Author: William A. Stein
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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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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
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$$113 0 \rightarrow {\cA^{\circ}_f}_{/\Fp} 114 \rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p} 115 \rightarrow 0 116$$
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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 $A_f^{\vee} \rta 136 A_f$, 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~$\T$
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(see, e.g.,~\cite{ribet:modreps}).
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Moreover, there exists a bilinear pairing
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$$160 \langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z 161$$
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called the {\em monodromy pairing} such that
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$$164 \Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)). 165$$
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Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$
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for~$t$
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in $I_f$, and let
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$$170 \alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z) 171$$
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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
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$$179 \#\Phi_{A_f,p} 180 = \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}} 181 {\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,. 182$$
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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 $(t.\varphi)(x) = \varphi(tx)$.
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We have a natural restriction map
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$$200 r_f:\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow 201 \Hom(H_1(X_0(N),\Z)[I_f],\Z). 202$$
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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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$$=.5cm{ 222 0\ar[d] & 0\ar[d] & 0\ar[d] \\ 223 H[I_f]\ar[d]\ar[r] & H\ar[d]\ar[r]&\Phi_f(H)\ar[d] \\ 224 \Hom(S,\C)[I_f]\ar[d]\ar[r] &\Hom(S,\C)\ar[d]\ar[r] &\Hom(S[I_f],\C)\ar[d]\\ 225 {A_f^{\vee}(\C)}\ar[d]\ar[r] 226 /_3.5pc/[rr]& J_0(N)(\C)\ar[d]\ar[r]& A_f(\C)\ar[d]\\ 227 0 & 0 & 0 \\ 228 }$$
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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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$$\Hom(H/\ker(\Phi_f),\Z) \isom \Hom(H,\Z)[I_f],$$
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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 $\Phi_{J,p} 266 \isom \coker(\cX(\cO) \rightarrow \Hom(\cX(\cO),\Z))$, 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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$$316 \begin{array}{lcl} 317 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 318 11 & 1 & 5\\ 319 14 & 1 & 6^*,3\\ 320 15 & 1 & 4^*,4\\ 321 17 & 1 & 4\\ 322 19 & 1 & 3\\ 323 20 & 1 & ?,2^*\\ 324 21 & 1 & 4,2^*\\ 325 23 & 2 & 11\\ 326 24 & 1 & ?,2^*\\ 327 26 & 1 & 3^*,3\\ 328 & 1 & 7,1^*\\ 329 27 & 1 & ?\\ 330 29 & 2 & 7\\ 331 30 & 1 & 4^*,3,1^*\\ 332 31 & 2 & 5\\ 333 32 & 1 & ?\\ 334 33 & 1 & 6^*,2\\ 335 34 & 1 & 6,1^*\\ 336 35 & 1 & 3^*,3\\ 337 & 2 & 8,4^*\\ 338 36 & 1 & ?,?\\ 339 37 & 1 & 1^*\\ 340 & 1 & 3\\ 341 38 & 1 & 9^*,3\\ 342 & 1 & 5,1^*\\ 343 39 & 1 & 2^*,2\\ 344 & 2 & 14,2^*\\ 345 40 & 1 & ?,2\\ 346 41 & 3 & 10\\ 347 42 & 1 & 8,2^*,1^*\\ 348 43 & 1 & 1^*\\ 349 & 2 & 7\\ 350 44 & 1 & ?,1^*\\ 351 45 & 1 & ?,1^*\\ 352 46 & 1 & 10^*,1\\ 353 47 & 4 & 23\\ 354 48 & 1 & ?,2\\ 355 49 & 1 & ?\\ 356 50 & 1 & 1^*,?\\ 357 & 1 & 5,?\\ 358 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^*,?\\ 367 & 1 & 3,?\\ 368 55 & 1 & 2,2^*\\ 369 & 2 & 14^*,2\\ 370 56 & 1 & ?,1\\ 371 & 1 & ?,1^*\\ 372 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^*\\ 395 69 & 1 & 2,1^*\\ 396 & 2 & 22^*,2\\ 397 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}$$
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\end{table}
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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}
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\label{table:lratios}
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\begin{center}
572
\caption{Large $L(A_f,1)/\Omega_{A_f}$}
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\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
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621
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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
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\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
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\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.
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C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the
640
modularity of elliptic curves over \protect{$\Q$}}, in preparation.
641
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\bibitem{neronmodels}
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S.~Bosch, W.~L{\"u}tkebohmert, and M.~Raynaud, \emph{N\'eron models},
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Springer-Verlag, Berlin, 1990.
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\bibitem{cremona:algs}
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J.\thinspace{}E. Cremona, \emph{Algorithms for modular elliptic curves},
648
second~ed., Cambridge University Press, Cambridge, 1997.
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651
P.~Deligne and M.~Rapoport, \emph{Les sch\'emas de modules de courbes
652
elliptiques}, In P.~Deligne and W.~Kuyk, eds.,
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Springer, Berlin, 1973, 143--316.
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F.~Diamond and J.~Im, \emph{Modular forms and modular curves},
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{L}ast {T}heorem}, Amer. Math. Soc., Providence, RI, 1995,
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\hbox{39--133}.
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A.~Grothendieck,
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670
D.~Kohel, \emph{Hecke module structure of quaternions},
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672
and Prospect}, The Advanced Studies in Pure Mathematics Series,
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Math Soc. Japan, to appear.
674
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W.~Bosma, J.~Cannon, and C.~Playoust.
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\emph{The {M}agma algebra system {I}: {T}he user language},
678
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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
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\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
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\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
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\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
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\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
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\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
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\bibitem{taylor-wiles}
712
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713
Hecke algebras}, Ann. of Math. {\bf 141} (1995), no.~3, 553--572.
714
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\bibitem{waterhouse}
716
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717
schemes}, Graduate Texts in Mathematics, {\bf 66},
718
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719
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721
A.~Wiles, \emph{Modular elliptic curves and Fermat's last theorem},
722
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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
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