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Author: William A. Stein
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% compgroup.tex
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\chapter{Component groups of optimal quotients}
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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~$K$,~$\m$
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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: 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 smooth schemes~$S$ over~$\O$.
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The closed fiber~$\A_k$ is a group scheme over~$k$,
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which need not be connected; denote by~$\A_k^0$ the
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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~$k$, i.e.,
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a finite abelian group
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equipped with an action of $\Gal(\kbar/k)$.
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\label{defn:componentgroup}
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In this chapter we study the groups~$\Phi_A$
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attached to quotients~$A$ of Jacobians of modular
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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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this 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 $A=A_f$ is a quotient of~$J$ corresponding
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to a newform~$f$, so the kernel of
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the map $\pi:J\ra A$ is connected.
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There is a natural map $\pi_*:\Phi_J\ra \Phi_A$.
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We describe how to compute
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the image and the order of the cokernel of $\pi_*$.
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We now state our main result more precisely.
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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 $\cL$ 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 $\alp:X_J\ra \Hom(\cL,\Z)$.
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Let $\Phi_X$ be the cokernel of $\alp$ and
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$m_X=[\alp(X_J):\alp(\cL)]$ be the order of the finite
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group $\alp(X_J)/\alp(\cL)$. 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 discuss the
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conjectures that they suggest.
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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 \defn{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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}$$
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Because $\theta_A$ is a polarization, $|\ker(\theta_A)|$ is
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a square \cite[Theorem 13.3]{milne:abvars}. The
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\defn{congruence modulus} is the integer
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$$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.$$
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The group scheme $\cA_k^0$ is an extension of an abelian variety~$\cB$
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of dimension~$a$ by a group scheme~$\cC$; we have a diagram
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$$\[email protected]=.3cm{
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&0\ar[d]\\
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&{\cT}\ar[d]\\
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0\ar[r]&{\cC}\ar[r]\ar[d]&{\cA_k^0}\ar[r]&{\cB}\ar[r]&0\\
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&{\cU}\ar[d]\\
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&0}$$
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with~$\cT$ a torus of dimension~$t$
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and of a~$\cU$ a unipotent group of dimension~$u$.
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The abelian variety~$A$ is said to have \defn{purely toric} reduction
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if $t=\dim A$, and is \defn{semistable} if $u=0$.
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The character group $X_A = \Hom(\cT,\Gm)$\label{defn:chargroup} is a
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free abelian group of rank~$t$ contravariantly associated to~$A$.
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If~$A$ is semistable there is a monodromy pairing
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$X_A\cross X_{A'}\ra \Z$ and an exact sequence
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$$0 \ra X_{A'} \ra \Hom(X_{A},\Z) \ra \Phi_A \ra 0.$$
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\section{Rigid uniformization}
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In this section we review the rigid analytic uniformization of
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a semistable abelian variety over a finite extension~$K$ of the
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maximal unramified extension $\Qp^{\ur}$ of $\Qp$. We use this to prove
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that if~$A$ is purely toric and $\phi:A'\ra A$ is a
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symmetric isogeny, then
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$$\deg(\phi) = (\# \coker(X_A\ra X_{A'}))^2.$$
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We also prove a some lemmas about character groups.
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\subsection{Raynaud and van der Put uniformization}\label{subsec:raynaud}
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\begin{theorem}[Raynaud]\label{raynaud}
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If~$A$ is a semistable Abelian variety, its universal
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covering is isomorphic to an extension~$G$ of an abelian
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variety~$B$ with good reduction by a torus~$T$, the
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covering map from~$G$ to~$A$ is a homomorphism and
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its kernel is a twisted free Abelian group~$\Gamma$ of finite rank.
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\end{theorem}
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This may be summarized by the following diagram,
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$$\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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}$$
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which we call the \defn{uniformization cross} of~$A$.
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The group~$\Gamma$ can be identified with the character
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group $X_A$ of the previous section.
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The uniformization cross of the dual abelian variety
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$A'$ is
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$$\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, e.g., \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,~$\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{
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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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}$$
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Because~$\pi$ is optimal,~$B$ is connected so~$M$ must also be connected.
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Since~$M$ is discrete it follows that $M=0$.
