CoCalc Shared Fileswww / cuny / vismw.tex
Author: William A. Stein
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5\title{\huge\sf Visibility of Galois Cohomology and\\ Mordell-Weil Groups}
6
7\date{November 20, 2003}
8
9\author{William Stein}
10\begin{document}
11\maketitle
12\tableofcontents
13
14\section{Visibility of $\H^1(K,A)$}
15Let $K$ be a number field.  (There should be a similar theory for
16function fields over a finite field.)
17\subsection{Motivation and Philosophy}
18Suppose
19$20 0 \to A \to B \to C \to 0 21$
22is an exact sequence of abelian varieties over~$K$.  (Thus each
23of~$A$,~$B$, and~$C$ is a complete group variety over~$K$, whose group
24is automatically abelian.)  Then there is a corresponding long exact
25sequence of cohomology for the group $\Gal(\Qbar/K)$:
26$27 0 \to A(K) \to B(K) \to C(K) \to \H^1(K,A) \to \H^1(K,B)\to 28 \H^1(K,C) \to \cdots 29$
30
31The study of the Mordell-Weil group $C(K)=\H^0(K,C)$ is popular in
32arithmetic geometry.  For example, the Birch and Swinnerton-Dyer
33conjecture (BSD conjecture), which is one of the million dollar Clay
34Math Problems, asserts that the dimension of $C(K)\tensor\Q$ equals
35the ordering vanishing of $L(C,s)$ at $s=1$.
36
37The group $\H^1(K,A)$ is also of interest in connection with
38the BSD conjecture, because it contains
39the Shafarevich-Tate group
40$41 \Sha(A) = \Sha(A/K) = 42 \Ker\left(\H^1(K,A)\to \bigoplus_{v} \H^1(K_v,A)\right) \subset \H^1(K,A), 43$
44where the sum is over all places~$v$ of~$K$ (e.g., when $K=\Q$, the
45fields $K_v$ are $\Q_p$ for all prime numbers~$p$ and
46$\Q_{\infty}=\R$).
47
48The group $A(K)$ is {\em fundamentally different} than $\H^1(K,C)$.  The
49Mordell-Weil group $A(K)$ is finitely generated, whereas the first
50Galois cohomology $\H^1(K,C)$ is far from being finitely
51generated---in fact, every element has finite order and there are
52infinitely many elements of any given order.
53
54This talk is about dimension shifting'', i.e., relating information
55about $\H^0(K,C)$ to information about $\H^1(K,A)$.
56
57\subsection{Definitions}
58\subsubsection{What are Elements of Galois Cohomology?}
59Elements of $\H^0(K,C)$ are simply points, i.e., elements of $C(K)$,
60so they are relatively easy to visualize''.  In contrast, elements
61of $\H^1(K, A)$ are Galois cohomology classes, i.e., equivalence
62classes of set-theoretic (continuous) maps $f:\Gal(\Qbar/K)\to 63A(\Qbar)$ such that $f(\sigma\tau) = f(\sigma) + \sigma f(\tau)$.  Two
64maps are equivalent if their difference is a map of the form
65$\sigma\mapsto \sigma(P)-P$ for some fixed $P\in A(\Qbar)$.  From this
66point of view $\H^1$ is more mysterious than $\H^0$.
67
68\subsubsection{Principal Homogeneous Spaces}
69There is an alternative way to view elements of $\H^1(K,A)$. The WC
70group of~$A$ is the group of isomorphism classes of principal
71homogeneous spaces for~$A$, where a principal homogeneous space is a
72variety~$X$ and a map $A\times X\to X$ that satisfies the same axioms
73as those for a simply transitive group action.  Thus~$X$ is a twist as
74variety of~$A$, but $X(K)=\emptyset$, unless~$X\ncisom A$.  Also, the
75nontrivial elements of $\Sha(A)$ correspond to the classes of~$X$ that
76have a $K_v$-rational point for all places~$v$, but no $K$-rational
77point.
78
79\subsubsection{Visibility of $\H^1(K,A)$}
80Barry Mazur introduced the following definition in order to help unify
81diverse constructions of principal homogeneous spaces:
82\begin{definition}
83The {\em visible subgroup} of $\H^1(K,A)$ in~$B$ is
84\begin{align*}
85  \Vis_B\H^1(K,A) &= \Ker(\H^1(K,A)\to \H^1(K,B))\\
86                  &= \Coker(B(K)\to C(K)).
