Sharedwww / cuny / vismw.texOpen in CoCalc
Author: William A. Stein
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\documentclass[11pt]{article}
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\usepackage[active]{srcltx}
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\title{\huge\sf Visibility of Galois Cohomology and\\ Mordell-Weil Groups}
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\date{November 20, 2003}
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\author{William Stein}
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\begin{document}
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\maketitle
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\tableofcontents
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\section{Visibility of $\H^1(K,A)$}
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Let $K$ be a number field. (There should be a similar theory for
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function fields over a finite field.)
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\subsection{Motivation and Philosophy}
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Suppose
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\[
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0 \to A \to B \to C \to 0
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\]
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is an exact sequence of abelian varieties over~$K$. (Thus each
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of~$A$,~$B$, and~$C$ is a complete group variety over~$K$, whose group
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is automatically abelian.) Then there is a corresponding long exact
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sequence of cohomology for the group $\Gal(\Qbar/K)$:
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\[
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0 \to A(K) \to B(K) \to C(K) \to \H^1(K,A) \to \H^1(K,B)\to
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\H^1(K,C) \to \cdots
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\]
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The study of the Mordell-Weil group $C(K)=\H^0(K,C)$ is popular in
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arithmetic geometry. For example, the Birch and Swinnerton-Dyer
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conjecture (BSD conjecture), which is one of the million dollar Clay
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Math Problems, asserts that the dimension of $C(K)\tensor\Q$ equals
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the ordering vanishing of $L(C,s)$ at $s=1$.
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The group $\H^1(K,A)$ is also of interest in connection with
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the BSD conjecture, because it contains
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the Shafarevich-Tate group
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\[
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\Sha(A) = \Sha(A/K) =
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\Ker\left(\H^1(K,A)\to \bigoplus_{v} \H^1(K_v,A)\right) \subset \H^1(K,A),
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\]
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where the sum is over all places~$v$ of~$K$ (e.g., when $K=\Q$, the
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fields $K_v$ are $\Q_p$ for all prime numbers~$p$ and
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$\Q_{\infty}=\R$).
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The group $A(K)$ is {\em fundamentally different} than $\H^1(K,C)$. The
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Mordell-Weil group $A(K)$ is finitely generated, whereas the first
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Galois cohomology $\H^1(K,C)$ is far from being finitely
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generated---in fact, every element has finite order and there are
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infinitely many elements of any given order.
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This talk is about ``dimension shifting'', i.e., relating information
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about $\H^0(K,C)$ to information about $\H^1(K,A)$.
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\subsection{Definitions}
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\subsubsection{What are Elements of Galois Cohomology?}
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Elements of $\H^0(K,C)$ are simply points, i.e., elements of $C(K)$,
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so they are relatively easy to ``visualize''. In contrast, elements
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of $\H^1(K, A)$ are Galois cohomology classes, i.e., equivalence
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classes of set-theoretic (continuous) maps $f:\Gal(\Qbar/K)\to
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A(\Qbar)$ such that $f(\sigma\tau) = f(\sigma) + \sigma f(\tau)$. Two
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maps are equivalent if their difference is a map of the form
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$\sigma\mapsto \sigma(P)-P$ for some fixed $P\in A(\Qbar)$. From this
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point of view $\H^1$ is more mysterious than $\H^0$.
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\subsubsection{Principal Homogeneous Spaces}
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There is an alternative way to view elements of $\H^1(K,A)$. The WC
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group of~$A$ is the group of isomorphism classes of principal
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homogeneous spaces for~$A$, where a principal homogeneous space is a
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variety~$X$ and a map $A\times X\to X$ that satisfies the same axioms
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as those for a simply transitive group action. Thus~$X$ is a twist as
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variety of~$A$, but $X(K)=\emptyset$, unless~$X\ncisom A$. Also, the
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nontrivial elements of $\Sha(A)$ correspond to the classes of~$X$ that
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have a $K_v$-rational point for all places~$v$, but no $K$-rational
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point.
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\subsubsection{Visibility of $\H^1(K,A)$}
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Barry Mazur introduced the following definition in order to help unify
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diverse constructions of principal homogeneous spaces:
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\begin{definition}
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The {\em visible subgroup} of $\H^1(K,A)$ in~$B$ is
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\begin{align*}
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\Vis_B\H^1(K,A) &= \Ker(\H^1(K,A)\to \H^1(K,B))\\
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&= \Coker(B(K)\to C(K)).
