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 -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.
Keywords
Cite
@article{arxiv.1311.7008,
title = {Mixed Tate motives and the unit equation},
author = {Ishai Dan-Cohen and Stefan Wewers},
journal= {arXiv preprint arXiv:1311.7008},
year = {2015}
}
Comments
Longer introduction. Some new material comparing the various notions of motivic iterated integral appearing in the literature. Some clarifying remarks regarding our hoped-for algorithm. In a new appendix we produce a second proof of the motivic identity for Li_3(1/2) via complex polylogarithms given to us by one of the referees