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# memo_has_moved_text();Level one algebraic cusp forms of classical groups of small rank

Gaëtan Chenevier and David Renard

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: http://dx.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

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Chapters

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