PIMS Number Theory CRG Distinguished Lecture: Noam Elkies (Harvard)

Diophantine equations, one of the oldest topics of mathematical research, remain the object of intense and fruitful study. A rational solution to a system of algebraic equations is tantamount to a point with rational coordinates (briefly, a "rational point") on the corresponding algebraic variety V. Already for V of dimension 1 (an "algebraic curve"), many natural theoretical and computational questions remain open, especially when the genus g of V exceeds 1. (The genus is a natural measure of the complexity of V; for example, if P is a nonconstant polynomial without repeated roots then the equation y^2 = P(x) gives a curve of genus g iff P has degree 2g+1 or 2g+2.) Faltings famously proved that if g>1 then the set of rational points is finite (Mordell's conjecture), but left open the question of how its size can vary with V, even for fixed g. Even for g=2 there are curves with literally hundreds of points; is the number unbounded?

We briefly review the structure of rational points on curves of genus 0 and 1, and then report on relevant work since Faltings on points on curves of given genus g>1.