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Author: William A. Stein
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\title{\sc
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Visualizing Mordell-Weil Groups of Elliptic Curves Using
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Shafarevich-Tate Groups}
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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~$\Q$. We prove that a very
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plausible conjecture about nonvanishing of prime-degree twists of
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$L(E,s)$ implies that for all but finitely many primes~$p$ there is a
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twist~$A$ of $E^{\times (p-1)}$ such that $E(\Q)/p E(\Q)$ is ``visible''
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as a subgroup of $\Sha(A/\Q)$.
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When $E$ is the elliptic curve of conductor~$43$, our
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construction yields a twist of $E\cross E$
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such that the Birch and Swinnerton-Dyer conjecture predicts that
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$\Sha(A/\Q)[3]$ has order~$3$.
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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~$\Q$. We prove that a very
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plausible conjecture about nonvanishing of prime-degree twists of
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$L(E,s)$ implies that for all but finitely many primes~$p$ there is a
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twist~$A$ of $E^{\times (p-1)}$ such that $E(\Q)/p E(\Q)$ is ``visible''
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as a subgroup of $\Sha(A/\Q)$.
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When $E$ is the elliptic curve of conductor~$43$, our
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construction yields a twist of $E\cross E$
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such that the Birch and Swinnerton-Dyer conjecture predicts that
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$\Sha(A/\Q)[3]$ has order~$3$.
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This paper is organized out as follows.
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In Section~\ref{sec:terminology} we define twisted powers, Tamagawa
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numbers, and rigid primes. We recall in
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Section~\ref{sec:restriction_of_scalars} the definition of the
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restriction of scalars of an elliptic curve and prove a proposition
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about a map induced by trace. In Section~\ref{sec:shaexists} we
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recall a construction of the author and Amod Agashe of visible
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subgroups of Shafarevich-Tate groups. We state a conjecture about
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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, we discuss the extent to
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which the order of $\Sha$ can fail to be square, and describe a
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connection with the Birch and Swinnerton-Dyer conjecture.
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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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\begin{definition}[Twisted Powers]
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A {\em twisted power} of an elliptic curve~$E$ over a field~$K$ is an
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abelian variety $A$ over~$K$ that is isomorphic over $\Kbar$ to
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$E^{\times n}$ 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~$\Q$ with N\'eron model $\cA$ over~$\Z$, and
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let~$p$ be a prime.
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The component group of~$A$ at~$p$ is the finite group scheme
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$\Phi_{A,p} = \cA_{\F_p}/\cA_{\F_p}^0$, where $\cA_{\F_p}^0$ is the
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identity component of $\cA_{\F_p}$. The
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{\em Tamagawa number} of~$A$ at~$p$ is
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$c_{A,p} = \#\Phi_{A,p}(\F_p)$.
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We also set $\cbar_{A,p} = \#\Phi_{A,p}(\Fbar_p)$.
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\end{definition}
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\begin{definition}[Rigid Primes]
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Let $E$ be an elliptic curve over~$\Q$.
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A prime~$p$ is {\em rigid} for~$E$ if~$p$ does not divide
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$2\cdot N_E \cdot \prod_{\ell\mid N_E} \cbar_{E,\ell}$ and the
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representation $\rho_{E,p}:\Gal(\Qbar/\Q)\ra \Aut(E[p])$
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is irreducible. Here $N_E$ is the conductor of~$E$ and
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$\overline{c}_{E,p} = \#\Phi_{E,p}(\Fpbar)$ is the
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order of the component group of~$E$ at~$p$.
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\end{definition}
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\section{Restriction of Scalars}\label{sec:restriction_of_scalars}
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Let~$E$ be an elliptic curve over~$\Q$, and let~$K$ be Galois
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over~$\Q$. We recall the definition of restriction of scalars,
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and prove that the kernel of a certain morphism induced by
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$\tr_{K/\Q}$ is geometrically connected.
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The restriction of scalars $R=\Res_{K/\Q}(E_K)$ is an abelian variety
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over~$\Q$ of dimension $[K:\Q]$, 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 $\Q$-schemes.
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More explicitly, as $\Gal(\Qbar/\Q)$-modules
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we have
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$$
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R(\Qbar) = E(\Qbar \tensor K) \isom E(\Qbar)\tensor_{\Z} \Z[\Gal(K/\Q)],
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$$
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where $\tau\in \Gal(\Qbar/\Q)$ acts on
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$\sum P_\sigma\tensor \sigma \in E(\Qbar)\tensor_{\Z}\Z[\Gal(K/\Q)]$ 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}
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The identity map induces a closed immerion $\iota: E\hookrightarrow
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R$, and the trace $\tr:K\ra \Q$ induces a surjection $\tr:R\ra E$
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whose kernel is geometrically connected.
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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 to~$\Qbar$. First, note that
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$$R_{\Qbar} \ncisom E_{\Qbar}\cross \cdots \cross E_{\Qbar}.$$
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After this base extension, the trace map is the summation map:
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$$+: E_{\Qbar} \cross \cdots \cross E_{\Qbar}
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\longrightarrow E_{\Qbar}.$$
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Let $n=[K:\Q]$. 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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$E_{\Qbar}^{\times (n-1)}$ to $\ker(+)=\ker(\tr_{\Qbar})$.
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Thus $\ker(\tr_{\Qbar})$ is a product of connected varieties,
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hence connected.
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\end{proof}
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\begin{corollary}
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Let $n=[K:\Q]$. Then
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$$(\iota(E)\intersect \ker(\tr))(\Qbar) \isom E[n](\Qbar)\ncisom (\Z/n\Z)^2.$$
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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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$$\Q\hookrightarrow K\xrightarrow{\tr} \Q$$ 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 $E[n](\Qbar)\ncisom (\Z/n\Z)^2$,
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where we have, of course, forgotten the action of $\Gal(\Qbar/\Q)$.
