[Paper Review] Mixed Tate motives and the unit equation II (formerly titled Explicit motivic Chabauty-Kim theory III)

Ishai Dan‐Cohen|arXiv (Cornell University)|Oct 5, 2015

Algebraic Geometry and Number Theory24 references4 citations

TL;DR

This paper advances Kim's non-abelian Chabauty method by constructing an algorithm that provably computes the set of integral points on the thrice-punctured line over the rationals, using explicit motivic iterated integrals and conjectural finiteness assumptions; it achieves full effectivity in this case and extends the framework to higher number fields.

ABSTRACT

Over the past twelve 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.

Motivation & Objective

• To realize the effectivity of Kim's motivic Chabauty-Kim method in the case of the thrice-punctured line over the rationals.
• To construct a concrete algorithm whose output is provably the full set of integral points, assuming standard conjectures.
• To extend the framework of motivic non-abelian Chabauty to higher number fields beyond the rationals.
• To provide a computational realization of the motivic iterated integral approach in explicit arithmetic geometry.
• To bridge the gap between conjectural finiteness and effective computation of integral points on hyperbolic curves.

Proposed method

• Utilizes explicit motivic iterated integrals to construct p-adic period maps encoding arithmetic information.
• Applies the non-abelian Chabauty method via the fundamental group of the thrice-punctured line to constrain integral points.
• Employs a filtration on the motivic Lie algebra to define a system of p-adic equations whose solutions correspond to integral points.
• Relies on conjectural finiteness of certain Selmer varieties to ensure the algorithm halts after finitely many steps.
• Integrates tools from mixed Tate motives to compute the relevant cohomological invariants in the motivic setting.
• Adapts the framework to number fields beyond Q by extending the motivic and p-adic structures to general rings of S-integers.

Experimental results

Research questions

• RQ1 Can Kim's non-abelian Chabauty-Kim method be made fully effective for computing integral points on the thrice-punctured line over Q?
• RQ2 What explicit algorithmic structure can be derived from motivic iterated integrals to bound integral points?
• RQ3 How can the conjectural finiteness of Selmer varieties be leveraged to ensure halting of the algorithm?
• RQ4 To what extent can the method be generalized to higher number fields beyond the rationals?
• RQ5 What role do mixed Tate motives play in constructing effective p-adic period maps for integral points?

Key findings

• An explicit algorithm is constructed whose output is provably the full set of integral points on the thrice-punctured line over Q.
• The algorithm's halting is conditional on standard conjectures, particularly the finiteness of certain Selmer varieties.
• The method achieves full effectivity in the case of the thrice-punctured line, realizing a key goal of Kim's program.
• The framework is extended to higher number fields, opening the door to applications beyond the rationals.
• The use of motivic iterated integrals provides a concrete computational pathway to non-abelian p-adic period maps.
• The results represent a major step toward making the non-abelian Chabauty method fully algorithmic and effective in arithmetic geometry.