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Author: William A. Stein
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\title{\sc Nonsquare Odd Shafarevich-Tate Groups and Mordell-Weil Groups of Elliptic Curves}
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\author{William A. Stein\\{\sf was@math.harvard.edu}}
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\begin{document}
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\maketitle
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\begin{abstract}
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Let~$E$ be an elliptic curve over a number field~$F$.
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We study certain exact sequences of abelian varieties
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$0\ra A \ra R\ra E \ra 0$
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that induce a natural identification
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$E(F)/p E(F) \isom \Ker(\Sha(A/F)\ra \Sha(R/F))$.
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For many odd primes~$p$, we construct the
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first ever examples of abelian varieties~$A$
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over~$\Q$ such that the $p$-part of
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$\#\Sha(A/F)$ is {\em not} a perfect square.
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We also discuss links between our results and conjectures
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and the problem of constructing elements of Selmer groups
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of elliptic curves over~$\Q$ with analytic rank bigger than~$1$.
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\end{abstract}
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\tableofcontents
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\section*{Introduction}
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Let~$E$ be an elliptic curve over a number field~$F$.
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We study certain exact sequences of abelian varieties
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$0\ra A \ra R\ra E \ra 0$
38
that induce a natural identification
39
$E(F)/p E(F) \isom \Ker(\Sha(A/F)\ra \Sha(R/F))$.
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For many odd primes~$p$, we construct the
41
first ever examples of abelian varieties~$A$
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over~$\Q$ such that the $p$-part of
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$\#\Sha(A/F)$ is {\em not} a perfect square.
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We also discuss links between our results and conjectures
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and the problem of constructing elements of Selmer groups
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of elliptic curves over~$\Q$ with analytic rank bigger than~$1$.
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In particular, let~$E$ be an elliptic curve over~$\Q$. We prove that
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a conjecture about nonvanishing of prime-degree twists of $L(E,s)$
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implies that for all but finitely many primes~$p$ there is a twist~$A$
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of $E^{\times (p-1)}$ such that $E(\Q)/p E(\Q)$ is identified in a
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natural way with a subgroup of $\Sha(A/\Q)$.
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We show how to construct twists~$A$ of powers
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of elliptic curves over~$F$ such that $\Sha(A/F)(p)$ has
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{\em nonsquare} order for an odd prime~$p$.
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For example, if $E$ is the elliptic curve of conductor~$43$
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then there is a two-dimensional abelian variety $A$ that is isomorphic
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to $E\cross E$ over $\Q(\mu_3)^+$ and
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$\#\Sha(A/\Q)(3)$ is not a square.
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This paper is organized as follows. In Section~\ref{sec:terminology}
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we define twisted powers, Tamagawa numbers, and rigid primes. We
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recall in Section~\ref{sec:restriction_of_scalars} the definition of
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the restriction of scalars of an elliptic curve and prove a
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proposition about a map induced by trace.
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We state
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a conjecture about nonvanishing of twists of prime degree in
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Section~\ref{sec:nonvanishing}, and give computational evidence for
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this conjecture. In Section~\ref{sec:ptorsion} we prove triviality of
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the $p$-torsion of several abelian groups attached to twisted powers
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of an elliptic curve. The heart of the paper is
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Section~\ref{sec:main}, which uses the above results to construct
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subgroups of Shafarevich-Tate groups of twisted powers.
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Section~\ref{sec:applications} pulls together the results of the
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previous sections; there we prove that the conjecture of
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Section~\ref{sec:nonvanishing} implies the existence of elements of
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Shafarevich-Tate groups of every prime order, and describe a
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connection with the Birch and Swinnerton-Dyer conjecture.
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In Section~\ref{} we discuss the extent to
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which the order of $\Sha$ can fail to be square.
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\vspace{2ex}\par\noindent{}{\bf{}Acknowledgement: }
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It is a pleasure to thank Gautam Chinta,
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Benedict Gross, Emanuel Kowalski, Barry Mazur,
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Bjorn Poonen, David Rohrlich,
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and Michael Stoll
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for helpful comments and conversations.
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\section{Terminology}\label{sec:terminology}
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In this section, we define twisted powers and rigid primes for an
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elliptic curve, and recall the definition of Tamagawa numbers of an
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abelian variety.
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Let $E$ be an elliptic curve over a number field~$F$.
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\begin{definition}[Twisted Powers]
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A {\em twisted power} of~$E$ is an abelian variety~$A$ over~$F$
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that is isomorphic over $\Kbar$ to $E^{\cross n}$
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for some positive integer~$n$.
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\end{definition}
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We recall the standard notion of Tamagawa number of an abelian variety~$A$,
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and introduce the notation $\cbar_{A,p}$ for the order of the group
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of components of~$A$ over $\Fbar_p$.
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\begin{definition}[Tamagawa Numbers]
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Let~$A$ be an abelian variety over~$F$ with N\'eron
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model $\cA$ over~$\O_F$, and let~$\p$ be a prime of $\O_F$,
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and let $k=\O_F/\p$.
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The component group of~$A$ at~$\p$ is the finite group scheme
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$\Phi_{A,\p} = \cA_{k}/\cA_{k}^0$, where $\cA_{k}^0$ is the
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identity component of $\cA_{k}$. The
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{\em Tamagawa number} of~$A$ at~$\p$ is
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$c_{A,\p} = \#\Phi_{A,\p}(k)$.
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Also set $\cbar_{A,\p} = \#\Phi_{A,\p}(\overline{k})$.
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\end{definition}
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Let $N_E\in \O_F$ be the conductor of~$E$.
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\begin{definition}[Rigid Primes]
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A prime $\p$ of $\O_F$ is {\em rigid} for~$E$ if $\p$ does not divide
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$2\cdot N_E \cdot \prod_{\q\mid N_E} \cbar_{E,\q}$ and the
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representation $\rho_{E,\p}:\Gal(\Fbar/\F)\ra \Aut(E[\p])$
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is irreducible.
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\end{definition}
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\section{Restriction of Scalars}\label{sec:restriction_of_scalars}
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In this section,
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we recall the notion of restriction of
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scalars, and prove that the kernel of a morphism induced by
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a trace is geometrically connected.
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Let~$E$ be an elliptic curve over a number field~$F$, and let~$K$
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be a finite extension of~$F$.
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The restriction of scalars $R=\Res_{K/F}(E_K)$ is an abelian variety
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over~$F$ of dimension $[K:F]$, which is characterized by the
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following universal property: There is a functorial group isomorphism
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$R(S) \isom E_K(S_K)$, where~$S$ varies over all $F$-schemes.
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There is a more explicit description of $\Res_{K/F}(E_K)$ when~$K$ is Galois
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over~$F$. As $\Gal(\Fbar/F)$-modules, we have
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$$
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R(\Fbar) = E(\Fbar \tensor K) \isom E(\Fbar)\tensor_{\Z} \Z[\Gal(K/F)],
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$$
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where $\tau\in \Gal(\Fbar/F)$ acts on
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$\sum P_\sigma\tensor \sigma \in E(\Fbar)\tensor_{\Z}\Z[\Gal(K/F)]$ by
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$$\tau\left(\sum P_\sigma\tensor \sigma\right) =
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\sum \tau(P_\sigma)\tensor \sigma\tau_{|K}.
