Explicit surjectivity of Galois representations attached to abelian surfaces and $\operatorname{GL}_2$-varieties

TL;DR

This paper establishes explicit bounds on primes for which the Galois representations attached to certain abelian varieties are surjective, confirming maximal Galois image for large primes under specific conditions.

Contribution

It provides explicit bounds for primes ensuring the Galois representations of abelian surfaces and GL2-type varieties are as large as possible, advancing understanding of their Galois images.

Findings

Explicit bound (A,K) for surjectivity of Galois representations

Maximal Galois image for primes (A,K)

Applicable to abelian surfaces and (GL_2)-type varieties

Abstract

Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$ such that, for every prime $\ell>\ell_0(A,K)$, the image of the absolute Galois group of $K$ in $\operatorname{Aut}(T_\ell(A))$ is as large as it is allowed to be by endomorphisms and polarizations.