Explicit open image theorems for abelian varieties with trivial endomorphism ring

TL;DR

This paper establishes explicit bounds for the Galois representations associated with abelian varieties over number fields, linking the bounds to height and residue field data, and proposes an algorithm for improved bounds in specific cases.

Contribution

It provides a semi-effective bound for the Galois image of abelian varieties with trivial endomorphism ring, connecting it to height and residue field, and introduces an algorithm for better bounds in certain cases.

Findings

Bound depends on Faltings height and residue field size.

Galois image equals symplectic group for large primes beyond the bound.

Algorithmic approach improves bounds for abelian threefolds over \\mathbb{Q}.

Abstract

Let $K$ be a number field and $A/K$ be an abelian variety of dimension $g$. Assuming that the image $G_{\ell^\infty}$ of the natural Galois representation attached to the Tate module $T_\ell(A)$ is $\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all sufficiently large primes $\ell$, we provide a semi-effective bound $\ell_0(A/K)$ such that $G_{\ell^\infty}=\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all primes $\ell > \ell_0(A/K)$. The bound is given in terms of the Faltings height of $A$ and of the cardinality of the residue field at a suitably generic place of $K$. We also describe an algorithmic approach to obtain better bounds for abelian threefolds over $\mathbb{Q}$.