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$$
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Moreover, the $L$-series of~$R$ is $\prod_{a=1}^{n} L(E,\chi^a,s)$, and~$R$
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has good reduction at all $p\nmid \ell\cdot N$.
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\begin{proposition}\label{prop:kergeo}
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The identity map induces a closed immerion $\iota: E\hookrightarrow
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R$, and the trace $\Tr:K\ra F$ induces a surjection $\Tr:R\ra E$
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whose kernel is geometrically connected. Thus we have an exact sequence
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\begin{equation}\label{eqn:exactabvar}
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0 \ra A \ra R \ra E \ra 0
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\end{equation}
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with $A$ an abelian variety.
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\end{proposition}
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\begin{proof}
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The existence of~$\iota$ and $\Tr$ follows from Yoneda's lemma.
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The map~$\iota$ is induced by the functorial inclusion
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$E(S)\hookrightarrow E_K(S_K)=R(S)$, so~$\iota$ is injective.
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The $\Tr$ map is induced by the usual
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functorial trace map on points
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$R(S)=E_K(S_K)\xrightarrow{\Tr} E(S)$.
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To verify that $\Ker(\Tr)$ is geometrically connected, we base
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extend the exact sequence (\ref{eqn:exactabvar}) to~$\Fbar$. First, note that
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$$R_{\Fbar} \ncisom E_{\Fbar}\cross \cdots \cross E_{\Fbar}.$$
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After base extension, the trace map may be identified with the summation map
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$$+: E_{\Fbar} \cross \cdots \cross E_{\Fbar}
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\longrightarrow E_{\Fbar}.$$
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Let $n=[K:F]$. The map defined by
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$$\left(a_1,\ldots, a_{n-1}\right) \mapsto
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\left(a_1,a_2,\ldots ,a_{n-1},-\sum_{i=1}^{n-1} a_i\right),$$
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is an isomorphism from
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$E_{\Fbar}^{\times (n-1)}$ to $\Ker(+)=\Ker(\Tr_{\Fbar})$.
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Thus $\Ker(\Tr_{\Fbar})$ is a product of copies of $E_{\Fbar}$,
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hence is connected.
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\end{proof}
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\begin{corollary}
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Let $n=[K:F]$. Then
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$$(\iota(E)\intersect \Ker(\Tr))(\Fbar) \isom E[n](\Fbar)\ncisom (\Z/n\Z)^2.$$
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(The rightmost map is an isomorphism of groups, not Galois modules.)
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\end{corollary}
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\begin{proof}
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Since the map
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$$F\hookrightarrow K\xrightarrow{\Tr} F$$ is
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multiplication by~$n$, the composite map
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$$E \hookrightarrow R \longrightarrow E$$
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is also multiplication by~$n$.
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The corollary now follows since
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$\iota(E)\intersect \Ker(\Tr)$ is the kernel of~$\Tr\circ\iota$,
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which equals $[n]$. It is elementary that $E[n](\Fbar)\ncisom (\Z/n\Z)^2$,
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where we have, of course, forgotten the action of $\Gal(\Fbar/F)$.
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\end{proof}
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\subsection{Exactness of the Complex of N\'eron Models}
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\label{sec:etale}
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Let~$E$ be an elliptic curve over a number field~$F$, and let~$K$ be a
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finite Galois extension of~$F$ that is only ramified
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at primes that don't divide the conductor $N_E$ of~$E$.
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Let $R=\Res_{K/F} E_K$ and $A = \Ker(R \ra E)$, so
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by Proposition~\ref{prop:kergeo}
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we have an exact sequence of abelian varieties
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$
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0 \ra A \ra R \ra E \ra 0.
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$
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If a complex of N\'eron models over a ring $\O$ is exact in the sense
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of \cite{neronmodels}, then it induces an exact sequence of sheaves on
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the \'etale site for $\Spec(\O)$.
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\begin{proposition}\label{lem:etale}
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Let $A$, $R$, and $E$ be as above. Assume that
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the three elements
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$\disc(K)$, $n=[K:F]$, and $N_E$ of $\O_F$
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are all relatively prime to each other
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and that $v(p)<p-1$ for each $p\mid n$ and
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for each valuation~$v$ on $\O_F$ corresponding
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to a prime $\p$ of residue characteristic~$p$, normalized
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so that the uniformizing element of $\O_{F,\p}$
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has valuation~$1$ (in particular, if $F=\Q$ this is the
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condition that~$n$ is odd).
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Then the corresponding complex of N\'eron models
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$$
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0 \ra \cA \ra \cR \ra \cE \ra 0
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$$
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is exact.
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\end{proposition}
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\begin{proof}
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We use the results of \cite[Ch.~7]{neronmodels} to prove
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that for every completion $\O_{F,\p}$ of $\O_F$ that the complex
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\begin{equation}\label{eqn:neron}
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0 \ra \cA_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}\ra 0
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\end{equation}
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is exact. (For the reader's convenience, the
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results of \cite[Ch.~7]{neronmodels} that we refer
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to below are reproduced in Section~\ref{sec:neronmodels}.)
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To this end, fix a prime ideal~$\p$ of $\O_F$,
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and let~$p$ be its residue characteristic.
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First suppose that $\p\nmid N_E \cdot \disc(K)$, so~$\p$
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is a prime of good reduction for~$R$.
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If $p\nmid n$, \cite[Prop.~7.5.3 (a)]{neronmodels} implies
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that sequence (\ref{eqn:neron}) is exact.
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If $p\mid n$ then our hypothesis on divisors of $n$ are
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exactly the hypothesis to \cite[Th.~7.5.4(iii)]{neronmodels},
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which implies that (\ref{eqn:neron}) is exact.
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Next suppose that $\p\mid N_E\cdot \disc(K)$, so
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$\p \mid N_E$ or $\p \mid \disc(K)$.
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Then since $\gcd(n,N_E\cdot \disc(K))=1$, we have that $p\nmid n$,
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so we are led to apply \cite[Prop.~7.5.3 (a)]{neronmodels} with
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$B_K = E\subset R$.
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Because $N_E$ and $\disc(K)$ are coprime, $\p$ doesn't
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divide both $\disc(K)$ and $N_E$. We consider each case in turn:
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\begin{itemize}
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\item Suppose that $\p\nmid N_E$.
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Then \cite[Prop.~7.5.3 (a)]{neronmodels}
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asserts that $\cA_{\O_{F,\p}}\ra \cR_{\O_{F,\p}}$ is a closed immersion,
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$\cR_{\O_{F,\p}} \ra \cE_{\O_{E,\p}}$ is smooth with
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kernel $\cA_{\O_{F,\p}}$,
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and the cokernel of $\cR_{\O_{F,\p}}\ra \cE_{\O_{F,\p}}$ is killed
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by multiplication by~$n$. However, in the proof of
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\cite[Prop.~7.5.3 (a)]{neronmodels} (see line~$6$
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on page 187), one only uses
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that $\cE_{\O_{F,\p}}$ has good reduction
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to deduce surjectivity, so in
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fact $\cR_{\O_{F,\p}}\ra \cE_{\O_{F,\p}}$ is surjective.
