Level one algebraic cusp forms of classical groups of small rank - Chenevier, Gaëtan; Renard, David (Memoirs of the American Mathematical Society, 2015)
The authors determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over \mathbb Q of any given infinitesimal character, for essentially all n ≤ 8. For this, they compute the dimensions of spaces of level 1 automorphic forms for certain semisimple \mathbb Z-forms of the compact groups SO_7, SO_8, 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 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.