\documentclass[11pt]{article}1\bibliographystyle{amsplain}2\include{macros}3\usepackage[all]{xy}4\title{\sc Nonsquare Odd Shafarevich-Tate Groups and Mordell-Weil Groups of Elliptic Curves}5\author{William A. Stein\\{\sf was@math.harvard.edu}}6\hoffset=-0.066\textwidth7\textwidth=1.133\textwidth8\voffset=-0.065\textheight9\textheight=1.13\textheight10\newcommand{\PP}{\mathfrak{P}}11\renewcommand{\q}{\mathfrak{q}}12\renewcommand{\Fbar}{\overline{F}}13\begin{document}14\maketitle1516\begin{abstract}17Let~$E$ be an elliptic curve over a number field~$F$.18We study certain exact sequences of abelian varieties19$0\ra A \ra R\ra E \ra 0$20that induce a natural identification21$E(F)/p E(F) \isom \Ker(\Sha(A/F)\ra \Sha(R/F))$.22For many odd primes~$p$, we construct the23first ever examples of abelian varieties~$A$24over~$\Q$ such that the $p$-part of25$\#\Sha(A/F)$ is {\em not} a perfect square.26We also discuss links between our results and conjectures27and the problem of constructing elements of Selmer groups28of elliptic curves over~$\Q$ with analytic rank bigger than~$1$.29\end{abstract}3031\tableofcontents3233\section*{Introduction}34Let~$E$ be an elliptic curve over a number field~$F$.35We study certain exact sequences of abelian varieties36$0\ra A \ra R\ra E \ra 0$37that induce a natural identification38$E(F)/p E(F) \isom \Ker(\Sha(A/F)\ra \Sha(R/F))$.39For many odd primes~$p$, we construct the40first ever examples of abelian varieties~$A$41over~$\Q$ such that the $p$-part of42$\#\Sha(A/F)$ is {\em not} a perfect square.43We also discuss links between our results and conjectures44and the problem of constructing elements of Selmer groups45of elliptic curves over~$\Q$ with analytic rank bigger than~$1$.4647In particular, let~$E$ be an elliptic curve over~$\Q$. We prove that48a conjecture about nonvanishing of prime-degree twists of $L(E,s)$49implies that for all but finitely many primes~$p$ there is a twist~$A$50of $E^{\times (p-1)}$ such that $E(\Q)/p E(\Q)$ is identified in a51natural way with a subgroup of $\Sha(A/\Q)$.5253We show how to construct twists~$A$ of powers54of elliptic curves over~$F$ such that $\Sha(A/F)(p)$ has55{\em nonsquare} order for an odd prime~$p$.56For example, if $E$ is the elliptic curve of conductor~$43$57then there is a two-dimensional abelian variety $A$ that is isomorphic58to $E\cross E$ over $\Q(\mu_3)^+$ and59$\#\Sha(A/\Q)(3)$ is not a square.6061This paper is organized as follows. In Section~\ref{sec:terminology}62we define twisted powers, Tamagawa numbers, and rigid primes. We63recall in Section~\ref{sec:restriction_of_scalars} the definition of64the restriction of scalars of an elliptic curve and prove a65proposition about a map induced by trace.66We state67a conjecture about nonvanishing of twists of prime degree in68Section~\ref{sec:nonvanishing}, and give computational evidence for69this conjecture. In Section~\ref{sec:ptorsion} we prove triviality of70the $p$-torsion of several abelian groups attached to twisted powers71of an elliptic curve. The heart of the paper is72Section~\ref{sec:main}, which uses the above results to construct73subgroups of Shafarevich-Tate groups of twisted powers.74Section~\ref{sec:applications} pulls together the results of the75previous sections; there we prove that the conjecture of76Section~\ref{sec:nonvanishing} implies the existence of elements of77Shafarevich-Tate groups of every prime order, and describe a78connection with the Birch and Swinnerton-Dyer conjecture.79In Section~\ref{} we discuss the extent to80which the order of $\Sha$ can fail to be square.8182\vspace{2ex}\par\noindent{}{\bf{}Acknowledgement: }83It is a pleasure to thank Gautam Chinta,84Benedict Gross, Emanuel Kowalski, Barry Mazur,85Bjorn Poonen, David Rohrlich,86and Michael Stoll87for helpful comments and conversations.888990\section{Terminology}\label{sec:terminology}91In this section, we define twisted powers and rigid primes for an92elliptic curve, and recall the definition of Tamagawa numbers of an93abelian variety.9495Let $E$ be an elliptic curve over a number field~$F$.9697\begin{definition}[Twisted Powers]98A {\em twisted power} of~$E$ is an abelian variety~$A$ over~$F$99that is isomorphic over $\Kbar$ to $E^{\cross n}$100for some positive integer~$n$.101\end{definition}102103We recall the standard notion of Tamagawa number of an abelian variety~$A$,104and introduce the notation $\cbar_{A,p}$ for the order of the group105of components of~$A$ over $\Fbar_p$.106\begin{definition}[Tamagawa Numbers]107Let~$A$ be an abelian variety over~$F$ with N\'eron108model $\cA$ over~$\O_F$, and let~$\p$ be a prime of $\O_F$,109and let $k=\O_F/\p$.110The component group of~$A$ at~$\p$ is the finite group scheme111$\Phi_{A,\p} = \cA_{k}/\cA_{k}^0$, where $\cA_{k}^0$ is the112identity component of $\cA_{k}$. The113{\em Tamagawa number} of~$A$ at~$\p$ is114$c_{A,\p} = \#\Phi_{A,\p}(k)$.115Also set $\cbar_{A,\p} = \#\Phi_{A,\p}(\overline{k})$.116\end{definition}117118Let $N_E\in \O_F$ be the conductor of~$E$.119120\begin{definition}[Rigid Primes]121A prime $\p$ of $\O_F$ is {\em rigid} for~$E$ if $\p$ does not divide122$2\cdot N_E \cdot \prod_{\q\mid N_E} \cbar_{E,\q}$ and the123representation $\rho_{E,\p}:\Gal(\Fbar/\F)\ra \Aut(E[\p])$124is irreducible.125\end{definition}126127128\section{Restriction of Scalars}\label{sec:restriction_of_scalars}129In this section,130we recall the notion of restriction of131scalars, and prove that the