69 & 1 & 2,1^*\\
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& 2 & 22^*,2\\
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70 & 1 & 4,2^*,1^*\\
331
71 & 3 & 5\\
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& 3 & 7\\
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72 & 1 & ?,?\\
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73 & 1 & 2\\
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& 2 & 1^*\\
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& 2 & 3\\
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74 & 2 & 9^*,3\\
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& 2 & 95,1^*\\
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75 & 1 & 1^*,?\\
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& 1 & 1,?\\
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& 1 & 5,?\\
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\end{array}\quad
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\begin{array}{lcl}
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N & \, d \, & \, \#\Phi_{A_f,p}\, \\
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76 & 1 & ?,1^*\\
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77 & 1 & 2^*,1^*\\
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& 1 & 3^*,2\\
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& 1 & 6,3^*\\
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& 2 & 2,2^*\\
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78 & 1 & 16^*,5^*,1\\
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79 & 1 & 1^*\\
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& 5 & 13\\
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80 & 1 & ?,2\\
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& 1 & ?,2^*\\
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81 & 2 & ?\\
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82 & 1 & 2^*,1^*\\
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& 2 & 28,1^*\\
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83 & 1 & 1^*\\
359
& 6 & 41\\
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84 & 1 & ?,1^*,2^*\\
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& 1 & ?,3,2\\
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85 & 1 & 2^*,1\\
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& 2 & 2^*,1^*\\
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& 2 & 6,1^*\\
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86 & 2 & 21^*,3\\
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& 2 & 55,1^*\\
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87 & 2 & 5,1^*\\
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& 3 & 92^*,4\\
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88 & 1 & ?,1^*\\
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& 2 & ?,2^*\\
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89 & 1 & 1^*\\
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& 1 & 2\\
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& 5 & 11\\
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90 & 1 & 2^*,?,3\\
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& 1 & 6,?,1^*\\
376
& 1 & 4,?,1\\
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91 & 1 & 1^*,1^*\\
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& 1 & 1,1\\
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& 2 & 7,1^*\\
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& 3 & 4^*,8\\
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92 & 1 & ?,1^*\\
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& 1 & ?,1\\
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93 & 2 & 4^*,1^*\\
384
& 3 & 64,2^*\\
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94 & 1 & 2,1^*\\
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& 2 & 94^*,1\\
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95 & 3 & 10,2^*\\
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& 4 & 54^*,6\\
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\end{array}\quad
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\begin{array}{lcl}
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N & \, d \, & \, \#\Phi_{A_f,p}\, \\
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96 & 1 & ?,2\\
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& 1 & ?,2^*\\
394
97 & 3 & 1^*\\
395
& 4 & 8\\
396
98 & 1 & 2^*,?\\
397
& 2 & 14,?\\
398
99 & 1 & ?,1^*\\
399
& 1 & ?,1\\
400
& 1 & ?,1^*\\
401
& 1 & ?,1^*\\
