Motivic fundamental groups and integral points

Abstract

We use motivic fundamental groups to show that $S$-integral points on a unirational variety over a totally real number field whose fundamental group is nonabelian enough in a certain sense can be covered by zero loci of finitely many nonzero $p$-adic analytic functions. In particular, in the $1$-dimensional case we obtain a motivic proof of finiteness of $S$-integral points of punctured projective line over totally real number fields, which gives as a special case a motivic proof of Siegel’s theorem over $\mathbb{Q}$ and totally real quadratic number fields.