kernel of a morphism induced by132a trace is geometrically connected.133134Let~$E$ be an elliptic curve over a number field~$F$, and let~$K$135be a finite extension of~$F$.136The restriction of scalars $R=\Res_{K/F}(E_K)$ is an abelian variety137over~$F$ of dimension $[K:F]$, which is characterized by the138following universal property: There is a functorial group isomorphism139$R(S) \isom E_K(S_K)$, where~$S$ varies over all $F$-schemes.140141There is a more explicit description of $\Res_{K/F}(E_K)$ when~$K$ is Galois142over~$F$. As $\Gal(\Fbar/F)$-modules, we have143$$144R(\Fbar) = E(\Fbar \tensor K) \isom E(\Fbar)\tensor_{\Z} \Z[\Gal(K/F)],145$$146where $\tau\in \Gal(\Fbar/F)$ acts on147$\sum P_\sigma\tensor \sigma \in E(\Fbar)\tensor_{\Z}\Z[\Gal(K/F)]$ by148$$\tau\left(\sum P_\sigma\tensor \sigma\right) =149\sum \tau(P_\sigma)\tensor \sigma\tau_{|K}.150$$151Moreover, the $L$-series of~$R$ is $\prod_{a=1}^{n} L(E,\chi^a,s)$, and~$R$152has good reduction at all $p\nmid \ell\cdot N$.153154155\begin{proposition}\label{prop:kergeo}156The identity map induces a closed immerion $\iota: E\hookrightarrow157R$, and the trace $\Tr:K\ra F$ induces a surjection $\Tr:R\ra E$158whose kernel is geometrically connected. Thus we have an exact sequence159\begin{equation}\label{eqn:exactabvar}1600 \ra A \ra R \ra E \ra 0161\end{equation}162with $A$ an abelian variety.163\end{proposition}164\begin{proof}165The existence of~$\iota$ and $\Tr$ follows from Yoneda's lemma.166The map~$\iota$ is induced by the functorial inclusion167$E(S)\hookrightarrow E_K(S_K)=R(S)$, so~$\iota$ is injective.168169The $\Tr$ map is induced by the usual170functorial trace map on points171$R(S)=E_K(S_K)\xrightarrow{\Tr} E(S)$.172To verify that $\Ker(\Tr)$ is geometrically connected, we base173extend the exact sequence (\ref{eqn:exactabvar}) to~$\Fbar$. First, note that174$$R_{\Fbar} \ncisom E_{\Fbar}\cross \cdots \cross E_{\Fbar}.$$175After base extension, the trace map may be identified with the summation map176$$+: E_{\Fbar} \cross \cdots \cross E_{\Fbar}177\longrightarrow E_{\Fbar}.$$178Let $n=[K:F]$. The map defined by179$$\left(a_1,\ldots, a_{n-1}\right) \mapsto180\left(a_1,a_2,\ldots ,a_{n-1},-\sum_{i=1}^{n-1} a_i\right),$$181is an isomorphism from182$E_{\Fbar}^{\times (n-1)}$ to $\Ker(+)=\Ker(\Tr_{\Fbar})$.183Thus $\Ker(\Tr_{\Fbar})$ is a product of copies of $E_{\Fbar}$,184hence is connected.185\end{proof}186187\begin{corollary}188Let $n=[K:F]$. Then189$$(\iota(E)\intersect \Ker(\Tr))(\Fbar) \isom E[n](\Fbar)\ncisom (\Z/n\Z)^2.$$190(The rightmost map is an isomorphism of groups, not Galois modules.)191\end{corollary}192\begin{proof}193Since the map194$$F\hookrightarrow K\xrightarrow{\Tr} F$$ is195multiplication by~$n$, the composite map196$$E \hookrightarrow R \longrightarrow E$$197is also multiplication by~$n$.198The corollary now follows since199$\iota(E)\intersect \Ker(\Tr)$ is the kernel of~$\Tr\circ\iota$,200which equals $[n]$. It is elementary that $E[n](\Fbar)\ncisom (\Z/n\Z)^2$,201where we have, of course, forgotten the action of $\Gal(\Fbar/F)$.202\end{proof}203204\subsection{Exactness of the Complex of N\'eron Models}205\label{sec:etale}206207Let~$E$ be an elliptic curve over a number field~$F$, and let~$K$ be a208finite Galois extension of~$F$ that is only ramified209at primes that don't divide the conductor $N_E$ of~$E$.210Let $R=\Res_{K/F} E_K$ and $A = \Ker(R \ra E)$, so211by Proposition~\ref{prop:kergeo}212we have an exact sequence of abelian varieties213$2140 \ra A \ra R \ra E \ra 0.215$216217If a complex of N\'eron models over a ring $\O$ is exact in the sense218of \cite{neronmodels}, then it induces an exact sequence of sheaves on219the \'etale site for $\Spec(\O)$.220221\begin{proposition}\label{lem:etale}222Let $A$, $R$, and $E$ be as above. Assume that223the three elements224$\disc(K)$, $n=[K:F]$, and $N_E$ of $\O_F$225are all relatively prime to each other226and that $v(p)<p-1$ for each $p\mid n$ and227for each valuation~$v$ on $\O_F$ corresponding228to a prime $\p$ of residue characteristic~$p$, normalized229so that the uniformizing element of $\O_{F,\p}$230has valuation~$1$ (in particular, if $F=\Q$ this is the231condition that~$n$ is odd).232Then the corresponding complex of N\'eron models233$$2340 \ra \cA \ra \cR \ra \cE \ra 0235$$236is exact.237\end{proposition}238\begin{proof}239We use the results of \cite[Ch.~7]{neronmodels} to prove240that for every completion $\O_{F,\p}$ of $\O_F$ that the complex241\begin{equation}\label{eqn:neron}2420 \ra \cA_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}\ra 0243\end{equation}244is exact. (For the reader's convenience, the245results of \cite[Ch.~7]{neronmodels} that we refer246to below are reproduced in Section~\ref{sec:neronmodels}.)247To this end, fix a prime ideal~$\p$ of $\O_F$,248and let~$p$ be its residue characteristic.249250First suppose that $\p\nmid N_E \cdot \disc(K)$, so~$\p$251is a prime of good reduction for~$R$.252If $p\nmid n$, \cite[Prop.~7.5.3 (a)]{neronmodels} implies253that sequence (\ref{eqn:neron}) is exact.254If $p\mid n$ then our hypothesis on divisors of $n$ are255exactly the hypothesis to \cite[Th.~7.5.4(iii)]{neronmodels},256which implies that (\ref{eqn:neron}) is exact.257258Next suppose that $\p\mid N_E\cdot \disc(K)$, so259$\p \mid N_E$ or $\p \mid \disc(K)$.260Then since $\gcd(n,N_E\cdot \disc(K))=1$, we have that $p\nmid n$,261so we are led to apply \cite[Prop.