Sharedwww / Tables / compgroup.texOpen in CoCalc
Author: William A. Stein
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\begin{document}
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\par\noindent
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{Preprint (\today), Version 0.5}
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\vspace{15ex}
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\par\noindent
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{\bf \LARGE Component groups of optimal quotients\\
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of Jacobians}\\
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\vspace{3ex}
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\par\noindent
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{\large W.A. Stein}\\
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{\small Department of Mathematics, University of California, Berkeley,
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CA 94720, USA}
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\section*{Introduction}
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Let $A$ be an abelian variety over a finite extension $K$ of the
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$p$-adic numbers $\Qp$. Let $\O$ be the ring of integers of $K$,
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$\m$ its maximal ideal and $k=\O/\m$ the residue class field.
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The N\'{e}ron model of $A$ is a smooth commutative group scheme $\A$
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over $\O$ such that $A$ is its generic fiber and satisfying the
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property:\\
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the restriction map
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$$\Hom_\O(S,\A)\lra \Hom_K(S/K,A)$$
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is bijective for all schemes $S$ over $\O$.
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The special fiber $\A_k$ is a group scheme over $k$,
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which need not be connected. Denote by
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$\A_k^0$ the connected component containing the identity.
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There is an exact sequence
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$$0\lra\A_k^0\lra\A_k\lra\Phi_A\lra0$$
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with $\Phi_A$ a finite \'{e}tale group scheme over $k$,
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i.e., a finite abelian group equipped with an action of
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$\Gal(\kbar/k)$.
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In this paper we study the group $\Phi_A$
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with particular emphasis on quotients $A$ of Jacobians
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of modular curves $X_0(N)$. When $A$ has semistable reduction
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Grothendieck described the component group in terms
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of a monodromy pairing on certain free abelian groups.
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When $A=J$ is the Jacobian of $X_0(N)$, this pairing
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can be explicitly computed, hence so can $\Phi_J$,
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as has been done in many cases in
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\cite{mazur:eisenstein} and \cite{edixhoven:eisen}.
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Suppose now that $A$ is a simple quotient of $J$ and
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that the kernel of the map $J\ra A$ is connected.
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There is a natural map $\Phi_J\ra \Phi_A$. In this
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paper we give a formula which can be used to compute
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the image and the order of the cokernel.
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We now state our main result in more precise language.
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Suppose $\pi:J\ra A$ is an optimal quotient,
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with $J$ a semistable Jacobian and $A$ purely toric.
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We express the component group of $A$ in terms of
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the monodromy pairing associated to $J$.
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Let $m_A=\sqrt{\deg(\theta_A)}$ where
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$\theta_A:A'\ra A$ is induced by
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the canonical principal polarization of $J$.
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Let $X_J$ be the character group of the toric part
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of the special fiber of $J$. Let $\L$ be the saturation
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of the image of $X_A$ in $X_J$. The monodromy pairing
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defines a map $\alp:X_J\ra \Hom(\L,\Z)$.
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Let $\Phi_X$ be the cokernel of $\alp$ and
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$m_X=[\alp(X_J):\alp(\L)]$ be the order of the finite
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group $\alp(X_J)/\alp(\L)$. We prove that
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$$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_X|}{m_X}.$$
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More precisely, $\Phi_X$ is the image of $\Phi_J \ra \Phi_A$
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and the cokernel of $\Phi_J \ra \Phi_A$ has order $m_A/m_X$.
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If the optimal quotient $J\ra A$ arises from a modular
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form on $\Gamma_0(N)$,
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then the quantities $m_A$, $m_X$ and $\Phi_X$ can
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be explicitly computed, hence so can $|\Phi_A|$.
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Having done this, we present some tables and conjectures
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which they suggest.
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{\bf Acknowledgement: }
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I am deeply grateful for conversations with
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A. Agashe, R. Coleman, B. Edixhoven, D. Lorenzini,
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B. Mazur, L. Merel, K. Ribet, and S. Takahashi.
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\section{Optimal quotients of jacobians}
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Let $J$ be a Jacobian equipped with its canonical principal
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polarization $\theta_J$.
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An {\bf optimal quotient} of $J$ is an
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abelian variety $A$ and a surjective
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map $\pi: J \ra A$ whose
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kernel is an abelian subvariety $B$ of $J$.
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Denote by $J'$ and $A'$ the abelian varieties dual to $J$ and $A$,
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respectively.
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Dualizing $\pi$ and composing with $\theta_J'=\theta_J$
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we obtain a map
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$A'\xrightarrow{\pi'} J'\xrightarrow{\theta_J} J$.
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\begin{proposition}
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$A'\ra J$ is injective.
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\end{proposition}
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\begin{proof}
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Since $\theta_J$ is an isomorphism it suffices to prove
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that $\pi'$ is injective.
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Since the dual of $\pi'$ is $(\pi')'=\pi$ and $\pi$ is surjective,
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$\pi'$ must have finite kernel.
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Thus $A' \ra C=\im(\pi')$ is
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an isogeny. Let $G$ denote the kernel,
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and dualize. By \cite[\S11]{milne:abvars} we have have
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$$\xymatrix{
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G\ar[r] & A'\[email protected]{->>}[r] \ar[dr]_{\pi'}
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& C\ar[d]\\
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&& J'
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}\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad
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\xymatrix{
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A & C\ar[l] & G'\ar[l] \\
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& J\ar[u]_{\vphi}\ar[ul]^{\pi}
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}$$
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with $G'$ the Cartier dual of $G$.
