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arXiv · 1910.10752

Geometric quadratic Chabauty

Bas Edixhoven·Guido Lido
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Abstract

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both 3). This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only `simple algebraic geometry' (line bundles over the jacobian and models over the integers).

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Bas Edixhoven, Guido Lido. 2019-10-23. Geometric quadratic Chabauty. https://doi.org/10.1017/s1474748021000244

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