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Local-global compatibility of mod p Langlands program for certain Shimura varieties

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Abstract

We generalize the local-global compatibility result in as reported by Scholze (Appendix by Michael Rapoport, Annales de l’ENS., 2018) to higher dimensional cases, by examining the relation between Scholze’s functor and cohomology of Kottwitz-Harris-Taylor type Shimura varieties. Along the way we prove a cuspidality criterion from type theory. We also deal with compatibility for torsion classes in the case of semisimple mod p Galois representations with distinct irreducible components under certain flatness hypotheses.

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Acknowledgements

The author would like to thank Pascal Boyer and Stefano Morra, for suggesting this project and for helpful discussions. It is also his pleasure to thank Zicheng Qian, Vincent Sécherre and Zhixiang Wu for several helpful discussions and for their feedbacks, and thank the anonymous referee for many helpful comments and suggestions. This article is part of the author’s PhD thesis and he wishes to thank ED Galilée of Université Paris 13 and also the ANR grant ANR-14-CE25-0002-01 PerCoLaTor for their support.

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  1. Institut Galilée, Université Sorbonne Paris Nord, 99 avenue J.B. Clément, 93430, Villetaneuse, France

    Kegang Liu

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Liu, K. Local-global compatibility of mod p Langlands program for certain Shimura varieties. manuscripta math. 172, 375–403 (2023). https://doi.org/10.1007/s00229-022-01410-1

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