CoCalc Public Fileswww / tables / compgroup.tex
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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$$\xymatrix{ 124 G\ar[r] & A'{->>}[r] \ar[dr]_{\pi'} 125 & C\ar[d]\\ 126 && J' 127 }\qquad \quad \xrightarrow{\textstyle \text{dualize}} \quad \qquad 128 \xymatrix{ 129 A & C\ar[l] & G'\ar[l] \\ 130 & J\ar[u]_{\vphi}\ar[ul]^{\pi} 131 }$$
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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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$$\xymatrix{ 143 A'\intersect B\ar[r]\ar[d] & B\ar[d] \\ 144 A'{^(->}[r]^{\pi'}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\ 145 & A 146 }$$
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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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$$0\ra \cU\cross \cT \ra \cA_k^0 \ra \cB \ra 0.$$
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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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$$\xymatrix{ 213 &\Gamma\ar[d] \\ 214 T\ar[r] & G\ar[r]\ar[d] & B\\ 215 & A 216 }$$
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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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$$\xymatrix{ 223 &\Gamma'\ar[d] \\ 224 T'\ar[r] & G'\ar[r]\ar[d] & B'\\ 225 & A' 226 }$$
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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
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a surjective map from $B=\ker(\pi)$ to
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$M=\coker(\Gamma_J\ra\Gamma_A)$.
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$$\xymatrix{ 257 & \Gamma_J\ar[r]\ar[d] & \Gamma_A\ar[r]\ar[d] & M\ar[r]\ar[d]& 0 \\ 258 & G_J\ar[r]\ar[d] & T_A\ar[d]\ar[r] & 0 \\ 259 B\ar[r] & J\ar[r]^{\pi}& A 260 }$$
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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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$$\xymatrix{ 281 0\ar[r]\ar[d] & \Gamma'\ar[r]\ar[d] & \Gamma\ar[r]\ar[d] 282 & \coker(\vphi_a)\ar[d]\\ 283 \ker(\vphi_t)\ar[r]\ar[d]& T'\ar[d]\ar[r] & T\ar[r]\ar[d] & 0\\ 284 \ker(\vphi)\ar[r] & A'\ar[r]^{\vphi} & A}$$
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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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$$\xymatrix{ 288 & \Gamma'\ar[r]\ar[d] & \Gamma\ar[d]\ar[r] & \coker(\vphi_a) \\ 289 \ker(\vphi_t)\ar[r] & T'\ar[r]^{\vphi} & T}$$
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is the diagram
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$$\xymatrix{ 292 & T & T'\ar[l]_{\vphi'} & \coker(\vphi_a)'\ar[l]\\ 293 \ker(\vphi_t)'& \Gamma\ar[l]\ar[u] &\Gamma'\ar[l]\ar[u] 294 }$$
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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 $J$
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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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$$=3pc{A' {^(->}[r]^{\pi'}\ar[dr]_{\theta} 337 & J {->>}[d]^{\pi}\\ 338 &A} 339 \qquad\qquad\qquad 340 =3pc{X_A {^(->}[r]^{\pi^*} \ar[dr]^{\theta^*} 341 & X_J {->>}[d]^{\pi_*} \\ 342 & X_{A'}/^1.5pc/[ul]^{\theta_*}} 343$$
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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
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$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$
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and multiplication by $\deg(\theta)$ on a free abelian
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group is injective.
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Let
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$$\alp:X_J\ra \Hom(\pi^* X_A,\Z)$$
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be the map defined by the monodromy pairing restricted
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to $X_J\cross \pi^* X_A$.
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\begin{lemma}\label{lem:twokers}
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$\ker(\pi_*) = \ker(\alp)$
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\end{lemma}
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\begin{proof}
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Suppose $x\in \ker(\pi_*)$ and let $y=\pi^* z$ with
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$z\in X_A$. Then
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$$\langle x, y \rangle = \langle x, \pi^* z \rangle 361 = \langle \pi_* x, z \rangle = 0$$
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so $x\in\ker(\alp)$.
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Next let $x\in\ker(\alp)$.
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Then for all $z\in X_A$,
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$$0 = \langle x, \pi^* z\rangle = \langle \pi_* x, z\rangle$$
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so $\pi_* x$ is in the kernel of the
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monodromy map
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$$X_{A'} \ra \Hom(X_A,\Z).$$
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Since $X_{A'}$ and $\Hom(X_A,\Z)$ are free of the same rank
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and the cokernel is torsion, the monodromy map is injective.
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Thus $\pi_* x=0$ and $x\in\ker(\pi_*)$.
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\end{proof}
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\begin{lemma}\label{lem:compphi}
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There is an exact sequence
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$$X_J \ra \Hom(\pi^* X_A,\Z) \ra \Phi_A \ra 0.$$
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\end{lemma}
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\begin{proof}
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Lemma~\ref{lem:twokers} gives the following
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commutative diagram with exact rows
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$$\xymatrix{0\ar[r] 382 & X_J/\ker(\alp)\ar[d]^{\isom} \ar[r] 383 & {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\ 384 0\ar[r] & X_{A'}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0}$$
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By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
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The second is an isomorphism because it is induced by the
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isomorphism $\pi^*:X_A\ra \pi^* X_A$. It follows that
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$\coker(\alp)\isom \Phi_A$, as claimed.
