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Level One Algebraic Cusp Forms of Classical Groups of Small Rank
 
Gaëtan Chenevier Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France
David A. Renard Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France
Level One Algebraic Cusp Forms of Classical Groups of Small Rank
eBook ISBN:  978-1-4704-2509-8
Product Code:  MEMO/237/1121.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Level One Algebraic Cusp Forms of Classical Groups of Small Rank
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Level One Algebraic Cusp Forms of Classical Groups of Small Rank
Gaëtan Chenevier Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France
David A. Renard Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France
eBook ISBN:  978-1-4704-2509-8
Product Code:  MEMO/237/1121.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2372015; 122 pp
    MSC: Primary 11; Secondary 22;

    The authors determine the number of level \(1\), polarized, algebraic regular, cuspidal automorphic representations of \(\mathrm{GL}_n\) over \(\mathbb Q\) of any given infinitesimal character, for essentially all \(n \leq 8\). For this, they compute the dimensions of spaces of level \(1\) automorphic forms for certain semisimple \(\mathbb Z\)-forms of the compact groups \(\mathrm{SO}_7\), \(\mathrm{SO}_8\), \(\mathrm{SO}_9\) (and \({\mathrm G}_2\)) and determine Arthur's endoscopic partition of these spaces in all cases. They also give applications to the \(121\) even lattices of rank \(25\) and determinant \(2\) found by Borcherds, to level one self-dual automorphic representations of \(\mathrm{GL}_n\) with trivial infinitesimal character, and to vector valued Siegel modular forms of genus \(3\). A part of the authors' results are conditional to certain expected results in the theory of twisted endoscopy.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Polynomial invariants of finite subgroups of compact connected Lie groups
    • 3. Automorphic representations of classical groups : review of Arthur’s results
    • 4. Determination of $\Pi _{\rm alg}^\bot ({\rm PGL}_n)$ for $n\leq 5$
    • 5. Description of $\Pi _{\rm disc}({\rm SO}_7)$ and $\Pi _{\rm alg}^{\rm s}({\rm PGL}_6)$
    • 6. Description of $\Pi _{\rm disc}({\rm SO}_9)$ and $\Pi _{\rm alg}^{\rm s}({\rm PGL}_8)$
    • 7. Description of $\Pi _{\rm disc}({\rm SO}_8)$ and $\Pi _{\rm alg}^{\rm o}({\rm PGL}_8)$
    • 8. Description of $\Pi _{\rm disc}({\rm G}_2)$
    • 9. Application to Siegel modular forms
    • A. Adams-Johnson packets
    • B. The Langlands group of $\mathbb {Z}$ and Sato-Tate groups
    • C. Tables
    • D. The $121$ level $1$ automorphic representations of ${\rm SO}_{25}$ with trivial coefficients
  • Requests
     
     
    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
Volume: 2372015; 122 pp
MSC: Primary 11; Secondary 22;

The authors determine the number of level \(1\), polarized, algebraic regular, cuspidal automorphic representations of \(\mathrm{GL}_n\) over \(\mathbb Q\) of any given infinitesimal character, for essentially all \(n \leq 8\). For this, they compute the dimensions of spaces of level \(1\) automorphic forms for certain semisimple \(\mathbb Z\)-forms of the compact groups \(\mathrm{SO}_7\), \(\mathrm{SO}_8\), \(\mathrm{SO}_9\) (and \({\mathrm G}_2\)) and determine Arthur's endoscopic partition of these spaces in all cases. They also give applications to the \(121\) even lattices of rank \(25\) and determinant \(2\) found by Borcherds, to level one self-dual automorphic representations of \(\mathrm{GL}_n\) with trivial infinitesimal character, and to vector valued Siegel modular forms of genus \(3\). A part of the authors' results are conditional to certain expected results in the theory of twisted endoscopy.

  • Chapters
  • 1. Introduction
  • 2. Polynomial invariants of finite subgroups of compact connected Lie groups
  • 3. Automorphic representations of classical groups : review of Arthur’s results
  • 4. Determination of $\Pi _{\rm alg}^\bot ({\rm PGL}_n)$ for $n\leq 5$
  • 5. Description of $\Pi _{\rm disc}({\rm SO}_7)$ and $\Pi _{\rm alg}^{\rm s}({\rm PGL}_6)$
  • 6. Description of $\Pi _{\rm disc}({\rm SO}_9)$ and $\Pi _{\rm alg}^{\rm s}({\rm PGL}_8)$
  • 7. Description of $\Pi _{\rm disc}({\rm SO}_8)$ and $\Pi _{\rm alg}^{\rm o}({\rm PGL}_8)$
  • 8. Description of $\Pi _{\rm disc}({\rm G}_2)$
  • 9. Application to Siegel modular forms
  • A. Adams-Johnson packets
  • B. The Langlands group of $\mathbb {Z}$ and Sato-Tate groups
  • C. Tables
  • D. The $121$ level $1$ automorphic representations of ${\rm SO}_{25}$ with trivial coefficients
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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