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

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 3.235.228.219 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.