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\end{proof}
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\section{Visibility Theory}\label{sec:shaexists}
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%The following theorem is \cite[Thm.~3.1]{agashe-stein:visibility},
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%and the group $\Vis_R(\Sha(A))$ below is a subgroup of $\Sha(A)$.
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Suppose $\iota: A\hookrightarrow{}R$ is a closed immersion of abelian varieties
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over a number field~$L$. The {\em visible subgroup} of $\Sha(A/L)$ with
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respect to~$\iota$ is
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$$
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\Vis_R(\Sha(A/L)) = \ker(\Sha(A/L) \ra \Sha(R/L)).
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$$
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The following theorem can sometimes be used to prove that
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$\Vis_R(\Sha(A/L))$ is nontrivial. We will use it later to construct
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nontrivial elements of Shafarevich-Tate groups in
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Section~\ref{sec:main}. For an alternative approach which uses
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\'etale cohomology, see Section~\ref{sec:etale}.
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\begin{theorem}\label{thm:shaexists}
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Let~$A$ and~$B$ be abelian subvarieties of an abelian
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variety~$R$ over a number field~$L$ such that $A\intersect B$ is finite,
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and suppose that $A(L)$ is finite.
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Let~$N$ be an integer divisible by the residue characteristics
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of primes of bad reduction for~$R$.
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Suppose~$p$ is a prime such that $e<p-1$, where~$e$ is
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the largest ramification of any
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prime of~$L$ lying over~$p$, and that
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$$p\nmid N \cdot \#(R/B)(L)\cdot\#B(L)_{\tor}\cdot
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\prod_{v} c_{A,v}\cdot c_{B,v},$$
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where $c_{A,v} = \#\Phi_{A,v}(\F_\ell)$ (resp., $c_{B,\ell}$) is
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the Tamagawa number of~$A$ (resp., $B$)
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at~$v$. Suppose furthermore that
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$B[p] \subset A$, where both are viewed as subgroups of $R(\Lbar)$.
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Then there is an isomorphism
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$$B(L)/pB(L)\isom \Vis_R(\Sha(A/L)[p]).$$
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\end{theorem}
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\begin{remark}
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Everything divides infinity, so if $(R/B)(\Q)$ is infinite, then
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no primes satisfy the hypothesis of the theorem.
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\end{remark}
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\begin{proof}
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\cite[Thm.~3.1]{agashe-stein:visibility} produces an embedding
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$$
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B(L)/p B(L)\hra \Vis_R(\Sha(A)[p]).
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$$
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That theorem is proved using a standard snake lemma argument to show
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that $B(L)/p B(L)\hookrightarrow \Vis_R(H^1(L,A))$, combined with a
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local analysis at each prime to see that $B(L)$ lands in $\Sha(A/L)$.
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To prove that this embedding is surjective, note that the index of
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$E(\Q)$ in $R(\Q)$ is finite and coprime to~$p$.
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\end{proof}
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\section{A Conjecture About Nonvanishing of Twists}\label{sec:nonvanishing}
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We state a conjecture about nonvanishing at~$1$ of certain
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prime-degree twists of the $L$-function attached to an elliptic curve,
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provide extensive computational evidence for the conjecture, and give
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an example which suggests that vanishing twists are very rare.
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\subsection{The Conjecture}
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Let~$E$ be an elliptic curve over~$\Q$, and
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suppose $p$ is a rigid prime for~$E$.
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For every prime $\ell\con 1\pmod{p}$, let
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$\chi_{p,\ell} : (\Z/\ell\Z)^* \onto \bmu_p$ be
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one of the Galois-conjugate characters
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of order~$p$ and modulus~$\ell$.
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\begin{conjecture}\label{conj:nonvanishing}
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There exists a prime~$\ell\nmid N_E$ such that
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$$L(E,\chi_{p,\ell},1)\neq 0
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\,\,\,\text{ and }\,\,\,
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a_\ell(E) \not\con 2\pmod{p}.$$
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\end{conjecture}
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The condition $a_\ell(E) \not\con 2\pmod{p}$ requires elaboration.
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Since $\ell\con 1\pmod{p}$, this condition can be rewritten
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$a_\ell(E)\not\con \ell+1\pmod{p}$, which is a ``familiar'' condition
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to impose. We demand that $a_\ell(E)\not\con \ell+1\pmod{p}$ because
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then the characteristic polynomial $x^2 + a_\ell x +\ell\in \F_p[x]$
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of $\Frob_\ell$ on $E[p]$ does not have $+1$ as an eigenvalue. This
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is a key hypothesis in Section~\ref{sec:ptorsion}.
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\begin{table}
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\caption{Evidence for Conjecture~\ref{conj:nonvanishing}\label{tbl:evidence}}
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\noindent%\hspace{-5ex}
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\begin{tabular} {|c|cccccccccccccc|}\hline
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$\,\,\,E$ & 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\\
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\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\\
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\nf{57A} & -&11&29&23&53&103&-&47&59&311&149&83&173&283\\
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\nf{58A} & 7&11&29&23&53&103&191&47&-&311&149&83&173&283\\
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\nf{61A} & 7&31&29&67&53&103&191&47&59&311&149&83&173&283\\
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\nf{65A} & 19&-&43&23&-&137&191&47&59&311&149&83&173&659\\
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\nf{77A} & 19&11&-&-&53&103&191&47&59&311&149&83&173&283\\
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\nf{79A} & 13&11&43&67&53&103&191&47&59&311&149&83&173&283\\
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\nf{82A} & 13&41&29&23&53&103&191&47&59&311&149&-&173&283\\
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\nf{83A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
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\nf{88A} & 7&11&29&-&131&103&191&47&59&311&149&83&173&283\\
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\nf{89A} & 19&11&29&67&53&103&191&47&59&311&149&83&173&283\\
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\nf{91A} & 31&11&-&23&-&103&191&47&59&311&149&83&173&283\\
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\nf{91B} & -&11&-&23&-&103&191&47&59&311&149&83&173&283\\
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\nf{92B} & 13&61&29&23&79&103&229&-&59&311&149&83&173&283\\
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\nf{99A} & -&11&29&-&53&103&191&47&59&311&149&83&173&283\\
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\nf{101A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
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\nf{102A} & -&11&29&23&53&-&191&47&59&311&149&83&173&283\\
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\nf{106B} & 7&11&29&23&53&137&191&47&59&311&149&83&431&283\\\hline
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\nf{389A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
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\nf{433A} & 7&11&43&23&53&103&191&47&59&311&149&83&173&283\\\hline
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\end{tabular}
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\end{table}
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\subsection{Computational Evidence for the Conjecture}
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Using a \magma{} program (see~\cite{magma}), the author's computer
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verified Conjecture~\ref{conj:nonvanishing} for every $p<50$ for the
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first $20$ optimal elliptic curve quotients of $R_0(N)$ of rank~$1$
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and the first~$2$ elliptic curve quotients of rank~$2$.
