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arXiv · 2609.01128

Nonabelian Chabauty for the Thrice-punctured Line over Cyclotomic Fields

Minhyong Kim·Xiang Li·Martin Lüdtke
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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 $K=\mathbb{Q}(\zeta_8)$ and $S=\left\{(1-\zeta_8)\right\}$, 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 $\mathbb{Q}_p$.

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Minhyong Kim, Xiang Li, Martin Lüdtke. 2026-09-01. Nonabelian Chabauty for the Thrice-punctured Line over Cyclotomic Fields. https://arxiv.org/abs/2609.01128

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