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Patching and the $p$ -adic Langlands program for $\operatorname{GL}_{2}(\mathbb{Q}_{p})$

Part of: Lie groups Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  01 December 2017

Ana Caraiani ,
Matthew Emerton ,
Toby Gee ,
David Geraghty ,
Vytautas Paškūnas  and
Sug Woo Shin
Ana Caraiani
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email a.caraiani@imperial.ac.uk
Matthew Emerton
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA email emerton@math.uchicago.edu
Toby Gee
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email toby.gee@imperial.ac.uk
David Geraghty
Affiliation:
Department of Mathematics, 301 Carney Hall, Boston College, Chestnut Hill, MA 02467, USA email david.geraghty@bc.edu
Vytautas Paškūnas
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany email paskunas@uni-due.de
Sug Woo Shin
Affiliation:
Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA email sug.woo.shin@berkeley.edu Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
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Abstract

We present a new construction of the $p$ -adic local Langlands correspondence for $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ via the patching method of Taylor–Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and the global aspects of the $p$ -adic Langlands program; in particular, it gives a new proof of many cases of the second author’s local–global compatibility theorem and relaxes a hypothesis on the local mod  $p$ representation in that theorem.

Keywords

p-adic Langlandslocal–global compatibilityTaylor–Wiles patching

MSC classification

Primary: 11S37: Langlands-Weil conjectures, nonabelian class field theory 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 3 , March 2018 , pp. 503 - 548
Copyright
© The Authors 2017 

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