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\end{align*}
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\end{definition}
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\begin{remark}
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Note that $\Vis_B\H^1(K,A)$ {\em does} depend on the embedding of~$A$
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into~$B$. For example, suppose $B=B_1\times A$. Then there could be
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nonzero visible elements if~$A$ is embedding into the first factor,
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but there will be no nonzero visible elements if~$A$ is embedded into
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the second factor. Here we are using that $\H^1(K, B_1\times A) =
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\H^1(K,B_1)\oplus \H^1(K,A).$
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\end{remark}
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The connection with the WC group of~$A$ is as follows.
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Suppose
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\[
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0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0
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\]
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is an exact sequence of abelian varieties and that $c\in \H^1(K,A)$ is
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visible in~$B$. Thus there exists $x\in C(K)$ such that $\delta(x) =
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c$. Then $X=\pi^{-1}(x)\subset B$ is a translate of $A$ in~$B$, so
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the group law on~$B$ gives~$X$ the structure of principal homogeneous
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space for~$A$, and one can show that the class of~$X$ in the WC group
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of~$A$ corresponds to~$c$.
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\subsubsection{Finiteness of the Visible Subgroup}
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\begin{lemma}
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The group $\Vis_B\H^1(K,A)$ is finite.
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\end{lemma}
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\begin{proof}
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By the Mordell-Weil theorem $C(K)$ is finitely generated. The group
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$\Vis_B\H^1(K,A)$ is a homomorphic image of $C(K)$ so it is finitely
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generated. On the other hand, it is a subgroup of $\H^1(K,A)$, so it
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is a torsion group. The lemma follows since a finitely
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generated torsion abelian group is finite.
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\end{proof}
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\subsection{Every Element of $\H^1(K,A)$ is Visible Somewhere}
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\begin{proposition}\label{prop:allvish1}
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Let $c\in\H^1(K,A)$. Then there exists an abelian variety~$B=B_c$ and
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an embedding $A\hra B$ such that~$c$ is visible in~$B$. Moreover,~$B$
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can be chosen to be a twist of a power of~$A$.
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\end{proposition}
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\begin{proof}
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By definition of Galois cohomology, there is a finite extension~$L$
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of~$K$ such that $\res_L(c)=0$. Thus $c$ maps to $0$ in
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$\H^1(L,A_L)$. By a slight generalization of the Shapiro Lemma from
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group cohomology (which is proved by dimension shifting; see, e.g.,
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Atiyah-Wall in Cassels-Frohlich), there is a canonical isomorphism
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\[
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\H^1(L,A_L) \isom \H^1(K,\Res_{L/K}(A_L)) = \H^1(K,B),
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\]
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where $B=\Res_{L/K}(A_L)$ is the Weil restriction of scalars of $A_L$
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back down to~$K$. The restriction of scalars~$B$ is an abelian
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variety of dimension $[L:K]\cdot \dim A$ that is characterized by the
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existence of functorial isomorphisms
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\[
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\Mor_K(S,B) \isom \Mor_L(S_L, A_L),
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\]
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for any $K$-scheme~$S$, i.e., $B(S)=A_L(S_L)$. In particular,
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setting~$S=A$ we find that the identity map $A_L\to A_L$ corresponds
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to an injection $A\hra B$. Moreover, $c\mapsto\res_L(c)=0\in\H^1(K,B)$.
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The assertion about the structure of~$B$ follows from general facts
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about restriction of scalars, which won't be proved here.
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\end{proof}
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\subsection{Other Results in the Context of Modularity}
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Usually one focuses on visibility of elements in $\Sha(A)$. There are
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a number of other results about visibility in various special cases,
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and large tables of examples in the context of elliptic curves and
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modular abelian varieties. There are also interesting modularity
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questions/conjectures in this context. I will not go into these
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further right now, except to note one example.
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Motivated by the notion of visibility, I developed (with input from
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Mazur, Cremona, and Agashe) computational techniques for
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unconditionally constructing Shafarevich-Tate groups of modular
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abelian varieties $A\subset J_1(N)$. For example, if
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$A\subset J_0(389)$ is the $20$-dimensional simple factor, then
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\[
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\Z/5\Z\times \Z/5\Z\subset \Sha(A),
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\]
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as predicted by the Birch and Swinnerton-Dyer conjecture. I found a
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few dozen other examples like this, where the computational
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construction of the Shafarevich-Tate group would be hopeless using any
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other known technique. See \cite{agashe-stein:bsd,
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agashe-stein:visibility} for more details, and \cite{cremona-mazur}
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for examples when $\dim A=1$.
