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Module MA342P: Elliptic Curves

Credit weighting (ECTS)
5 credits
Semester/term taught
Hilary term 2015-16
Contact Hours
11 weeks, 3 lectures including tutorials per week
Prof. Tim Murphy
Learning Outcomes
On successful completion of this module, students will be able to:
  • Determine if a cubic curve is non-singular, and reduce it to standard form if it is.
  • Determine the abelian group of an elliptic curve over a finite field.
  • Determine the abelian group of an elliptic curve over the reals and complex numbers.
  • Determine the points of finite order on an elliptic curve over the rationals.
  • Be able to compute with p-adic numbers.
  • Prove Fermat's Last Theorem when n=4.
  • Understand how elliptic curves are used in cryptopgraphy.
  • Understand the theory of doubly-periodic complex functions.

Module Content
  • An elliptic curve is a non-singular cubic curve. A non-singular quadratic curve is a conic so elliptic curves are the next step in algebraic geometry after the conic sections studied by the ancient Greeks.
  • An elliptic curve can be brought to Weiersrass standard form $y^2 = x^3 + bx + c$, at least if the ground field does not have characteristic 2 or 3
  • Elliptic curves have a natural abelian group structure; if P, Q are two points on the curve then the line PQ cuts the curve in a third point R, and P +Q +R = 0
  • This allows elliptic curves to be studied over all fields, in particular finite fields, providing the basis for elliptic curve crytography (ECC) the standard encryption technique today.
  • Elliptic curves provided the central tool in Wiles 'proof of Fermat's Last Theorem', that the equations $x^n + y^n = z^n$ has no non-trivial integer solution if $n > 2$. They are also the idea behind Fernat's proof by infinite descent in the case $n = 4$.
  • There is an entirely different way of approaching elliptic curves, through elliptic functions, and the course will follow both approaches. An elliptic function $f(z)$ is a doubly-periodic complex function, and can be thought of as the complex analogue of the real trigonometric functions $\cos x, \sin x $, etc.
  • These functions, and the related modular forms are one of the most active areas in mathematics today, at the intersection of number theory, algebraic geometry and complex analysis (Ramanujan's Conjectures on modular forms, published 100 years ago, remain a lively topic in number theory.)

Course Notes -

Module Prerequisite

MA2215 Fields, Rings and Modules, MA2328 Complex Analysis

Assessment Detail
This module will be examined jointly in a 2-hour examination in Trinity term. Continuous assessment will contribute 20% to the final annual grade. Supplementals if required will consist of 100% exam.