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\end{proof}
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Let $\L$ be the {\bf saturation} of $\pi^* X_A$ in $X_J$, i.e.,
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$[\L:\pi^*X_A]$ is finite and $X_J/\L$ is torsion free.
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Suppose $L$ is of finite index in $\L$.
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Define the {\bf congruence modulus} of $L$
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$$m_L = [\alp(X_J):\alp(L)]$$
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and the {\bf component group} by
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$$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
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When $L=\L$ we often set $m_X=m_\L$ and $\Phi_X=\Phi_\L$
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and think of $m_X$ and $\Phi_X$ as the character group
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congruence modulus and component group.''
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\begin{lemma}\label{lem:homog}
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The rational number $\ds \frac{|\Phi_L|}{m_L}$ does not
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depend on the choice of $L$.
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\end{lemma}
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\begin{proof}
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If $L'$ is another choice let $n=[L:L']\in\Q$.
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Then since $\alp$ is injective when restricted to $\L$,
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$$m_{L'} = [\alp(X_J):\alp(L')] 410 = [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n$$
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and similarly $|\Phi_{L'}| = |\Phi_L|\cdot n$.
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\end{proof}
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Recall that we defined
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\begin{eqnarray*}
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m_A &=& \sqrt{\deg(\theta)}\\
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\Phi_A &=& \coker(X_{A'}\ra \Hom(X_A,\Z))
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\end{eqnarray*}
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\begin{theorem}\label{formula}
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For any $L$ of finite index in $\L$
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the following relation holds:
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$$\frac{|\Phi_A|}{m_A} = \frac{|\Phi_L|}{m_L}.$$
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\end{theorem}
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\begin{proof}
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By Lemma~\ref{lem:homog} we may assume that $L=\pi^*X_A$.
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With this choice of $L$, Lemma~\ref{lem:compphi} says that
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$\Phi_L \isom \Phi_A$.
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By Lemma~\ref{lem:twokers}, properties of the index,
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and Lemma~\ref{lem:isogcoker} we have
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\begin{eqnarray*}
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m_L&=&[\alp(X_J):\alp(L)] \\
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&=& [\pi_*(X_J):\pi_*(L)]\\
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&=& [X_{A'}:\pi_*(\pi^*X_A)]\\
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&=& [X_{A'}:\theta^* X_A]\\
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&=& \coker(\theta^*) \\
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&=& \sqrt{\deg(\theta)} = m_A.
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\end{eqnarray*}
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\end{proof}
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\begin{proposition}\label{prop:compim}
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$$\image(\Phi_J\ra\Phi_A) \isom \Phi_\L.$$
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\end{proposition}
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\begin{proof}
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Because $\pi^*X_A\subset \L \subset X_J$, by
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Lemma~\ref{lem:compphi} we obtain a commutative diagram
447
with exact rows
448
$$\xymatrix{ 449 X_J\ar[r]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\ 450 X_J\ar[r]{=}[d]& \Hom(\L,\Z)\ar[r]\ar[d]& \Phi_\L \ar[r]\ar[d] & 0\\ 451 X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0 452 }$$
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The map $\Hom(\L,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism
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so $\Phi_\L\ra\Phi_A$ is injective, hence
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$$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\L).$$
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The cokernel of $\Hom(X_J,\Z)\ra\Hom(\L,\Z)$
457
surjects onto the cokernel of $\Phi_J\ra \Phi_\L$.
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Using the exact sequence
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$$0\ra \L \ra X_J \ra X_J/\L \ra 0,$$
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we find that
461
$$\coker(\Hom(X_J,\Z)\ra\Hom(\L,\Z)) \subset \Ext^1(X_J/\L,\Z)=0,$$
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where $\Ext^1$ vanishes because $\L$ is saturated
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so that $X_J/\L$ is torsion free. Thus the cokernel of
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$\Phi_J\ra\Phi_\L$ is $0$, from which the proposition follows.
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\end{proof}
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The following corollary
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follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
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\begin{corollary}\label{cor:div}
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$$|\coker(X_J\ra X_A)| = \frac{m_A}{m_\L}.$$
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
$$\xymatrix{ 506 H[I_f] {^(->}[r]{^(->}[d] & H {->>}[r]{^(->}[d] 507 & \alp(H)\ar[d]{^(->}[d]\\ 508 \Hom_\C(S,\C)[I_f]{^(->}[r]{->>}[d] & \Hom_\C(S,\C){->>}[r]{->>}[d] 509 &\Hom_\C(S[I_f],\C){->>}[d]\\ 510 A_f'(\C){^(->}[r]/_2pc/_{\theta_A}[rr] & J(\C) {->>}[r] & A_f(\C) \\ 511 }$$
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
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\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