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Rigid Character Groups, Lubin-Tate Theory, and $(\varphi ,\Gamma )$-Modules
About this Title
Laurent Berger, UMPA de l’ENS de Lyon, UMR 5669 du CNRS, IUF, Lyon, France, Peter Schneider, Universität Münster, Mathematisches Institut, Münster, Germany and Bingyong Xie, Department of Mathematics, East China Normal University, Shanghai, PR China
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 263, Number 1275
ISBNs: 978-1-4704-4073-2 (print); 978-1-4704-5658-0 (online)
DOI: https://doi.org/10.1090/memo/1275
Published electronically: February 24, 2020
MSC: Primary 11S31, 11S37, 14G22, 22E50, 46S10; Secondary 11F80, 11S20, 12H25, 13F05, 13J05
Table of Contents
Chapters
- Introduction
- 1. Lubin-Tate theory and the character variety
- 2. The boundary of $\mathfrak {X}$ and $(\varphi _L,\Gamma _L)$-modules
- 3. Construction of $(\varphi _L,\Gamma _L)$-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 $(\varphi ,\Gamma )$-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 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.- James Ax, Zeros of polynomials over local fields—The Galois action, J. Algebra 15 (1970), 417–428. MR 263786, DOI 10.1016/0021-8693(70)90069-4
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