Author: William A. Stein
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5\title{\sc Nonsquare Odd Shafarevich-Tate Groups and Mordell-Weil Groups of Elliptic Curves}
6\author{William A. Stein\\{\sf was@math.harvard.edu}}
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14\begin{document}
15\maketitle
16
17\begin{abstract}
18Let~$E$ be an elliptic curve over a number field~$F$.
19We study certain exact sequences of abelian varieties
20     $0\ra A \ra R\ra E \ra 0$
21that induce a natural identification
22$E(F)/p E(F) \isom \Ker(\Sha(A/F)\ra \Sha(R/F))$.
23For many odd primes~$p$, we construct the
24first ever examples of abelian varieties~$A$
25over~$\Q$ such that the $p$-part of
26$\#\Sha(A/F)$ is {\em not} a perfect square.
27We also discuss links between our results and conjectures
28and the problem of constructing elements of Selmer groups
29of elliptic curves over~$\Q$ with analytic rank bigger than~$1$.
30\end{abstract}
31
32\tableofcontents
33
34\section*{Introduction}
35Let~$E$ be an elliptic curve over a number field~$F$.
36We study certain exact sequences of abelian varieties
37     $0\ra A \ra R\ra E \ra 0$
38that induce a natural identification
39$E(F)/p E(F) \isom \Ker(\Sha(A/F)\ra \Sha(R/F))$.
40For many odd primes~$p$, we construct the
41first ever examples of abelian varieties~$A$
42over~$\Q$ such that the $p$-part of
43$\#\Sha(A/F)$ is {\em not} a perfect square.
44We also discuss links between our results and conjectures
45and the problem of constructing elements of Selmer groups
46of elliptic curves over~$\Q$ with analytic rank bigger than~$1$.
47
48In particular, let~$E$ be an elliptic curve over~$\Q$.  We prove that
49a conjecture about nonvanishing of prime-degree twists of $L(E,s)$
50implies that for all but finitely many primes~$p$ there is a twist~$A$
51of $E^{\times (p-1)}$ such that $E(\Q)/p E(\Q)$ is identified in a
52natural way with a subgroup of $\Sha(A/\Q)$.
53
54We show how to construct twists~$A$ of powers
55of elliptic curves over~$F$ such that $\Sha(A/F)(p)$ has
56{\em nonsquare} order for an odd prime~$p$.
57For example, if $E$ is the elliptic curve of conductor~$43$
58then there is a two-dimensional abelian variety $A$ that is isomorphic
59to $E\cross E$ over $\Q(\mu_3)^+$ and
60$\#\Sha(A/\Q)(3)$ is not a square.
61
62This paper is organized as follows.  In Section~\ref{sec:terminology}
63we define twisted powers, Tamagawa numbers, and rigid primes.  We
64recall in Section~\ref{sec:restriction_of_scalars} the definition of
65the restriction of scalars of an elliptic curve and prove a
66proposition about a map induced by trace.
67We state
68a conjecture about nonvanishing of twists of prime degree in
69Section~\ref{sec:nonvanishing}, and give computational evidence for
70this conjecture.  In Section~\ref{sec:ptorsion} we prove triviality of
71the $p$-torsion of several abelian groups attached to twisted powers
72of an elliptic curve.  The heart of the paper is
73Section~\ref{sec:main}, which uses the above results to construct
74subgroups of Shafarevich-Tate groups of twisted powers.
75Section~\ref{sec:applications} pulls together the results of the
76previous sections; there we prove that the conjecture of
77Section~\ref{sec:nonvanishing} implies the existence of elements of
78Shafarevich-Tate groups of every prime order, and describe a
79connection with the Birch and Swinnerton-Dyer conjecture.
80In Section~\ref{} we discuss the extent to
81which the order of $\Sha$ can fail to be square.
82
83\vspace{2ex}\par\noindent{}{\bf{}Acknowledgement: }
84It is a pleasure to thank Gautam Chinta,
85Benedict Gross, Emanuel Kowalski, Barry Mazur,
86Bjorn Poonen, David Rohrlich,
87and Michael Stoll
89
90
91\section{Terminology}\label{sec:terminology}
92In this section, we define twisted powers and rigid primes for an
93elliptic curve, and recall the definition of Tamagawa numbers of an
94abelian variety.
95
96Let $E$ be an elliptic curve over a number field~$F$.
97
98\begin{definition}[Twisted Powers]
99A {\em twisted power} of~$E$ is an abelian variety~$A$ over~$F$
100that is isomorphic over $\Kbar$ to $E^{\cross n}$
101for some positive integer~$n$.
102\end{definition}
103
104We recall the standard notion of Tamagawa number of an abelian variety~$A$,
105and introduce the notation $\cbar_{A,p}$ for the order of the group
106of components of~$A$ over $\Fbar_p$.
107\begin{definition}[Tamagawa Numbers]
108Let~$A$ be an abelian variety over~$F$ with N\'eron
109model $\cA$ over~$\O_F$, and let~$\p$ be a prime of $\O_F$,
110and let $k=\O_F/\p$.
111The component group of~$A$ at~$\p$ is the finite group scheme
112$\Phi_{A,\p} = \cA_{k}/\cA_{k}^0$, where $\cA_{k}^0$ is the
113identity component of $\cA_{k}$.  The
114{\em Tamagawa number} of~$A$ at~$\p$ is
115$c_{A,\p} = \#\Phi_{A,\p}(k)$.
116Also set $\cbar_{A,\p} = \#\Phi_{A,\p}(\overline{k})$.
117\end{definition}
118
119Let $N_E\in \O_F$ be the conductor of~$E$.
120
121\begin{definition}[Rigid Primes]
122A prime $\p$ of $\O_F$ is {\em rigid} for~$E$ if $\p$ does not divide
123$2\cdot N_E \cdot \prod_{\q\mid N_E} \cbar_{E,\q}$ and the
124representation $\rho_{E,\p}:\Gal(\Fbar/\F)\ra \Aut(E[\p])$
125is irreducible.
126\end{definition}
127
128
129\section{Restriction of Scalars}\label{sec:restriction_of_scalars}
130In this section,
131we recall the notion of restriction of
132scalars, and prove that the kernel of a morphism induced by
133a trace is geometrically connected.
134
135Let~$E$ be an elliptic curve over a number field~$F$, and let~$K$
136be a finite extension of~$F$.
137The restriction of scalars $R=\Res_{K/F}(E_K)$ is an abelian variety
138over~$F$ of dimension $[K:F]$, which is characterized by the
139following universal property: There is a functorial group isomorphism
140$R(S) \isom E_K(S_K)$, where~$S$ varies over all $F$-schemes.
