\documentclass[11pt]{article}1\usepackage[active]{srcltx}2\include{macros}34\title{\huge\sf Visibility of Galois Cohomology and\\ Mordell-Weil Groups}56\date{November 20, 2003}78\author{William Stein}9\begin{document}10\maketitle11\tableofcontents1213\section{Visibility of $\H^1(K,A)$}14Let $K$ be a number field. (There should be a similar theory for15function fields over a finite field.)16\subsection{Motivation and Philosophy}17Suppose18\[190 \to A \to B \to C \to 020\]21is an exact sequence of abelian varieties over~$K$. (Thus each22of~$A$,~$B$, and~$C$ is a complete group variety over~$K$, whose group23is automatically abelian.) Then there is a corresponding long exact24sequence of cohomology for the group $\Gal(\Qbar/K)$:25\[260 \to A(K) \to B(K) \to C(K) \to \H^1(K,A) \to \H^1(K,B)\to27\H^1(K,C) \to \cdots28\]2930The study of the Mordell-Weil group $C(K)=\H^0(K,C)$ is popular in31arithmetic geometry. For example, the Birch and Swinnerton-Dyer32conjecture (BSD conjecture), which is one of the million dollar Clay33Math Problems, asserts that the dimension of $C(K)\tensor\Q$ equals34the ordering vanishing of $L(C,s)$ at $s=1$.3536The group $\H^1(K,A)$ is also of interest in connection with37the BSD conjecture, because it contains38the Shafarevich-Tate group39\[40\Sha(A) = \Sha(A/K) =41\Ker\left(\H^1(K,A)\to \bigoplus_{v} \H^1(K_v,A)\right) \subset \H^1(K,A),42\]43where the sum is over all places~$v$ of~$K$ (e.g., when $K=\Q$, the44fields $K_v$ are $\Q_p$ for all prime numbers~$p$ and45$\Q_{\infty}=\R$).4647The group $A(K)$ is {\em fundamentally different} than $\H^1(K,C)$. The48Mordell-Weil group $A(K)$ is finitely generated, whereas the first49Galois cohomology $\H^1(K,C)$ is far from being finitely50generated---in fact, every element has finite order and there are51infinitely many elements of any given order.5253This talk is about ``dimension shifting'', i.e., relating information54about $\H^0(K,C)$ to information about $\H^1(K,A)$.5556\subsection{Definitions}57\subsubsection{What are Elements of Galois Cohomology?}58Elements of $\H^0(K,C)$ are simply points, i.e., elements of $C(K)$,59so they are relatively easy to ``visualize''. In contrast, elements60of $\H^1(K, A)$ are Galois cohomology classes, i.e., equivalence61classes of set-theoretic (continuous) maps $f:\Gal(\Qbar/K)\to62A(\Qbar)$ such that $f(\sigma\tau) = f(\sigma) + \sigma f(\tau)$. Two63maps are equivalent if their difference is a map of the form64$\sigma\mapsto \sigma(P)-P$ for some fixed $P\in A(\Qbar)$. From this65point of view $\H^1$ is more mysterious than $\H^0$.6667\subsubsection{Principal Homogeneous Spaces}68There is an alternative way to view elements of $\H^1(K,A)$. The WC69group of~$A$ is the group of isomorphism classes of principal70homogeneous spaces for~$A$, where a principal homogeneous space is a71variety~$X$ and a map $A\times X\to X$ that satisfies the same axioms72as those for a simply transitive group action. Thus~$X$ is a twist as73variety of~$A$, but $X(K)=\emptyset$, unless~$X\ncisom A$. Also, the74nontrivial elements of $\Sha(A)$ correspond to the classes of~$X$ that75have a $K_v$-rational point for all places~$v$, but no $K$-rational76point.7778\subsubsection{Visibility of $\H^1(K,A)$}79Barry Mazur introduced the following definition in order to help unify80diverse constructions of principal homogeneous spaces:81\begin{definition}82The {\em visible subgroup} of $\H^1(K,A)$ in~$B$ is83\begin{align*}84\Vis_B\H^1(K,A) &= \Ker(\H^1(K,A)\to \H^1(K,B))\\85&= \Coker(B(K)\to C(K)).86\end{align*}87\end{definition}8889\begin{remark}90Note that $\Vis_B\H^1(K,A)$ {\em does} depend on the embedding of~$A$91into~$B$. For example, suppose $B=B_1\times A$. Then there could be92nonzero visible elements if~$A$ is embedding into the first factor,93but there will be no nonzero visible elements if~$A$ is embedded into94the second factor. Here we are using that $\H^1(K, B_1\times A) =95\H^1(K,B_1)\oplus \H^1(K,A).