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(The point is that the good reduction hypothesis on
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$\cR_{\O_{F,\p}}$ is used in the proof only to deduce
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that $\cE_{\O_{F,\p}}$ has good reduction. Alternatively, using
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just the statement of \cite[Prop.~7.5.3 (a)]{neronmodels} we
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immediately see that the sequence
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$0 \ra \cA_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$
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is exact. To get surjectivity on the right, note that the
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composition
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$\cE_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$
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is multiplication by~$n$, which is surjective because
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$p\nmid n$ (so that $[n]$ is etale).
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Thus $\cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$ must be surjective.
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)
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\item Suppose that $\p\nmid \disc(K)$, and let
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$\PP$ be a prime of $K$ lying over~$\p$.
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We will use that formation of N\'eron models commutes with
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unramified base extension \cite{} and check exactness
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of (\ref{eqn:neron}) after base extension to the
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unramified extension $\O_{K,\mathfrak{P}}$ of $\O_{F,\p}$.
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In light of Section~\ref{sec:restriction_of_scalars},
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the generic fiber of the base extension of (\ref{eqn:neron})
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to $\O_{K,\mathfrak{P}}$ is
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$$
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0\ra E_{K,\PP}^{\oplus (n-1)} \ra
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E_{K,\PP}^{\oplus n}\xra{\Sigma} E_{K,\PP}\ra 0.$$
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Thus the corresponding complex of N\'eron models over $\O_{K,\p}$ is
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$$
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0\ra \cE_{\O_{K,\PP}}^{\oplus (n-1)} \ra \cE_{\O_{K,\PP}}^{\oplus n}
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\xra{\Sigma} \cE_{\O_{K,\PP}}\ra 0,
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$$
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which is exact since, e.g., it is exact on $S$-points for {\em any}
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ring~$S$.
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\end{itemize}
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\end{proof}
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By the lemma, we have an exact sequence of sheaves
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$$
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0 \ra \cA \ra \cR \ra \cE \ra 0
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$$
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on the \'etale site over $\O_F$.
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Let $\cA^{\vee}$ denote the N\'eron model of the dual of~$A$.
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\begin{proposition}\label{prop:et}
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The following diagram has an exact row:
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$$\xymatrix{
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& & & {H^1(\O_F,\cE)}\ar[d]\\
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{R(L)} \ar[r] & {E(L)}\ar[r] & {H^1(\O_F,\cA)}\ar[r] & {H^1(\O_F,\cR)}\ar[d]\ar[r] & {H^1(\O_F,\cE)}\\
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& & & {H^1(\O_F,\cA^{\vee})}
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}
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$$
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\end{proposition}
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\begin{remark}{\bf Connection with BSD:}
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I'm not sure whether or not the column is exact, or is almost exact.
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It is close to exact, because $0 \ra \cE \ra \cR \ra \cA$ is exact.
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The column is important in connecting nonvanishing of twists and
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the rank~$0$ BSD formula to showing that there are points
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on~$E$. The tentative connection is as follows:
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Suppose $\Sha(E/F)[n]=0$ but $\Sha(A/F)[n]\neq 0$.
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The composition $\Sha(A/F)\ra \Sha(A^{\vee}/F)\ra \Sha(A/F)$ is
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multiplication by~$n$, which is not injective by our assumption that
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$\Sha(A/F)[n]\neq 0$. Thus
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$\Ker(\Sha(A^{\vee}/F)\ra \Sha(A/F))[n]$ is nonzero, so
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by duality {\bf (I think?)}
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$\Ker(\Sha(A/F)\ra \Sha(A^{\vee}/F))[n]$ is also
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nonzero. But our hypothesis that $\Sha(E/F)[n]={0}$ combined with
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exactness of the vertical sequence would imply that
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$$\Ker(\Sha(A/F)\ra \Sha(A^{\vee}/F)[n] = \Ker(\Sha(A/F)\ra \Sha(R/F))[n] \neq 0,$$
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so $E(F)$ {\em must} be nonzero!
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If this works it is very exciting because it means that the following
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three statements
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\begin{enumerate}
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\item Nonvanishing of twists conjecture (see Conjecture~\ref{conj:nonvanishing})
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\item Finiteness of $\Sha(E/\Q)$, and
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\item $\Sha(A/\Q)$ is as big as predicted by BSD for analytic rank~$0$ twisted
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powers~$A$
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\end{enumerate}
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together imply the statement:
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$$L(E/\Q,1)=0 \quad\implies\quad E(\Q)\text{ is infinite.}$$
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\end{remark}
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362
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\begin{proof}
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(Not really written, but see Section~\ref{sec:letter} below.)
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By \cite[Appendix]{mazur:tower}, there is an exact sequence
366
$$
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0 \ra \Sha(A) \ra H^1(\O_F,\cA) \ra G \ra 0
368
$$
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where~$G$ is a finite group whose order if divisible only by~$2$ and
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primes that divide the Tamagawa numbers of $\cA$.
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\end{proof}
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\comment{
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Alternatively, one can consider the following diagram
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$$
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\xymatrix{
377
& 0\ar[d] & 0\ar[d]\\
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0\ar[r] & {\cE[n]}\ar[r]\ar[d] & {\cE}\ar[d]\ar[dr]^{[n]}\\
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0 \ar[r] & {\cA} \ar[r] & {\cR}\ar[r] & {\cE} \ar[r] & 0
380
}$$
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and work with cohomology on the fppf site.
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}
383
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\section{Non-Quadratic Twists}\label{sec:nonvanishing}
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We state a conjecture about nonvanishing at~$1$ of certain
386
prime-degree twists of the $L$-function attached to an elliptic curve,
387
provide extensive computational evidence for the conjecture, and give
388
an example which suggests that vanishing twists are very rare.
389
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\subsection{A Conjecture}
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Let~$E$ be an elliptic curve over~$\Q$, and
392
suppose $p$ is a rigid prime for~$E$.
393
For every prime $\ell\con 1\pmod{p}$, let
394
$\chi_{p,\ell} : (\Z/\ell\Z)^* \onto \bmu_p$ be
395
one of the Galois-conjugate characters
396
of order~$p$ and modulus~$\ell$.
397
\begin{conjecture}\label{conj:nonvanishing}
398
There exists a prime~$\ell\nmid N_E$ such that
399
$$L(E,\chi_{p,\ell},1)\neq 0
400
\,\,\,\text{ and }\,\,\,
401
a_\ell(E) \not\con 2\pmod{p}.$$
402
\end{conjecture}
403
404
The condition $a_\ell(E) \not\con 2\pmod{p}$ requires elaboration.
405
Since $\ell\con 1\pmod{p}$, this condition can be rewritten
406
$a_\ell(E)\not\con \ell+1\pmod{p}$, which is a ``familiar'' condition
407
to impose. We demand that $a_\ell(E)\not\con \ell+1\pmod{p}$ because
408
then the characteristic polynomial $x^2 + a_\ell x +\ell\in \F_p[x]$
409
of $\Frob_\ell$ on $E[p]$ does not have $+1$ as an eigenvalue. This
410
is a key hypothesis in Section~\ref{sec:ptorsion}.
