Prime geodesic theorem for the Picard manifold
| Thesis title in Czech: | Prvočíselná věta pro geodesiky na Picardově varietě |
|---|---|
| Thesis title in English: | Prime geodesic theorem for the Picard manifold |
| Key words: | prvočíselná věta pro geodesiky|Picardova varieta|hyperbolická geometrie |
| English key words: | prime geodesic theorem|Picard variety|hyperbolic geometry |
| Academic year of topic announcement: | 2021/2022 |
| Thesis type: | diploma thesis |
| Thesis language: | angličtina |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | Giacomo Cherubini, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 24.01.2022 |
| Date of assignment: | 24.01.2022 |
| Confirmed by Study dept. on: | 31.01.2022 |
| Date and time of defence: | 06.09.2022 09:00 |
| Date of electronic submission: | 20.07.2022 |
| Date of submission of printed version: | 25.07.2022 |
| Date of proceeded defence: | 06.09.2022 |
| Opponents: | Matteo Bordignon, Ph.D. |
| Advisors: | doc. Mgr. Vítězslav Kala, Ph.D. |
| Guidelines |
| The prime geodesic theorem describes the asymptotic distribution of primitive geodesics on the modular surface -- the quotient of the hyperbolic plane by the modular group PSL(2,Z). The thesis aims to prove the analogous result in the case of the Picard manifold, associated to the group PSL(2,Z[i]).
The student will cover the main properties of the geometry in three dimensional hyperbolic space and the action of PSL(2,Z[i]) and will apply them to the classification of elements in PSL(2,Z[i]). He will further apply the Selberg trace formula to deduce the asymptotic result in the prime geodesic theorem. Finally, he may possibly study a first moment estimate for the remainder. |
| References |
| [1]. Balog, A., Biró, A., Cherubini G., Laaksonen N.: Bykovskii-Type Theorem for the Picard Manifold, I.M.R.N. 2020, https://doi.org/10.1093/imrn/rnaa128
[2]. Cherubini, G., Guerreiro, J.: Mean square in the prime geodesic theorem. Algebra Number Theory 12 (2018), no. 3, 571–597. [3]. Elstrodt J., Grunewald F., Mennicke J.: Groups acting on hyperbolic space. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. [4]. Phillips, R.; Conjugacy classes of Γ(2) and spectral rigidity, Math. Comp. 64 (1995), no. 211, 1287-1306. [5]. Yılmaz, N., Cangul, I. N.: Conjugacy classes of elliptic elements in the Picard group, Turkish J. Math. 24 (2000), no. 2, 209–220. |