Motivic Fundamental Groups and Integral Points

Abstract

We give a motivic proof of finiteness of S-integral points on punctured projective line. We do this by studying torsors over different notions of unipotent fundamental groups attached to an open curve defined over a number field and the algebraic spaces which parametrize these torsors. This reduces finiteness of integral points of such curves to a strict inequality between some global and local Galois cohomology groups. When the curve is a punctured projective line, we use abelian categories of mixed Tate motives over the base number field and localizations of its ring of integers to replace the global cohomology groups by algebraic K-groups of the base number field. Finally for totally real number fields, we use Borel's explicit calculations to conclude finiteness of S-integral points.