CoCalc Shared Fileswww / papers / nonsquaresha / final2.texOpen in CoCalc with one click!
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}
100
Let~$A$ denote an abelian variety over~$\Q$.
101
We give the first known examples in which $\#\Sha(A/\Q)$
102
is neither a square nor twice a
103
square. For example, let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of
104
conductor~$37$. We prove that for every odd prime
105
$p< \testbnd$ (with $p\neq 37$), there is a twist~$A$
106
of $E\cross\cdots \cross E$ ($p-1$ copies)
107
such that $\#\Sha(A/\Q)=p n^2$ for some integer~$n$.
108
We prove this by showing under certain hypothesis on~$E$
109
and~$p$ that there is an exact sequence
110
$$
111
0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(E/K)[p^\infty]\ra
112
\Sha(E/\Q)[p^\infty] \ra 0,
113
$$
114
where $K$ is a certain abelian extension of $\Q$ of degree $p$.
115
\end{abstract}
116
117
\maketitle
118
119
\section{Introduction}
120
The Shafarevich--Tate group of
121
an abelian variety~$A$ over a number field~$F$ is
122
$$
123
\Sha(A/F) := \Ker\left(H^1(F,A) \ra \bigoplus_{\text{all $v$}} H^1(F_v,A)\right).
124
$$
125
What are the possibilities for the group structure of $\Sha(A/F)$?
126
It is conjectured that $\Sha(A/F)$ is finite and this
127
is known in some cases.
128
\begin{theorem}[Kato, Kolyvagin, Wiles, et al.]\label{thm:finite}
129
Suppose~$A$ 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
132
extension~$K$ of~$\Q$ and
133
$L(A,\chi,1)\neq 0$, then the
134
$\chi$-component of $\Sha(A/K)\tensor_\Z\Z[\chi]$
135
is finite. (Here $\Z[\chi]$ is generated by
136
the image of~$\chi$.)
137
\end{theorem}
138
139
The Cassels--Tate pairing $\Sha(A/F)\cross \Sha(A^{\vee}/F)\ra \Q/\Z$
140
imposes strong constraints on the structure of $\Sha(A/F)$.
141
\begin{theorem}[Tate, Flach]\label{thm:tate}
142
Let~$p$ be a prime and suppose that there is a polarization
143
$\lambda : A \ra A^{\vee}$ of degree coprime to~$p$.
144
If $p=2$ assume also that~$\lambda$ arises from an $F$-rational
145
divisor on~$A$ (this hypothesis is automatic if~$A$ is
146
an elliptic curve, but can fail in general).
147
If $\Sha(A/F)[p^\infty]$ is finite then $\#\Sha(A/F)[p^\infty]$
148
is 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
161
It is tempting to
162
conjecture that $\#\Sha(A/F)$ is always a perfect square. Perhaps
163
squareness is a fundamental property of Shafarevich--Tate groups?
164
While implementing algorithms based on \cite{poonen-schaefer} for
165
computing with Jacobians of hyperelliptic curves, M.~Stoll was shocked to
166
discover an example of an abelian variety of dimension two such
167
that $\#\Sha(A/F)[2^{\infty}]=2$. This was surprising because,
168
for example, one finds in the literature \cite[pg.149]{sd:bsd}
169
the following statement:
170
``[The group $\Sha(A/F)$] is conjectured to be finite,
171
and Tate [26] has shown that if it is finite its order
172
is a perfect square.''
173
Stoll and B.~Poonen discovered what hid behind
174
this and other similar examples in which $\#\Sha(A/F)$ is twice
175
a perfect square.
176
177
An algebraic curve~$X$ of genus~$g$ over a local
178
field~$k$ is {\em deficient} if~$X$ has no $k$-rational divisor
179
of degree $g-1$.
180
\begin{theorem}[Poonen-Stoll \cite{poonen-stoll}]\label{thm:ps}
181
Suppose~$A$ is the Jacobian of an algebraic curve over~$F$ that
182
is deficient at an odd number of places. If $\#\Sha(A/F)$ is
183
finite, then $\#\Sha(A/F)$ is twice a square.
184
\end{theorem}
185
For example, they prove that the
186
Jacobian~$J$ of the nonsingular projective curve defined by
187
$$
188
y^2 = -3(x^2+1)(x^2-6x+1)(x^2+6x+1)
189
$$
190
has Shafarevich--Tate group of order~$2$
191
(to see that $\#\Sha(J)\mid 2$ they observe that~$J$ is isogenous
192
to a product of CM elliptic curves and apply a theorem of Rubin;
193
see~\cite[Prop.~27]{poonen-stoll} for details).
194
Also, Jordan and Livn\'e
195
\cite{jordan-livne:sha} give an infinite
196
family of Atkin--Lehner quotients of Shimura curves which are deficient
197
at an odd number of places.
198
199
Though $\#\Sha(A/F)$ need not be square, one might still be tempted to
200
conjecture that $\Sha(A/F)$ must have order
201
either a square or twice a square. Let~$p$ be an odd prime.