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Table~\ref{tbl:evidence} contains, for each $p < 50$, the smallest
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prime~$\ell$ satisfying the conditions of
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Conjecture~\ref{conj:nonvanishing}. The elliptic curves are labeled
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as in Cremona. The curves \nf{389A} and \nf{433A} both have rank~$2$,
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and all others have rank~$1$. A dash (-) in the table indicates that the
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corresponding prime is not rigid, so the conjecture does not apply.
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In all cases the first prime $\ell\nmid N_E$ with
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$\ell\con 1\pmod{p}$ with $a_\ell(E)\not \con 2\pmod{p}$
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satisfied $L(E,\chi_{p,\ell},1)\neq 0$, except
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for \nf{61A} with $p=5$, \nf{79A} with $p=7$,
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\nf{82A} with $p=5$, \nf{89A} with $p=11$,
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and \nf{92B} with $p=5$. In every one of these~$5$ exceptional
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cases, the second prime that we tried
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satisfied the conclusion of
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Conjecture~\ref{conj:nonvanishing}.
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\subsection{The Density}
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The following conjecture is not mentioned
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elsewhere in this paper.
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\begin{conjecture}
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Let~$p$ be a rigid prime for an elliptic curve~$E$.
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The set of primes
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$$ \left\{\ell \,\,:\,\, \ell \con 1\!\!\!\!\!\pmod{p}\text{ and }
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L(E,\chi_{p,\ell},1)=0\right\}
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$$
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has Dirichlet density~$0$.
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\end{conjecture}
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The following numerical example gives evidence for this conjecture.
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\begin{example}
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Let~$E$ be \nf{37A} and let $p=5$. Then the only $\ell<1000$ (with
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$\ell\con 1\pmod{5}$) for which $L(E,\chi_{5,\ell},1)=0$ is $\ell=41$.
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% 4 minutes to compute
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\end{example}
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\section{$p$-Torsion of Twisted Powers}\label{sec:ptorsion}
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Let~$p$ and~$\ell$ be as in Conjecture~\ref{conj:nonvanishing}.
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In order to apply Theorem~\ref{thm:shaexists}, it is necessary
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to know that~$p$ does not divide the orders of certain groups.
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In this section, we use that $a_\ell(E)\not\con 2\pmod{p}$ to
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deduce that certain groups do not have any~$p$ torsion.
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The key idea is that the condition on $a_\ell(E)$ implies
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that~$+1$ is not an eigenvalue of $\Frob_\ell$ on the
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$p$-adic Tate module attached to~$E$.
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First, we recall that certain torsion points on the closed
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fiber of a N\'eron model lift to the generic fiber. Let~$K$ be a
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finite extension of~$\Q_\ell$ with ring of integers~$\O$ and residue
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class field~$k$.
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\begin{lemma}\label{lem:red_mod_n}
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Let $A$ be an abelian variety over~$K$ with N\'eron model $\cA$ over~$\O$.
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Then for every integer $n$ not divisible by~$\ell$, there is an isomorphism
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$$A(K)[n] \xrightarrow{\,\,\isom\,\,} \cA(k)[n].$$
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\end{lemma}
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\begin{proof}
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This is a standard fact, whose proof we recall for the convenience
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of the reader.
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Let $A^{1}(K)$ denote the kernel of the natural reduction
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map $r:A(K)\ra \cA(k)$. Because $A^{1}(K)$ is a formal group,
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it is pro-$p$, so $[n]:A^{1}(K)\ra{}A^{1}(K)$ is an isomorphism.
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Since $\cA$ is smooth over~$\O$,
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Hensel's lemma (see BLR) implies that the reduction map
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is surjective, so the following sequence is exact:
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$$0\ra A^1(K) \ra A(K) \ra \cA(k) \ra 0.$$
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The snake lemma applied to the multiplication by~$n$ diagram
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attached to this exact sequence yields the following
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exact sequence:
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$$0\ra0\ra A(K)[n]\ra \cA(k)[n] \ra 0 \ra A(K)/n A(K) \ra \cA(k)/n\cA(k)\ra0,$$
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which proves the proposition.
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\end{proof}
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Let~$E$ be an elliptic curve over~$\Q$ with associated newform
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$f = \sum a_n q^n$, and fix a prime~$p$ that is rigid for~$E$.
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Suppose~$K$ is the extension of~$\Q$ corresponding
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to a surjective Dirichlet character
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$\chi: (\Z/\ell\Z)^* \onto \bmu_p$
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of prime conductor; then~$K$ is
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the subfield of $\Q(\bmu_\ell)$ fixed by $\ker(\chi)$,
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so it is of degree~$p$, is totally ramified
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at~$\ell$, and is unramified outside~$\ell$.
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Let~$A=\ker(\tr : \Res_{K/\Q} E_K \ra E)$.
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We next compute the Tamagawa number
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$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$
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and the $p$-torsion of several abelian varieties.
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\begin{proposition}\label{prop:ptorsion}
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Let~$E$, $\chi$, $K$, and~$A$ be as above and suppose
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that $\ell\nmid N_E$ and $a_\ell \not\equiv 2\pmod{p}$.