141
142There is a more explicit description of $\Res_{K/F}(E_K)$ when~$K$ is Galois
143over~$F$.  As $\Gal(\Fbar/F)$-modules, we have
144$$145 R(\Fbar) = E(\Fbar \tensor K) \isom E(\Fbar)\tensor_{\Z} \Z[\Gal(K/F)], 146$$
147where $\tau\in \Gal(\Fbar/F)$ acts on
148$\sum P_\sigma\tensor \sigma \in E(\Fbar)\tensor_{\Z}\Z[\Gal(K/F)]$ by
149$$\tau\left(\sum P_\sigma\tensor \sigma\right) = 150 \sum \tau(P_\sigma)\tensor \sigma\tau_{|K}. 151$$
152Moreover, the $L$-series of~$R$ is $\prod_{a=1}^{n} L(E,\chi^a,s)$, and~$R$
153has good reduction at all $p\nmid \ell\cdot N$.
154
155
156\begin{proposition}\label{prop:kergeo}
157The identity map induces a closed immerion $\iota: E\hookrightarrow 158R$, and the trace $\Tr:K\ra F$ induces a surjection $\Tr:R\ra E$
159whose kernel is geometrically connected.  Thus we have an exact sequence
160\begin{equation}\label{eqn:exactabvar}
161  0 \ra A \ra R \ra E \ra 0
162\end{equation}
163with $A$ an abelian variety.
164\end{proposition}
165\begin{proof}
166The existence of~$\iota$ and $\Tr$ follows from Yoneda's lemma.
167The map~$\iota$ is induced by the functorial inclusion
168$E(S)\hookrightarrow E_K(S_K)=R(S)$, so~$\iota$ is injective.
169
170The $\Tr$ map is induced by the usual
171functorial trace map on points
172$R(S)=E_K(S_K)\xrightarrow{\Tr} E(S)$.
173To verify that $\Ker(\Tr)$ is geometrically connected, we base
174extend the exact sequence (\ref{eqn:exactabvar}) to~$\Fbar$.  First, note that
175$$R_{\Fbar} \ncisom E_{\Fbar}\cross \cdots \cross E_{\Fbar}.$$
176After base extension, the trace map may be identified with the summation map
177$$+: E_{\Fbar} \cross \cdots \cross E_{\Fbar} 178 \longrightarrow E_{\Fbar}.$$
179Let $n=[K:F]$.  The map defined by
180$$\left(a_1,\ldots, a_{n-1}\right) \mapsto 181 \left(a_1,a_2,\ldots ,a_{n-1},-\sum_{i=1}^{n-1} a_i\right),$$
182 is an isomorphism from
183$E_{\Fbar}^{\times (n-1)}$ to $\Ker(+)=\Ker(\Tr_{\Fbar})$.
184Thus $\Ker(\Tr_{\Fbar})$ is a product of copies of $E_{\Fbar}$,
185hence is connected.
186\end{proof}
187
188\begin{corollary}
189Let $n=[K:F]$.  Then
190$$(\iota(E)\intersect \Ker(\Tr))(\Fbar) \isom E[n](\Fbar)\ncisom (\Z/n\Z)^2.$$
191(The rightmost map is an isomorphism of groups, not Galois modules.)
192\end{corollary}
193\begin{proof}
194Since the map
195$$F\hookrightarrow K\xrightarrow{\Tr} F$$ is
196multiplication by~$n$, the composite map
197$$E \hookrightarrow R \longrightarrow E$$
198is also multiplication by~$n$.
199The corollary now follows since
200$\iota(E)\intersect \Ker(\Tr)$ is the kernel of~$\Tr\circ\iota$,
201which equals $[n]$.  It is elementary that $E[n](\Fbar)\ncisom (\Z/n\Z)^2$,
202where we have, of course, forgotten the action of $\Gal(\Fbar/F)$.
203\end{proof}
204
205\subsection{Exactness of the Complex of N\'eron Models}
206\label{sec:etale}
207
208Let~$E$ be an elliptic curve over a number field~$F$, and let~$K$ be a
209finite Galois extension of~$F$ that is only ramified
210at primes that don't divide the conductor $N_E$ of~$E$.
211Let $R=\Res_{K/F} E_K$ and $A = \Ker(R \ra E)$, so
212by Proposition~\ref{prop:kergeo}
213we have an exact sequence of abelian varieties
214$215 0 \ra A \ra R \ra E \ra 0. 216$
217
218If a complex of N\'eron models over a ring $\O$ is exact in the sense
219of \cite{neronmodels}, then it induces an exact sequence of sheaves on
220the \'etale site for $\Spec(\O)$.
221
222\begin{proposition}\label{lem:etale}
223Let $A$, $R$, and $E$ be as above.  Assume that
224the three elements
225$\disc(K)$, $n=[K:F]$, and $N_E$ of $\O_F$
226are all relatively prime to each other
227and that $v(p)<p-1$ for each $p\mid n$ and
228for each valuation~$v$ on $\O_F$ corresponding
229to a prime $\p$ of residue characteristic~$p$, normalized
230so that the uniformizing element of $\O_{F,\p}$
231has valuation~$1$ (in particular, if $F=\Q$ this is the
232condition that~$n$ is odd).
233Then the corresponding complex of N\'eron models
234$$235 0 \ra \cA \ra \cR \ra \cE \ra 0 236$$
237is exact.
238\end{proposition}
239\begin{proof}
240We use the results of \cite[Ch.~7]{neronmodels} to prove
241that for every completion $\O_{F,\p}$ of $\O_F$ that the complex
242\begin{equation}\label{eqn:neron}
243   0 \ra \cA_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}\ra 0
244\end{equation}
245is exact.  (For the reader's convenience, the
246results of \cite[Ch.~7]{neronmodels} that we refer
247to below are reproduced in Section~\ref{sec:neronmodels}.)
248To this end, fix a prime ideal~$\p$ of $\O_F$,
249and let~$p$ be its residue characteristic.
250
251First suppose that $\p\nmid N_E \cdot \disc(K)$, so~$\p$
252is a prime of good reduction for~$R$.
253If $p\nmid n$, \cite[Prop.~7.5.3 (a)]{neronmodels} implies
254that sequence (\ref{eqn:neron}) is exact.
255If $p\mid n$ then our hypothesis on divisors of $n$ are
256exactly the hypothesis to \cite[Th.~7.5.4(iii)]{neronmodels},
257which implies that (\ref{eqn:neron}) is exact.
258
259Next suppose that $\p\mid N_E\cdot \disc(K)$, so
260$\p \mid N_E$ or $\p \mid \disc(K)$.
261Then since $\gcd(n,N_E\cdot \disc(K))=1$, we have that $p\nmid n$,
262so we are led to apply \cite[Prop.~7.5.3 (a)]{neronmodels} with
263$B_K = E\subset R$.
264Because $N_E$ and $\disc(K)$ are coprime, $\p$ doesn't
265divide both $\disc(K)$ and $N_E$. We consider each case in turn:
266\begin{itemize}
267
268\item Suppose that $\p\nmid N_E$.
