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Softcover ISBN:  9781470440732 
Product Code:  MEMO/263/1275 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
eBook ISBN:  9781470456580 
Product Code:  MEMO/263/1275.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
Softcover ISBN:  9781470440732 
eBook ISBN:  9781470456580 
Product Code:  MEMO/263/1275.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $102.00 $76.50 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 263; 2020; 75 ppMSC: Primary 11; 14; 22; 46; Secondary 12; 13
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 LubinTate \((\varphi ,\Gamma )\)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 LubinTate group. The main idea of this article is to carry out a LubinTate 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.

Table of Contents

Chapters

Introduction

1. LubinTate 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


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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 LubinTate \((\varphi ,\Gamma )\)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 LubinTate group. The main idea of this article is to carry out a LubinTate 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.

Chapters

Introduction

1. LubinTate 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