Quadratic Chabauty for (bi)elliptic curves and Kim’s conjecture. (English) Zbl 07275231

Bianchi, Francesca

Algebra Number Theory 14, No. 9, 2369-2416 (2020).

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

Keywords:

quadratic Chabauty; p-adic heights; integral points on hyperbolic curves