Mixed Tate motives and the unit equation II
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 in the last installment, we construct an algorithm whose output upon halting is provably the set of integral points, and whose halting would follow from conjectures. Our results go a long way towards achieving our goals over the rationals, while broaching the topic of higher number fields.
Keywords
Cite
@article{arxiv.1510.01362,
title = {Mixed Tate motives and the unit equation II},
author = {Ishai Dan-Cohen},
journal= {arXiv preprint arXiv:1510.01362},
year = {2020}
}
Comments
The "realization algorithm" of previous versions contained an error, so has been substantially reworked in section 3.7. Numerous inaccuracies in my account of the numerical approximation algorithm have been corrected in section 6. Finally, the Hasse principle in Galois cohomology of Condition 2.2.13 is now shown to follow from a conjecture due to Jannsen. To appear in Algebra and Number Theory