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\end{proof}
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\subsubsection{Purely toric abelian varieties}
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Assume that~$A$ is purely toric. Then
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$B=0$, and the uniformization cross becomes
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$$\xymatrix{\Gamma\ar[d]\\ T\ar[d] \\ A}$$
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Let $\vphi:A'\ra A$ be a \defn{symmetric isogeny}, i.e.,
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$\vphi':A'\ra (A')'=A$ is equal to~$\vphi$.
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Denote by $\vphi_t:T'\ra T$ and $\vphi_a:\Gamma'\ra\Gamma$ the
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induced maps.
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\begin{proposition}\label{prop:kerphi}
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There is an exact sequence
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$$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.$$
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If~$J$ is a Jacobian then~$J$ is canonically self-dual so
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the monodromy pairing on~$J$
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can be viewed as a pairing $X_J\cross X_J \ra \Z$ and
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there is an exact sequence
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$$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).$$
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Thus $\Phi_E$ is cyclic of order $-\ord_p(j)$.
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\end{example}
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282
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\subsubsection{Proof of the main theorem}
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We now prove the main theorem.
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The key diagrams are
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$$\[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_*}}
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$$
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The surjectivity of $\pi_*$ was proved in Lemma~\ref{lem:surj}.
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The injectivity of $\pi^*$ follows because
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$$\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
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= \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]
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& X_J/\ker(\alp)\ar[d]^{\isom} \ar[r]
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& {\Hom(\pi^* X_A,\Z)}\ar[r] \ar[d]^{\isom} & \coker(\alp)\ar[r]\ar[d] & 0\\
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0\ar[r] & X_{A'}\ar[r] & {\Hom(X_A,\Z)}\ar[r] & {\Phi_A}\ar[r] & 0}$$
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By Lemma~\ref{lem:twokers}, the first vertical map is an isomorphism.
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The second is an isomorphism because it is induced by the
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isomorphism $\pi^*:X_A\ra \pi^* X_A$. It follows that
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$\coker(\alp)\isom \Phi_A$, as claimed.
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\end{proof}
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Let $\cL$ be the \defn{saturation} of $\pi^* X_A$ in $X_J$, i.e.,
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$[\cL:\pi^*X_A]$ is finite and $X_J/\cL$ is torsion free.
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Suppose $L$ is of finite index in $\cL$.
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Define the \defn{congruence modulus} of $L$
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$$m_L = [\alp(X_J):\alp(L)]$$
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and the \defn{component group} by
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$$\Phi_L = \coker( X_J \ra \Hom(L,\Z)).$$
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When $L=\cL$ we often set $m_X=m_\cL$ and $\Phi_X=\Phi_\cL$
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and think of $m_X$ and $\Phi_X$ as the character group
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``congruence modulus and component group.''
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\begin{lemma}\label{lem:homog}
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The rational number $\ds \frac{|\Phi_L|}{m_L}$ does not
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depend on the choice of~$L$.
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\end{lemma}
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\begin{proof}
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If $L'$ is another choice let $n=[L:L']\in\Q$.
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Then since $\alp$ is injective when restricted to $\cL$,
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$$m_{L'} = [\alp(X_J):\alp(L')]
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= [\alp(X_J):\alp(L)]\cdot[\alp(L):\alp(L')] = m_L\cdot n$$
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and similarly $|\Phi_{L'}| = |\Phi_L|\cdot n$.
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\end{proof}
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Recall that we defined
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\begin{eqnarray*}
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m_A &=& \sqrt{\deg(\theta)}\\
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\Phi_A &=& \coker(X_{A'}\ra \Hom(X_A,\Z))
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\end{eqnarray*}
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\begin{theorem}\label{formula}
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For any $L$ of finite index in $\cL$
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the following relation holds:
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$$\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_\cL.$$
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\end{proposition}
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\begin{proof}
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Because $\pi^*X_A\subset \cL \subset X_J$, by
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Lemma~\ref{lem:compphi} we obtain a commutative diagram
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with exact rows
398
$$\xymatrix{
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X_J\ar[r]\[email protected]{=}[d]& \Hom(X_J,\Z)\ar[r]\ar[d]& \Phi_J \ar[r]\ar[d]& 0\\
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X_J\ar[r]\[email protected]{=}[d]& \Hom(\cL,\Z)\ar[r]\ar[d]& \Phi_\cL \ar[r]\ar[d] & 0\\
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X_J\ar[r]& \Hom(\pi^*X_A,\Z)\ar[r]& \Phi_A \ar[r]& 0
402
}$$
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The map $\Hom(\cL,\Z)\ra\Hom(\pi^*X_A,\Z)$ is an isomorphism
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so $\Phi_\cL\ra\Phi_A$ is injective, hence
405
$$\image(\Phi_J\ra\Phi_A) \isom \image(\Phi_J\ra \Phi_\cL).$$
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The cokernel of $\Hom(X_J,\Z)\ra\Hom(\cL,\Z)$
407
surjects onto the cokernel of $\Phi_J\ra \Phi_\cL$.
