Refined Counting of Geodesic Segments in the Hyperbolic Plane
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911195730018304 |
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| author | Voskou, Marios |
| author_facet | Voskou, Marios |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_03134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Refined Counting of Geodesic Segments in the Hyperbolic Plane Voskou, Marios Number Theory 11F72 (Primary) 11N45, 11N36, 11N75 (Secondary) 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. |
| title | Refined Counting of Geodesic Segments in the Hyperbolic Plane |
| topic | Number Theory 11F72 (Primary) 11N45, 11N36, 11N75 (Secondary) |
| url | https://arxiv.org/abs/2407.03134 |