Visibility Theory

Visibility theory has been developed by Barry Mazur, Amod Agashe, and myself, with periodic help from Brian Conrad.

Let $A\hookrightarrow J$ be a closed immersion of abelian varieties. Then

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

Theorem 2.1   Suppose $A, B\subset J$, and $(A\cap B)(\overline{\mathbb{Q}})$ is finite. If $p$ is a prime such that $B[p]\subset A$ and

$$p\nmid 2\cdot N_J \cdot \char93 (J/B)(\mathbb{Q})_{\tor}\cdot \ch... ...i_{A,\ell}(\mathbb{F}_\ell)\cdot\char93 \Phi_{B,\ell}(\mathbb{F}_\ell)\right),$$

then

$$B(\mathbb{Q})/p B(\mathbb{Q})\cong \Vis_J({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)[p]).$$

For the proof, look at [Agashe-Stein, Visibility of Shafarevich-Tate Groups of Abelian Varieties]. It uses the snake lemma, and a careful local analysis at each prime that uses standard arithmetic geometry tools.