The affine Chabauty method

Given a hyperbolic curve $Y$ defined over the integers and a finite set of primes $S$, the set of $S$-integral points $Y(\mathbb{Z}_S)$ is finite by theorems of Siegel, Mahler, and Faltings. Determining this set in practice is a difficult problem for which no general method is known. In this talk I report on joint work in progress with Martin Lüdtke in which we develop a Chabauty--Coleman method for finding $S$-integral points on affine curves. We achieve this by bounding the image of $Y(\mathbb{Z}_S)$ in the Mordell--Weil group of the generalised Jacobian using arithmetic intersection theory on a regular model.