Congruence testing for odd subgroups of the modular group

TL;DR

This paper presents a practical method to determine if a finite-index subgroup of SL(2, Z) is a congruence subgroup, expanding previous work to include subgroups of SL(2, Z).

Contribution

It introduces an effective computational criterion for congruence testing applicable to subgroups of SL(2, Z), extending prior results for PSL(2, Z).

Findings

Provides a new criterion for congruence subgroup testing.

Extends earlier work from PSL(2, Z) to SL(2, Z).

Enables practical verification of subgroup properties.

Abstract

We give a computationally effective criterion for determining whether a finite-index subgroup of SL(2, Z) is a congruence subgroup, extending earlier work of Hsu for subgroups of PSL(2, Z).