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Drinfeld's lemma for F-isocrystals, II: Tannakian approach

Part of: Families, fibrations (Co)homology theory

Published online by Cambridge University Press:  01 December 2023

Kiran S. Kedlaya Open the ORCID record for Kiran S. Kedlaya [Opens in a new window]  and
Daxin Xu
Kiran S. Kedlaya
Affiliation:
Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA [email protected]
Daxin Xu
Affiliation:
Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China [email protected]
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Abstract

We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from $\ell$-adic to $p$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.

Keywords

Drinfeld's lemmaoverconvergent isocrystalsp-adic cohomology

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 14D24: Geometric Langlands program 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 1 , January 2024 , pp. 90 - 119
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The first author is supported by NSF grants DMS-1802161 and DMS-2053473 and the UC San Diego Warschawski Professorship. The second author is supported by the National Natural Science Foundation of China grants 12222118 and 12288201 and the CAS Project for Young Scientists in Basic Research grant YSBR-033. D.X. is grateful to Xinwen Zhu for suggesting this question and helpful discussions.

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