Mathematics and Statistics
Dr. Noam Elkies, Harvard University
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.