Sharedwww / tables / Notes / refineogg.texOpen in CoCalc
Author: William A. Stein
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\title{\bf\Huge \mbox{A refinement of Ogg's conjecture}\vspace{5ex}}
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\author{\LARGE William A. Stein}
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\date{\Large \today \vspace{2ex}\\}
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\pagestyle{myheadings}
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\markboth{}{W.\thinspace{}A. Stein, A refinement of Ogg's conjecture}
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\S\thePagecount. #1}\vspace{1ex}}
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\newtheorem{thm}{Theorem}
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\newtheorem{prp}[thm]{Proposition}
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\newtheorem{conj}[thm]{Conjecture}
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\DeclareMathOperator{\numer}{numer}
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\newcommand{\eisen}{\numer\left(\frac{p-1}{12}\right)}
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\begin{document}
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\Large
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\maketitle
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\pagenumbering{Roman}
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\setcounter{page}{0}
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\Page{The conjecture}
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\begin{bulletlist}
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\item $p$ prime
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\item $f = q+a_2 q^2 + \cdots \in S_2(\Gamma_0(p);\C)$
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eigenform for $\T=\Z[\ldots T_n\ldots]$
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\item $I_f := \ker(\T \xrightarrow{T_p \mapsto a_p} K_f=\Q(\ldots a_n\ldots))$
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\item $A_f := J_0(p)/I_f J_0(p)$, abelian variety, dimension $d:=[K_f:\Q]$
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\item $\Adual_f :=$ dual of $A_f$
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\item $\pi: J_0(p)\ra A_f$
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\end{bulletlist}
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{\bf Associated finite groups:}
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\begin{bulletlist}
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\item $A_f(\Q)_{\tor}$ rational torsion
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\item $\Phi_{A_f}(\Fpbar)$ component group of closed fiber of N\'{e}ron model
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\end{bulletlist}
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{\bf Motivation:} Both quantities are closely related to quantities that
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appear in the conjecture of Birch and Swinnerton-Dyer, so we must
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understand them completely.
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\begin{conj}
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\begin{itemize}
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\item[(a)] We have
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$$ \#\Phi_{A_f}(\Fpbar) = \#A_f(\Q)_{\tor}
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% = \#\Adual_f(\Q)_{\tor}
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% = \#\Phi_{A_f}(\Fp)
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= \order(\pi(0-\infty))$$
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\item[(b)] (More tentative)
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We have
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$$\prod \#A_f(\Q)_{\tor} =
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\numer\left(\frac{p-1}{12}\right),$$
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where the product is over the Galois conjugacy classes.
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\end{itemize}
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\end{conj}
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{\bf Why?}
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\begin{bulletlist}
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\item It is the nicest possible answer.
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\item All of my computer computations suggest it.
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%\item Part (a) is true for elliptic curves (Mestre-Oesterle, 1989)
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\end{bulletlist}
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{\bf Remark:} Totally false when the level
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is composite! Counterexamples abound, e.g., $E=${\bf 33A},
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$E(\Q)=\Z/2\cross\Z/2$, $\Phi_E(\Fbar_3)=\Z/6\Z$.
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\Page{Computational evidence}
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\begin{bulletlist}
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\item Conjecture~1 true for all $305$ of the $A_f$ of conductor $p\leq 631$.
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\item For all $666$ of the $A_f$ of conductor $p\leq 1571$ the conjecture
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is true, up to powers of $2$.
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I haven't checked the $2$-power. Same statement for $880$ of the
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abelian varieties up to level $2411$ (some levels missed).
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\end{bulletlist}
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{\bf Example 1:}
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$$\begin{array}{lrrrrr}
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J_0(389) \sim& A_1 &\cross A_2&\cross A_3 &\cross A_6 &\cross A_{20}\\
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\#A_f(\Q)_{\tor}& 1 & 1& 1&1&97\\
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\#\Phi_f(\Fpbar)&1&1&1&1&97\\
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\frac{L(1)}{\Omega}&0&0&0&0&\frac{2^{11}\cdot 5^2}{97}
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\end{array}$$
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$$\numer\left(\frac{389-1}{12}\right) = 97$$
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{\bf Example 2:}
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$$\begin{array}{lrrrr}
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J_0(113) \sim&A_1 &\cross A_2&\cross A_3 & \cross B_3\\
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\#A_f(\Q)_{\tor}&2&2&1&7\\
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\#\Phi_f(\Fpbar)&2&2&1&7\\
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\frac{L(1)}{\Omega}&\frac{1}{2}&\frac{1}{2}&0&\frac{8}{7}
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\end{array}$$
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$$\numer\left(\frac{113-1}{12}\right) = 28$$
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(Remark: Probably $\text{Sha}(B_3)$ has order $8$,
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and is explained by the Mordell-Weil group of $A_3$.)
