Mixed Tate motives and the unit equation

Abstract

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.

Mixed Tate Motives and the Unit Equation

“…1.1. This is the third installment in a series [DCW1,DCW2] 1 devoted to what may reasonably be described as explicit motivic Chabauty-Kim theory. 'Chabauty-Kim theory' refers to a framework developed by Minhyong Kim for making effective use of the fundamental group to bound, or conjecturally compute, integral solutions to hyperbolic equations.…”

“…2 We also obtain an algorithmic solution to the unit equation over totally real fields obeying the same condition. Specializing to the case of the rationals, we obtain the algorithm envisioned in Dan-Cohen-Wewers [DCW2].…”

“…In the construction that follows, we attempt to bridge this gap; we fail in many ways, but are nevertheless able to make the error incurred arbitrarily small. This work does not have significant logical dependence on its predecessors [DCW1,DCW2]. The reason for this, in part, is that I found it preferable to modify portions of the work done in [DCW2] in preparation for the construction of the algorithm.…”

“…This work does not have significant logical dependence on its predecessors [DCW1,DCW2]. The reason for this, in part, is that I found it preferable to modify portions of the work done in [DCW2] in preparation for the construction of the algorithm. For instance, our use of the map "ev Everywhere " here (along with our acceptance of the p-adic period conjecture as yet another condition for halting) allows us to carry out the geometric part of the computation in a single step over the rationals and in a manner entirely divorced from arithmetic.…”

Rational motivic path spaces and Kim's relative unipotent section conjecture

“…We do not carry out the task of comparing our unipotent fundamental group with the one constructed by Deligne-Goncharov[24]. We note however that our group has all the needed properties (comparison with Tannakian étale and de Rham fundamental groups, relationship with motivic Ext groups) to carry out "motivic Chabauty-Kim theory", as in Dan-Cohen-Wewers[23] and Dan-Cohen[20].…”