408
Using the exact sequence
409
$$0\ra \cL \ra X_J \ra X_J/\cL \ra 0,$$
410
we find that
411
$$\coker(\Hom(X_J,\Z)\ra\Hom(\cL,\Z)) \subset \Ext^1(X_J/\cL,\Z)=0,$$
412
where $\Ext^1$ vanishes because $\cL$ is saturated
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so that $X_J/\cL$ is torsion free. Thus the cokernel of
414
$\Phi_J\ra\Phi_\cL$ is $0$, from which the proposition follows.
415
\end{proof}
416
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The following corollary
418
follows from Theorem~\ref{formula} and Proposition~\ref{prop:compim}.
419
\begin{corollary}\label{cor:div}
420
$$|\coker(X_J\ra X_A)| = \frac{m_A}{m_\cL}.$$
421
As a consequence, $m_\cL | m_A.$
422
\end{corollary}
423
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\section{Optimal quotients of $J_0(N)$}
425
Let $X_0(N)/\Q$ be the modular curve associated to the congruence
426
subgroup $\Gamma_0(N)\subset\sltwoz$ of matrices which are upper
427
triangular modulo $N$. Let $p$ be a prime divisor of $N$ which is
428
coprime to $M=N/p$. The Jacobian $J=J_0(N)$ of $X_0(N)$ has semistable
429
reduction at $p$. The Hecke algebra
430
$$\T=\Z[\ldots T_n\ldots]\subset\End(J)$$
431
is a commutative ring of endomorphisms of $J$ of $\Z$-rank $=\dim J$.
432
The character group $X_J$ is equipped with a
433
functorial action of $\T$.
434
The Hecke algebra $\T$ also act on the cusp
435
forms $$S = S_2(\Gamma_0(N),\C).$$
436
A newform $f$ is an eigenform normalized so that the coefficient
437
of $q$ in the expansion of $f$ at the cusp $\infty$ is $1$, and
438
such that $f$ does not occur at any level $N'\mid N$ with $N'\neq N$.
439
If $f$ is a newform, let $I_f$ be the ideal in $\T$ of
440
elements which annihilate $f$. Then $\O_f=\T/I_f$ is an
441
order in the ring of integers of the totally real number field
442
$K_f$ obtained by adjoining the Fourier coefficients of $f$ to $\Q$.
443
The quotient
444
$$A_f = J_0(N)/ I_f J_0(N)$$
445
is a purely toric optimal quotient of dimension $[K_f:\Q]$.
446
447
Let $H=H_1(X_0(N),\Z)$ be the integral homology of the
448
complex algebraic curve $X_0(N)$. Integration defines a
449
$\T$-equivariant nondegenerate
450
pairing $S \cross H \ra \C$ which we view as a map
451
$\alp: H \ra \Hom_\C(S,\C)$.
452
453
\begin{theorem}\label{Af}
454
We have the following commutative diagram of $\T$-modules:
455
$$\xymatrix{
456
H[I_f] \[email protected]{^(->}[r]\[email protected]{^(->}[d] & H \[email protected]{->>}[r]\[email protected]{^(->}[d]
457
& \alp(H)\ar[d]\[email protected]{^(->}[d]\\
458
\Hom_\C(S,\C)[I_f]\[email protected]{^(->}[r]\[email protected]{->>}[d] & \Hom_\C(S,\C)\[email protected]{->>}[r]\[email protected]{->>}[d]
459
&\Hom_\C(S[I_f],\C)\[email protected]{->>}[d]\\
460
A_f'(\C)\[email protected]{^(->}[r]\[email protected]/_2pc/_{\theta_A}[rr] & J(\C) \[email protected]{->>}[r] & A_f(\C) \\
461
}$$
462
\end{theorem}
463
\begin{proof}
464
This can be deduced from \cite{shimura:factors}.
