Applications of modular functions in algebraic number theory

Bc. Ondrej Bínovský

Master's thesis

Abstract:

V tejto diplomovej práci skúmame grupu racionálnych bodov na eliptickej krivke tvaru x^3+y^3=2p, respektíve x^3+y^3=2p^2, kde p je nepárne prvočíslo, ktoré spĺňa istú podmienku modulo 9. Zkonštruujeme modulárnu parametrizáciu homogénneho priestoru prislúchajúcemu danej eliptickej krivke a použitím Šimurovej reciprocity zkonštruujeme racionálny bod na tomto homogénnom priestore. Nakoniec použitím metódy zostupu na našej eliptickej krivke ukážeme, že jej rank je rovný jednej.

Abstract:

In this thesis we study the group of rational points on the elliptic curve of the form x^3+y^3=2p and x^3+y^3=2p^2 where p is an odd prime satisfying a certain condition modulo 9. We will construct a modular parametrization of a homogeneous space associated with the elliptic curve in question, and using Shimura reciprocity we will construct a rational point on this homogeneous space. Finally, using descent on the elliptic curve, we will show that the its rank is equal to one.

Keywords

Eliptická krivka, Modulárna funkcia, Súčet dvoch tretích mocnín racionálnych čísel, Zostup na eliptickej krivke, Rank eliptickej krivky, Mordellova-Weilova veta, Šimurova reciprocita, Elliptic curve, Modular function, Sum of two rational cubes, Descent on an elliptic curve, Rank of an elliptic curve, Mordell-Weil theorem, Shimura reciprocity

Language used: English

Date on which the thesis was submitted / produced: 7. 5. 2024

Thesis defence

• Date of defence: 17. 6. 2024
• Supervisor: prof. RNDr. Radan Kučera, DSc.
• Reader: doc. Lukáš Vokřínek, PhD.

Masaryk University

Faculty of Science

Master programme / field:
Mathematics / Mathematics