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Then the following groups have no nontrivial $p$-torsion:
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$$A(K), \quad A(\Q_\ell),\quad R(\Q_\ell),\quad
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(R/E)(\Q_\ell),\quad \text{and}\quad \Phi_{A,\ell}(\F_\ell).$$
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%$$A(K)[p]=A(\Q_\ell)[p] = R(\Q_\ell)[p] =
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%(R/E)(\Q_\ell)[p]=\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$$
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\end{proposition}
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\begin{proof}
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The reason the $p$-torsion vanishes in all these cases
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is that the condition $a_\ell \not\equiv 2\pmod{p}$ implies
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in each case that $\Frob_\ell$ has no $+1$ eigenvalue.
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The details are as follows.
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We first show that $R(\Q_\ell)[p]=\{0\}$, where $R=\Res_{K/\Q}E_K$.
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By definition,
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$$R(\Q_\ell) = E_K(\Q_\ell\tensor_\Q K) =
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E(K_v)\cross \cdots \cross E(K_v) \quad \text{($p$ copies)},$$
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where $K_v$ is the completion of~$K$ at the unique prime of~$K$
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lying over~$\ell$.
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The action of $\Frob_\ell\in \Gal(\Q_\ell^{\ur}/\Q_\ell)$
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on $E[p](\Q_\ell^{\ur})=E[p](\Qbar_\ell)$ has characteristic
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polynomial
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$F(x) = x^2-a_\ell x + \ell \in \F_p[x]$.
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Since $a_\ell \not\equiv 2\pmod{p}$ and $\ell\equiv 1\pmod{p}$, it
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follows that $\Frob_\ell$ does not have
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$+1$ as an eigenvalue, so $E(\Q_\ell)[p]=\{0\}$.
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If $z\in E(K_v)[p]$, then the field $L=\Q_\ell(z)$ is an unramified
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subfield of the totally ramified field $K_v$, so $z\in E(\Q_\ell)[p]=\{0\}$.
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Thus $E(K_v)[p]=\{0\}$, which implies that $E(K)[p]=\{0\}$ and
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$R(\Q_\ell)[p]=\{0\}$.
418
Since $R_K/E_K \isom E_K \cross \cdots \cross E_K$ ($p-1$ times),
419
we see that
420
$$(R/E)(\Q_\ell)[p]\subset (R/E)(K_v)[p] =
421
(E(K_v)\cross \cdots \cross E(K_v))[p] = \{0\}.$$
422
423
Finally, we turn to the component group $\Phi_{A,\ell}$.
424
Let $\cA$ denote the N\'eron
425
model of~$A$. By Lang's Lemma the natural map $\cA(\F_\ell) \ra
426
\Phi_{A,\ell}(\F_\ell)$ is surjective. Thus if
427
$\Phi_{A,\ell}(\F_\ell)[p]\neq \{0\}$, then $\cA(\F_\ell)[p]\neq
428
\{0\}$. However, by Lemma~\ref{lem:red_mod_n} and observation of the
429
previous paragraph,
430
$$\cA(\F_\ell)[p] = A(\Q_\ell)[p]\subset R(\Q_\ell)[p]=\{0\},$$
431
so $\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$, as claimed.
432
\end{proof}
433
434
435
\subsection{The Tamagawa Number of $A$ at $\ell$}
436
In this section, the notation and hypothesis are as in
437
Proposition~\ref{prop:ptorsion}.
438
That proposition implies that the Tamagawa number
439
$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$ is coprime
440
to~$n$. In this section we use Remark~5.4 of \cite{edixhoven:tame} to
441
prove that in fact $c_{A,\ell}=1$.
442
443
Let $\lambda$ be the prime of~$K$ lying over $\ell$, and
444
let $K_{\lambda}$ denote the completion of~$K$ at~$\lambda$,
445
so $K_{\lambda}$ is totally and tamely ramified over $\Q_\ell$.
446
Since
447
$$A_{K} \isom \ker(\Sigma: E_{K}^{\oplus n}
448
\ra E_{K}),$$
449
and $E_{K_\lambda}$ has good reduction,
450
the geometric closed fiber of the N\'eron model of $A_{K_{\lambda}}$ is
451
$
452
A'_{\kbar}\isom \ker(\Sigma : E_{\kbar}^{\oplus n} \ra E_{\kbar}).
453
$
454
In the notation of \cite{edixhoven:tame},
455
$\mu_n$ acts on $A'_{\kbar}$ by the action
456
it induces by cyclically permuting the factors of
457
$E_{\kbar}^{\oplus n}$. Thus
458
$A_{\kbar}'(\kbar)^{\mu_n}$ is the set of
459
$\sum P_\sigma\tensor\sigma \in E(\kbar)^{\oplus n}$
460
such that all $P_\sigma$ are equal and $\sum P_\sigma = 0$,
461
i.e.,
462
$$
463
A_{\kbar}'(\kbar)^{\mu_n} \isom E(\kbar)[n]\ncisom (\Z/n\Z)^2.
464
$$
465
Thus Remark~5.4 in \cite{edixhoven:tame} implies that
466
$\Phi_{A,\ell}(\kbar) \ncisom E(\kbar)[n]$.
467
By Proposition~\ref{prop:ptorsion},
468
$\Phi_{A,\ell}(k)$ has no elements of order dividing $n$,
469
so $\Phi_{A,\ell}(k)=0$.
470
471
\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}
472
Let $E$ be an elliptic curve over~$\Q$.
473
In this section, we prove that Conjecture~\ref{conj:nonvanishing}
474
implies if $p$ is a rigid prime, then
475
$E(\Q)/p E(\Q)$ is canonical isomorphic to the
476
elements of order $p$ in the visible Shafarevich-Tate
477
group of a rank~$0$ twisted power of~$E$.