465
\end{proof}
466
467
\begin{corollary}\label{moduluscomp}
468
$m_A^2 = [\alp(H) : \alp(H[I_f])]$.
469
\end{corollary}
470
\begin{proof}
471
Recall that $m_A$ is defined to be $\sqrt{\deg(\theta_A)}$.
472
The kernel of an isogeny of complex tori is
473
isomorphic to the cokernel of the induced map
474
on lattices. The corollary now follows from
475
the diagram of Theorem~\ref{Af}
476
which indicates that the index $[\alp(H):\alp(H[I_f])]$
477
is the cokernel of the map $H[I_f]\ra \alp(H).$
478
\end{proof}
479
480
Let $\Frob_p:X_J\ra X_J$ denote the map induced by Frobenius.
481
One has $\Frob_p=-W_p$, where $W_p$ is the map induced
482
by the Atkin-Lehner involution on $J_0(p)$.
483
Let $f$ be a newform, $A=A_f$ the corresponding optimal
484
quotient, and $w_p$ the sign of the eigenvalue of
485
$W_p$ on $f$.
486
\begin{proposition}
487
$$\Phi_A(\Fp)
488
= \begin{cases}
489
\Phi_A(\Fpbar) & \text{if $w_p=-1$},\\
490
\Phi_A(\Fpbar)[2] & \text{if $w_p=1$.}
491
\end{cases}$$
492
\end{proposition}
493
\begin{proof}
494
If $w_p=-1$, then $\Frob_p=1$ and the $\Gal(\Fpbar/\Fp)$-action
495
of $\Phi_A(\Fpbar)$ is trivial. Thus in this case,
496
$\Phi(\Fp)=\Phi(\Fpbar)$.
497
Next suppose $w_p=1$. We have an exact sequence
498
$$0\ra X_{A'} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0.$$
499
Since $W_p$ acts as $+1$ on $f$ it acts as $+1$ on
500
$A$, on $X_A$, on $\Hom(X_A,\Z)$, and on $\Phi_A$.
501
Thus $\Frob_p=-W_p$ acts as $-1$ on $\Phi_A$. The $2$-torsion
502
in a finite abelian group equals the fixed points under $-1$.
503
\end{proof}
504
505
{\bf WARNING:} When extending this result to the whole
506
of $J_0(N)$ be careful! The action of $\Frob_p=T_p$ is
507
not by $\pm 1$, even though it must be by an involution
508
of order $2$. For example, the component group of
509
$J_0(65)/\F_5$ is cyclic of order $42$. The action
510
of $\Frob_5$ is by multiplication by $-13$. Note that
511
$(-13)^2 = 1 \pmod{42}$. The fixed points of
512
multipliction by $-13$ is the order $14$ subgroup
513
generated by $3$.
514
515
\subsubsection{Computation}
516
Using the algorithms of Chapter~\ref{chap:computing},
517
we can enumerate the optimal
518
quotients $A_f$ and compute $m_A$.
519
The method of graphs \cite{mestre:graphs} and \cite{kohel:hecke}
520
can be used to compute $X=X_{J_0(N)}$ with its $\T$-action
521
and the monodromy pairing. We can then compute
522
$$\cL=\bigcap_{t\in I_f} \ker(t|_X),$$
523
$m_X:=m_\cL$, and $\Phi_X:=\Phi_\cL$.
524
By Theorem~\ref{formula} we can now compute
525
$$|\Phi_A| = |\Phi_X| \cdot \frac{m_A}{m_X}.$$
526
We have computed $\Phi_A$ in a number of cases. In the
527
next subsection we discuss two conjectures suggested by
528
our numerical computations.
529
530
\subsection{Conjectures}
531
Our numerical computations suggest the following conjectures.
532
Suppose that $N=pM$ with $(p,M)=1$.
533
Let
534
$$H_{\new} =
535
\ker\,\Bigl( H_1(X_0(N),\Z)\lra
536
(H_1(X_0(M),\Z)\oplus H_1(X_0(M),\Z)\Bigr),$$
537
where the map is induced by the two natural
538
degeneracy maps $X_0(N)\ra X_0(M)$.
