Level one algebraic cusp forms of classical groups of small ranks
Abstract
We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over Q of any given infinitesimal character, for essentially all n <= 8. For this, we compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO_7, SO_8, SO_9 (and 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 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.
Keywords
Cite
@article{arxiv.1207.0724,
title = {Level one algebraic cusp forms of classical groups of small ranks},
author = {Gaetan Chenevier and David Renard},
journal= {arXiv preprint arXiv:1207.0724},
year = {2013}
}
Comments
119 pages, 15 tables, v3 : The introduction has been mostly rewritten and expanded. Several improvements in the exposition, references added, as well as an appendix on the Langlands group of Z and Sato-Tate groups