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 \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 $(\varphi,\Gamma)$-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 our 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, we work over a character variety, that parameterizes the locally $L$-analytic characters on $o_L$. We study $(\varphi,\Gamma)$-modules in this setting, and relate some of them to what was known previously.