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 $\mathcal {O}_K$ be the ring of integers of $K$. The quotient $[\mathbb {H}^2]^r\times [\mathbb {H}^3]^s/\mathrm {PSL}_2(\mathcal {O}_K)$ 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 $\mathbb {Q}$ or an imaginary quadratic field and if $i \not \in K$, then $\mathrm {PSL}_2(\mathcal {O}_K)$ has infinitely many maximal subgroups with $h_K$ cusps. A key element in the proof is a connection to Artin’s Primitive Root Conjecture.