Let K be a number field, let S be a finite set of finite places of K, and let X=P^1smallsetminus0,1,∞. Kim's conjecture asserts that the motivic Chabauty--Kim loci of X eventually determine exactly the S-integral points: there exists n_0 such that, for every depth n≥ n_0, X^CK_n(K,S)=X(O_K,S), where X^CK_n(K,S) denotes the depth-n Chabauty--Kim locus.

Kim’s conjecture for S-integral points on the thrice-punctured line

The conjecture concerns determining the points integral outside a finite set of primes on the thrice-punctured projective line. The new work advances the method over one cyclotomic field, but does not address the conjecture in full generality.

Known results

$S=\varnothing$ , depth $2$ , for $p<10^5$ (Balakrishnan, Dan-Cohen, Kim, and Wewers, 2018).
$S=\{2\}$ , depth $4$ , for $3\le p\le29$ (Dan-Cohen and Wewers, 2016).
$S=\{3\}$ , depth $4$ , for $p\in\{5,7\}$ (Corwin and Dan-Cohen, 2020).
For $S=\{2,3\}$ , depth $4$ verifies the refined conjecture for $5\le p<10{,}000$ , with points $\{-3,-1,3,9\}$ (2024).

September 2026 cyclotomic development

A preprint develops the motivic Chabauty–Kim method to depth four over $K=\mathbb{Q}(\zeta_8)$ for a specified set $S$ , reports verification at several primes, and identifies exceptional $p$ -adic roots-of-unity points. This is claimed progress, not a verified resolution: it treats one field and a polylogarithmic quotient rather than the general conjecture.

Current status (as of September 2026): Special refined cases are established, and a new cyclotomic computation is reported, but the general conjecture for $S$ -integral points remains open.

Kim’s conjecture for S-integral points on the thrice-punctured line

Primary source

Additional references

Progress summary

Solutions 0