On the local-global principle for isogenies of abelian surfaces

Abstract

Let $\ell$ be a prime number. We classify the subgroups $G$ of ${\mathrm{Sp}}_4({\mathbb{F}}_{\ell })$ and ${\mathrm{GSp}}_4({\mathbb{F}}_{\ell })$ that act irreducibly on ${\mathbb{F}}_{\ell }^4$, but such that every element of $G$ fixes an ${\mathbb{F}}_{\ell }$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface $A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.

Minor changes. Final version of the paper. 64 pages

References in corpus (1)

• A Local-global principle for isogenies of composite degree