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Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula

Abstract : We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we explicitly compute the Euler-Poincaré characteristic of the level one discrete automorphic spectrum of $\mathbf{G}$ with respect to a finite-dimensional representation of $\mathbf{G}(\mathbb{R})$. For such a group $\mathbf{G}$, Arthur's endoscopic classification of the discrete spectrum allows to analyse precisely this Euler-Poincaré characteristic. For example one can deduce the number of everywhere unramified automorphic representations $\pi$ of $\mathbf{G}$ such that $\pi_{\infty}$ is isomorphic to a given discrete series representation of $\mathbf{G}(\mathbb{R})$. Dimension formulae for the spaces of vector-valued Siegel modular forms are easily derived.
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https://hal.archives-ouvertes.fr/hal-01007666
Contributor : Olivier Taïbi Connect in order to contact the contributor
Submitted on : Monday, June 16, 2014 - 11:53:37 PM
Last modification on : Thursday, March 17, 2022 - 10:08:17 AM
Long-term archiving on: : Tuesday, September 16, 2014 - 11:45:28 AM

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  • HAL Id : hal-01007666, version 1
  • ARXIV : 1406.4247

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Olivier Taïbi. Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula. 2014. ⟨hal-01007666⟩

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