478
479
\begin{theorem}\label{thm:main}
480
Assume that Conjecture~\ref{conj:nonvanishing} is true.
481
If~$E$ is an elliptic curve over~$\Q$, then for every
482
rigid prime~$p$, there is a degree~$p$ abelian extension $K$
483
of~$\Q$ such that
484
$$E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p]),$$
485
where $R=\Res_{K/\Q}(E_K)$ and~$A\subset R$ has
486
dimension~$p-1$ and rank~$0$.
487
\end{theorem}
488
The proof divides naturally into three steps. First, we use
489
Conjecture~\ref{conj:nonvanishing} to construct~$A$. The we
490
use a theorem of Kato and that formation of N\'eron models
491
commutes with unramified base change to prove that $A$ has
492
rank~$0$ and that~$p$ does not torsion of Tamagawa numbers
493
of~$A$. Next, we apply the visibility Theorem~\ref{thm:shaexists}
494
to obtain an isomorphism $E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p])$.
495
496
\begin{proof}
497
Conjecture~\ref{conj:nonvanishing} implies that there exists
498
a prime $\ell\nmid N_E$
499
with $\ell\con 1\pmod{p}$ such that $L(E,\chi_{p,\ell},1)\neq 0$ and
500
$a_\ell(E)\not\con 2\pmod{p}$. Let~$\ell$ be such a prime, and
501
let~$K$ be the abelian extension of~$\Q$ corresponding to
502
a surjective character $\chi_{p,\ell} : (\Z/\ell\Z)^* \ra \bmu_p$.
503
Recall from Section~\ref{sec:restriction_of_scalars} that
504
the restriction of scalars $R = \Res_{K/\Q}(E_K)$ is an
505
abelian variety over~$\Q$ of dimension~$p$, and
506
we have a commutative diagram
507
%$$\xymatrix{
508
% & {A}\[email protected]{^(->}[rd]\\
509
%{E[p]\,\,}\[email protected]{^(->}[ur] \[email protected]{^(->}[dr] & & {R}\ar[dr]^{\tr}\\
510
% & {E}\[email protected]{^(->}[ur]\ar[rr]^{[p]} & & {E}
511
%}$$
512
$$\[email protected]=3pc{
513
{E[p]\,\,}\ar[r]\ar[d] & {E} \ar[dr]^{[p]}\ar[d] \\
514
{A}\ar[r] & {R} \ar[r]^{\tr} & {E,}
515
}$$
516
where $A=\ker(\tr)$ is an abelian variety.
517
518
%Since $\rho_{E,p}$ is irreducible, $p\nmid E(\Q)_{\tor}$.
519
%We have $A(\Q)\subset R(\Q)=E(K)$. Since~$K$ is totally ramified
520
%at~$\ell$, unramified outside~$\ell$, and $\ell\nmid p N_E$, $E(K)[p]$
521
%equals $E(\Q)[p]$, which is $\{0\}$ because $\rho_{E,p}$ is assumed
522
%irreducible. Thus $p\nmid \#A(\Q)$.
523
524
Since $L(A,s)=\prod L(E,\chi_{p,\ell}^\sigma,s)$, and $L(E,\chi,1)\neq
525
0$, Kato's work on Euler systems\edit{Reference?} implies that $A(\Q)$ is finite.
526
Proposition~\ref{prop:ptorsion} implies that
527
$p\nmid \#A(\Q)\cdot \#(R/E)(\Q)$.
528
Next suppose that~$q$ is a prime of bad reduction for~$A$. If
529
$q\not=\ell$, then $K/\Q$ is unramified at~$q$.
530
The formation of N\'eron models commutes with unramified base
531
change\edit{Reference?} and $A_K=E^{\times(p-1)}$, so $c_{A,q}$ divides
532
$\cbar_{E,q}$, which is not divisible by~$p$ since~$p$ is rigid
533
for~$E$. If $q=\ell$, Proposition~\ref{prop:ptorsion} asserts that
534
$p\nmid c_{A,q}$.
535
536
The previous paragraph combined with Proposition~\ref{prop:ptorsion}
537
shows that the hypothesis of
538
Theorem~\ref{thm:shaexists} are satisfied with $A=A$, $B=E$,
539
$R=R$, and $L=\Q$. Thus there is an injective map
540
$$E(\Q)/p E(\Q) \hookrightarrow \Vis_R(\Sha(A/\Q))\subset \Sha(A/\Q).$$
541
542
To prove surjectivity, note that by definition every element of
543
$\Vis_R(\Sha(A/\Q)[p])$ is the image of an element of $R(\Q)$ and
544
by Proposition~\ref{prop:ptorsion} the index of $E(\Q)$ in $R(\Q)$ is
545
finite and coprime to~$p$.
546
\end{proof}
547
548
549
550
\section{Applications}\label{sec:applications}
551
We apply the above results to prove that
552
Conjecture~\ref{conj:nonvanishing} implies the existence of
553
elements of Shafarevich-Tate groups of twisted powers of
554
elliptic curves of every prime order. We also construct an
555
abelian variety~$A$ over~$\Q$ such that the Birch
556
and Swinnerton-Dyer conjecture predicts that $\Sha(A/\Q)[3]=\Z/3\Z$ and
557
that $\#\Sha(A/\Q)$ is not a square or twice a square.
558
559
\subsection{Existence of Elements of $\Sha$ of all Prime Orders}
560
\begin{proposition}\label{prop:all_prime_orders}
561
Let~$p$ be a prime number. Then Conjecture~\ref{conj:nonvanishing}
562
implies that there exists infinitely many twisted powers~$A$ of some
563
elliptic curve such that $\Sha(A/\Q)[p]\neq \{0\}$.
564
\end{proposition}
565
\begin{proof}
566
Most of the proposition can be proved using a single elliptic curve.
