K_1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_1$$\end{document}

-Invariants in the Mod p Cohomology of U(3) Arithmetic Manifolds

Abstract

Let

F/F^{+}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F/F^+$$\end{document}

be a CM extension and

H_{/F^{+}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{/F^+}$$\end{document}

a definite unitary group in three variables that splits over F. We describe Hecke isotypic components of mod p algebraic modular forms on H at first principal congruence level at p and “minimal” level away from p in terms of the restrictions of the associated Galois representation to decomposition groups at p when these restrictions are tame and sufficiently generic. This confirms an expectation of local–global compatibility in the mod p Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support.

Keywords

Mod p local Langlands program / Mod p cohomology of arithmetic manifolds / Deformation rings of Galois representations / Mod p representation of GL3(Fq) / 11F33 / 11F80 / 22E50

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Funding

National Science Foundation(DMS-1952678)

Institut Universitaire de France(DMS-2302619)

RIGHTS & PERMISSIONS

Peking University

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