Refined Counting of Geodesic Segments in the Hyperbolic Plane

TL;DR

This paper develops a new trace formula to refine the counting of geodesic segments in the hyperbolic plane, providing improved error bounds and connections to number theory for arithmetic groups.

Contribution

It introduces a novel relative trace formula that yields sharper error estimates and links geometric counting problems to ideal counting in number fields.

Findings

Error bound improved to O(X^{2/3})

Mean square error bound established as O(X^{1/2} log X)

Connections made between geometric counts and ideal norms in number fields

Abstract

For $\Gamma$ 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.