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\Page{Theoretical evidence}
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{\bf Ribet and Mestre-Oesterl\'{e} (J. Reine Angew. Math., 1989):}
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If $A_f$ is an elliptic curve, then Conjecture~1(a) true.
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{\bf Mazur:}
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\begin{bulletlist}
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\item (Ogg's conjecture)
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$$\# J_0(p)(\Q)_{\tor} = \# \Phi_{J_0(p)}(\Fpbar)
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= \numer\left(\frac{p-1}{12}\right).$$
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So my conjecture can be viewed as a refinement of Ogg's.
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\item If $\ell \mid\# A_f(\Q)$ then
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$\ell \mid \eisen$\\(Mazur's {\em Modular curves}\dots, page 141).
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\item Conjecture 1(b): Mazur's comment on the conjecture is:
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\begin{quote}
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What you are conjecturing is something I don't think I ever thought
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of---at least in those terms---It has some implications which are interesting
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(to me). For example suppose that a prime q divides the numerator of
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$(p-1)/12$ but $q^2$ does not; let $A$ be the $q$-Eisenstein
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quotient. Then only
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one optimal quotient of $A$ would be allowed to have nontrivial $q$-torsion,
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and the other(s), if there are any, would, I guess, then have to settle for
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a $\mu_q$ subgroup. I suppose, though, that it is very often the case that
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(under the above hypotheses of $q$) the $q$-Eisenstein quotient is simple.
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\end{quote}
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\end{bulletlist}
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{\bf Ribet:}
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If $\ell\mid \#\Phi_{A_f}(\Fpbar)$ then $\rho_{f,\m}$ {\em tends}
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to be finite for some $\m$ containing $\ell$. If so, and $\rho_{f,\m}$
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is irreducible, then Ribet's level lowering leads to a contradiction.
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{\bf Remark:} If $\Phi_J\ra \Phi_{A_f}$ is surjective, then
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the primes on the left hand side of Conjecture 1b divide the right
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hand side. So we make the following weaker conjecture.
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\begin{conj}
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The natural map $\Phi_J \into \Phi_{A_f}$ is surjective.
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\end{conj}
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\Page{Grothendieck's monodromy pairing}
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\vspace{1ex}
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\begin{bulletlist}
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\item $J:=J_0(p)$.
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\item $A:=A_f$.
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\item $\cA:=$ N\'{e}ron model of $A$.
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\item $T_A := \text{(torus of closed fiber of $\cA$)} = \cA_{\Fp}^o$,
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since $A$ is purely toric.
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\item $X_A := \Hom(T_A,\Gm)$; $A\mapsto X_A$ contravariant;
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$X_A$ free abelian of rank $X_A=\dim A$.
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\end{bulletlist}
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The key diagrams are as follows.
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$$\[email protected]=6pc{\Adual \[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]=6pc{X_{A} \[email protected]{^(->}[r]^{\pi^*} \ar[dr]^{\theta^*}
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& X_{J} \[email protected]{->>}[d]^{\pi_*} \\
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& X_{\Adual}\[email protected]/^1.5pc/[ul]^{\theta_*}}
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$$
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{\bf Modular degree:} $m_A = \sqrt{\deg \theta}$.
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{\bf Monodromy:}
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$$\langle \quad , \quad \rangle : X_J \cross X_J \ra \Z$$
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$$\langle \quad , \quad \rangle : X_A \cross X_{\Adual} \ra \Z$$
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{\bf Component groups:}
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$$0\ra X_J \ra \Hom(X_J,\Z) \ra \Phi_J \ra 0$$
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$$0\ra X_{\Adual} \ra \Hom(X_A,\Z) \ra \Phi_A \ra 0$$
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\Page{Formulas for $\Phi_A$}
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$S := \text{ saturation of $\pi^* X_A$ }$\\
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For any $L\subset S$ of finite index, define
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\begin{eqnarray*}
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m_L &:=& \#(X_{\Adual}/\pi_* L)\\
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\Phi_L &:=& \coker(X_J \ra \Hom(L,\Z))
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\end{eqnarray*}
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\begin{thm}[-]
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Let $L\subset S$ be of finite index; then
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\begin{enumerate}
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\item \vspace{-2ex}$\Phi_A = \Phi_{\pi^* X_A}$, i.e.,
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$X_J \ra \Hom(\pi^* X_A, \Z) \ra \Phi_A \ra 0$
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is exact;
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\item $\displaystyle \frac{\#\Phi_A}{m_A} = \frac{\#\Phi_L}{m_L}$;
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\item $\image(\Phi_{J}\ra\Phi_A) \isom \Phi_S$.