539
The Hecke algebra $\T$ acts on $H_{\new}$,
540
and on the submodule $H_{\new}[I_f]$ of elements annihilated
541
by $I_f$. Integration defines a map
542
$\alp: H_{\new}\ra \Hom(S[I_f],\C)$.
543
Define the homology congruence modulus $m_H$ by
544
$$m_H^2 = [\alp(H_{\new}) : \alp(H_{\new}[I_f])].$$
545
We expect that there is a very close relationship
546
between $m_X$ and $m_H$.
547
\begin{conjecture}\label{conj:deg}
548
$m_X = m_H.$
549
\end{conjecture}
550
551
When $N=p$ is prime we make the following conjecture.
552
\begin{conjecture}\label{conj:iso}
553
Let~$p$ be a prime and let $f_1,\ldots,f_n$ be a set of
554
representatives for the Galois conjugacy classes of newforms
555
in $S_2(\Gamma_0(p))$. Let $A_1,\ldots, A_n$ be the corresponding
556
optimal quotients. Then
557
$\#A_i(\Q)=\#\Phi_{A_i}$ for each~$i$ and
558
$\#\Phi_{J_0(p)}= \prod_{i=1}^d \#\Phi_{A_i}$.
559
\end{conjecture}
560
Note, the natural map
561
$\Phi_{J_0(113)}\ra \prod_{i=1}^4 \Phi_{A_i}$
562
is not an isomorphism because two of the $\Phi_{A_i}$
563
have order two, so the product is not cyclic.
564
565
\section{Tables}
566
We computed several component groups of optimal quotients
567
$A_f$ of $J_0(N)$ associated to newforms $f$.
568
We denote such an optimal quotient by
569
\begin{center}
570
{\bf N\, isogeny-class\, dimension}
571
\end{center}
572
The dimension frequently determines the factor, so it
573
is included in the notation.
574
575
576
\subsection{Table 1: Some large component groups predicted by
577
the Birch and Swinnerton-Dyer conjecture}
578
Using the algorithm described in \cite{stein:vissha} we computed
579
the special value $L(A,1)/\Omega$ (up to a Manin constant)
580
for every optimal quotient $A=A_f$ of level $\leq 1500$.
581
We found exactly five for which the numerator of
582
$L(A,1)/\Omega$ is nonzero and divisible by a
583
prime number $>10^9$.
584
These are given below.
585
$$\begin{array}{|lcc|}\hline
586
$A$ & N & \text{\qquad $L(A,1)/\Omega\cdot \text{Manin constant}$\qquad }\\\hline
587
\text{\bf 1154E20}&2\cdot 577 & 2^?\cdot 85495047371/17^2\\
588
\text{\bf 1238G19}& 2\cdot 619 & 2^?\cdot 7553329019/5\cdot 31\\
589
\text{\bf 1322E21}& 2\cdot 661 & 2^?\cdot 57851840099/331\\
590
\text{\bf 1382D20}& 2\cdot 691 & 2^?\cdot 37 \cdot 1864449649 /173\\
591
\text{\bf 1478J20}& 2\cdot 739 & 2^?\cdot 7\cdot 29\cdot 1183045463 / 5\cdot 37\\\hline
592
\end{array}$$
593
The Birch and Swinnerton-Dyer conjecture predicts that these large
594
prime divisors must divide either $|\Phi_A|$ or
595
the Shafarevich-Tate group of $A$. We computed $\Phi_A$ and
596
found that this was the case.
