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Quadratic Chabauty for (bi)elliptic curves and Kim’s conjecture. (English) Zbl 07275231
Summary: 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 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.
##### MSC:
 11D45 Counting solutions of Diophantine equations 11G50 Heights 11Y50 Computer solution of Diophantine equations 14H52 Elliptic curves
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