Quadratic Chabauty and rational points I: p-adic heights

Abstract

We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$-adic heights in explicit nonabelian Chabauty.

arXiv comment45 pages, including two appendices; Appendix B by J. Steffen Mueller

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Cited by in corpus (17)

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• Geometric quadratic Chabauty
• Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture
• The local theory of unipotent Kummer maps and refined Selmer schemes
• Explicit isogenies of prime degree over quadratic fields
• Quadratic Chabauty for Modular Curves
• Explicit Coleman Integration in Larger Characteristic
• Geometric quadratic Chabauty and $p$-adic heights
• Geometric quadratic Chabauty over number fields
• p-adic adelic metrics and Quadratic Chabauty I
• A geometric linear Chabauty comparison theorem
• Algorithms for $p$-adic Heights on Hyperelliptic Curves of Arbitrary Reduction
• Quadratic Chabauty and $p$-adic Gross-Zagier
• $M_{0,5}$: Towards the Chabauty-Kim method in higher dimensions
• The Chabauty-Kim Method for Relative Completions
• Linear and quadratic Chabauty for affine hyperbolic curves
• A naive p-adic height on the Jacobians of curves of genus 2