202
In this paper, we construct (under certain hypotheses that are
203
satisfied for $p<25000$) abelian varieties~$A$ such
204
that $\#\Sha(A/\Q)=pn^2$ for some integer~$n$. For
205
example (see Section~\ref{sec:ex}):
206
\begin{theorem}
207
Let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of conductor~$37$.
208
For 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
210
some 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
220
This paper was originally motivated by the problem of relating the
221
conjecture of Birch and Swinnerton-Dyer about the ranks of elliptic
222
curves~$E$ to the Birch and Swinnerton-Dyer formula for the orders
223
$\#\Sha(A)$ for abelian varieties~$A$ of analytic rank~$0$.
224
225
Let~$p$ be a prime. Under suitable hypotheses, we construct an abelian
226
variety~$A$ 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
228
subgroup of order~$p$, and no other natural subgroup of order~$p$
229
presents itself. Moreover, when $E$ is defined by $y^2+y=x^3-x$,
230
the Birch and Swinnerton-Dyer formula predicts that $\Sha(A/\Q)[3]$ is
231
of 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
236
to thank Kevin Buzzard, Frank Calegari,
237
Sol Friedberg, Benedict Gross, Emmanuel Kowalski, Barry Mazur, Bjorn
238
Poonen, and David Rohrlich for their helpful comments, and in particular
239
Michael Stoll for Lemma~\ref{lem:red_mod_n} and Cristian Gonz\'{a}lez
240
for carefully reading this paper, making many comments, and
241
sending me a proof of Proposition~\ref{prop:h2}.
242
243
\subsection{Notation}
244
If~$G$ 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~$n$.
247
We refer to elliptic curves using the notation of \cite{cremona:algs}.
248
249
\section{Construction of Nonsquare Shafarevich--Tate Groups}
250
For the rest of this paper we will work with an elliptic curve~$E$
251
over~$\Q$. Aside from the significant use of known cases of the Birch
252
and Swinnerton-Dyer conjecture below, much of the construction
253
should generalize to the situation when~$E$ is replaced by a
254
principally polarized abelian variety over a global field.
255
256
For the rest of this section, fix an elliptic curve~$E$ over~$\Q$.
257
By \cite{breuil-conrad-diamond-taylor}, $E$ is modular so
258
there is a newform $f=\sum_{n=1}^{\infty} a_n q^n$ of level
259
equal to the conductor~$N=N_E$ of~$E$ such that $L(E,s)=L(f,s)$.
260
For each prime $q\mid N$,
261
the Tamagawa number~$c_q$ of~$E$ at~$q$ is the order of the group of
262
rational components of the special fiber of the N\'eron model of~$E$
263
at~$q$.
264
265
266
\subsection{Twisting By Characters of Prime Order}
267
Let $p$ be a prime number.
268
For any prime $\ell \con 1\pmod{p}$, let
269
$$
270
\chi_{p,\ell} : (\Z/\ell\Z)^* \to \mu_p \subset \C^*
271
$$
272
be one of the $p-1$ Galois-conjugate Dirichlet characters of
273
order~$p$ and conductor~$\ell$.
274
275
\begin{conjecture}\label{conj:nonvanish}
276
Suppose~$p$ is a prime such that
277
$\rho_{E,p}:\Gal(\Qbar/\Q)\ra \Aut(E[p])$ is surjective. Then there exists a
278
prime $\ell\nmid N$ 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
285
and only if $L(E,\chi_{p,\ell}^\sigma,1)\neq 0$ for any $\Gal(\Qbar/\Q)$-conjugate~$\chi_{p,\ell}^\sigma$
286
of~$\chi_{p,\ell}$.
287
\item
288
J.~Fearnley proved related nonvanishing results when
289
$L(E,1)\neq 0$ in \cite{fearnley:phd}.
290
\item
291
If~$E$ is the elliptic curve $y^2+y=x^3-x$ of conductor~$37$ and rank~$1$,
292
then $\ell=41$ is the only $\ell\con 1\pmod{5}$ with $\ell<1000$
293
for which $L(E,\chi_{5,\ell},1)=0$.
294
295
\end{enumerate}
296
\end{remarks}
297
298
The following proposition gives evidence for
299
Conjecture~\ref{conj:nonvanish} for the lowest-conductor elliptic
300
curves of ranks $1$, $2$, and $3$.
301
\begin{proposition}\label{prop:conjtest}
302
Conjecture~\ref{conj:nonvanish} is true for
303
the rank~$1$ elliptic curve \nf{37A} for
304
every odd $p<\testbnd$ (with $p\neq 37$).
305
The conjecture is true for the rank~$2$ curve
306
\nf{389A} for every odd $p<1000$ (with $p\neq 389$).
307
The conjecture is true for the rank~$3$ curve
308
\nf{5077A} for every odd $p<1000$.
309
\end{proposition}
310
\begin{proof}
311
Consider the modular symbol
312
$$
313
e_{p,\ell} = \sum_{a\in (\Z/\ell\Z)^*} \chi_{p,\ell}(a)
314
\cdot \left\{0,\,\, \frac{a}{\ell}\right\}
315
\in H_1(X_0(N),\Q(\zeta_p)).