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\section{Visibility of Mordell-Weil Groups}
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\subsection{Motivation and Philosophy}
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The previous section was about understanding elements of $\H^1$ in
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terms of Mordell-Weil groups. The BSD conjecture implies the
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following conjecture:
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\begin{conjecture}\label{conj:inf}
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If $L(C,1)=0$, then $C(\Q)$ is infinite.
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\end{conjecture}
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We know by the Gross-Zagier formula that if~$C$ is an elliptic curves
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over~$\Q$ and $\ord_{s=1} L(C,s)=1$, then $C(\Q)$ is infinite, but
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little more is known toward Conjecture~\ref{conj:inf}. More
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generally, the conjecture is known when $C\subset J_0(N)$ and
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$\ord_{s=1} L(C,s) = \dim(C)$, and there are other results over
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totally real number fields. People also seem to have a reasonable
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(but not good enough!) understanding of $\Sha(C)$ when $L(C,1)\neq
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0$.
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\subsubsection{Rank $>1$: A New Idea is Needed}
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Suppose~$C$ is an elliptic curve over~$\Q$ and $\ord_{s=1} L(C,s)=2$.
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Conjecture~\ref{conj:inf} asserts that $C(\Q)$ is infinite, but this
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is currently a difficult open problem. Nick Katz told me at dinner
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once that ``a new idea is needed.'' It seems that nobody knows a good
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analogue of Gross-Zagier for rank two elliptic curves. (I've noticed
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that Mazur has been working on this question, in one way or another,
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since I've been at Harvard...)
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Visibility of Mordell-Weil groups is an idea I came up which might
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have some relevance.
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\subsection{Definition}
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Suppose
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\[
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0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0
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\]
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is an exact sequence of abelian varieties over a number field~$K$,
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with corresponding long exact sequence
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\[
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0 \to A(K) \to B(K) \to C(K) \xrightarrow{\delta} \H^1(K,A) \to\cdots
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\]
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of $\Gal(\Qbar/K)$-cohomology.
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\begin{definition}
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Let $x \in C(K)$ and suppose~$m\in\Z_{>0}$ is a divisor of $\order(x)$
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(everything divides $\infty$). Then~$x$ is {\em $m$-visible in
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$\H^1(K,A)$} if the order of $\delta(x)\in \H^1(K,A)$ is divisible
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by~$m$.
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\end{definition}
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Motivated by Proposition~\ref{prop:allvish1}, I made the following
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conjecture at a talk at MSRI in August 2000.
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\begin{conjecture}[Stein]\label{conj:allvish1}
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Suppose $x \in C(K)$ and $m\mid \order(x)$. Then there exists an
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exact sequence $0\to A\to B \to C\to 0$ such that $x$ is $m$-visible
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in $\H^1(K,A)$.
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\end{conjecture}
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\subsection{Visibility for Elliptic Curves over $\Q$}
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The following theorem provides evidence for the conjecture in general.
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\begin{theorem}
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Let $C$ be an elliptic curve over~$\Q$. Then
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Conjecture~\ref{conj:allvish1} is true when~$m$ a prime power.
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\end{theorem}
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\begin{proof}
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Suppose $m$ is a power of a prime~$p$. Let $\Q_{\infty}$ be the
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cyclotomic $\Z_p$ extension of $\Q$, so $\Q_{\infty}$ is the Galois
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subfield of $\Q(\zeta_{p^n}, n\geq 1)$ of index $p-1$. By
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\cite{breuil-conrad-diamond-taylor},~$C$ is a modular elliptic curve.
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Rohrlich \cite{rohrlich:cyclo} proved that all but finitely many
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special values $L(C,\chi,1)$ are nonzero, where $\chi$ varies over
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Dirichlet characters of $p$-power order. Kato recently proved using
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his Euler system (see, e.g., \cite{scholl:kato}) that if
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$L(C,\chi,1)\neq 0$, then the $\chi$ part of $C(\Q)\tensor\Q$ is $0$.
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Combining these two results, we see that $C(\Q_\infty)$ is finitely
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generated.
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Because $C(\Q_\infty)$ is finitely generated, there is an integer~$n$
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such that $C(\Q_{\infty}) = C(\Q_n)$. Let
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\[
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B= \Res_{\Q_n/\Q}(C_{\Q_n}).
