  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.
