Seminar in Algebraic Geometry and Number Theory

Abstract: I will give a gentle introuction to the theory of elliptic divisibility sequences, and discuss some of the problems to which it can be applied.

Let E be an elliptic curve defined by a Weierstrass equation with integer coefficients. Any rational point on E other than the identity is of the form (x/z²,y/z³) where x,y∈Z and z∈N and in addition both gcd(x,z)=1 and gcd(y,z)=1 hold. Taking multiples nP of a fixed point P gives a sequence of denominators (zₙ).

A central problem is to understand the distribution of integer points, those with z=1. Related problems concern points with z divisible only by primes in a fixed set S, points with z prime, points with z being a perfect power, and points for which z has no 'primitive divisor'.

The theory of elliptic divisibility sequences is a powerful tool for understanding this sequence (zₙ), with applications to the problems mentioned above.