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Since $G'$ is finite, $\ker(\vphi)\subset\ker(\pi)$
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is of finite index.
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Since $\ker(\pi)$ is an abelian variety it is divisible.
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Thus $\ker(\vphi)=\ker(\pi)$ so $G'=0$ and hence $G=0$.
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\end{proof}
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We denote the map $A'\ra J$ by $\pi'$.
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The kernel of $\theta_A$ measures the intersection of
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$A'$ and $B=\ker(\theta_A)$ inside of $J$
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as shown in the following diagram.
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$$\xymatrix{
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A'\intersect B\ar[r]\ar[d] & B\ar[d] \\
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A'\[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\
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& A
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}$$
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Because $\theta_A$ is a polarization, $|\ker(\theta_A)|$ is
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a square \cite[Theorem 13.3]{milne:abvars}. The
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{\bf congruence modulus} is the integer
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$$m_A=\sqrt{|\ker(\theta_A)|}.$$
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\section{The special fiber of the N\'{e}ron model}
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Let $K$ be a finite extension of $\Qp$ with ring of integers $\O$
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and residue class field $k$.
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Let $A/K$ be an abelian variety and denote its N\'{e}ron model
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by $\cA$.
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Let $\Phi_A$ be the group of connected components of
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the special fiber $\cA_k$. This group
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is a finite \'{e}tale group scheme over $k$, i.e.,
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a finite abelian group equipped with an action of
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$\Gal(\kbar/k)$. There is an exact sequence of group schemes
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$$0 \ra \cA_k^0 \ra \cA_k \ra \Phi_A \ra 0.$$
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The group $\cA_k^0$ is an extension of an abelian variety
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$\cB$ of dimension $a$ by the product of a torus $\cT$ of
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dimension $t$ and of a unipotent group $\cU$ of dimension $u$:
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$$0\ra \cU\cross \cT \ra \cA_k^0 \ra \cB \ra 0.$$
168
The abelian variety $A$ is said to have {\bf purely toric} reduction
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if $t=\dim A$, and is {\bf semistable} if $u=0$.
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The character group $X_A = \Hom(\cT,\Gm)$ is a free abelian
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group of rank $t$ contravariantly associated to $A$.
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If $A$ is semistable there is a monodromy pairing
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$X_A\cross X_{A'}\ra \Z$ and an exact sequence
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$$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
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\comment{
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Suppose again that $J\ra A$ is a symmetric optimal quotient, that $J$
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has semistable reduction and $A$ has purely toric reduction.
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Since $J$ is canonically self-dual the monodromy pairing
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defines a map $X=X_J \ra \Hom(X,\Z)$. By functoriality there
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is a map $X\ra X_{A'}$.
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Let $\alpha:X\ra \Hom(X_{A'},\Z)$ be the resulting map.
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Let $L \subset X$ be the saturation of the image of $X_{A}$
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in $X$, i.e.,the image of $X_{A}$ in $L$ has finite index
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and $X/L$ is torsion free. The {\bf character group congruence modulus}
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of the optimal quotient $J\ra A$ is the integer
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$$m_x=[\alpha(X):\alpha(L)].$$
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We now state our main result.
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$$|\Phi_A| = |\image(\Phi_J\ra\Phi_A)|\cdot \frac{m_\theta}{m_x}$$
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This is an expression for the cokernel of the map $\Phi_J\ra \Phi_A$
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as the quotient of two congruence moduli.}
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\section{Rigid uniformization}
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In this section we review the rigid analytic uniformization of
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a semistable abelian variety over a finite extension $K$ of the
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maximal unramified extension $\Qp^{\ur}$ of $\Qp$. We use this
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to prove that if $A$ is purely toric, and $\phi:A'\ra A$ is an
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isogeny, then
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$$\deg(\phi) = |\coker(X_A\ra X_{A'})|^2.$$
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We also prove a few lemmas about the character groups $X_A$.
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\subsection{Raynaud-van der Put uniformization}\label{subsec:raynaud}
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\begin{theorem}[Raynaud, van der Put]\label{raynaud}
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If $A$ is a semistable Abelian variety, its universal
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covering is isomorphic to an extension $G$ of an abelian
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variety $B$ with good reduction by a torus $T$, the
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covering map from $G$ to $A$ is a homomorphism and
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its kernel is a twisted free Abelian group $\Gamma$ of finite rank.
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\end{theorem}
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This may be summarized by the following diagram,
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$$\xymatrix{
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&\Gamma\ar[d] \\
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T\ar[r] & G\ar[r]\ar[d] & B\\
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& A
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}$$
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which we call the {\bf uniformization cross} of $A$.
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The group $\Gamma$ can be identified with the character
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group $X_A$ of the previous section.
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The uniformization cross of the dual abelian variety
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$A'$ is
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$$\xymatrix{
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&\Gamma'\ar[d] \\
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T'\ar[r] & G'\ar[r]\ar[d] & B'\\
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& A'
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}$$
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where $\Gamma'=\Hom(T,\Gm)$ and $T'=\Hom(\Gamma,\Gm)$
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and the morphisms $\Gamma'\ra G'$ and $T'\ra G'$ are the
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one-motive duals of the morphisms $T\ra G$ and $\Gamma\ra G$,
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respectively.
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For more details see \cite{coleman:monodromy}.
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\begin{example}[Tate curve]
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If $E/\Qp$ is an elliptic curve with multiplicative reduction
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then the uniformization is $E=\Gm/q^\Z$ where $q=q(j)$ is
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obtained by inverting the expression for $j$ as a function of
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$q(z)=e^{2\pi iz}$.
