DOI: 10.1515/forum-2026-0040 ISSN: 0933-7741

A relative trace formula and counting geodesic arcs in the hyperbolic plane

Dimitrios Lekkas, Yiannis N. Petridis
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Abstract

We study a modification of the hyperbolic circle problem: instead of all elements of a Fuchsian group Γ, we consider the double cosets by two hyperbolic subgroups. This has a geometric interpretation in terms of the number of common perpendiculars between two closed geodesics for

Γ \ ℍ {\Gamma\backslash\mathbb{H}}
. We prove an explicit relative trace formula, which is flexible for the counting problem. Using a large sieve inequality developed by the first author and Voskou, we prove a new bound in mean square for the error term of order
O ⁢ ( X 1 2 ⁢ log ⁡ X ) {O(X^{\frac{1}{2}}\log X)}
. We conjecture that the correct order of growth is
O ⁢ ( X 1 2 + ϵ ) {O(X^{\frac{1}{2}+\epsilon})}
. Along the way, we provide a new proof of the pointwise error bound
O ⁢ ( X 2 3 ) {O(X^{\frac{2}{3}})}
, originally proved by Good.

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