#### Title

A Brief Exploration of Rational Points on Elliptic Curves

#### Date

5-31-2018 2:00 PM

#### End Time

31-5-2018 2:20 PM

#### Location

HWC 204

#### Session Chair

Cheryl Beaver

#### Session Chair

Leanne Merrill

#### Session Title

Mathematics Capstone Project Presentations

#### Presentation Type

Presentation

#### Faculty Sponsor(s)

Leanne Merrill

#### Abstract

Combining abstract algebra, algebraic geometry and number theory, the study of elliptic curves can yield interesting results and applications. Though the subject can become very involved, we provide an introduction to the study of elliptic curves and the group of rational points on elliptic curves. We derive and prove the structure of the group of rational points on elliptic curves, adhering to the work of Silverman and Tate, and prove that the group operation is Abelian. We also consider the work of Rienzo by working out examples over finite fields. We explore the general geometric structure of elliptic curves, Weierstrass normal form and the different types of curves associated with the Weierstrass form of a curve, and finally, provide a possible proof of the conjecture given by Rienzo in "Elliptic Curves Over Local Fields."

#### Recommended Citation

Sosa-Vázquez, José, "A Brief Exploration of Rational Points on Elliptic Curves" (2018). *Academic Excellence Showcase Schedule*. 204.

https://digitalcommons.wou.edu/aes_event/2018/all/204

A Brief Exploration of Rational Points on Elliptic Curves

HWC 204

Combining abstract algebra, algebraic geometry and number theory, the study of elliptic curves can yield interesting results and applications. Though the subject can become very involved, we provide an introduction to the study of elliptic curves and the group of rational points on elliptic curves. We derive and prove the structure of the group of rational points on elliptic curves, adhering to the work of Silverman and Tate, and prove that the group operation is Abelian. We also consider the work of Rienzo by working out examples over finite fields. We explore the general geometric structure of elliptic curves, Weierstrass normal form and the different types of curves associated with the Weierstrass form of a curve, and finally, provide a possible proof of the conjecture given by Rienzo in "Elliptic Curves Over Local Fields."