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

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 $\mathrm{\#}S$.