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.

中文翻译：

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（双）椭圆曲线的二次 Chabauty 和 Kim 猜想

我们探讨了与用于确定双曲线上积分点的二次 Chabauty 方法相关的许多问题。我们在包含积分点 $\mathcal{X}(\mathbb{Z})$ 的二次 Chabauty 集 $\mathcal{X}(\mathbb{Z}_p)_2$ 的描述中删除了半稳定性假设。秩的椭圆曲线至多 $1$。受 Kim 猜想的启发，我们从理论上和计算上研究了集合论差异 $\mathcal{X}(\mathbb{Z}_p)_2\setminus \mathcal{X}(\mathbb{Z})$。我们还考虑了由 Balakrishnan 产生的一些算法问题——Dogra 对属二双椭圆曲线的有理点的显式二次 Chabauty。作为一个例子，我们提供了一个新的解决方案来解决一个丢番图问题，这个问题首先由 Wetherell 解决。在计算上，