{"volume":30,"intvolume":"        30","year":"2024","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.","publication":"Selecta Mathematica","author":[{"full_name":"Lombardo, Davide","first_name":"Davide","last_name":"Lombardo"},{"id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","full_name":"Verzobio, Matteo","first_name":"Matteo","last_name":"Verzobio","orcid":"0000-0002-0854-0306"}],"doi":"10.1007/s00029-023-00908-0","publication_identifier":{"eissn":["1420-9020"],"issn":["1022-1824"]},"external_id":{"arxiv":["2206.15240"]},"article_type":"original","_id":"12312","article_number":"18","language":[{"iso":"eng"}],"date_created":"2023-01-16T11:45:53Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","oa":1,"scopus_import":"1","publisher":"Springer Nature","status":"public","quality_controlled":"1","title":"On the local-global principle for isogenies of abelian surfaces","OA_place":"repository","OA_type":"green","day":"26","type":"journal_article","date_updated":"2024-09-19T11:36:51Z","month":"01","article_processing_charge":"Yes (via OA deal)","issue":"2","date_published":"2024-01-26T00:00:00Z","publication_status":"epub_ahead","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 $\\ell$."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2206.15240"}],"department":[{"_id":"TiBr"}]}