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

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 $\mathcal{𝒳}{\left({\mathbb{Z}}_p\right)}_2$ containing the integral points $\mathcal{𝒳}\left(\mathbb{Z}\right)$ 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 $\mathcal{𝒳}{\left({\mathbb{Z}}_p\right)}_2\setminus \mathcal{𝒳}\left(\mathbb{Z}\right)$. 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.