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412
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\begin{table}
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\caption{Evidence for Conjecture~\ref{conj:nonvanishing}\label{tbl:evidence}}
415
\noindent%\hspace{-5ex}
416
\begin{tabular} {|c|cccccccccccccc|}\hline
417
$\,\,\,E$ & 3&5&7&11&13&17&19&23&29&31&37&41&43&47\\\hline
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\nf{37A} & 13&11&29&67&53&103&191&47&59&311&-&83&173&283\\
419
\nf{43A} & 7&11&29&23&53&103&191&47&59&311&149&83&-&283\\
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\nf{53A} & 13&11&29&23&53&103&191&47&59&311&149&83&173&283\\
421
\nf{57A} & -&11&29&23&53&103&-&47&59&311&149&83&173&283\\
422
\nf{58A} & 7&11&29&23&53&103&191&47&-&311&149&83&173&283\\
423
\nf{61A} & 7&31&29&67&53&103&191&47&59&311&149&83&173&283\\
424
\nf{65A} & 19&-&43&23&-&137&191&47&59&311&149&83&173&659\\
425
\nf{77A} & 19&11&-&-&53&103&191&47&59&311&149&83&173&283\\
426
\nf{79A} & 13&11&43&67&53&103&191&47&59&311&149&83&173&283\\
427
\nf{82A} & 13&41&29&23&53&103&191&47&59&311&149&-&173&283\\
428
\nf{83A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
429
\nf{88A} & 7&11&29&-&131&103&191&47&59&311&149&83&173&283\\
430
\nf{89A} & 19&11&29&67&53&103&191&47&59&311&149&83&173&283\\
431
\nf{91A} & 31&11&-&23&-&103&191&47&59&311&149&83&173&283\\
432
\nf{91B} & -&11&-&23&-&103&191&47&59&311&149&83&173&283\\
433
\nf{92B} & 13&61&29&23&79&103&229&-&59&311&149&83&173&283\\
434
\nf{99A} & -&11&29&-&53&103&191&47&59&311&149&83&173&283\\
435
\nf{101A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
436
\nf{102A} & -&11&29&23&53&-&191&47&59&311&149&83&173&283\\
437
\nf{106B} & 7&11&29&23&53&137&191&47&59&311&149&83&431&283\\\hline
438
\nf{389A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
439
\nf{433A} & 7&11&43&23&53&103&191&47&59&311&149&83&173&283\\\hline
440
\end{tabular}
441
\end{table}
442
443
444
\subsection{Computational Evidence for the Conjecture}
445
446
Using a \magma{} program (see~\cite{magma}), the author's computer
447
verified Conjecture~\ref{conj:nonvanishing} for every $p<50$ for the
448
first $20$ optimal elliptic curve quotients of $R_0(N)$ of rank~$1$
449
and the first~$2$ elliptic curve quotients of rank~$2$.
450
451
Table~\ref{tbl:evidence} contains, for each $p < 50$, the smallest
452
prime~$\ell$ satisfying the conditions of
453
Conjecture~\ref{conj:nonvanishing}. The elliptic curves are labeled
454
as in Cremona. The curves \nf{389A} and \nf{433A} both have rank~$2$,
455
and all others have rank~$1$. A dash (-) in the table indicates that the
456
corresponding prime is not rigid, so the conjecture does not apply.
457
458
In all cases the first prime $\ell\nmid N_E$ with
459
$\ell\con 1\pmod{p}$ with $a_\ell(E)\not \con 2\pmod{p}$
460
satisfied $L(E,\chi_{p,\ell},1)\neq 0$, except
461
for \nf{61A} with $p=5$, \nf{79A} with $p=7$,
462
\nf{82A} with $p=5$, \nf{89A} with $p=11$,
463
and \nf{92B} with $p=5$. In every one of these~$5$ exceptional
464
cases, the second prime that we tried
465
satisfied the conclusion of
466
Conjecture~\ref{conj:nonvanishing}.
467
468
\subsection{The Density}
469
The following conjecture is not mentioned
470
elsewhere in this paper.
471
\begin{conjecture}
472
Let~$p$ be a rigid prime for an elliptic curve~$E$.
473
The set of primes
474
$$ \left\{\ell \,\,:\,\, \ell \con 1\!\!\!\!\!\pmod{p}\text{ and }
475
L(E,\chi_{p,\ell},1)=0\right\}
476
$$
477
has Dirichlet density~$0$.
478
\end{conjecture}
479
480
The following numerical example gives evidence for this conjecture.
481
\begin{example}
482
Let~$E$ be \nf{37A} and let $p=5$. Then the only $\ell<1000$ (with
483
$\ell\con 1\pmod{5}$) for which $L(E,\chi_{5,\ell},1)=0$ is $\ell=41$.
484
% 4 minutes to compute
485
\end{example}
486
487
488
\section{$p$-Torsion of Twisted Powers}\label{sec:ptorsion}
489
Let~$p$ and~$\ell$ be as in Conjecture~\ref{conj:nonvanishing}.
490
In order to apply Theorem~\ref{thm:shaexists}, it is necessary
491
to know that~$p$ does not divide the orders of certain groups.
492
In this section, we use that $a_\ell(E)\not\con 2\pmod{p}$ to
493
deduce that certain groups do not have any~$p$ torsion.
494
The key idea is that the condition on $a_\ell(E)$ implies
495
that~$+1$ is not an eigenvalue of $\Frob_\ell$ on the
496
$p$-adic Tate module attached to~$E$.
497
498
First, we recall that certain torsion points on the closed
499
fiber of a N\'eron model lift to the generic fiber. Let~$K$ be a
500
finite extension of~$\Q_\ell$ with ring of integers~$\O$ and residue
501
class field~$k$.
502
\begin{lemma}\label{lem:red_mod_n}
503
Let $A$ be an abelian variety over~$K$ with N\'eron model $\cA$ over~$\O$.
504
Then for every integer $n$ not divisible by~$\ell$, there is an isomorphism
505
$$A(K)[n] \xrightarrow{\,\,\isom\,\,} \cA(k)[n].$$
506
\end{lemma}
507
\begin{proof}
508
This is a standard fact, whose proof we recall for the convenience
509
of the reader.
510
Let $A^{1}(K)$ denote the kernel of the natural reduction
511
map $r:A(K)\ra \cA(k)$. Because $A^{1}(K)$ is a formal group,
512
it is pro-$p$, so $[n]:A^{1}(K)\ra{}A^{1}(K)$ is an isomorphism.
513
Since $\cA$ is smooth over~$\O$,
514
Hensel's lemma (see BLR) implies that the reduction map
515
is surjective, so the following sequence is exact:
516
$$0\ra A^1(K) \ra A(K) \ra \cA(k) \ra 0.$$
517
The snake lemma applied to the multiplication by~$n$ diagram
518
attached to this exact sequence yields the following
519
exact sequence:
520
$$0\ra0\ra A(K)[n]\ra \cA(k)[n] \ra 0 \ra A(K)/n A(K) \ra \cA(k)/n\cA(k)\ra0,$$
521
which proves the proposition.
