Explicit surjectivity of Galois representations attached to abelian surfaces and ${\mathrm{GL}}_2$-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 ${\mathrm{GL}}_2$-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 $\mathrm{Aut}(T_{\ell }(A))$ is as large as it is allowed to be by endomorphisms and polarizations.

arXiv commentv3: added a discussion of GL_2-varieties. Title changed accordingly

References in corpus (1)

• On the image of l-adic Galois representations for abelian varieties of type I and II

Cited by in corpus (1)

• On the local-global divisibility of torsion points on elliptic curves and ${\mathrm{GL}}_2$-type varieties