How many points can a curve have? - PIMS Voyageur Colloquium

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.