$96\end{remark}979899100The connection with the WC group of~$A$ is as follows.101Suppose102\[1030 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0104\]105is an exact sequence of abelian varieties and that $c\in \H^1(K,A)$ is106visible in~$B$. Thus there exists $x\in C(K)$ such that $\delta(x) =107c$. Then $X=\pi^{-1}(x)\subset B$ is a translate of $A$ in~$B$, so108the group law on~$B$ gives~$X$ the structure of principal homogeneous109space for~$A$, and one can show that the class of~$X$ in the WC group110of~$A$ corresponds to~$c$.111112\subsubsection{Finiteness of the Visible Subgroup}113\begin{lemma}114The group $\Vis_B\H^1(K,A)$ is finite.115\end{lemma}116\begin{proof}117By the Mordell-Weil theorem $C(K)$ is finitely generated. The group118$\Vis_B\H^1(K,A)$ is a homomorphic image of $C(K)$ so it is finitely119generated. On the other hand, it is a subgroup of $\H^1(K,A)$, so it120is a torsion group. The lemma follows since a finitely121generated torsion abelian group is finite.122\end{proof}123124\subsection{Every Element of $\H^1(K,A)$ is Visible Somewhere}125126\begin{proposition}\label{prop:allvish1}127Let $c\in\H^1(K,A)$. Then there exists an abelian variety~$B=B_c$ and128an embedding $A\hra B$ such that~$c$ is visible in~$B$. Moreover,~$B$129can be chosen to be a twist of a power of~$A$.130\end{proposition}131\begin{proof}132By definition of Galois cohomology, there is a finite extension~$L$133of~$K$ such that $\res_L(c)=0$. Thus $c$ maps to $0$ in134$\H^1(L,A_L)$. By a slight generalization of the Shapiro Lemma from135group cohomology (which is proved by dimension shifting; see, e.g.,136Atiyah-Wall in Cassels-Frohlich), there is a canonical isomorphism137\[138\H^1(L,A_L) \isom \H^1(K,\Res_{L/K}(A_L)) = \H^1(K,B),139\]140where $B=\Res_{L/K}(A_L)$ is the Weil restriction of scalars of $A_L$141back down to~$K$. The restriction of scalars~$B$ is an abelian142variety of dimension $[L:K]\cdot \dim A$ that is characterized by the143existence of functorial isomorphisms144\[145\Mor_K(S,B) \isom \Mor_L(S_L, A_L),146\]147for any $K$-scheme~$S$, i.e., $B(S)=A_L(S_L)$. In particular,148setting~$S=A$ we find that the identity map $A_L\to A_L$ corresponds149to an injection $A\hra B$. Moreover, $c\mapsto\res_L(c)=0\in\H^1(K,B)$.150151The assertion about the structure of~$B$ follows from general facts152about restriction of scalars, which won't be proved here.153\end{proof}154155\subsection{Other Results in the Context of Modularity}156Usually one focuses on visibility of elements in $\Sha(A)$. There are157a number of other results about visibility in various special cases,158and large tables of examples in the context of elliptic curves and159modular abelian varieties. There are also interesting modularity160questions/conjectures in this context. I will not go into these161further right now, except to note one example.162163Motivated by the notion of visibility, I developed (with input from164Mazur, Cremona, and Agashe) computational techniques for165unconditionally constructing Shafarevich-Tate groups of modular166abelian varieties $A\subset J_1(N)$. For example, if167$A\subset J_0(389)$ is the $20$-dimensional simple factor, then168\[169\Z/5\Z\times \Z/5\Z\subset \Sha(A),170\]171as predicted by the Birch and Swinnerton-Dyer conjecture. I found a172few dozen other examples like this, where the computational173construction of the Shafarevich-Tate group would