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Construction of Visible Elements

The goal of this section is to state and prove the main result of this paper, which we use to construct visible elements of Shafarevich-Tate groups and sometimes give a nontrivial lower bound for the order of the Shafarevich-Tate group of an abelian variety, thus providing new evidence for the conjecture of Birch and Swinnerton-Dyer (see Section 4.1 and [AS02]). The Tamagawa numbers $ c_{A,v}$ and $ c_{B,v}$ will be defined in Section 3.1 below.

Theorem 3.1   Let $ A$ and $ B$ be abelian subvarieties of an abelian variety $ J$ over a number field $ K$ such that $ A\cap B$ is finite. Let $ N$ be an integer divisible by the residue characteristics of primes of bad reduction for $ B$. Suppose $ n$ is an integer such that for each prime $ p\mid n$, we have $ e_p<p-1$ where $ e_p$ is the largest ramification of any prime of $ K$ lying over $ p$, and that

$\displaystyle \gcd\left(n, \,\,N \cdot \char93 (J/B)(K)_{\tor}\cdot\char93 B(K)... ...t \prod_{\text{\rm all places $v$}} \left(c_{A,v}\cdot c_{B,v}\right)\right)=1,$

where $ c_{A,v} = \char93 \Phi_{A,v}(\mathbf{F}_\ell)$ (resp., $ c_{B,\ell}$) is the Tamagawa number of $ A$ (resp., $ B$) at $ v$ (see Section 3.1 for the definition of $ \Phi_{A,v}$). Suppose furthermore that $ B[n] \subset A$ as subgroup schemes of $ J$. Then there is a natural map

$\displaystyle \varphi :B(K)/nB(K)\rightarrow \Vis_J({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)), $

such that $ \ker(\varphi )\subset J(K)/(B(K)+A(K))$. If $ A(K)$ has rank 0, then $ \ker(\varphi )=0$ (more generally, $ \ker(\varphi )$ has order at most $ n^r$ where $ r$ is the rank of $ A(K)$).

Remark 3.2   Mazur has proved similar results for elliptic curves using flat cohomology (unpublished), and discussions with him motivated this theorem.

In Section 3.1 we recall a definition of the Tamagawa numbers of an abelian variety. In Section 3.2 we prove a lemma, which gives a condition under which there is an unramified $ n$th root of an unramified point. In Section 3.3, we use the snake lemma to produce a map

$\displaystyle B(K)/n B(K)\hookrightarrow \Vis_J(H^1(K,A)) $

with bounded kernel. Finally, in Section 3.4, we use a local analysis at each place of $ K$ to show that the image of the above map lies in $ {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$.


Subsections
next up previous
Next: Tamagawa Numbers Up: Visibility of Shafarevich-Tate Groups Previous: The Visibility Dimension for
William A Stein 2002-02-27