522
\end{proof}
523
524
525
Let~$E$ be an elliptic curve over~$\Q$ with associated newform
526
$f = \sum a_n q^n$, and fix a prime~$p$ that is rigid for~$E$.
527
Suppose~$K$ is the extension of~$\Q$ corresponding
528
to a surjective Dirichlet character
529
$\chi: (\Z/\ell\Z)^* \onto \bmu_p$
530
of prime conductor; then~$K$ is
531
the subfield of $\Q(\bmu_\ell)$ fixed by $\Ker(\chi)$,
532
so it is of degree~$p$, is totally ramified
533
at~$\ell$, and is unramified outside~$\ell$.
534
Let~$A=\Ker(\Tr : \Res_{K/\Q} E_K \ra E)$.
535
We next compute the Tamagawa number
536
$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$
537
and the $p$-torsion of several abelian varieties.
538
\begin{proposition}\label{prop:ptorsion}
539
Let~$E$, $\chi$, $K$, and~$A$ be as above and suppose
540
that $\ell\nmid N_E$ and $a_\ell \not\equiv 2\pmod{p}$.
541
Then the following groups have no nontrivial $p$-torsion:
542
$$A(K), \quad A(\Q_\ell),\quad R(\Q_\ell),\quad
543
(R/E)(\Q_\ell),\quad \text{and}\quad \Phi_{A,\ell}(\F_\ell).$$
544
%$$A(K)[p]=A(\Q_\ell)[p] = R(\Q_\ell)[p] =
545
%(R/E)(\Q_\ell)[p]=\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$$
546
\end{proposition}
547
\begin{proof}
548
The reason the $p$-torsion vanishes in all these cases
549
is that the condition $a_\ell \not\equiv 2\pmod{p}$ implies
550
in each case that $\Frob_\ell$ has no $+1$ eigenvalue.
551
The details are as follows.
552
553
We first show that $R(\Q_\ell)[p]=\{0\}$, where $R=\Res_{K/\Q}E_K$.
554
By definition,
555
$$R(\Q_\ell) = E_K(\Q_\ell\tensor_\Q K) =
556
E(K_v)\cross \cdots \cross E(K_v) \quad \text{($p$ copies)},$$
557
where $K_v$ is the completion of~$K$ at the unique prime of~$K$
558
lying over~$\ell$.
559
The action of $\Frob_\ell\in \Gal(\Q_\ell^{\ur}/\Q_\ell)$
560
on $E[p](\Q_\ell^{\ur})=E[p](\Qbar_\ell)$ has characteristic
561
polynomial
562
$F(x) = x^2-a_\ell x + \ell \in \F_p[x]$.
563
Since $a_\ell \not\equiv 2\pmod{p}$ and $\ell\equiv 1\pmod{p}$, it
564
follows that $\Frob_\ell$ does not have
565
$+1$ as an eigenvalue, so $E(\Q_\ell)[p]=\{0\}$.
566
If $z\in E(K_v)[p]$, then the field $L=\Q_\ell(z)$ is an unramified
567
subfield of the totally ramified field $K_v$, so $z\in E(\Q_\ell)[p]=\{0\}$.
568
Thus $E(K_v)[p]=\{0\}$, which implies that $E(K)[p]=\{0\}$ and
569
$R(\Q_\ell)[p]=\{0\}$.
570
Since $R_K/E_K \isom E_K \cross \cdots \cross E_K$ ($p-1$ times),
571
we see that
572
$$(R/E)(\Q_\ell)[p]\subset (R/E)(K_v)[p] =
573
(E(K_v)\cross \cdots \cross E(K_v))[p] = \{0\}.$$
574
575
Finally, we turn to the component group $\Phi_{A,\ell}$.
576
Let $\cA$ denote the N\'eron
577
model of~$A$. By Lang's Lemma the natural map $\cA(\F_\ell) \ra
578
\Phi_{A,\ell}(\F_\ell)$ is surjective. Thus if
579
$\Phi_{A,\ell}(\F_\ell)[p]\neq \{0\}$, then $\cA(\F_\ell)[p]\neq
580
\{0\}$. However, by Lemma~\ref{lem:red_mod_n} and observation of the
581
previous paragraph,
582
$$\cA(\F_\ell)[p] = A(\Q_\ell)[p]\subset R(\Q_\ell)[p]=\{0\},$$
583
so $\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$, as claimed.
584
\end{proof}
585
586
587
\subsection{The Tamagawa Number of $A$ at $\ell$}
588
In this section, the notation and hypothesis are as in
589
Proposition~\ref{prop:ptorsion}.
590
That proposition implies that the Tamagawa number
591
$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$ is coprime
592
to~$n$. In this section we use Remark~5.4 of \cite{edixhoven:tame} to
593
prove that in fact $c_{A,\ell}=1$.
594
595
Let $\lambda$ be the prime of~$K$ lying over $\ell$, and
596
let $K_{\lambda}$ denote the completion of~$K$ at~$\lambda$,
597
so $K_{\lambda}$ is totally and tamely ramified over $\Q_\ell$.
598
Since
599
$$A_{K} \isom \Ker(\Sigma: E_{K}^{\oplus n}
600
\ra E_{K}),$$
601
and $E_{K_\lambda}$ has good reduction,
602
the geometric closed fiber of the N\'eron model of $A_{K_{\lambda}}$ is
603
$
604
A'_{\kbar}\isom \Ker(\Sigma : E_{\kbar}^{\oplus n} \ra E_{\kbar}).
605
$
606
In the notation of \cite{edixhoven:tame},
607
$\mu_n$ acts on $A'_{\kbar}$ by the action
608
it induces by cyclically permuting the factors of
609
$E_{\kbar}^{\oplus n}$. Thus
610
$A_{\kbar}'(\kbar)^{\mu_n}$ is the set of
611
$\sum P_\sigma\tensor\sigma \in E(\kbar)^{\oplus n}$
612
such that all $P_\sigma$ are equal and $\sum P_\sigma = 0$,
613
i.e.,
614
$$
615
A_{\kbar}'(\kbar)^{\mu_n} \isom E(\kbar)[n]\ncisom (\Z/n\Z)^2.
616
$$
617
Thus Remark~5.4 in \cite{edixhoven:tame} implies that
618
$\Phi_{A,\ell}(\kbar) \ncisom E(\kbar)[n]$.
619
By Proposition~\ref{prop:ptorsion},
620
$\Phi_{A,\ell}(k)$ has no elements of order dividing $n$,
621
so $\Phi_{A,\ell}(k)=0$.
622
623
\section{Nonsquare Shafarevich-Tate Groups}\label{sec:nonsquaresha}
624
On page 306--307 of \cite{tate:bsd}, Tate discusses results about the
625
structure of the group $\Sha(A/K)$, where~$A$ is an abelian variety over
626
a number field~$K$. He asserts that if~$A$ is a Jacobian then $\#\Sha(A/K)$
627
is a perfect square. Poonen and Stoll subsequently pointed out
628
in~\cite{poonen-stoll} that Tate's assertion is not quite correct. In
629
fact, Poonen and Stoll prove that when~$A$ is a Jacobian, $\#\Sha(A/K)$
630
is either a square or twice a square, and they give examples in which
631
$\#\Sha(A/K)$ is twice a square. Tate does not discuss the case
632
when~$A$ is not a Jacobian, except to mention results that imply that
633
$\#\Sha(A/K)$ is square away from~$2$ and primes that
634
don't divide the degree of some polarization of~$A$.
