Vol. 14, No. 5, 2020

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Mixed Tate motives and the unit equation II

Ishai Dan-Cohen

Vol. 14 (2020), No. 5, 1175–1237
Abstract

Over the past fifteen years or so, Minhyong Kim has developed a framework for making effective use of the fundamental group to bound (or even compute) integral points on hyperbolic curves. This is the third installment in a series whose goal is to realize the potential effectivity of Kim’s approach in the case of the thrice punctured line. As envisioned by Dan-Coehn and Wewers (2016), we construct an algorithm whose output upon halting is provably the set of integral points, and whose halting would follow from certain natural conjectures. Our results go a long way towards achieving our goals over the rationals, while broaching the topic of higher number fields.

Keywords
mixed Tate motives, unipotent fundamental group, p-adic periods, polylogarithms, unit equation, integral points
Mathematical Subject Classification 2010
Primary: 11G55
Secondary: 11D45, 14F30, 14F35, 14F42, 14G05
Milestones
Received: 10 October 2018
Revised: 16 September 2019
Accepted: 27 November 2019
Published: 13 July 2020
Authors
Ishai Dan-Cohen
Department of Mathematics
Ben Gurion University
Be’er Sheva
Israel