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100 & 1 & ?,?\\
403
101 & 1 & 1^*\\
404
& 7 & 25\\
405
102 & 1 & 2^*,2^*,1^*\\
406
& 1 & 6^*,6,1^*\\
407
& 1 & 8,4,1\\
408
103 & 2 & 1^*\\
409
& 6 & 17\\
410
104 & 1 & ?,1^*\\
411
& 2 & ?,2\\
412
105 & 1 & 1,1,1\\
413
& 2 & 10^*,2^*,2\\
414
106 & 1 & 4^*,1^*\\
415
& 1 & 5^*,1\\
416
& 1 & 24,1^*\\
417
& 1 & 3,1^*\\
418
107 & 2 & 1^*\\
419
& 7 & 53\\
420
108 & 1 & ?,?\\
421
109 & 1 & 1\\
422
& 3 & 1^*\\
423
& 4 & 9\\
424
110 & 1 & 7^*,1^*,3\\
425
& 1 & 3,1^*,1^*\\
426
& 1 & 5,5,1\\
427
& 2 & 16^*,3,1^*\\
428
111 & 3 & 10^*,2\\
429
& 4 & 266,2^*\\
430
112 & 1 & ?,1^*\\
431
& 1 & ?,1\\
432
& 1 & ?,1^*\\
433
113 & 1 & 2\\
434
& 2 & 2\\
435
& 3 & 1^*\\
436
\end{array}\quad
437
\begin{array}{lcl}
438
N & \, d \, & \, \#\Phi_{A_f,p}\, \\
439
& 3 & 7\\
440
114 & 1 & 2^*,5^*,1\\
441
& 1 & 20,3^*,1^*\\
442
& 1 & 6,3,1\\
443
115 & 1 & 5^*,1\\
444
& 2 & 4^*,1^*\\
445
& 4 & 32,4^*\\
446
116 & 1 & ?,1^*\\
447
& 1 & ?,2^*\\
448
& 1 & ?,1^*\\
449
117 & 1 & ?,1\\
450
& 2 & ?,3\\
451
& 2 & ?,1^*\\
452
118 & 1 & 2^*,1^*\\
453
& 1 & 19^*,1\\
454
& 1 & 10,1^*\\
455
& 1 & 1,1^*\\
456
119 & 4 & 9,3^*\\
457
& 5 & 48^*,8\\
458
120 & 1 & ?,1,1^*\\
459
& 1 & ?,2,1\\
460
121 & 1 & ?\\
461
& 1 & ?\\
462
& 1 & ?\\
463
& 1 & ?\\
464
122 & 1 & 4^*,1^*\\
465
& 2 & 39^*,3\\
466
& 3 & 248,1^*\\
467
123 & 1 & 1^*,1^*\\
468
& 1 & 5,1\\
469
& 2 & 7,1^*\\
470
& 3 & 184^*,4\\
471
124 & 1 & ?,1^*\\
472
& 1 & ?,1\\
473
125 & 2 & ?\\
474
& 2 & ?\\
475
& 4 & ?\\
476
126 & 1 & 8^*,?,1^*\\
477
& 1 & 2,?,1\\
478
127 & 3 & 1^*\\
479
& 7 & 21\\
480
&&\\
481
&&\\
482
&&\\
483
\end{array}$$
484
\end{table}
485
486
487
488
\subsection{Some large component groups}
489
490
Let $\Omega_{A_f}$ be the real period of $A_f$, as defined by
491
Tate~\cite{tate:bsd} in his extension of the Birch and Swinnerton-Dyer
492
conjecture to dimension greater than one. The second author computed
493
$L(A_f,1)/\Omega_{A_f}$ for every newform~$f$ of level $N\leq 1500$.
494
The four ratios with the largest prime divisors in the numerator
495
are given in Table~\ref{table:lratios}.
496
The Birch and Swinnerton-Dyer conjecture predicts that the large
497
prime divisors of the numerators of these special values must divide
498
either the order of some $\Phi_{A_f,p}$ or of the Shafarevich-Tate
499
group of~$A_f$. In each instance this is the case.
500
501
\begin{table}
502
\label{table:lratios}
503
\begin{center}
504
\caption{Large $L(A_f,1)/\Omega_{A_f}$}
505
\end{center}
506
$$\begin{array}{ccll}
507
\qquad N \qquad\quad &
508
\quad \dim \quad &
509
\quad\quad L(A_f,1)/\Omega_{A_f} \qquad\qquad &
510
\qquad \#\Phi_{A_f,p} \\
511
1154=2\tdot 577 & 20 & 2^?\tdot85495047371/17^2
512
& 2^?\tdot 17^2 \tdot 85495047371, 2^? \\
513
1238=2\tdot 619 & 19 & 2^?\tdot 7553329019/5\tdot 31
514
& 2^?\tdot 5\tdot31\tdot7553329019 , 2^?\\
515
1322=2\tdot 661 & 21 & 2^?\tdot 57851840099/331
516
& 2^?\tdot 331 \tdot 57851840099, 2^?\\
517
1382=2\tdot 691 & 20 & 2^?\tdot 37 \tdot 1864449649 /173
518
& 2^?\tdot 37\tdot173\tdot 1864449649, 2^?\\
519
1478=2\tdot 739 & 20 &
520
2^?\tdot 7\tdot 29\tdot 1183045463 / 5\tdot 37
521
& 2^?\tdot5\tdot 7\tdot29\tdot37\tdot1183045463, 2^?\\
522
\end{array}$$
523
\end{table}
524
525
\section{Further directions}
526
527
Further considerations are needed to compute the {\em group}
528
structure of $\Phi_{A_f,p}$.
529
Once the group structure is determined, however, since the
530
action of Frobenius is known, the structure as a group scheme
531
is immediate.
532
533
An equivalence with quaternion divisor groups is not known to
534
hold for the character group at~$p$ when $p^2$ divides~$N$.
535
Thus our methods say nothing about the component group at primes
536
whose {\em square} divides the level. The free abelian group
537
on classes of nonmaximal
538
orders of index~$p$ at a ramified prime give a well-defined
539
divisor group. Do the resulting Hecke modules determine the
540
component groups for quotients of level $p^2M$?
541
542
Is it possible to define quantities as in Theorem~\ref{thm:main}
543
even when the weight of~$f$ is {\em greater than~$2$}?
544
If so, how are the resulting quantities related to the Bloch-Kato
545
Tamagawa numbers (see~\cite{bloch-kato}) of the higher weight motive
546
attached to~$f$?
547
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\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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