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



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)


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 $p\geq5$.


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

2010 Mathematics Subject Classification

Primary 11S37. Secondary 11F80, 22E50.

Full Text (PDF format)