~7.5.3 (a)]{neronmodels} with262$B_K = E\subset R$.263Because $N_E$ and $\disc(K)$ are coprime, $\p$ doesn't264divide both $\disc(K)$ and $N_E$. We consider each case in turn:265\begin{itemize}266267\item Suppose that $\p\nmid N_E$.268Then \cite[Prop.~7.5.3 (a)]{neronmodels}269asserts that $\cA_{\O_{F,\p}}\ra \cR_{\O_{F,\p}}$ is a closed immersion,270$\cR_{\O_{F,\p}} \ra \cE_{\O_{E,\p}}$ is smooth with271kernel $\cA_{\O_{F,\p}}$,272and the cokernel of $\cR_{\O_{F,\p}}\ra \cE_{\O_{F,\p}}$ is killed273by multiplication by~$n$. However, in the proof of274\cite[Prop.~7.5.3 (a)]{neronmodels} (see line~$6$275on page 187), one only uses276that $\cE_{\O_{F,\p}}$ has good reduction277to deduce surjectivity, so in278fact $\cR_{\O_{F,\p}}\ra \cE_{\O_{F,\p}}$ is surjective.279(The point is that the good reduction hypothesis on280$\cR_{\O_{F,\p}}$ is used in the proof only to deduce281that $\cE_{\O_{F,\p}}$ has good reduction. Alternatively, using282just the statement of \cite[Prop.~7.5.3 (a)]{neronmodels} we283immediately see that the sequence284$0 \ra \cA_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$285is exact. To get surjectivity on the right, note that the286composition287$\cE_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$288is multiplication by~$n$, which is surjective because289$p\nmid n$ (so that $[n]$ is etale).290Thus $\cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$ must be surjective.291)292293\item Suppose that $\p\nmid \disc(K)$, and let294$\PP$ be a prime of $K$ lying over~$\p$.295We will use that formation of N\'eron models commutes with296unramified base extension \cite{} and check exactness297of (\ref{eqn:neron}) after base extension to the298unramified extension $\O_{K,\mathfrak{P}}$ of $\O_{F,\p}$.299In light of Section~\ref{sec:restriction_of_scalars},300the generic fiber of the base extension of (\ref{eqn:neron})301to $\O_{K,\mathfrak{P}}$ is302$$3030\ra E_{K,\PP}^{\oplus (n-1)} \ra304E_{K,\PP}^{\oplus n}\xra{\Sigma} E_{K,\PP}\ra 0.$$305Thus the corresponding complex of N\'eron models over $\O_{K,\p}$ is306$$3070\ra \cE_{\O_{K,\PP}}^{\oplus (n-1)} \ra \cE_{\O_{K,\PP}}^{\oplus n}308\xra{\Sigma} \cE_{\O_{K,\PP}}\ra 0,309$$310which is exact since, e.g., it is exact on $S$-points for {\em any}311ring~$S$.312\end{itemize}313314\end{proof}315316By the lemma, we have an exact sequence of sheaves317$$3180 \ra \cA \ra \cR \ra \cE \ra 0319$$320on the \'etale site over $\O_F$.321Let $\cA^{\vee}$ denote the N\'eron model of the dual of~$A$.322\begin{proposition}\label{prop:et}323The following diagram has an exact row:324$$\xymatrix{325& & & {H^1(\O_F,\cE)}\ar[d]\\326{R(L)} \ar[r] & {E(L)}\ar[r] & {H^1(\O_F,\cA)}\ar[r] & {H^1(\O_F,\cR)}\ar[d]\ar[r] & {H^1(\O_F,\cE)}\\327& & & {H^1(\O_F,\cA^{\vee})}328}329$$330\end{proposition}331\begin{remark}{\bf Connection with BSD:}332I'm not sure whether or not the column is exact, or is almost exact.333It is close to exact, because $0 \ra \cE \ra \cR \ra \cA$ is exact.334The column is important in connecting nonvanishing of twists and335the rank~$0$ BSD formula to showing that there are points336on~$E$. The tentative connection is as follows:337Suppose $\Sha(E/F)[n]=0$ but $\Sha(A/F)[n]\neq 0$.338The composition $\Sha(A/F)\ra \Sha(A^{\vee}/F)\ra \Sha(A/F)$ is339multiplication by~$n$, which is not injective by our assumption that340$\Sha(A/F)[n]\neq 0$. Thus341$\Ker(\Sha(A^{\vee}/F)\ra \Sha(A/F))[n]$ is nonzero, so342by duality {\bf (I think?)}343$\Ker(\Sha(A/F)\ra \Sha(A^{\vee}/F))[n]$ is also344nonzero. But our hypothesis that $\Sha(E/F)[n]={0}$ combined with345exactness of the vertical sequence would imply that346$$\Ker(\Sha(A/F)\ra \Sha(A^{\vee}/F)[n] = \Ker(\Sha(A/F)\ra \Sha(R/F))[n] \neq 0,$$347so $E(F)$ {\em must} be nonzero!348349If this works it is very exciting because it means that the following350three statements351\begin{enumerate}352\item Nonvanishing of twists conjecture (see Conjecture~\ref{conj:nonvanishing})353\item Finiteness of $\Sha(E/\Q)$, and354\item $\Sha(A/\Q)$ is as big as predicted by BSD for analytic rank~$0$ twisted355powers~$A$356\end{enumerate}357together imply the statement:358$$L(E/\Q,1)=0 \quad\implies\quad E(\Q)\text{ is infinite.}$$359\end{remark}360361362\begin{proof}363(Not really written, but see Section~\ref{sec:letter} below.)364By \cite[Appendix]{mazur:tower}, there is an exact sequence365$$3660 \ra \Sha(A) \ra H^1(\O_F,\cA) \ra G \ra 0367$$368where~$G$ is a finite group whose order if divisible only by~$2$ and369primes that divide the Tamagawa numbers of $\cA$.370\end{proof}371372\comment{373Alternatively, one can consider the following diagram374$$375\xymatrix{376& 0\ar[d] & 0\ar[d]\\3770\ar[r] & {\cE[n]}\ar[r]\ar[d] & {\cE}\ar[d]\ar[dr]^{[n]}\\3780 \ar[r] & {\cA} \ar[r] & {\cR}\ar[r] & {\cE} \ar[r] & 0379}$$380and work with cohomology on the fppf site.381}382383\section{Non-Quadratic Twists}\label{sec:nonvanishing}384We state a conjecture about nonvanishing at~$1$ of certain385prime-degree twists of the $L$-function attached to an elliptic curve,386provide extensive computational evidence for the conjecture, and give387an example which suggests that vanishing twists are very rare.388389\subsection{A Conjecture}390Let~$E$ be an elliptic curve over~$\Q$, and391suppose $p$ is a rigid prime for~$E$.392For every prime $\ell\con 1\pmod{p}$, let393$\chi_{p,\ell} : (\Z/\ell\Z)^* \onto \bmu_p$ be394one of the Galois-conjugate characters395of order~$p$ and modulus~$\ell$.396\begin{conjecture}\label{conj:nonvanishing}397There exists a prime~$\ell\nmid N_E$ such that398$$L(E,\chi_{p,\ell},1)\neq 0399\,\,\,\text{ and }\,\,\,400a_\ell(E) \not\con 2\pmod{p}.