CoCalc Shared Fileswww / papers / nonsquaresha / bigsha.tex
Author: William A. Stein
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5\title{\sc
6Visualizing Mordell-Weil Groups of Elliptic Curves Using
7Shafarevich-Tate Groups}
8\author{William A. Stein\\{\sf was@math.harvard.edu}}
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13\begin{document}
14\maketitle
15
16\begin{abstract}
17Let $E$ be an elliptic curve over~$\Q$.  We prove that a very
18plausible conjecture about nonvanishing of prime-degree twists of
19$L(E,s)$ implies that for all but finitely many primes~$p$ there is a
20twist~$A$ of $E^{\times (p-1)}$ such that $E(\Q)/p E(\Q)$ is visible''
21as a subgroup of $\Sha(A/\Q)$.
22When $E$ is the elliptic curve of conductor~$43$, our
23construction yields a twist of $E\cross E$
24such that the Birch and Swinnerton-Dyer conjecture predicts that
25$\Sha(A/\Q)[3]$ has order~$3$.
26\end{abstract}
27
28\tableofcontents
29
30\section*{Introduction}
31Let~$E$ be an elliptic curve over~$\Q$.  We prove that a very
32plausible conjecture about nonvanishing of prime-degree twists of
33$L(E,s)$ implies that for all but finitely many primes~$p$ there is a
34twist~$A$ of $E^{\times (p-1)}$ such that $E(\Q)/p E(\Q)$ is visible''
35as a subgroup of $\Sha(A/\Q)$.
36When $E$ is the elliptic curve of conductor~$43$, our
37construction yields a twist of $E\cross E$
38such that the Birch and Swinnerton-Dyer conjecture predicts that
39$\Sha(A/\Q)[3]$ has order~$3$.
40
41
42This paper is organized out as follows.
43In Section~\ref{sec:terminology} we define twisted powers, Tamagawa
44numbers, and rigid primes.  We recall in
45Section~\ref{sec:restriction_of_scalars} the definition of the
46restriction of scalars of an elliptic curve and prove a proposition
47about a map induced by trace.  In Section~\ref{sec:shaexists} we
48recall a construction of the author and Amod Agashe of visible
49subgroups of Shafarevich-Tate groups.  We state a conjecture about
50nonvanishing of twists of prime degree in
51Section~\ref{sec:nonvanishing}, and give computational evidence for
52this conjecture.  In Section~\ref{sec:ptorsion} we prove triviality of
53the $p$-torsion of several abelian groups attached to twisted powers
54of an elliptic curve.  The heart of the paper is
55Section~\ref{sec:main}, which uses the above results to construct
56subgroups of Shafarevich-Tate groups of twisted powers.
57Section~\ref{sec:applications} pulls together the results of the
58previous sections; there we prove that the conjecture of
59Section~\ref{sec:nonvanishing} implies the existence of elements of
60Shafarevich-Tate groups of every prime order, we discuss the extent to
61which the order of $\Sha$ can fail to be square, and describe a
62connection with the Birch and Swinnerton-Dyer conjecture.
63
64\vspace{2ex}\par\noindent{}{\bf{}Acknowledgement: }
65It is a pleasure to thank Gautam Chinta,
66Benedict Gross, Emanuel Kowalski, Barry Mazur,
67Bjorn Poonen, David Rohrlich,
68and Michael Stoll
69for helpful comments and conversations.
70
71
72\section{Terminology}\label{sec:terminology}
73In this section, we define twisted powers and rigid primes for an
74elliptic curve, and recall the definition of Tamagawa numbers of an
75abelian variety.
76
77\begin{definition}[Twisted Powers]
78A {\em twisted power} of an elliptic curve~$E$ over a field~$K$ is an
79abelian variety $A$ over~$K$ that is isomorphic over $\Kbar$ to
80$E^{\times n}$ for some positive integer~$n$.
81\end{definition}
82
83We recall the standard notion of Tamagawa number of an abelian variety~$A$,
84and introduce the notation $\cbar_{A,p}$ for the order of the group
85of components of~$A$ over $\Fbar_p$.
86\begin{definition}[Tamagawa Numbers]
87Let~$A$ be an abelian variety over~$\Q$ with N\'eron model $\cA$ over~$\Z$, and
88let~$p$ be a prime.
89The component group of~$A$ at~$p$ is the finite group scheme
90$\Phi_{A,p} = \cA_{\F_p}/\cA_{\F_p}^0$, where $\cA_{\F_p}^0$ is the
91identity component of $\cA_{\F_p}$.  The
92{\em Tamagawa number} of~$A$ at~$p$ is
93$c_{A,p} = \#\Phi_{A,p}(\F_p)$.
94We also set $\cbar_{A,p} = \#\Phi_{A,p}(\Fbar_p)$.
95\end{definition}
96
97
98\begin{definition}[Rigid Primes]
99Let $E$ be an elliptic curve over~$\Q$.
100A prime~$p$ is {\em rigid} for~$E$ if~$p$ does not divide
101$2\cdot N_E \cdot \prod_{\ell\mid N_E} \cbar_{E,\ell}$ and the
102representation $\rho_{E,p}:\Gal(\Qbar/\Q)\ra \Aut(E[p])$
103is irreducible.  Here $N_E$ is the conductor of~$E$ and
104$\overline{c}_{E,p} = \#\Phi_{E,p}(\Fpbar)$ is the
105order of the component group of~$E$ at~$p$.
106\end{definition}
107
108
109\section{Restriction of Scalars}\label{sec:restriction_of_scalars}
110Let~$E$ be an elliptic curve over~$\Q$, and let~$K$ be Galois
111over~$\Q$.  We recall the definition of restriction of scalars,
112and prove that the kernel of a certain morphism induced by
113$\tr_{K/\Q}$ is geometrically connected.
114
115The restriction of scalars $R=\Res_{K/\Q}(E_K)$ is an abelian variety
116over~$\Q$ of dimension $[K:\Q]$, which is characterized by the
117following universal property: There is a functorial group isomorphism
118$R(S) \isom E_K(S_K)$, where~$S$ varies over all $\Q$-schemes.
