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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: https://dx.doi.org/10.4310/CJM.2014.v2.n1.a1
Authors
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.
Published 19 June 2014