Counting elements of the congruence subgroup

TL;DR

This paper derives uniform asymptotic formulas for counting matrices in the congruence subgroup (Q) with height up to X, covering a wide range of Q and X values.

Contribution

It provides the first uniform asymptotic estimates for the number of matrices in (Q) with bounded height, extending previous non-uniform results.

Findings

Asymptotic formulas valid for broad Q and X ranges

Uniform estimates improve understanding of matrix distribution in congruence subgroups

Results applicable to various problems in number theory and automorphic forms

Abstract

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