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

TL;DR

This paper provides an explicit, effective bound for the open image theorem concerning the Galois representations of products of principally polarized abelian varieties over number fields, assuming GRH.

Contribution

It makes the open image theorem for such abelian varieties effective by deriving explicit bounds under GRH, based on invariants of the varieties and the number field.

Findings

Derived explicit bounds for the constant c(A) under GRH.

Bound depends on invariants of the number field and abelian varieties.

Enhanced understanding of Galois representations of product abelian varieties.

Abstract

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