Counting elements of the congruence subgroup

Abstract

We obtain asymptotic formulas for the number of matrices in the congruence subgroup \[ Γ_0(Q) = \left\{ A\in\mathrm{SL}_2(\mathbb Z):~c \equiv 0 \pmod Q\right\}, \] which are of naive height at most $X$. Our result is uniform in a very broad range of values $Q$ and $X$.