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OverviewThe 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. Full Product DetailsAuthor: Gaetan Chenevier , David A. RenardPublisher: American Mathematical Society Imprint: American Mathematical Society Volume: 1121 Weight: 0.203kg ISBN: 9781470410940ISBN 10: 147041094 Pages: 122 Publication Date: 30 September 2015 Audience: College/higher education , Postgraduate, Research & Scholarly Format: Paperback Publisher's Status: Active Availability: Out of stock  The supplier is temporarily out of stock of this item. It will be ordered for you on backorder and shipped when it becomes available. Table of ContentsIntroduction Polynomial invariants of finite subgroups of compact connected Lie groups Automorphic representations of classical groups : review of Arthur's results Determination of $\Pi_\mathrm{alg}^\bot(\mathrm{PGL}_n)$ for $n\leq 5$ Description of $\Pi_\mathrm{disc}(\mathrm{SO}_7)$ and $\Pi_\mathrm{alg}^{\mathrm s}(\mathrm{PGL}_6)$ Description of $\Pi_\mathrm{disc}({\mathrm SO}_9)$ and $\Pi_\mathrm{alg}^{\mathrm s}(\mathrm{PGL}_8)$ Description of $\Pi_\mathrm{disc}(\mathrm{SO}_8)$ and $\Pi_\mathrm{alg}^{\mathrm o}(\mathrm{PGL}_8)$ Description of $\Pi_\mathrm{disc}({\mathrm G}_2)$ Application to Siegel modular forms Appendix A. Adams-Johnson packets Appendix B. The Langlands group of $\mathbb Z$ and Sato-Tate groups Appendix C. Tables Appendix D. The $121$ level $1$ automorphic representations of ${\mathrm SO}_{25}$ with trivial coefficients BibliographyReviewsAuthor InformationGaetan Chenevier and David A. Renard, Centre de Mathematiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France. Tab Content 6Author Website:Countries AvailableAll regions |
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