@article{12312,
  abstract     = {Let $\ell$ be a prime number. We classify the subgroups $G$ of $\operatorname{Sp}_4(\mathbb{F}_\ell)$ and $\operatorname{GSp}_4(\mathbb{F}_\ell)$ that act irreducibly on $\mathbb{F}_\ell^4$, but such that every element of $G$ fixes an $\mathbb{F}_\ell$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface
$A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.},
  author       = {Lombardo, Davide and Verzobio, Matteo},
  issn         = {1420-9020},
  journal      = {Selecta Mathematica},
  number       = {2},
  publisher    = {Springer Nature},
  title        = {{On the local-global principle for isogenies of abelian surfaces}},
  doi          = {10.1007/s00029-023-00908-0},
  volume       = {30},
  year         = {2024},
}