Chebotarev geodesic theorem: non-split case

Abstract

We study the prime geodesic theorem for congruence subgroups of indefinite quaternion orders. We prove that the geodesic analogue of the Chebotarev density theorem for these groups holds with exponent $25/36 + \varepsilon$. In particular, we deduce that the prime geodesic theorem holds with exponent $25/36 + \varepsilon$ for any congruence subgroup of any indefinite quaternion order over $\mathbb{Q}$. The idea of the proof is to reduce the problem to the split case, which has been handled previously by the author.