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

Bianchi, Francesca

doi:10.2140/ant.2020.14.2369

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

Francesca Bianchi

Vol. 14 (2020), No. 9, 2369–2416

DOI: 10.2140/ant.2020.14.2369

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 $𝒳 {(ℤ_{p})}_{2}$ containing the integral points $𝒳 (ℤ)$ 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 $𝒳 {(ℤ_{p})}_{2} ∖ 𝒳 (ℤ)$ . 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

Vol. 14, No. 9, 2020