Constantinescu, Petru;
(2021)
Distribution results for automorphic forms, their periods and masses.
Doctoral thesis (Ph.D), UCL (University College London).
Preview |
Text
thesis_final.pdf - Accepted Version Download (907kB) | Preview |
Abstract
We explore a variety of topics in the analytic theory of automorphic forms. The main results of this thesis are about the arithmetic statistics of periods of automorphic forms and the distribution of masses of automorphic forms in the context of Quantum Chaos. We introduce a new technique for the study of the distribution of modular symbols. Answering an average version of a conjecture due to Mazur and Rubin for $\Gamma_0(N)$ and recovering results of Petridis and Risager using a different method, we show that modular symbols are asymptotically normally distributed, We apply our technique to obtain new results for congruence subgroups of Bianchi groups. Our novel insight is to use the behaviour of the smallest eigenvalue of the Laplace operator for twisted spaces. Our approach also recovers the first and the second moment of the distribution. In work joint with Asbjørn Nordentoft, we introduce an automorphic method for studying the residual distribution of modular symbols modulo primes. We obtain a refinement of a result of Lee and Sun, which solved an average version of another conjecture of Mazur and Rubin. In addition, we solve the full conjecture in some special cases. Furthermore, we generalise the results to quotients of general hyperbolic spaces. Lastly, we obtain a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms, as proved by Holowinsky and Soundararajan. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Distribution results for automorphic forms, their periods and masses |
| Event: | UCL |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2021. 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/10137024 |
Archive Staff Only
 |
View Item |
