Vol. 14, No. 9, 2020

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Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture

Francesca Bianchi

Vol. 14 (2020), No. 9, 2369–2416
Abstract

We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets 𝒳(p)2 containing the integral points 𝒳() of an elliptic curve of rank at most 1. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference 𝒳(p)2 𝒳(). We also consider some algorithmic questions arising from Balakrishnan and Dogra’s explicit quadratic Chabauty for the rational points of a genus-two bielliptic curve. As an example, we provide a new solution to a problem of Diophantus which was first solved by Wetherell.

Computationally, the main difference from the previous approach to quadratic Chabauty is the use of the p-adic sigma function in place of a double Coleman integral.

Keywords
quadratic Chabauty, p-adic heights, integral points on hyperbolic curves
Mathematical Subject Classification 2010
Primary: 11D45
Secondary: 11G50, 11Y50, 14H52
Milestones
Received: 12 April 2019
Revised: 1 February 2020
Accepted: 23 April 2020
Published: 13 October 2020
Authors
Francesca Bianchi
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence
University of Groningen
Groningen
Netherlands