597
$$\begin{array}{|lccccc|}\hline
598
A & p & w & |\Phi_X| & m_X & |\Phi_A| \\\hline
599
\text{\bf 1154E20}&2 & - & 17^2 & 2^{24}
600
& 2^?\cdot 17^2 \cdot 85495047371 \\
601
&577& + & 1 & 2^{26}\cdot85495047371
602
& 2^? \\
603
\vspace{-1ex}&&&&&\\
604
\text{\bf 1238G19}& 2 &- & 5\cdot31 & 2^{26} &2^?\cdot 5\cdot31\cdot7553329019 \\
605
& 619 &+ & 1 & 2^{28}\cdot 7553329019 & 2^? \\
606
\vspace{-1ex}&&&&&\\
607
\text{\bf 1322E21}& 2 & - & 331 & 2^{28} & 2^?\cdot 331 \cdot 57851840099\\
608
& 661& + & 1 & 2^{32}\cdot 57851840099 & 2^?\\
609
\vspace{-1ex}&&&&&\\
610
\text{\bf 1382D20}& 2 & - & 173 & 2^{29} & 2^?\cdot 37\cdot173\cdot 1864449649 \\
611
& 691 & + & 1 & 2^{31}\cdot 37\cdot 1864449649 & 2^?\\
612
\vspace{-1ex}&&&&&\\
613
\text{\bf 1478J20}& 2 & - & 5\cdot37 &2^{31}
614
& 2^?\cdot5\cdot 7\cdot29\cdot37\cdot1183045463 \\
615
& 739 & + & 1 &2^{33}\cdot 7\cdot 29\cdot 1183045463
616
& 2^? \\
617
\hline
618
\end{array}$$
619
\vfill
620
621
\subsection{Table 2: Some quotients of $J_0(N)$}
622
In this table we give the invariants defined above for
623
the optimal quotients of levels $65$, $66$, $68$, and $69$.
624
$$\begin{array}{|lccccccc|}\hline
625
A & p & w & |\Phi_X| & m_X & m_H & m_A & |\Phi_A| \\\hline
626
\text{\bf 65A1} & 5 & +& 1 &2 & ? & 2 & 1\\
627
& 13 &+& 1 & 2& ? & & 1\\
628
629
\text{\bf 65B2} & 5 &+& 3 & 2^2&? & 2^2 & 3\\
630
& 13 &- & 3 & 2^2&? & & 3\\
631
632
\text{\bf 65C2} & 5 &-& 7 & 2^2&? & 2^2 &7 \\
633
& 13 &+ & 1 & 2^2&? & & 1\\
634
635
\vspace{-1ex} & & & & & & & \\
636
\text{\bf 66A1}& 2 &+ & 1 &2 & ?& 2^2&2 \\
637
& 3 &- & 3 &2^2 & ?& & 3\\
638
& 11 &+ & 1 &2^2 & ?& &1 \\
639
640
\text{\bf 66B1}& 2 &- & 2 &2 & ?& 2^2& 2^2\\
641
& 3 &+ & 1 &2^2& ?& & 1\\
642
& 11 &+ & 1 &2^2 &? & & 1\\
643
644
\text{\bf 66C1}& 2 & -& 1 & 2&? & 2^2\cdot 5& 2\cdot5\\
645
& 3 &- & 1 & 2^2&? & & 5\\
646
& 11 &- & 1 & 2^2\cdot5&? & &1 \\
647
648
\vspace{-1ex} & & & & & & & \\
649
\text{\bf 68A2} &17&+&2 &2\cdot3 &? &2\cdot3 & 2 \\
650
651
\vspace{-1ex} & & & & & & & \\
652
\text{\bf 69A1} &3 &-&2 &2 & ?&2& 2\\
653
&23 &+& 1&2 &? & & 1\\
654
655
\text{\bf 69B2} &3 &+&2 &2 &? &2\cdot11& 2\cdot11 \\
656
&23 &-&2 &2\cdot11 &? && 2 \\
657
658
\hline
659
\end{array}$$
660
661
662
\subsection{Table 3: Some quotients of $J_0(p)$}
663
Using the method of graphs and modular symbols we computed
664
the quantities $m_A$, $m_L$ and $\Phi_L$ for each abelian
665
variety $A=A_f$ associated to a newform of prime level
666
$p\leq 757$. The results were as follows:
667
\begin{enumerate}
668
\item In all cases $m_A=m_L$, so the map $\Phi_J\ra \Phi_A$
669
is surjective.
670
\item $\Phi_A=1$ whenever the sign of the Atkin-Lehner involution
671
$w_p$ on $A$ is $1$.
672
\item $\prod |\Phi_A| = |\Phi_J|$
673
\end{enumerate}
674
Table 1 lists those $A$ of level $\leq 631$ for which $w_p=-1$, along with
675
the order of the component group.
