Explicit Chabauty-Kim theory for the thrice punctured line in depth 2

Abstract

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.

Mixed Tate Motives and the Unit Equation

“…The commutativity of the upper square is discussed at length in Dan-Cohen-Wewers [DCW1] for n = 2; the same discussion applies here, without change.…”

“…An important point is that step one will not depend on making effective use of Borel's computations of higher K-groups. In this sense, our construction here is less precise than the one made in [DCW1] where we made effective use of the vanishing of K 2 (Q) ⊗ Q to construct a preferred basis of A(S) 2 .…”

“…A range of special cases which may be approached by the methods of nonabelian Galois cohomology are summarized in [BDCKW]. The present article is the second in a series, begun in [DCW1], whose goal, as we now see it, is to approach the case of the thrice punctured line…”

Explicit Chabauty–Kim for the split Cartan modular curve of level 13

“…As in the method of Chabauty and Coleman, one hopes to be able to translate Kim's approach into a practical explicit method for computing (a finite set of p-adic points containing) X(Q) in practice for a given curve X/ Q having r ≥ g. However, in part due to the technical nature of the objects involved, this is a rather delicate task. Kim's results [Kim05] on integral points on P 1 \ {0, 1, ∞} have been made explicit by Dan-Cohen and Wewers [DCW15] and used to develop an algorithm to solve the S-unit equation [DCW16,DC17] using iterated p-adic integrals. The work [BDCKW] of the first author with Dan-Cohen, Kim and Wewers contains explicit results for integral points on elliptic curves of ranks 0 and 1.…”

$M_{0,5}$: Towards the Chabauty-Kim method in higher dimensions

“…The methods of Dan-Cohen-Wewers [DCW1,DCW2,DC] and Corwin-Dan-Cohen [CDC1,CDC2], while so far limited to the simplest of all cases (X " P 1 zt0, 1, 8u), have been particularly successful in going beyond the quadratic level. These articles incorporate the methods of mixed Tate motives and motivic iterated integrals (see, for instance, [DG,Gon,Bro1,Bro3]).…”