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\]
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Then trace induces an exact sequence
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\[
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0 \to A \to B \xrightarrow{f} C\to 0,
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\]
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with~$A$ an abelian variety. Then for any integer $j\geq n$ we have
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\begin{align*}
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\Im\left(\delta:C(\Q)\to\H^1(\Q,A)\right) &\isom C(\Q)/f(B(\Q)) \\
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&= C(\Q)/\Tr_{\Q_j/\Q}(C(\Q_j)) \\
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& = C(\Q) / p^{j-n} \Tr_{\Q_n/\Q}(C(\Q_{n})) \\
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& \onto C(\Q) / p^{j-n} C(\Q),
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\end{align*}
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where the last map is a surjection since
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\[
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\Tr_{\Q_n/\Q}(C(\Q_{n})) \subset C(\Q).
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\]
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Suppose $x\in C(\Q)$ has order divisible by $m=p^r$. Then for $j$
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sufficiently large the image of $x$ in $C(\Q)/p^{j-n} C(\Q)$ will have
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order order divisible by~$m$, which proves the theorem.
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\end{proof}
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\begin{remark}
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This theorem is probably true with the same proof with $C$ replaced by
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any modular abelian variety over~$\Q$, i.e., quotient of $J_1(N)$.
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However, I'm not certain the details of the relevant theorems by Kato
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and Rohrlich have all been written down in this more general
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case. Also, one should investigate conjectures of Mazur about finite
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generatedness of $C(\Q_\infty)$ for general $C$ (see \cite{mazur:towers}).
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\end{remark}
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\subsection{Visibility of Mordell-Weil in Shafarevich-Tate Groups}
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Let $0\to A \to B\to \C\to 0$ be an exact sequence of abelian
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varieties.
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\begin{definition}
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Let $x \in C(K)$ and suppose~$m\in\Z_{>0}$ is a divisor of
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$\order(x)$. Then~$x$ is {\em $m$-visible in $\Sha(A)$} if
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$\delta(x)\in\Sha(A)$ and the order of $\delta(x)\in \H^1(K,A)$ is
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divisible by~$m$.
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\end{definition}
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The following conjecture strengthens Conjecture~\ref{conj:allvish1}.
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\begin{conjecture}[Stein]\label{conj:allvissha}
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Suppose $x \in C(K)$ and $m\mid \order(x)$. Then there exists an
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exact sequence $0\to A\to B \to C\to 0$ such that~$x$ is $m$-visible
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in $\Sha(A)$.
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\end{conjecture}
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\subsubsection{Spiced Up Version of the Conjecture}
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We spice the conjecture up a little by requiring in addition that $A$
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be modular and $L(A,1)\neq 0$, motivated by the fact that this is the
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most general class of abelian varieties for which $\Sha(A)$ is known
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to be finite (by work of Kato).
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\begin{conjecture}[Stein]\label{conj:strongvismssha}
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Suppose $C$ is a modular abelian variety (i.e., $C$ is a quotient of
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$J_1(N)$ for some~$N$). Suppose $x \in C(K)$ and $m\mid \order(x)$.
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Then there exists a modular abelian variety~$A$ with $L(A,1)\neq 0$
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and an exact sequence $0\to A\to B \to C\to 0$ such that~$x$ is
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$m$-visible in $\Sha(A)$.
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\end{conjecture}
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We offer the following evidence for the conjecture, which I prove
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in \cite{stein:nonsquaresha}.
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\begin{theorem}
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Let $C$ be the rank~$1$ elliptic curve $y(y+1)=x(x-1)(x+1)$ of
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conductor $37$, and let $x$ be a generator of $C(\Q)$.
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Then for all {\em primes} $m<25000$ with $m\neq 2, 37$,
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Conjecture~\ref{conj:strongvismssha} is true.
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\end{theorem}
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Let $f=\sum a_n q^n$ be the newform associated to~$C$. Suppose $m$ is
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one of the primes in the theorem. Then there exists a surjective
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Dirichlet character $\chi:(\Z/\ell\Z)^*\to \mu_m$ such that
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$L(f\tensor\chi, 1)\neq 0$. Moreover, the $A$ of the theorem is the
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(up to isogeny) abelian variety $A_{f\tensor\chi}$ associated to
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$f\tensor\chi$ by Shimura, which has dimension $m-1$.
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\subsubsection{Nonsquare Shafarevich-Tate Groups}
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A surprising observation that comes out of the proof is that
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\[
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\# \Sha(A) = m \cdot \text{(perfect square)},
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\]
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so we obtain the first ever examples of abelian varieties whose
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Shafarevich-Tate groups have order neither a square nor twice a
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square.
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\bibliographystyle{amsalpha} \bibliography{biblio}
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\end{document}