Elliptic Curves

Elliptic curves are the simplest type of equation that we still do not fully understand; equations as simple as \[y^2 = x^3 - 2.\] They have many beautiful geometric and algebraic properties, and are one of the central objects of modern number theory. The solutions to (or points on) an elliptic curve naturally have the structure of a group, and this is known to always have the form \[T \times \mathbb{Z}^r\] for some integer \(r\) and finite group \(T\). However, determining \(r\) and finding generators for the group is still a big mystery in general. This topic has lots of scope for numerical examples and computer investigations.

The project will begin with the definition and geometry of elliptic curves, the group structure on their points, their points over the complex numbers, their points over finite fields, and methods for determining their rational points. Possible topics for further investigation could be:

• The Hasse bound and the Sato--Tate conjecture
• Complex multiplication
• \(L\)-functions and the Birch and Swinnerton--Dyer conjecture
• Integral points
• Statistics
• Galois representations
• Tate curves: \(p\)-adic uniformization
• Modularity
• ...

Prerequisites:

Algebra II. Elementary Number Theory II is also recommended. Number Theory III, Galois Theory III, and Codes and Cryptography III would help.

Corequisites:

Topics in Algebra and Geometry IV is relevant but not essential.

Textbooks:

• J. W. S. Cassels, Lectures on elliptic curves, LMS Student Texts 24, Cambridge University Press, 1991
• J. Silverman and J. Tate, Rational points on elliptic curves, 2nd ed., Undergraduate Texts in Mathematics, Springer, 2015.
• J. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, Springer, 2009.
• L. Washington, Elliptic Curves: number theory and cryptography , 2nd ed., Chapmand and Hall, 2008.