Lekkas, Dimitrios;
(2023)
A relative trace formula and counting geodesic segments in the hyperbolic plane.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
In this work we study a modification of the hyperbolic circle problem, which is one of the problems originally studied by A. Good. We consider the orbit of double cosets of a Fuchsian group Γ by two hyperbolic subgroups H_1, H_2 in the hyperbolic plane. We use a relative trace formula with suitable test functions for the counting of lengths of geodesic segments perpendicular to the closed geodesics corresponding to H_1 and H_2. We present an elementary proof providing the main term in the asymptotics and an error termof order O(X^{2/3}). We study themean square of the error termand prove that it is consistent with the conjectural optimal error term O(X^{1/2+ϵ}). To apply the relative trace formula we develop a large sieve inequality for periods ofMaass forms. This requires a more subtle understanding of Huber’s transform, which is a special case of the Jacobi transform studied by Flensted- Jensen and Koornwinder. Our counting problem is a special case of counting in the orthospectrum. We are motivated by previous work on geodesic segments between a point and a closed geodesic, studied by Huber and Chatzakos–Petridis.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | A relative trace formula and counting geodesic segments in the hyperbolic plane |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2023. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10171116 |
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