Mathematics > Number Theory
[Submitted on 17 Jun 2014]
Title:Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula
View PDFAbstract: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.
Submission history
From: Olivier Taibi [view email] [via CCSD proxy][v1] Tue, 17 Jun 2014 06:25:16 UTC (83 KB)
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