$$401\end{conjecture}402403The condition $a_\ell(E) \not\con 2\pmod{p}$ requires elaboration.404Since $\ell\con 1\pmod{p}$, this condition can be rewritten405$a_\ell(E)\not\con \ell+1\pmod{p}$, which is a ``familiar'' condition406to impose. We demand that $a_\ell(E)\not\con \ell+1\pmod{p}$ because407then the characteristic polynomial $x^2 + a_\ell x +\ell\in \F_p[x]$408of $\Frob_\ell$ on $E[p]$ does not have $+1$ as an eigenvalue. This409is a key hypothesis in Section~\ref{sec:ptorsion}.410411412\begin{table}413\caption{Evidence for Conjecture~\ref{conj:nonvanishing}\label{tbl:evidence}}414\noindent%\hspace{-5ex}415\begin{tabular} {|c|cccccccccccccc|}\hline416$\,\,\,E$ & 3&5&7&11&13&17&19&23&29&31&37&41&43&47\\\hline417\nf{37A} & 13&11&29&67&53&103&191&47&59&311&-&83&173&283\\418\nf{43A} & 7&11&29&23&53&103&191&47&59&311&149&83&-&283\\419\nf{53A} & 13&11&29&23&53&103&191&47&59&311&149&83&173&283\\420\nf{57A} & -&11&29&23&53&103&-&47&59&311&149&83&173&283\\421\nf{58A} & 7&11&29&23&53&103&191&47&-&311&149&83&173&283\\422\nf{61A} & 7&31&29&67&53&103&191&47&59&311&149&83&173&283\\423\nf{65A} & 19&-&43&23&-&137&191&47&59&311&149&83&173&659\\424\nf{77A} & 19&11&-&-&53&103&191&47&59&311&149&83&173&283\\425\nf{79A} & 13&11&43&67&53&103&191&47&59&311&149&83&173&283\\426\nf{82A} & 13&41&29&23&53&103&191&47&59&311&149&-&173&283\\427\nf{83A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\428\nf{88A} & 7&11&29&-&131&103&191&47&59&311&149&83&173&283\\429\nf{89A} & 19&11&29&67&53&103&191&47&59&311&149&83&173&283\\430\nf{91A} & 31&11&-&23&-&103&191&47&59&311&149&83&173&283\\431\nf{91B} & -&11&-&23&-&103&191&47&59&311&149&83&173&283\\432\nf{92B} & 13&61&29&23&79&103&229&-&59&311&149&83&173&283\\433\nf{99A} & -&11&29&-&53&103&191&47&59&311&149&83&173&283\\434\nf{101A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\435\nf{102A} & -&11&29&23&53&-&191&47&59&311&149&83&173&283\\436\nf{106B} & 7&11&29&23&53&137&191&47&59&311&149&83&431&283\\\hline437\nf{389A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\438\nf{433A} & 7&11&43&23&53&103&191&47&59&311&149&83&173&283\\\hline439\end{tabular}440\end{table}441442443\subsection{Computational Evidence for the Conjecture}444445Using a \magma{} program (see~\cite{magma}), the author's computer446verified Conjecture~\ref{conj:nonvanishing} for every $p<50$ for the447first $20$ optimal elliptic curve quotients of $R_0(N)$ of rank~$1$448and the first~$2$ elliptic curve quotients of rank~$2$.449450Table~\ref{tbl:evidence} contains, for each $p < 50$, the smallest451prime~$\ell$ satisfying the conditions of452Conjecture~\ref{conj:nonvanishing}. The elliptic curves are labeled453as in Cremona. The curves \nf{389A} and \nf{433A} both have rank~$2$,454and all others have rank~$1$. A dash (-) in the table indicates that the455corresponding prime is not rigid, so the conjecture does not apply.456457In all cases the first prime $\ell\nmid N_E$ with458$\ell\con 1\pmod{p}$ with $a_\ell(E)\not \con 2\pmod{p}$459satisfied $L(E,\chi_{p,\ell},1)\neq 0$, except460for \nf{61A} with $p=5$, \nf{79A} with $p=7$,461\nf{82A} with $p=5$, \nf{89A} with $p=11$,462and \nf{92B} with $p=5$. In every one of these~$5$ exceptional463cases, the second prime that we tried464satisfied the conclusion of465Conjecture~\ref{conj:nonvanishing}.466467\subsection{The Density}468The following conjecture is not mentioned469elsewhere in this paper.470\begin{conjecture}471Let~$p$ be a rigid prime for an elliptic curve~$E$.472The set of primes473$$ \left\{\ell \,\,:\,\, \ell \con 1\!\!\!\!\!\pmod{p}\text{ and }474L(E,\chi_{p,\ell},1)=0\right\}475$$476has Dirichlet density~$0$.477\end{conjecture}478479The following numerical example gives evidence for this conjecture.480\begin{example}481Let~$E$ be \nf{37A} and let $p=5$. Then the only $\ell<1000$ (with482$\ell\con 1\pmod{5}$) for which $L(E,\chi_{5,\ell},1)=0$ is $\ell=41$.483% 4 minutes to compute484\end{example}485486487\section{$p$-Torsion of Twisted Powers}\label{sec:ptorsion}488Let~$p$ and~$\ell$ be as in Conjecture~\ref{conj:nonvanishing}.489In order to apply Theorem~\ref{thm:shaexists}, it is necessary490to know that~$p$ does not divide the orders of certain groups.491In this section, we use that $a_\ell(E)\not\con 2\pmod{p}$ to492deduce that certain groups do not have any~$p$ torsion.493The key idea is that the condition on $a_\ell(E)$ implies494that~$+1$ is not an eigenvalue of $\Frob_\ell$ on the495$p$-adic Tate module attached to~$E$.496497First, we recall that certain torsion points on the closed498fiber of a N\'eron model lift to the generic fiber. Let~$K$ be a499finite extension of~$\Q_\ell$ with ring of integers~$\O$ and residue500class field~$k$.501\begin{lemma}\label{lem:red_mod_n}502Let $A$ be an abelian variety over~$K$ with N\'eron model $\cA$ over~$\O$.503Then for every integer $n$ not divisible by~$\ell$, there is an isomorphism504$$A(K)[n] \xrightarrow{\,\,\isom\,\,} \cA(k)[n].$$505\end{lemma}506\begin{proof}507This is a standard fact, whose proof we recall for the convenience508of the reader.509Let $A^{1}(K)$ denote the kernel of the natural reduction510map $r:A(K)\ra \cA(k)$. Because $A^{1}(K)$ is a formal group,511it is pro-$p$, so $[n]:A^{1}(K)\ra{}A^{1}(K)$ is an isomorphism.512Since $\cA$ is smooth over~$\O$,513Hensel's lemma (see BLR) implies that the reduction map514is surjective, so the following sequence is exact:515$$0\ra A^1(K) \ra A(K) \ra \cA(k) \ra 0.