be hopeless using any174other known technique. See \cite{agashe-stein:bsd,175agashe-stein:visibility} for more details, and \cite{cremona-mazur}176for examples when $\dim A=1$.177178\section{Visibility of Mordell-Weil Groups}179\subsection{Motivation and Philosophy}180The previous section was about understanding elements of $\H^1$ in181terms of Mordell-Weil groups. The BSD conjecture implies the182following conjecture:183\begin{conjecture}\label{conj:inf}184If $L(C,1)=0$, then $C(\Q)$ is infinite.185\end{conjecture}186187We know by the Gross-Zagier formula that if~$C$ is an elliptic curves188over~$\Q$ and $\ord_{s=1} L(C,s)=1$, then $C(\Q)$ is infinite, but189little more is known toward Conjecture~\ref{conj:inf}. More190generally, the conjecture is known when $C\subset J_0(N)$ and191$\ord_{s=1} L(C,s) = \dim(C)$, and there are other results over192totally real number fields. People also seem to have a reasonable193(but not good enough!) understanding of $\Sha(C)$ when $L(C,1)\neq1940$.195196\subsubsection{Rank $>1$: A New Idea is Needed}197Suppose~$C$ is an elliptic curve over~$\Q$ and $\ord_{s=1} L(C,s)=2$.198Conjecture~\ref{conj:inf} asserts that $C(\Q)$ is infinite, but this199is currently a difficult open problem. Nick Katz told me at dinner200once that ``a new idea is needed.'' It seems that nobody knows a good201analogue of Gross-Zagier for rank two elliptic curves. (I've noticed202that Mazur has been working on this question, in one way or another,203since I've been at Harvard...)204205Visibility of Mordell-Weil groups is an idea I came up which might206have some relevance.207208\subsection{Definition}209Suppose210\[2110 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0212\]213is an exact sequence of abelian varieties over a number field~$K$,214with corresponding long exact sequence215\[2160 \to A(K) \to B(K) \to C(K) \xrightarrow{\delta} \H^1(K,A) \to\cdots217\]218of $\Gal(\Qbar/K)$-cohomology.219220\begin{definition}221Let $x \in C(K)$ and suppose~$m\in\Z_{>0}$ is a divisor of $\order(x)$222(everything divides $\infty$). Then~$x$ is {\em $m$-visible in223$\H^1(K,A)$} if the order of $\delta(x)\in \H^1(K,A)$ is divisible224by~$m$.225\end{definition}226227Motivated by Proposition~\ref{prop:allvish1}, I made the following228conjecture at a talk at MSRI in August 2000.229\begin{conjecture}[Stein]\label{conj:allvish1}230Suppose $x \in C(K)$ and $m\mid \order(x)$. Then there exists an231exact sequence $0\to A\to B \to C\to 0$ such that $x$ is $m$-visible232in $\H^1(K,A)$.233\end{conjecture}234235\subsection{Visibility for Elliptic Curves over $\Q$}236The following theorem provides evidence for the conjecture in general.237\begin{theorem}238Let $C$ be an elliptic curve over~$\Q$. Then239Conjecture~\ref{conj:allvish1} is true when~$m$ a prime power.240\end{theorem}241\begin{proof}242Suppose $m$ is a power of a prime~$p$. Let $\Q_{\infty}$ be the243cyclotomic $\Z_p$ extension of $\Q$, so $\Q_{\infty}$ is the Galois244subfield of $\Q(\zeta_{p^n}, n\geq 1)$ of index $p-1$. By245\cite{breuil-conrad-diamond-taylor},~$C$ is a modular elliptic curve.246Rohrlich \cite{rohrlich:cyclo} proved that all but finitely many247special values $L(C,\chi,1)$ are nonzero, where $\chi$ varies over248Dirichlet characters of $p$-power order. Kato recently proved using249his Euler system (see, e.g., \cite{scholl:kato}) that if250$L(C,\chi,1)\neq 0$, then the $\chi$ part of $C(\Q)\tensor\Q$ is $0$.251Combining these two results, we see that $C(\Q_\infty)$ is finitely252generated.253254Because $C(\Q_\infty)$ is finitely generated, there is an integer~$n$255such that $C(\Q_{\infty}) = C(\Q_n)$. Let256\[257B= \Res_{\Q_n/\Q}(C_{\Q_n}).258\]259Then