119More explicitly, as $\Gal(\Qbar/\Q)$-modules
120we have
121$$122 R(\Qbar) = E(\Qbar \tensor K) \isom E(\Qbar)\tensor_{\Z} \Z[\Gal(K/\Q)], 123$$
124where $\tau\in \Gal(\Qbar/\Q)$ acts on
125$\sum P_\sigma\tensor \sigma \in E(\Qbar)\tensor_{\Z}\Z[\Gal(K/\Q)]$ by
126$$\tau\left(\sum P_\sigma\tensor \sigma\right) = 127 \sum \tau(P_\sigma)\tensor \sigma\tau_{|K}. 128$$
129Moreover, the $L$-series of~$R$ is $\prod_{a=1}^{n} L(E,\chi^a,s)$, and~$R$
130has good reduction at all $p\nmid \ell\cdot N$.
131
132
133\begin{proposition}
134The identity map induces a closed immerion $\iota: E\hookrightarrow 135R$, and the trace $\tr:K\ra \Q$ induces a surjection $\tr:R\ra E$
136whose kernel is geometrically connected.
137\end{proposition}
138\begin{proof}
139The existence of~$\iota$ and $\tr$ follows from Yoneda's lemma.
140The map~$\iota$ is induced by the functorial inclusion
141$E(S)\hookrightarrow E_K(S_K)=R(S)$, so~$\iota$ is injective.
142The $\tr$ map is induced by the usual
143functorial trace map on points
144$R(S)=E_K(S_K)\xrightarrow{\tr} E(S)$.
145
146To verify that $\ker(\tr)$ is geometrically connected, we base
147extend to~$\Qbar$.  First, note that
148$$R_{\Qbar} \ncisom E_{\Qbar}\cross \cdots \cross E_{\Qbar}.$$
149After this base extension, the trace map is the summation map:
150$$+: E_{\Qbar} \cross \cdots \cross E_{\Qbar} 151 \longrightarrow E_{\Qbar}.$$
152Let $n=[K:\Q]$.  The map defined by
153$$\left(a_1,\ldots a_{n-1}\right) \mapsto 154 \left(a_1,a_2,\ldots a_{n-1},-\sum_{i=1}^{n-1} a_i\right),$$
155 is an isomorphism from
156$E_{\Qbar}^{\times (n-1)}$ to $\ker(+)=\ker(\tr_{\Qbar})$.
157Thus $\ker(\tr_{\Qbar})$ is a product of connected varieties,
158hence connected.
159\end{proof}
160
161\begin{corollary}
162Let $n=[K:\Q]$.  Then
163$$(\iota(E)\intersect \ker(\tr))(\Qbar) \isom E[n](\Qbar)\ncisom (\Z/n\Z)^2.$$
164(The rightmost map is an isomorphism of groups, not Galois modules.)
165\end{corollary}
166\begin{proof}
167Since the map
168$$\Q\hookrightarrow K\xrightarrow{\tr} \Q$$ is
169multiplication by~$n$, the composite map
170$$E \hookrightarrow R \longrightarrow E$$
171is also multiplication by~$n$.
172The corollary now follows since
173$\iota(E)\intersect \ker(\tr)$ is the kernel of~$\tr\circ\iota$,
174which equals $[n]$.  It is elementary that $E[n](\Qbar)\ncisom (\Z/n\Z)^2$,
175where we have, of course, forgotten the action of $\Gal(\Qbar/\Q)$.
176\end{proof}
177
178
179
180\section{Visibility Theory}\label{sec:shaexists}
181%The following theorem is \cite[Thm.~3.1]{agashe-stein:visibility},
182%and the group $\Vis_R(\Sha(A))$ below is a subgroup of $\Sha(A)$.
183
184Suppose $\iota: A\hookrightarrow{}R$ is a closed immersion of abelian varieties
185over a number field~$L$.  The {\em visible subgroup} of $\Sha(A/L)$ with
186respect to~$\iota$ is
187$$188 \Vis_R(\Sha(A/L)) = \ker(\Sha(A/L) \ra \Sha(R/L)). 189$$
190The following theorem can sometimes be used to prove that
191$\Vis_R(\Sha(A/L))$ is nontrivial.  We will use it later to construct
192nontrivial elements of Shafarevich-Tate groups in
193Section~\ref{sec:main}.  For an alternative approach which uses
194\'etale cohomology, see Section~\ref{sec:etale}.
195
196\begin{theorem}\label{thm:shaexists}
197Let~$A$ and~$B$ be abelian subvarieties of an abelian
198variety~$R$ over a number field~$L$ such that $A\intersect B$ is finite,
199and suppose that $A(L)$ is finite.
200Let~$N$ be an integer divisible by the residue characteristics
201of primes of bad reduction for~$R$.
202Suppose~$p$ is a prime such that $e<p-1$, where~$e$ is
203the largest ramification of any
204prime of~$L$ lying over~$p$, and that
205$$p\nmid N \cdot \#(R/B)(L)\cdot\#B(L)_{\tor}\cdot 206 \prod_{v} c_{A,v}\cdot c_{B,v},$$
207where $c_{A,v} = \#\Phi_{A,v}(\F_\ell)$  (resp., $c_{B,\ell}$)  is
208the Tamagawa number of~$A$ (resp., $B$)
209at~$v$.  Suppose furthermore that
210$B[p] \subset A$, where both are viewed as subgroups of $R(\Lbar)$.
211Then there is an isomorphism
212         $$B(L)/pB(L)\isom \Vis_R(\Sha(A/L)[p]).$$
213\end{theorem}
214\begin{remark}
215Everything divides infinity, so if $(R/B)(\Q)$ is infinite, then
216no primes satisfy the hypothesis of the theorem.
217\end{remark}
218
219\begin{proof}
220\cite[Thm.~3.1]{agashe-stein:visibility} produces an embedding
221$$222 B(L)/p B(L)\hra \Vis_R(\Sha(A)[p]). 223$$
224That theorem is proved using a standard snake lemma argument to show
225that $B(L)/p B(L)\hookrightarrow \Vis_R(H^1(L,A))$, combined with a
226local analysis at each prime to see that $B(L)$ lands in $\Sha(A/L)$.
227To prove that this embedding is surjective, note that the index of
228$E(\Q)$ in $R(\Q)$ is finite and coprime to~$p$.
