Sharedwww / tables / ants.texOpen in CoCalc
Author: William A. Stein
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\documentclass[11pt]{llncs}
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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{\C}{\mathbf{C}}
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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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\usepackage[all]{xy}
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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~$A_f$
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be the corresponding optimal abelian variety quotient of $J_0(N)$.
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We describe an algorithm to compute the order of the component group
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of $A_f$ 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 five
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examples in which the component group is very large, as predicted
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by 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 $K_f$.
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The kernel $I_f$ of the map $\T \rta R_f$ sending $T_n$ to~$a_n$
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is a prime ideal.
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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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for elliptic curves over $\Q$, were greatly influenced by
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computations.
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The scale of these computations was extended and systematized
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by 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
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to establish Fermat's Last Theorem. The full Shimura-Taniyama
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conjecture was later proved by Breuil, Conrad, Diamond, and Taylor
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in~\cite{breuil-conrad-diamond-taylor}.
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This illustrates the central role 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
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definition a smooth commutative group scheme over~$\Z$ with
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generic fiber~$A$ such that for any smooth scheme~$S$ over~$\Z$,
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the restriction map $$\Hom_\Z(S,\cA) \rta \Hom_\Q(S_\Q,A)$$ is
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a bijection. For more details, including a proof of existence,
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see, e.g.,~\cite{neronmodels}.
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Suppose that $A_f$ is a quotient of $J_0(N)$ corresponding to a
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newform~$f$ on $\Gamma_0(N)$,
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and let $\cA_f$ be the N\'eron model of~$A_f$.
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For any prime divisor~$p$ of~$N$, the closed fiber~${\cA_f}_{/\Fp}$
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is a group scheme over~$\Fp$, which need not be connected.
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Denote the connected component of the identity
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by~${\cA^{\circ}_f}_{/\Fp}$. There is an exact sequence
112
$$
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0 \rightarrow {\cA^{\circ}_f}_{/\Fp}
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\rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p}
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\rightarrow 0
116
$$
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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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The category of
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finite \'etale group schemes over $\Fp$ is
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equivalent to the category of
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finite groups equipped with an action of
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$\Gal(\Fpbar/\Fp)$ (see, e.g., \cite[\S6.4]{waterhouse}).
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The {\em order} of an \'etale group scheme $G/\Fp$ is defined
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to be the order of the group $G(\Fpbar)$.
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In this paper we describe an algorithm for computing the order of
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$\Phi_{A_f,p}$, when~$p$ exactly divides~$N$.
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\section{The Algorithm}
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Let~$J = J_0(N)$, fix a newform~$f$ of weight-two for $\Gamma_0(N)$,
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and let $A_f$ be the corresponding quotient of~$J$. Because~$J$ is
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the Jacobian of a curve, it is canonically isomorphic to its dual, so
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the projection $J \rta A_f$ induces a polarization $A_f^{\vee} \rta
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A_f$, where $A_f^{\vee}$ denotes the abelian variety dual of $A_f$.
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We define the {\em modular degree} $\delta_{A_f}$ of $A_f$ to be the
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positive square root of the degree of this polarization. This agrees
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with the usual notion of modular degree when $A_f$ is an elliptic curve.
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A {\it torus} $T$ over a field $k$ is a group scheme whose base
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extension to the separable closure $k_s$ of $k$ is a finite
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product of copies of $\G_m$. Every commutative algebraic group
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over~$k$ admits a unique maximal subtorus, defined over~$k$,
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whose formation commutes with base extension (see IX \S2.1
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of~\cite{groth:sga7}). The {\it character group}
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of a torus~$T$ is the group $\cX = \Hom_{k_s}(T,\G_m)$ which is
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a free abelian group of finite rank together with an action of
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$\Gal(k_s/k)$ (see, e.g.,~\cite[\S7.3]{waterhouse}).
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We apply this construction to our setting as follows.
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The closed fiber of the N\'eron model of $J$ at~$p$ is a group
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scheme over $\Fp$, whose maximal torus we denote by $T_{J,p}$.
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We define $\cX_{J,p}$ to be the character group of $T_{J,p}$.
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Then $\cX_{J,p}$ is a free abelian group equipped with an
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action of both $\Gal(\Fpbar/\Fp)$ and the Hecke algebra~$\T$
157
(see, e.g.,~\cite{ribet:modreps}).