87\end{align*}
88\end{definition}
89
90\begin{remark}
91Note that $\Vis_B\H^1(K,A)$ {\em does} depend on the embedding of~$A$
92into~$B$.  For example, suppose $B=B_1\times A$.  Then there could be
93nonzero visible elements if~$A$ is embedding into the first factor,
94but there will be no nonzero visible elements if~$A$ is embedded into
95the second factor.  Here we are using that $\H^1(K, B_1\times A) = 96\H^1(K,B_1)\oplus \H^1(K,A).$
97\end{remark}
98
99
100
101The connection with the WC group of~$A$ is as follows.
102Suppose
103$104 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 105$
106is an exact sequence of abelian varieties and that $c\in \H^1(K,A)$ is
107visible in~$B$.  Thus there exists $x\in C(K)$ such that $\delta(x) = 108c$.  Then $X=\pi^{-1}(x)\subset B$ is a translate of $A$ in~$B$, so
109the group law on~$B$ gives~$X$ the structure of principal homogeneous
110space for~$A$, and one can show that the class of~$X$ in the WC group
111of~$A$ corresponds to~$c$.
112
113\subsubsection{Finiteness of the Visible Subgroup}
114\begin{lemma}
115The group $\Vis_B\H^1(K,A)$ is finite.
116\end{lemma}
117\begin{proof}
118By the Mordell-Weil theorem $C(K)$ is finitely generated.  The group
119$\Vis_B\H^1(K,A)$ is a homomorphic image of $C(K)$ so it is finitely
120generated.  On the other hand, it is a subgroup of $\H^1(K,A)$, so it
121is a torsion group.  The lemma follows since a finitely
122generated torsion abelian group is finite.
123\end{proof}
124
125\subsection{Every Element of $\H^1(K,A)$ is Visible Somewhere}
126
127\begin{proposition}\label{prop:allvish1}
128Let $c\in\H^1(K,A)$.  Then there exists an abelian variety~$B=B_c$ and
129an embedding $A\hra B$ such that~$c$ is visible in~$B$.  Moreover,~$B$
130can be chosen to be a twist of a power of~$A$.
131\end{proposition}
132\begin{proof}
133By definition of Galois cohomology, there is a finite extension~$L$
134of~$K$ such that $\res_L(c)=0$.  Thus $c$ maps to $0$ in
135$\H^1(L,A_L)$.  By a slight generalization of the Shapiro Lemma from
136group cohomology (which is proved by dimension shifting; see, e.g.,
137Atiyah-Wall in Cassels-Frohlich), there is a canonical isomorphism
138$139 \H^1(L,A_L) \isom \H^1(K,\Res_{L/K}(A_L)) = \H^1(K,B), 140$
141where $B=\Res_{L/K}(A_L)$ is the Weil restriction of scalars of $A_L$
142back down to~$K$.  The restriction of scalars~$B$ is an abelian
143variety of dimension $[L:K]\cdot \dim A$ that is characterized by the
144existence of functorial isomorphisms
145$146 \Mor_K(S,B) \isom \Mor_L(S_L, A_L), 147$
148for any $K$-scheme~$S$, i.e., $B(S)=A_L(S_L)$.  In particular,
149setting~$S=A$ we find that the identity map $A_L\to A_L$ corresponds
150to an injection $A\hra B$.  Moreover, $c\mapsto\res_L(c)=0\in\H^1(K,B)$.
151
152The assertion about the structure of~$B$ follows from general facts
153about restriction of scalars, which won't be proved here.
154\end{proof}
155
156\subsection{Other Results in the Context of Modularity}
157Usually one focuses on visibility of elements in $\Sha(A)$.  There are
158a number of other results about visibility in various special cases,
159and large tables of examples in the context of elliptic curves and
160modular abelian varieties.  There are also interesting modularity
161questions/conjectures in this context.  I will not go into these
162further right now, except to note one example.
163
164Motivated by the notion of visibility, I developed (with input from
165Mazur, Cremona, and Agashe) computational techniques for
166unconditionally constructing Shafarevich-Tate groups of modular
167abelian varieties $A\subset J_1(N)$.  For example, if
168$A\subset J_0(389)$ is the $20$-dimensional simple factor, then
169$170 \Z/5\Z\times \Z/5\Z\subset \Sha(A), 171$
172as predicted by the Birch and Swinnerton-Dyer conjecture.  I found a
173few dozen other examples like this, where the computational
174construction of the Shafarevich-Tate group would be hopeless using any
175other known technique.  See \cite{agashe-stein:bsd,
176agashe-stein:visibility} for more details, and \cite{cremona-mazur}
177for examples when $\dim A=1$.