567
When ordered by conductor, the first elliptic curve~$E$ over $\Q$ with
568
positive rank has prime conductor~$37$ and is defined by the
569
Weierstrass equation $y^2 + y = x^3 - x$. Table~1 of
570
\cite{cremona:algs} shows that~$E$ is isolated in its isogeny class,
571
so \cite[Exercise~4]{ribet-stein:serre} implies that representations
572
$\rho_{E,p}$ are irreducible. Since
573
$\ord_{37}(j(E))=-1$, $\cbar_{37}=1$. Thus all odd primes $p\neq 37$
574
are rigid for~$E$. The proposition then follows for all odd primes
575
$p\neq 37$ by Theorem~\ref{thm:main}.
576
577
We complete the proof as follows. Exactly the same argument applied
578
to the unique elliptic curve of conductor~$43$ proves the proposition
579
for all odd primes $p\neq 43$. Finally, B\"olling proved in
580
\cite{bolling:sha} that for every $j\in\Q$ there is an elliptic
581
curve~$E$ with $j$-invariant~$j$ such that infinitely many twists~$E'$
582
of $E$ have $\Sha(E'/\Q)[2]\neq\{0\}$.
583
\end{proof}
584
585
\subsection{The Possible Orders of Shafarevich-Tate Groups}
586
On page 306--307 of \cite{tate:bsd}, Tate discusses results about the
587
structure of the group $\Sha(A/K)$, where~$A$ is an abelian variety over
588
a number field~$K$. He asserts that if~$A$ is a Jacobian then $\#\Sha(A/K)$
589
is a perfect square. Poonen and Stoll subsequently pointed out
590
in~\cite{poonen-stoll} that Tate's assertion is not quite correct. In
591
fact, Poonen and Stoll prove that when~$A$ is a Jacobian, $\#\Sha(A/K)$
592
is either a square or twice a square, and they give examples in which
593
$\#\Sha(A/K)$ is twice a square. Tate does not discuss the case
594
when~$A$ is not a Jacobian, except to mention results that imply that
595
$\#\Sha(A/K)$ is square away from~$2$ and primes that
596
don't divide the degree of some polarization of~$A$.
597
598
Now suppose~$A$ is an arbitrary abelian variety over a number field~$K$.
599
In this case, it has remained an unresolved problem during the last
600
$35$ years to decide whether or not $\#\Sha(A/K)$ is a square or twice a
601
square. Let~$E$ be an elliptic curve over~$\Q$ of rank~$1$. Then
602
the construction of the present paper gives, for suitable primes~$p$,
603
an injection
604
$$\Z/p\Z \ncisom E(\Q)/p E(\Q) \hookrightarrow \Sha(A/\Q),$$ where~$A$
605
is an abelian variety over~$\Q$ which is a twist of $E^{\times p-1}$.
606
Thus $\Sha(A/\Q)[p]$ has a ``natural'' subgroup of order~$p$; moreover,
607
no other natural subgroup of order~$p$ presents itself. Is
608
$\#\Sha(A/\Q)[p]\ncisom\Z/p\Z$? If the answer is yes for even a
609
single~$p>2$, then the question of whether or not $\#\Sha(A/\Q)$ must be
610
a square or twice a square is finally resolved.
611
We make the following contribution toward settling this problem.
612
613
\begin{proposition}
614
Let~$E$ be the unique elliptic curve over~$\Q$ of conductor~$43$.
615
Let $K=\Q(\mu_7)^+$ be the real subfield of $\Q(\mu_7)$, let
616
$R=\Res_{K/\Q}E_K$, and
617
let $A = \ker\left(R \ra E\right).$
618
Then $3\mid \#\Sha(A/\Q)$ and the Birch and Swinnerton-Dyer conjecture
619
implies that $\#\Sha(A/\Q)[3]=3$ and $\#\Sha(R/\Q)[3]=1$.
620
\end{proposition}
621
\begin{proof}
622
We first verify that the BSD conjecture predicts that
623
$\#\Sha(E/K)[3]=1$.
624
Because $K/\Q$ is abelian,
625
$$L(E_K,s) = L(E,s)\cdot L(E,\chi,s)\cdot L(E,\chi^{-1},s),$$
626
where $\chi:(\Z/7\Z)^*\ra \mu_3$ is a Dirichlet character of order~$3$.
627
For each of the three real places~$v$ of~$K$, we have
628
$\Omega_{E,v}=\Omega_{E/\Q}$. Next observe that
629
$$E(\Q)\tensor\Z_3 \ra E(K)\tensor \Z_3$$
630
is surjective, because
631
$$\frac{E(K)}{E(\Q)} = \frac{R(\Q)}{E(\Q)}
632
\hookrightarrow (R/E)(\Q)_{\tor}$$
633
and $(R/E)(\Q)[3]=\{0\}$ by Proposition~\ref{prop:ptorsion}.
634
If $P\in E(\Q)$ then
635
$$\langle P, P\rangle_\Q = \frac{1}{[K:\Q]} \langle P, P \rangle_K,$$
636
so $\Reg(E/K) \sim 3\cdot \Reg(E/\Q)$, where~$\sim$ denotes
637
``equality up to a number coprime to~$3$''.
638
By Proposition~\ref{prop:ptorsion}, $E(K)_{\tor}[3]=\{0\}$ and
639
$3\nmid c_v$ for places~$v$ of~$\Q$ (because this is true
640
for~$R$).
641
Finally,
642
\begin{align*}
643
\#\Sha(E/K) &=
644
\frac{L'(E_K,1)\cdot \#E(K)_{\tor}^2}
645
{\Reg(E/K)\cdot\Omega_{E/\Q}^3\cdot \prod_{v} c_{v}}\\
646
&\sim \frac{L'(E,1)}{3\cdot \Reg(E/\Q) \cdot \Omega_E}
647
\cdot \Norm_{\Q(\mu_3)/\Q}\left(\frac{L(E,\chi,1)}{\Omega_E}\right)\\
648
&= \frac{1}{3}\cdot 3=1.
649
\end{align*}
650
We verified the last nontrivial equality with a computer using standard
651
modular symbols techniques.
