On a result of Petersson concerning the modular group

Abstract

SynopsisLet N(u, g, h) denote the number of subgroups of index u, genus g and parabolic class number h in the classical modular group. In 1974, Petersson proved that, for g = 0 and h = 1, 2, N(u, g, h) grows exponentially with u. We obtain a formula valid for all g and h. From this we derive an asymptotic estimate, showing that Petersson's result gives the correct type of growth. The proofs use coset diagrams.