$$516The snake lemma applied to the multiplication by~$n$ diagram517attached to this exact sequence yields the following518exact sequence:519$$0\ra0\ra A(K)[n]\ra \cA(k)[n] \ra 0 \ra A(K)/n A(K) \ra \cA(k)/n\cA(k)\ra0,$$520which proves the proposition.521\end{proof}522523524Let~$E$ be an elliptic curve over~$\Q$ with associated newform525$f = \sum a_n q^n$, and fix a prime~$p$ that is rigid for~$E$.526Suppose~$K$ is the extension of~$\Q$ corresponding527to a surjective Dirichlet character528$\chi: (\Z/\ell\Z)^* \onto \bmu_p$529of prime conductor; then~$K$ is530the subfield of $\Q(\bmu_\ell)$ fixed by $\Ker(\chi)$,531so it is of degree~$p$, is totally ramified532at~$\ell$, and is unramified outside~$\ell$.533Let~$A=\Ker(\Tr : \Res_{K/\Q} E_K \ra E)$.534We next compute the Tamagawa number535$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$536and the $p$-torsion of several abelian varieties.537\begin{proposition}\label{prop:ptorsion}538Let~$E$, $\chi$, $K$, and~$A$ be as above and suppose539that $\ell\nmid N_E$ and $a_\ell \not\equiv 2\pmod{p}$.540Then the following groups have no nontrivial $p$-torsion:541$$A(K), \quad A(\Q_\ell),\quad R(\Q_\ell),\quad542(R/E)(\Q_\ell),\quad \text{and}\quad \Phi_{A,\ell}(\F_\ell).$$543%$$A(K)[p]=A(\Q_\ell)[p] = R(\Q_\ell)[p] =544%(R/E)(\Q_\ell)[p]=\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$$545\end{proposition}546\begin{proof}547The reason the $p$-torsion vanishes in all these cases548is that the condition $a_\ell \not\equiv 2\pmod{p}$ implies549in each case that $\Frob_\ell$ has no $+1$ eigenvalue.550The details are as follows.551552We first show that $R(\Q_\ell)[p]=\{0\}$, where $R=\Res_{K/\Q}E_K$.553By definition,554$$R(\Q_\ell) = E_K(\Q_\ell\tensor_\Q K) =555E(K_v)\cross \cdots \cross E(K_v) \quad \text{($p$ copies)},$$556where $K_v$ is the completion of~$K$ at the unique prime of~$K$557lying over~$\ell$.558The action of $\Frob_\ell\in \Gal(\Q_\ell^{\ur}/\Q_\ell)$559on $E[p](\Q_\ell^{\ur})=E[p](\Qbar_\ell)$ has characteristic560polynomial561$F(x) = x^2-a_\ell x + \ell \in \F_p[x]$.562Since $a_\ell \not\equiv 2\pmod{p}$ and $\ell\equiv 1\pmod{p}$, it563follows that $\Frob_\ell$ does not have564$+1$ as an eigenvalue, so $E(\Q_\ell)[p]=\{0\}$.565If $z\in E(K_v)[p]$, then the field $L=\Q_\ell(z)$ is an unramified566subfield of the totally ramified field $K_v$, so $z\in E(\Q_\ell)[p]=\{0\}$.567Thus $E(K_v)[p]=\{0\}$, which implies that $E(K)[p]=\{0\}$ and568$R(\Q_\ell)[p]=\{0\}$.569Since $R_K/E_K \isom E_K \cross \cdots \cross E_K$ ($p-1$ times),570we see that571$$(R/E)(\Q_\ell)[p]\subset (R/E)(K_v)[p] =572(E(K_v)\cross \cdots \cross E(K_v))[p] = \{0\}.$$573574Finally, we turn to the component group $\Phi_{A,\ell}$.575Let $\cA$ denote the N\'eron576model of~$A$. By Lang's Lemma the natural map $\cA(\F_\ell) \ra577\Phi_{A,\ell}(\F_\ell)$ is surjective. Thus if578$\Phi_{A,\ell}(\F_\ell)[p]\neq \{0\}$, then $\cA(\F_\ell)[p]\neq579\{0\}$. However, by Lemma~\ref{lem:red_mod_n} and observation of the580previous paragraph,581$$\cA(\F_\ell)[p] = A(\Q_\ell)[p]\subset R(\Q_\ell)[p]=\{0\},$$582so $\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$, as claimed.583\end{proof}584585586\subsection{The Tamagawa Number of $A$ at $\ell$}587In this section, the notation and hypothesis are as in588Proposition~\ref{prop:ptorsion}.589That proposition implies that the Tamagawa number590$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$ is coprime591to~$n$. In this section we use Remark~5.4 of \cite{edixhoven:tame} to592prove that in fact $c_{A,\ell}=1$.593594Let $\lambda$ be the prime of~$K$ lying over $\ell$, and595let $K_{\lambda}$ denote the completion of~$K$ at~$\lambda$,596so $K_{\lambda}$ is totally and tamely ramified over $\Q_\ell$.597Since598$$A_{K} \isom \Ker(\Sigma: E_{K}^{\oplus n}599\ra E_{K}),$$600and $E_{K_\lambda}$ has good reduction,601the geometric closed fiber of the N\'eron model of $A_{K_{\lambda}}$ is602$603A'_{\kbar}\isom \Ker(\Sigma : E_{\kbar}^{\oplus n} \ra E_{\kbar}).604$605In the notation of \cite{edixhoven:tame},606$\mu_n$ acts on $A'_{\kbar}$ by the action607it induces by cyclically permuting the factors of608$E_{\kbar}^{\oplus n}$. Thus609$A_{\kbar}'(\kbar)^{\mu_n}$ is the set of610$\sum P_\sigma\tensor\sigma \in E(\kbar)^{\oplus n}$611such that all $P_\sigma$ are equal and $\sum P_\sigma = 0$,612i.e.,613$$614A_{\kbar}'(\kbar)^{\mu_n} \isom E(\kbar)[n]\ncisom (\Z/n\Z)^2.615$$616Thus Remark~5.4 in \cite{edixhoven:tame} implies that617$\Phi_{A,\ell}(\kbar) \ncisom E(\kbar)[n]$.618By Proposition~\ref{prop:ptorsion},619$\Phi_{A,\ell}(k)$ has no elements of order dividing $n$,620so $\Phi_{A,\ell}(k)=0$.621622\section{Nonsquare Shafarevich-Tate Groups}\label{sec:nonsquaresha}623On page 306--307 of \cite{tate:bsd}, Tate discusses results about the624structure of the group $\Sha(A/K)$, where~$A$ is an abelian variety over625a number field~$K$. He asserts that if~$A$ is a Jacobian then $\#\Sha(A/K)$626is a perfect square. Poonen and Stoll subsequently pointed out627in~\cite{poonen-stoll} that Tate's assertion is not quite correct. In628fact, Poonen and Stoll prove that when~$A$ is a Jacobian, $\#\Sha(A/K)$629is either a square or twice a square, and they give examples in which630$\#\Sha(A/K)$ is twice a square. Tate does not discuss the case631when~$A$ is not a Jacobian, except to