Marius Leonhardt (Universität Frankfurt): Affine abelian non-abelian Chabauty

Abstract: The central question of this talk is how to find all integral points on affine hyperbolic curves. For example, which integers x,y satisfy y^2 = x^3 + x^2 + x + 1?
We approach this problem by introducing Kim’s non-abelian Chabauty method, which constructs p-adic analytic functions that have the integral points among their zeroes. In joint work with M. Lütdke and J.S. Müller, we showed that the abelian version of this method succeeds if the curve satisfies a certain inequality involving its genus and the Mordell-Weil rank of its Jacobian.
This talk is an introduction to the Chabauty–Kim method. I will present the above result and report on work in progress with M. Lüdtke that turns it into an algorithm determining the integral points on the curve.