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Level one algebraic cusp forms of classical groups of small rank

About this Title

Gaëtan Chenevier, Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau Cedex, France and David Renard, Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 237, Number 1121
ISBNs: 978-1-4704-1094-0 (print); 978-1-4704-2509-8 (online)
DOI: https://doi.org/10.1090/memo/1121
Published electronically: January 22, 2015
Keywords: Automorphic representations, classical groups, compact groups, conductor one, dimension formulas, endoscopy, invariants of finite groups, Langlands group of ${\mathbb {Z}}$, euclidean lattices, Sato-Tate groups, vector-valued Siegel modular forms.
MSC: Primary 11FXX; Secondary 11F46, 11F55, 11F70, 11F72, 11F80, 11G40, 11H06, 11R39, 11Y55, 22C05.

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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 $\Pi _\textrm {alg}^\bot (\textrm {PGL}_n)$ for $n\leq 5$
  • 5. Description of $\Pi _\textrm {disc}(\textrm {SO}_7)$ and $\Pi _\textrm {alg}^\textrm {s}(\textrm {PGL}_6)$
  • 6. Description of $\Pi _\textrm {disc}(\textrm {SO}_9)$ and $\Pi _\textrm {alg}^\textrm {s}(\textrm {PGL}_8)$
  • 7. Description of $\Pi _\textrm {disc}(\textrm {SO}_8)$ and $\Pi _\textrm {alg}^\textrm {o}(\textrm {PGL}_8)$
  • 8. Description of $\Pi _\textrm {disc}(\textrm {G}_2)$
  • 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 $\textrm {SO}_{25}$ with trivial coefficients

Abstract

We determine the number of level $1$, polarized, algebraic regular, cuspidal automorphic representations of $\textrm {GL}_n$ over $\mathbb {Q}$ of any given infinitesimal character, for essentially all $n \leq 8$. For this, we compute the dimensions of spaces of level $1$ automorphic forms for certain semisimple $\mathbb {Z}$-forms of the compact groups $\textrm {SO}_7$, $\textrm {SO}_8$, $\textrm {SO}_9$ (and $\textrm {G}_2$) and determine Arthur’s endoscopic partition of these spaces in all cases. We 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 $\textrm {GL}_n$ with trivial infinitesimal character, and to vector valued Siegel modular forms of genus $3$. A part of our results are conditional to certain expected results in the theory of twisted endoscopy.

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