Level One Algebraic Cusp Forms of Classical Groups of Small Rank

The authors determine the number of level $1$, polarized, algebraic regular, cuspidal automorphic representations of $mathrmGLn$ 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 $mathrmSO7$, $mathrmSO8$, $mathrmSO9$ (and $mathrm G2$) 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 $mathrmGLn$ 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.