# Cambridge Journal of Mathematics

## Volume 4 (2016)

### Number 2

### Patching and the $p$-adic local Langlands correspondence

Pages: 197 – 287

DOI: http://dx.doi.org/10.4310/CJM.2016.v4.n2.a2

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Authors

Ana Caraiani (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

Matthew Emerton (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

Toby Gee (Department of Mathematics, Imperial College London, United Kingdom)

David Geraghty (Department of Mathematics, Boston College, Chestnut Hill, Massachusetts, U.S.A.)

Vytautas Paškūnas (Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany)

Sug Woo Shin (Department of Mathematics, University of California at Berkeley; and Korea Institute for Advanced Study, Seoul, Korea)

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Abstract

We use the patching method of Taylor–Wiles and Kisin to construct a candidate for the $p$-adic local Langlands correspondence for $\mathrm{GL}_n (F)$, $F$ a finite extension of $\mathbb{Q}_p$. We use our construction to prove many new cases of the Breuil–Schneider conjecture.

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