269Then \cite[Prop.~7.5.3 (a)]{neronmodels}
270asserts that $\cA_{\O_{F,\p}}\ra \cR_{\O_{F,\p}}$ is a closed immersion,
271$\cR_{\O_{F,\p}} \ra \cE_{\O_{E,\p}}$ is smooth with
272kernel $\cA_{\O_{F,\p}}$,
273and the cokernel of $\cR_{\O_{F,\p}}\ra \cE_{\O_{F,\p}}$ is killed
274by multiplication by~$n$.  However, in the proof of
275\cite[Prop.~7.5.3 (a)]{neronmodels} (see line~$6$
276on page 187), one only uses
277that $\cE_{\O_{F,\p}}$ has good reduction
278to deduce surjectivity, so in
279fact $\cR_{\O_{F,\p}}\ra \cE_{\O_{F,\p}}$ is surjective.
280(The point is that the good reduction hypothesis on
281$\cR_{\O_{F,\p}}$ is used in the proof only to deduce
282that $\cE_{\O_{F,\p}}$ has good reduction.  Alternatively, using
283just the statement of \cite[Prop.~7.5.3 (a)]{neronmodels} we
284immediately see that the sequence
285$0 \ra \cA_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$
286is exact.  To get surjectivity on the right, note that the
287composition
288$\cE_{\O_{F,\p}} \ra \cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$
289is multiplication by~$n$, which is surjective because
290$p\nmid n$ (so that $[n]$ is etale).
291Thus $\cR_{\O_{F,\p}} \ra \cE_{\O_{F,\p}}$ must be surjective.
292)
293
294\item Suppose that $\p\nmid \disc(K)$, and let
295$\PP$ be a prime of $K$ lying over~$\p$.
296We will use that formation of N\'eron models commutes with
297unramified base extension \cite{} and check exactness
298of (\ref{eqn:neron}) after base extension to the
299unramified extension $\O_{K,\mathfrak{P}}$ of $\O_{F,\p}$.
300In light of Section~\ref{sec:restriction_of_scalars},
301the generic fiber of the base extension of (\ref{eqn:neron})
302to $\O_{K,\mathfrak{P}}$ is
303$$304 0\ra E_{K,\PP}^{\oplus (n-1)} \ra 305 E_{K,\PP}^{\oplus n}\xra{\Sigma} E_{K,\PP}\ra 0.$$
306Thus the corresponding complex of N\'eron models over $\O_{K,\p}$ is
307$$308 0\ra \cE_{\O_{K,\PP}}^{\oplus (n-1)} \ra \cE_{\O_{K,\PP}}^{\oplus n} 309 \xra{\Sigma} \cE_{\O_{K,\PP}}\ra 0, 310$$
311which is exact since, e.g., it is exact on $S$-points for {\em any}
312ring~$S$.
313\end{itemize}
314
315\end{proof}
316
317By the lemma, we have an exact sequence of sheaves
318$$319 0 \ra \cA \ra \cR \ra \cE \ra 0 320$$
321on the \'etale site over $\O_F$.
322Let $\cA^{\vee}$ denote the N\'eron model of the dual of~$A$.
323\begin{proposition}\label{prop:et}
324The following diagram has an exact row:
325$$\xymatrix{ 326 & & & {H^1(\O_F,\cE)}\ar[d]\\ 327{R(L)} \ar[r] & {E(L)}\ar[r] & {H^1(\O_F,\cA)}\ar[r] & {H^1(\O_F,\cR)}\ar[d]\ar[r] & {H^1(\O_F,\cE)}\\ 328 & & & {H^1(\O_F,\cA^{\vee})} 329} 330$$
331\end{proposition}
332\begin{remark}{\bf Connection with BSD:}
333I'm not sure whether or not the column is exact, or is almost exact.
334It is close to exact, because $0 \ra \cE \ra \cR \ra \cA$ is exact.
335The column is important in connecting nonvanishing of twists and
336the rank~$0$ BSD formula to showing that there are points
337on~$E$.  The tentative connection is as follows:
338Suppose $\Sha(E/F)[n]=0$ but $\Sha(A/F)[n]\neq 0$.
339The composition $\Sha(A/F)\ra \Sha(A^{\vee}/F)\ra \Sha(A/F)$ is
340multiplication by~$n$, which is not injective by our assumption that
341$\Sha(A/F)[n]\neq 0$. Thus
342$\Ker(\Sha(A^{\vee}/F)\ra \Sha(A/F))[n]$ is nonzero, so
343by duality {\bf (I think?)}
344$\Ker(\Sha(A/F)\ra \Sha(A^{\vee}/F))[n]$ is also
345nonzero.  But our hypothesis that $\Sha(E/F)[n]={0}$ combined with
346exactness of the vertical sequence would imply that
347$$\Ker(\Sha(A/F)\ra \Sha(A^{\vee}/F)[n] = \Ker(\Sha(A/F)\ra \Sha(R/F))[n] \neq 0,$$
348so $E(F)$ {\em must} be nonzero!
349
350If this works it is very exciting because it means that the following
351three statements
352\begin{enumerate}
353\item Nonvanishing of twists conjecture (see Conjecture~\ref{conj:nonvanishing})
354\item Finiteness of $\Sha(E/\Q)$, and
355\item $\Sha(A/\Q)$ is as big as predicted by BSD for analytic rank~$0$ twisted
356powers~$A$
357\end{enumerate}
358together imply the statement:
359$$L(E/\Q,1)=0 \quad\implies\quad E(\Q)\text{ is infinite.}$$
360\end{remark}
361
362
363\begin{proof}
364(Not really written, but see Section~\ref{sec:letter} below.)
365By \cite[Appendix]{mazur:tower}, there is an exact sequence
366$$367 0 \ra \Sha(A) \ra H^1(\O_F,\cA) \ra G \ra 0 368$$
369where~$G$ is a finite group whose order if divisible only by~$2$ and
370primes that divide the Tamagawa numbers of $\cA$.
371\end{proof}
372
373\comment{
374Alternatively, one can consider the following diagram
375$$376\xymatrix{ 377 & 0\ar[d] & 0\ar[d]\\ 378 0\ar[r] & {\cE[n]}\ar[r]\ar[d] & {\cE}\ar[d]\ar[dr]^{[n]}\\ 379 0 \ar[r] & {\cA} \ar[r] & {\cR}\ar[r] & {\cE} \ar[r] & 0 380}$$
381and work with cohomology on the fppf site.
382}
383
385We state a conjecture about nonvanishing at~$1$ of certain
386prime-degree twists of the $L$-function attached to an elliptic curve,
387provide extensive computational evidence for the conjecture, and give
388an example which suggests that vanishing twists are very rare.
389
390\subsection{A Conjecture}
391Let~$E$ be an elliptic curve over~$\Q$, and
392suppose $p$ is a rigid prime for~$E$.
393For every prime $\ell\con 1\pmod{p}$, let
394$\chi_{p,\ell} : (\Z/\ell\Z)^* \onto \bmu_p$ be
395one of the Galois-conjugate characters
396of order~$p$ and modulus~$\ell$.
397\begin{conjecture}\label{conj:nonvanishing}
398There exists a prime~$\ell\nmid N_E$ such that
399$$L(E,\chi_{p,\ell},1)\neq 0 400\,\,\,\text{ and }\,\,\, 401a_\ell(E) \not\con 2\pmod{p}.$$
402\end{conjecture}
403
404The condition $a_\ell(E) \not\con 2\pmod{p}$ requires elaboration.
