## PIMS Number Theory CRG Distinguished Lecture: Noam Elkies (Harvard)

- Date: 01/14/2011

Noam Elkies, Harvard University

University of Calgary

How many points can a curve have?

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.

3:00pm-4:00pm, ICT 114

For further information, please visit the webpage at http://math.ucalgary.ca/news-events/events/pims-events/pims/crg/how-many-points-can-curve-have