316
$$
317
Then $L(E,\chi_{p,\ell},1)\neq 0$ if and only if the
318
image of $e_{p,\ell}$ under
319
$$
320
H_1(X_0(N),\Q(\zeta_p)) \ra H_1(E,\Q(\zeta_p))
321
$$
322
is nonzero. In any particular case, we can use modular
323
symbols to determine whether or not this image is nonzero.
324
325
When~$p$ is large, it is difficult to compute in the
326
field $\Q(\zeta_p)$, so instead we compute in the residue class
327
field $\Fell=\Z[\zeta_p]/\m\isom Z/\ell\Z$, where~$\m$ is one of
328
the maximal ideals of $\Z[\zeta_p]$ that lies over~$\ell$.
329
(Note that $\ell$ splits completely in $\Z[\zeta_p]$
330
because $\ell\con 1\pmod{p}$.) After reducing modulo~$\m$,
331
we compute the image of
332
$$
333
\overline{e}_{p,\ell} = \sum_{a\in (\Z/\ell\Z)^*} a^{(\ell-1)/p}
334
\cdot \left\{0, \,\,\frac{a}{\ell}\right\}
335
\in H_1(X_0(N),\Fell)
336
$$
337
in $H_1(E,\Fell)$.
338
If it is nonzero, then the image of $e_{p,\ell}$
339
in $H_1(E,\Q(\zeta_p))$ is nonzero.
340
341
A big computation (that takes hundreds of hours using
342
{\sc Magma} \cite{magma})
343
shows that the image of $\overline{e}_{p,\ell}$
344
is nonzero in the cases asserted by the proposition.
345
So the reader can carry out similar computations,
346
we include the following {\sc Magma} V2.10-6 code,
347
which illustrates verification of the proposition
348
for {\bf 37A} for $p<100$:
349
\begin{verbatim}
350
procedure 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;
377
end procedure;
378
379
E := EllipticCurve([0,0,1,-1,0]); // 37A
380
for p in [q : q in [3..100] | IsPrime(q) and q ne 37] do
381
VerifyConjecture(E,p);
382
end for;
383
\end{verbatim}
384
The above input results in the following abbreviated output:
385
\begin{verbatim}
386
p=3: trying ell=7... success! (0.021 seconds)
387
p=5: trying ell=11... success! (0.039 seconds)
388
p=7: trying ell=29... success! (0.121 seconds)
389
...
390
p=89: trying ell=179... success! (0.739 seconds)
391
p=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}
398
As above, $E$ is an elliptic curve over~$\Q$. Let~$p$
399
be any prime (note that $p=2$ is allowed).
400
Suppose $\ell\con 1\pmod{p}$ is another prime and that
401
$\ell\nmid N_E$.
402
Let $K\subset \Q(\mu_\ell)$ be the abelian
403
extension of~$\Q$ that corresponds to~$\chi_{p,\ell}$
404
(thus~$K$ is the unique subfield of $\Q(\mu_\ell)$ of
405
degree~$p$).
406
407
Let $R = \Res_{K/\Q}(E_K)$ be the restriction of scalars down
408
to~$\Q$ of~$E$ viewed as an elliptic curve over~$K$. Thus~$R$
409
is an abelian variety over~$\Q$ of dimension $p=[K:\Q]$.
410
It is characterized by the fact that it
411
represents the following functor on $\Q$-schemes~$S$:
412
$$
413
S \mapsto E_K(S_K).
414
$$
415
As a Galois module,
416
$$
417
R(\Qbar) = E(\Qbar)\tensor_\Z \Z[\Gal(K/\Q)],
418
$$
419
where
420
$\tau\in \Gal(\Qbar/\Q)$ acts on
421
$\sum P_{\sigma} \tensor \sigma$ by
422
$$\tau\left(\sum P_\sigma\tensor \sigma\right) =
423
\sum \tau(P_\sigma)\tensor \tau_{|K}\cdot\sigma,
424
$$
425
where $\tau_{|K}$ is the image of~$\tau$ in $\Gal(K/\Q)$.
426
427
\begin{proposition}\label{prop:exactabvar}
428
The identity map induces a closed immerion $\iota: E\hookrightarrow
429
R$, and the trace $\Tr:K\ra \Q$ induces a surjection $\Tr:R\ra E$
430
whose kernel is geometrically connected. Thus we have an exact
431
sequence 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}
437
The existence of~$\iota$ and $\Tr$ follows from Yoneda's lemma.
438
The map~$\iota$ is induced by the functorial inclusion
439
$E(S)\hookrightarrow E_K(S_K)=R(S)$, so~$\iota$ is injective.
440
The $\Tr$ map is induced by the functorial trace map on points
441
$R(S)=E_K(S_K)\xrightarrow{\Tr} E(S)$.
