Weak Weyl's Law For Congruence Subgroups

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.