Level one algebraic cusp forms of classical groups of small rank

We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over double-struck 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 double-struck Z-forms of the compact groups SO₇, SO₈, SO₉ (and G₂) 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.