Level one algebraic cusp forms of classical groups of small rank
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Bibliographic Information
Level one algebraic cusp forms of classical groups of small rank
（Memoirs of the American Mathematical Society, v. 237,
American Mathematical Society, 2015
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Note
Includes bibliographical references (p. 117-122)
Description and Table of Contents
Description
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 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.
Table of Contents
Introduction Polynomial invariants of finite subgroups of compact connected Lie groups
Automorphic representations of classical groups : review of Arthur's results
Determination of $\Pi_\mathrm{alg}^\bot(\mathrm{PGL}_n)$ 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
Bibliography
by "Nielsen BookData"