442
443
To verify that $\Ker(\Tr)$ is geometrically connected, we base
444
extend the exact sequence (\ref{eqn:exactabvar}) to~$\Qbar$. First,
445
note that there is an isomorphism
446
$$
447
R_{\Qbar} \isom E_{\Qbar}\cross \cdots \cross E_{\Qbar}.
448
$$
449
After base extension, we identify
450
the trace map with the summation map
451
$$
452
+: E_{\Qbar} \cross \cdots \cross E_{\Qbar}
453
\longrightarrow E_{\Qbar}.
454
$$
455
Let $n=[K:\Q]$. 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 $\Ker(+)=\Ker(\Tr_{\Qbar})$.
460
Thus $\Ker(\Tr_{\Qbar})$ is isomorphic to
461
a 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}
471
The composition
472
$\Q\hookrightarrow K\xrightarrow{\Tr} \Q$
473
is multiplication by~$p$, so
474
the composition
475
$E \xra{\,\,\iota\,\,} R \xra{\Tr} E$
476
is also multiplication by~$p$.
477
Since
478
$\iota(E)\intersect \Ker(\Tr)$ is the kernel of~$\Tr\circ\iota = [p]$,
479
it equals $E[p]$.
480
\end{proof}
481
482
\begin{lemma}\label{lem:powers}
483
The abelian varieties
484
$A_K$, $R_K$, and $(R/\iota(E))_K$ are all isomorphic to
485
a product of copies of $E_K$.
486
\end{lemma}
487
\comment{
488
\begin{proof}
489
Base extending (\ref{eqn:eraprime}) to~$K$
490
we have an exact sequence
491
$$
492
0 \ra E_K \xra{\Delta} E_K^{\times p} \ra A'_K \ra 0.
493
$$
494
Embed $E_K^{\times (p-1)}$ in $E_K^{\times p}$ by
495
$(a_1,\ldots, a_{p-1}) \mapsto (a_1,\ldots, a_{p-1},0)$.
496
Then $E_K^{\times (p-1)}$ maps injectively to $A'_K$, since
497
$E_K^{\times (p-1)}$ has $0$ intersection with the diagonal,
498
so $A'_K \ncisom E_K^{\times (p-1)}$.
499
\end{proof}
500
}
501
502
503
504
\begin{proposition}\label{prop:exactneron}
505
The exact sequence $0\ra A \ra R \ra E\ra 0$ of
506
Proposition~\ref{prop:exactabvar} extends to
507
an exact sequence
508
$ 0 \ra \cA \ra \cR \ra \cE \ra 0$
509
of N\'eron models over~$\Z$.
510
\end{proposition}
511
\begin{proof}
512
We use results of \cite[Ch.~7]{neronmodels} and the
513
fact that formation of N\'eron models commutes with unramified base
514
change (see \cite[\S1.2, Prop.~2]{neronmodels})
515
to 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}
519
is exact.
520
521
First suppose that $q\neq \ell$, and let~$\q$ be a prime
522
of~$K$ lying over~$q$. We use the fact that formation of N\'eron models
523
commutes with unramified base extension and check exactness
524
of (\ref{eqn:neron}) after base extension to the
525
unramified extension $\O_{K,\q}$ of $\Z_q$.
526
By Lemma~\ref{lem:powers}, the generic fiber of the base
527
extension of (\ref{eqn:neron}) to $\O_{K,\q}$ is
528
$$
529
0\ra E_{K,\q}^{\oplus (n-1)} \ra
530
E_{K,\q}^{\oplus n}\xra{\Sigma} E_{K,\q}\ra 0.
531
$$
532
Thus the corresponding complex of N\'eron models over $\O_{K,\q}$ is
533
$$
534
0\ra \cE_{\O_{K,\q}}^{\oplus (n-1)} \ra \cE_{\O_{K,\q}}^{\oplus n}
535
\xra{\Sigma} \cE_{\O_{K,\q}}\ra 0,
536
$$
537
which is exact, since it is exact on $S$-points for {\em any}
538
ring~$S$.
539
540
Suppose that $q=\ell$. Since $p\neq \ell$,
541
\cite[Prop.~7.5.3 (a)]{neronmodels} asserts
542
that the sequence
543
$0 \ra \cA_{\Z_q} \ra \cR_{\Z_q} \ra \cE_{\Z_q}$
544
is exact.
545
Since $p\neq q$,
546
the map $[p]: \cE_{\Z_q} \to \cE_{\Z_q}$ is
547
an \'etale morphism of smooth schemes.
548
Since~$E$ has good reduction at~$q$, we also
549
know that the fibers of $\cE_{\Z_q}$ are geometrically connected,
550
so $[p]$ is surjective (for more details, see the proof
551
of~\cite[Lem.~3.2]{agashe-stein:visibility}).
552
It follows that $\cR_{\Z_q} \ra \cE_{\Z_q}$ is surjective.
553
554
\end{proof}
555
556
557
558
\subsection{The Cokernel of Trace}
559
Let~$\ell$ be a prime as in Conjecture~\ref{conj:nonvanish}.
