Level One Algebraic Cusp Forms of Classical Groups of Small Rank

Gaetan Chenevier, David A. Renard (Autoren)

Buch | Softcover

122 Seiten

2015
American Mathematical Society (Verlag)
978-1-4704-1094-0 (ISBN)

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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.

Gaetan Chenevier and David A. Renard, Centre de Mathematiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France.

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

Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	203 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-1094-X / 147041094X
ISBN-13	978-1-4704-1094-0 / 9781470410940
Zustand	Neuware