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Moreover, there exists a bilinear pairing
159
$$
160
\langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z
161
$$
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called the {\em monodromy pairing} such that
163
$$
164
\Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)).
165
$$
166
Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$
167
for~$t$
168
in $I_f$, and let
169
$$
170
\alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z)
171
$$
172
be the map induced by the monodromy pairing.
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The following theorem of the second author~\cite{stein:phd}, provides the
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basis for the computation of orders of component groups.
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\begin{theorem}\label{thm:main}
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With the notation as above, we have the equality
178
$$
179
\#\Phi_{A_f,p}
180
= \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}}
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{\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,.
182
$$
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\end{theorem}
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\subsection{Computing the modular degree $\delta_{A,f}$}
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Using modular symbols (see, e.g.,~\cite{cremona:algs}),
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we first compute the homology group $H_1(X_0(N),\Q;\mbox{\rm cusps})$.
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Using lattice reduction, we compute the $\Z$-submodule
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$H_1(X_0(N),\Z;\mbox{\rm cusps})$
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generated by all Manin symbols $(c,d)$. Then
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$H_1(X_0(N),\Z)$ is the
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{\em integer} kernel of the boundary map.
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The Hecke ring $\T$ acts on $H_1(X_0(N),\Z)$ and also on
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the linear dual $\Hom(H_1(X_0(N),\Z),\Z)$,
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where $t \in \T$ acts on $\varphi \in \Hom(H_1(X_0(N),\Z),\Z)$
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by $(t.\varphi)(x) = \varphi(tx)$.
198
We have a natural restriction map
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$$
200
r_f:\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow
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\Hom(H_1(X_0(N),\Z)[I_f],\Z).
202
$$
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\begin{proposition}
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The cokernel of $r_f$ is isomorphic to the kernel
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of the polarization $A_f^{\vee}\rightarrow A_f$
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induced by the map $J_0(N)\rightarrow A_f$.
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\end{proposition}
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Thus the order of the cokernel of $r_f$ is the square of the modular
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degree~$\delta_f$. We pause to note that the degree of any
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polarization is a square; see, e.g.,~\cite[Thm.~13.3]{milne:abvar}.
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\begin{proof}
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Let $S = S_2(\Gamma_0(N),\C)$ be the complex vector space of
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weight-two modular forms of level~$N$, and set $H = H_1(X_0(N),Z)$.
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The integration pairing $S\times H \rightarrow \C$
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induces a natural map
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$$\Phi_f:H\rightarrow \Hom(S[I_f],\C).$$
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Using the classical
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Abel-Jacobi theorem, we deduce the following commutative diagram,
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which has exact columns, but whose rows are not exact.
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$$\[email protected]=.5cm{
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0\ar[d] & 0\ar[d] & 0\ar[d] \\
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H[I_f]\ar[d]\ar[r] & H\ar[d]\ar[r]&\Phi_f(H)\ar[d] \\
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\Hom(S,\C)[I_f]\ar[d]\ar[r] &\Hom(S,\C)\ar[d]\ar[r] &\Hom(S[I_f],\C)\ar[d]\\
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{A_f^{\vee}(\C)}\ar[d]\ar[r]
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\[email protected]/_3.5pc/[rr]& J_0(N)(\C)\ar[d]\ar[r]& A_f(\C)\ar[d]\\
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0 & 0 & 0 \\
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}$$
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By the snake lemma,
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the kernel of $A_f^{\vee}(\C)\rightarrow A_f(\C)$ is isomorphic to the
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cokernel of the map $H[I_f] \rightarrow \Phi_f(H)$.
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Since
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$$\Hom(H/\ker(\Phi_f),\Z) \isom \Hom(H,\Z)[I_f],$$
234
the $\Hom(-,\Z)$ dual of the map $H[I_f] \rightarrow \Phi_f(H)=H/\ker(\Phi_f)$
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is $r_f$, which proves the proposition.
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\end{proof}
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\subsection{Computing the character group $\cX_{J,p}$}
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Let $N = Mp$, where $M$ and $p$ are coprime. If~$M$ is small,
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then the algorithm of Mestre and Oesterl\'e~\cite{mestre:graphs}
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can be used to compute $\cX_{J,p}$. This algorithm constructs
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the graph of isogenies between $\Fpbar$-isomorphism classes of
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pairs consisting of a supersingular elliptic curve and a cyclic
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$M$-torsion subgroup. In particular, the method is elementary
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to apply when $X_0(M)$ has genus~$0$.
