On the local-global principle for isogenies of abelian surfaces

Let ℓ be a prime number. We classify the subgroups G of Sp 4(Fℓ) and GSp 4(Fℓ) that act irreducibly on Fℓ4 , but such that every element of G fixes an Fℓ -vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree ℓ between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and ℓ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ℓ for which some abelian surface A/ Q fails the local-global principle for isogenies of degree ℓ .