229\end{proof}
230
231
232\section{A Conjecture About Nonvanishing of Twists}\label{sec:nonvanishing}
233We state a conjecture about nonvanishing at~$1$ of certain
234prime-degree twists of the $L$-function attached to an elliptic curve,
235provide extensive computational evidence for the conjecture, and give
236an example which suggests that vanishing twists are very rare.
237
238\subsection{The Conjecture}
239Let~$E$ be an elliptic curve over~$\Q$, and
240suppose $p$ is a rigid prime for~$E$.
241For every prime $\ell\con 1\pmod{p}$, let
242$\chi_{p,\ell} : (\Z/\ell\Z)^* \onto \bmu_p$ be
243one of the Galois-conjugate characters
244of order~$p$ and modulus~$\ell$.
245\begin{conjecture}\label{conj:nonvanishing}
246There exists a prime~$\ell\nmid N_E$ such that
247$$L(E,\chi_{p,\ell},1)\neq 0 248\,\,\,\text{ and }\,\,\, 249a_\ell(E) \not\con 2\pmod{p}.$$
250\end{conjecture}
251
252The condition $a_\ell(E) \not\con 2\pmod{p}$ requires elaboration.
253Since $\ell\con 1\pmod{p}$, this condition can be rewritten
254$a_\ell(E)\not\con \ell+1\pmod{p}$, which is a familiar'' condition
255to impose.  We demand that $a_\ell(E)\not\con \ell+1\pmod{p}$ because
256then the characteristic polynomial $x^2 + a_\ell x +\ell\in \F_p[x]$
257of $\Frob_\ell$ on $E[p]$ does not have $+1$ as an eigenvalue.  This
258is a key hypothesis in Section~\ref{sec:ptorsion}.
259
260
261\begin{table}
262\caption{Evidence for Conjecture~\ref{conj:nonvanishing}\label{tbl:evidence}}
263\noindent%\hspace{-5ex}
264\begin{tabular} {|c|cccccccccccccc|}\hline
265  $\,\,\,E$ & 3&5&7&11&13&17&19&23&29&31&37&41&43&47\\\hline
266\nf{37A} & 13&11&29&67&53&103&191&47&59&311&-&83&173&283\\
267\nf{43A} & 7&11&29&23&53&103&191&47&59&311&149&83&-&283\\
268\nf{53A} & 13&11&29&23&53&103&191&47&59&311&149&83&173&283\\
269\nf{57A} & -&11&29&23&53&103&-&47&59&311&149&83&173&283\\
270\nf{58A} & 7&11&29&23&53&103&191&47&-&311&149&83&173&283\\
271\nf{61A} & 7&31&29&67&53&103&191&47&59&311&149&83&173&283\\
272\nf{65A} & 19&-&43&23&-&137&191&47&59&311&149&83&173&659\\
273\nf{77A} & 19&11&-&-&53&103&191&47&59&311&149&83&173&283\\
274\nf{79A} & 13&11&43&67&53&103&191&47&59&311&149&83&173&283\\
275\nf{82A} & 13&41&29&23&53&103&191&47&59&311&149&-&173&283\\
276\nf{83A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
277\nf{88A} & 7&11&29&-&131&103&191&47&59&311&149&83&173&283\\
278\nf{89A} & 19&11&29&67&53&103&191&47&59&311&149&83&173&283\\
279\nf{91A} & 31&11&-&23&-&103&191&47&59&311&149&83&173&283\\
280\nf{91B} & -&11&-&23&-&103&191&47&59&311&149&83&173&283\\
281\nf{92B} & 13&61&29&23&79&103&229&-&59&311&149&83&173&283\\
282\nf{99A} & -&11&29&-&53&103&191&47&59&311&149&83&173&283\\
283\nf{101A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
284\nf{102A} & -&11&29&23&53&-&191&47&59&311&149&83&173&283\\
285\nf{106B} & 7&11&29&23&53&137&191&47&59&311&149&83&431&283\\\hline
286\nf{389A} & 7&11&29&23&53&103&191&47&59&311&149&83&173&283\\
287\nf{433A} & 7&11&43&23&53&103&191&47&59&311&149&83&173&283\\\hline
288\end{tabular}
289\end{table}
290
291
292\subsection{Computational Evidence for the Conjecture}
293
294Using a \magma{} program (see~\cite{magma}), the author's computer
295verified Conjecture~\ref{conj:nonvanishing} for every $p<50$ for the
296first $20$ optimal elliptic curve quotients of $R_0(N)$ of rank~$1$
297and the first~$2$ elliptic curve quotients of rank~$2$.
298
299Table~\ref{tbl:evidence} contains, for each $p < 50$, the smallest
300prime~$\ell$ satisfying the conditions of
301Conjecture~\ref{conj:nonvanishing}.  The elliptic curves are labeled
302as in Cremona.  The curves \nf{389A} and \nf{433A} both have rank~$2$,
303and all others have rank~$1$.  A dash (-) in the table indicates that the
304corresponding prime is not rigid, so the conjecture does not apply.
305
306In all cases the first prime $\ell\nmid N_E$ with
307$\ell\con 1\pmod{p}$ with $a_\ell(E)\not \con 2\pmod{p}$
308satisfied $L(E,\chi_{p,\ell},1)\neq 0$, except
309for \nf{61A} with $p=5$, \nf{79A} with $p=7$,
310\nf{82A} with $p=5$, \nf{89A} with $p=11$,
311and \nf{92B} with $p=5$.  In every one of these~$5$ exceptional
312cases, the second prime that we tried
313satisfied the conclusion of
314Conjecture~\ref{conj:nonvanishing}.
315
316\subsection{The Density}
317The following conjecture is not mentioned
318elsewhere in this paper.
319\begin{conjecture}
320Let~$p$ be a rigid prime for an elliptic curve~$E$.
321The set of primes
322$$\left\{\ell \,\,:\,\, \ell \con 1\!\!\!\!\!\pmod{p}\text{ and } 323 L(E,\chi_{p,\ell},1)=0\right\} 324$$
325has Dirichlet density~$0$.
326\end{conjecture}
327
328The following numerical example gives evidence for this conjecture.
