Séance Séminaire

Séminaire de Théorie des Nombres de Montpellier

Level one algebraic cusp forms of classical groups of small ranks.

We determine the number of level 1, self-dual, half-algebraic regular, cuspidal automorphic representations of GL(n) over Q of any given infinitesimal character, and essentially all n less than or equal to 8. For this, we compute the dimensions of the 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.