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

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 $o_L$.

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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.