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\end{example}
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\subsection{Some lemmas}
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Let $\pi:J\ra A$ be an optimal quotient,
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with $J$ semistable and $A$ purely toric.
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\begin{lemma}\label{lem:surj}
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The map $\Gamma_J\ra \Gamma_A$ induced by $\pi$ is surjective.
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\end{lemma}
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\begin{proof}
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Because $G_J$ is simply connected,
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$\pi$ induces a map
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$G_J\ra T_A$ and a map $\Gamma_J\ra \Gamma_A$.
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Because $\pi$ is surjective and $T(A)$ is a
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torus, the map $G_J\ra T_A$ is surjective.
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The snake lemma applied to the following diagram gives
254
a surjective map from $B=\ker(\pi)$ to
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$M=\coker(\Gamma_J\ra\Gamma_A)$.
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$$\xymatrix{
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& \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\
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& G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\
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B\ar[r] & J\ar[r]^{\pi}& A
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}$$
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Because $\pi$ is optimal, $B$ is connected so $M$ must also be connected.
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Since $M$ is discrete it follows that $M=0$.
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\end{proof}
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\subsection{Purely toric abelian varieties}
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Assume that $A$ is purely toric. Then
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$B=0$, and the uniformization cross becomes
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$$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A}$$
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Let $\vphi:A'\ra A$ be a {\bf symmetric isogeny}, i.e.,
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$\vphi':A'\ra (A')'=A$ is equal to $\vphi$.
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Denote by $\vphi_t:T'\ra T$ and $\vphi_a:\Gamma'\ra\Gamma$ the
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induced maps.
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\begin{proposition}\label{prop:kerphi}
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There is an exact sequence
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$$0\ra\ker(\vphi_t) \ra \ker(\vphi) \ra \coker(\vphi_a)\ra 0,$$
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and $\ker(\vphi_t)$ is dual to $\coker(\vphi_a)$.
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\end{proposition}
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\begin{proof}
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Since $\vphi$ is an isogeny we have the following diagram:
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$$\xymatrix{
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0\ar[r]\ar[d] & \Gamma'\ar[r]\ar[d] & \Gamma\ar[r]\ar[d]
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& \coker(\vphi_a)\ar[d]\\
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\ker(\vphi_t)\ar[r]\ar[d]& T'\ar[d]\ar[r] & T\ar[r]\ar[d] & 0\\
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\ker(\vphi)\ar[r] & A'\ar[r]^{\vphi} & A}$$
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The snake lemma then gives the claimed exact sequence.
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For the second assertion observe that the one-motive dual of the diagram
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$$\xymatrix{
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& \Gamma'\ar[r]\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\
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\ker(\vphi_t)\ar[r] & T'\ar[r]^{\vphi} & T}$$
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is the diagram
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$$\xymatrix{
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& T & T'\ar[l]_{\vphi'} & \coker(\vphi_a)'\ar[l]\\
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\ker(\vphi_t)'& \Gamma\ar[l]\ar[u] &\Gamma'\ar[l]\ar[u]
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}$$
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Since $\vphi$ is symmetric, $\vphi'=\vphi$ and so
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$$\ker(\vphi_t) = \coker(\vphi_a)'.$$
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\end{proof}
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\begin{lemma}\label{lem:isogcoker}
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$|\ker(\vphi)|=|\coker(\vphi_a)|^2$
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\end{lemma}
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\begin{proof}
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The order of a finite group scheme equals the order of its
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dual.
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\end{proof}
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\section{The main theorem}
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Let $\pi:J\ra A$ be an optimal quotient,
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with $J$ a semistable Jacobian and $A$ purely toric.
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Let $X_A$, $X_{A'}$, and $X_J$ denote the
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character groups of the toric parts of the
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special fibers.
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\subsection{Monodromy description of the component group}
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There is a pairing
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$X_A\cross X_{A'}\ra \Z$ called
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the monodromy pairing. We have an exact sequence
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$$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
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If $J$ is a Jacobian then $J$ is canonically self-dual so
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the monodromy pairing on $J$
321
can be viewed as a pairing $X_J\cross X_J \ra \Z$ and
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there is an exact sequence
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$$0 \ra X_{J} \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0.$$
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\begin{example}[Tate curve]
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Suppose $E=\Gm/q^{\Z}$ is a Tate curve.
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The monodromy pairing on $X_E=q^{\Z}$ is
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$$\langle q, q\rangle = \ord_p(q)=-\ord_p(j).$$
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Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
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\end{example}
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\subsection{Proof of the main theorem}
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We now prove the main theorem.
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The key diagrams are
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$$\[email protected]=3pc{A' \[email protected]{^(->}[r]^{\pi'}\ar[dr]_{\theta}
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& J \[email protected]{->>}[d]^{\pi}\\
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&A}
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\qquad\qquad\qquad
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\[email protected]=3pc{X_A \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*}
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& X_J \[email protected]{->>}[d]^{\pi_*} \\
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& X_{A'}\[email protected]/^1.5pc/[ul]^{\theta_*}}
343
$$
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The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
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The injectivity of $\pi^*$ follows because
346
$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
347
and multiplication by $\deg(\theta)$ on a free abelian
348
group is injective.
349
350
Let
351
$$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
352
be the map defined by the monodromy pairing restricted
353
to $X_J\cross \pi^* X_A$.
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\begin{lemma}\label{lem:twokers}
355
$\ker(\pi_*) = \ker(\alp)$
356
\end{lemma}
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\begin{proof}
358
Suppose $x\in \ker(\pi_*)$ and let $y=\pi^* z$ with
359
$z\in X_A$. Then
360
$$\langle x, y \rangle = \langle x, \pi^* z \rangle
361
= \langle \pi_* x, z \rangle = 0$$
362
so $x\in\ker(\alp)$.
