Visibility Theory

Suppose $\iota: A\hookrightarrow{}R$ is a closed immersion of abelian varieties over a number field $L$. The visible subgroup of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/L)$ with respect to $\iota$ is

$$\Vis_R({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}... ...ding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(R/L)).$$

The following theorem can sometimes be used to prove that $\Vis_R({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/L))$ is nontrivial. We will use it later to construct nontrivial elements of Shafarevich-Tate groups in Section 6. For an alternative approach which uses étale cohomology, see Section 8.

Theorem 3.1   Let $A$ and $B$ be abelian subvarieties of an abelian variety $R$ over a number field $L$ such that $A\cap B$ is finite, and suppose that $A(L)$ is finite. Let $N$ be an integer divisible by the residue characteristics of primes of bad reduction for $R$. Suppose $p$ is a prime such that $e<p-1$, where $e$ is the largest ramification of any prime of $L$ lying over $p$, and that

$$p\nmid N \cdot \char93 (R/B)(L)\cdot\char93 B(L)_{\tor}\cdot \prod_{v} c_{A,v}\cdot c_{B,v},$$

where $c_{A,v} = \char93 \Phi_{A,v}(\mathbb{F}_\ell)$ (resp., $c_{B,\ell}$) is the Tamagawa number of $A$ (resp., $B$) at $v$. Suppose furthermore that $B[p] \subset A$, where both are viewed as subgroups of $R(\overline{L})$. Then there is an isomorphism

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

Remark 3.2   Everything divides infinity, so if $(R/B)(\mathbb{Q})$ is infinite, then no primes satisfy the hypothesis of the theorem.

Proof. [1, Thm. 3.1] produces an embedding

$$B(L)/p B(L)\hookrightarrow \Vis_R({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)[p]).$$

That theorem is proved using a standard snake lemma argument to show that $B(L)/p B(L)\hookrightarrow \Vis_R(H^1(L,A))$, combined with a local analysis at each prime to see that $B(L)$ lands in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/L)$. To prove that this embedding is surjective, note that the index of $E(\mathbb{Q})$ in $R(\mathbb{Q})$ is finite and coprime to $p$. $\qedsymbol$