Equivariant Visibility

Let $K$ be a number field, let $A_{/K}$ and $B_{/K}$ be abelian subvarieties of an abelian variety $C_{/K}$ , such that $C=A+B$ and $A \cap B$ is finite. Let $Q_{/K}$ denotes the quotient $C/B$ . Let $N$ be a positive integer divisible by all primes of bad reduction for $C$ .

Let $\ell$ be a prime such that $B[\ell]\subset A$ and $e < \ell-1$ , where $e$ is the largest ramification index of any prime of $K$ lying over $\ell$ . Suppose that

$$\ell \nmid N \cdot \char93 B(K)_{{\mathrm{tor}}} \cdot \char93 Q(K)_{{\mathrm{tor}}} \cdot \prod_{v\mid N} c_{A,v} c_{B,v}.$$

Under those conditions, Agashe and Stein (see [AS02, Thm. 3.1]) construct a homomorphism $B(K)/\ell B(K) \rightarrow {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, A)[\ell]$ whose kernel has $\mathbb{F}_\ell$ -dimension bounded by the Mordell-Weil rank of $A(K)$ .

In this paper, we refine [AS02, Prop. 1.3] by taking into account the algebraic structure coming from the endomorphism ring ${\mathrm{End}}_K(C)$ . In particular, when we apply the theory to modular abelian varieties, we would like to use the additional structure coming from the Hecke algebra. There are numerous example (see [AS05]) where [AS02, Prop. 1.3] does not apply, but nevertheless, we can use our refinement to prove existence of visible elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f)$ at higher level (e.g., see Propositions 6.1.3 and 6.2.1 below).