Rank 2 local systems and abelian varieties

Abstract

Let X/mathbb {F}_{q} be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of text {GL}_2-type on X, modeled after a theorem of Simpson. Inspired by work of Corlette-Simpson over mathbb {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.