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Rigid Character Groups, Lubin-Tate Theory, and $(\varphi,\Gamma)$-Modules
 
Laurent Berger UMPA ENS de Lyon, Lyon, France
Peter Schneider Universität Múnster, Münster, Germany
Bingyong Xie East China Normal University, Shanghai, People’s Republic of China
Rigid Character Groups, Lubin-Tate Theory, and (varphi,Gamma)-Modules
Softcover ISBN:  978-1-4704-4073-2
Product Code:  MEMO/263/1275
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
eBook ISBN:  978-1-4704-5658-0
Product Code:  MEMO/263/1275.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
Softcover ISBN:  978-1-4704-4073-2
eBook: ISBN:  978-1-4704-5658-0
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
Rigid Character Groups, Lubin-Tate Theory, and (varphi,Gamma)-Modules
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Rigid Character Groups, Lubin-Tate Theory, and $(\varphi,\Gamma)$-Modules
Laurent Berger UMPA ENS de Lyon, Lyon, France
Peter Schneider Universität Múnster, Münster, Germany
Bingyong Xie East China Normal University, Shanghai, People’s Republic of China
Softcover ISBN:  978-1-4704-4073-2
Product Code:  MEMO/263/1275
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
eBook ISBN:  978-1-4704-5658-0
Product Code:  MEMO/263/1275.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
Softcover ISBN:  978-1-4704-4073-2
eBook ISBN:  978-1-4704-5658-0
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 Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2632020; 75 pp
    MSC: 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 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.

  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
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Volume: 2632020; 75 pp
MSC: 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 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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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