560
This section is devoted to computing the cokernel
561
of the trace map $R(\Q) \ra E(\Q)$. Note that $R(\Q)=E(K)$, so
562
this cokernel is also $E(\Q)/\Tr_{K/\Q}(E(K))$.
563
564
\begin{lemma}\label{lem:kellsize}
565
Let $K_\ell$ denote the completion of $K$ at the totally ramified
566
prime of~$K$ lying over~$\ell$. Then $E(K)[p] = E(K_{\ell})[p]=0$.
567
\end{lemma}
568
\begin{proof}
569
The characteristic polynomial of
570
$\Frob_{\ell}\in\Gal(\Q_\ell^{\ur}/\Q_\ell)$
571
on $E[p] = E(\Q_\ell^{\ur})[p]$ is $x^2 - a_\ell x + \ell \in \F_p[x]$.
572
By 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
$$
575
E(\Q_\ell)[p] = E(\Q_\ell^{\ur})[p]^{\Frob_{\ell}-1} = 0.
576
$$
577
Since~$K$ is totally ramified at~$\ell$ and~$E$ has
578
good reduction at~$\ell$, $E(K_\ell)[p]=0$ as well,
579
so $E(K)[p]=0$, 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}
588
By Corollary~\ref{cor:intersection} the
589
the image of $\iota(E(\Q))\subset R(\Q)$ in $E(\Q)$
590
is $p E(\Q)$, so the
591
cokernel of $R(\Q)\ra E(\Q)$
592
is a quotient of $E(\Q)/p E(\Q)$.
593
Thus it suffices to prove that $R(\Q)/\iota(E(\Q))$
594
is {\em finite} of order coprime to~$p$.
595
596
We have an exact sequence
597
$0 \ra E \ra R \ra A' \ra 0$,
598
with $A'$ an abelian variety that is isogenous to~$A$
599
(in fact, $A'$ is the abelian variety dual of~$A$ since~$R$
600
is self dual, but we will not use this fact.)
601
The $L$-series of $A'$ is
602
$\prod_{i=1}^{p-1} L(E,\chi_{p,\ell}^i,s)$,
603
so by hypothesis $L(A',1)\neq 0$ and
604
it follows from Kato's theorem (see \cite[\S8.1]{rubin:kato})
605
that $A'(\Q)$ is finite.
606
Thus $R(\Q)/\iota(E(\Q))$ is finite since $R(\Q)/\iota(E(\Q))\subset A'(\Q)$.
607
By Lemma~\ref{lem:powers}, $A'_K \ncisom E_K^{\times (p-1)}$
608
and 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}
613
Fix an elliptic curve~$E$ over~$\Q$ and a prime $p\nmid \prod c_{E,q}$.
614
615
In this section, we use results mostly due to Mazur to relate the
616
Shafarevich--Tate groups of~$A$,~$R$, and~$E$ to certain \'etale
617
cohomology groups.
618
We maintain the notation and assumptions
619
of the previous sections, except
620
that we abuse notation slightly and let $\cA$, $\cR$, and $\cE$ also
621
denote the \'etale sheaves on $\Spec(\Z)$ defined by
622
the N\'eron models $\cA$, $\cR$, and $\cE$.
623
Let $\cB$ be either
624
$\cA$, $\cR$, or $\cE$ and let $B=\cB_\Q$
625
be the corresponding abelian variety.
626
Let $H^q(\Z,\cB)$ be the $q$th \'etale cohomology group of~$\cB$.
627
628
\begin{lemma}\label{lem:red_mod_n}
629
There is an isomorphism
630
$B(\Q_\ell)[p] \isom \cB(\Fell)[p].$
631
\end{lemma}
632
\begin{proof}
633
This follows from
634
\cite[Lem.~2, pg.~495]{serre-tate},
635
but we sketch a proof for the convenience of the reader.
636
Let $B^{1}(\Q_\ell)$ denote the kernel of the natural reduction
637
map $r:B(\Q_\ell)\ra \cB(\Fell)$. Using formal groups and
638
that $p\neq \ell$,
639
one sees that $[p]:B^{1}(\Q_\ell)\ra{}B^{1}(\Q_\ell)$ is an isomorphism.
640
Since $\cB$ is smooth over~$\Q_\ell$,
641
Hensel's lemma (see \cite[\S2.3~Prop.~5]{neronmodels})
642
implies that the reduction map
643
is surjective, so we obtain an exact sequence
644
$$
645
0\ra B^1(\Q_\ell) \ra B(\Q_\ell) \ra \cB(\Fell) \ra 0.
646
$$
647
The snake lemma applied to the multiplication-by-$p$ diagram
648
attached to this exact sequence yields the
649
exact sequence
650
$$0\ra B(\Q_\ell)[p]\ra \cB(\Fell)[p] \ra 0 \ra B(\Q_\ell)/p B(\Q_\ell)
651
\ra \cB(\Fell)/p\cB(\Fell)\ra0,$$
652
which proves the lemma.
653
\end{proof}
654
655
The {\em Tamagawa number} of~$B$ at a prime~$q$
656
is $c_{B,q}=\#\Phi_{B,q}(\F_q)$, where $\Phi_{B,q}$
657
is the component group of the closed fiber
658
of 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}
665
First suppose $q=\ell$.
