CoCalc Shared Fileswww / papers / nonsquaresha / final2.tex
Author: William A. Stein
1\documentclass{birkart}
2\usepackage{amssymb}
3
4\newcommand{\testbnd}{25000}
5
6%\hoffset=-0.06\textwidth
7%\textwidth=1.12\textwidth
8%\voffset=-0.05\textheight
9%\textheight=1.10\textheight
10%\bibliographystyle{amsalpha}
11
12%%%% Theoremstyles
13\theoremstyle{plain}
14\newtheorem{theorem}{Theorem}[section]
15\newtheorem{proposition}[theorem]{Proposition}
16\newtheorem{corollary}[theorem]{Corollary}
17\newtheorem{claim}[theorem]{Claim}
18\newtheorem{lemma}[theorem]{Lemma}
19\newtheorem{conjecture}[theorem]{Conjecture}
20
21\theoremstyle{remark}
22\newtheorem{remark}[theorem]{Remark}
23\newtheorem{remarks}[theorem]{Remarks}
24
25% ---- SHA ----
26\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
27  \newcommand{\textcyr}[1]{%
28    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
29     \selectfont #1}}
30\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
31
32\newcommand{\cbarq}{\overline{c}_q}
33\newcommand{\q}{\mathfrak{q}}
34\newcommand{\nf}[1]{{\bf #1}}
35\newcommand{\e}{\mathbf{e}}
36\newcommand{\cA}{\mathcal{A}}
37\newcommand{\ds}{\displaystyle}
38\newcommand{\M}{\mathcal{M}}
39\newcommand{\intersect}{\cap}
40\newcommand{\cross}{\times}
41\newcommand{\ra}{\rightarrow}
42\newcommand{\xra}[1]{\xrightarrow{#1}}
43\newcommand{\hra}{\hookrightarrow}
44\newcommand{\la}{\leftarrow}
45\newcommand{\con}{\equiv}
46\newcommand{\tensor}{\otimes}
47\newcommand{\comment}[1]{}
48\newcommand{\Q}{\mathbb{Q}}
49\newcommand{\R}{\mathbb{R}}
50\newcommand{\D}{{\mathbb D}}
51\newcommand{\K}{{\mathbb K}}
52\newcommand{\C}{\mathbb{C}}
53\newcommand{\Qbar}{\overline{\Q}}
54\newcommand{\T}{\mathbb{T}}
55\newcommand{\Z}{\mathbb{Z}}
56\newcommand{\F}{\mathbb{F}}
57\newcommand{\Fl}{\F_{\ell}}
58\newcommand{\Fell}{\Fl}
59\newcommand{\Fbar}{\overline{\F}}
60\newcommand{\A}{\mathcal{A}}
61\newcommand{\isom}{\cong}
62\newcommand{\ncisom}{\approx}
63\newcommand{\cB}{\mathcal{B}}
64\newcommand{\cE}{\mathcal{E}}
65\newcommand{\cR}{\mathcal{R}}
66\newcommand{\h}{\mathfrak{h}}
67\newcommand{\p}{\mathfrak{p}}
68\newcommand{\m}{\mathfrak{m}}
69\newcommand{\cbar}{\overline{c}}
70\renewcommand{\Re}{\mbox{\rm Re}}
71\renewcommand{\l}{\ell}
72\renewcommand{\t}{\tau}
73\renewcommand{\P}{\mathbb{P}}
74\renewcommand{\O}{\mathcal{O}}
75\renewcommand{\a}{\mathfrak{a}}
76\DeclareMathOperator{\Res}{Res}
77\DeclareMathOperator{\new}{new}
78\DeclareMathOperator{\Spec}{Spec}
79\DeclareMathOperator{\Ker}{Ker}
80\DeclareMathOperator{\Coker}{Coker}
81\DeclareMathOperator{\Aut}{Aut}
82\DeclareMathOperator{\Frob}{Frob}
83\DeclareMathOperator{\Fr}{Fr}
84\DeclareMathOperator{\ord}{ord}
85\DeclareMathOperator{\Gal}{Gal}
86\DeclareMathOperator{\ur}{ur}
87\DeclareMathOperator{\Tr}{Tr}
88
89
90
91\begin{document}
92
93\title{Shafarevich--Tate Groups of Nonsquare Order}
94\author[W. A. Stein]{William A. Stein}
95\address{515 Science Center\\ Department of Mathematics\\ Harvard University\\ }
96\email{was@math.harvard.edu}
97
98
99\begin{abstract}
100Let~$A$ denote an abelian variety over~$\Q$.
101We give the first known examples in which $\#\Sha(A/\Q)$
102is neither a square nor twice a
103square.  For example, let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of
104conductor~$37$. We prove that for every odd prime
105$p< \testbnd$ (with $p\neq 37$),  there is a twist~$A$
106of $E\cross\cdots \cross E$ ($p-1$ copies)
107such that $\#\Sha(A/\Q)=p n^2$ for some integer~$n$.
108We prove this by showing under certain hypothesis on~$E$
109and~$p$ that there is  an exact sequence
110$$111 0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(E/K)[p^\infty]\ra 112 \Sha(E/\Q)[p^\infty] \ra 0, 113$$
114where $K$ is a certain abelian extension of $\Q$ of degree $p$.
115\end{abstract}
116
117\maketitle
118
119\section{Introduction}
120The Shafarevich--Tate group of
121an abelian variety~$A$ over a number field~$F$ is
122$$123 \Sha(A/F) := \Ker\left(H^1(F,A) \ra \bigoplus_{\text{all v}} H^1(F_v,A)\right). 124$$
125What are the possibilities for the group structure of $\Sha(A/F)$?
126It is conjectured that $\Sha(A/F)$ is finite and this
127is known in some cases.
128\begin{theorem}[Kato, Kolyvagin, Wiles, et al.]\label{thm:finite}
129Suppose~$A$ is an elliptic curve over~$\Q$.
130(1) If $\ord_{s=1}L(A,s)\leq 1$, then $\Sha(A/\Q)$ is finite.
131(2) If~$\chi$ is a character of the Galois group of an abelian
132extension~$K$ of~$\Q$ and
133$L(A,\chi,1)\neq 0$, then the
134$\chi$-component of $\Sha(A/K)\tensor_\Z\Z[\chi]$
135is finite.  (Here $\Z[\chi]$ is generated by
136the image of~$\chi$.)
137\end{theorem}
138
139The Cassels--Tate pairing $\Sha(A/F)\cross \Sha(A^{\vee}/F)\ra \Q/\Z$
140imposes strong constraints on the structure of $\Sha(A/F)$.
141\begin{theorem}[Tate, Flach]\label{thm:tate}
142Let~$p$ be a prime and suppose that there is a polarization
143$\lambda : A \ra A^{\vee}$ of degree coprime to~$p$.
144If $p=2$ assume also that~$\lambda$ arises from an $F$-rational
145divisor on~$A$ (this hypothesis is automatic if~$A$ is
146an elliptic curve, but can fail in general).
147If $\Sha(A/F)[p^\infty]$ is finite then $\#\Sha(A/F)[p^\infty]$
148is a perfect square.
