eBook

Rigid Character Groups, Lubin-Tate Theory, and (𝜑,Γ)-Modules

Xie, Bingyong, Schneider, Peter, Berger, Laurent

2020

Overview

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 $(\\varphi ,\\Gamma )$-modules. Here 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 $(\\varphi ,\\Gamma )$-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 this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic $(\\varphi ,\\Gamma )$-modules in a different fashion. Instead of the $p$-adic open unit disk, the authors work over a character variety that parameterizes the locally $L$-analytic characters on $o_L$. They study $(\\varphi ,\\Gamma )$-modules in this setting and relate some of them to what was known previously.

• Publisher: American Mathematical Society
• ISBN: 9781470440732, 1470440733