On the local-global principle for isogenies of abelian surfaces

Let \(\ell \) be a prime number. We classify the subgroups G of \({\text {Sp}}_4({\mathbb {F}}_\ell )\) and \({\text {GSp}}_4({\mathbb {F}}_\ell )\) 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 a 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 \(\ell \).