666
The cokernel of
667
$\cB(\F_\ell) \ra \Phi_{B,\ell}(\F_\ell)$
668
is contained in $H^1(\F_\ell,\cB^0)$, which
669
is~$0$ by Lang's theorem (see \cite{lang:finitefields} or
670
\cite[\S{}VI.4]{serre:alggroups}),
671
so if $\Phi_{B,\ell}(\F_\ell)[p]\neq 0$ then $\cB(\F_\ell)[p]\neq 0$. But by
672
Lemmas~\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
678
Next suppose that $q\neq \ell$. Since formation of N\'eron models
679
commutes with unramified base extension, we have
680
$$
681
\Phi_{B,q}(\Fbar_q)[p] \isom
682
\Phi_{E,q}(\Fbar_q)[p] \cross \cdots \cross \Phi_{E,q}(\Fbar_q)[p] = 0,
683
$$
684
by our hypotheses on~$p$.
685
\end{proof}
686
687
Following the appendix to \cite{mazur:tower}, let
688
$$
689
\Sigma(B/\Q) = \ker\left(H^1(\Q,B)\ra \bigoplus_{\text{all finite $q$}}
690
H^1(\Q_q, B)\right),
691
$$
692
where the sum is over all finite primes~$q$ of~$\Q$.
693
If~$p$ is an odd prime, then
694
$\Sigma(B/\Q)[p^\infty] = \Sha(B/\Q)[p^\infty]$;
695
also one can see easily using Tate cohomology for the cyclic
696
group $\Gal(\C/\R)$ that
697
$$
698
\Sigma(B/\Q)[2]/\Sha(B/\Q)[2]\subset
699
H^1(\R,B(\C)) \isom B(\R)/B(\R)^0,
700
$$
701
where $B(\R)/B(\R)^0$ has order $2^e$ for some $e\leq \dim B$.
702
\begin{proposition}[Mazur]\label{prop:shah1}
703
Suppose that $a_\ell\not\con \ell+1\pmod{p}$.
704
If $p$ is odd, then
705
$$\ds
706
H^1(\Z,\cB)[p^{\infty}] \isom \Sha(B/\Q)[p^\infty].
707
$$
708
Also,
709
$\#H^1(\Z,\cB)[2^{\infty}] / \Sha(B/\Q)[2^\infty]$
710
divides $\#(B(\R)/B(\R)^0)$.
711
\end{proposition}
712
\begin{proof}
713
It follows from the appendix to \cite{mazur:tower} that there is an
714
exact 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}
719
where $\Phi_{B,q}$ is the component group of the fiber of $\cB$
720
at~$q$.
721
By \cite[VIII.4.8]{serre:localfields},
722
$$
723
\#H^1(\F_q,\Phi_{B,q}(\Fbar_q)) = \#\Phi_{B,q}(\F_q) = c_{B,q},
724
$$
725
so 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}
732
We apply the lemmas in \cite[\S{}III.6]{schneider:iwasawa}.
733
Note that~$A$ has good reduction at~$p$ by \cite[Prop.~1]{milne:bsdres},
734
and $H^1(\Z,\cA)[p^\infty]$ is finite by Kato's theorem
735
(see \cite[\S8.1]{rubin:kato}) and Proposition~\ref{prop:shah1}.
736
In the proof of Proposition~\ref{prop:coker}, we showed that
737
$A'(\Q)$ is finite of order coprime to~$p$, where $A'$
738
is the abelian variety dual of~$A$. We now use\footnote{Note that
739
the proof of Lemma~7 of \cite[\S{}III.6]{schneider:iwasawa}
740
relies on a theorem of
741
Artin and Mazur whose proof they never published;
742
generalizations of this theorem have been published by
743
McCallum \cite[\S5]{mccallum:duality} and
744
Milne \cite[\S{}III.3.4]{milne:duality}, and Mazur assures the author
745
that he and Milne both know the proof of Artin-Mazur duality well.}
746
Lemma~7 of \cite[\S{}III.6]{schneider:iwasawa}, which because
747
$A'(\Q)[p]=0$
748
implies that $H^2(\Z,\cA[p^\infty]) = 0$
749
(Schneider uses $H^q_{\text{fpqf}}$, but this is not a problem
750
since \'etale and \text{fpqf} cohomology agree on the smooth
751
scheme $\cA$.)
752
It is easy to see (see, e.g., the proof of Lemma~6
753
of \cite[\S{}III.6]{schneider:iwasawa}) that
754
the natural map $H^q(\Z,\cA[p^\infty]) \ra H^q(\Z,\cA)[p^\infty]$
755
is surjective for any $q>0$, in particular, for $q=2$,
756
so $H^2(\Z,\cA)[p^\infty]=0$ which proves the proposition.