363
Next let $x\in\ker(\alp)$.
364
Then for all $z\in X_A$,
365
$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle $$
366
so $\pi_* x$ is in the kernel of the
367
monodromy map
368
$$X_{A'} \ra \Hom(X_A,\Z).$$
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Since $X_{A'}$ and $\Hom(X_A,\Z)$ are free of the same rank
370
and the cokernel is torsion, the monodromy map is injective.
371
Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
372
\end{proof}
373
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\begin{lemma}\label{lem:compphi}
375
There is an exact sequence
376
$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
377
\end{lemma}
378
\begin{proof}
379
Lemma~\ref{lem:twokers} gives the following
380
commutative diagram with exact rows
381
$$\xymatrix{0\ar[r]
382
& X_J/\ker(\alp)\ar[d]^{\isom} \ar[r]
383
& {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\
384
0\ar[r] & X_{A'}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0}$$
385
By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
386
The second is an isomorphism because it is induced by the
387
isomorphism $\pi^*:X_A\ra \pi^* X_A$. It follows that
388
$\coker(\alp)\isom \Phi_A$, as claimed.
389
\end{proof}
390
391
Let $\L$ be the {\bf saturation} of $\pi^* X_A$ in $X_J$, i.e.,
392
$[\L:\pi^*X_A]$ is finite and $X_J/\L$ is torsion free.
393
Suppose $L$ is of finite index in $\L$.
394
Define the {\bf congruence modulus} of $L$
395
$$m_L = [\alp(X_J):\alp(L)]$$
396
and the {\bf component group} by
397
$$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
398
When $L=\L$ we often set $m_X=m_\L$ and $\Phi_X=\Phi_\L$
399
and think of $m_X$ and $\Phi_X$ as the character group
400
``congruence modulus and component group.''
401
402
\begin{lemma}\label{lem:homog}
403
The rational number $\ds \frac{|\Phi_L|}{m_L}$ does not
404
depend on the choice of $L$.
405
\end{lemma}
406
\begin{proof}
407
If $L'$ is another choice let $n=[L:L']\in\Q$.
408
Then since $\alp$ is injective when restricted to $\L$,
409
$$m_{L'} = [\alp(X_J):\alp(L')]
410
= [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n$$
411
and similarly $|\Phi_{L'}| = |\Phi_L|\cdot n$.
412
\end{proof}
413
414
Recall that we defined
415
\begin{eqnarray*}
416
m_A &=& \sqrt{\deg(\theta)}\\
417
\Phi_A &=& \coker(X_{A'}\ra \Hom(X_A,\Z))
418
\end{eqnarray*}
419
420
\begin{theorem}\label{formula}
421
For any $L$ of finite index in $\L$
422
the following relation holds:
423
$$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_L|}{m_L}.$$
424
\end{theorem}
425
\begin{proof}
426
By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
427
With this choice of $L$, Lemma~\ref{lem:compphi} says that
428
$\Phi_L \isom \Phi_A$.
429
By Lemma~\ref{lem:twokers}, properties of the index,
430
and Lemma~\ref{lem:isogcoker} we have
431
\begin{eqnarray*}
432
m_L&=&[\alp(X_J):\alp(L)] \\
433
&=& [\pi_*(X_J):\pi_*(L)]\\
434
&=& [X_{A'}:\pi_*(\pi^*X_A)]\\
435
&=& [X_{A'}:\theta^* X_A]\\
436
&=& \coker(\theta^*) \\
437
&=& \sqrt{\deg(\theta)} = m_A.
438
\end{eqnarray*}
439
\end{proof}
440
441
\begin{proposition}\label{prop:compim}
442
$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\L.$$
443
\end{proposition}
444
\begin{proof}
445
Because $\pi^*X_A\subset \L \subset X_J$, by
446
Lemma~\ref{lem:compphi} we obtain a commutative diagram
447
with exact rows
448
$$\xymatrix{
449
X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\
450
X_J\ar[r]\[email protected]{=}[d]& \Hom(\L,\Z)\ar[r]\ar[d]& \Phi_\L \ar[r]\ar[d] & 0\\
451
X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0
452
}$$
453
The map $\Hom(\L,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism
454
so $\Phi_\L\ra\Phi_A$ is injective, hence
455
$$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\L).$$
456
The cokernel of $\Hom(X_J,\Z)\ra\Hom(\L,\Z)$
457
surjects onto the cokernel of $\Phi_J\ra \Phi_\L$.
458
Using the exact sequence
459
$$0\ra \L \ra X_J \ra X_J/\L \ra 0,$$
460
we find that
461
$$\coker(\Hom(X_J,\Z)\ra\Hom(\L,\Z)) \subset \Ext^1(X_J/\L,\Z)=0,$$
462
where $\Ext^1$ vanishes because $\L$ is saturated
463
so that $X_J/\L$ is torsion free. Thus the cokernel of
464
$\Phi_J\ra\Phi_\L$ is $0$, from which the proposition follows.
