Visibility Theory

We briefly recall visibility theory, which we will use to construct elements of Shafarevich-Tate groups.

Definition 3.12   Let $\iota:A\hookrightarrow J$ be an embedding of abelian varities over  $\mathbf{Q}$. The visible subgroup of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ with respect to the embedding $\iota$ is

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

The following is a special case of Theorem 3.1 of [AS02b].

Theorem 3.13   Let $A$ and $B$ be abelian subvarieties of an abelian variety $J$ over  $\mathbf{Q}$ such that $A(\overline{\mathbf{Q}})\cap B(\overline{\mathbf{Q}})$ is finite. Let $N$ be an integer divisible by the residue characteristics of primes of bad reduction for $J$ (e.g., the conductor of $J$). Suppose $p$ is a prime such that

$$p\nmid 2\cdot N \cdot \char93 (J/B)(\mathbf{Q})_{\tor}\cdot\char93 B(\mathbf{Q})_{\tor}\cdot \prod_{\ell} c_{A,\ell}\cdot c_{B,\ell},$$

where $c_{A,\ell} = \char93 \Phi_{A,\ell}(\mathbf{F}_\ell)$ (resp., $c_{B,\ell}$) is the Tamagawa number of $A$ (resp., $B$) at $\ell$. Suppose furthermore that $B[p](\overline{\mathbf{Q}}) \subset A(\overline{\mathbf{Q}})$ as subgroups of $J(\overline{\mathbf{Q}})$. Then there is a natural map

$$\varphi :B(\mathbf{Q})/pB(\mathbf{Q})\rightarrow \Vis_J({\mbox{{\... ...ncoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A))$$

such that $\dim_{\mathbf{F}_p} \ker(\varphi ) \leq \dim_{\mathbf{Q}} A(\mathbf{Q})\otimes \mathbf{Q}$.