Abstract:
The prime geodesic theorems (PGT) are analogues of the prime number theorem (PNT), in which we count the primitive closed geodesics in Γ\H instead of prime numbers. Here Γ<PSL2(R) is a lattice, and H is the Poincare upper half plane. The race is about the upper bound of the error term as O(xθ ). In the setting of co-compact lattices constructed from quaternion algebras, the record is kept by Koyama for full level subgroups at Luo–Sarnak’s level θ= 7/10. Koyama’s method is an exploitation of the Jacquet-Langlands correspondences on the spectral side.
We generalize Koyama’s 7/10 bound 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. We shall discuss the possibility and difficulty for further generalization to other congruence subgroups, as well as other possible projects around this topic.
This is a joint work with three undergraduate students (Chenhao Tang, Jie Yang and Wenyan Yang) during the event of the 2025 summer school entitled “Algebra and Number Theory” held at the Chinese Academy of Sciences.
