Counting cusps of subgroups of $\mathrm {PSL}_2(\mathcal {O}_K)$

Abstract

Let K be a number field with r real places and s complex places, and let O K be the ring of integers of K. The quotient [H 2 ] r X [H 3 ] s /PSL 2 (O K ) has h K cusps, where h K is the class number of K. We show that under the assumption of the generalized Riemann hypothesis that if K is not Q or an imaginary quadratic field and if i ∉ K, then PSL 2 (O K ) has infinitely many maximal subgroups with h K cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.