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

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.