Weight filtrations on Selmer schemes and the effective Chabauty–Kim method

Abstract

We develop an effective version of the Chabauty–Kim method which gives explicit upper bounds on the number of$S$-integral points on a hyperbolic curve in terms of dimensions of certain Bloch–Kato Selmer groups. Using this, we give a new ‘motivic’ proof that the number of solutions to the$S$-unit equation is bounded uniformly in terms of$\#S$.