Posloupnosti aritmeticko-geometrických průměrů a eliptické křivky nad konečnými tělesy

• Thesis title in Czech: Posloupnosti aritmeticko-geometrických průměrů a eliptické křivky nad konečnými tělesy
• Thesis title in English: Arithmetic-geometric mean sequences and elliptic curves over finite fields
• Academic year of topic announcement: 2023/2024
• Thesis type: Bachelor's thesis
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Mgr. Vítězslav Kala, Ph.D.
• Author: Natália Bátorová - 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
• Advisors: Stevan Gajović, Ph.D.

Guidelines

Arithmetic-geometric mean (AGM) sequences have a remarkable property of fast approximation of real numbers, for example, Euler used them to quickly compute digits of pi. However, these sequences make sense over other fields as well. For example, Griffin, Ono, Saikia, and Tsai [GOST] consider such sequences over finite fields F_p, where p is a prime number congruent to 3 modulo 4. They relate these sequences and their adjoined graphs to isogenies of certain elliptic curves and the corresponding graphs, which are used to give bounds on the number of components of these graphs. The student will explain AGM sequences, very briefly introduce elliptic curves (for example, from [Sil], [ST]), and explain the connection between them. Furthermore, the student may give some analogous results over F_p, for a different family of primes p, for example, for primes p congruent to 5 modulo 8.

References

[GOST] Griffin, M., Ono, K., Saikia, N., and Tsai, W-L. (2023). AGM and Jellyfish Swarms of Elliptic Curves. American Mathematical Monthly (130). 355-369.
[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.