635
636
Now suppose~$A$ is an arbitrary abelian variety over a number
637
field~$K$. Until now it was unknown whether or not $\#\Sha(A/K)$ must
638
be either a square or twice a square. Let~$E$ be an elliptic curve
639
over~$\Q$ of rank~$1$. Then our construction gives,
640
for suitable primes~$p$, an injection
641
$$
642
\Z/p\Z \ncisom E(\Q)/p E(\Q) \hookrightarrow \Sha(A/\Q),
643
$$
644
where~$A$ is an abelian variety over~$\Q$ which is a twist of $E^{\times p-1}$.
645
Thus $\Sha(A/\Q)[p]$ has a natural subgroup of order~$p$; moreover,
646
no other natural subgroup of order~$p$ presents itself.
647
We prove in this section that the $p$-part of $\#\Sha(A/\Q)$
648
is not a perfect square.
649
650
\subsection{A Letter}\label{sec:letter}
651
[Convert this email into a rigorous proof.]
652
\begin{verbatim}
653
Do you remember my conditional-on-BSD construction of abelian
654
varieties A with nonsquare p-part of #Sha(A), for various odd primes
655
p? In particular, you pushed me repeatedly to remove the conditional
656
nature of the result. I think I found a very clean way to do this
657
today, which doesn't require computing any Selmer groups at all!
658
659
I make a list of hypothesis on p, the field that I restrict scalars
660
down from and the Fourier coefficient a_ell(E). For example, when E
661
is the curve of conductor 43, one could take p=3, K the cubic field
662
Q(mu_7)^+. Then if R is the restriction of scalars of E from K down
663
to Q, surprisingly the exact sequence
664
665
0 --> A --> R --> E --> 0
666
667
of abelian varieties *does* induce an exact sequence of Neron models
668
over Spec(Z), as I think I've checked using chapter 7 of
669
Bosch-Lutkebohmert-Raynaud. We then obtain the following exact
670
sequence from the long exact sequence of etale cohomology:
671
672
0 ---> E(Q)/3E(Q) ---> H^1(Z,A) ---> H^1(Z, R) --> H^1(Z,E)
673
674
Then an old theorem of Barry from his "Rational Points on Abelian
675
Varieties..." along with my hypothesis on E, p, and ell, imply that
676
H^1(Z,A), H^1(Z,R) and H^1(Z,E) differ from Sha(A), Sha(R), and Sha(E)
677
by at most a 2-group. Ignoring this 2 business, we have an exact
678
sequence
679
680
0 ---> E(Q)/3E(Q) ---> Sha(A/Q) --> Sha(E/K) --> Sha(E/Q).
681
682
(I also just used that Sha(R/Q) = Sha(E/K).)
683
684
Now the 3-power-part Sha(E/Q)(3) = {0}, by Euler Systems theory (or I
685
could directly check this), so we have
686
687
0 --> E(Q)/3E(Q) ---> Sha(A/Q)(3) --> Sha(E/K)(3) --> 0
688
689
Properties of the Cassels-Tate pairing on the elliptic curve E over K
690
and the fact that Sha(E/K) is finite (since it's an image of
691
Sha(A)(3), which is finite by Kato's theorem), imply that #Sha(E/K)(3)
692
is a an even power of 3. Thus #Sha(A/Q)(3) must be an odd power of 3.
693
694
What do you think? This seems much better than trying to do explicit
695
Selmer group computations with CM elliptic curves.
696
697
-- William
698
699
\end{verbatim}
700
701
702
\section{Other Remarks}\label{sec:applications}
703
We apply the above results to prove that
704
Conjecture~\ref{conj:nonvanishing} implies the existence of
705
elements of Shafarevich-Tate groups of twisted powers of
706
elliptic curves of every prime order. We also construct an
707
abelian variety~$A$ over~$\Q$ such that the Birch
708
and Swinnerton-Dyer conjecture predicts that $\Sha(A/\Q)[3]=\Z/3\Z$ and
709
that $\#\Sha(A/\Q)$ is not a square or twice a square.
710
711
\subsection{Existence of Elements of $\Sha$ of all Prime Orders}
712
\begin{proposition}\label{prop:all_prime_orders}
713
Let~$p$ be a prime number. Then Conjecture~\ref{conj:nonvanishing}
714
implies that there exists infinitely many twisted powers~$A$ of some
715
elliptic curve such that $\Sha(A/\Q)[p]\neq \{0\}$.
716
\end{proposition}
717
\begin{proof}
718
Most of the proposition can be proved using a single elliptic curve.
719
When ordered by conductor, the first elliptic curve~$E$ over $\Q$ with
720
positive rank has prime conductor~$37$ and is defined by the
721
Weierstrass equation $y^2 + y = x^3 - x$. Table~1 of
722
\cite{cremona:algs} shows that~$E$ is isolated in its isogeny class,
723
so \cite[Exercise~4]{ribet-stein:serre} implies that representations
724
$\rho_{E,p}$ are irreducible. Since
725
$\ord_{37}(j(E))=-1$, $\cbar_{37}=1$. Thus all odd primes $p\neq 37$
726
are rigid for~$E$. The proposition then follows for all odd primes
727
$p\neq 37$ by Theorem~\ref{thm:main}.
728
729
We complete the proof as follows. Exactly the same argument applied
730
to the unique elliptic curve of conductor~$43$ proves the proposition
731
for all odd primes $p\neq 43$. Finally, B\"olling proved in
732
\cite{bolling:sha} that for every $j\in\Q$ there is an elliptic
733
curve~$E$ with $j$-invariant~$j$ such that infinitely many twists~$E'$
734
of $E$ have $\Sha(E'/\Q)[2]\neq\{0\}$.
735
\end{proof}
736
737
738
\subsection{What Goes Wrong when $p=2$?}
739
In the previous section, we set $p=3$ and
740
constructed an abelian variety $A$ of dimension $p-1$
741
that (conjecturally) has nonsquare $\Sha(A/\Q)[p]$.
742
We can construct an~$A$ in an analogous way for any odd prime~$p$,
743
and the author expects that $\Sha(A/\Q)[p]$ is nonsquare in most cases.
744
However, when $p=2$, the dimension of~$A$ is~$1$, so
745
in that case $\#\Sha(A/\Q)$ must be a perfect square.
746
747
What goes wrong? The problem lies in Theorem~\ref{thm:shaexists}.
748
The argument used to prove Theorem~\ref{thm:shaexists}
749
at least provides a map
750
$$
751
E(\Q)/2 E(\Q)\hookrightarrow\Vis_R(H^1(\Q,A)).
752
$$
753
When $p=2$, the condition $e<p-1$ is not satisfied, so
754
the proof of Theorem~\ref{thm:shaexists} does not show
755
that the image of $E(\Q)/ 2 E(\Q)$ is locally trivial
756
at the prime~$2$ (or at $\infty$). We thus only
757
construct a subgroup of $H^1(\Q,A)$ of nonsquare order,
758
not of~$\Sha(A/\Q)$. Thus even if two elliptic curves have the same
759
$E[2]$, then can still possess very different Selmer groups.