465
\end{proof}
466
467
The following corollary
468
follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
469
\begin{corollary}\label{cor:div}
470
$$|\coker(X_J\ra X_A)| = \frac{m_A}{m_\L}.$$
471
As a consequence, $m_\L | m_A.$
472
\end{corollary}
473
474
\section{Optimal quotients of $J_0(N)$}
475
Let $X_0(N)/\Q$ be the modular curve associated to the congruence
476
subgroup $\Gamma_0(N)\subset\sltwoz$ of matrices which are upper
477
triangular modulo $N$. Let $p$ be a prime divisor of $N$ which is
478
coprime to $M=N/p$. The Jacobian $J=J_0(N)$ of $X_0(N)$ has semistable
479
reduction at $p$. The Hecke algebra
480
$$\T=\Z[\ldots T_n\ldots]\subset\End(J)$$
481
is a commutative ring of endomorphisms of $J$ of $\Z$-rank $=\dim J$.
482
The character group $X_J$ is equipped with a
483
functorial action of $\T$.
484
The Hecke algebra $\T$ also act on the cusp
485
forms $$S = S_2(\Gamma_0(N),\C).$$
486
A newform $f$ is an eigenform normalized so that the coefficient
487
of $q$ in the expansion of $f$ at the cusp $\infty$ is $1$, and
488
such that $f$ does not occur at any level $N'\mid N$ with $N'\neq N$.
489
If $f$ is a newform, let $I_f$ be the ideal in $\T$ of
490
elements which annihilate $f$. Then $\O_f=\T/I_f$ is an
491
order in the ring of integers of the totally real number field
492
$K_f$ obtained by adjoining the Fourier coefficients of $f$ to $\Q$.
493
The quotient
494
$$A_f = J_0(N)/ I_f J_0(N)$$
495
is a purely toric optimal quotient of dimension $[K_f:\Q]$.
496
497
Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
498
complex algebraic curve $X_0(N)$. Integration defines a
499
$\T$-equivariant nondegenerate
500
pairing $S \cross H \ra \C$ which we view as a map
501
$\alp: H \ra \Hom_\C(S,\C)$.
502
503
\begin{theorem}\label{Af}
504
We have the following commutative diagram of $\T$-modules:
505
$$\xymatrix{
506
H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d]
507
& \alp(H)\ar[d]\[email protected]{^(->}[d]\\
508
\Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d]
509
&\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\
510
A_f'(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\
511
}$$
512
\end{theorem}
513
\begin{proof}
514
This can be deduced from \cite{shimura:factors}.
515
\end{proof}
516
517
\begin{corollary}\label{moduluscomp}
518
$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
519
\end{corollary}
520
\begin{proof}
521
Recall that $m_A$ is defined to be $\sqrt{\deg(\theta_A)}$.
522
The kernel of an isogeny of complex tori is
523
isomorphic to the cokernel of the induced map
524
on lattices. The corollary now follows from
525
the diagram of Theorem~\ref{Af}
526
which indicates that the index $[\alp(H):\alp(H[I_f])]$
527
is the cokernel of the map $H[I_f]\ra \alp(H).$
528
\end{proof}
529
530
Let $\Frob_p:X_J\ra X_J$ denote the map induced by Frobenius.
531
One has $\Frob_p=-W_p$, where $W_p$ is the map induced
532
by the Atkin-Lehner involution on $J_0(p)$.
533
Let $f$ be a newform, $A=A_f$ the corresponding optimal
534
quotient, and $w_p$ the sign of the eigenvalue of
535
$W_p$ on $f$.
536
\begin{proposition}
537
$$\Phi_A(\Fp)
538
= \begin{cases}
539
\Phi_A(\Fpbar) & \text{if $w_p=-1$},\\
540
\Phi_A(\Fpbar)[2] & \text{if $w_p=1$.}
541
\end{cases}$$
542
\end{proposition}
543
\begin{proof}
544
If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
545
of $\Phi_A(\Fpbar)$ is trivial. Thus in this case,
546
$\Phi(\Fp)=\Phi(\Fpbar)$.
547
Next suppose $w_p=1$. We have an exact sequence
548
$$0\ra X_{A'} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
549
Since $W_p$ acts as $+1$ on $f$ it acts as $+1$ on
550
$A$, on $X_A$, on $\Hom(X_A,\Z)$, and on $\Phi_A$.
551
Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$. The $2$-torsion
552
in a finite abelian group equals the fixed points under $-1$.
553
\end{proof}
554
555
\subsection{Computation}
556
Suitable generalizations of the algorithms described in
557
\cite{cremona:algs} can be used to enumerate the optimal
558
quotients $A_f$ and to compute $m_A$. These will be
559
described in the author's Berkeley
560
Ph.D. thesis \cite{stein:phd}.
561
The method of graphs \cite{mestre:graphs} and \cite{kohel:hecke}
562
can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
563
and the monodromy pairing. We can then compute
564
$$\L=\bigcap_{t\in I_f} \ker(t|_X),$$
565
$m_X:=m_\L$, and $\Phi_X:=\Phi_\L$.
566
By Theorem~\ref{formula} we can now compute
567
$$|\Phi_A| = |\Phi_X| \cdot \frac{m_A}{m_X}.$$
568
We have computed $\Phi_A$ in a number of cases. In the
569
next subsection we discuss two conjectures suggested by
570
our numerical computations.
571
572
\subsection{Conjectures}
573
Our numerical computations suggest the following conjectures.
574
Suppose that $N=pM$ with $(p,M)=1$.
575
Let
576
$$H_{\new} =
577
\ker\,\Bigl( H_1(X_0(N),\Z)\lra
578
(H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
579
where the map is induced by the two natural
580
degeneracy maps $X_0(N)\ra X_0(M)$.
581
The Hecke algebra $\T$ acts on $H_{\new}$,
582
and on the submodule $H_{\new}[I_f]$ of elements annihilated
583
by $I_f$. Integration defines a map
584
$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
585
Define the homology congruence modulus $m_H$ by
586
$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
587
We expect that there is a very close relationship
588
between $m_X$ and $m_H$.
