Quadratic Chabauty and rational points. I: \(p\)-adic heights. (English) Zbl 1401.14123

Balakrishnan, Jennifer S.; Dogra, Netan

Duke Math. J. 167, No. 11, 1981-2038 (2018).

Let \(X\) be a smooth projective curve of genus \(g\geq 2\) defined over a number field \(K\). The \(p\)-adic method of Chabauty and Coleman gives an effective way of computing the finite set \(X(K)\), under some conditions, the main one being that the Mordell-Weil rank of the Jacobian be less than \(g\). Kim’s non abelian approach replaces the Jacobian by the Selmer variety [M. Kim, Invent. Math. 161, No. 3, 629–656 (2005; Zbl 1090.14006)]. In the paper under review, the authors introduce new techniques for studying Selmer varieties which enable them to produce new methods for determining the rational points of a variety over \({\mathbb{Q}}\) or over a quadratic field when the Mordell-Weil rank is equal to \(g\). The methods generalize those which were used for the study of integral points on hyperelliptic curves using \(p\)-adic heights in [J. S. Balakrishnan et al., J. Reine Angew. Math. 720, 51–79 (2016; Zbl 1350.11067)]. They also use the results of the PhD Thesis of the second author [Topics in the theory of Selmer varieties. Oxford: Oxford University (2015)]. A crucial role in the proof is played by Nekovář’s approach to the \(p\)-adic height pairing [J. Nekovář, Prog. Math. 108, 127–202 (1993; Zbl 0859.11038)]. As an example, the authors show how to compute explicitly a finite set containing the rational points over \({\mathbb{Q}}\) or over a quadratic number field for a genus \(2\) bielliptic curve of Mordell-Weil rank \(2\). The example of \(X_0(37)\) over \({\mathbb{Q}}(i)\) is given; in an appendix to this paper, Stephen Müller carries out the Mordell-Weil sieve computations.

Reviewer: Michel Waldschmidt (Paris)

MSC:

14G05	Rational points
11G50	Heights
14G40	Arithmetic varieties and schemes; Arakelov theory; heights

Keywords:

Chabauty method; Chabauty-Coleman method; Chabauty-Kim method; Mordell’s conjecture; Faltings’s theorem; rational points; \(p\)-adic height; Selmer varieties; hyperelliptic curves; bielliptic curves; conjectures of Bloch and Kato; Nekovář’s \(p\)-adic height pairing; mixed extensions; Mordell-Weil sieve

Citations:

Zbl 1090.14006; Zbl 1350.11067; Zbl 0859.11038

Software:

SageMath; GitHub; QCI

References:

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