149\end{theorem}
150\begin{proof}
151  If~$\lambda$ is $F$-rational, the
152  Cassels--Tate pairing on $\Sha(A/F)[p^\infty]$ (induced by~$\lambda)$
153  is nondegenerate and alternating (see \cite{tate:duality}),
154  so $\#\Sha(A/F)[p^\infty]$ is a
155  perfect square.  Even when~$\lambda$ is not $F$-rational, the
156  Cassels--Tate pairing is nondegenerate and antisymmetric (see
157  \cite{flach:pairing}), which when~$p$ is odd implies that
158  $\#\Sha(A/F)[p^\infty]$ is a perfect square.
159\end{proof}
160
161It is tempting to
162conjecture that $\#\Sha(A/F)$ is always a perfect square.  Perhaps
163squareness is a fundamental property of Shafarevich--Tate groups?
164While implementing  algorithms based on \cite{poonen-schaefer} for
165computing with Jacobians of hyperelliptic curves, M.~Stoll was shocked to
166discover an example of an abelian variety of dimension two such
167that $\#\Sha(A/F)[2^{\infty}]=2$.  This was surprising because,
168for example, one finds in the literature \cite[pg.149]{sd:bsd}
169the following statement:
170[The group $\Sha(A/F)$] is conjectured to be finite,
171and Tate [26] has shown that if it is finite its order
172is a perfect square.''
173Stoll and B.~Poonen discovered what hid behind
174this and other similar examples in which $\#\Sha(A/F)$ is twice
175a perfect square.
176
177An algebraic curve~$X$ of genus~$g$ over a local
178field~$k$ is {\em deficient} if~$X$ has no $k$-rational divisor
179of degree $g-1$.
180\begin{theorem}[Poonen-Stoll \cite{poonen-stoll}]\label{thm:ps}
181Suppose~$A$ is the Jacobian of an algebraic curve over~$F$ that
182is deficient at an odd number of places.  If $\#\Sha(A/F)$ is
183finite, then $\#\Sha(A/F)$ is twice a square.
184\end{theorem}
185For example, they prove that the
186Jacobian~$J$ of the nonsingular projective curve defined by
187$$188 y^2 = -3(x^2+1)(x^2-6x+1)(x^2+6x+1) 189$$
190has Shafarevich--Tate group of order~$2$
191(to see that $\#\Sha(J)\mid 2$ they observe that~$J$ is isogenous
192to a product of CM elliptic curves and apply a theorem of Rubin;
193see~\cite[Prop.~27]{poonen-stoll} for details).
194Also, Jordan and Livn\'e
195\cite{jordan-livne:sha} give an infinite
196family of Atkin--Lehner quotients of Shimura curves which are deficient
197at an odd number of places.
198
199Though $\#\Sha(A/F)$ need not be square, one might still be tempted to
200conjecture that $\Sha(A/F)$ must have order
201either a square or twice a square.  Let~$p$ be an odd prime.
202In this paper, we construct (under certain hypotheses that are
203satisfied for $p<25000$) abelian varieties~$A$ such
204that $\#\Sha(A/\Q)=pn^2$ for some integer~$n$.  For
205example (see Section~\ref{sec:ex}):
206\begin{theorem}
207Let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of conductor~$37$.
208For every odd prime $p<\testbnd$ (with $p\neq 37$), there is a twist~$A$ of
209$E^{\cross (p-1)}$ such that $\#\Sha(A/\Q)=pn^2$ for
210some integer~$n$.
211\end{theorem}
212%The abelian variety $A$ of the theorem is the kernel of the
213%trace map $R\ra E$, where $R=\Res_{K/\Q}(E_K)$ is the restriction
214%of scalars of $E$ from an abelian extension~$K$ of~$\Q$.
215%Since $\#\Sha(R)$ is a perfect square,
216%our construction also gives an example of an isogeny $\phi : A \ra R$
217%of abelian varieties such that $\#\Sha(A)$ and $\#\Sha(B)$ do not
218%differ by a perfect square.
219
220This paper was originally motivated by the problem of relating the
221conjecture of Birch and Swinnerton-Dyer about the ranks of elliptic
222curves~$E$ to the Birch and Swinnerton-Dyer formula for the orders
223$\#\Sha(A)$ for abelian varieties~$A$ of analytic rank~$0$.
224
225Let~$p$ be a prime.  Under suitable hypotheses, we construct an abelian
226variety~$A$ and a natural map $E(\Q)/p E(\Q) \hra 227\Sha(A/\Q)$.  Thus if $E(\Q)\isom \Z$ then $\Sha(A/\Q)$ has a natural
228subgroup of order~$p$, and no other natural subgroup of order~$p$
229presents itself.  Moreover, when $E$ is defined by $y^2+y=x^3-x$,
230the Birch and Swinnerton-Dyer formula predicts that $\Sha(A/\Q)[3]$ is
231of order~$3$. Further investigation led to the results of this paper.
232
233
234\vspace{2ex}
235\par\noindent{}{\bf{}Acknowledgement: } It is a pleasure
236to thank Kevin Buzzard, Frank Calegari,
237Sol Friedberg, Benedict Gross, Emmanuel Kowalski, Barry Mazur, Bjorn
239Michael Stoll for Lemma~\ref{lem:red_mod_n} and Cristian Gonz\'{a}lez
241sending me a proof of Proposition~\ref{prop:h2}.
242
243\subsection{Notation}
244If~$G$ is an abelian group and $n$ is an integer, then
245$G[n]$ denotes the subgroup of elements of order~$n$ and
246$G[n^\infty]$ is the subgroup of elements of order any power of~$n$.
247We refer to elliptic curves using the notation of \cite{cremona:algs}.
248
249\section{Construction of Nonsquare Shafarevich--Tate Groups}
250For the rest of this paper we will work with an elliptic curve~$E$
251over~$\Q$.  Aside from the significant use of known cases of the Birch
252and Swinnerton-Dyer conjecture below, much of the construction
253should generalize to the situation when~$E$ is replaced by a
254principally polarized abelian variety over a global field.
255
256For the rest of this section, fix an elliptic curve~$E$ over~$\Q$.
257By \cite{breuil-conrad-diamond-taylor}, $E$ is modular so
258there is a newform $f=\sum_{n=1}^{\infty} a_n q^n$ of level
259equal to the conductor~$N=N_E$ of~$E$ such that $L(E,s)=L(f,s)$.
260For each prime $q\mid N$,
261the Tamagawa number~$c_q$ of~$E$ at~$q$ is the order of the group of
262rational components of the special fiber of the N\'eron model of~$E$
263at~$q$.
264
265
266\subsection{Twisting By Characters of Prime Order}
267Let $p$ be a prime number.
268For any prime $\ell \con 1\pmod{p}$, let
269$$270 \chi_{p,\ell} : (\Z/\ell\Z)^* \to \mu_p \subset \C^* 271$$
272be one of the $p-1$ Galois-conjugate Dirichlet characters of
273order~$p$ and conductor~$\ell$.
274
275\begin{conjecture}\label{conj:nonvanish}
276Suppose~$p$ is a prime such that
277$\rho_{E,p}:\Gal(\Qbar/\Q)\ra \Aut(E[p])$ is surjective.  Then there exists a
278prime $\ell\nmid N$ such that $L(E,\chi_{p,\ell},1)\neq 0$,
279$\ell\con 1\pmod{p}$ and $a_\ell\not\con \ell+1 \pmod{p}$.