329\begin{example}
330Let~$E$ be \nf{37A} and let $p=5$.  Then the only $\ell<1000$ (with
331$\ell\con 1\pmod{5}$) for which $L(E,\chi_{5,\ell},1)=0$ is $\ell=41$.
332% 4 minutes to compute
333\end{example}
334
335
336\section{$p$-Torsion of Twisted Powers}\label{sec:ptorsion}
337Let~$p$ and~$\ell$ be as in Conjecture~\ref{conj:nonvanishing}.
338In order to apply Theorem~\ref{thm:shaexists}, it is necessary
339to know that~$p$ does not divide the orders of certain groups.
340In this section, we use that $a_\ell(E)\not\con 2\pmod{p}$ to
341deduce that certain groups do not have any~$p$ torsion.
342The key idea is that the condition on $a_\ell(E)$ implies
343that~$+1$ is not an eigenvalue of $\Frob_\ell$ on the
344$p$-adic Tate module attached to~$E$.
345
346First, we recall that certain torsion points on the closed
347fiber of a N\'eron model lift to the generic fiber.  Let~$K$ be a
348finite extension of~$\Q_\ell$ with ring of integers~$\O$ and residue
349class field~$k$.
350\begin{lemma}\label{lem:red_mod_n}
351Let $A$ be an abelian variety over~$K$ with N\'eron model $\cA$ over~$\O$.
352Then for every integer $n$ not divisible by~$\ell$, there is an isomorphism
353$$A(K)[n] \xrightarrow{\,\,\isom\,\,} \cA(k)[n].$$
354\end{lemma}
355\begin{proof}
356This is a standard fact, whose proof we recall for the convenience
357of the reader.
358Let $A^{1}(K)$ denote the kernel of the natural reduction
359map $r:A(K)\ra \cA(k)$.  Because $A^{1}(K)$ is a formal group,
360it is pro-$p$, so $[n]:A^{1}(K)\ra{}A^{1}(K)$ is an isomorphism.
361Since $\cA$ is smooth over~$\O$,
362Hensel's lemma (see BLR) implies that the reduction map
363is surjective, so the following sequence is exact:
364       $$0\ra A^1(K) \ra A(K) \ra \cA(k) \ra 0.$$
365The snake lemma applied to the multiplication by~$n$ diagram
366attached to this exact sequence yields the following
367exact sequence:
368$$0\ra0\ra A(K)[n]\ra \cA(k)[n] \ra 0 \ra A(K)/n A(K) \ra \cA(k)/n\cA(k)\ra0,$$
369which proves the proposition.
370\end{proof}
371
372
373Let~$E$ be an elliptic curve over~$\Q$ with  associated newform
374$f = \sum a_n q^n$, and fix a prime~$p$ that is rigid for~$E$.
375Suppose~$K$ is the extension of~$\Q$ corresponding
376to a surjective Dirichlet character
377$\chi: (\Z/\ell\Z)^* \onto \bmu_p$
378of prime conductor; then~$K$ is
379the subfield of $\Q(\bmu_\ell)$ fixed by $\ker(\chi)$,
380so it is of degree~$p$, is totally ramified
381at~$\ell$, and is unramified outside~$\ell$.
382Let~$A=\ker(\tr : \Res_{K/\Q} E_K \ra E)$.
383We next compute the Tamagawa number
384$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$
385and the $p$-torsion of several abelian varieties.
386\begin{proposition}\label{prop:ptorsion}
387Let~$E$, $\chi$, $K$, and~$A$ be as above and suppose
388that $\ell\nmid N_E$ and $a_\ell \not\equiv 2\pmod{p}$.
389Then the following groups have no nontrivial $p$-torsion:
390$$A(K), \quad A(\Q_\ell),\quad R(\Q_\ell),\quad 391(R/E)(\Q_\ell),\quad \text{and}\quad \Phi_{A,\ell}(\F_\ell).$$
392%$$A(K)[p]=A(\Q_\ell)[p] = R(\Q_\ell)[p] = 393%(R/E)(\Q_\ell)[p]=\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$$
394\end{proposition}
395\begin{proof}
396The reason the $p$-torsion vanishes in all these cases
397is that the condition $a_\ell \not\equiv 2\pmod{p}$ implies
398in each case that $\Frob_\ell$ has no $+1$ eigenvalue.
399The details are as follows.
400
401We first show that $R(\Q_\ell)[p]=\{0\}$, where $R=\Res_{K/\Q}E_K$.
402By definition,
403 $$R(\Q_\ell) = E_K(\Q_\ell\tensor_\Q K) = 404 E(K_v)\cross \cdots \cross E(K_v) \quad \text{(p copies)},$$
405where $K_v$ is the completion of~$K$ at the unique prime of~$K$
406lying over~$\ell$.
407The action of $\Frob_\ell\in \Gal(\Q_\ell^{\ur}/\Q_\ell)$
408on $E[p](\Q_\ell^{\ur})=E[p](\Qbar_\ell)$ has characteristic
409polynomial
410$F(x) = x^2-a_\ell x + \ell \in \F_p[x]$.
411Since $a_\ell \not\equiv 2\pmod{p}$ and $\ell\equiv 1\pmod{p}$, it
412follows that $\Frob_\ell$ does not have
413$+1$ as an eigenvalue, so $E(\Q_\ell)[p]=\{0\}$.
414If $z\in E(K_v)[p]$, then the field $L=\Q_\ell(z)$ is an unramified
415subfield of the totally ramified field $K_v$, so $z\in E(\Q_\ell)[p]=\{0\}$.
416Thus $E(K_v)[p]=\{0\}$, which implies that $E(K)[p]=\{0\}$ and
417$R(\Q_\ell)[p]=\{0\}$.
418Since $R_K/E_K \isom E_K \cross \cdots \cross E_K$ ($p-1$ times),
419we 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
423Finally, we turn to the component group $\Phi_{A,\ell}$.
424Let $\cA$ denote the N\'eron
425model 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
429previous paragraph,
430  $$\cA(\F_\ell)[p] = A(\Q_\ell)[p]\subset R(\Q_\ell)[p]=\{0\},$$
431so $\Phi_{A,\ell}(\F_\ell)[p]=\{0\}$, as claimed.