589
\begin{conjecture}\label{conj:deg}
590
Up to powers of $2$,
591
$$m_X = m_H.$$
592
\end{conjecture}
593
594
When $N=p$ is prime we make the following conjecture.
595
\begin{conjecture}\label{conj:iso}
596
Let $p$ be a prime and let $f_1,\ldots,f_n$ be a set of
597
representatives of the Galois conjugacy classes for newforms
598
in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the corresponding
599
optimal quotients. Then the natural maps
600
\begin{eqnarray*}
601
\Phi_{J_0(p)} &\lra& \prod_{i=1}^n \Phi_{A_i}\\
602
J_0(p)(\Q) &\lra& \prod_{i=1}^n A_i(\Q)
603
\end{eqnarray*}
604
are isomorphisms.
605
\end{conjecture}
606
We offer the tables in the next section as evidence that
607
an assertion such as the above two conjectures may be true.
608
609
\section{Tables}
610
We computed several component groups of optimal quotients
611
$A_f$ of $J_0(N)$ associated to newforms $f$.
612
We denote such an optimal quotient by
613
\begin{center}
614
{\bf N\, isogeny-class\, dimension}
615
\end{center}
616
The dimension frequently determines the factor, so it
617
is included in the notation.
618
619
620
\subsection{Table 1: Some large component groups predicted by
621
the Birch and Swinnerton-Dyer conjecture}
622
Using the algorithm described in \cite{stein:vissha} we computed
623
the special value $L(A,1)/\Omega$ (up to a Manin constant)
624
for every optimal quotient $A=A_f$ of level $\leq 1500$.
625
We found exactly five for which the numerator of
626
$L(A,1)/\Omega$ is nonzero and divisible by a
627
prime number $>10^9$.
628
These are given below.
629
$$\begin{array}{|lcc|}\hline
630
$A$ & N & \text{\qquad $L(A,1)/\Omega\cdot \text{Manin constant}$\qquad }\\\hline
631
\text{\bf 1154E20}&2\cdot 577 & 2^?\cdot 85495047371/17^2\\
632
\text{\bf 1238G19}& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\
633
\text{\bf 1322E21}& 2\cdot 661 & 2^?\cdot 57851840099/331\\
634
\text{\bf 1382D20}& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\
635
\text{\bf 1478J20}& 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\\hline
636
\end{array}$$
637
The Birch and Swinnerton-Dyer conjecture predicts that these large
638
prime divisors must divide either $|\Phi_A|$ or
639
the Shafarevich-Tate group of $A$. We computed $\Phi_A$ and
640
found that this was the case.
641
642
$$\begin{array}{|lccccc|}\hline
643
A & p & w & |\Phi_X| & m_X & |\Phi_A| \\\hline
644
\text{\bf 1154E20}&2 & - & 17^2 & 2^{24}
645
& 2^?\cdot 17^2 \cdot 85495047371 \\
646
&577& + & 1 & 2^{26}\cdot85495047371
647
& 2^? \\
648
\vspace{-1ex}&&&&&\\
649
\text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\
650
& 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\
651
\vspace{-1ex}&&&&&\\
652
\text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\
653
& 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\
654
\vspace{-1ex}&&&&&\\
655
\text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\
656
& 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\
657
\vspace{-1ex}&&&&&\\
658
\text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31}
659
& 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\
660
& 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463
661
& 2^? \\
662
\hline
663
\end{array}$$
664
665
666
\subsection{Table 3: Some quotients of $J_0(N)$}
667
In this table we give the invariants defined above for
668
the optimal quotients of levels $65$, $66$, $68$, and $69$.
669
$$\begin{array}{|lccccccc|}\hline
670
A & p & w & |\Phi_X| & m_X & m_H & m_A & |\Phi_A| \\\hline
671
\text{\bf 65A1} & 5 & +& 1 &2 & ? & 2 & 1\\
672
& 13 &+& 1 & 2& ? & & 1\\
673
674
\text{\bf 65B2} & 5 &+& 3 & 2^2&? & 2^2 & 3\\
675
& 13 &- & 3 & 2^2&? & & 3\\
676
677
\text{\bf 65C2} & 5 &-& 7 & 2^2&? & 2^2 &7 \\
678
& 13 &+ & 1 & 2^2&? & & 1\\
679
680
\vspace{-1ex} & & & & & & & \\
681
\text{\bf 66A1}& 2 &+ & 1 &2 & ?& 2^2&2 \\
682
& 3 &- & 3 &2^2 & ?& & 3\\
683
& 11 &+ & 1 &2^2 & ?& &1 \\
684
685
\text{\bf 66B1}& 2 &- & 2 &2 & ?& 2^2& 2^2\\
686
& 3 &+ & 1 &2^2& ?& & 1\\
687
& 11 &+ & 1 &2^2 &? & & 1\\
688
689
\text{\bf 66C1}& 2 & -& 1 & 2&? & 2^2\cdot 5& 2\cdot5\\
690
& 3 &- & 1 & 2^2&? & & 5\\
691
& 11 &- & 1 & 2^2\cdot5&? & &1 \\
692
693
\vspace{-1ex} & & & & & & & \\
694
\text{\bf 68A2} &17&+&2 &2\cdot3 &? &2\cdot3 & 2 \\
695
696
\vspace{-1ex} & & & & & & & \\
697
\text{\bf 69A1} &3 &-&2 &2 & ?&2& 2\\
698
&23 &+& 1&2 &? & & 1\\
699
700
\text{\bf 69B2} &3 &+&2 &2 &? &2\cdot11& 2\cdot11 \\
701
&23 &-&2 &2\cdot11 &? && 2 \\
702
703
\hline
704
\end{array}$$
705
706
707
\subsection{Table 3: Some quotients of $J_0(p)$}
708
Using the method of graphs and modular symbols we computed
709
the quantities $m_A$, $m_L$ and $\Phi_L$ for each abelian
710
variety $A=A_f$ associated to a newform of prime level
711
$p\leq 757$. The results were as follows:
712
\begin{enumerate}
713
\item In all cases $m_A=m_L$, so the map $\Phi_J\ra \Phi_A$
714
is surjective.
