Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves

TL;DR

This paper establishes a link between rank 2 local systems and Barsotti-Tate groups on curves over finite fields, proposing a criterion for identifying Shimura curves and exploring related conjectures in the Langlands program.

Contribution

It introduces a descent criterion for K-linear abelian categories and constructs a correspondence between rank 2 local systems and Barsotti-Tate groups, advancing understanding of Shimura curves over finite fields.

Findings

A new descent criterion for K-linear abelian categories.

A correspondence between rank 2 local systems and Barsotti-Tate groups.

A criterion for recognizing Shimura curves over finite fields.

Abstract

We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.