Refined Counting of Geodesic Segments in the Hyperbolic Plane

TL;DR

This work develops a refined counting framework for oriented geodesic segments in the hyperbolic plane by introducing a new relative trace formula tailored to hyperbolic-hyperbolic configurations. It establishes a concrete $O(X^{2/3})$ error bound and an averaged $O(X^{1/2}\log X)$ mean-square bound for the associated counting problems, with explicit spectral terms involving Maaß periods and small eigenvalues. The authors provide a precise equivalent of the main counting results via a modified trace formula and carefully chosen test functions, enabling a sharp decomposition into main terms and spectral contributions. They also connect these analytic results to arithmetic by interpreting certain counts in terms of ideals in number fields, via Jacquet–Langlands correspondences and quaternion algebra groups, yielding new correlation sums and conditional refinements under Selberg's conjecture. Overall, the paper advances both the analytic and arithmetic understanding of refined geodesic counting in the hyperbolic plane and offers tools for averaged error control in related automorphic counting problems.

Abstract

For $Γ$ a cofinite Fuchsian group, and $l$ a fixed closed geodesic, we study the asymptotics of the number of those images of $l$ that have a prescribed orientation and distance from $l$ less than or equal to $X$. Using a new relative trace formula that we develop, we give a new concrete proof of the error bound $O(X^{2/3})$ that appears in the works of Good and Hejhal. Furthermore, we prove a new bound $O(X^{1/2}\log{X})$ for the mean square of the error. For particular arithmetic groups, we provide interpretations in terms of correlation sums of the number of ideals of norm at most $X$ in associated number fields, generalizing previous examples due to Hejhal.

Figures (2)

• Figure 1: Geometric interpretation of the four cases that correspond to the four possible choices of $(\mu,\mu')$.
• Figure 2: Effect of perturbation of $l_1$ and $l_2$ on Figure 1.

Theorems & Definitions (56)

• Theorem 1: Selberg selberg, Günther gunther, Good good
• Theorem 2: Good good, Lekkas lekkas
• Theorem 3
• Conjecture 1
• Theorem 4
• Theorem 5
• Remark 1
• Remark 2
• Theorem 6
• Remark 3