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In general, the above category of ``enhanced'' supersingular
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elliptic curves can be replaced by one of left (or right) ideals
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of a quaternion order~$\cO$ of level~$M$ in the quaternion
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algebra over~$\Q$ ramified at~$p$.
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This gives an equivalent category, in which the computation of
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homomorphisms is efficient. The character group $\cX_{J,p}$ is
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known by Deligne-Rapoport~\cite{deligne-rapoport} to be canonically
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isomorphic 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.
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Moreover, this isomorphism is compatible with the operation of
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Hecke operators, which are effectively computable in $\cX(\cO)$
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in terms of ideal homomorphisms.
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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 isomorphism, 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 arithmetic of quaternions has been implemented in
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{\sc Magma}~\cite{magma} by the first author. Additional details
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and the application to Shimura curves, generalizing $X_0(N)$,
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can 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$.
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However, $\Frob_p$ acts on $\Phi_{A_f,p}$ in the same way
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as $-W_p$, where $W_p$ is the
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$p$th Atkin-Lehner involution, which can be computed using modular
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symbols. Since~$f$ is an eigenform, the involution $W_p$ acts
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as either $+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 it can be determined explicitly
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on the character group.
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\section{Tables}
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The main computational results of this work are presented below
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in two tables. The relevant algorithms have been implemented in
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{\sc Magma} and will be made part of a future release.
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They can also be obtained from the second author.
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\subsection{Component groups at low level}
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Table~\ref{tbl:lowlevel} gives the component groups of the
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quotients $A_f$ of $J_0(N)$ for $N\leq 127$.
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The column labeled $d$ contains the
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dimensions of the $A_f$,
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and the column labeled $\#\Phi_{A_f,p}$ contains a list
