Rigid character groups, Lubin-Tate theory, and $(\phi ,\Gamma )$-modules

Abstract

The construction of the $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2({\mathbf{Q}}_p)$ uses in an essential way Fontaine's theory of cyclotomic $(\phi ,\Gamma )$-modules. Here \emph{cyclotomic} means that $\Gamma =\mathrm{Gal}({\mathbf{Q}}_p({\mu }_{p^{\infty }})/{\mathbf{Q}}_p)$ is the Galois group of the cyclotomic extension of ${\mathbf{Q}}_p$. In order to generalize the $p$-adic local Langlands correspondence to ${\mathrm{GL}}_2(L)$, where $L$ is a finite extension of ${\mathbf{Q}}_p$, it seems necessary to have at our disposal a theory of Lubin-Tate $(\phi ,\Gamma )$-modules. Such a generalization has been carried out to some extent, by working over the $p$-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of our article is to carry out a Lubin-Tate generalization of the theory of cyclotomic $(\phi ,\Gamma )$-modules in a different fashion. Instead of the $p$-adic open unit disk, we work over a character variety, that parameterizes the locally $L$-analytic characters on $o_L$. We study $(\phi ,\Gamma )$-modules in this setting, and relate some of them to what was known previously.

arXiv comment69 pages