Unlikely intersections and the Chabauty-Kim method over number fields

TL;DR

This paper advances the understanding of the Chabauty--Kim method over number fields by establishing foundational results, including unlikely intersection theorems and analysis of Selmer schemes, with implications for rational point detection.

Contribution

It provides new foundational results on the Chabauty--Kim method over number fields, including unlikely intersection results and analysis of Selmer schemes, addressing Siksek's question.

Findings

Established unlikely intersection results for zeroes of iterated integrals.

Analyzed intersections of Selmer schemes with unipotent Albanese varieties.

Provided a counterexample to the strong form of Siksek's question.

Abstract

The Chabauty--Kim method is a tool for finding the integral or rational points on varieties over number fields via certain transcendental $p$-adic analytic functions arising from certain Selmer schemes associated to the unipotent fundamental group of the variety. In this paper we establish several foundational results on the Chabauty--Kim method for curves over number fields. The two main ingredients in the proof of these results are an unlikely intersection result for zeroes of iterated integrals, and a careful analysis of the intersection of the Selmer scheme of the original curve with the unipotent Albanese variety of certain $\mathbf{Q} _p $-subvarieties of the restriction of scalars of the curve. The main theorem also gives a partial answer to a question of Siksek on Chabauty's method over number fields, and an explicit counterexample is given to the strong form of Siksek's question.