280\end{conjecture}
281\begin{remarks}\mbox{}\vspace{-4ex}\\
282\begin{enumerate}
283\item Formulas involving  modular symbols imply that
284$L(E,\chi_{p,\ell},1)\neq 0$ if
285and only if $L(E,\chi_{p,\ell}^\sigma,1)\neq 0$ for any $\Gal(\Qbar/\Q)$-conjugate~$\chi_{p,\ell}^\sigma$
286of~$\chi_{p,\ell}$.
287\item
288J.~Fearnley proved related nonvanishing results when
289$L(E,1)\neq 0$ in \cite{fearnley:phd}.
290\item
291If~$E$ is the elliptic curve $y^2+y=x^3-x$ of conductor~$37$ and rank~$1$,
292then $\ell=41$ is the only $\ell\con 1\pmod{5}$ with $\ell<1000$
293for which $L(E,\chi_{5,\ell},1)=0$.
294
295\end{enumerate}
296\end{remarks}
297
298The following proposition gives evidence for
299Conjecture~\ref{conj:nonvanish} for the lowest-conductor elliptic
300curves of ranks $1$, $2$, and $3$.
301\begin{proposition}\label{prop:conjtest}
302Conjecture~\ref{conj:nonvanish} is true for
303the rank~$1$ elliptic curve \nf{37A} for
304every odd $p<\testbnd$ (with $p\neq 37$).
305The conjecture is true for the rank~$2$ curve
306\nf{389A} for every odd $p<1000$ (with $p\neq 389$).
307The conjecture is true for the rank~$3$ curve
308\nf{5077A} for every odd $p<1000$.
309\end{proposition}
310\begin{proof}
311Consider the modular symbol
312$$313 e_{p,\ell} = \sum_{a\in (\Z/\ell\Z)^*} \chi_{p,\ell}(a) 314 \cdot \left\{0,\,\, \frac{a}{\ell}\right\} 315 \in H_1(X_0(N),\Q(\zeta_p)). 316$$
317Then $L(E,\chi_{p,\ell},1)\neq 0$ if and only if the
318image of $e_{p,\ell}$ under
319$$320 H_1(X_0(N),\Q(\zeta_p)) \ra H_1(E,\Q(\zeta_p)) 321$$
322is nonzero.  In any particular case, we can use modular
323symbols to determine whether or not this image is nonzero.
324
325When~$p$ is large, it is difficult to compute in the
326field $\Q(\zeta_p)$, so instead we compute in the residue class
327field $\Fell=\Z[\zeta_p]/\m\isom Z/\ell\Z$, where~$\m$ is one of
328the maximal ideals of $\Z[\zeta_p]$ that lies over~$\ell$.
329(Note that $\ell$ splits completely in $\Z[\zeta_p]$
330because $\ell\con 1\pmod{p}$.)  After reducing modulo~$\m$,
331we compute the image of
332$$333 \overline{e}_{p,\ell} = \sum_{a\in (\Z/\ell\Z)^*} a^{(\ell-1)/p} 334 \cdot \left\{0, \,\,\frac{a}{\ell}\right\} 335 \in H_1(X_0(N),\Fell) 336$$
337in $H_1(E,\Fell)$.
338If it is nonzero, then the image of $e_{p,\ell}$
339in $H_1(E,\Q(\zeta_p))$ is nonzero.
340
341A big computation (that takes hundreds of hours using
342{\sc Magma} \cite{magma})
343shows that the image of $\overline{e}_{p,\ell}$
344is nonzero in the cases asserted by the proposition.
345So the reader can carry out similar computations,
346we include the following {\sc Magma} V2.10-6 code,
347which illustrates verification of the proposition
348for {\bf 37A} for $p<100$:
349\begin{verbatim}
350procedure VerifyConjecture(E, p)
351   assert Type(E) eq CrvEll;
352   assert Type(p) eq RngIntElt and IsPrime(p) and IsOdd(p);
353   N := Conductor(E);
354   assert N mod p ne 0;
355   M := ModularSymbols(E,+1);  // takes a long time if N large!
356   ell := 3; t := Cputime();
357   printf "p=%o: ", p;
358   while true do
359      while (ell mod p ne 1)  or  (N mod ell eq 0) or
360       TraceOfFrobenius(ChangeRing(E,GF(ell))) mod p eq (ell+1) do
361         ell := NextPrime(ell);
362      end while;
363      k := FiniteField(ell);
364      printf "trying ell=%o...",ell;
365      psi := DirichletGroup(ell,k).1;
366      eps := psi^(Order(psi) div p);  // order p character
367      M_k := BaseExtend(M,k);
368      phi := RationalMapping(M_k);
369      e := TwistedWindingElement(M_k,1,eps);
370      if phi(e) ne 0 then
371         printf " success! (%o seconds)\n", Cputime(t);
372         return;
373      end if;
374      printf "failed. ";
375      ell := NextPrime(ell);
376   end while;
377end procedure;
378
379E := EllipticCurve([0,0,1,-1,0]);  // 37A
380for p in [q : q in [3..100] | IsPrime(q) and q ne 37] do
381   VerifyConjecture(E,p);
382end for;
383\end{verbatim}
384The above input results in the following abbreviated output:
385\begin{verbatim}
386p=3: trying ell=7... success! (0.021 seconds)
387p=5: trying ell=11... success! (0.039 seconds)
388p=7: trying ell=29... success! (0.121 seconds)
389...
390p=89: trying ell=179... success! (0.739 seconds)
391p=97: trying ell=389... success! (1.491 seconds)
392\end{verbatim}
393\end{proof}
394
395
396
397\subsection{A Restriction of Scalars Exact Sequence}
398As above, $E$ is an elliptic curve over~$\Q$.  Let~$p$
399be any prime (note that $p=2$ is allowed).
400Suppose $\ell\con 1\pmod{p}$ is another prime and that
401$\ell\nmid N_E$.
402Let $K\subset \Q(\mu_\ell)$ be the abelian
403extension of~$\Q$ that corresponds to~$\chi_{p,\ell}$
404(thus~$K$ is the unique subfield of $\Q(\mu_\ell)$ of
405degree~$p$).
406
407Let $R = \Res_{K/\Q}(E_K)$ be the restriction of scalars down
408to~$\Q$ of~$E$ viewed as an elliptic curve over~$K$.  Thus~$R$
409is an abelian variety over~$\Q$ of dimension $p=[K:\Q]$.
410It is characterized by the fact that it
411represents the following functor on $\Q$-schemes~$S$:
412 $$413 S \mapsto E_K(S_K). 414$$
415As a Galois module,
416$$417 R(\Qbar) = E(\Qbar)\tensor_\Z \Z[\Gal(K/\Q)], 418$$
419where
420$\tau\in \Gal(\Qbar/\Q)$ acts on
421$\sum P_{\sigma} \tensor \sigma$ by
422$$\tau\left(\sum P_\sigma\tensor \sigma\right) = 423 \sum \tau(P_\sigma)\tensor \tau_{|K}\cdot\sigma, 424$$
425where $\tau_{|K}$ is the image of~$\tau$ in $\Gal(K/\Q)$.
426
427\begin{proposition}\label{prop:exactabvar}
428The identity map induces a closed immerion $\iota: E\hookrightarrow 429R$, and the trace $\Tr:K\ra \Q$ induces a surjection $\Tr:R\ra E$
430whose kernel is geometrically connected.  Thus we have an exact
431sequence of abelian varieties
432\begin{equation}\label{eqn:exactabvar}
433  0 \ra A \ra R \xra{\Tr} E \ra 0.
