**Duration:** 21 weeks

**Number of lectures per week:** 3

**Assessment:**

**End-of-year Examination:** Probably examined in 3 parts, one examination covering each term's work

**Description: **
Two recent events have elevated the theory of elliptic curves
from a somewhat specialized topic
to the centre of mathematical development.

Firstly, Andrew Wiles' proof of Fermat's Last Theorem in 1995 was set in the context of elliptic curves over the rationals \mathbbQ.

Secondly, elliptic curve cryptography -
based on elliptic curves over finite fields \mathbbF_{q} -
is rapidly becoming the standard encryption algorithm.

Elliptic curves are the next most simple curves after straight lines and conics. They are particularly interesting because each elliptic curve carries a natural structure as an abelian group; geometrically, if A, B, C are 3 collinear points on the curve then A + B + C = 0. This group structure is the basis for virtually all applications of elliptic curves.

The theory of elliptic curves grew out of Weierstrass' study of doubly-periodic complex functions. (The term `elliptic' arose from the rôle of such functions in computing the arc-length around an ellipse.)

Today the theory has become an exciting amalgam of complex function theory, algebraic geometry and arithmetic (or number theory).

Oct 1, 1999