Kim’s Conjecture for the full Chabauty–Kim tower

Establish that, for the thrice-punctured line over a number field with a finite set of excluded primes, the Chabauty–Kim locus associated with sufficiently large motivic fundamental-group quotients equals the set of S-integral points; in the notation of the paper, prove that X(O_K\otimes Q_p)_{S,\Pi}=X(O_{K,S}) for sufficiently large quotients \Pi.

Background

The paper studies the Chabauty–Kim loci X(O_K\otimes Q_p){S,N}, which form a descending sequence of subsets of the p-adic points and all contain the S-integral points X(O{K,S}). Kim’s Conjecture predicts that sufficiently deep quotients of the motivic fundamental group impose enough conditions to recover exactly the S-integral points.

For K=Q(ζ_8) and S={(1−ζ_8)}, the paper verifies the conjecture at depth 4 after applying S_3-symmetrisation for several split primes p<200. The general assertion for sufficiently large quotients remains unresolved, and the authors emphasize that the polylogarithmic quotient alone cannot establish it because certain root-of-unity points persist in every depth.