434\end{equation}
435\end{proposition}
436\begin{proof}
437The existence of~$\iota$ and $\Tr$ follows from Yoneda's lemma.
438The map~$\iota$ is induced by the functorial inclusion
439$E(S)\hookrightarrow E_K(S_K)=R(S)$, so~$\iota$ is injective.
440The $\Tr$ map is induced by the  functorial trace map on points
441$R(S)=E_K(S_K)\xrightarrow{\Tr} E(S)$.
442
443To verify that $\Ker(\Tr)$ is geometrically connected, we base
444extend the exact sequence (\ref{eqn:exactabvar}) to~$\Qbar$.  First,
445note that there is an isomorphism
446$$447 R_{\Qbar} \isom E_{\Qbar}\cross \cdots \cross E_{\Qbar}. 448$$
449After base extension, we identify
450the trace map with the summation map
451$$452 +: E_{\Qbar} \cross \cdots \cross E_{\Qbar} 453 \longrightarrow E_{\Qbar}. 454$$
455Let $n=[K:\Q]$.  The map defined by
456$$\left(a_1,\ldots, a_{n-1}\right) \mapsto 457 \left(a_1,a_2,\ldots ,a_{n-1},-\sum_{i=1}^{n-1} a_i\right),$$
458 is an isomorphism from
459$E_{\Qbar}^{\times (n-1)}$ to $\Ker(+)=\Ker(\Tr_{\Qbar})$.
460Thus $\Ker(\Tr_{\Qbar})$ is isomorphic to
461a product of copies of $E_{\Qbar}$,  and hence is connected.
462\end{proof}
463
464
465\begin{corollary}\label{cor:intersection}
466$467 \ds\iota(E)\intersect \Ker(\Tr) = \iota(E)[p]. 468$
469\end{corollary}
470\begin{proof}
471The composition
472$\Q\hookrightarrow K\xrightarrow{\Tr} \Q$
473is multiplication by~$p$, so
474the composition
475$E \xra{\,\,\iota\,\,} R \xra{\Tr} E$
476is also multiplication by~$p$.
477Since
478$\iota(E)\intersect \Ker(\Tr)$ is the kernel of~$\Tr\circ\iota = [p]$,
479it equals $E[p]$.
480\end{proof}
481
482\begin{lemma}\label{lem:powers}
483The abelian varieties
484$A_K$, $R_K$, and $(R/\iota(E))_K$ are all isomorphic to
485a product of copies of $E_K$.
486\end{lemma}
487\comment{
488\begin{proof}
489Base extending (\ref{eqn:eraprime}) to~$K$
490we have an exact sequence
491$$492 0 \ra E_K \xra{\Delta} E_K^{\times p} \ra A'_K \ra 0. 493$$
494Embed $E_K^{\times (p-1)}$ in $E_K^{\times p}$ by
495$(a_1,\ldots, a_{p-1}) \mapsto (a_1,\ldots, a_{p-1},0)$.
496Then $E_K^{\times (p-1)}$ maps injectively to $A'_K$, since
497$E_K^{\times (p-1)}$ has $0$ intersection with the diagonal,
498so $A'_K \ncisom E_K^{\times (p-1)}$.
499\end{proof}
500}
501
502
503
504\begin{proposition}\label{prop:exactneron}
505The exact sequence $0\ra A \ra R \ra E\ra 0$ of
506Proposition~\ref{prop:exactabvar} extends to
507an exact sequence
508$0 \ra \cA \ra \cR \ra \cE \ra 0$
509of N\'eron models over~$\Z$.
510\end{proposition}
511\begin{proof}
512We use results of \cite[Ch.~7]{neronmodels} and the
513fact that formation of N\'eron models commutes with unramified base
514change (see \cite[\S1.2, Prop.~2]{neronmodels})
515to prove that for every prime~$q$, the complex
516\begin{equation}\label{eqn:neron}
517   0 \ra \cA_{\Z_q} \ra \cR_{\Z_q} \ra \cE_{\Z_q}\ra 0
518\end{equation}
519is exact.
520
521First suppose that $q\neq \ell$, and let~$\q$ be a prime
522of~$K$ lying over~$q$.  We use the fact that formation of N\'eron models
523commutes with unramified base extension and check exactness
524of (\ref{eqn:neron}) after base extension to the
525unramified extension $\O_{K,\q}$ of $\Z_q$.
526By Lemma~\ref{lem:powers}, the generic fiber of the base
527extension of (\ref{eqn:neron}) to $\O_{K,\q}$ is
528$$529 0\ra E_{K,\q}^{\oplus (n-1)} \ra 530 E_{K,\q}^{\oplus n}\xra{\Sigma} E_{K,\q}\ra 0. 531$$
532Thus the corresponding complex of N\'eron models over $\O_{K,\q}$ is
533$$534 0\ra \cE_{\O_{K,\q}}^{\oplus (n-1)} \ra \cE_{\O_{K,\q}}^{\oplus n} 535 \xra{\Sigma} \cE_{\O_{K,\q}}\ra 0, 536$$
537which is exact, since it is exact on $S$-points for {\em any}
538ring~$S$.
539
540Suppose that $q=\ell$.  Since $p\neq \ell$,
541\cite[Prop.~7.5.3 (a)]{neronmodels} asserts
542that the sequence
543$0 \ra \cA_{\Z_q} \ra \cR_{\Z_q} \ra \cE_{\Z_q}$
544is exact.
545Since $p\neq q$,
546the map $[p]: \cE_{\Z_q} \to \cE_{\Z_q}$ is
547an \'etale morphism of smooth schemes.
548Since~$E$ has good reduction at~$q$, we also
549know that the fibers of $\cE_{\Z_q}$ are geometrically connected,
550so $[p]$ is surjective (for more details, see the proof
551of~\cite[Lem.~3.2]{agashe-stein:visibility}).
552It follows that $\cR_{\Z_q} \ra \cE_{\Z_q}$ is surjective.
553
554\end{proof}
555
556
557
558\subsection{The Cokernel of Trace}
559Let~$\ell$ be a prime as in Conjecture~\ref{conj:nonvanish}.
560This section is devoted to computing the cokernel
561of the trace map $R(\Q) \ra E(\Q)$.  Note that $R(\Q)=E(K)$, so
562this cokernel is also $E(\Q)/\Tr_{K/\Q}(E(K))$.
563
564\begin{lemma}\label{lem:kellsize}
565Let $K_\ell$ denote the completion of $K$ at the totally ramified
566prime of~$K$ lying over~$\ell$.  Then $E(K)[p] = E(K_{\ell})[p]=0$.
567\end{lemma}
568\begin{proof}
569The characteristic polynomial of
570$\Frob_{\ell}\in\Gal(\Q_\ell^{\ur}/\Q_\ell)$
571on $E[p] = E(\Q_\ell^{\ur})[p]$ is $x^2 - a_\ell x + \ell \in \F_p[x]$.
572By hypothesis $a_\ell \not\con \ell+1\pmod{p}$, so
573$+1$ is not a root of $x^2 - a_\ell x + \ell$ hence
574$$575 E(\Q_\ell)[p] = E(\Q_\ell^{\ur})[p]^{\Frob_{\ell}-1} = 0. 576$$
577Since~$K$ is totally ramified at~$\ell$ and~$E$ has
578good reduction at~$\ell$, $E(K_\ell)[p]=0$ as well,
579so $E(K)[p]=0$, as required.
