Unlikely intersections and the Chabauty–Kim method over number fields

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 $${\mathbb {Q}}_p $$ 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.

M0, 5: Toward the Chabauty–Kim method in higher dimensions

“…We may then ask if the vanishing locus of 𝐹 BC and its conjugates is finite, or even equal to the set of integral points. Precedent for computations of this sort may be found in [3,23,30]. Unfortunately, the large size of 𝐹 BC presents a hurdle to computation.…”

Variations on the method of Chabauty and Coleman

“…While these equations often cut out a finite set in practice, this is not always the case. For instance, if X is defined over Q and does not satisfy the Chabauty rank condition over Q, this method cannot succeed over any number field K, even if X satisfies the rank condition over K. In addition, the method sometimes does not work for more subtle reasons, as explained in [Dog23]. For more work on Chabauty methods based on restriction of scalars, see [Tri21].…”