Rigid Character Groups, Lubin-Tate Theory, and $(\varphi ,\Gamma )$-Modules

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.