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 $\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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2019
- DOI:
- 10.48550/arXiv.1903.05032
- arXiv:
- arXiv:1903.05032
- Bibcode:
- 2019arXiv190305032D
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Algebraic Geometry
- E-Print:
- Several changes due to an error in Lemma 2.2 and Lemma 5.4 from the previous version. 50 pages, comments welcome