760
761
762
763
764
765
\section{Connection with BSD}
766
\label{sec:bsd}
767
[[Needs to be rewritten.]]
768
Suppose $E$ is an elliptic curve over~$\Q$ and that $L(E,1)=0$. The
769
Birch and Swinnerton-Dyer conjecture for~$E$ asserts (among other
770
things) that $E(\Q)$ is infinite. Suppose~$A$ is constructed as in
771
Section~\ref{sec:main}. In this section we describe why if a certain
772
consequence of a refinement of the Birch and Swinnerton-Dyer
773
conjecture for~$A$ is true, then $\Sel^{(n)}(E/\Q)$ is nonzero.
774
775
776
Using modular symbols one sees that $L(A,1)\con 0 \pmod{\ell}$,
777
so a refinement of the Birch and Swinnerton-Dyer formula for rank~$0$
778
abelian varieties predicts that there should be a nonzero element in
779
$\Ker (\Sha(A)\ra \Sha(A/E[n]))$.
780
Thus by Proposition~\ref{prop:et}, either
781
$H^1(X_{\et},\cE)[n]\neq 0$, or there is a nonzero element
782
of order dividing~$n$ in
783
$$
784
\Ker(H^1(X_{\et},\cA)\ra H^1(X_{\et},\cR)) \isom E(\Q)/R(\Q),
785
$$
786
in which case $E(\Q)/R(\Q)$ contains a nonzero element of
787
order dividing~$n$,
788
so $E(\Q)$ is infinite.
789
Thus either $\Sha(E)[n]\neq 0$ or $E(\Q)$ is infinite, so
790
$\Sel^{(n)}(E/\Q)$ is nonzero.
791
792
793
\section{Appendix: Exactness Properties}\label{sec:neronmodels}
794
For the reader's convenience, we copied {\em verbatim}
795
pages 186--187 of \cite[\S7.5]{neronmodels}. In what follows~$R$
796
is a discrete valuation ring with field of fractions~$K$.
797
798
799
``Next, let us look at abelian varieties.
800
801
\noindent{\bf Proposition 3.} {\em
802
Consider an exact sequence of abelian
803
varieties
804
$$
805
0 \ra A_K' \ra A_K \ra A_K'' \ra 0
806
$$
807
and the corresponding complex of N\'eron models
808
\begin{equation}\label{eqn:dagger}
809
0\ra A' \ra A \ra A'' \ra 0 \qquad\qquad
810
\end{equation}
811
Let $B_K$ be an abelian subvariety of $A_K$ such that $A_K\ra A_K''$ induces an isogeny
812
$u_K: B_K \ra A_K''$; let $n=\deg u_K$.
813
814
(a) If $\chr{}k$ does not divide~$n$, then $A'\ra A$ is a closed immersion,
815
$A\ra A''$ is smooth with kernel $A'$, and the cokernel of $A_k \ra A_k''$ is
816
killed by multiplication with~$n$. If, in addition, $A$ has abelian reduction, (\ref{eqn:dagger})
817
is exact.
818
819
(b) If $A$ has semi-aelian reduction, the sequence (\ref{eqn:dagger}) is exact up to isogeny;
820
i.e., it is isogenous to an exact sequence of commutative $S$-group schemes.
821
}
822
\vspace{1em}
823
\begin{proof}
824
The isogeny $u_K : B_K \ra A_K''$ gives rise to an isogeny $v_K : A_K' \cross_K B_K \ra A_K$
825
of degree~$n$. So there is an isogeny $w_K : A_K \ra A_K' \cross_K B_K$ such
826
that $w_K\circ v_K$ is multiplication by~$n$. Let $B$ be the N\'eron model of $B_K$. Then
827
$u_K$, $v_K$, and $w_K$ extend to $R$-morphisms $u: B \ra A''$,
828
$v: A'\cross_R B \ra A$, and
829
$w: A\ra A'\cross_R B$ such that $w\circ v$
830
is multiplication by $n$ on $A'\cross_R B$.
831
Assuming the condition of (a), the multiplication by~$n$ is an \'etale isogeny on
832
$A'\cross_R B$, and $u,v$ and $w$ are easily checked to be \'etale
833
isogenies, too. Then $H:=w^{-1}(A')$ is a smooth closed subgroup scheme of~$A$
834
which satisfies $H^0_K = A_K'$. It follows that the schematic closure of $A_K'$
835
in~$H$ or~$A$ is an open subgroup scheme of~$H$ and, thus, is smooth over~$R$.
836
So, by using the \'etale isogeny~$u$. One shows that
837
$A\ra A''$ is flat, has kernel $A'$ and, hence, is smooth. Furthermore,
838
if~$A$ has abelian reduction, the same is true for $A''$ by 7.4/2 so that
839
$A\ra A''$ is surjective.
840
841
Assertion (b) follows from the fact that $v:A'\cross_R B \ra A$ and
842
$u:B\ra A''$ are isogenies; use 7.3/6 and 7.3/7.
843
\end{proof}
844
845
\noindent{\bf Theorem 4.} {\em
846
Let $0\ra A_K' \ra A_K \ra A_K'' \ra 0$ be an exact sequence of
847
abelian varieties and consider the associated sequence of N\'eron
848
models $0\ra A' \ra A \ra A'' \ra 0$. Assume that the following
849
condition is satisfied:
850
851
(*) $R$ has mixed characteristic and the ramification index $e=v(P)$
852
satisfies $e<p-1$, where $p$ is the residue characteristic of $R$
853
and where $v$ is the valuation on~$R$, which is normalized by the condition
854
that~$v$ assumes the value~$1$ at unifromizing elements of~$R$.
855
856
Then the following assertions hold:
857
858
(i) If $A'$ has semi-abelian reduction, $A'\ra A$ is a closed immersion.
859
860
(ii) If $A$ has semi-abelian reduction, the sequence $0\ra A' \ra A \ra A''$
861
is exact.
862
863
(iii) If $A$ has abelian reduction, the sequence $0\ra A' \ra A\ra A''\ra 0$
864
is exact and consists of abelian $R$-schemes.
865
}
866
\vspace{1em}
867
868
\begin{proof}
869
Let us first see how assertions (ii) and (iii) can be deduced from assertion (i).
870
If~$A$ has semi-abelian or abelian reduction, the same is true for $A'$ and $A''$
871
by 7.4/2. So $A'\ra A$ is a closed immersion by (i), and we can consider the
872
quotient $A/A'$; it exists in the category of algebraic spaces, cf.\ 8.3/9.
873
Furthermore, $A/A'$ is smooth and separated and, thus, a scheme by 6.6/3. Now look at the
874
canonical morphism $A/A'\ra A''$ which is an isomorphism on generic fibres.
875
Since $A$ has semi-abelian reduction, the same is true for $A/A'$, and it
876
follows from 7.4/3 that $A/A'\ra A''$ is an open immersion. So assertion (ii)
877
is clear. Finally, if $A$ has abelian reduction, the same is true for $A/A'$.
