Multivariable (φ,Γ)-modules and locally analytic vectors

Abstract

Let $K$ be a finite extension of ${\mathbf{Q}}_p$, and let $G_K=\mathrm{Gal}({\overline{\mathbf{Q}}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(\phi ,\Gamma )$-modules (cyclotomic means that $\Gamma =\mathrm{Gal}(K_{\infty }/K)$ where $K_{\infty }$ is the cyclotomic extension of $K$). One particularly convenient feature of the cyclotomic theory is the fact that the $(\phi ,\Gamma )$-module attached to any $p$-adic representation is overconvergent.

Questions pertaining to the $p$-adic local Langlands correspondence lead us to ask for a generalization of the theory of $(\phi ,\Gamma )$-modules, with the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie extension $K_{\infty }/K$. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely, the case of a Lubin–Tate extension, most $(\phi ,\Gamma )$-modules fail to be overconvergent.

In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of $\Gamma$ inside some big modules defined using Fontaine’s rings of periods. We show that, in the cyclotomic case, we recover the usual overconvergent $(\phi ,\Gamma )$-modules. In the Lubin–Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that $(\phi ,\Gamma )$-modules attached to $F$-analytic representations are overconvergent.