A relative trace formula and counting geodesic arcs in the hyperbolic plane
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909791242158080 |
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| author | Lekkas, Dimitrios Petridis, Yiannis |
| author_facet | Lekkas, Dimitrios Petridis, Yiannis |
| contents | 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 $Γ\backslash\mathbb{H}$. 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 $O(X^{1/2}\log X)$. We conjecture that this is the correct order of growth. Along the way we provide a new proof of the pointwise error bound $O(X^{2/3})$, originally proved by Good. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12902 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A relative trace formula and counting geodesic arcs in the hyperbolic plane Lekkas, Dimitrios Petridis, Yiannis Number Theory 11J71, 11N45, 11N36 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 $Γ\backslash\mathbb{H}$. 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 $O(X^{1/2}\log X)$. We conjecture that this is the correct order of growth. Along the way we provide a new proof of the pointwise error bound $O(X^{2/3})$, originally proved by Good. |
| title | A relative trace formula and counting geodesic arcs in the hyperbolic plane |
| topic | Number Theory 11J71, 11N45, 11N36 |
| url | https://arxiv.org/abs/2509.12902 |