---
OA_place: repository
OA_type: green
_id: '12312'
abstract:
- lang: eng
  text: "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\r\n$A/\\mathbb{Q}$ fails the local-global principle for isogenies of degree
    $\\ell$."
acknowledgement: "It is a pleasure to thank Samuele Anni for his interest in this
  project and for several discussions on the topic of this paper, which led in particular
  to Remark 6.30 and to a better understanding of the difficulties with [6]. We also
  thank John Cullinan for correspondence about [6] and Barinder Banwait for his many
  insightful comments on the first version of this paper. Finally, we thank the referee
  for their thorough reading of the manuscript.\r\nOpen access funding provided by
  Università di Pisa within the CRUI-CARE Agreement. The authors have been partially
  supported by MIUR (Italy) through PRIN 2017 “Geometric, algebraic and analytic methods
  in arithmetic\" and PRIN 2022 “Semiabelian varieties, Galois representations and
  related Diophantine problems\", and by the University of Pisa through PRA 2018-19
  and 2022 “Spazi di moduli, rappresentazioni e strutture combinatorie\". The first
  author is a member of the INdAM group GNSAGA."
article_number: '18'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Davide
  full_name: Lombardo, Davide
  last_name: Lombardo
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
date_created: 2023-01-16T11:45:53Z
date_published: 2024-01-26T00:00:00Z
date_updated: 2025-02-13T11:47:12Z
day: '26'
department:
- _id: TiBr
doi: 10.1007/s00029-023-00908-0
external_id:
  arxiv:
  - '2206.15240'
intvolume: '        30'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2206.15240
month: '01'
oa: 1
oa_version: Preprint
publication: Selecta Mathematica
publication_identifier:
  eissn:
  - 1420-9020
  issn:
  - 4321-1234
  issnl:
  - 1022-1824
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the local-global principle for isogenies of abelian surfaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 30
year: '2024'
...