eBook ISBN:  9781470425098 
Product Code:  MEMO/237/1121.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470425098 
Product Code:  MEMO/237/1121.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 237; 2015; 122 ppMSC: 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 selfdual 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. AdamsJohnson packets

B. The Langlands group of $\mathbb {Z}$ and SatoTate groups

C. Tables

D. The $121$ level $1$ automorphic representations of ${\rm SO}_{25}$ with trivial coefficients


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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 selfdual 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. AdamsJohnson packets

B. The Langlands group of $\mathbb {Z}$ and SatoTate groups

C. Tables

D. The $121$ level $1$ automorphic representations of ${\rm SO}_{25}$ with trivial coefficients