Cambridge Journal of Mathematics

Volume 2 (2014)

Number 1

The $p$-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$

Pages: 1 – 47

DOI: http://dx.doi.org/10.4310/CJM.2014.v2.n1.a1

Authors

Pierre Colmez (C.N.R.S., Institut de mathématiques de Jussieu, Paris, France)

Gabriel Dospinescu (UMPA, École Normale Supérieure de Lyon, France)

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

Abstract

The $p$-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$ is given by an exact functor from unitary Banach representations of ${\rm GL}_2(\mathbb{Q}_p)$ to representations of the absolute Galois group $\mathcal G_{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. We prove, using characteristic $0$ methods, that this correspondence induces a bijection between absolutely irreducible non-ordinary representations of ${\rm GL}_2(\mathbb{Q}_p)$ and absolutely irreducible $2$-dimensional representations of $\mathcal G_{\mathbb{Q}_p}$. This had already been proved, by characteristic $p$ methods, but only for $p\geq5$.

Keywords

$p$-adic Langlands, $(\varphi, \gamma)$-modules, Banach space representations

2010 Mathematics Subject Classification

Primary 11S37. Secondary 11F80, 22E50.

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