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of the orders of the component groups of $A_f$,
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one for each divisor~$p$ of~$N$, ordered by increasing~$p$.
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An entry of ``?'' indicates that $p^2\mid N$, so our algorithm
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does not apply.
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A component group order is starred if the $\Gal(\Fpbar/\Fp)$-action is
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nontrivial. More data along these lines can be obtained from
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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 & ?\\
334
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^*\\
348
43 & 1 & 1^*\\
349
& 2 & 7\\
350
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^*,?\\
357
& 1 & 5,?\\
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51 & 1 & 3,1^*\\
359
& 2 & 16^*,4\\
360
52 & 1 & ?,2^*\\
361
53 & 1 & 1^*\\
362
\end{array}\quad
363
\begin{array}{lcl}
364
N & \, d \, & \, \#\Phi_{A_f,p}\, \\
365
& 3 & 13\\
366
54 & 1 & 3^*,?\\
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& 1 & 3,?\\
368
55 & 1 & 2,2^*\\
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& 2 & 14^*,2\\
370
56 & 1 & ?,1\\
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& 1 & ?,1^*\\
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57 & 1 & 2^*,1^*\\
373
& 1 & 2,2^*\\
374
& 1 & 10,1^*\\
375
58 & 1 & 2^*,1^*\\
376
& 1 & 10,1^*\\
377
59 & 5 & 29\\
378
61 & 1 & 1^*\\
379
& 3 & 5\\
380
62 & 1 & 4,1^*\\
381
& 2 & 66^*,3\\
382
63 & 1 & ?,1^*\\
383
& 2 & ?,3\\
384
64 & 1 & ?\\
385
65 & 1 & 1^*,1^*\\
386
& 2 & 3^*,3\\
387
& 2 & 7,1^*\\
388
66 & 1 & 2^*,3,1^*\\
389
& 1 & 4,1^*,1^*\\
390
& 1 & 10,5,1\\
391
67 & 1 & 1\\
392
& 2 & 1^*\\
393
& 2 & 11\\
394
68 & 2 & ?,2^*\\
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69 & 1 & 2,1^*\\
396
& 2 & 22^*,2\\
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70 & 1 & 4,2^*,1^*\\
398
71 & 3 & 5\\
399
& 3 & 7\\
400
72 & 1 & ?,?\\
401
73 & 1 & 2\\
402
& 2 & 1^*\\
403
& 2 & 3\\
404
74 & 2 & 9^*,3\\
405
& 2 & 95,1^*\\
406
75 & 1 & 1^*,?\\
407
& 1 & 1,?\\
408
& 1 & 5,?\\
409
\end{array}\quad
410
\begin{array}{lcl}
411
N & \, d \, & \, \#\Phi_{A_f,p}\, \\
412
76 & 1 & ?,1^*\\
413
77 & 1 & 2^*,1^*\\
414
& 1 & 3^*,2\\
415
& 1 & 6,3^*\\
416
& 2 & 2,2^*\\
417
78 & 1 & 16^*,5^*,1\\
418
79 & 1 & 1^*\\
419
& 5 & 13\\
420
80 & 1 & ?,2\\
421
& 1 & ?,2^*\\
422
81 & 2 & ?\\
423
82 & 1 & 2^*,1^*\\
424
& 2 & 28,1^*\\
425
83 & 1 & 1^*\\
426
& 6 & 41\\
427
84 & 1 & ?,1^*,2^*\\
428
& 1 & ?,3,2\\
429
85 & 1 & 2^*,1\\
430
& 2 & 2^*,1^*\\
431
& 2 & 6,1^*\\
432
86 & 2 & 21^*,3\\
433
& 2 & 55,1^*\\
434
87 & 2 & 5,1^*\\
435
& 3 & 92^*,4\\
436
88 & 1 & ?,1^*\\
437
& 2 & ?,2^*\\
438
89 & 1 & 1^*\\
439
& 1 & 2\\
440
& 5 & 11\\
441
90 & 1 & 2^*,?,3\\
442
& 1 & 6,?,1^*\\
443
& 1 & 4,?,1\\
444
91 & 1 & 1^*,1^*\\
445
& 1 & 1,1\\
446
& 2 & 7,1^*\\
447
& 3 & 4^*,8\\
448
92 & 1 & ?,1^*\\
449
& 1 & ?,1\\
450
93 & 2 & 4^*,1^*\\
451
& 3 & 64,2^*\\
452
94 & 1 & 2,1^*\\
453
& 2 & 94^*,1\\
454
95 & 3 & 10,2^*\\
455
& 4 & 54^*,6\\
456
\end{array}\quad
457
\begin{array}{lcl}
458
N & \, d \, & \, \#\Phi_{A_f,p}\, \\
459
96 & 1 & ?,2\\
460
& 1 & ?,2^*\\
461
97 & 3 & 1^*\\
462
& 4 & 8\\
463
98 & 1 & 2^*,?\\
464
& 2 & 14,?\\
465
99 & 1 & ?,1^*\\
466
& 1 & ?,1\\
467
& 1 & ?,1^*\\
