Counting elements of the congruence subgroup

Abstract

Abstract We obtain asymptotic formulas for the number of matrices in the congruence subgroup $$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$ which are of naive height at most X. Our result is uniform in a very broad range of values Q and X.