676
677
\newpage
678
Table 3: Some quotients of $J_0(p)$~%
679
$$
680
\begin{array}{|lc}\hline
681
\vspace{-2ex}\\
682
A & |\Phi_A| \\
683
\vspace{-2ex}\\\hline
684
11\text{A}1&5\\
685
17\text{A}1&2^2\\
686
19\text{A}1&3\\
687
23\text{A}2&11\\
688
\vspace{-2ex} &\\
689
29\text{A}2&7\\
690
31\text{A}2&5\\
691
37\text{B}1&3\\
692
41\text{A}3&2\cdot5\\
693
\vspace{-2ex} &\\
694
43\text{B}2&7\\
695
47\text{A}4&23\\
696
53\text{B}3&13\\
697
59\text{A}5&29\\
698
\vspace{-2ex} &\\
699
61\text{B}3&5\\
700
67\text{A}1&1\\
701
67\text{C}2&11\\
702
71\text{A}3&5\\
703
\vspace{-2ex} &\\
704
71\text{B}3&7\\
705
73\text{A}1&2\\
706
73\text{C}2&3\\
707
79\text{B}5&13\\
708
\vspace{-2ex} &\\
709
83\text{B}6&41\\
710
89\text{B}1&2\\
711
89\text{C}5&11\\
712
97\text{B}4&2^3\\
713
\vspace{-2ex} &\\
714
101\text{B}7&5^2\\
715
103\text{B}6&17\\
716
107\text{B}7&53\\
717
109\text{A}1&1\\
718
\vspace{-2ex} &\\
719
109\text{C}4&3^2\\
720
113\text{A}1&2\\
721
113\text{B}2&2\\
722
113\text{D}3&7\\
723
\vspace{-2ex} &\\
724
127\text{B}7&3\cdot7\\
725
131\text{B}10&5\cdot13\\
726
137\text{B}7&2\cdot17\\
727
139\text{A}1&1\\
728
\vspace{-2ex} &\\
729
139\text{C}7&23\\
730
149\text{B}9&37\\
731
151\text{B}3&1\\
732
151\text{C}6&5^2\\
733
\hline\end{array}
734
\begin{array}{lc}\hline
735
\vspace{-2ex}\\
736
A & |\Phi_A| \\
737
\vspace{-2ex}\\\hline
738
157\text{B}7&13\\
739
163\text{C}7&3^3\\
740
167\text{B}12&83\\
741
173\text{B}10&43\\
742
\vspace{-2ex} &\\
743
179\text{A}1&1\\
744
179\text{C}11&89\\
745
181\text{B}9&3\cdot5\\
746
191\text{B}14&5\cdot19\\
747
\vspace{-2ex} &\\
748
193\text{C}8&2^4\\
749
197\text{C}10&7^2\\
750
199\text{A}2&1\\
751
199\text{C}10&3\cdot11\\
752
\vspace{-2ex} &\\
753
211\text{A}2&5\\
754
211\text{D}9&7\\
755
223\text{C}12&37\\
756
227\text{B}2&1\\
757
\vspace{-2ex} &\\
758
227\text{C}2&1\\
759
227\text{E}10&113\\
760
229\text{C}11&19\\
761
233\text{A}1&2\\
762
\vspace{-2ex} &\\
763
233\text{C}11&29\\
764
239\text{B}17&7\cdot17\\
765
241\text{B}12&2^2\cdot5\\
766
251\text{B}17&5^3\\
767
\vspace{-2ex} &\\
768
257\text{B}14&2^6\\
769
263\text{B}17&131\\
770
269\text{C}16&67\\
771
271\text{B}16&3^2\cdot5\\
772
\vspace{-2ex} &\\
773
277\text{B}3&1\\
774
277\text{D}9&23\\
775
281\text{B}16&2\cdot5\cdot7\\
776
283\text{B}14&47\\
777
\vspace{-2ex} &\\
778
293\text{B}16&73\\
779
307\text{A}1&1\\
780
307\text{B}1&1\\
781
307\text{C}1&1\\
782
\vspace{-2ex} &\\
783
307\text{D}1&1\\
784
307\text{E}2&3\\
785
307\text{F}9&17\\
786
311\text{B}22&5\cdot31\\
787
\hline\end{array}
788
\begin{array}{lc}\hline
789
\vspace{-2ex}\\
790
A & |\Phi_A| \\
