Congruence Subgroups of the Modular Group

The congruence subgroups of the classical modular group which can be defined as the automorphs modulo q of some fixed matrix are studied, and their genera determined. Let T = SL{2, Z). A congruence subgroup of T is any subgroup containing a principal congruence subgroup T^), defined as the set of elements A of T such that A = I mod q, where q is a positive integer. Of these one of the most important is the group ro(<7), defined as the set of elements (" *) belonging to T such that c = 0 mod q. It is known that (r : nq)) = q3 U (1 Up2), (T : T0{q)) = q ft (1 + i/p), p\q p\q where p runs over the distinct primes dividing q. Let C= {1,-1} be the center of T, and F = T/C = PSL{2, Z). Then f is the classical modular group, and we will be interested here in the congruence subgroups of T, which are the subgroups of T corresponding to the congruence subgroups of T under the natural homomorphism ^ of F onto T. In particular we will study those congruence subgroups which can be defined as the set of automorphs modulo q of some fixed 2x2 matrix over Z. If SI is a subgroup of T, then SI will denote the subgroup of T corresponding to SI under y. It is more convenient to study the problem for T and its subgroups, and then make the transition to V by means of Theorem 1 below. Also, we will choose q to be a prime > 3 for simplicity of exposition. We first prove Theorem 1. Let SI be a subgroup of T. Then the subgroup SI of T corresponding to SI under the natural homomorphism ip is Ü = {Sí, I}/C. Thus Í(r : si), -ie si, &(T:Í2), -IsfSl. Proof. If G is any group and N a normal subgroup, then any subgroup H Received February 15, 1974. AMS (MOS) subject classifications (1970). Primary 10D05, 15A36, 20H05.