Counting congruence subroups

Abstract

Let $\Gamma$ denote the modular group $SL(2,\mathbb{Z})$ and $C_n(\Gamma )$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\underset{n\to \infty }{\lim }\frac{\log C_n(\Gamma )}{(\log n{)}^2/\log \log n}=\frac{3-2\sqrt{2}}{4}.$ We also present a very general conjecture giving an asymptotic estimate for $C_n(\Gamma )$ for general arithmetic groups. The lower bound of the conjecture is proved modulo the generalized Riemann hypothesis for Artin-Hecke L-functions, and in many cases is also proved unconditionally.

30 pages