An effective open image theorem for products of principally polarized abelian varieties

Abstract

Let $A={\prod }_{1\le i\le n}{A}_i$ be the product of principally polarized abelian varieties $A_1,\dots ,A_n$ of dimensions $g_1,\dots ,g_n$, respectively, each defined over a number field K, and pairwise nonisogenous over $\bar{K}$. We make effective an open image theorem for A due to Hindry and Ratazzi. More specifically, we give an explicit bound of the constant $c\left(A\right)$ under GRH, in terms of standard invariants of K and each $A_i$, where $c\left(A\right)$ is defined to be the smallest positive integer such that for any prime $\ell >c\left(A\right)$, the image of the ℓ-adic Galois representation of A is “as large as possible” in a suitable sense.