#### Vol. 14, No. 9, 2020

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

### Francesca Bianchi

Vol. 14 (2020), No. 9, 2369–2416
##### 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 $\mathsc{𝒳}{\left({ℤ}_{p}\right)}_{2}$ containing the integral points $\mathsc{𝒳}\left(ℤ\right)$ 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 $\mathsc{𝒳}{\left({ℤ}_{p}\right)}_{2}\setminus \mathsc{𝒳}\left(ℤ\right)$. 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.

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##### Mathematical Subject Classification 2010
Primary: 11D45
Secondary: 11G50, 11Y50, 14H52
##### Milestones
Revised: 1 February 2020
Accepted: 23 April 2020
Published: 13 October 2020
##### Authors
 Francesca Bianchi Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence University of Groningen Groningen Netherlands