## PIMS Voyageur Colloquium

- Date: 01/14/2011

University of Calgary

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.