Description
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.
In the first term we 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.
For the second term, there are many possible directions, including:
- The congruent number problem (see [K]).
- The Hasse bound and the Sato--Tate conjecture.
- Complex multiplication (see [ATAEC]).
- \(L\)-functions and the Birch and Swinnerton--Dyer conjecture.
- Integral points (see [AEC Chapter IX]).
- Statistics (average ranks, `murmurations', ...).
- Galois representations.
- Bad reduction: Tate curves, Néron models. (see [AEC Chapter VII, ATAEC Chapters IV, V])
- Connections with modular forms (modular curves, modularity). (See [DS] or [ATAEC Chapter I]. Would benefit from Topics in Algebra and Geometry IV).
Mode of Operation and Evidence of Learning
The project will revolve around learning through reading, examples, and problem solving, with focus on mathematical rigour and independent development of a specialised topic. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.
Prerequisites and Co-requisites
Prerequisites : Algebra II. Number Theory III or Galois Theory III would be useful for some directions.
Co-requisites : None required; TAG IV would be useful for some directions.
Additional information
Feel free to contact me with any questions at jack.g.shotton@durham.ac.uk.
- [C] J. W. S. Cassels: Lectures on Elliptic Curves.
- [ST] J. Silverman and J. Tate: Rational Points on Elliptic Curves.
- [AEC] J. Silverman: The Arithmetic of Elliptic Curves.
- [ATAEC] J. Silverman: Advanced Topics in the Arithmetic of Elliptic Curves.
- [K] N. Koblitz: Introduction to Elliptic Curves and Modular Forms.
- [W] L. Washington: Elliptic Curves: Number Theory and Cryptography.
- [C] D. Cox: Primes of the Form \(x^2 + ny^2\).
- [DS] F. Diamond and J. Shurman: A First Course in Modular Forms.
- [LMFDB] The LMFDB, for examples and data on elliptic curves.