Rank 2 local systems and abelian varieties

Abstract

Let $X/{\mathbb{F}}_q$ be a smooth geometrically connected variety. 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. When $X$ is a projective variety, we prove a Lefschetz-style theorem for abelian schemes of ${\text{GL}}_2$-type on $X$, modeled after a theorem of Simpson. If one assumes a strong form of Deligne's ($p$-adic) \emph{companions conjecture} from Weil II, this implies that our conjecture for projective varieties also reduces to the case of projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their $p$-divisible groups.

29 pages, comments very welcome. v3: completely reorganized, minor errors fixed. v4: final version

References in corpus (1)

• Independence of ell of Monodromy Groups

Cited by in corpus (2)

• Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves
• Rank 2 local systems and abelian varieties II