432\end{proof}
433
434
435\subsection{The Tamagawa Number of $A$ at $\ell$}
436In this section, the notation and hypothesis are as in
437Proposition~\ref{prop:ptorsion}.
438That proposition implies that the Tamagawa number
439$c_{A,\ell}=\#\Phi_{A,\ell}(\F_\ell)$ of~$A$ at~$\ell$ is coprime
440to~$n$.  In this section we use Remark~5.4 of \cite{edixhoven:tame} to
441prove that in fact $c_{A,\ell}=1$.
442
443Let $\lambda$ be the prime of~$K$ lying over $\ell$, and
444let $K_{\lambda}$ denote the completion of~$K$ at~$\lambda$,
445so $K_{\lambda}$ is totally and tamely ramified over $\Q_\ell$.
446Since
447$$A_{K} \isom \ker(\Sigma: E_{K}^{\oplus n} 448 \ra E_{K}),$$
449and $E_{K_\lambda}$ has good reduction,
450the geometric closed fiber of the  N\'eron model of $A_{K_{\lambda}}$ is
451$452 A'_{\kbar}\isom \ker(\Sigma : E_{\kbar}^{\oplus n} \ra E_{\kbar}). 453$
454In the notation of \cite{edixhoven:tame},
455$\mu_n$ acts on $A'_{\kbar}$ by the action
456it induces by cyclically permuting the factors of
457$E_{\kbar}^{\oplus n}$.  Thus
458$A_{\kbar}'(\kbar)^{\mu_n}$ is the set of
459$\sum P_\sigma\tensor\sigma \in E(\kbar)^{\oplus n}$
460such that all $P_\sigma$ are equal and $\sum P_\sigma = 0$,
461i.e.,
462$$463 A_{\kbar}'(\kbar)^{\mu_n} \isom E(\kbar)[n]\ncisom (\Z/n\Z)^2. 464$$
465Thus Remark~5.4 in \cite{edixhoven:tame} implies that
466$\Phi_{A,\ell}(\kbar) \ncisom E(\kbar)[n]$.
467By Proposition~\ref{prop:ptorsion},
468$\Phi_{A,\ell}(k)$ has no elements of order dividing $n$,
469so $\Phi_{A,\ell}(k)=0$.
470
471\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}
472Let $E$ be an elliptic curve over~$\Q$.
473In this section, we prove that Conjecture~\ref{conj:nonvanishing}
474implies if $p$ is a rigid prime, then
475$E(\Q)/p E(\Q)$ is canonical isomorphic to the
476elements of order $p$ in the visible Shafarevich-Tate
477group of a rank~$0$ twisted power of~$E$.
478
479\begin{theorem}\label{thm:main}
480Assume that Conjecture~\ref{conj:nonvanishing} is true.
481If~$E$ is an elliptic curve over~$\Q$, then for every
482rigid prime~$p$, there is a degree~$p$ abelian extension $K$
483of~$\Q$ such that
484$$E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p]),$$
485where $R=\Res_{K/\Q}(E_K)$ and~$A\subset R$ has
486dimension~$p-1$ and rank~$0$.
487\end{theorem}
488The proof divides naturally into three steps.  First, we use
489Conjecture~\ref{conj:nonvanishing} to construct~$A$.  The we
490use a theorem of Kato and that formation of N\'eron models
491commutes with unramified base change to prove that $A$ has
492rank~$0$ and that~$p$ does not torsion of Tamagawa numbers
493of~$A$.  Next, we apply the visibility Theorem~\ref{thm:shaexists}
494to obtain an isomorphism $E(\Q)/p E(\Q)\isom \Vis_R(\Sha(A/\Q)[p])$.
495
496\begin{proof}
497Conjecture~\ref{conj:nonvanishing} implies that there exists
498a prime $\ell\nmid N_E$
499with $\ell\con 1\pmod{p}$ 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
501let~$K$ be the abelian extension of~$\Q$ corresponding to
502a surjective character $\chi_{p,\ell} : (\Z/\ell\Z)^* \ra \bmu_p$.
503Recall from Section~\ref{sec:restriction_of_scalars} that
504the restriction of scalars $R = \Res_{K/\Q}(E_K)$ is an
505abelian variety over~$\Q$ of dimension~$p$, and
506we 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}$$
516where $A=\ker(\tr)$ is an abelian variety.
517
518%Since $\rho_{E,p}$ is irreducible, $p\nmid E(\Q)_{\tor}$.
519%We have $A(\Q)\subset R(\Q)=E(K)$.  Since~$K$ is totally ramified
520%at~$\ell$, unramified outside~$\ell$, and $\ell\nmid p N_E$, $E(K)[p]$
521%equals $E(\Q)[p]$, which is $\{0\}$ because $\rho_{E,p}$ is assumed
522%irreducible.  Thus $p\nmid \#A(\Q)$.
523
524Since $L(A,s)=\prod L(E,\chi_{p,\ell}^\sigma,s)$, and $L(E,\chi,1)\neq 5250$, Kato's work on Euler systems\edit{Reference?} implies that $A(\Q)$ is finite.
526Proposition~\ref{prop:ptorsion} implies that
527$p\nmid \#A(\Q)\cdot \#(R/E)(\Q)$.
528Next suppose that~$q$ is a prime of bad reduction for~$A$.  If
529$q\not=\ell$, then $K/\Q$ is unramified at~$q$.
530The formation of N\'eron models commutes with unramified base
531change\edit{Reference?} and $A_K=E^{\times(p-1)}$, so $c_{A,q}$ divides
532$\cbar_{E,q}$, which is not divisible by~$p$ since~$p$ is rigid
533for~$E$.  If $q=\ell$, Proposition~\ref{prop:ptorsion} asserts that
534$p\nmid c_{A,q}$.
535
536The previous paragraph combined with Proposition~\ref{prop:ptorsion}
537shows that the hypothesis of
538Theorem~\ref{thm:shaexists} are satisfied with $A=A$, $B=E$,
539$R=R$, and $L=\Q$.  Thus there is an injective map
540      $$E(\Q)/p E(\Q) \hookrightarrow \Vis_R(\Sha(A/\Q))\subset \Sha(A/\Q).$$
541
542To 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
544by Proposition~\ref{prop:ptorsion} the index of $E(\Q)$ in $R(\Q)$ is
545finite and coprime to~$p$.