405Since $\ell\con 1\pmod{p}$, this condition can be rewritten
406$a_\ell(E)\not\con \ell+1\pmod{p}$, which is a familiar'' condition
407to impose.  We demand that $a_\ell(E)\not\con \ell+1\pmod{p}$ because
408then the characteristic polynomial $x^2 + a_\ell x +\ell\in \F_p[x]$
409of $\Frob_\ell$ on $E[p]$ does not have $+1$ as an eigenvalue.  This
410is a key hypothesis in Section~\ref{sec:ptorsion}.
411
412
413\begin{table}
414\caption{Evidence for Conjecture~\ref{conj:nonvanishing}\label{tbl:evidence}}
415\noindent%\hspace{-5ex}
416\begin{tabular} {|c|cccccccccccccc|}\hline
417  $\,\,\,E$ & 3&5&7&11&13&17&19&23&29&31&37&41&43&47\\\hline
418\nf{37A} & 13&11&29&67&53&103&191&47&59&311&-&83&173&283\\
419\nf{43A} & 7&11&29&23&53&103&191&47&59&311&149&83&-&283\\
420\nf{53A} & 13&11&29&23&53&103&191&47&59&311&149&83&173&283\\
421\nf{57A} & -&11&29&23&53&103&-&47&59&311&149&83&173&283\\
422\nf{58A} & 7&11&29&23&53&103&191&47&-&311&149&83&173&283\\
423\nf{61A} & 7&31&29&67&53&103&191&47&59&311&149&83&173&283\\
424\nf{65A} & 19&-&43&23&-&137&191&47&59&311&149&83&173&659\\
425\nf{77A} & 19&11&-&-&53&103&191&47&59&311&149&83&173&283\\
426\nf{79A} & 13&11&43&67&53&103&191&47&59&311&149&83&173&283\\
427\nf{82A} & 13&41&29&23&53&103&191&47&59&311&149&-&173&283\\
428\nf{83A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
429\nf{88A} & 7&11&29&-&131&103&191&47&59&311&149&83&173&283\\
430\nf{89A} & 19&11&29&67&53&103&191&47&59&311&149&83&173&283\\
431\nf{91A} & 31&11&-&23&-&103&191&47&59&311&149&83&173&283\\
432\nf{91B} & -&11&-&23&-&103&191&47&59&311&149&83&173&283\\
433\nf{92B} & 13&61&29&23&79&103&229&-&59&311&149&83&173&283\\
434\nf{99A} & -&11&29&-&53&103&191&47&59&311&149&83&173&283\\
435\nf{101A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
436\nf{102A} & -&11&29&23&53&-&191&47&59&311&149&83&173&283\\
437\nf{106B} & 7&11&29&23&53&137&191&47&59&311&149&83&431&283\\\hline
438\nf{389A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
439\nf{433A} & 7&11&43&23&53&103&191&47&59&311&149&83&173&283\\\hline
440\end{tabular}
441\end{table}
442
443
444\subsection{Computational Evidence for the Conjecture}
445
446Using a \magma{} program (see~\cite{magma}), the author's computer
447verified Conjecture~\ref{conj:nonvanishing} for every $p<50$ for the
448first $20$ optimal elliptic curve quotients of $R_0(N)$ of rank~$1$
449and the first~$2$ elliptic curve quotients of rank~$2$.
450
451Table~\ref{tbl:evidence} contains, for each $p < 50$, the smallest
452prime~$\ell$ satisfying the conditions of
453Conjecture~\ref{conj:nonvanishing}.  The elliptic curves are labeled
454as in Cremona.  The curves \nf{389A} and \nf{433A} both have rank~$2$,
455and all others have rank~$1$.  A dash (-) in the table indicates that the
456corresponding prime is not rigid, so the conjecture does not apply.
457
458In all cases the first prime $\ell\nmid N_E$ with
459$\ell\con 1\pmod{p}$ with $a_\ell(E)\not \con 2\pmod{p}$
460satisfied $L(E,\chi_{p,\ell},1)\neq 0$, except
461for \nf{61A} with $p=5$, \nf{79A} with $p=7$,
462\nf{82A} with $p=5$, \nf{89A} with $p=11$,
463and \nf{92B} with $p=5$.  In every one of these~$5$ exceptional
464cases, the second prime that we tried
465satisfied the conclusion of
466Conjecture~\ref{conj:nonvanishing}.
467
468\subsection{The Density}
469The following conjecture is not mentioned
470elsewhere in this paper.
471\begin{conjecture}
472Let~$p$ be a rigid prime for an elliptic curve~$E$.
473The set of primes
474$$\left\{\ell \,\,:\,\, \ell \con 1\!\!\!\!\!\pmod{p}\text{ and } 475 L(E,\chi_{p,\ell},1)=0\right\} 476$$
477has Dirichlet density~$0$.
478\end{conjecture}
479
480The following numerical example gives evidence for this conjecture.
481\begin{example}
482Let~$E$ be \nf{37A} and let $p=5$.  Then the only $\ell<1000$ (with
483$\ell\con 1\pmod{5}$) for which $L(E,\chi_{5,\ell},1)=0$ is $\ell=41$.
484% 4 minutes to compute
485\end{example}
486
487
488\section{$p$-Torsion of Twisted Powers}\label{sec:ptorsion}
489Let~$p$ and~$\ell$ be as in Conjecture~\ref{conj:nonvanishing}.
490In order to apply Theorem~\ref{thm:shaexists}, it is necessary
491to know that~$p$ does not divide the orders of certain groups.
492In this section, we use that $a_\ell(E)\not\con 2\pmod{p}$ to
493deduce that certain groups do not have any~$p$ torsion.
494The key idea is that the condition on $a_\ell(E)$ implies
495that~$+1$ is not an eigenvalue of $\Frob_\ell$ on the
496$p$-adic Tate module attached to~$E$.
497
498First, we recall that certain torsion points on the closed
499fiber of a N\'eron model lift to the generic fiber.  Let~$K$ be a
500finite extension of~$\Q_\ell$ with ring of integers~$\O$ and residue
501class field~$k$.
502\begin{lemma}\label{lem:red_mod_n}
503Let $A$ be an abelian variety over~$K$ with N\'eron model $\cA$ over~$\O$.
504Then for every integer $n$ not divisible by~$\ell$, there is an isomorphism
505$$A(K)[n] \xrightarrow{\,\,\isom\,\,} \cA(k)[n].$$
506\end{lemma}
507\begin{proof}
508This is a standard fact, whose proof we recall for the convenience
510Let $A^{1}(K)$ denote the kernel of the natural reduction
511map $r:A(K)\ra \cA(k)$.  Because $A^{1}(K)$ is a formal group,
512it is pro-$p$, so $[n]:A^{1}(K)\ra{}A^{1}(K)$ is an isomorphism.
