Nonabelian Chabauty for the Thrice-punctured Line over Cyclotomic Fields

In this paper we study the motivic Chabauty–Kim method, which aims to determine the set of $S$-integral points of ${\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$, over cyclotomic fields. We focus on the case $K=\mathbb{Q}({\zeta }_8)$ and $S=\{(1-{\zeta }_8)\}$, 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 ${\mathbb{Q}}_p$.