Mixed Tate motives and the unit equation II

Dan-Cohen, Ishai

doi:10.2140/ant.2020.14.1175

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Mixed Tate motives and the unit equation II

Ishai Dan-Cohen

Vol. 14 (2020), No. 5, 1175–1237

DOI: 10.2140/ant.2020.14.1175

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

Vol. 14, No. 5, 2020