On the local-global principle for isogenies of abelian surfaces

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