Nonabelian Chabauty for the Thrice-punctured Line over Cyclotomic Fields

Abstract

In this paper we study the motivic Chabauty--Kim method, which aims to determine the set of $S$-integral points of $\mathbb{P}^1\smallsetminus \{0,1,\infty\}$, over cyclotomic fields. We focus on the case $K=\mathbb{Q}(ζ_8)$ and $S=\left\{(1-ζ_8)\right\}$, where we obtain explicit polylogarithmic Kim functions up to depth $4$ and verify Kim's Conjecture for several primes. We also observe and explain that the Chabauty--Kim locus for the polylogarithmic quotient contains, in addition to the $S$-integral points, certain exceptional points arising from roots of unity in $\mathbb{Q}_p$.