Let X denote a hyperbolic curve over Q and let p denote a prime of good reduction. The third author's approach to integral points, introduced in [Kim2] and [Kim3], endows X(Zp) with a nested sequence of subsets X(Zp)n which contain X(Z). These sets have been computed in a range of special cases [Kim4, BKK, DCW2, DCW3]; there is good reason to believe them to be practically computable in general. In 2012, the third author announced the conjecture that for n sufficiently large, X(Z) = X(Zp)n. This conjecture may be seen as a sort of compromise between the abelian confines of the BSD conjecture and the profinite world of the Grothendieck section conjecture. After stating the conjecture and explaining its relationship to these other conjectures, we explore a range of special cases in which the new conjecture can be verified.

Abstract. This is the second installment in a sequence of articles devoted to "explicit ChabautyKim theory" for the thrice punctured line. Its ultimate goal is to construct an algorithmic solution to the unit equation whose halting will be conditional on Goncharov's conjecture about exhaustion of mixed Tate motives by motivic iterated integrals (refined somewhat with respect to ramification), and on Kim's conjecture about the determination of integral points via p-adic iterated integrals. In this installment we explain what this means while developing basic tools for the construction of the algorithm. We also work out an elaborate example, which goes beyond the cases that were understood before, and allows us to verify Kim's conjecture in a range of new cases.

Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of [Formula: see text]. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim’s conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim’s conjecture.

Let X=double-struckP1∖{0,1,∞}, and let S denote a finite set of prime numbers. In an article of 2005, Kim gave a new proof of Siegel's theorem for X: the set X(Z[S−1]) of S‐integral points of X is finite. The proof relies on a ‘nonabelian’ version of the classical Chabauty method. At its heart is a modular interpretation of unipotent p‐adic Hodge theory, given by a tower of morphisms hn between certain Qp‐varieties. We set out to obtain a better understanding of h2. Its mysterious piece is a polynomial in 2|S| variables. Our main theorem states that this polynomial is quadratic, and gives a procedure for writing its coefficients in terms of p‐adic logarithms and dilogarithms.

This is the second installment in a sequence of articles devoted to "explicit Chabauty-Kim theory" for the thrice punctured line. Its ultimate goal is to construct an algorithmic solution to the unit equation whose halting will be conditional on Goncharov's conjecture about exhaustion of mixed Tate motives by motivic iterated integrals (refined somewhat with respect to ramification), and on Kim's conjecture about the determination of integral points via p-adic iterated integrals. In this installment we explain what this means while developing basic tools for the construction of the algorithm. We also work out an elaborate example, which goes beyond the cases that were understood before, and allows us to verify Kim's conjecture in a range of new cases.