546\end{proof}
547
548
549
550\section{Applications}\label{sec:applications}
551We apply the above results to prove that
552Conjecture~\ref{conj:nonvanishing} implies the existence of
553elements of Shafarevich-Tate groups of twisted powers of
554elliptic curves of every prime order.  We also construct an
555abelian variety~$A$ over~$\Q$ such that the Birch
556and Swinnerton-Dyer conjecture predicts that $\Sha(A/\Q)[3]=\Z/3\Z$ and
557that $\#\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}
561Let~$p$ be a prime number.  Then Conjecture~\ref{conj:nonvanishing}
562implies that there exists infinitely many twisted powers~$A$ of some
563elliptic curve such that $\Sha(A/\Q)[p]\neq \{0\}$.
564\end{proposition}
565\begin{proof}
566Most of the proposition can be proved using a single elliptic curve.
567When ordered by conductor, the first elliptic curve~$E$ over $\Q$ with
568positive rank has prime conductor~$37$ and is defined by the
569Weierstrass equation $y^2 + y = x^3 - x$.  Table~1 of
570\cite{cremona:algs} shows that~$E$ is isolated in its isogeny class,
571so \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$
574are rigid for~$E$.  The proposition then follows for all odd primes
575$p\neq 37$ by Theorem~\ref{thm:main}.
576
577We complete the proof as follows.  Exactly the same argument applied
578to the unique elliptic curve of conductor~$43$ proves the proposition
579for 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
581curve~$E$ with $j$-invariant~$j$ such that infinitely many twists~$E'$
582of $E$ have $\Sha(E'/\Q)[2]\neq\{0\}$.
583\end{proof}
584
585\subsection{The Possible Orders of Shafarevich-Tate Groups}
586On page 306--307 of \cite{tate:bsd}, Tate discusses results about the
587structure of the group $\Sha(A/K)$, where~$A$ is an abelian variety over
588a number field~$K$.  He asserts that if~$A$ is a Jacobian then $\#\Sha(A/K)$
589is a perfect square.  Poonen and Stoll subsequently pointed out
590in~\cite{poonen-stoll} that Tate's assertion is not quite correct.  In
591fact, Poonen and Stoll prove that when~$A$ is a Jacobian, $\#\Sha(A/K)$
592is 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
594when~$A$ is not a Jacobian, except to mention results that imply that
595$\#\Sha(A/K)$ is square away from~$2$ and primes that
596don't divide the degree of some polarization of~$A$.
597
598Now suppose~$A$ is an arbitrary abelian variety over a number field~$K$.
599In 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
601square.  Let~$E$ be an elliptic curve over~$\Q$ of rank~$1$.  Then
602the construction of the present paper gives, for suitable primes~$p$,
603an injection
604 $$\Z/p\Z \ncisom E(\Q)/p E(\Q) \hookrightarrow \Sha(A/\Q),$$ where~$A$
605is an abelian variety over~$\Q$ which is a twist of $E^{\times p-1}$.
606Thus $\Sha(A/\Q)[p]$ has a natural'' subgroup of order~$p$; moreover,
607no other natural subgroup of order~$p$ presents itself.  Is
608$\#\Sha(A/\Q)[p]\ncisom\Z/p\Z$?  If the answer is yes for even a
609single~$p>2$, then the question of whether or not $\#\Sha(A/\Q)$ must be
610a square or twice a square is finally resolved.
611We make the following contribution toward settling this problem.
612
613\begin{proposition}
614Let~$E$ be the unique elliptic curve over~$\Q$ of conductor~$43$.
615Let $K=\Q(\mu_7)^+$ be the real subfield of $\Q(\mu_7)$, let
616$R=\Res_{K/\Q}E_K$, and
617let $A = \ker\left(R \ra E\right).$
618Then $3\mid \#\Sha(A/\Q)$ and the Birch and Swinnerton-Dyer conjecture
619implies that $\#\Sha(A/\Q)[3]=3$ and $\#\Sha(R/\Q)[3]=1$.
620\end{proposition}
621\begin{proof}
622We first verify that the BSD conjecture predicts that
623$\#\Sha(E/K)[3]=1$.
624Because $K/\Q$ is abelian,
625$$L(E_K,s) = L(E,s)\cdot L(E,\chi,s)\cdot L(E,\chi^{-1},s),$$
626where $\chi:(\Z/7\Z)^*\ra \mu_3$ is a Dirichlet character of order~$3$.
627For each of the three real places~$v$ of~$K$, we have
628$\Omega_{E,v}=\Omega_{E/\Q}$.  Next observe that
629$$E(\Q)\tensor\Z_3 \ra E(K)\tensor \Z_3$$
630is surjective, because
631$$\frac{E(K)}{E(\Q)} = \frac{R(\Q)}{E(\Q)} 632\hookrightarrow (R/E)(\Q)_{\tor}$$
633and $(R/E)(\Q)[3]=\{0\}$ by Proposition~\ref{prop:ptorsion}.
634If $P\in E(\Q)$ then
635 $$\langle P, P\rangle_\Q = \frac{1}{[K:\Q]} \langle P, P \rangle_K,$$
636so $\Reg(E/K) \sim 3\cdot \Reg(E/\Q)$, where~$\sim$ denotes
637equality up to a number coprime to~$3$''.
638By Proposition~\ref{prop:ptorsion}, $E(K)_{\tor}[3]=\{0\}$ and
639$3\nmid c_v$ for places~$v$ of~$\Q$ (because this is true
640for~$R$).
641Finally,
642\begin{align*}
643\#\Sha(E/K) &=
644   \frac{L'(E_K,1)\cdot \#E(K)_{\tor}^2}
645        {\Reg(E/K)\cdot\Omega_{E/\Q}^3\cdot \prod_{v} c_{v}}\\
646   &\sim \frac{L'(E,1)}{3\cdot \Reg(E/\Q) \cdot \Omega_E}
647    \cdot \Norm_{\Q(\mu_3)/\Q}\left(\frac{L(E,\chi,1)}{\Omega_E}\right)\\
648   &= \frac{1}{3}\cdot 3=1.
649\end{align*}
650We verified the last nontrivial equality with a computer using standard
651modular symbols techniques.