757
\comment{
758
As discussed in [], there is an exact sequence
759
$$0\ra \cB^0 \ra \cB \ra \bigoplus_{\text{primes }q} \Phi_{B,q} \ra 0,$$
760
which leads to the exact sequence
761
$$ H^2(\Z,\cB^0)[p^\infty] \ra H^2(\Z,\cB)[p^\infty]
762
\ra \bigoplus_{\text{primes $q$}} H^2\left(\F_q, \Phi_{B,q}(\Fbar_q)\right)[p^\infty].$$
763
By \cite[\S{}VIII.4]{serre:localfields} and Lemma~\ref{lem:boundcq},
764
$$
765
\#H^2(\F_q,\Phi_{B,q})[p] = \#H^0(\F_q,\Phi_{B,q})[p] = 1,
766
$$
767
so it suffices to show that $H^2(\Z,\cB^0)[p]=0$.
768
}
769
\end{proof}
770
771
772
773
\subsection{The Main Theorem}
774
Fix an elliptic curve~$E$ over~$\Q$ and a prime
775
$p\nmid \prod c_{E,q}$
776
such that $\rho_{E,p}:G_\Q\to \Aut(E[p])$ is
777
surjective.
778
If $p=2$ assume also that $E(\R)$ is connected.
779
Assume that~$\ell$ is one of the primes whose existence
780
is predicted by Conjecture~\ref{conj:nonvanish}.
781
Let~$A$ and~$R$ be the corresponding abelian varieties, which fit
782
into an exact sequence $0\to A\to R \to E \to 0$, and recall
783
that $L(A,1)\neq 0$ 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}
787
There is an exact sequence
788
$$
789
0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(E/K)[p^\infty]\ra
790
\Sha(E/\Q)[p^\infty] \ra 0.
791
$$
792
In particular, if~$E$ has odd rank and $\Sha(E/\Q)[p^\infty]$ is
793
finite, then $\#\Sha(A/\Q)[p^\infty]$
794
is not a perfect square.
795
\end{theorem}
796
\begin{proof}
797
By Proposition~\ref{prop:exactneron} we have an
798
exact sequence of \'etale sheaves
799
$$
800
0 \ra \cA \ra \cR \ra \cE \ra 0,
801
$$
802
which gives rise to an exact
803
sequence of \'etale cohomology groups
804
$$
805
H^0(\Z,\cR) \ra H^0(\Z,\cE)
806
\ra H^1(\Z,\cA) \ra H^1(\Z,\cR) \ra H^1(\Z,\cE) \ra H^2(\Z,\cA).
807
$$
808
We have
809
$$
810
H^0(\Z, \cR) = \cR(\Z) = R(\Q)$$
811
and likewise for $\cE$, so by Propositions~\ref{prop:coker},~\ref{prop:shah1},
812
and \ref{prop:h2} we obtain an exact sequence
813
$$
814
0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(R/\Q)[p^\infty] \ra
815
\Sha(E/\Q)[p^\infty] \ra 0.
816
$$
817
By 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
819
claimed exact sequence.
820
821
Kato's theorem (\cite[\S8.1]{rubin:kato} and \cite[Cor.~14.3]{kato:secret})
822
implies that $\Sha(E/K)[p^\infty]$ is
823
finite (for the trivial character use our hypothesis
824
that $\Sha(E/\Q)[p^\infty]$
825
is finite, and for the nontrivial characters use our hypothesis
826
that $L(E,\chi_{p,\ell},1)\neq 0$).
827
Theorem~\ref{thm:tate} then implies that $\#\Sha(E/K)[p^\infty]$ is a
828
perfect square. If $E(\Q)$ has odd rank then $\#(E(\Q)/p E(\Q))$
829
is 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}
835
In the language of visibility of Shafarevich-Tate
836
groups (see~\cite{cremona-mazur}),
837
Theorem~\ref{thm:nonsquare}
838
asserts that the
839
visible subgroup of $\Sha(A)$ with respect to the
840
embedding $A\hra R$ is canonically isomorphic to
841
the Mordell-Weil quotient $E(\Q)/p E(\Q)$.
842
\end{remark}
843
844
845
\begin{proposition}\label{prop:away_from_p}
846
If $q\neq p$ 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}
850
In particular, if $\Sha(E/\Q)[q^\infty]$ is finite,
851
then $\Sha(A/\Q)[q^\infty]$ has order a perfect square.
852
\end{proposition}
853
\begin{proof}
854
The intersection of~$E$ and~$A$ in~$R$ is $E[p]$, so
855
the summation map $E\times A \to R$ is an isogeny with kernel $E[p]$.
856
Considering 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}
861
and likewise for any completion $\Q_v$ of~$\Q$.
862
We then obtain (\ref{eqn:away_from_p}) by
863
combining (\ref{eqn:qinfpart})
864
with the fact that cohomology commutes with products and
865
that $H^1(\Q,R)\isom H^1(K,E)$.
866
867
If $\Sha(E/\Q)[q^\infty]$ is finite, then since $\Sha(A/\Q)[q^\infty]$
868
is 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
870
by 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
873
a perfect square.
