The Generalized Artin Conjecture and Arithmetic Orbifolds

M. Ram Murty, Kathleen L. Petersen

Let $K$ be a number field with $r_1$ real places and $r_2$ complex places, and class number $h_K$. The quotient $[\Hyp^2]^{r_1}\times [\Hyp^3]^{r_2}/\mathrm{PSL}_2(\Ox_K)$ has $h_K$ cusps. For a prime ideal $\pi$ of $\Ox_K$ of norm $q$, let $\varphi_{\pi}$ denote the reduction modulo $\pi$ map. We provide an un-conditional proof that if $K$ is Galois with unit rank greater than three, then there are infinitely many such $\pi$ with the property that $\varphi_{\pi}(\Ox_K^{\times})=(\Ox_K/\pi)^{\times}.$ Our result establishes an un-conditional proof that for such $K$, $\mathrm{PSL}_2(\Ox_K)$ has infinitely many maximal subgroups, $\Gamma,$ such that the quotient $[\Hyp^2]^{r_1}\times [\Hyp^3]^{r_2}/\Gamma$ has exactly $h_K$ cusps. This was previously known under the assumption of the Generalized Riemann Hypothesis.