mention results that imply that632$\#\Sha(A/K)$ is square away from~$2$ and primes that633don't divide the degree of some polarization of~$A$.634635Now suppose~$A$ is an arbitrary abelian variety over a number636field~$K$. Until now it was unknown whether or not $\#\Sha(A/K)$ must637be either a square or twice a square. Let~$E$ be an elliptic curve638over~$\Q$ of rank~$1$. Then our construction gives,639for suitable primes~$p$, an injection640$$641\Z/p\Z \ncisom E(\Q)/p E(\Q) \hookrightarrow \Sha(A/\Q),642$$643where~$A$ is an abelian variety over~$\Q$ which is a twist of $E^{\times p-1}$.644Thus $\Sha(A/\Q)[p]$ has a natural subgroup of order~$p$; moreover,645no other natural subgroup of order~$p$ presents itself.646We prove in this section that the $p$-part of $\#\Sha(A/\Q)$647is not a perfect square.648649\subsection{A Letter}\label{sec:letter}650[Convert this email into a rigorous proof.]651\begin{verbatim}652Do you remember my conditional-on-BSD construction of abelian653varieties A with nonsquare p-part of #Sha(A), for various odd primes654p? In particular, you pushed me repeatedly to remove the conditional655nature of the result. I think I found a very clean way to do this656today, which doesn't require computing any Selmer groups at all!657658I make a list of hypothesis on p, the field that I restrict scalars659down from and the Fourier coefficient a_ell(E). For example, when E660is the curve of conductor 43, one could take p=3, K the cubic field661Q(mu_7)^+. Then if R is the restriction of scalars of E from K down662to Q, surprisingly the exact sequence6636640 --> A --> R --> E --> 0665666of abelian varieties *does* induce an exact sequence of Neron models667over Spec(Z), as I think I've checked using chapter 7 of668Bosch-Lutkebohmert-Raynaud. We then obtain the following exact669sequence from the long exact sequence of etale cohomology:6706710 ---> E(Q)/3E(Q) ---> H^1(Z,A) ---> H^1(Z, R) --> H^1(Z,E)672673Then an old theorem of Barry from his "Rational Points on Abelian674Varieties..." along with my hypothesis on E, p, and ell, imply that675H^1(Z,A), H^1(Z,R) and H^1(Z,E) differ from Sha(A), Sha(R), and Sha(E)676by at most a 2-group. Ignoring this 2 business, we have an exact677sequence6786790 ---> E(Q)/3E(Q) ---> Sha(A/Q) --> Sha(E/K) --> Sha(E/Q).680681(I also just used that Sha(R/Q) = Sha(E/K).)682683Now the 3-power-part Sha(E/Q)(3) = {0}, by Euler Systems theory (or I684could directly check this), so we have6856860 --> E(Q)/3E(Q) ---> Sha(A/Q)(3) --> Sha(E/K)(3) --> 0687688Properties of the Cassels-Tate pairing on the elliptic curve E over K689and the fact that Sha(E/K) is finite (since it's an image of690Sha(A)(3), which is finite by Kato's theorem), imply that #Sha(E/K)(3)691is a an even power of 3. Thus #Sha(A/Q)(3) must be an odd power of 3.692693What do you think? This seems much better than trying to do explicit694Selmer group computations with CM elliptic curves.695696-- William697698\end{verbatim}699700701\section{Other Remarks}\label{sec:applications}702We apply the above results to prove that703Conjecture~\ref{conj:nonvanishing} implies the existence of704elements of Shafarevich-Tate groups of twisted powers of705elliptic curves of every prime order. We also construct an706abelian variety~$A$ over~$\Q$ such that the Birch707and Swinnerton-Dyer conjecture predicts that $\Sha(A/\Q)[3]=\Z/3\Z$ and708that $\#\Sha(A/\Q)$ is not a square or twice a square.709710\subsection{Existence of Elements of $\Sha$ of all Prime Orders}711\begin{proposition}\label{prop:all_prime_orders}712Let~$p$ be a prime number. Then Conjecture~\ref{conj:nonvanishing}713implies that there exists infinitely many twisted powers~$A$ of some714elliptic curve such that $\Sha(A/\Q)[p]\neq \{0\}$.715\end{proposition}716\begin{proof}717Most of the proposition can be proved using a single elliptic curve.718When ordered by conductor, the first elliptic curve~$E$ over $\Q$ with719positive rank has prime conductor~$37$ and is defined by the720Weierstrass equation $y^2 + y = x^3 - x$. Table~1 of721\cite{cremona:algs} shows that~$E$ is isolated in its isogeny class,722so \cite[Exercise~4]{ribet-stein:serre} implies that representations723$\rho_{E,p}$ are irreducible. Since724$\ord_{37}(j(E))=-1$, $\cbar_{37}=1$. Thus all odd primes $p\neq 37$725are rigid for~$E$. The proposition then follows for all odd primes726$p\neq 37$ by Theorem~\ref{thm:main}.727728We complete the proof as follows. Exactly the same argument applied729to the unique elliptic curve of conductor~$43$ proves the proposition730for all odd primes $p\neq 43$. Finally, B\"olling proved in731\cite{bolling:sha} that for every $j\in\Q$ there is an elliptic732curve~$E$ with $j$-invariant~$j$ such that infinitely many twists~$E'$733of $E$ have $\Sha(E'/\Q)[2]\neq\{0\}$.734\end{proof}735736737\subsection{What Goes Wrong when $p=2$?