On the local-global principle for isogenies of abelian surfaces

Abstract

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 .