580\end{proof}
581
582\begin{proposition}\label{prop:coker}
583$\ds 584 \Coker(R(\Q)\ra E(\Q)) \isom E(\Q)/p E(\Q). 585$
586\end{proposition}
587\begin{proof}
588By Corollary~\ref{cor:intersection} the
589the image of $\iota(E(\Q))\subset R(\Q)$ in $E(\Q)$
590is $p E(\Q)$, so the
591cokernel of $R(\Q)\ra E(\Q)$
592is a quotient of $E(\Q)/p E(\Q)$.
593Thus it suffices to prove that $R(\Q)/\iota(E(\Q))$
594is {\em finite} of order coprime to~$p$.
595
596We have an exact sequence
597$0 \ra E \ra R \ra A' \ra 0$,
598with $A'$ an abelian variety that is isogenous to~$A$
599(in fact, $A'$ is the abelian variety dual of~$A$ since~$R$
600is self dual, but we will not use this fact.)
601The $L$-series of $A'$ is
602$\prod_{i=1}^{p-1} L(E,\chi_{p,\ell}^i,s)$,
603so by hypothesis $L(A',1)\neq 0$ and
604it follows from Kato's theorem (see \cite[\S8.1]{rubin:kato})
605that $A'(\Q)$ is finite.
606Thus $R(\Q)/\iota(E(\Q))$ is finite since $R(\Q)/\iota(E(\Q))\subset A'(\Q)$.
607By Lemma~\ref{lem:powers}, $A'_K \ncisom E_K^{\times (p-1)}$
608and by Lemma~\ref{lem:kellsize} $E(K)[p]=0$, so
609$A'(\Q)[p]=0$, which proves the proposition.
610\end{proof}
611
612\subsection{\'Etale Cohomology and Shafarevich--Tate Groups}\label{sec:etale}
613Fix an elliptic curve~$E$ over~$\Q$ and a prime $p\nmid \prod c_{E,q}$.
614
615In this section, we use results mostly due to Mazur to relate the
616Shafarevich--Tate groups of~$A$,~$R$, and~$E$ to certain \'etale
617cohomology groups.
618We maintain the notation and assumptions
619of the previous sections, except
620that we abuse notation slightly and let $\cA$, $\cR$, and $\cE$ also
621denote the \'etale sheaves on $\Spec(\Z)$ defined by
622the N\'eron models $\cA$, $\cR$, and $\cE$.
623Let $\cB$ be either
624$\cA$, $\cR$, or $\cE$ and let $B=\cB_\Q$
625be the corresponding abelian variety.
626Let $H^q(\Z,\cB)$ be the $q$th \'etale cohomology group of~$\cB$.
627
628\begin{lemma}\label{lem:red_mod_n}
629There is an isomorphism
630$B(\Q_\ell)[p] \isom \cB(\Fell)[p].$
631\end{lemma}
632\begin{proof}
633This follows from
634\cite[Lem.~2, pg.~495]{serre-tate},
635but we sketch a proof for the convenience of the reader.
636Let $B^{1}(\Q_\ell)$ denote the kernel of the natural reduction
637map $r:B(\Q_\ell)\ra \cB(\Fell)$.  Using formal groups and
638that $p\neq \ell$,
639one sees that  $[p]:B^{1}(\Q_\ell)\ra{}B^{1}(\Q_\ell)$ is an isomorphism.
640Since $\cB$ is smooth over~$\Q_\ell$,
641Hensel's lemma (see \cite[\S2.3~Prop.~5]{neronmodels})
642implies that the reduction map
643is surjective, so we obtain an exact sequence
644$$645 0\ra B^1(\Q_\ell) \ra B(\Q_\ell) \ra \cB(\Fell) \ra 0. 646$$
647The snake lemma applied to the multiplication-by-$p$ diagram
648attached to this exact sequence yields the
649exact sequence
650$$0\ra B(\Q_\ell)[p]\ra \cB(\Fell)[p] \ra 0 \ra B(\Q_\ell)/p B(\Q_\ell) 651 \ra \cB(\Fell)/p\cB(\Fell)\ra0,$$
652which proves the lemma.
653\end{proof}
654
655The {\em Tamagawa number} of~$B$ at a prime~$q$
656is $c_{B,q}=\#\Phi_{B,q}(\F_q)$, where $\Phi_{B,q}$
657is the component group of the closed fiber
658of the N\'eron model of~$B$ at~$q$.
659%The {\em geometric Tamagawa number} is  $\cbar_{B,q} = \#\Phi_{B,q}(\Fbar_q)$.
660
661\begin{lemma}\label{lem:boundcq}
662$p\nmid c_{B,q}$.
663\end{lemma}
664\begin{proof}
665First suppose $q=\ell$.
666The cokernel of
667$\cB(\F_\ell) \ra \Phi_{B,\ell}(\F_\ell)$
668is contained in $H^1(\F_\ell,\cB^0)$, which
669is~$0$ by Lang's theorem (see \cite{lang:finitefields} or
670\cite[\S{}VI.4]{serre:alggroups}),
671so if $\Phi_{B,\ell}(\F_\ell)[p]\neq 0$ then $\cB(\F_\ell)[p]\neq 0$.  But by
672Lemmas~\ref{lem:powers}, \ref{lem:kellsize}, and~\ref{lem:red_mod_n},
673$$674 \cB(\F_\ell)[p] \isom \cB(\Q_\ell)[p] \subset \cB(K_\ell)[p] \isom 675 E(K_\ell)[p]\cross \cdots \cross E(K_\ell)[p] = 0. 676$$
677
678Next suppose that $q\neq \ell$.   Since formation of N\'eron models
679commutes with unramified base extension, we have
680$$681 \Phi_{B,q}(\Fbar_q)[p] \isom 682 \Phi_{E,q}(\Fbar_q)[p] \cross \cdots \cross \Phi_{E,q}(\Fbar_q)[p] = 0, 683$$
684by our hypotheses on~$p$.
685\end{proof}
686
687Following the appendix to \cite{mazur:tower}, let
688$$689 \Sigma(B/\Q) = \ker\left(H^1(\Q,B)\ra \bigoplus_{\text{all finite q}} 690 H^1(\Q_q, B)\right), 691$$
692where the sum is over all finite primes~$q$ of~$\Q$.
693If~$p$ is an odd prime, then
694$\Sigma(B/\Q)[p^\infty] = \Sha(B/\Q)[p^\infty]$;
695also one can see easily using Tate cohomology for the cyclic
696group $\Gal(\C/\R)$ that
697$$698\Sigma(B/\Q)[2]/\Sha(B/\Q)[2]\subset 699H^1(\R,B(\C)) \isom B(\R)/B(\R)^0, 700$$
701where $B(\R)/B(\R)^0$ has order $2^e$ for some $e\leq \dim B$.
702\begin{proposition}[Mazur]\label{prop:shah1}
703Suppose that $a_\ell\not\con \ell+1\pmod{p}$.
704If $p$ is odd, then
705$$\ds 706 H^1(\Z,\cB)[p^{\infty}] \isom \Sha(B/\Q)[p^\infty]. 707$$
708Also,
709$\#H^1(\Z,\cB)[2^{\infty}] / \Sha(B/\Q)[2^\infty]$
710divides $\#(B(\R)/B(\R)^0)$.
