#### Vol. 16, No. 2, 2022

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Rank 2 local systems, Barsotti–Tate groups, and Shimura curves

### Raju Krishnamoorthy

Vol. 16 (2022), No. 2, 231–259
##### Abstract

We construct a descent-of-scalars criterion for $K\phantom{\rule{-0.17em}{0ex}}$-linear abelian categories. Using 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 ${\mathbb{𝔽}}_{\phantom{\rule{-0.17em}{0ex}}q}$. Along the way we formulate a conjecture on the field-of-coefficients of certain compatible systems.

##### Keywords
Shimura curves, Barsotti–Tate groups, local systems
##### Mathematical Subject Classification 2010
Primary: 14G15
Secondary: 14G35, 14H25