MA426 Elliptic curves
=====================
Aims
====
Elliptic curves live in several different worlds of math (analysis and
complex function theory, geometry, algebraic geometry, number theory). The
course aims to give a basic overview of some of the main lines of study of
elliptic curves, building on the student's knowledge of undergraduate
algebra and complex analysis, and filling in background material where
required (especially in number theory and geometry). Particular aims are
to establish the link between doubly periodic functions, Riemann surfaces
of genus 1, plane cubic curves, and associated Diophantine problems.
At the cultural level, the final section of the course aspires to link
undergraduate material with more advanced research, and in particular, to
give finallist students a taste of the material surrounding Wiles' proof
of Fermat's last theorem in nontechnical or "popular science" language.
Objectives
==========
The student should be in a position to appreciate the following points and
to be able to combine them in simple applications:
The link between complex analysis and global meromorphic functions on the
Riemann sphere, for example, the number of zeros and poles, and the space
of functions with given poles.
The parallel between the geometry of an algebraic curve in the plane and
its rational function field.
The Weierstrass p-function as a sum over a lattice, and its application to
construct elliptic functions; the number of zeros and poles, and the space
of functions with given poles.
The geometric treatment of the group law on the cubic curve, its relation
with rational functions, and its applications to Diophantine problems.
The idea of infinite descent in number theory, and its applications,
including (special cases of) the Mordell--Weil theorem.
The parameter on which an elliptic curve depends (modulus), the idea of a
modular form in analysis.
The cultural background to Wiles' work on the Taniyama--Shimura conjecture
and Fermat's last theorem.