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

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.

23 pages. Part of author's PhD thesis. Comments welcome!

References in corpus (6)

• A finiteness theorem for Galois representations of function fields over finite fields (after Deligne)
• Independence of ell of Monodromy Groups
• A Lefschetz theorem for overconvergent isocrystals with Frobenius structure
• Independence of l in Lafforgue's theorem
• On the deformation of a Barsotti-Tate group over a curve
• Rank 2 local systems and abelian varieties

Cited by in corpus (1)

• Etale and crystalline companions, I