652
653We have an exact sequence
654$$0 \ra \Vis(\Sha(A/\Q)[3]) \ra \Sha(A/\Q)[3] \ra 655 \Sha(R/\Q)[3].$$
656Since $$\Sha(R/\Q)[3]=\Sha(E/K)[3]=\{0\},$$
657$$E(\Q)/3 E(\Q)\hookrightarrow \Vis(\Sha(A/\Q)[3]),$$
658and $\Vis(\Sha(A/\Q)[3])$ is a surjective image of
659$R(\Q)\tensor\Z_3 = E(K)\tensor\Z_3$, which,
660as mentioned above, is a surjective image of $E(\Q)\tensor\Z_3=\Z_3$,
661it follows that
662$$\Vis(\Sha(A/\Q)[3])=\Z/3\Z.$$
663\end{proof}
664
665\subsection{What Goes Wrong when $p=2$?}
666In the previous section, we set $p=3$ and
667constructed an abelian variety $A$ of dimension $p-1$
668that (conjecturally) has nonsquare $\Sha(A/\Q)[p]$.
669We can construct an~$A$ in an analogous way for any odd prime~$p$,
670and the author expects that $\Sha(A/\Q)[p]$ is nonsquare in most cases.
671However, when $p=2$, the dimension of~$A$ is~$1$, so
672in that case $\#\Sha(A/\Q)$ must be a perfect square.
673
674What goes wrong?   The problem lies in Theorem~\ref{thm:shaexists}.
675The argument used to prove Theorem~\ref{thm:shaexists}
676at least provides a map
677$$678 E(\Q)/2 E(\Q)\hookrightarrow\Vis_R(H^1(\Q,A)). 679$$
680When $p=2$, the condition $e<p-1$ is not satisfied, so
681the proof of Theorem~\ref{thm:shaexists} does not show
682that the image of $E(\Q)/ 2 E(\Q)$ is locally trivial
683at the prime~$2$ (or at $\infty$).  We thus only
684construct a subgroup of $H^1(\Q,A)$ of nonsquare order,
685not of~$\Sha(A/\Q)$.  Thus even if two elliptic curves have the same
686$E[2]$, then can still possess very different Selmer groups.
687
688
689\subsection{A Connection with the BSD Conjecture}
690\label{sec:bsd}
691There is heuristic reason why equalities like the one that we proved
692with a computer computation in the previous section should frequently
693be true.
694First, the Birch and Swinnerton-Dyer conjecture predicts that
695$$\#\Sha(E/\Q)=\frac{L'(E,1)}{\Reg(E/\Q) \cdot \Omega_E},$$ and in
696fact one knows that $\Sha(E/\Q)[3]=\{1\}$. Second, because~$\chi$ is
697congruent to the identity character modulo~$3$,
698    $$L(E,\chi,1)/\Omega_E \con L(E,1)=0\pmod{3},$$
699suitably interpreted,
700and there is no reason why this congruence should hold modulo $3^2$.
701Thus, in the situation of Theorem~\ref{thm:main}, the author expects
702that usually $\Sha(A/\Q)[p]\isom E(\Q)/p E(\Q)$.  More generally,
703one suspects that usually
704         $$\Sel^{(p)}(A/\Q) \isom \Sel^{(p)}(E/\Q).$$
705
706\begin{conjecture}\label{conj:strong_nonvanishing}
707Let the notation be as in Conjecture~\ref{conj:nonvanishing}.
708Then there exists a rigid prime~$p$ and a prime~$\ell\nmid N_E$ such
709that  $L(E,\chi_{p,\ell},1)\neq 0$, $a_\ell(E) \not\con 2\pmod{p}$,
710and $\Sha(E/K)[p]=\{0\}$, where~$K$ corresponds
711to $\chi_{p,\ell}$.
712\end{conjecture}
713\begin{proposition}
714Assume Conjecture~\ref{conj:strong_nonvanishing} and the following
715weak consequence of the Birch and Swinnerton-Dyer conjecture: if~$A$
716is a twisted power of an elliptic curve of analytic
717rank~$0$ over~$\Q$ and $p\mid L(A,1)/\Omega_A$, then $p\mid 718\#\Sha(A/\Q)\cdot \prod c_q$.  If~$E$ is an elliptic curve over~$\Q$
719and $L(E,1)=0$, then $E(\Q)$ is infinite.
720\end{proposition}
721
722
723\section{Etale Cohomology Approach to Constructing $\Sha(A)$}
724\label{sec:etale}
725
726Let~$E$ be an elliptic curve over a number field~$L$, let~$K$ be a
727finite Galois extension of~$L$ that is only ramified
728at primes that don't divide the conductor $N_E$ of~$E$.
729Let $R=\Res_{K/L} E_K$ and $A = \ker(R \ra E)$, so we have
730an exact sequence
731$$732 0 \ra A \ra R \ra E \ra 0 733$$
734of abelian varieties.
735\begin{lemma}\label{lem:etale}
736Let $A$, $R$, and $E$ be as above.  Assume that
737$\disc(K)$, $n=[K:L]$, and $N_E$ have no pairwise common
738factors and that $v(p)<p-1$
739for each $p\mid n$ and for each valuation~$v$ on
740$\O_L$ corresponding to a prime of residue characteristic~$p$
741(thus, e.g., $n$ must be odd).
742Then the corresponding complex of N\'eron models
743$$744 0 \ra \cA \ra \cR \ra \cE \ra 0 745$$
746is exact.
747\end{lemma}
748\begin{proof}
749We must show that for every prime~$\p$ of~$L$
750the complex
751\begin{equation}\label{eqn:neron}
752   0 \ra \cA_{\O_\p} \ra \cR_{\O_\p} \ra \cE_{\O_\p}\ra 0
753\end{equation}
754is exact, where $\O_\p$ denotes the completion of $\O_L$ at~$\p$.
755Suppose~$\p$ is a prime of~$L$.
756By the criterion of Neron-Ogg-Shafarevich,
757the only possible primes of bad reduction for~$R$ are those
758that divide $N_E$ and those that ramify in~$K$.