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\end{enumerate}
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\end{thm}
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\begin{cor}
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$$\#\coker(\Phi_J\ra\Phi_A) = \frac{\#\Phi_A}{\#\Phi_S} = \frac{m_A}{m_S},$$
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in particular, $m_S \mid m_A$.
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\end{cor}
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{\bf Conclusion:}
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$$\[email protected]=1.5cm{
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& *++++[F-,]{\Phi_J\onto \Phi_A} \[email protected]{<=>}[dl]_{\txt{easy }}\[email protected]{<=>}[dr]^{\txt{ hard}}\\
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\mbox{$\pi^* X_A \text{ is saturated}$} & &{m_A=m_S}}$$
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{\bf Observation:}
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\begin{eqnarray*}
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m_S &=& \#(X_{\Adual}/\pi_*(X_J[I_f]))\\
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m_A &=& \#(H_1(A,\Z)^+/\pi_*(H_1(J,\Z)^+[I_f]))\cdot (2-\text{power})
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\end{eqnarray*}
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so if $H_1(J,\Z)^+\tensor\Zp \isom X_J\tensor\Zp$ as $\T\tensor\Zp$-modules,
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then $$p\nmid \#\coker(\Phi_J\ra\Phi_A).$$
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\Page{Level lowering}
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The component group $\Phi_A$ of $A$ ``frequently forces'' various
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$A[\m]$ to arise from a finite flat group scheme.
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Ribet's theorem implies such~$\m$ are reducible, and then
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Mazur's theorem implies such $\m$ Eisenstein.
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Suppose $\ell\mid\#\Phi_A$. Does $\ell\mid \numer(\frac{p-1}{12})$?\\
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$X:=X_A, \quad Y:=X_{\Adual}$
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$$\xymatrix{
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& 0\ar[r]\ar[d]& Y\ar[d]\ar[rr]^{\ell}& & Y\ar[d]\ar[r]& Y/\ell Y\ar[r]\ar[d]& 0\\
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0\ar[r]& {\Hom(X/\ell X,\mu_\ell)}\ar[d]\ar[r]
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& T\ar[d]\ar[rr]^{\cdot^\ell}&& T\ar[d]\ar[r] & 0\ar[d]\\
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0\ar[r]& {A[\ell]}\ar[r]& A\ar[rr]^{\ell}&& A\ar[r]& 0}$$
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Monodromy: $0 \ra Y \ra \Hom(X,\Z) \ra \Phi_A \ra 0$, gives
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$$0 \ra \Phi_A[\ell] \ra Y\tensor \Z/\ell\Z \ra \Hom(X,\Z/\ell\Z)
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\ra \Phi_A \tensor\Z/\ell\Z\ra 0.$$
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Snake gives: $0 \ra \Hom(X/\ell X,\mu_{\ell})
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\ra A[\ell] \xrightarrow{\,f\,} Y/\ell Y \ra 0.$\\
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So:
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$$\xymatrix{
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& & f^{-1}(\Phi_A[\ell])\[email protected]{^(->}[d]
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& {\Phi_A[\ell]} \[email protected]{^(->}[d]\\
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0\ar[r]&{\Hom(X/\ell X,\mu_{\ell})}
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\ar[r]& {A[\ell]}\ar[r] & {Y/\ell Y}\ar[r]&0.
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}$$
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The maximal finite part of $A[\ell]$ is $f^{-1}(\Phi_A[\ell])$,
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which is {\em bigger} than $\Hom(X/\ell X,\mu_{\ell})$.
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Is it possible to show that the $f^{-1}(\Phi_A[\ell])$
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in $A[\ell]$ forces some $A[\m]$ to be finite?
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Example: If $\dim A=1$ then $Y/\ell Y$ has dimension one
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and the sequence becomes
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$$0 \ra \Hom(X/\ell X,\mu_{\ell})
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\ra A[\ell] \ra \Phi_A[\ell] \ra 0$$
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so $A[\ell]$ is finite, so $\ell\nmid\eisen$.
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\end{document}
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