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

Krishnamoorthy, Raju

doi:10.2140/ant.2022.16.231

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

Raju Krishnamoorthy

Vol. 16 (2022), No. 2, 231–259

DOI: 10.2140/ant.2022.16.231

Abstract

We construct a descent-of-scalars criterion for $K$ -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 $𝔽_{q}$ . 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

Milestones

Received: 20 March 2019

Revised: 2 June 2021

Accepted: 7 July 2021

Published: 27 April 2022

Authors

Raju Krishnamoorthy
Bergische Universität Wuppertal Germany

Vol. 16, No. 2, 2022