468
& 1 & ?,1^*\\
469
100 & 1 & ?,?\\
470
101 & 1 & 1^*\\
471
& 7 & 25\\
472
102 & 1 & 2^*,2^*,1^*\\
473
& 1 & 6^*,6,1^*\\
474
& 1 & 8,4,1\\
475
103 & 2 & 1^*\\
476
& 6 & 17\\
477
104 & 1 & ?,1^*\\
478
& 2 & ?,2\\
479
105 & 1 & 1,1,1\\
480
& 2 & 10^*,2^*,2\\
481
106 & 1 & 4^*,1^*\\
482
& 1 & 5^*,1\\
483
& 1 & 24,1^*\\
484
& 1 & 3,1^*\\
485
107 & 2 & 1^*\\
486
& 7 & 53\\
487
108 & 1 & ?,?\\
488
109 & 1 & 1\\
489
& 3 & 1^*\\
490
& 4 & 9\\
491
110 & 1 & 7^*,1^*,3\\
492
& 1 & 3,1^*,1^*\\
493
& 1 & 5,5,1\\
494
& 2 & 16^*,3,1^*\\
495
111 & 3 & 10^*,2\\
496
& 4 & 266,2^*\\
497
112 & 1 & ?,1^*\\
498
& 1 & ?,1\\
499
& 1 & ?,1^*\\
500
113 & 1 & 2\\
501
& 2 & 2\\
502
& 3 & 1^*\\
503
\end{array}\quad
504
\begin{array}{lcl}
505
N & \, d \, & \, \#\Phi_{A_f,p}\, \\
506
& 3 & 7\\
507
114 & 1 & 2^*,5^*,1\\
508
& 1 & 20,3^*,1^*\\
509
& 1 & 6,3,1\\
510
115 & 1 & 5^*,1\\
511
& 2 & 4^*,1^*\\
512
& 4 & 32,4^*\\
513
116 & 1 & ?,1^*\\
514
& 1 & ?,2^*\\
515
& 1 & ?,1^*\\
516
117 & 1 & ?,1\\
517
& 2 & ?,3\\
518
& 2 & ?,1^*\\
519
118 & 1 & 2^*,1^*\\
520
& 1 & 19^*,1\\
521
& 1 & 10,1^*\\
522
& 1 & 1,1^*\\
523
119 & 4 & 9,3^*\\
524
& 5 & 48^*,8\\
525
120 & 1 & ?,1,1^*\\
526
& 1 & ?,2,1\\
527
121 & 1 & ?\\
528
& 1 & ?\\
529
& 1 & ?\\
530
& 1 & ?\\
531
122 & 1 & 4^*,1^*\\
532
& 2 & 39^*,3\\
533
& 3 & 248,1^*\\
534
123 & 1 & 1^*,1^*\\
535
& 1 & 5,1\\
536
& 2 & 7,1^*\\
537
& 3 & 184^*,4\\
538
124 & 1 & ?,1^*\\
539
& 1 & ?,1\\
540
125 & 2 & ?\\
541
& 2 & ?\\
542
& 4 & ?\\
543
126 & 1 & 8^*,?,1^*\\
544
& 1 & 2,?,1\\
545
127 & 3 & 1^*\\
546
& 7 & 21\\
547
&&\\
548
&&\\
549
&&\\
550
\end{array}$$
551
\end{table}
552
553
554
\subsection{Examples of large component groups}
555
556
Let $\Omega_{A_f}$ be the real period of $A_f$, as defined by
557
J.~Tate in~\cite{tate:bsd}. The second author computed
558
the rational numbers $L(A_f,1)/\Omega_{A_f}$ for every
559
newform~$f$ of level $N\leq 1500$.
560
The five largest prime divisors occur in the ratios
561
given in Table~\ref{table:lratios}.
562
The Birch and Swinnerton-Dyer conjecture predicts that the large
563
prime divisor of the numerator of each special value must
564
divide the order either of some component group $\Phi_{A_f,p}$ or of the
565
Shafarevich-Tate group of~$A_f$. In each instance
566
$\Phi_{A_f,2}$ is divisible by the large prime divisor, as
567
predicted.
568
569
\begin{table}
570
\label{table:lratios}
571
\begin{center}
572
\caption{Large $L(A_f,1)/\Omega_{A_f}$}
573
\end{center}
574
$$\begin{array}{ccll}
575
\qquad N \qquad\quad &
576
\quad \dim \quad &
577
\quad\quad L(A_f,1)/\Omega_{A_f} \qquad\qquad &
578
\qquad \#\Phi_{A_f,p} \\
579
1154=2\tdot 577 & 20 & 2^?\tdot85495047371/17^2
580
& 2^?\tdot 17^2 \tdot 85495047371, 2^? \\
581
1238=2\tdot 619 & 19 & 2^?\tdot 7553329019/5\tdot 31
582
& 2^?\tdot 5\tdot31\tdot7553329019 , 2^?\\
583
1322=2\tdot 661 & 21 & 2^?\tdot 57851840099/331
584
& 2^?\tdot 331 \tdot 57851840099, 2^?\\
585
1382=2\tdot 691 & 20 & 2^?\tdot 37 \tdot 1864449649 /173
586
& 2^?\tdot 37\tdot173\tdot 1864449649, 2^?\\
587
1478=2\tdot 739 & 20 &
588
2^?\tdot 7\tdot 29\tdot 1183045463 / 5\tdot 37
589
& 2^?\tdot5\tdot 7\tdot29\tdot37\tdot1183045463, 2^?\\
590
\end{array}$$
591
\end{table}
592
593
\section{Further directions}
594
595
Further considerations are needed to compute the {\em group}
596
structure of $\Phi_{A_f,p}$. However, since the action of Frobenius
597
is known, computing the group structure of $\Phi_{A_f,p}$ suffices
598
to determine its structure as a group scheme.