513Since $\cA$ is smooth over~$\O$,
514Hensel's lemma (see BLR) implies that the reduction map
515is surjective, so the following sequence is exact:
516       $$0\ra A^1(K) \ra A(K) \ra \cA(k) \ra 0.$$
517The snake lemma applied to the multiplication by~$n$ diagram
518attached to this exact sequence yields the following
519exact sequence:
520$$0\ra0\ra A(K)[n]\ra \cA(k)[n] \ra 0 \ra A(K)/n A(K) \ra \cA(k)/n\cA(k)\ra0,$$
521which proves the proposition.
522\end{proof}
523
524
525Let~$E$ be an elliptic curve over~$\Q$ with  associated newform
526$f = \sum a_n q^n$, and fix a prime~$p$ that is rigid for~$E$.
527Suppose~$K$ is the extension of~$\Q$ corresponding
528to a surjective Dirichlet character
529$\chi: (\Z/\ell\Z)^* \onto \bmu_p$
530of prime conductor; then~$K$ is
531the subfield of $\Q(\bmu_\ell)$ fixed by $\Ker(\chi)$,
532so it is of degree~$p$, is totally ramified
533at~$\ell$, and is unramified outside~$\ell$.
534Let~$A=\Ker(\Tr : \Res_{K/\Q} E_K \ra E)$.
535We next compute the Tamagawa number
536$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$
537and the $p$-torsion of several abelian varieties.
538\begin{proposition}\label{prop:ptorsion}
539Let~$E$, $\chi$, $K$, and~$A$ be as above and suppose
540that $\ell\nmid N_E$ and $a_\ell \not\equiv 2\pmod{p}$.
541Then the following groups have no nontrivial $p$-torsion:
542$$A(K), \quad A(\Q_\ell),\quad R(\Q_\ell),\quad 543(R/E)(\Q_\ell),\quad \text{and}\quad \Phi_{A,\ell}(\F_\ell).$$
544%$$A(K)[p]=A(\Q_\ell)[p] = R(\Q_\ell)[p] = 545%(R/E)(\Q_\ell)[p]=\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$$
546\end{proposition}
547\begin{proof}
548The reason the $p$-torsion vanishes in all these cases
549is that the condition $a_\ell \not\equiv 2\pmod{p}$ implies
550in each case that $\Frob_\ell$ has no $+1$ eigenvalue.
551The details are as follows.
552
553We first show that $R(\Q_\ell)[p]=\{0\}$, where $R=\Res_{K/\Q}E_K$.
554By definition,
555 $$R(\Q_\ell) = E_K(\Q_\ell\tensor_\Q K) = 556 E(K_v)\cross \cdots \cross E(K_v) \quad \text{(p copies)},$$
557where $K_v$ is the completion of~$K$ at the unique prime of~$K$
558lying over~$\ell$.
559The action of $\Frob_\ell\in \Gal(\Q_\ell^{\ur}/\Q_\ell)$
560on $E[p](\Q_\ell^{\ur})=E[p](\Qbar_\ell)$ has characteristic
561polynomial
562$F(x) = x^2-a_\ell x + \ell \in \F_p[x]$.
563Since $a_\ell \not\equiv 2\pmod{p}$ and $\ell\equiv 1\pmod{p}$, it
564follows that $\Frob_\ell$ does not have
565$+1$ as an eigenvalue, so $E(\Q_\ell)[p]=\{0\}$.
566If $z\in E(K_v)[p]$, then the field $L=\Q_\ell(z)$ is an unramified
567subfield of the totally ramified field $K_v$, so $z\in E(\Q_\ell)[p]=\{0\}$.
568Thus $E(K_v)[p]=\{0\}$, which implies that $E(K)[p]=\{0\}$ and
569$R(\Q_\ell)[p]=\{0\}$.
570Since $R_K/E_K \isom E_K \cross \cdots \cross E_K$ ($p-1$ times),
571we see that
572$$(R/E)(\Q_\ell)[p]\subset (R/E)(K_v)[p] = 573 (E(K_v)\cross \cdots \cross E(K_v))[p] = \{0\}.$$
574
575Finally, we turn to the component group $\Phi_{A,\ell}$.
576Let $\cA$ denote the N\'eron
577model of~$A$.  By Lang's Lemma the natural map $\cA(\F_\ell) \ra 578\Phi_{A,\ell}(\F_\ell)$ is surjective.  Thus if
579$\Phi_{A,\ell}(\F_\ell)[p]\neq \{0\}$, then $\cA(\F_\ell)[p]\neq 580\{0\}$.  However, by Lemma~\ref{lem:red_mod_n} and observation of the
581previous paragraph,
582  $$\cA(\F_\ell)[p] = A(\Q_\ell)[p]\subset R(\Q_\ell)[p]=\{0\},$$
583so $\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$, as claimed.
584\end{proof}
585
586
587\subsection{The Tamagawa Number of $A$ at $\ell$}
588In this section, the notation and hypothesis are as in
589Proposition~\ref{prop:ptorsion}.
590That proposition implies that the Tamagawa number
591$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$ is coprime
592to~$n$.  In this section we use Remark~5.4 of \cite{edixhoven:tame} to
593prove that in fact $c_{A,\ell}=1$.
594
595Let $\lambda$ be the prime of~$K$ lying over $\ell$, and
596let $K_{\lambda}$ denote the completion of~$K$ at~$\lambda$,
597so $K_{\lambda}$ is totally and tamely ramified over $\Q_\ell$.
598Since
599$$A_{K} \isom \Ker(\Sigma: E_{K}^{\oplus n} 600 \ra E_{K}),$$
601and $E_{K_\lambda}$ has good reduction,
602the geometric closed fiber of the  N\'eron model of $A_{K_{\lambda}}$ is
603$604 A'_{\kbar}\isom \Ker(\Sigma : E_{\kbar}^{\oplus n} \ra E_{\kbar}). 605$
606In the notation of \cite{edixhoven:tame},
607$\mu_n$ acts on $A'_{\kbar}$ by the action
608it induces by cyclically permuting the factors of
609$E_{\kbar}^{\oplus n}$.  Thus
610$A_{\kbar}'(\kbar)^{\mu_n}$ is the set of
611$\sum P_\sigma\tensor\sigma \in E(\kbar)^{\oplus n}$
612such that all $P_\sigma$ are equal and $\sum P_\sigma = 0$,
613i.e.,
614$$615 A_{\kbar}'(\kbar)^{\mu_n} \isom E(\kbar)[n]\ncisom (\Z/n\Z)^2. 616$$
617Thus Remark~5.4 in \cite{edixhoven:tame} implies that
618$\Phi_{A,\ell}(\kbar) \ncisom E(\kbar)[n]$.
619By Proposition~\ref{prop:ptorsion},
620$\Phi_{A,\ell}(k)$ has no elements of order dividing $n$,
621so $\Phi_{A,\ell}(k)=0$.