715
\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
716
$w_p$ on $A$ is $1$.
717
\item $\prod |\Phi_A| = |\Phi_J|$
718
\end{enumerate}
719
Table 1 lists those $A$ of level $\leq 631$ for which $w_p=-1$, along with
720
the order of the component group.
721
722
\newpage
723
Table 1: Some quotients of $J_0(p)$~%
724
$$
725
\begin{array}{|lc}\hline
726
\vspace{-2ex}\\
727
A & |\Phi_A| \\
728
\vspace{-2ex}\\\hline
729
11\text{A}1&5\\
730
17\text{A}1&2^2\\
731
19\text{A}1&3\\
732
23\text{A}2&11\\
733
\vspace{-2ex} &\\
734
29\text{A}2&7\\
735
31\text{A}2&5\\
736
37\text{B}1&3\\
737
41\text{A}3&2\cdot5\\
738
\vspace{-2ex} &\\
739
43\text{B}2&7\\
740
47\text{A}4&23\\
741
53\text{B}3&13\\
742
59\text{A}5&29\\
743
\vspace{-2ex} &\\
744
61\text{B}3&5\\
745
67\text{A}1&1\\
746
67\text{C}2&11\\
747
71\text{A}3&5\\
748
\vspace{-2ex} &\\
749
71\text{B}3&7\\
750
73\text{A}1&2\\
751
73\text{C}2&3\\
752
79\text{B}5&13\\
753
\vspace{-2ex} &\\
754
83\text{B}6&41\\
755
89\text{B}1&2\\
756
89\text{C}5&11\\
757
97\text{B}4&2^3\\
758
\vspace{-2ex} &\\
759
101\text{B}7&5^2\\
760
103\text{B}6&17\\
761
107\text{B}7&53\\
762
109\text{A}1&1\\
763
\vspace{-2ex} &\\
764
109\text{C}4&3^2\\
765
113\text{A}1&2\\
766
113\text{B}2&2\\
767
113\text{D}3&7\\
768
\vspace{-2ex} &\\
769
127\text{B}7&3\cdot7\\
770
131\text{B}10&5\cdot13\\
771
137\text{B}7&2\cdot17\\
772
139\text{A}1&1\\
773
\vspace{-2ex} &\\
774
139\text{C}7&23\\
775
149\text{B}9&37\\
776
151\text{B}3&1\\
777
151\text{C}6&5^2\\
778
\hline\end{array}
779
\begin{array}{lc}\hline
780
\vspace{-2ex}\\
781
A & |\Phi_A| \\
782
\vspace{-2ex}\\\hline
783
157\text{B}7&13\\
784
163\text{C}7&3^3\\
785
167\text{B}12&83\\
786
173\text{B}10&43\\
787
\vspace{-2ex} &\\
788
179\text{A}1&1\\
789
179\text{C}11&89\\
790
181\text{B}9&3\cdot5\\
791
191\text{B}14&5\cdot19\\
792
\vspace{-2ex} &\\
793
193\text{C}8&2^4\\
794
197\text{C}10&7^2\\
795
199\text{A}2&1\\
796
199\text{C}10&3\cdot11\\
797
\vspace{-2ex} &\\
798
211\text{A}2&5\\
799
211\text{D}9&7\\
800
223\text{C}12&37\\
801
227\text{B}2&1\\
802
\vspace{-2ex} &\\
803
227\text{C}2&1\\
804
227\text{E}10&113\\
805
229\text{C}11&19\\
806
233\text{A}1&2\\
807
\vspace{-2ex} &\\
808
233\text{C}11&29\\
809
239\text{B}17&7\cdot17\\
810
241\text{B}12&2^2\cdot5\\
811
251\text{B}17&5^3\\
812
\vspace{-2ex} &\\
813
257\text{B}14&2^6\\
814
263\text{B}17&131\\
815
269\text{C}16&67\\
816
271\text{B}16&3^2\cdot5\\
817
\vspace{-2ex} &\\
818
277\text{B}3&1\\
819
277\text{D}9&23\\
820
281\text{B}16&2\cdot5\cdot7\\
821
283\text{B}14&47\\
822
\vspace{-2ex} &\\
823
293\text{B}16&73\\
824
307\text{A}1&1\\
825
307\text{B}1&1\\
826
307\text{C}1&1\\
827
\vspace{-2ex} &\\
828
307\text{D}1&1\\
829
307\text{E}2&3\\
830
307\text{F}9&17\\
831
311\text{B}22&5\cdot31\\
832
\hline\end{array}
833
\begin{array}{lc}\hline
834
\vspace{-2ex}\\
835
A & |\Phi_A| \\
836
\vspace{-2ex}\\\hline
837
313\text{A}2&1\\
838
313\text{C}12&2\cdot13\\
839
317\text{B}15&79\\
840
331\text{D}16&5\cdot11\\
841
\vspace{-2ex} &\\
842
337\text{B}15&2^2\cdot7\\
843
347\text{D}19&173\\
844
349\text{B}17&29\\
845
353\text{A}1&2\\
846
\vspace{-2ex} &\\
847
353\text{B}3&2\\
848
353\text{D}14&2\cdot11\\
849
359\text{D}24&179\\
850
367\text{B}19&61\\
851
\vspace{-2ex} &\\
852
373\text{C}17&31\\
853
379\text{B}18&3^2\cdot7\\
854
383\text{C}24&191\\
855
389\text{A}1&1\\
