Rank 2 local systems and abelian varieties

• Raju Krishnamoorthy ^[1] ; Ambrus Pál ^[2]
  1. [1] Bergische Universität, Alemania
  2. [2] Imperial College, London, Reino Unido
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 4, 2021
• Idioma: inglés
• DOI: 10.1007/s00029-021-00669-8
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• Resumen

  • Let X/Fq be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of GL2-type on X, modeled after a theorem of Simpson. Inspired by work of Corlette-Simpson over C, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on X come from families of abelian varieties. We have the following application of our main result. If one assumes a strong form of Deligne’s (p-adic) companions conjecture from Weil II, then our conjecture for projective varieties reduces to the conjecture for projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their p-divisible groups.