622
623\section{Nonsquare Shafarevich-Tate Groups}\label{sec:nonsquaresha}
624On page 306--307 of \cite{tate:bsd}, Tate discusses results about the
625structure of the group $\Sha(A/K)$, where~$A$ is an abelian variety over
626a number field~$K$.  He asserts that if~$A$ is a Jacobian then $\#\Sha(A/K)$
627is a perfect square.  Poonen and Stoll subsequently pointed out
628in~\cite{poonen-stoll} that Tate's assertion is not quite correct.  In
629fact, Poonen and Stoll prove that when~$A$ is a Jacobian, $\#\Sha(A/K)$
630is either a square or twice a square, and they give examples in which
631$\#\Sha(A/K)$ is twice a square.  Tate does not discuss the case
632when~$A$ is not a Jacobian, except to mention results that imply that
633$\#\Sha(A/K)$ is square away from~$2$ and primes that
634don't divide the degree of some polarization of~$A$.
635
636Now suppose~$A$ is an arbitrary abelian variety over a number
637field~$K$.  Until now it was unknown whether or not $\#\Sha(A/K)$ must
638be either a square or twice a square.  Let~$E$ be an elliptic curve
639over~$\Q$ of rank~$1$.  Then our construction gives,
640for suitable primes~$p$, an injection
641 $$642 \Z/p\Z \ncisom E(\Q)/p E(\Q) \hookrightarrow \Sha(A/\Q), 643$$
644where~$A$ is an abelian variety over~$\Q$ which is a twist of $E^{\times p-1}$.
645Thus $\Sha(A/\Q)[p]$ has a natural subgroup of order~$p$; moreover,
646no other natural subgroup of order~$p$ presents itself.
647We prove in this section that the $p$-part of $\#\Sha(A/\Q)$
648is not a perfect square.
649
650\subsection{A Letter}\label{sec:letter}
651[Convert this email into a rigorous proof.]
652\begin{verbatim}
653Do you remember my conditional-on-BSD construction of abelian
654varieties A with nonsquare p-part of #Sha(A), for various odd primes
655p?  In particular, you pushed me repeatedly to remove the conditional
656nature of the result.  I think I found a very clean way to do this
657today, which doesn't require computing any Selmer groups at all!
658
659I make a list of hypothesis on p, the field that I restrict scalars
660down from and the Fourier coefficient a_ell(E).  For example, when E
661is the curve of conductor 43, one could take p=3, K the cubic field
662Q(mu_7)^+.  Then if R is the restriction of scalars of E from K down
663to Q, surprisingly the exact sequence
664
665             0 --> A --> R --> E --> 0
666
667of abelian varieties *does* induce an exact sequence of Neron models
668over Spec(Z), as I think I've checked using chapter 7 of
669Bosch-Lutkebohmert-Raynaud.   We then obtain the following exact
670sequence from the long exact sequence of etale cohomology:
671
672         0  --->  E(Q)/3E(Q) ---> H^1(Z,A) ---> H^1(Z, R) --> H^1(Z,E)
673
674Then an old theorem of Barry from his "Rational Points on Abelian
675Varieties..." along with my hypothesis on E, p, and ell, imply that
676H^1(Z,A), H^1(Z,R) and H^1(Z,E) differ from Sha(A), Sha(R), and Sha(E)
677by at most a 2-group.  Ignoring this 2 business, we have an exact
678sequence
679
680     0  --->  E(Q)/3E(Q) ---> Sha(A/Q) --> Sha(E/K) --> Sha(E/Q).
681
682(I also just used that Sha(R/Q) = Sha(E/K).)
683
684Now the 3-power-part Sha(E/Q)(3) = {0}, by Euler Systems theory (or I
685could directly check this), so we have
686
687     0  -->  E(Q)/3E(Q) ---> Sha(A/Q)(3) --> Sha(E/K)(3) --> 0
688
689Properties of the Cassels-Tate pairing on the elliptic curve E over K
690and the fact that Sha(E/K) is finite (since it's an image of
691Sha(A)(3), which is finite by Kato's theorem), imply that #Sha(E/K)(3)
692is a an even power of 3.  Thus #Sha(A/Q)(3) must be an odd power of 3.
693
694What do you think?  This seems much better than trying to do explicit
695Selmer group computations with CM elliptic curves.
696
697 -- William
698
699\end{verbatim}
700
701
702\section{Other Remarks}\label{sec:applications}
703We apply the above results to prove that
704Conjecture~\ref{conj:nonvanishing} implies the existence of
705elements of Shafarevich-Tate groups of twisted powers of
706elliptic curves of every prime order.  We also construct an
707abelian variety~$A$ over~$\Q$ such that the Birch
708and Swinnerton-Dyer conjecture predicts that $\Sha(A/\Q)=\Z/3\Z$ and
709that $\#\Sha(A/\Q)$ is not a square or twice a square.
710
711\subsection{Existence of Elements of $\Sha$ of all Prime Orders}
712\begin{proposition}\label{prop:all_prime_orders}
713Let~$p$ be a prime number.  Then Conjecture~\ref{conj:nonvanishing}
714implies that there exists infinitely many twisted powers~$A$ of some
715elliptic curve such that $\Sha(A/\Q)[p]\neq \{0\}$.
716\end{proposition}
717\begin{proof}
718Most of the proposition can be proved using a single elliptic curve.
719When ordered by conductor, the first elliptic curve~$E$ over $\Q$ with
720positive rank has prime conductor~$37$ and is defined by the
721Weierstrass equation $y^2 + y = x^3 - x$.  Table~1 of
722\cite{cremona:algs} shows that~$E$ is isolated in its isogeny class,
723so \cite[Exercise~4]{ribet-stein:serre} implies that representations
724$\rho_{E,p}$ are irreducible.  Since
725$\ord_{37}(j(E))=-1$, $\cbar_{37}=1$.  Thus all odd primes $p\neq 37$
726are rigid for~$E$.  The proposition then follows for all odd primes
727$p\neq 37$ by Theorem~\ref{thm:main}.
728
729We complete the proof as follows.  Exactly the same argument applied
730to the unique elliptic curve of conductor~$43$ proves the proposition
731for all odd primes $p\neq 43$.  Finally, B\"olling proved in
732\cite{bolling:sha} that for every $j\in\Q$ there is an elliptic
733curve~$E$ with $j$-invariant~$j$ such that infinitely many twists~$E'$
734of $E$ have $\Sha(E'/\Q)\neq\{0\}$.
735\end{proof}
736
737
738\subsection{What Goes Wrong when $p=2$?}
739In the previous section, we set $p=3$ and
740constructed an abelian variety $A$ of dimension $p-1$
741that (conjecturally) has nonsquare $\Sha(A/\Q)[p]$.
742We can construct an~$A$ in an analogous way for any odd prime~$p$,
743and the author expects that $\Sha(A/\Q)[p]$ is nonsquare in most cases.
744However, when $p=2$, the dimension of~$A$ is~$1$, so
745in that case $\#\Sha(A/\Q)$ must be a perfect square.
746
747What goes wrong?   The problem lies in Theorem~\ref{thm:shaexists}.
