Principal Congruence Subgroups of the Hecke Groups Original Research Article

Let q be an odd integer >3 and let Gq be the Hecke group associated to q. Let (τ) be a prime ideal of image[λq] and G(q, τ) the principal congruence subgroup of Gq associated to τ. We give a formula for [Gq : G(q, τ)], the index of the principal congruence subgroup G(q, τ) in Gq. We also give formulae for the indices [G1(q, τ), G(q, τ)] and [G0(q, τ), G1(q, τ)]. Finally, we give a formula for the geometric invariants of G(q, τ) when q is a rational prime.