Explicit open image theorems for abelian varieties with trivial endomorphism ring

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 ${\mathrm{GSp}}_{2g}({\mathbb{Z}}_{\ell })$ for all sufficiently large primes $\ell$, we provide a semi-effective bound ${\ell }_0(A/K)$ such that $G_{{\ell }^{\infty }}={\mathrm{GSp}}_{2g}({\mathbb{Z}}_{\ell })$ 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 $\mathbb{Q}$.

v3: added new author, streamlined exposition, several new results. v2: the result now applies to abelian varieties of arbitrary dimension