Determining the rational points on curves

Some of the oldest problems in mathematics amount to finding solutions in rational numbers to equations describing curves. Faltings established the landmark result that curves of general type have only finitely many rational points, but his method of proof is ineffective. The problem of deciding if an arbitrary curve has any rational points at all remains unsolved.

I will discuss some of the methods that can, under some conditions, decide if a curve has no rational points, and determine the rational points otherwise. While we know that for each of these methods there exist curves for which the methods fail, we know thanks to work by Bhargava et al. that at least for hyperelliptic curves the methods do work for the majority of them.