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

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

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

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

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Remark 3.2   Mazur has proved similar results for elliptic curves using flat cohomology (unpublished), and discussions with him motivated this theorem.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

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

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 ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$.