Prime Geodesic Theorem for Arithmetic Compact Surfaces: Principal Congruence Case

We generalize Koyama’s $7\mathrm{/}10$ bound of the error term in the prime geodesic theorems to the principal congruence subgroups for quaternion algebras. Our method avoids the spectral side of the Jacquet–Langlands correspondences, and relates the counting function directly to those for the principal congruence subgroups of Eichler orders of level less than one.