Level One Algebraic Cusp Forms of Classical Groups of Small Rank

ISBN-13: 9781470410940 / Angielski

Gaetan Chenevier;David A. Renard

Opis

The authors determine the number of level $1$, polarized, algebraic regular, cuspidal automorphic representations of $mathrm_n$ 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 $mathrm_7$, $mathrm_8$, $mathrm_9$ (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 $mathrm_n$ 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.

Spis treści

• Introduction Polynomial invariants of finite subgroups of compact connected Lie groups
• Automorphic representations of classical groups : review of Arthur's results
• Determination of $\Pi_\mathrm^\bot(\mathrm_n)$ for $n\leq 5$
• Description of $\Pi_\mathrm(\mathrm_7)$ and $\Pi_\mathrm^{\mathrm s}(\mathrm_6)$
• Description of $\Pi_\mathrm({\mathrm SO}_9)$ and $\Pi_\mathrm^{\mathrm s}(\mathrm_8)$
• Description of $\Pi_\mathrm(\mathrm_8)$ and $\Pi_\mathrm^{\mathrm o}(\mathrm_8)$
• Description of $\Pi_\mathrm({\mathrm G}_2)$
• 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 ${\mathrm SO}_$ with trivial coefficients
• Bibliography