Rank 2 local systems and abelian varieties

TL;DR

This paper explores the relationship between rank 2 local systems with infinite monodromy and abelian varieties over finite fields, proving new theorems and connecting conjectures in algebraic geometry.

Contribution

It formulates a conjecture linking rank 2 local systems to abelian varieties and proves a Lefschetz-style theorem for abelian schemes of $ ext{GL}_2$-type, advancing understanding in the field.

Findings

Proved a Lefschetz-style theorem for abelian schemes of $ ext{GL}_2$-type.

Formulated a conjecture relating rank 2 local systems to abelian varieties.

Connected the conjecture to Deligne's companions conjecture, reducing the problem to curves.

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.