Mathematics and Statistics

University of Calgary

Submitted by ccameron on Fri, 01/07/2011 - 12:17pm.

Jan 14 2011 - 3:00pm

Jan 14 2011 - 4:00pm

Speaker:

Dr. Noam Elkies, Harvard University

Location:

ICT 114**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.