599
600
%An equivalence with quaternion divisor groups is not known to
601
%hold for the character group at~$p$ when $p^2$ divides~$N$.
602
%Thus
603
Our methods say nothing about the component group at primes
604
whose {\em square} divides the level. The free abelian group
605
on classes of nonmaximal orders of index~$p$ at a ramified prime
606
gives a well-defined divisor group.
607
Do the resulting Hecke modules determine the component groups
608
for quotients of level $p^2M$?
609
610
Is it possible to define quantities as in Theorem~\ref{thm:main}
611
even when the weight of~$f$ is {\em greater than~$2$}?
612
If so, how are the resulting quantities related to the Bloch-Kato
613
Tamagawa numbers (see~\cite{bloch-kato}) of the higher weight
614
motive attached to~$f$?
615
616
\newpage
617
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
618
\begin{thebibliography}{1}
619
620
\bibitem{birch-swd-I}
621
B.\thinspace{}J.~Birch, and
622
H.\thinspace{}P.\thinspace{}F.~Swinnerton-Dyer,
623
\emph{Notes on elliptic curves, I}, J.~Reine Angew. Math.
624
{\bf 212} (1963), 7--25.
625
626
\bibitem{birch-swd-II}
627
B.\thinspace{}J.~Birch and
628
H.\thinspace{}P.\thinspace{}F.~Swinnerton-Dyer,
629
\emph{Notes on elliptic curves, II}, J.~Reine Angew. Math.
630
{\bf 218} (1965), 79--108.
631
632
\bibitem{bloch-kato}
633
S.~Bloch and K.~Kato, \emph{\protect{${L}$}-functions and
634
\protect{T}amagawa numbers of motives}, The Grothendieck
635
Festschrift, Vol. \protect{I}, Birkh\"auser Boston, Boston,
636
MA, 1990, 333--400.
637
638
\bibitem{breuil-conrad-diamond-taylor}
639
C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the
640
modularity of elliptic curves over \protect{$\Q$}}, in preparation.
641
642
\bibitem{neronmodels}
643
S.~Bosch, W.~L{\"u}tkebohmert, and M.~Raynaud, \emph{N\'eron models},
644
Springer-Verlag, Berlin, 1990.
645
646
\bibitem{cremona:algs}
647
J.\thinspace{}E. Cremona, \emph{Algorithms for modular elliptic curves},
648
second~ed., Cambridge University Press, Cambridge, 1997.
649
650
\bibitem{deligne-rapoport}
651
P.~Deligne and M.~Rapoport, \emph{Les sch\'emas de modules de courbes
652
elliptiques}, In P.~Deligne and W.~Kuyk, eds.,
653
\emph{Modular functions of one variable,
654
Vol.~\protect{II}}, Lecture Notes in Math., {\bf 349},
655
Springer, Berlin, 1973, 143--316.
656
657
\bibitem{diamond-im}
658
F.~Diamond and J.~Im, \emph{Modular forms and modular curves},
659
In V.\thinspace{}K.~Murty,~ed., \emph{Seminar on {F}ermat's
660
{L}ast {T}heorem}, Amer. Math. Soc., Providence, RI, 1995,
661
\hbox{39--133}.
662
663
\bibitem{groth:sga7}
664
A.~Grothendieck,
665
\emph{S\'eminaire de g\'eom\'etrie alg\'ebrique du Bois-Marie
666
1967--1969 (SGA 7 I)}, Lecture Notes in Mathematics, {\bf 288},
667
Springer-Verlag, Berlin-New York, 1972
668
669
\bibitem{kohel}
670
D.~Kohel, \emph{Hecke module structure of quaternions},
671
In K. Miyake, ed., \emph{Class Field Theory -- Its Centenary
672
and Prospect}, The Advanced Studies in Pure Mathematics Series,
673
Math Soc. Japan, to appear.
674
675
\bibitem{magma}
676
W.~Bosma, J.~Cannon, and C.~Playoust.
677
\emph{The {M}agma algebra system {I}: {T}he user language},
678
J. Symb. Comp., {\bf 24} (1997), no. 3-4, 235--265.
679
680
\bibitem{mestre:graphs}
681
J.-F.~Mestre, \emph{La m\'ethode des graphes. \protect{E}xemples
682
et applications}, In \emph{Proceedings of the international
683
conference on class numbers and fundamental units of algebraic
684
number fields}, Nagoya University, Nagoya, 1986, 217--242.
