Exploring the Existence of Rational Points on Elliptic Curves: What is the Point?
| dc.contributor.advisor | Ostrik, Victor | |
| dc.contributor.author | Garcia-Perez, Salvador | |
| dc.date.accessioned | 2023-09-28T20:27:03Z | |
| dc.date.available | 2023-09-28T20:27:03Z | |
| dc.date.issued | 2023-05 | |
| dc.description | 44 pages | en_US |
| dc.description.abstract | When introduced with any equation, whether it is given by a polynomial or cubic polynomial, the first question we typically ask ourselves is, what is the solution? In many cases, like in the equation 𝑥𝑥+2=5, we can easily determine that the solution is 𝑥𝑥=3. But, in other scenarios, like finding rational or integer solutions to certain cubic equations, the answer may be difficult to find, or, in some cases, there may not be a definitive way to find an answer at all. It is especially the case for elliptic curves. Elliptic curves of the Weierstrass form are equations of the form 𝑦𝑦2=𝑓𝑓(𝑥𝑥)=𝑥𝑥3+𝐴𝐴𝐴 +𝐵𝐵, where 𝐴𝐴,𝐵𝐵∈ℤ. Although solutions to cubic equations are well-understood with real numbers, the challenge appears when we try to find integers and rational solutions in cubic equations, which are not yet well understood. Independent rational solutions on elliptic curves, defined by the rank of a curve, have proven difficult to uncover. This thesis is interested in developing a better understanding of how to find rational solutions and understanding the ranks of elliptic curves. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/28933 | |
| dc.language.iso | en | en_US |
| dc.publisher | University of Oregon | en_US |
| dc.rights | Creative Commons BY-NC-ND 4.0-US | en_US |
| dc.subject | elliptic curves | en_US |
| dc.subject | rank | en_US |
| dc.subject | Weierstrass | en_US |
| dc.title | Exploring the Existence of Rational Points on Elliptic Curves: What is the Point? | en_US |
| dc.type | Thesis / Dissertation | en_US |