On the local-global principle for isogenies of abelian surfaces

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