On the local-global principle for isogenies of abelian surfaces

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