Eigenvarieties and invariant norms

Claus M. Sorensen

Vol. 275 (2015), No. 1, 191–230

DOI: 10.2140/pjm.2015.275.191

Abstract

We give a proof of the Breuil–Schneider conjecture in a large number of cases, which complement the indecomposable case, which we dealt with earlier. In this paper, we view the conjecture from a broader global perspective. If $U_{∕F}$ is any definite unitary group, which is an inner form of $GL\left(n\right)$ over $\mathcal{K}$, we point out how the eigenvariety $\mathbb{X}\left(K^p\right)$ parametrizes a global $p$-adic Langlands correspondence between certain $n$-dimensional $p$-adic semisimple representations $\rho$ of $Gal\left(\overline{\mathbb{Q}}|\mathcal{K}\right)$ (or what amounts to the same, pseudorepresentations) and certain Banach–Hecke modules $\mathcal{ℬ}$ with an admissible unitary action of $U\left(F\otimes {\mathbb{Q}}_p\right)$, when $p$ splits. We express the locally regular-algebraic vectors of $\mathcal{ℬ}$ in terms of the Breuil–Schneider representation of $\rho$. As an application, we give a weak form of local–global compatibility in the crystalline case, showing that the Banach space representations $B_{\xi ,\zeta }$ of Schneider and Teitelbaum fit the picture as predicted.

Keywords

Galois representations, p-adic Langlands, eigenvarieties, automorphic forms

Mathematical Subject Classification 2010

Primary: 11F33

Milestones

Received: 7 May 2014

Accepted: 21 October 2014

Published: 12 April 2015