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 $\#S$.