856
\vspace{-2ex} &\\
857
389\text{E}20&97\\
858
397\text{B}2&1\\
859
397\text{C}5&11\\
860
397\text{D}10&3\\
861
\vspace{-2ex} &\\
862
401\text{B}21&2^2\cdot5^2\\
863
409\text{B}20&2\cdot17\\
864
419\text{B}26&11\cdot19\\
865
421\text{B}19&5\cdot7\\
866
\vspace{-2ex} &\\
867
431\text{B}1&1\\
868
431\text{D}3&1\\
869
431\text{F}24&5\cdot43\\
870
433\text{A}1&1\\
871
\vspace{-2ex} &\\
872
433\text{B}3&1\\
873
433\text{D}16&2^2\cdot3^2\\
874
439\text{C}25&73\\
875
443\text{C}1&1\\
876
\vspace{-2ex} &\\
877
443\text{E}22&13\cdot17\\
878
449\text{B}23&2^4\cdot7\\
879
457\text{C}20&2\cdot19\\
880
461\text{D}26&5\cdot23\\
881
\vspace{-2ex} &\\
882
463\text{B}22&7\cdot11\\
883
467\text{C}26&233\\
884
479\text{B}32&239\\
885
487\text{A}2&1\\
886
\hline\end{array}
887
\begin{array}{lc|}\hline
888
\vspace{-2ex}&\\
889
A & |\Phi_A| \\
890
\vspace{-2ex}&\\\hline
891
487\text{B}2&3\\
892
487\text{C}3&1\\
893
487\text{D}16&3^3\\
894
491\text{C}29&5\cdot7^2\\
895
\vspace{-2ex} &\\
896
499\text{C}23&83\\
897
503\text{B}1&1\\
898
503\text{C}1&1\\
899
503\text{D}3&1\\
900
\vspace{-2ex} &\\
901
503\text{F}26&251\\
902
509\text{B}28&127\\
903
521\text{B}29&2\cdot5\cdot13\\
904
523\text{C}26&3\cdot29\\
905
\vspace{-2ex} &\\
906
541\text{B}24&3^2\cdot5\\
907
547\text{C}25&7\cdot13\\
908
557\text{B}1&1\\
909
557\text{D}26&139\\
910
\vspace{-2ex} &\\
911
563\text{A}1&1\\
912
563\text{E}31&281\\
913
569\text{B}31&2\cdot71\\
914
571\text{A}1&1\\
915
\vspace{-2ex} &\\
916
571\text{B}1&1\\
917
571\text{C}2&1\\
918
571\text{D}2&1\\
919
571\text{F}4&1\\
920
\vspace{-2ex} &\\
921
571\text{I}18&5\cdot19\\
922
577\text{A}2&3\\
923
577\text{B}2&1\\
924
577\text{C}3&1\\
925
\vspace{-2ex} &\\
926
577\text{D}18&2^4\\
927
587\text{C}31&293\\
928
593\text{B}1&2\\
929
593\text{C}2&1\\
930
\vspace{-2ex} &\\
931
593\text{E}27&2\cdot37\\
932
599\text{C}37&13\cdot23\\
933
601\text{B}29&2\cdot5^2\\
934
607\text{D}31&101\\
935
\vspace{-2ex} &\\
936
613\text{C}27&3\cdot17\\
937
617\text{B}28&2\cdot7\cdot11\\
938
619\text{B}30&103\\
939
631\text{B}32&3\cdot5\cdot7\\
940
\hline\end{array}
941
$$
942
943
944
\bibliography{biblio}
945
946
\end{document}
947
948
949
950
951
$$\begin{array}{|lcccccc|}\hline
952
A & p & w & |\Phi_X| & m_X & m_A & |\Phi_A| \\\hline
953
\text{\bf 1154E20}&2 & - & 17^2 & 2^{24}& 2^?\cdot 85495047371
954
& 2^?\cdot 17^2 \cdot 85495047371 \\
955
&577& + & 1 & 2^{26}\cdot85495047371 &
956
& 2^? \\
957
\vspace{-1ex}&&&&&&\\
958
\text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26}& 2^?\cdot7553329019 &2^?\cdot 5\cdot31\cdot7553329019 \\
959
& 619 &+ & 1 & 2^{28}\cdot 7553329019 & & 2^? \\
960
\vspace{-1ex}&&&&&&\\
961
\text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot57851840099 & 2^?\cdot 331 \cdot 57851840099\\
962
& 661& + & 1 & 2^{32}\cdot 57851840099 & & 2^?\\
963
\vspace{-1ex}&&&&&&\\
964
\text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot37\cdot1864449649 & 2^?\cdot 37\cdot173\cdot 1864449649 \\
965
& 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & & 2^?\\
966
\vspace{-1ex}&&&&&&\\
967
\text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31}
968
& 2^?\cdot7\cdot29\cdot1183045463 & 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\
969
& 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463
970
& & 2^? \\
971
\hline
972
\end{array}$$
973