Fundamental groups, p-adic integrals, and rational points on the cursed curve
Finding all rational or integral solutions to a system of polynomial equations is an ancient mathematical problem, which is in general unsolvable. My talk will be about the simplest non-trivial case: One equation in two variables, defining the geometric object of an algebraic curve. Under certain assumptions about the curve, the method of Chabauty from the 1940's provides an incredibly efficient algorithm for finding the rational points of the curve inside the $p$-adic points in terms of the vanishing of certain additional functions. More recently, Minhyong Kim initiated a program for extending this method, and in particular to try to remove the restrictions of the original method. This program is called non-abelian Chabauty as it relies on a study of the fundamental group of the curve, which is in general non-abelian. In this talk I will survey the ideas of Chabaugy and Kim and describe a closely related extension, called Quadratic Chabauty, which is both practical and far easier to explain.
I will then survey a recent success of these methods: the determination of rational points on the so called cursed curve. I will explain the history of the curve and its relation with a famous conjecture of Serre, as well as the new ideas involved in its study.