652
653
We have an exact sequence
654
$$0 \ra \Vis(\Sha(A/\Q)[3]) \ra \Sha(A/\Q)[3] \ra
655
\Sha(R/\Q)[3].$$
656
Since $$\Sha(R/\Q)[3]=\Sha(E/K)[3]=\{0\},$$
657
$$E(\Q)/3 E(\Q)\hookrightarrow \Vis(\Sha(A/\Q)[3]),$$
658
and $\Vis(\Sha(A/\Q)[3])$ is a surjective image of
659
$R(\Q)\tensor\Z_3 = E(K)\tensor\Z_3$, which,
660
as mentioned above, is a surjective image of $E(\Q)\tensor\Z_3=\Z_3$,
661
it follows that
662
$$\Vis(\Sha(A/\Q)[3])=\Z/3\Z.$$
663
\end{proof}
664
665
\subsection{What Goes Wrong when $p=2$?}
666
In the previous section, we set $p=3$ and
667
constructed an abelian variety $A$ of dimension $p-1$
668
that (conjecturally) has nonsquare $\Sha(A/\Q)[p]$.
669
We can construct an~$A$ in an analogous way for any odd prime~$p$,
670
and the author expects that $\Sha(A/\Q)[p]$ is nonsquare in most cases.
671
However, when $p=2$, the dimension of~$A$ is~$1$, so
672
in that case $\#\Sha(A/\Q)$ must be a perfect square.
673
674
What goes wrong? The problem lies in Theorem~\ref{thm:shaexists}.
675
The argument used to prove Theorem~\ref{thm:shaexists}
676
at least provides a map
677
$$
678
E(\Q)/2 E(\Q)\hookrightarrow\Vis_R(H^1(\Q,A)).
679
$$
680
When $p=2$, the condition $e<p-1$ is not satisfied, so
681
the proof of Theorem~\ref{thm:shaexists} does not show
682
that the image of $E(\Q)/ 2 E(\Q)$ is locally trivial
683
at the prime~$2$ (or at $\infty$). We thus only
684
construct a subgroup of $H^1(\Q,A)$ of nonsquare order,
685
not of~$\Sha(A/\Q)$. Thus even if two elliptic curves have the same
686
$E[2]$, then can still possess very different Selmer groups.
687
688
689
\subsection{A Connection with the BSD Conjecture}
690
\label{sec:bsd}
691
There is heuristic reason why equalities like the one that we proved
692
with a computer computation in the previous section should frequently
693
be true.
694
First, the Birch and Swinnerton-Dyer conjecture predicts that
695
$$\#\Sha(E/\Q)=\frac{L'(E,1)}{\Reg(E/\Q) \cdot \Omega_E},$$ and in
696
fact one knows that $\Sha(E/\Q)[3]=\{1\}$. Second, because~$\chi$ is
697
congruent to the identity character modulo~$3$,
698
$$L(E,\chi,1)/\Omega_E \con L(E,1)=0\pmod{3},$$
699
suitably interpreted,
700
and there is no reason why this congruence should hold modulo $3^2$.
701
Thus, in the situation of Theorem~\ref{thm:main}, the author expects
702
that usually $\Sha(A/\Q)[p]\isom E(\Q)/p E(\Q)$. More generally,
703
one suspects that usually
704
$$\Sel^{(p)}(A/\Q) \isom \Sel^{(p)}(E/\Q).$$
705
706
\begin{conjecture}\label{conj:strong_nonvanishing}
707
Let the notation be as in Conjecture~\ref{conj:nonvanishing}.
708
Then there exists a rigid prime~$p$ and a prime~$\ell\nmid N_E$ such
709
that $L(E,\chi_{p,\ell},1)\neq 0$, $a_\ell(E) \not\con 2\pmod{p}$,
710
and $\Sha(E/K)[p]=\{0\}$, where~$K$ corresponds
711
to $\chi_{p,\ell}$.
712
\end{conjecture}
713
\begin{proposition}
714
Assume Conjecture~\ref{conj:strong_nonvanishing} and the following
715
weak consequence of the Birch and Swinnerton-Dyer conjecture: if~$A$
716
is a twisted power of an elliptic curve of analytic
717
rank~$0$ over~$\Q$ and $p\mid L(A,1)/\Omega_A$, then $p\mid
718
\#\Sha(A/\Q)\cdot \prod c_q$. If~$E$ is an elliptic curve over~$\Q$
719
and $L(E,1)=0$, then $E(\Q)$ is infinite.
720
\end{proposition}
721
722
723
\section{Etale Cohomology Approach to Constructing $\Sha(A)$}
724
\label{sec:etale}
725
726
Let~$E$ be an elliptic curve over a number field~$L$, let~$K$ be a
727
finite Galois extension of~$L$ that is only ramified
728
at primes that don't divide the conductor $N_E$ of~$E$.
729
Let $R=\Res_{K/L} E_K$ and $A = \ker(R \ra E)$, so we have
730
an exact sequence
731
$$
732
0 \ra A \ra R \ra E \ra 0
733
$$
734
of abelian varieties.
735
\begin{lemma}\label{lem:etale}
736
Let $A$, $R$, and $E$ be as above. Assume that
737
$\disc(K)$, $n=[K:L]$, and $N_E$ have no pairwise common
738
factors and that $v(p)<p-1$
739
for each $p\mid n$ and for each valuation~$v$ on
740
$\O_L$ corresponding to a prime of residue characteristic~$p$
741
(thus, e.g., $n$ must be odd).
742
Then the corresponding complex of N\'eron models
743
$$
744
0 \ra \cA \ra \cR \ra \cE \ra 0
745
$$
746
is exact.
747
\end{lemma}
748
\begin{proof}
749
We must show that for every prime~$\p$ of~$L$
750
the complex
751
\begin{equation}\label{eqn:neron}
752
0 \ra \cA_{\O_\p} \ra \cR_{\O_\p} \ra \cE_{\O_\p}\ra 0
753
\end{equation}
754
is exact, where $\O_\p$ denotes the completion of $\O_L$ at~$\p$.
