Authors | ## Gaetan Chenevier, David Renard |
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Year | 2015 |

Pages | 122 |

Publisher | Amer Mathematical Society |

Language | en |

ISBN | 9781470410940 |

File Size | 993.12 KB |

File Format | |

Download Counter | 84 |

Amazon | Link |

The authors determine the number of level $1$, polarized, algebraic regular, cuspidal automorphic representations of $\mathrm{GL}_n$ 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 $\mathrm{SO}_7$, $\mathrm{SO}_8$, $\mathrm{SO}_9$ (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 $\mathrm{GL}_n$ 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.