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

TL;DR

This paper introduces an effective approach to the Chabauty--Kim method, providing explicit bounds on $S$-integral points on hyperbolic curves using Selmer groups, and offers a new proof for bounds on solutions to the $S$-unit equation.

Contribution

It develops an effective version of the Chabauty--Kim method with explicit bounds, and presents a new motivic proof for uniform bounds on $S$-unit solutions.

Findings

Explicit upper bounds on $S$-integral points derived

New motivic proof for uniform bounds on $S$-unit solutions

Connection established between Selmer groups and Diophantine bounds

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 $\#S$.