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

## TL;DR

This paper advances the quadratic Chabauty method for hyperbolic curves, removing semistability assumptions, exploring set differences related to Kim's conjecture, and applying new computational techniques to solve classical Diophantine problems.

## Contribution

It removes semistability assumptions in quadratic Chabauty, investigates set differences related to Kim's conjecture, and introduces a new computational approach using the p-adic sigma function.

## Key findings

- Removed semistability assumption in quadratic Chabauty sets
- Analyzed the set-theoretic difference related to Kim's conjecture
- Provided a new solution to a classical Diophantine problem

## 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 $\mathcal{X}(\mathbb{Z}_p)_2$ containing the integral points $\mathcal{X}(\mathbb{Z})$ 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 $\mathcal{X}(\mathbb{Z}_p)_2\setminus \mathcal{X}(\mathbb{Z})$. We also consider some algorithmic questions arising from Balakrishnan--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.