711\end{proposition}
712\begin{proof}
713It follows from the appendix to \cite{mazur:tower} that there is an
714exact sequence
715\begin{equation}
716  0 \ra \Sigma(B)[p^\infty] \ra H^1(\Z,\cB)[p^\infty] \ra
717        \bigoplus_{\text{all finite $q$}} H^1\left(\F_q, \Phi_{B,q}(\Fbar_q)\right)[p^\infty],
718\end{equation}
719where $\Phi_{B,q}$ is the component group of the fiber of $\cB$
720at~$q$.
721By \cite[VIII.4.8]{serre:localfields},
722$$723 \#H^1(\F_q,\Phi_{B,q}(\Fbar_q)) = \#\Phi_{B,q}(\F_q) = c_{B,q}, 724$$
725so the proposition follows from Lemma~\ref{lem:boundcq}.
726\end{proof}
727
728\begin{proposition}\label{prop:h2}
729$H^2(\Z,\cA)[p] = 0$.
730\end{proposition}
731\begin{proof}
732We apply the lemmas in \cite[\S{}III.6]{schneider:iwasawa}.
733Note that~$A$ has good reduction at~$p$ by \cite[Prop.~1]{milne:bsdres},
734and $H^1(\Z,\cA)[p^\infty]$ is finite by Kato's theorem
735(see \cite[\S8.1]{rubin:kato}) and Proposition~\ref{prop:shah1}.
736In the proof of Proposition~\ref{prop:coker}, we showed that
737$A'(\Q)$ is finite of order coprime to~$p$, where $A'$
738is the abelian variety dual of~$A$.  We now use\footnote{Note that
739the proof of Lemma~7 of \cite[\S{}III.6]{schneider:iwasawa}
740relies on a theorem of
741Artin and Mazur whose proof they never published;
743McCallum \cite[\S5]{mccallum:duality} and
744Milne \cite[\S{}III.3.4]{milne:duality}, and Mazur assures the author
745that he and Milne both know the proof of Artin-Mazur duality well.}
746Lemma~7 of \cite[\S{}III.6]{schneider:iwasawa}, which because
747$A'(\Q)[p]=0$
748implies that $H^2(\Z,\cA[p^\infty]) = 0$
749(Schneider uses $H^q_{\text{fpqf}}$, but this is not a problem
750since \'etale and \text{fpqf} cohomology agree on the smooth
751scheme $\cA$.)
752It is easy to see (see, e.g., the proof of  Lemma~6
753of \cite[\S{}III.6]{schneider:iwasawa}) that
754the natural map $H^q(\Z,\cA[p^\infty]) \ra H^q(\Z,\cA)[p^\infty]$
755is surjective for any $q>0$, in particular, for $q=2$,
756so $H^2(\Z,\cA)[p^\infty]=0$ which proves the proposition.
757\comment{
758As discussed in [], there is an exact sequence
759$$0\ra \cB^0 \ra \cB \ra \bigoplus_{\text{primes }q} \Phi_{B,q} \ra 0,$$
760which leads to the exact sequence
761$$H^2(\Z,\cB^0)[p^\infty] \ra H^2(\Z,\cB)[p^\infty] 762 \ra \bigoplus_{\text{primes q}} H^2\left(\F_q, \Phi_{B,q}(\Fbar_q)\right)[p^\infty].$$
763By \cite[\S{}VIII.4]{serre:localfields} and Lemma~\ref{lem:boundcq},
764$$765 \#H^2(\F_q,\Phi_{B,q})[p] = \#H^0(\F_q,\Phi_{B,q})[p] = 1, 766$$
767so it suffices to show that $H^2(\Z,\cB^0)[p]=0$.
768}
769\end{proof}
770
771
772
773\subsection{The Main Theorem}
774Fix an elliptic curve~$E$ over~$\Q$ and a prime
775$p\nmid \prod c_{E,q}$
776such that $\rho_{E,p}:G_\Q\to \Aut(E[p])$ is
777surjective.
778If $p=2$ assume also that $E(\R)$ is connected.
779Assume that~$\ell$ is one of the primes whose existence
780is predicted by Conjecture~\ref{conj:nonvanish}.
781Let~$A$ and~$R$ be the corresponding abelian varieties, which fit
782into an exact sequence $0\to A\to R \to E \to 0$, and recall
783that $L(A,1)\neq 0$ so $A(\Q)$ and $\Sha(A/\Q)$ are both finite
784(by \cite[\S8.1]{rubin:kato} and \cite[Cor.~14.3]{kato:secret}).
785
786\begin{theorem}\label{thm:nonsquare}
787There is an exact sequence
788$$789 0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(E/K)[p^\infty]\ra 790 \Sha(E/\Q)[p^\infty] \ra 0. 791$$
792In particular, if~$E$ has odd rank and $\Sha(E/\Q)[p^\infty]$ is
793finite, then $\#\Sha(A/\Q)[p^\infty]$
794is not a perfect square.
795\end{theorem}
796\begin{proof}
797By Proposition~\ref{prop:exactneron} we have an
798exact sequence of \'etale sheaves
799$$800 0 \ra \cA \ra \cR \ra \cE \ra 0, 801$$
802which gives rise to an exact
803sequence of \'etale cohomology groups
804$$805 H^0(\Z,\cR) \ra H^0(\Z,\cE) 806 \ra H^1(\Z,\cA) \ra H^1(\Z,\cR) \ra H^1(\Z,\cE) \ra H^2(\Z,\cA). 807$$
808We have
809$$810 H^0(\Z, \cR) = \cR(\Z) = R(\Q)$$
811and likewise for $\cE$, so by Propositions~\ref{prop:coker},~\ref{prop:shah1},
812and \ref{prop:h2} we obtain an exact sequence
813$$814 0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(R/\Q)[p^\infty] \ra 815\Sha(E/\Q)[p^\infty] \ra 0. 816$$
817By Shapiro's lemma, there is an isomorphism $\Sha(R/\Q)\isom \Sha(E/K)$
818(see \cite[\S1.3]{agashe-stein:visibility}), which yields the
819claimed exact sequence.
820
821Kato's theorem (\cite[\S8.1]{rubin:kato} and \cite[Cor.~14.3]{kato:secret})
822implies that $\Sha(E/K)[p^\infty]$ is
823finite (for the trivial character use our hypothesis
824 that $\Sha(E/\Q)[p^\infty]$
825is finite, and for the nontrivial characters use our hypothesis
826that $L(E,\chi_{p,\ell},1)\neq 0$).
827Theorem~\ref{thm:tate} then implies that $\#\Sha(E/K)[p^\infty]$ is a
828perfect square.  If $E(\Q)$ has odd rank then $\#(E(\Q)/p E(\Q))$
829is an odd power of~$p$
830(since $E[p]$ is irreducible), so
831$\#\Sha(A/\Q)[p^\infty]$ cannot be a perfect square.
832\end{proof}
833
834\begin{remark}
835In the language of visibility of Shafarevich-Tate
836groups (see~\cite{cremona-mazur}),
837Theorem~\ref{thm:nonsquare}
838asserts that the
839visible subgroup of $\Sha(A)$ with respect to the
840embedding $A\hra R$ is canonically isomorphic to
841the Mordell-Weil quotient $E(\Q)/p E(\Q)$.
