Pacific Institute for the Mathematical Sciences - PIMS

How many points can a curve have?

Abstract:  

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.