Congruent numbers, elliptic curves, and L-functions
| Thesis title in Czech: | Kongruentní čísla, eliptické křivky a L-funkce |
|---|---|
| Thesis title in English: | Congruent numbers, elliptic curves, and L-functions |
| Key words: | kongruentní čísla|eliptické křivky|Zeta-funkce|L-funkce |
| English key words: | congruent numbers|elliptic curves|Zeta-functions|L-functions |
| Academic year of topic announcement: | 2023/2024 |
| Thesis type: | Bachelor's thesis |
| Thesis language: | angličtina |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | doc. Mgr. Vítězslav Kala, Ph.D. |
| Author: | Jan Kotyk - assigned and confirmed by the Study Dept. |
| Date of registration: | 21.02.2024 |
| Date of assignment: | 21.02.2024 |
| Confirmed by Study dept. on: | 21.02.2024 |
| Date of electronic submission: | 07.05.2024 |
| Opponents: | doc. Mgr. Pavel Příhoda, Ph.D. |
| Advisors: | Stevan Gajović, Ph.D. |
| Guidelines |
| It is a well-known problem to classify all congruent numbers, i.e., positive integers n such that there is a right triangle with rational sides and area n. This problem is closely related to the ranks of certain elliptic curves over Q. It is, for example, explained in [Kob]. Also, via the famous Birch and Swinnerton-Dyer conjecture, the rank of an elliptic curve is equal to an order of vanishing of the L-function of the elliptic curve. The student will very briefly introduce elliptic curves (besides [Kob], using, for example, [Sil], [ST]), with special attention on elliptic curves over finite fields. Then, the student will introduce L-functions of elliptic curves. In conclusion, the student will compute several ranks of elliptic curves and several L-functions and as a consequence determine some congruent and non-congruent numbers. |
| References |
| [Kob] Koblitz, N. (1993). Introduction to elliptic curves and modular forms. 2nd ed. Springer (Graduate Texts in Mathematics, 97).
[Sil] Silverman, J. H. (2009). The Arithmetic of Elliptic Curves. 2nd ed. Springer-Verlag. [ST] Silverman, J. H., Tate, J. T. (2015). Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics. 2nd ed. Cham: Springer. |