On the local-global principle for isogenies of abelian surfaces

• Davide Lombardo ^[1] ; Matteo Verzobio ^[2]
  1. [1] University of Pisa

     University of Pisa

     Pisa, Italia

     
  2. [2] IST Austria, Austria
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 2, 2024, págs. 1-68
• Idioma: inglés
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• Resumen

  • Let ɭ a prime number. We classify the subgroups G of Sp4(F) and GSp4(F) that act irreducibly on F4 , but such that every element of G fixes an F-vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree ɭ between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and ɭ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ɭ for which some abelian surface A/Q fails the local-global principle for isogenies of degree ɭ.