842\end{remark}
843
844
845\begin{proposition}\label{prop:away_from_p}
846If $q\neq p$ is a prime, then
847\begin{equation}\label{eqn:away_from_p}
848 \Sha(E/K)[q^\infty]\isom \Sha(E/\Q)[q^\infty] \oplus \Sha(A/\Q)[q^\infty].
849\end{equation}
850In particular, if $\Sha(E/\Q)[q^\infty]$ is finite,
851then $\Sha(A/\Q)[q^\infty]$ has order a perfect square.
852\end{proposition}
853\begin{proof}
854The intersection of~$E$ and~$A$ in~$R$ is $E[p]$, so
855the summation map $E\times A \to R$ is an isogeny with kernel $E[p]$.
856Considering the long exact sequence associated to
857$0\to E[p]\to E\times A\to R\to 0$, we see that
858\begin{equation}\label{eqn:qinfpart}
859  H^1(\Q,E\times A)[q^\infty] \isom H^1(\Q,R)[q^\infty],
860\end{equation}
861and likewise for any completion $\Q_v$ of~$\Q$.
862We then obtain (\ref{eqn:away_from_p}) by
863combining (\ref{eqn:qinfpart})
864with the fact that cohomology commutes with products and
865that $H^1(\Q,R)\isom H^1(K,E)$.
866
867If $\Sha(E/\Q)[q^\infty]$ is finite, then since $\Sha(A/\Q)[q^\infty]$
868is finite (since $L(A,1)\neq 0$, by construction), it follows from
869(\ref{eqn:away_from_p}) that $\Sha(E/K)[q^\infty]$ is finite.  We have
870by Theorem~\ref{thm:tate} that both $\Sha(E/K)[q^\infty]$ and
871$\Sha(E/\Q)[q^\infty]$ have order a perfect square, so
872(\ref{eqn:away_from_p}) implies that $\Sha(A/\Q)[q^\infty]$ has order
873a perfect square.
874\end{proof}
875
876
877
878
879\section{An Example}\label{sec:ex}
880Combining Proposition~\ref{prop:conjtest}, Theorem~\ref{thm:nonsquare},
881and Proposition~\ref{prop:away_from_p} yields the following theorem.
882\begin{theorem}\label{thm:37}
883Let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of conductor~$37$.
884For every odd prime $p<\testbnd$ (with $p\neq 37$), there is a twist~$A$ of
885$E^{\cross (p-1)}$ such that $\#\Sha(A/\Q)=pn^2$
886for some integer~$n$.
887\end{theorem}
888\begin{remark}
889Using the elliptic curve of conductor $43$ in place of~$E$
890one can construct an abelian variety $A$ with $\Sha(A/\Q)=37n^2$
891for some integer~$n$.
892\end{remark}
893
894Though unnecessary for Theorem~\ref{thm:37}, we
895prove below that $\Sha(E/\Q)=0$, which removes our
896dependence on Proposition~\ref{prop:h2}.
897We show that $\Sha(E/\Q)[p^\infty]=0$ for all odd~$p$
898using \cite[Thm.~A]{kolyvagin:euler_systems}, and
899we use a $2$-descent (with \cite{mwrank}) to see that $\Sha(E/\Q)[2]=0$.
900\begin{theorem}[Kolyvagin]\label{thm:kolybound}
901Let~$E$ be an elliptic curve
902and let $L=\Q(\sqrt{-D})$ be an imaginary quadratic
903field of odd discriminant $-D$, where all primes dividing the conductor
904of~$E$ split, and assume
905that $D\neq 3,4$.  If the Heegner point $y_L\in E(L)$ has infinite order (equivalently,
906by \cite{gross-zagier}, $L'(E/L,1)\neq 0$),
907then $\#\Sha(E/L)\mid t\cdot [E(L): \Z y_L]^2$,
908where the only primes that
909divide~$t$ are~$2$ or primes where $\rho_{E,p}$ is not surjective.
910\end{theorem}
911
912By \cite{cremona:algs},~$E$ is isolated in its isogeny class,
913so $\rho:\Gal(\Qbar/\Q)\ra \Aut(E[p])$ is surjective for all primes~$p$
914(see \cite[\S1.4]{ribet-stein:serre})
915hence~$t$ is a power of~$2$.
916Let $L = \Q(\sqrt{-7})$.
917To compute $[E(L):\Z{}y_L]$ up to a power of~$2$
918we use the Gross-Zagier formula and the fact that $[E(L) : E(\Q)+ E^D(\Q)]$
919is a power of~$2$. By \cite[Thm.~6.3]{gross-zagier},
920$$921 h(y_L) = \frac{u^2|D|^{\frac{1}{2}}}{\|\omega_f\|} L'(E,1)L(E^D,1), 922$$
923where $D=-7$, $u=1$, and $\|\omega_f\|$
924is the Peterson norm of the newform~$f$ corresponding to~$E$.
925Generators for the period lattice of~$E$ are
926$\omega_1 \sim 2.993459$ and
927$\omega_2 \sim 2.451389i$, so $\|\omega_f\|\sim 7.338133$.
928The quadratic twist $E^D$ is the curve \nf{1813B1} in \cite{cremona:onlinetables},
929and $E^D(\Q)=0$.  From \cite{cremona:onlinetables} we find that
930$L'(E,1) \sim 0.306000$
931and $L(E^D,1) \sim 1.853076$,
932so $h(y_L)\sim 0.204446$.
933The height of a generator of $E(\Q)$ is
934$\sim 0.051111\sim h(y_L)/4$, so
935$[E(L):\Z{}y_L]$ is a power of~$2$.
936(As a double check, and to avoid dependence on
937the Gross-Zagier formula, we wrote a program using
938\cite{magma} to compute Heegner points and
939found that $y_L=(0,0)$, which is a generator for $E(\Q)$.)
940Thus $\#\Sha(E/L)$ is a power of~$2$.
941
942To connect $\Sha(E/L)$ with $\Sha(E/\Q)$,
943use the inflation-restriction exact sequence
944$$945 0 \ra H^1(L/\Q, E(L)) \ra H^1(\Q,E(\Qbar)) \ra H^1(L,E(\Qbar)). 946$$
947Let~$p$ be an odd prime.
948Since $H^1(L/\Q,E(L))$ is a $2$-group, the above sequence leads
949to an injective map
950$$951 H^1(\Q,E(\Qbar))[p] \hra H^1(L, E(\Qbar))[p], 952$$
953which induces an inclusion
954$$955 \Sha(E/\Q)[p] \hra \Sha(E/L)[p] = 0. 956$$
957
958
959\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
960\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
961% \MRhref is called by the amsart/book/proc definition of \MR.
962\providecommand{\MRhref}[2]{%
963  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
964}
965\providecommand{\href}[2]{#2}
966\begin{thebibliography}{BCDT00}
967
968\bibitem[AS02]{agashe-stein:visibility}
969A.~Agashe and W.\thinspace{}A. Stein, \emph{Visibility of {S}hafarevich-{T}ate
970  {G}roups of {A}belian {V}arieties}, J. of Number Theory,
971\textbf{97} (2002), no.~1, 171--184.
972
974C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the modularity of
975  elliptic curves over \protect{$\Q$}: {W}ild $3$-adic exercises},
976  J.~Amer.~Math.~Soc. \textbf{15} (2001), no.~4, 843--939.
977%  \\\protect{\sf
978%  http://www.math.harvard.edu/HTML/Individuals/Richard\_Taylor.html}.