trace induces an exact sequence260\[2610 \to A \to B \xrightarrow{f} C\to 0,262\]263with~$A$ an abelian variety. Then for any integer $j\geq n$ we have264\begin{align*}265\Im\left(\delta:C(\Q)\to\H^1(\Q,A)\right) &\isom C(\Q)/f(B(\Q)) \\266&= C(\Q)/\Tr_{\Q_j/\Q}(C(\Q_j)) \\267& = C(\Q) / p^{j-n} \Tr_{\Q_n/\Q}(C(\Q_{n})) \\268& \onto C(\Q) / p^{j-n} C(\Q),269\end{align*}270where the last map is a surjection since271\[272\Tr_{\Q_n/\Q}(C(\Q_{n})) \subset C(\Q).273\]274Suppose $x\in C(\Q)$ has order divisible by $m=p^r$. Then for $j$275sufficiently large the image of $x$ in $C(\Q)/p^{j-n} C(\Q)$ will have276order order divisible by~$m$, which proves the theorem.277\end{proof}278279\begin{remark}280This theorem is probably true with the same proof with $C$ replaced by281any modular abelian variety over~$\Q$, i.e., quotient of $J_1(N)$.282However, I'm not certain the details of the relevant theorems by Kato283and Rohrlich have all been written down in this more general284case. Also, one should investigate conjectures of Mazur about finite285generatedness of $C(\Q_\infty)$ for general $C$ (see \cite{mazur:towers}).286\end{remark}287288289\subsection{Visibility of Mordell-Weil in Shafarevich-Tate Groups}290Let $0\to A \to B\to \C\to 0$ be an exact sequence of abelian291varieties.292\begin{definition}293Let $x \in C(K)$ and suppose~$m\in\Z_{>0}$ is a divisor of294$\order(x)$. Then~$x$ is {\em $m$-visible in $\Sha(A)$} if295$\delta(x)\in\Sha(A)$ and the order of $\delta(x)\in \H^1(K,A)$ is296divisible by~$m$.297\end{definition}298299The following conjecture strengthens Conjecture~\ref{conj:allvish1}.300\begin{conjecture}[Stein]\label{conj:allvissha}301Suppose $x \in C(K)$ and $m\mid \order(x)$. Then there exists an302exact sequence $0\to A\to B \to C\to 0$ such that~$x$ is $m$-visible303in $\Sha(A)$.304\end{conjecture}305306\subsubsection{Spiced Up Version of the Conjecture}307We spice the conjecture up a little by requiring in addition that $A$308be modular and $L(A,1)\neq 0$, motivated by the fact that this is the309most general class of abelian varieties for which $\Sha(A)$ is known310to be finite (by work of Kato).311312\begin{conjecture}[Stein]\label{conj:strongvismssha}313Suppose $C$ is a modular abelian variety (i.e., $C$ is a quotient of314$J_1(N)$ for some~$N$). Suppose $x \in C(K)$ and $m\mid \order(x)$.315Then there exists a modular abelian variety~$A$ with $L(A,1)\neq 0$316and an exact sequence $0\to A\to B \to C\to 0$ such that~$x$ is317$m$-visible in $\Sha(A)$.318\end{conjecture}319320We offer the following evidence for the conjecture, which I prove321in \cite{stein:nonsquaresha}.322\begin{theorem}323Let $C$ be the rank~$1$ elliptic curve $y(y+1)=x(x-1)(x+1)$ of324conductor $37$, and let $x$ be a generator of $C(\Q)$.325Then for all {\em primes} $m<25000$ with $m\neq 2, 37$,326Conjecture~\ref{conj:strongvismssha} is true.327\end{theorem}328Let $f=\sum a_n q^n$ be the newform associated to~$C$. Suppose $m$ is329one of the primes in the theorem. Then there exists a surjective330Dirichlet character $\chi:(\Z/\ell\Z)^*\to \mu_m$ such that331$L(f\tensor\chi, 1)\neq 0$. Moreover, the $A$ of the theorem is the332(up to isogeny) abelian variety $A_{f\tensor\chi}$ associated to333$f\tensor\chi$ by Shimura, which has dimension $m-1$.334335\subsubsection{Nonsquare Shafarevich-Tate Groups}336A surprising observation that comes out of the proof is that337\[338\# \Sha(A) = m \cdot \text{(perfect square)},339\]340so we obtain the first ever examples of abelian varieties whose341Shafarevich-Tate groups have order neither a square nor twice a342square.343344\bibliographystyle{amsalpha} \bibliography{biblio}345346347\end{document}348