748The argument used to prove Theorem~\ref{thm:shaexists}
749at least provides a map
750$$751 E(\Q)/2 E(\Q)\hookrightarrow\Vis_R(H^1(\Q,A)). 752$$
753When $p=2$, the condition $e<p-1$ is not satisfied, so
754the proof of Theorem~\ref{thm:shaexists} does not show
755that the image of $E(\Q)/ 2 E(\Q)$ is locally trivial
756at the prime~$2$ (or at $\infty$).  We thus only
757construct a subgroup of $H^1(\Q,A)$ of nonsquare order,
758not of~$\Sha(A/\Q)$.  Thus even if two elliptic curves have the same
759$E$, then can still possess very different Selmer groups.
760
761
762
763
764
765\section{Connection with BSD}
766\label{sec:bsd}
767[[Needs to be rewritten.]]
768Suppose $E$ is an elliptic curve over~$\Q$ and that $L(E,1)=0$.  The
769Birch and Swinnerton-Dyer conjecture for~$E$ asserts (among other
770things) that $E(\Q)$ is infinite.  Suppose~$A$ is constructed as in
771Section~\ref{sec:main}.  In this section we describe why if a certain
772consequence of a refinement of the Birch and Swinnerton-Dyer
773conjecture for~$A$ is true, then $\Sel^{(n)}(E/\Q)$ is nonzero.
774
775
776Using modular symbols one sees that $L(A,1)\con 0 \pmod{\ell}$,
777so a refinement of the Birch and Swinnerton-Dyer formula for rank~$0$
778abelian varieties predicts that there should be a nonzero element in
779$\Ker (\Sha(A)\ra \Sha(A/E[n]))$.
780Thus by Proposition~\ref{prop:et}, either
781$H^1(X_{\et},\cE)[n]\neq 0$, or there is a nonzero element
782of order dividing~$n$ in
783$$784 \Ker(H^1(X_{\et},\cA)\ra H^1(X_{\et},\cR)) \isom E(\Q)/R(\Q), 785$$
786in which case $E(\Q)/R(\Q)$ contains a nonzero element of
787order dividing~$n$,
788so $E(\Q)$ is infinite.
789Thus either $\Sha(E)[n]\neq 0$ or $E(\Q)$ is infinite, so
790$\Sel^{(n)}(E/\Q)$ is nonzero.
791
792
793\section{Appendix: Exactness Properties}\label{sec:neronmodels}
794For the reader's convenience, we copied {\em verbatim}
795pages 186--187 of \cite[\S7.5]{neronmodels}.  In what follows~$R$
796is a discrete valuation ring with field of fractions~$K$.
797
798
799Next, let us look at abelian varieties.
800
801\noindent{\bf Proposition 3.} {\em
802Consider an exact sequence of abelian
803varieties
804$$805 0 \ra A_K' \ra A_K \ra A_K'' \ra 0 806$$
807and the corresponding complex of N\'eron models
808\begin{equation}\label{eqn:dagger}
810\end{equation}
811Let $B_K$ be an abelian subvariety of $A_K$ such that $A_K\ra A_K''$ induces an isogeny
812$u_K: B_K \ra A_K''$; let $n=\deg u_K$.
813
814(a) If $\chr{}k$ does not divide~$n$, then $A'\ra A$ is a closed immersion,
815$A\ra A''$ is smooth with kernel $A'$, and the cokernel of $A_k \ra A_k''$ is
816killed by multiplication with~$n$.  If, in addition, $A$ has abelian reduction, (\ref{eqn:dagger})
817is exact.
818
819(b) If $A$ has semi-aelian reduction, the sequence (\ref{eqn:dagger}) is exact up to isogeny;
820i.e., it is isogenous to an exact sequence of commutative $S$-group schemes.
821}
822\vspace{1em}
823\begin{proof}
824The isogeny $u_K : B_K \ra A_K''$ gives rise to an isogeny $v_K : A_K' \cross_K B_K \ra A_K$
825of degree~$n$.  So there is an isogeny $w_K : A_K \ra A_K' \cross_K B_K$ such
826that $w_K\circ v_K$ is multiplication by~$n$.  Let $B$ be the N\'eron model of $B_K$.  Then
827$u_K$, $v_K$, and $w_K$ extend to $R$-morphisms $u: B \ra A''$,
828$v: A'\cross_R B \ra A$, and
829$w: A\ra A'\cross_R B$ such that $w\circ v$
830is multiplication by $n$ on $A'\cross_R B$.
831Assuming the condition of (a), the multiplication by~$n$ is an \'etale isogeny on
832$A'\cross_R B$, and $u,v$ and $w$ are easily checked to be \'etale
833isogenies, too.  Then $H:=w^{-1}(A')$ is a smooth closed subgroup scheme of~$A$
834which satisfies $H^0_K = A_K'$.  It follows that the schematic closure of $A_K'$
835in~$H$ or~$A$ is an open subgroup scheme of~$H$ and, thus, is smooth over~$R$.
836So, by using the \'etale isogeny~$u$.  One shows that
837$A\ra A''$ is flat, has kernel $A'$ and, hence, is smooth.  Furthermore,
838if~$A$ has abelian reduction, the same is true for $A''$ by 7.4/2 so that
839$A\ra A''$ is surjective.
840
841Assertion (b) follows from the fact that $v:A'\cross_R B \ra A$ and
842$u:B\ra A''$ are isogenies; use 7.3/6 and 7.3/7.
843\end{proof}
844
845\noindent{\bf Theorem 4.} {\em
846Let $0\ra A_K' \ra A_K \ra A_K'' \ra 0$ be an exact sequence of
847abelian varieties and consider the associated sequence of N\'eron
848models $0\ra A' \ra A \ra A'' \ra 0$.  Assume that the following
849condition is satisfied:
850
851(*) $R$ has mixed characteristic and the ramification index $e=v(P)$
852satisfies $e<p-1$, where $p$ is the residue characteristic of $R$
853and where $v$ is the valuation on~$R$, which is normalized by the condition
854that~$v$ assumes the value~$1$ at unifromizing elements of~$R$.
855
856Then the following assertions hold:
857
858(i) If $A'$ has semi-abelian reduction, $A'\ra A$ is a closed immersion.
859
860(ii) If $A$ has semi-abelian reduction, the sequence $0\ra A' \ra A \ra A''$
861is exact.
862
863(iii) If $A$ has abelian reduction, the sequence $0\ra A' \ra A\ra A''\ra 0$
864is exact and consists of abelian $R$-schemes.
865}
866\vspace{1em}
867
868\begin{proof}
869Let us first see how assertions (ii) and (iii) can be deduced from assertion (i).
870If~$A$ has semi-abelian or abelian reduction, the same is true for $A'$ and $A''$
871by 7.4/2.  So $A'\ra A$ is a closed immersion by (i), and we can consider the
872quotient $A/A'$; it exists in the category of algebraic spaces, cf.\ 8.3/9.
873Furthermore, $A/A'$ is smooth and separated and, thus, a scheme by 6.6/3.  Now look at the
874canonical morphism $A/A'\ra A''$ which is an isomorphism on generic fibres.
875Since $A$ has semi-abelian reduction, the same is true for $A/A'$, and it
876follows from 7.4/3 that $A/A'\ra A''$ is an open immersion.  So assertion (ii)
877is clear.  Finally, if $A$ has abelian reduction, the same is true for $A/A'$.