979
980\bibitem[BCP97]{magma}
981W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
982  {T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
983  235--265, Computational algebra and number theory (London, 1993). \MR{1 484
984  478}
985
986\bibitem[BLR90]{neronmodels}
987S.~Bosch, W.~L{\"u}tkebohmert, and M.~Raynaud, \emph{N\'eron models},
988  Springer-Verlag, Berlin, 1990. \MR{91i:14034}
989
990\bibitem[C97]{cremona:algs}
991J.\thinspace{}E. Cremona,
992\emph{Algorithms for modular elliptic curves}, second ed., Cambridge
993  University Press, Cambridge, 1997.
994
995\bibitem[CrA]{cremona:onlinetables}
996\bysame, \emph{Elliptic Curve Data},\hfill\\
997{\sf http://www.maths.nott.ac.uk/personal/jec/ftp/data/}.
998
999\bibitem[CrB]{mwrank}
1000\bysame, \emph{{\tt mwrank} (computer software)}, \\{\tt
1001  http://www.maths.nott.ac.uk/personal/jec/ftp/progs/}.
1002
1003\bibitem[CM00]{cremona-mazur}
1004J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements in the
1005  {S}hafarevich-{T}ate group}, Experiment. Math. \textbf{9} (2000), no.~1,
1006  13--28. \MR{1 758 797}
1007
1008\bibitem[Fea01]{fearnley:phd}
1009J.~Fearnley, \emph{{V}anishing and {N}on-{V}anishing of {L}-series of
1010  {E}lliptic {C}urves {T}wisted by {D}irichlet {C}haracters}, Concordia Ph.D.
1011  thesis (2001).
1012
1013\bibitem[Fla90]{flach:pairing}
1014M.~Flach, \emph{A generalisation of the {C}assels-{T}ate pairing}, J. Reine
1015  Angew. Math. \textbf{412} (1990), 113--127. \MR{92b:11037}
1016
1017\bibitem[GZ86]{gross-zagier}
1018B.~Gross and D.~Zagier, \emph{Heegner points and derivatives of
1019  \protect{${L}$}-series}, Invent. Math. \textbf{84} (1986), no.~2, 225--320.
1020  \MR{87j:11057}
1021
1022\bibitem[JL99]{jordan-livne:sha}
1023B.\thinspace{}W. Jordan and R.~Livn{\'e}, \emph{On {A}tkin-{L}ehner quotients
1024  of {S}himura curves}, Bull. London Math. Soc. \textbf{31} (1999), no.~6,
1025  681--685. \MR{2000j:11090}
1026
1027\bibitem[Kat]{kato:secret}
1028K.~Kato, \emph{$p$-adic {H}odge theory and values of zeta functions of modular
1029  forms}, Preprint, 244 pages.
1030
1031\bibitem[Kol90]{kolyvagin:euler_systems}
1032V.~A. Kolyvagin, \emph{Euler systems}, The Grothendieck Festschrift, Vol.\ II,
1033  Birkh\"auser Boston, Boston, MA, 1990, pp.~435--483. \MR{92g:11109}
1034
1035\bibitem[Lan56]{lang:finitefields}
1036S.~Lang, \emph{Algebraic groups over finite fields}, Amer. J. Math. \textbf{78}
1037  (1956), 555--563. \MR{19,174a}
1038
1039\bibitem[Maz72]{mazur:tower}
1040B.~Mazur, \emph{Rational points of abelian varieties with values in towers of
1041  number fields}, Invent. Math. \textbf{18} (1972), 183--266.
1042
1043\bibitem[McC86]{mccallum:duality}
1044W.\thinspace{}G. McCallum, \emph{Duality theorems for {N}\'eron models}, Duke
1045  Math. J. \textbf{53} (1986), no.~4, 1093--1124. \MR{88c:14062}
1046
1047\bibitem[Mil72]{milne:bsdres}
1048J.\thinspace{}S. Milne, \emph{On the arithmetic of abelian varieties}, Invent.
1049  Math. \textbf{17} (1972), 177--190. \MR{48 \#8512}
1050
1051\bibitem[Mil86]{milne:duality}
1052\bysame, \emph{Arithmetic duality theorems}, Academic Press Inc., Boston,
1053  Mass., 1986.
1054
1055\bibitem[PS97]{poonen-schaefer}
1056B.~Poonen and E.\thinspace{}F. Schaefer, \emph{Explicit descent for {J}acobians
1057  of cyclic covers of the projective line}, J. Reine Angew. Math. \textbf{488}
1058  (1997), 141--188. \MR{98k:11087}
1059
1060\bibitem[PS99]{poonen-stoll}
1061B.~Poonen and M.~Stoll, \emph{The {C}assels-{T}ate pairing on polarized abelian
1062  varieties}, Ann. of Math. (2) \textbf{150} (1999), no.~3, 1109--1149.
1063  \MR{2000m:11048}
1064
1065\bibitem[RS01]{ribet-stein:serre}
1066K.\thinspace{}A. Ribet and W.\thinspace{}A. Stein, \emph{Lectures on {S}erre's
1067  conjectures}, Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park
1068  City Math. Ser., vol.~9, Amer. Math. Soc., Providence, RI, 2001,
1069  pp.~143--232. \MR{2002h:11047}
1070
1071\bibitem[Rub98]{rubin:kato}
1072K.~Rubin, \emph{Euler systems and modular elliptic curves}, Galois
1073  representations in arithmetic algebraic geometry (Durham, 1996), Cambridge
1074  Univ. Press, Cambridge, 1998, pp.~351--367. \MR{2001a:11106}
1075
1076\bibitem[Sch83]{schneider:iwasawa}
1077P.~Schneider, \emph{Iwasawa ${L}$-functions of varieties over algebraic number
1078  fields. {A} first approach}, Invent. Math. \textbf{71} (1983), no.~2,
1079  251--293. \MR{85d:11063}
1080
1081\bibitem[SD67]{sd:bsd}
1082P.~Swinnerton-Dyer, \emph{The conjectures of {B}irch and {S}winnerton-{D}yer,
1083  and of {T}ate}, Proc. Conf. Local Fields (Driebergen, 1966), Springer,
1084  Berlin, 1967, pp.~132--157. \MR{37 \#6287}
1085
1086\bibitem[Ser79]{serre:localfields}
1087J-P. Serre, \emph{Local fields}, Springer-Verlag, New York, 1979, Translated
1088  from the French by Marvin Jay Greenberg.
1089
1090\bibitem[Ser88]{serre:alggroups}
1091\bysame, \emph{Algebraic groups and class fields}, Springer-Verlag, New York,
1092  1988, Translated from the French.
1093
1094\bibitem[ST68]{serre-tate}
1095J-P. Serre and J.\thinspace{}T. Tate, \emph{Good reduction of abelian
1096  varieties}, Ann. of Math. (2) \textbf{88} (1968), 492--517.
1097
1098\bibitem[Tat63]{tate:duality}
1099J.~Tate, \emph{Duality theorems in {G}alois cohomology over number fields},
1100  Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst.
1101  Mittag-Leffler, Djursholm, 1963, pp.~288--295. \MR{31 \#168}
1102
1103\end{thebibliography}
1104
1105
1106
1107\end{document}
1108