178
179\section{Visibility of Mordell-Weil Groups}
180\subsection{Motivation and Philosophy}
181The previous section was about understanding elements of $\H^1$ in
182terms of Mordell-Weil groups.  The BSD conjecture implies the
183following conjecture:
184\begin{conjecture}\label{conj:inf}
185If $L(C,1)=0$, then $C(\Q)$ is infinite.
186\end{conjecture}
187
188We know by the Gross-Zagier formula that if~$C$ is an elliptic curves
189over~$\Q$ and $\ord_{s=1} L(C,s)=1$, then $C(\Q)$ is infinite, but
190little more is known toward Conjecture~\ref{conj:inf}.  More
191generally, the conjecture is known when $C\subset J_0(N)$ and
192$\ord_{s=1} L(C,s) = \dim(C)$, and there are other results over
193totally real number fields.  People also seem to have a reasonable
194(but not good enough!)  understanding of $\Sha(C)$ when $L(C,1)\neq 1950$.
196
197\subsubsection{Rank $>1$: A New Idea is Needed}
198Suppose~$C$ is an elliptic curve over~$\Q$ and $\ord_{s=1} L(C,s)=2$.
199Conjecture~\ref{conj:inf} asserts that $C(\Q)$ is infinite, but this
200is currently a difficult open problem.  Nick Katz told me at dinner
201once that a new idea is needed.''  It seems that nobody knows a good
202analogue of Gross-Zagier for rank two elliptic curves.  (I've noticed
203that Mazur has been working on this question, in one way or another,
204since I've been at Harvard...)
205
206Visibility of Mordell-Weil groups is an idea I came up which might
207have some relevance.
208
209\subsection{Definition}
210Suppose
211$212 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 213$
214is an exact sequence of abelian varieties over a number field~$K$,
215with corresponding long exact sequence
216$217 0 \to A(K) \to B(K) \to C(K) \xrightarrow{\delta} \H^1(K,A) \to\cdots 218$
219of $\Gal(\Qbar/K)$-cohomology.
220
221\begin{definition}
222Let $x \in C(K)$ and suppose~$m\in\Z_{>0}$ is a divisor of $\order(x)$
223(everything divides $\infty$).  Then~$x$ is {\em $m$-visible in
224$\H^1(K,A)$} if the order of $\delta(x)\in \H^1(K,A)$ is divisible
225by~$m$.
226\end{definition}
227
228Motivated by Proposition~\ref{prop:allvish1}, I made the following
229conjecture at a talk at MSRI in August 2000.
230\begin{conjecture}[Stein]\label{conj:allvish1}
231Suppose $x \in C(K)$ and $m\mid \order(x)$.  Then there exists an
232exact sequence $0\to A\to B \to C\to 0$ such that $x$ is $m$-visible
233in $\H^1(K,A)$.
234\end{conjecture}
235
236\subsection{Visibility for Elliptic Curves over $\Q$}
237The following theorem provides evidence for the conjecture in general.
238\begin{theorem}
239Let $C$ be an elliptic curve over~$\Q$.  Then
240Conjecture~\ref{conj:allvish1} is true when~$m$ a prime power.
241\end{theorem}
242\begin{proof}
243Suppose $m$ is a power of a prime~$p$.  Let $\Q_{\infty}$ be the
244cyclotomic $\Z_p$ extension of $\Q$, so $\Q_{\infty}$ is the Galois
245subfield of $\Q(\zeta_{p^n}, n\geq 1)$ of index $p-1$.  By
246\cite{breuil-conrad-diamond-taylor},~$C$ is a modular elliptic curve.
247Rohrlich \cite{rohrlich:cyclo} proved that all but finitely many
248special values $L(C,\chi,1)$ are nonzero, where $\chi$ varies over
249Dirichlet characters of $p$-power order.  Kato recently proved using
250his Euler system (see, e.g., \cite{scholl:kato}) that if
251$L(C,\chi,1)\neq 0$, then the $\chi$ part of $C(\Q)\tensor\Q$ is $0$.
252Combining these two results, we see that $C(\Q_\infty)$ is finitely
253generated.