755
Suppose~$\p$ is a prime of~$L$.
756
By the criterion of Neron-Ogg-Shafarevich,
757
the only possible primes of bad reduction for~$R$ are those
758
that divide $N_E$ and those that ramify in~$K$.
759
760
If $\gcd(\chr(\O/\p),n)=1$, then
761
\cite[Prop.~7.5.3 (a)]{neronmodels}, with
762
$B = E\hra R$, implies
763
that if $\p\nmid N_E\cdot \disc(K)$, then
764
(\ref{eqn:neron}) is exact, since $\cR$ has
765
good reduction at~$\p$. If $\p\nmid \disc(K)$,
766
then the base extension of (\ref{eqn:neron})
767
to $\O_{K,\p}$ is exact, because the formation of
768
N\'eron models commutes with unramified base extension
769
and over~$K$ the sequence of abelian varieties
770
is $0\ra E_K^{\oplus (n-1)} \ra E_K^{\oplus n}\xra{\Sigma} E_K\ra 0$.
771
If $\p\nmid N_E$, then \cite[Prop.~7.5.3 (a)]{neronmodels}
772
implies that $\cA_{\O_\p}\ra \cR_{\O_\p}$ is a closed immersion,
773
and the cokernel of $\cR_{\O_\p}\ra \cE_{\O_\p}$ is killed
774
by multiplication by~$n$; however, in the proof of
775
\cite[Prop.~7.5.3 (a)]{neronmodels} (see
776
the top of page 187), one only uses that $\cE_{\O_\p}$ has good
777
reduction to deduce surjectivity, so in
778
fact $\cR_{\O_\p}\ra \cE_{\O_\p}$ is surjective.
779
Since $\gcd(N_E,\disc(K))=1$, this completes the argument
780
when $\gcd(\chr(\O/\p),n) = 1$.
781
782
Next suppose $\gcd(\chr(\O/\p),n)\neq 1$.
783
Our hypothesis on~$n$ implies that $p=\chr(\O/\p)$ satisfies
784
$v(p)<p-1$ and $\p\nmid N_E\cdot \disc(K)$.
785
By \cite[Th.~7.5.4(iii)]{neronmodels},
786
the sequence (\ref{eqn:neron}) is exact.
787
\end{proof}
788
789
By the lemma, we have an exact sequence of sheaves
790
$$
791
0 \ra \cA \ra \cR \ra \cE \ra 0
792
$$
793
on the \'etale site over $X=\Spec(\O_L)$.
794
Let $\cA^{\vee}$ denote the N\'eron model of the dual of~$A$.
795
\begin{proposition}\label{prop:et}
796
The following diagram has exact rows and column:
797
$$\xymatrix{
798
& & & {H^1(X_{\et},\cE)}\ar[d]\\
799
{R(L)} \ar[r] & {E(L)}\ar[r] & {H^1(X_{\et},\cA)}\ar[r] & {H^1(X_{\et},\cR)}\ar[d]\ar[r] & {H^1(X_{\et},\cE)}\\
800
& & & {H^1(X_{\et},\cA^{\vee})}
801
}
802
$$
803
(Actually the vertical one might not be exact, but it is close
804
enough, because $0 \ra \cE \ra \cR \ra \cA$ is exact.)
805
\end{proposition}
806
By \cite[Appendix]{mazur:tower}, there is an exact sequence
807
$$
808
0 \ra \Sha(A) \ra H^1(X_{\et},\cA) \ra G \ra 0
809
$$
810
where~$G$ is a finite group whose order if divisible only by~$2$ and
811
primes that divide the Tamagawa numbers of $\cA$.
812
813
814
Alternatively, one can consider the following diagram
815
$$
816
\xymatrix{
817
& 0\ar[d] & 0\ar[d]\\
818
0\ar[r] & {\cE[n]}\ar[r]\ar[d] & {\cE}\ar[d]\ar[dr]^{[n]}\\
819
0 \ar[r] & {\cA} \ar[r] & {\cR}\ar[r] & {\cE} \ar[r] & 0
820
}$$
821
and work with cohomology on the fppf site.
822
823
\subsection{Connection with the Birch and Swinnerton-Dyer Conjecture}
824
Suppose $E$ is an elliptic curve over~$\Q$ and that $L(E,1)=0$. The
825
Birch and Swinnerton-Dyer conjecture for~$E$ asserts (among other
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things) that $E(\Q)$ is infinite. Suppose~$A$ is constructed as in
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Section~\ref{sec:main}. In this section we describe why if a certain
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consequence of a refinement of the Birch and Swinnerton-Dyer
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conjecture for~$A$ is true, then $\Sel^{(n)}(E/\Q)$ is nonzero.
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Using modular symbols one sees that $L(A,1)\con 0 \pmod{\ell}$,
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so a refinement of the Birch and Swinnerton-Dyer formula for rank~$0$
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abelian varieties predicts that there should be a nonzero element in
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$\ker (\Sha(A)\ra \Sha(A/E[n]))$.
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Thus by Proposition~\ref{prop:et}, either
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$H^1(X_{\et},\cE)[n]\neq 0$, or there is a nonzero element
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of order dividing~$n$ in
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$$
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\ker(H^1(X_{\et},\cA)\ra H^1(X_{\et},\cR)) \isom E(\Q)/R(\Q),
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$$
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in which case $E(\Q)/R(\Q)$ contains a nonzero element of
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order dividing~$n$,
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so $E(\Q)$ is infinite.
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Thus either $\Sha(E)[n]\neq 0$ or $E(\Q)$ is infinite, so
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$\Sel^{(n)}(E/\Q)$ is nonzero.
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\bibliography{biblio}
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\end{document}
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E := EC("37A");
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M := MS(E);
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K := CyclotomicField(5);
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M5 := BaseExtend(M,K);
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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
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[ <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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