}738In the previous section, we set $p=3$ and739constructed an abelian variety $A$ of dimension $p-1$740that (conjecturally) has nonsquare $\Sha(A/\Q)[p]$.741We can construct an~$A$ in an analogous way for any odd prime~$p$,742and the author expects that $\Sha(A/\Q)[p]$ is nonsquare in most cases.743However, when $p=2$, the dimension of~$A$ is~$1$, so744in that case $\#\Sha(A/\Q)$ must be a perfect square.745746What goes wrong? The problem lies in Theorem~\ref{thm:shaexists}.747The argument used to prove Theorem~\ref{thm:shaexists}748at least provides a map749$$750E(\Q)/2 E(\Q)\hookrightarrow\Vis_R(H^1(\Q,A)).751$$752When $p=2$, the condition $e<p-1$ is not satisfied, so753the proof of Theorem~\ref{thm:shaexists} does not show754that the image of $E(\Q)/ 2 E(\Q)$ is locally trivial755at the prime~$2$ (or at $\infty$). We thus only756construct a subgroup of $H^1(\Q,A)$ of nonsquare order,757not of~$\Sha(A/\Q)$. Thus even if two elliptic curves have the same758$E[2]$, then can still possess very different Selmer groups.759760761762763764\section{Connection with BSD}765\label{sec:bsd}766[[Needs to be rewritten.]]767Suppose $E$ is an elliptic curve over~$\Q$ and that $L(E,1)=0$. The768Birch and Swinnerton-Dyer conjecture for~$E$ asserts (among other769things) that $E(\Q)$ is infinite. Suppose~$A$ is constructed as in770Section~\ref{sec:main}. In this section we describe why if a certain771consequence of a refinement of the Birch and Swinnerton-Dyer772conjecture for~$A$ is true, then $\Sel^{(n)}(E/\Q)$ is nonzero.773774775Using modular symbols one sees that $L(A,1)\con 0 \pmod{\ell}$,776so a refinement of the Birch and Swinnerton-Dyer formula for rank~$0$777abelian varieties predicts that there should be a nonzero element in778$\Ker (\Sha(A)\ra \Sha(A/E[n]))$.779Thus by Proposition~\ref{prop:et}, either780$H^1(X_{\et},\cE)[n]\neq 0$, or there is a nonzero element781of order dividing~$n$ in782$$783\Ker(H^1(X_{\et},\cA)\ra H^1(X_{\et},\cR)) \isom E(\Q)/R(\Q),784$$785in which case $E(\Q)/R(\Q)$ contains a nonzero element of786order dividing~$n$,787so $E(\Q)$ is infinite.788Thus either $\Sha(E)[n]\neq 0$ or $E(\Q)$ is infinite, so789$\Sel^{(n)}(E/\Q)$ is nonzero.790791792\section{Appendix: Exactness Properties}\label{sec:neronmodels}793For the reader's convenience, we copied {\em verbatim}794pages 186--187 of \cite[\S7.5]{neronmodels}. In what follows~$R$795is a discrete valuation ring with field of fractions~$K$.796797798``Next, let us look at abelian varieties.799800\noindent{\bf Proposition 3.} {\em801Consider an exact sequence of abelian802varieties803$$8040 \ra A_K' \ra A_K \ra A_K'' \ra 0805$$806and the corresponding complex of N\'eron models807\begin{equation}\label{eqn:dagger}8080\ra A' \ra A \ra A'' \ra 0 \qquad\qquad809\end{equation}810Let $B_K$ be an abelian subvariety of $A_K$ such that $A_K\ra A_K''$ induces an isogeny811$u_K: B_K \ra A_K''$; let $n=\deg u_K$.812813(a) If $\chr{}k$ does not divide~$n$, then $A'\ra A$ is a closed immersion,814$A\ra A''$ is smooth with kernel $A'$, and the cokernel of $A_k \ra A_k''$ is815killed by multiplication with~$n$. If, in addition, $A$ has abelian reduction, (\ref{eqn:dagger})816is exact.817818(b) If $A$ has semi-aelian reduction, the sequence (\ref{eqn:dagger}) is exact up to isogeny;819i.e., it is isogenous to an exact sequence of commutative $S$-group schemes.820}821\vspace{1em}822\begin{proof}823The isogeny $u_K : B_K \ra A_K''$ gives rise to an isogeny $v_K : A_K' \cross_K B_K \ra A_K$824of degree~$n$. So there is an isogeny $w_K : A_K \ra A_K' \cross_K B_K$ such825that $w_K\circ v_K$ is multiplication by~$n$. Let $B$ be the N\'eron model of $B_K$. Then826$u_K$, $v_K$, and $w_K$ extend to $R$-morphisms $u: B \ra A''$,827$v: A'\cross_R B \ra A$, and828$w: A\ra A'\cross_R B$ such that $w\circ v$829is multiplication by $n$ on $A'\cross_R B$.830Assuming the condition of (a), the multiplication by~$n$ is an \'etale isogeny on831$A'\cross_R B$, and $u,v$ and $w$ are easily checked to be \'etale832isogenies, too. Then $H:=w^{-1}(A')$ is a smooth closed subgroup scheme of~$A$833which satisfies $H^0_K = A_K'$. It follows that the schematic closure of $A_K'$834in~$H$ or~$A$ is an open subgroup scheme of~$H$ and, thus, is smooth over~$R$.835So, by using the \'etale isogeny~$u$. One shows that836$A\ra A''$ is flat, has kernel $A'$ and, hence, is smooth. Furthermore,837if~$A$ has abelian reduction, the same is true for $A''$ by 7.4/2 so that838$A\ra A''$ is surjective.839840Assertion (b) follows from the fact that $v:A'\cross_R B \ra A$ and841$u:B\ra A''$ are isogenies; use 7.3/6 and 7.3/7.842\end{proof}843844\noindent{\bf Theorem 4.} {\em845Let $0\ra A_K' \ra A_K \ra A_K'' \ra 0$ be an exact sequence of846abelian varieties and consider the associated sequence of N\'eron847models $0\ra A' \ra A \ra A'' \ra 0$. Assume that the following848condition is satisfied:849850(*) $R$ has mixed characteristic and the ramification index $e=v(P)$851satisfies $e<p-1$, where $p$ is the residue characteristic of $R$852and where $v$ is the valuation on~$R$, which is normalized by the condition853that~$v$ assumes the value~$1$ at unifromizing elements of~$R$.854855Then the following assertions hold:856857(i) If $A'$ has semi-abelian reduction, $A'\ra A$ is a closed immersion.858859(ii) If $A$ has semi-abelian reduction, the sequence $0\ra A' \ra A \ra A''$860is exact.861862(iii) If $A$ has abelian reduction, the sequence $0\ra A' \ra A\ra A''\ra 0$863is exact and consists of abelian $R$-schemes.864}865\vspace{1em}866867\begin{proof}868Let us first see how assertions (ii) and (iii) can be deduced from assertion (i).869If~$A$ has semi-abelian or abelian reduction, the same is true for $A'$ and $A''$870by 7.4/2. So $A'\ra A$ is a closed immersion by (i), and we can consider the871quotient $A/A'$; it exists in the category of algebraic spaces, cf.