791
\vspace{-2ex}\\\hline
792
313\text{A}2&1\\
793
313\text{C}12&2\cdot13\\
794
317\text{B}15&79\\
795
331\text{D}16&5\cdot11\\
796
\vspace{-2ex} &\\
797
337\text{B}15&2^2\cdot7\\
798
347\text{D}19&173\\
799
349\text{B}17&29\\
800
353\text{A}1&2\\
801
\vspace{-2ex} &\\
802
353\text{B}3&2\\
803
353\text{D}14&2\cdot11\\
804
359\text{D}24&179\\
805
367\text{B}19&61\\
806
\vspace{-2ex} &\\
807
373\text{C}17&31\\
808
379\text{B}18&3^2\cdot7\\
809
383\text{C}24&191\\
810
389\text{A}1&1\\
811
\vspace{-2ex} &\\
812
389\text{E}20&97\\
813
397\text{B}2&1\\
814
397\text{C}5&11\\
815
397\text{D}10&3\\
816
\vspace{-2ex} &\\
817
401\text{B}21&2^2\cdot5^2\\
818
409\text{B}20&2\cdot17\\
819
419\text{B}26&11\cdot19\\
820
421\text{B}19&5\cdot7\\
821
\vspace{-2ex} &\\
822
431\text{B}1&1\\
823
431\text{D}3&1\\
824
431\text{F}24&5\cdot43\\
825
433\text{A}1&1\\
826
\vspace{-2ex} &\\
827
433\text{B}3&1\\
828
433\text{D}16&2^2\cdot3^2\\
829
439\text{C}25&73\\
830
443\text{C}1&1\\
831
\vspace{-2ex} &\\
832
443\text{E}22&13\cdot17\\
833
449\text{B}23&2^4\cdot7\\
834
457\text{C}20&2\cdot19\\
835
461\text{D}26&5\cdot23\\
836
\vspace{-2ex} &\\
837
463\text{B}22&7\cdot11\\
838
467\text{C}26&233\\
839
479\text{B}32&239\\
840
487\text{A}2&1\\
841
\hline\end{array}
842
\begin{array}{lc|}\hline
843
\vspace{-2ex}&\\
844
A & |\Phi_A| \\
845
\vspace{-2ex}&\\\hline
846
487\text{B}2&3\\
847
487\text{C}3&1\\
848
487\text{D}16&3^3\\
849
491\text{C}29&5\cdot7^2\\
850
\vspace{-2ex} &\\
851
499\text{C}23&83\\
852
503\text{B}1&1\\
853
503\text{C}1&1\\
854
503\text{D}3&1\\
855
\vspace{-2ex} &\\
856
503\text{F}26&251\\
857
509\text{B}28&127\\
858
521\text{B}29&2\cdot5\cdot13\\
859
523\text{C}26&3\cdot29\\
860
\vspace{-2ex} &\\
861
541\text{B}24&3^2\cdot5\\
862
547\text{C}25&7\cdot13\\
863
557\text{B}1&1\\
864
557\text{D}26&139\\
865
\vspace{-2ex} &\\
866
563\text{A}1&1\\
867
563\text{E}31&281\\
868
569\text{B}31&2\cdot71\\
869
571\text{A}1&1\\
870
\vspace{-2ex} &\\
871
571\text{B}1&1\\
872
571\text{C}2&1\\
873
571\text{D}2&1\\
874
571\text{F}4&1\\
875
\vspace{-2ex} &\\
876
571\text{I}18&5\cdot19\\
877
577\text{A}2&3\\
878
577\text{B}2&1\\
879
577\text{C}3&1\\
880
\vspace{-2ex} &\\
881
577\text{D}18&2^4\\
882
587\text{C}31&293\\
883
593\text{B}1&2\\
884
593\text{C}2&1\\
885
\vspace{-2ex} &\\
886
593\text{E}27&2\cdot37\\
887
599\text{C}37&13\cdot23\\
888
601\text{B}29&2\cdot5^2\\
889
607\text{D}31&101\\
890
\vspace{-2ex} &\\
891
613\text{C}27&3\cdot17\\
892
617\text{B}28&2\cdot7\cdot11\\
893
619\text{B}30&103\\
894
631\text{B}32&3\cdot5\cdot7\\
895
\hline\end{array}
896
$$
897
898