874
\end{proof}
875
876
877
878
879
\section{An Example}\label{sec:ex}
880
Combining Proposition~\ref{prop:conjtest}, Theorem~\ref{thm:nonsquare},
881
and Proposition~\ref{prop:away_from_p} yields the following theorem.
882
\begin{theorem}\label{thm:37}
883
Let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of conductor~$37$.
884
For every odd prime $p<\testbnd$ (with $p\neq 37$), there is a twist~$A$ of
885
$E^{\cross (p-1)}$ such that $\#\Sha(A/\Q)=pn^2$
886
for some integer~$n$.
887
\end{theorem}
888
\begin{remark}
889
Using the elliptic curve of conductor $43$ in place of~$E$
890
one can construct an abelian variety $A$ with $\Sha(A/\Q)=37n^2$
891
for some integer~$n$.
892
\end{remark}
893
894
Though unnecessary for Theorem~\ref{thm:37}, we
895
prove below that $\Sha(E/\Q)=0$, which removes our
896
dependence on Proposition~\ref{prop:h2}.
897
We show that $\Sha(E/\Q)[p^\infty]=0$ for all odd~$p$
898
using \cite[Thm.~A]{kolyvagin:euler_systems}, and
899
we use a $2$-descent (with \cite{mwrank}) to see that $\Sha(E/\Q)[2]=0$.
900
\begin{theorem}[Kolyvagin]\label{thm:kolybound}
901
Let~$E$ be an elliptic curve
902
and let $L=\Q(\sqrt{-D})$ be an imaginary quadratic
903
field of odd discriminant $-D$, where all primes dividing the conductor
904
of~$E$ split, and assume
905
that $D\neq 3,4$. If the Heegner point $y_L\in E(L)$ has infinite order (equivalently,
906
by \cite{gross-zagier}, $L'(E/L,1)\neq 0$),
907
then $\#\Sha(E/L)\mid t\cdot [E(L): \Z y_L]^2$,
908
where the only primes that
909
divide~$t$ are~$2$ or primes where $\rho_{E,p}$ is not surjective.
910
\end{theorem}
911
912
By \cite{cremona:algs},~$E$ is isolated in its isogeny class,
913
so $\rho:\Gal(\Qbar/\Q)\ra \Aut(E[p])$ is surjective for all primes~$p$
914
(see \cite[\S1.4]{ribet-stein:serre})
915
hence~$t$ is a power of~$2$.
916
Let $L = \Q(\sqrt{-7})$.
917
To compute $[E(L):\Z{}y_L]$ up to a power of~$2$
918
we use the Gross-Zagier formula and the fact that $[E(L) : E(\Q)+ E^D(\Q)]$
919
is a power of~$2$. By \cite[Thm.~6.3]{gross-zagier},
920
$$
921
h(y_L) = \frac{u^2|D|^{\frac{1}{2}}}{\|\omega_f\|} L'(E,1)L(E^D,1),
922
$$
923
where $D=-7$, $u=1$, and $\|\omega_f\|$
924
is the Peterson norm of the newform~$f$ corresponding to~$E$.
925
Generators for the period lattice of~$E$ are
926
$\omega_1 \sim 2.993459$ and
927
$\omega_2 \sim 2.451389i$, so $\|\omega_f\|\sim 7.338133$.
928
The quadratic twist $E^D$ is the curve \nf{1813B1} in \cite{cremona:onlinetables},
929
and $E^D(\Q)=0$. From \cite{cremona:onlinetables} we find that
930
$L'(E,1) \sim 0.306000$
931
and $L(E^D,1) \sim 1.853076$,
932
so $h(y_L)\sim 0.204446$.
933
The 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
937
the Gross-Zagier formula, we wrote a program using
938
\cite{magma} to compute Heegner points and
939
found that $y_L=(0,0)$, which is a generator for $E(\Q)$.)
940
Thus $\#\Sha(E/L)$ is a power of~$2$.
941
942
To connect $\Sha(E/L)$ with $\Sha(E/\Q)$,
943
use the inflation-restriction exact sequence
944
$$
945
0 \ra H^1(L/\Q, E(L)) \ra H^1(\Q,E(\Qbar)) \ra H^1(L,E(\Qbar)).
946
$$
947
Let~$p$ be an odd prime.
948
Since $H^1(L/\Q,E(L))$ is a $2$-group, the above sequence leads
949
to an injective map
950
$$
951
H^1(\Q,E(\Qbar))[p] \hra H^1(L, E(\Qbar))[p],
952
$$
953
which 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}
969
A.~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
973
\bibitem[BCDT01]{breuil-conrad-diamond-taylor}
974
C.~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}
981
W.~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}
987
S.~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}
991
J.\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}
1004
J.\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}
1009
J.~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}
1014
M.~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}
1018
B.~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}
1023
B.\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}
1028
K.~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}
1032
V.~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}
1036
S.~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}
1040
B.~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}
1044
W.\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}
1048
J.\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}
1056
B.~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}
1061
B.~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}
1066
K.\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}
1072
K.~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}
1077
P.~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}
1082
P.~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}
1087
J-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}
1095
J-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}
1099
J.~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