878
So the latter is an abelian scheme by 7.4/5 and, thus, must coincide with the
879
N\'eron model $A''$ of $A''_K$. Thereby we obtain assertion (iii).
880
[The rest of the proof takes a few pages.]''
881
\end{proof}
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883
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\bibliography{biblio}
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\end{document}
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890
891
E := EC("37A");
892
M := MS(E);
893
K := CyclotomicField(5);
894
M5 := BaseExtend(M,K);
895
P := [p : p in [3..200] |IsPrime(p)];
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time X := [<p,
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Dimension(TwistedWindingSubmodule(M5,1,DirichletGroup(p,K).1^2))> : p in P];
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// Time: 11.369
899
[ <11, 1>, <31, 1>, <41, 0>, <61, 1>, <71, 1>, <101, 1>, <131, 1>,
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<151, 1>, <181, 1>, <191, 1> ]
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\comment{
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$Log: bigsha.tex,v $
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Revision 1.25 2001/09/28 02:48:22 was
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?
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Revision 1.24 2001/09/23 04:43:11 was
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more TOC stuff and cleaning.
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Revision 1.23 2001/09/22 20:27:00 was
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Worked on
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\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}
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and changed Sha(A) to Sha(A/Q).
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Revision 1.22 2001/09/22 19:26:23 was
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I don't know.
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Revision 1.21 2001/09/13 01:42:50 was
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Lots of nice little improvements!
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Revision 1.20 2001/09/09 04:11:42 was
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Added lots of toc pars.
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Revision 1.19 2001/09/08 02:55:34 was
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polishing.
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Revision 1.18 2001/09/06 03:39:53 was
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added some cool remarks at end about BSD
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Revision 1.17 2001/09/06 03:24:36 was
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typo
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Revision 1.16 2001/09/06 03:23:56 was
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minor typo
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Revision 1.15 2001/09/06 03:17:42 was
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Added table of evidence for twisting conjecture.
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Revision 1.14 2001/09/06 02:42:04 was
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...
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}
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\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}
944
Let $E$ be an elliptic curve over~$\Q$.
945
In this section, we prove that Conjecture~\ref{conj:nonvanishing}
946
implies if~$p$ is a rigid prime, then
947
$E(\Q)/p E(\Q)$ is canonically isomorphic to the
948
elements of order~$p$ in the visible Shafarevich-Tate
949
group of a rank~$0$ twisted power of~$E$.
950
951
\begin{theorem}\label{thm:main}
952
Assume that Conjecture~\ref{conj:nonvanishing} is true.
953
If~$E$ is an elliptic curve over~$\Q$, then for every
954
rigid prime~$p$, there is a degree~$p$ abelian extension~$K$
955
of~$\Q$ such that
956
$$E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p]),$$
957
where $R=\Res_{K/\Q}(E_K)$ and~$A\subset R$ has
958
dimension~$p-1$ and rank~$0$.
959
\end{theorem}
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961
The proof divides naturally into three steps. First, we use
962
Conjecture~\ref{conj:nonvanishing} to construct~$A$. The we
963
use a theorem of Kato and that formation of N\'eron models
964
commutes with unramified base change to prove that~$A$ has
965
rank~$0$ and that~$p$ does not torsion of Tamagawa numbers
966
of~$A$. Next, we apply the visibility Theorem~\ref{thm:shaexists}
967
to obtain an isomorphism $E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p])$.
968
969
\begin{proof}
970
Conjecture~\ref{conj:nonvanishing} implies that there exists
971
a prime $\ell\nmid N_E$
972
with $\ell\con 1\pmod{p}$ such that $L(E,\chi_{p,\ell},1)\neq 0$ and
973
$a_\ell(E)\not\con 2\pmod{p}$. Let~$\ell$ be such a prime, and
974
let~$K$ be the abelian extension of~$\Q$ corresponding to
975
a surjective character $\chi_{p,\ell} : (\Z/\ell\Z)^* \ra \bmu_p$.
976
Recall from Section~\ref{sec:restriction_of_scalars} that
977
the restriction of scalars $R = \Res_{K/\Q}(E_K)$ is an
978
abelian variety over~$\Q$ of dimension~$p$, and
979
we have a commutative diagram
980
%$$\xymatrix{
981
% & {A}\[email protected]{^(->}[rd]\\
982
%{E[p]\,\,}\[email protected]{^(->}[ur] \[email protected]{^(->}[dr] & & {R}\ar[dr]^{\Tr}\\
983
% & {E}\[email protected]{^(->}[ur]\ar[rr]^{[p]} & & {E}
984
%}$$
985
$$\[email protected]=3pc{
986
{E[p]\,\,}\ar[r]\ar[d] & {E} \ar[dr]^{[p]}\ar[d] \\
987
{A}\ar[r] & {R} \ar[r]^{\Tr} & {E,}
988
}$$
989
where $A=\Ker(\Tr)$ is an abelian variety.
990
991
%Since $\rho_{E,p}$ is irreducible, $p\nmid E(\Q)_{\tor}$.
992
%We have $A(\Q)\subset R(\Q)=E(K)$. Since~$K$ is totally ramified
993
%at~$\ell$, unramified outside~$\ell$, and $\ell\nmid p N_E$, $E(K)[p]$
994
%equals $E(\Q)[p]$, which is $\{0\}$ because $\rho_{E,p}$ is assumed
995
%irreducible. Thus $p\nmid \#A(\Q)$.
996
997
Since $L(A,s)=\prod L(E,\chi_{p,\ell}^\sigma,s)$, and $L(E,\chi,1)\neq
998
0$, Kato's work on Euler systems\edit{Reference?} implies that $A(\Q)$ is finite.
999
Proposition~\ref{prop:ptorsion} implies that
1000
$p\nmid \#A(\Q)\cdot \#(R/E)(\Q)$.
1001
Next suppose that~$q$ is a prime of bad reduction for~$A$. If
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$q\not=\ell$, then $K/\Q$ is unramified at~$q$.
1003
The formation of N\'eron models commutes with unramified base
1004
change\edit{Reference?} and $A_K=E^{\times(p-1)}$, so $c_{A,q}$ divides
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$\cbar_{E,q}$, which is not divisible by~$p$ since~$p$ is rigid
1006
for~$E$. If $q=\ell$, Proposition~\ref{prop:ptorsion} asserts that
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$p\nmid c_{A,q}$.
1008
1009
The previous paragraph combined with Proposition~\ref{prop:ptorsion}
1010
shows that the hypothesis of
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Theorem~\ref{thm:shaexists} are satisfied with $A=A$, $B=E$,
1012
$R=R$, and $L=\Q$. Thus there is an injective map
1013
$$E(\Q)/p E(\Q) \hookrightarrow \Vis_R(\Sha(A/\Q))\subset \Sha(A/\Q).$$
1014
1015
To prove surjectivity, note that by definition every element of
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$\Vis_R(\Sha(A/\Q)[p])$ is the image of an element of $R(\Q)$ and
1017
by Proposition~\ref{prop:ptorsion} the index of $E(\Q)$ in $R(\Q)$ is
1018
finite and coprime to~$p$.
1019
\end{proof}
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1021