685
686
\bibitem{milne:abvar}
687
J.\thinspace{}S.~Milne, \emph{Abelian Varieties},
688
In G.~Cornell and J.~Silverman,~eds., \emph{Arithmetic geometry},
689
Springer, New York, 1986, 103--150,
690
691
\bibitem{ribet:modreps}
692
K.\thinspace{}A.~Ribet, \emph{On modular representations of
693
\protect{${\rm {G}al}(\overline{\bf{Q}}/{\bf {Q}})$} arising
694
from modular forms}, Invent. Math. \textbf{100} (1990), no.~2,
695
431--476.
696
697
\bibitem{shimura:factors}
698
G.~Shimura, \emph{On the factors of the jacobian variety of a
699
modular function field}, J. Math. Soc. Japan \textbf{25} (1973),
700
no.~3, 523--544.
701
702
\bibitem{stein:phd}
703
W.\thinspace{}A. Stein, \emph{Explicit approaches to modular abelian
704
varieties}, Ph.D. thesis, University of California, Berkeley, 2000.
705
706
\bibitem{tate:bsd}
707
J.~Tate, \emph{On the conjectures of {B}irch and {S}winnerton-{D}yer
708
and a geometric analog}, S\'eminaire Bourbaki, Vol.\ 9, Soc. Math.
709
France, Paris, 1995, Exp.\ No.\ 306, 415--440.
710
711
\bibitem{taylor-wiles}
712
R.~Taylor and A.~Wiles, \emph{Ring-theoretic properties of certain
713
Hecke algebras}, Ann. of Math. {\bf 141} (1995), no.~3, 553--572.
714
715
\bibitem{waterhouse}
716
W.\thinspace{}C.~Waterhouse, \emph{Introduction to affine group
717
schemes}, Graduate Texts in Mathematics, {\bf 66},
718
Springer-Verlag, New York-Berlin, 1979
719
720
\bibitem{wiles:fermat}
721
A.~Wiles, \emph{Modular elliptic curves and Fermat's last theorem},
722
Ann. of Math. {\bf 141} (1995), no.~3, 443--551.
723
724
\end{thebibliography}
725
726
\end{document}
727
728
729
Oh, and the tables now get pushed into the middle of
730
the references. If you have a quick and dirty way
731
of fixing this, then please do so as I find it ugly.
732
Hmmm, I played slightly with this to no success, but
733
also noted that I get no table numbers, at least as
734
it compiles on the system here. Could you look at
735
this? Sorry for leaving you this dirty work.
736
737
-----------------------------------------------------------------
738
739
Typographical errors or syntax:
740
741
- page 1: "-" between One and dimensional?
742
- Sec 2 line 1/2: mistaken \par inserted?
743
- Sec 3.1 before display: mistaken \par inserted?
744
- sentence on pagebreak 3/4: do you mean canonically isomorphic
745
instead of canonically equivalent? That would help to interpret the
746
next sentence.
747
- page 4 isomophism (maybe spell check the whole document?)
748
- Sec 3.3 last sentence: "so can" --> "so it can"
749
- Sec 4.2 "either the order of" --> "the order of either" ?
750
751
Corrected.
752
753
Comments of substance:
754
755
- last line of sec 2:
756
how can two objects in distinct categories be equivalent?
757
Maybe say that GIVING one is equivalent to GIVING the other.
758
759
We correct this by saying that the (implicit) parent categories
760
are equivalent rather than two objects in them.
761
762
- Section 3: can you explain or give a reference for what the
763
"toric part" is. The average ANTS reader would appreciate it.
764
It is not a subgroup or a quotient, but a subquotient, right?
765
766
We add a short paragraph to define a torus and the existence
767
of a maximal torus, with references.
768
769
- Sec 3.1 last sentence: You view abelian varieties as
770
complex lattices? That sounds like a stretch--or do you have
771
some equivalence in mind. Maybe view their fundamental groups
772
as complex lattices?
773
774
We rewrote this section for clarity, and added a proof of the
775
formula for the modular degree.
776
777
-----------------------------------------------------------------
778