TY  - UNPB
N1  - Copyright © The Author 2025. 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.
SP  - 1
AV  - public
Y1  - 2025/01/28/
EP  - 136
TI  - Distributional Results for Geodesic Segments in the Hyperbolic Plane
A1  - Voskou, Marios
M1  - Doctoral
PB  - UCL (University College London)
UR  - https://discovery.ucl.ac.uk/id/eprint/10203381/
N2  - For ? a cofinite Fuchsian group, and l a fixed closed geodesic, we study refined
asymptotics of the number of images of l over ? that have a distance from l less than or equal to X. In particular, we partition the images into four cases, according to orientation, and prove that they all contribute asymptotically one-fourth of the total. This was originally studied by A. Good. For ?_1 the stabilizer of l, this is equivalent to counting the number of double cosets in ?_1\?/?_1 with prescribed signs for its entries, according to a certain growth parameter. We achieve this by developing new modified relative trace formulae, as well as bounds for hyperbolic periods in mean square.  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) logX) for the mean square of the error. To that end, we obtain large sieve inequalities
with weights the hyperbolic periods of Maaß forms of even weight. This is inspired by work of Chamizo, who proved a large sieve inequality with weights the values of Maaß forms of weight 0. We also prove ? results, supporting the conjectural best error term O_?(X^(1/2+?))$. 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.
ID  - discovery10203381
ER  -