\ 8.3/9.872Furthermore, $A/A'$ is smooth and separated and, thus, a scheme by 6.6/3. Now look at the873canonical morphism $A/A'\ra A''$ which is an isomorphism on generic fibres.874Since $A$ has semi-abelian reduction, the same is true for $A/A'$, and it875follows from 7.4/3 that $A/A'\ra A''$ is an open immersion. So assertion (ii)876is clear. Finally, if $A$ has abelian reduction, the same is true for $A/A'$.877So the latter is an abelian scheme by 7.4/5 and, thus, must coincide with the878N\'eron model $A''$ of $A''_K$. Thereby we obtain assertion (iii).879[The rest of the proof takes a few pages.]''880\end{proof}881882883884885886\bibliography{biblio}887\end{document}888889890E := EC("37A");891M := MS(E);892K := CyclotomicField(5);893M5 := BaseExtend(M,K);894P := [p : p in [3..200] |IsPrime(p)];895time X := [<p,896Dimension(TwistedWindingSubmodule(M5,1,DirichletGroup(p,K).1^2))> : p in P];897// Time: 11.369898[ <11, 1>, <31, 1>, <41, 0>, <61, 1>, <71, 1>, <101, 1>, <131, 1>,899<151, 1>, <181, 1>, <191, 1> ]900901\comment{902$Log: bigsha.tex,v $903Revision 1.25 2001/09/28 02:48:22 was904?905906Revision 1.24 2001/09/23 04:43:11 was907more TOC stuff and cleaning.908909Revision 1.23 2001/09/22 20:27:00 was910Worked on911\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}912and changed Sha(A) to Sha(A/Q).913914Revision 1.22 2001/09/22 19:26:23 was915I don't know.916917Revision 1.21 2001/09/13 01:42:50 was918Lots of nice little improvements!919920Revision 1.20 2001/09/09 04:11:42 was921Added lots of toc pars.922923Revision 1.19 2001/09/08 02:55:34 was924polishing.925926Revision 1.18 2001/09/06 03:39:53 was927added some cool remarks at end about BSD928929Revision 1.17 2001/09/06 03:24:36 was930typo931932Revision 1.16 2001/09/06 03:23:56 was933minor typo934935Revision 1.15 2001/09/06 03:17:42 was936Added table of evidence for twisting conjecture.937938Revision 1.14 2001/09/06 02:42:04 was939...940941}942\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}943Let $E$ be an elliptic curve over~$\Q$.944In this section, we prove that Conjecture~\ref{conj:nonvanishing}945implies if~$p$ is a rigid prime, then946$E(\Q)/p E(\Q)$ is canonically isomorphic to the947elements of order~$p$ in the visible Shafarevich-Tate948group of a rank~$0$ twisted power of~$E$.949950\begin{theorem}\label{thm:main}951Assume that Conjecture~\ref{conj:nonvanishing} is true.952If~$E$ is an elliptic curve over~$\Q$, then for every953rigid prime~$p$, there is a degree~$p$ abelian extension~$K$954of~$\Q$ such that955$$E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p]),$$956where $R=\Res_{K/\Q}(E_K)$ and~$A\subset R$ has957dimension~$p-1$ and rank~$0$.958\end{theorem}959960The proof divides naturally into three steps. First, we use961Conjecture~\ref{conj:nonvanishing} to construct~$A$. The we962use a theorem of Kato and that formation of N\'eron models963commutes with unramified base change to prove that~$A$ has964rank~$0$ and that~$p$ does not torsion of Tamagawa numbers965of~$A$. Next, we apply the visibility Theorem~\ref{thm:shaexists}966to obtain an isomorphism $E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p])$.967968\begin{proof}969Conjecture~\ref{conj:nonvanishing} implies that there exists970a prime $\ell\nmid N_E$971with $\ell\con 1\pmod{p}$ such that $L(E,\chi_{p,\ell},1)\neq 0$ and972$a_\ell(E)\not\con 2\pmod{p}$. Let~$\ell$ be such a prime, and973let~$K$ be the abelian extension of~$\Q$ corresponding to974a surjective character $\chi_{p,\ell} : (\Z/\ell\Z)^* \ra \bmu_p$.975Recall from Section~\ref{sec:restriction_of_scalars} that976the restriction of scalars $R = \Res_{K/\Q}(E_K)$ is an977abelian variety over~$\Q$ of dimension~$p$, and978we have a commutative diagram979%$$\xymatrix{980% & {A}\[email protected]{^(->}[rd]\\981%{E[p]\,\,}\[email protected]{^(->}[ur] \[email protected]{^(->}[dr] & & {R}\ar[dr]^{\Tr}\\982% & {E}\[email protected]{^(->}[ur]\ar[rr]^{[p]} & & {E}983%}$$984$$\[email protected]=3pc{985{E[p]\,\,}\ar[r]\ar[d] & {E} \ar[dr]^{[p]}\ar[d] \\986{A}\ar[r] & {R} \ar[r]^{\Tr} & {E,}987}$$988where $A=\Ker(\Tr)$ is an abelian variety.989990%Since $\rho_{E,p}$ is irreducible, $p\nmid E(\Q)_{\tor}$.991%We have $A(\Q)\subset R(\Q)=E(K)$. Since~$K$ is totally ramified992%at~$\ell$, unramified outside~$\ell$, and $\ell\nmid p N_E$, $E(K)[p]$993%equals $E(\Q)[p]$, which is $\{0\}$ because $\rho_{E,p}$ is assumed994%irreducible. Thus $p\nmid \#A(\Q)$.995996Since $L(A,s)=\prod L(E,\chi_{p,\ell}^\sigma,s)$, and $L(E,\chi,1)\neq9970$, Kato's work on Euler systems\edit{Reference?} implies that $A(\Q)$ is finite.998Proposition~\ref{prop:ptorsion} implies that999$p\nmid \#A(\Q)\cdot \#(R/E)(\Q)$.1000Next suppose that~$q$ is a prime of bad reduction for~$A$. If1001$q\not=\ell$, then $K/\Q$ is unramified at~$q$.1002The formation of N\'eron models commutes with unramified base1003change\edit{Reference?} and $A_K=E^{\times(p-1)}$, so $c_{A,q}$ divides1004$\cbar_{E,q}$, which is not divisible by~$p$ since~$p$ is rigid1005for~$E$. If $q=\ell$, Proposition~\ref{prop:ptorsion} asserts that1006$p\nmid c_{A,q}$.10071008The previous paragraph combined with Proposition~\ref{prop:ptorsion}1009shows that the hypothesis of1010Theorem~\ref{thm:shaexists} are satisfied with $A=A$, $B=E$,1011$R=R$, and $L=\Q$. Thus there is an injective map1012$$E(\Q)/p E(\Q) \hookrightarrow \Vis_R(\Sha(A/\Q))\subset \Sha(A/\Q).$$10131014To prove surjectivity, note that by definition every element of1015$\Vis_R(\Sha(A/\Q)[p])$ is the image of an element of $R(\Q)$ and1016by Proposition~\ref{prop:ptorsion} the index of $E(\Q)$ in $R(\Q)$ is1017finite and coprime to~$p$.1018\end{proof}101910201021