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December, 2004 Weak Weyl's Law For Congruence Subgroups
Jean-Pierre LaBesse, Werner Muller
Asian J. Math. 8(4): 733-746 (December, 2004).

Abstract

A conjecture of Sarnak [Sa] states that the counting function fo the cuspidal spectrum satisfies Weyl's law. This conjecture has been established in some special cases. First of all, it was Selberg [Se] who proved it for congruence subgroups of $SL(2,Z)$ and $\sigma = 1$. Other cases for which the conjecture has been established are Hilbert modular groups [Ef], congruence subgroups of $SO(n,1)$ [Rez], $SL(3,Z)$ [Mil], and in particular, the conjecture was proved in [Mu2] for principal congruence subgroups of $SL(n,Z)$ and arbitrary $\sigma$ ... The purpose of this paper is to prove a weaker result which holds for every $G$. Recall that an upper bound, which holds for arbitrary $G$ and $\Gamma$, is already known thanks to Donnelly [Do] ... By working with a simple form of the trace formula, we shall get, for a general $G$, a lower bound that depends on the choice of a set $S$ of primes containing at least two finite primes. For every such set $S$ we shall define a certain constant $c_{S}(\Gamma}$ less than or equal to 1, which is non zero for $\Gamma$ deep enough.

Citation

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Jean-Pierre LaBesse. Werner Muller. "Weak Weyl's Law For Congruence Subgroups." Asian J. Math. 8 (4) 733 - 746, December, 2004.

Information

Published: December, 2004
First available in Project Euclid: 13 June 2005

zbMATH: 1162.11344
MathSciNet: MR2127945

Rights: Copyright © 2004 International Press of Boston

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Vol.8 • No. 4 • December, 2004
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