759
760If $\gcd(\chr(\O/\p),n)=1$, then
761\cite[Prop.~7.5.3 (a)]{neronmodels}, with
762$B = E\hra R$, implies
763that if $\p\nmid N_E\cdot \disc(K)$, then
764(\ref{eqn:neron}) is exact, since $\cR$ has
765good reduction at~$\p$.  If $\p\nmid \disc(K)$,
766then the base extension of (\ref{eqn:neron})
767to $\O_{K,\p}$ is exact, because the formation of
768N\'eron models commutes with unramified base extension
769and over~$K$  the sequence of abelian varieties
770is $0\ra E_K^{\oplus (n-1)} \ra E_K^{\oplus n}\xra{\Sigma} E_K\ra 0$.
771If $\p\nmid N_E$, then \cite[Prop.~7.5.3 (a)]{neronmodels}
772implies that $\cA_{\O_\p}\ra \cR_{\O_\p}$ is a closed immersion,
773and the cokernel of $\cR_{\O_\p}\ra \cE_{\O_\p}$ is killed
774by multiplication by~$n$; however, in the proof of
775\cite[Prop.~7.5.3 (a)]{neronmodels} (see
776the top of page 187), one only uses that $\cE_{\O_\p}$ has good
777reduction to deduce surjectivity, so in
778fact $\cR_{\O_\p}\ra \cE_{\O_\p}$ is surjective.
779Since $\gcd(N_E,\disc(K))=1$, this completes the argument
780when $\gcd(\chr(\O/\p),n) = 1$.
781
782Next suppose $\gcd(\chr(\O/\p),n)\neq 1$.
783Our hypothesis on~$n$ implies that $p=\chr(\O/\p)$ satisfies
784$v(p)<p-1$ and $\p\nmid N_E\cdot \disc(K)$.
785By \cite[Th.~7.5.4(iii)]{neronmodels},
786the sequence (\ref{eqn:neron}) is exact.
787\end{proof}
788
789By the lemma, we have an exact sequence of sheaves
790$$791 0 \ra \cA \ra \cR \ra \cE \ra 0 792$$
793on the \'etale site over $X=\Spec(\O_L)$.
794Let $\cA^{\vee}$ denote the N\'eron model of the dual of~$A$.
795\begin{proposition}\label{prop:et}
796The following diagram has exact rows and column:
797$$\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$$
803(Actually the vertical one might not be exact, but it is close
804enough, because $0 \ra \cE \ra \cR \ra \cA$ is exact.)
805\end{proposition}
806By \cite[Appendix]{mazur:tower}, there is an exact sequence
807$$808 0 \ra \Sha(A) \ra H^1(X_{\et},\cA) \ra G \ra 0 809$$
810where~$G$ is a finite group whose order if divisible only by~$2$ and
811primes that divide the Tamagawa numbers of $\cA$.
812
813
814Alternatively, one can consider the following diagram
815$$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}$$
821and work with cohomology on the fppf site.
822
823\subsection{Connection with the Birch and Swinnerton-Dyer Conjecture}
824Suppose $E$ is an elliptic curve over~$\Q$ and that $L(E,1)=0$.  The
825Birch and Swinnerton-Dyer conjecture for~$E$ asserts (among other
826things) that $E(\Q)$ is infinite.  Suppose~$A$ is constructed as in
827Section~\ref{sec:main}.  In this section we describe why if a certain
828consequence of a refinement of the Birch and Swinnerton-Dyer
829conjecture for~$A$ is true, then $\Sel^{(n)}(E/\Q)$ is nonzero.
830
831
832Using modular symbols one sees that $L(A,1)\con 0 \pmod{\ell}$,
833so a refinement of the Birch and Swinnerton-Dyer formula for rank~$0$
834abelian varieties predicts that there should be a nonzero element in
835$\ker (\Sha(A)\ra \Sha(A/E[n]))$.
836Thus by Proposition~\ref{prop:et}, either
837$H^1(X_{\et},\cE)[n]\neq 0$, or there is a nonzero element
838of order dividing~$n$ in
839$$840 \ker(H^1(X_{\et},\cA)\ra H^1(X_{\et},\cR)) \isom E(\Q)/R(\Q), 841$$
842in which case $E(\Q)/R(\Q)$ contains a nonzero element of
843order dividing~$n$,
844so $E(\Q)$ is infinite.
845Thus either $\Sha(E)[n]\neq 0$ or $E(\Q)$ is infinite, so
846$\Sel^{(n)}(E/\Q)$ is nonzero.
847
848
849\bibliography{biblio}
850\end{document}
851
852
853E := EC("37A");
854M := MS(E);
855K := CyclotomicField(5);
856M5 := BaseExtend(M,K);
857P := [p : p in [3..200] |IsPrime(p)];
858time X := [<p,
859 Dimension(TwistedWindingSubmodule(M5,1,DirichletGroup(p,K).1^2))> : p in P];
860// Time: 11.369
861[ <11, 1>, <31, 1>, <41, 0>, <61, 1>, <71, 1>, <101, 1>, <131, 1>,
862  <151, 1>, <181, 1>, <191, 1> ]
863
864\comment{
865$Log: bigsha.tex,v$
866Revision 1.25  2001/09/28 02:48:22  was
867?
868
869Revision 1.24  2001/09/23 04:43:11  was
870more TOC stuff and cleaning.
871
872Revision 1.23  2001/09/22 20:27:00  was
873Worked on
874\section{Shafarevich-Tate Groups of Twisted Powers}\label{sec:main}
875and changed Sha(A) to Sha(A/Q).
876
877Revision 1.22  2001/09/22 19:26:23  was
878I don't know.
879
880Revision 1.21  2001/09/13 01:42:50  was
881Lots of nice little improvements!
882
883Revision 1.20  2001/09/09 04:11:42  was
884Added lots of toc pars.
885
886Revision 1.19  2001/09/08 02:55:34  was
887polishing.
888
889Revision 1.18  2001/09/06 03:39:53  was
890added some cool remarks at end about BSD
891
892Revision 1.17  2001/09/06 03:24:36  was
893typo
894
895Revision 1.16  2001/09/06 03:23:56  was
896minor typo
897
898Revision 1.15  2001/09/06 03:17:42  was
899Added table of evidence for twisting conjecture.
900
901Revision 1.14  2001/09/06 02:42:04  was
902...
903
904}
905