Level one algebraic cusp forms of classical groups of small ranks

Gaetan Chenevier; David Renard

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arXiv · 1207.0724

Level one algebraic cusp forms of classical groups of small ranks

Gaetan Chenevier·David Renard

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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.

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Gaetan Chenevier, David Renard. 2013-07-18. Level one algebraic cusp forms of classical groups of small ranks. https://arxiv.org/abs/1207.0724

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