Rational points on curves

TL;DR

This paper reviews current methods and challenges in determining rational points on smooth projective curves over Q, especially for genus at least 2, emphasizing practical approaches for finite sets.

Contribution

It provides an overview of the state of the art and practical techniques for finding rational points on high-genus curves over Q.

Findings

Current methods effectively find rational points on certain curves.

Challenges remain for higher-genus curves with complex structures.

Practical approaches are discussed for finite rational point sets.

Abstract

This is an extended version of an invited lecture I gave at the Journees Arithmetiques in St. Etienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over Q. The focus is on practical aspects of this problem in the case that the genus of C is at least 2, and therefore the set of rational points is finite.