878So the latter is an abelian scheme by 7.4/5 and, thus, must coincide with the
879N\'eron model $A''$ of $A''_K$.  Thereby we obtain assertion (iii).
880[The rest of the proof takes a few pages.]''
881\end{proof}
882
883
884
885
886
887\bibliography{biblio}
888\end{document}
889
890
891E := EC("37A");
892M := MS(E);
893K := CyclotomicField(5);
894M5 := BaseExtend(M,K);
895P := [p : p in [3..200] |IsPrime(p)];
896time X := [<p,
897 Dimension(TwistedWindingSubmodule(M5,1,DirichletGroup(p,K).1^2))> : p in P];
898// Time: 11.369
899[ <11, 1>, <31, 1>, <41, 0>, <61, 1>, <71, 1>, <101, 1>, <131, 1>,
900  <151, 1>, <181, 1>, <191, 1> ]
901
902\comment{
903$Log: bigsha.tex,v$
904Revision 1.25  2001/09/28 02:48:22  was
905?
906
907Revision 1.24  2001/09/23 04:43:11  was
908more TOC stuff and cleaning.
909
910Revision 1.23  2001/09/22 20:27:00  was
911Worked on
912\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}
913and changed Sha(A) to Sha(A/Q).
914
915Revision 1.22  2001/09/22 19:26:23  was
916I don't know.
917
918Revision 1.21  2001/09/13 01:42:50  was
919Lots of nice little improvements!
920
921Revision 1.20  2001/09/09 04:11:42  was
923
924Revision 1.19  2001/09/08 02:55:34  was
925polishing.
926
927Revision 1.18  2001/09/06 03:39:53  was
929
930Revision 1.17  2001/09/06 03:24:36  was
931typo
932
933Revision 1.16  2001/09/06 03:23:56  was
934minor typo
935
936Revision 1.15  2001/09/06 03:17:42  was
937Added table of evidence for twisting conjecture.
938
939Revision 1.14  2001/09/06 02:42:04  was
940...
941
942}
943\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}
944Let $E$ be an elliptic curve over~$\Q$.
945In this section, we prove that Conjecture~\ref{conj:nonvanishing}
946implies if~$p$ is a rigid prime, then
947$E(\Q)/p E(\Q)$ is canonically isomorphic to the
948elements of order~$p$ in the visible Shafarevich-Tate
949group of a rank~$0$ twisted power of~$E$.
950
951\begin{theorem}\label{thm:main}
952Assume that Conjecture~\ref{conj:nonvanishing} is true.
953If~$E$ is an elliptic curve over~$\Q$, then for every
954rigid prime~$p$, there is a degree~$p$ abelian extension~$K$
955of~$\Q$ such that
956$$E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p]),$$
957where $R=\Res_{K/\Q}(E_K)$ and~$A\subset R$ has
958dimension~$p-1$ and rank~$0$.
959\end{theorem}
960
961The proof divides naturally into three steps.  First, we use
962Conjecture~\ref{conj:nonvanishing} to construct~$A$.  The we
963use a theorem of Kato and that formation of N\'eron models
964commutes with unramified base change to prove that~$A$ has
965rank~$0$ and that~$p$ does not torsion of Tamagawa numbers
966of~$A$.  Next, we apply the visibility Theorem~\ref{thm:shaexists}
967to obtain an isomorphism $E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p])$.
968
969\begin{proof}
970Conjecture~\ref{conj:nonvanishing} implies that there exists
971a prime $\ell\nmid N_E$
972with $\ell\con 1\pmod{p}$ such that $L(E,\chi_{p,\ell},1)\neq 0$ and
973$a_\ell(E)\not\con 2\pmod{p}$.  Let~$\ell$ be such a prime, and
974let~$K$ be the abelian extension of~$\Q$ corresponding to
975a surjective character $\chi_{p,\ell} : (\Z/\ell\Z)^* \ra \bmu_p$.
976Recall from Section~\ref{sec:restriction_of_scalars} that
977the restriction of scalars $R = \Res_{K/\Q}(E_K)$ is an
978abelian variety over~$\Q$ of dimension~$p$, and
979we have a commutative diagram
980%$$\xymatrix{ 981% & {A}\[email protected]{^(->}[rd]\\ 982%{E[p]\,\,}\[email protected]{^(->}[ur] \[email protected]{^(->}[dr] & & {R}\ar[dr]^{\Tr}\\ 983% & {E}\[email protected]{^(->}[ur]\ar[rr]^{[p]} & & {E} 984%}$$
985$$\[email protected]=3pc{ 986{E[p]\,\,}\ar[r]\ar[d] & {E} \ar[dr]^{[p]}\ar[d] \\ 987 {A}\ar[r] & {R} \ar[r]^{\Tr} & {E,} 988}$$
989where $A=\Ker(\Tr)$ is an abelian variety.
990
991%Since $\rho_{E,p}$ is irreducible, $p\nmid E(\Q)_{\tor}$.
992%We have $A(\Q)\subset R(\Q)=E(K)$.  Since~$K$ is totally ramified
993%at~$\ell$, unramified outside~$\ell$, and $\ell\nmid p N_E$, $E(K)[p]$
994%equals $E(\Q)[p]$, which is $\{0\}$ because $\rho_{E,p}$ is assumed
995%irreducible.  Thus $p\nmid \#A(\Q)$.
996
997Since $L(A,s)=\prod L(E,\chi_{p,\ell}^\sigma,s)$, and $L(E,\chi,1)\neq 9980$, Kato's work on Euler systems\edit{Reference?} implies that $A(\Q)$ is finite.
999Proposition~\ref{prop:ptorsion} implies that
1000$p\nmid \#A(\Q)\cdot \#(R/E)(\Q)$.
1001Next suppose that~$q$ is a prime of bad reduction for~$A$.  If
1002$q\not=\ell$, then $K/\Q$ is unramified at~$q$.
1003The formation of N\'eron models commutes with unramified base
1004change\edit{Reference?} and $A_K=E^{\times(p-1)}$, so $c_{A,q}$ divides
1005$\cbar_{E,q}$, which is not divisible by~$p$ since~$p$ is rigid
1006for~$E$.  If $q=\ell$, Proposition~\ref{prop:ptorsion} asserts that
1007$p\nmid c_{A,q}$.
1008
1009The previous paragraph combined with Proposition~\ref{prop:ptorsion}
1010shows that the hypothesis of
1011Theorem~\ref{thm:shaexists} are satisfied with $A=A$, $B=E$,
1012$R=R$, and $L=\Q$.  Thus there is an injective map
1013      $$E(\Q)/p E(\Q) \hookrightarrow \Vis_R(\Sha(A/\Q))\subset \Sha(A/\Q).$$
1014
1015To prove surjectivity, note that by definition every element of
1016$\Vis_R(\Sha(A/\Q)[p])$ is the image of an element of $R(\Q)$ and
1017by Proposition~\ref{prop:ptorsion} the index of $E(\Q)$ in $R(\Q)$ is
1018finite and coprime to~$p$.
1019\end{proof}
1020
1021