254
255Because $C(\Q_\infty)$ is finitely generated, there is an integer~$n$
256such that $C(\Q_{\infty}) = C(\Q_n)$.    Let
257$258 B= \Res_{\Q_n/\Q}(C_{\Q_n}). 259$
260Then trace induces an exact sequence
261$262 0 \to A \to B \xrightarrow{f} C\to 0, 263$
264with~$A$ an abelian variety.  Then for any integer $j\geq n$ we have
265\begin{align*}
266 \Im\left(\delta:C(\Q)\to\H^1(\Q,A)\right) &\isom C(\Q)/f(B(\Q)) \\
267 &= C(\Q)/\Tr_{\Q_j/\Q}(C(\Q_j)) \\
268 & = C(\Q) / p^{j-n} \Tr_{\Q_n/\Q}(C(\Q_{n})) \\
269 & \onto C(\Q) / p^{j-n} C(\Q),
270\end{align*}
271where the last map is a surjection since
272$273 \Tr_{\Q_n/\Q}(C(\Q_{n})) \subset C(\Q). 274$
275Suppose $x\in C(\Q)$ has order divisible by $m=p^r$. Then for $j$
276sufficiently large the image of $x$ in $C(\Q)/p^{j-n} C(\Q)$ will have
277order order divisible by~$m$, which proves the theorem.
278\end{proof}
279
280\begin{remark}
281This theorem is probably true with the same proof with $C$ replaced by
282any modular abelian variety over~$\Q$, i.e., quotient of $J_1(N)$.
283However, I'm not certain the details of the relevant theorems by Kato
284and Rohrlich have all been written down in this more general
285case. Also, one should investigate conjectures of Mazur about finite
286generatedness of $C(\Q_\infty)$ for general $C$ (see \cite{mazur:towers}).
287\end{remark}
288
289
290\subsection{Visibility of Mordell-Weil in Shafarevich-Tate Groups}
291Let $0\to A \to B\to \C\to 0$ be an exact sequence of abelian
292varieties.
293\begin{definition}
294 Let $x \in C(K)$ and suppose~$m\in\Z_{>0}$ is a divisor of
295 $\order(x)$.  Then~$x$ is {\em $m$-visible in $\Sha(A)$} if
296 $\delta(x)\in\Sha(A)$ and the order of $\delta(x)\in \H^1(K,A)$ is
297 divisible by~$m$.
298\end{definition}
299
300The following conjecture strengthens Conjecture~\ref{conj:allvish1}.
301\begin{conjecture}[Stein]\label{conj:allvissha}
302Suppose $x \in C(K)$ and $m\mid \order(x)$.  Then there exists an
303exact sequence $0\to A\to B \to C\to 0$ such that~$x$ is $m$-visible
304in $\Sha(A)$.
305\end{conjecture}
306
307\subsubsection{Spiced Up Version of the Conjecture}
308We spice the conjecture up a little by requiring in addition that $A$
309be modular and $L(A,1)\neq 0$, motivated by the fact that this is the
310most general class of abelian varieties for which $\Sha(A)$ is known
311to be finite (by work of Kato).
312
313\begin{conjecture}[Stein]\label{conj:strongvismssha}
314Suppose $C$ is a modular abelian variety (i.e., $C$ is a quotient of
315$J_1(N)$ for some~$N$).  Suppose $x \in C(K)$ and $m\mid \order(x)$.
316Then there exists a modular abelian variety~$A$ with $L(A,1)\neq 0$
317and an exact sequence $0\to A\to B \to C\to 0$ such that~$x$ is
318$m$-visible in $\Sha(A)$.
319\end{conjecture}
320
321We offer the following evidence for the conjecture, which I prove
322in \cite{stein:nonsquaresha}.
323\begin{theorem}
324Let $C$ be the rank~$1$ elliptic curve $y(y+1)=x(x-1)(x+1)$ of
325conductor $37$, and let $x$ be a generator of $C(\Q)$.
326Then for all {\em primes} $m<25000$ with $m\neq 2, 37$,
327Conjecture~\ref{conj:strongvismssha} is true.
328\end{theorem}
329Let $f=\sum a_n q^n$ be the newform associated to~$C$.  Suppose $m$ is
330one of the primes in the theorem.  Then there exists a surjective
331Dirichlet character $\chi:(\Z/\ell\Z)^*\to \mu_m$ such that
332$L(f\tensor\chi, 1)\neq 0$.  Moreover, the $A$ of the theorem is the
333(up to isogeny) abelian variety $A_{f\tensor\chi}$ associated to
334$f\tensor\chi$ by Shimura, which has dimension $m-1$.
335
336\subsubsection{Nonsquare Shafarevich-Tate Groups}
337A surprising observation that comes out of the proof is that
338$339 \# \Sha(A) = m \cdot \text{(perfect square)}, 340$
341so we obtain the first ever examples of abelian varieties whose
342Shafarevich-Tate groups have order neither a square nor twice a
343square.
344
345\bibliographystyle{amsalpha} \bibliography{biblio}
346
347
348\end{document}