Vol. 14, No. 9, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 2, 231–519
Issue 1, 1–230

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 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
This article is available for purchase or by subscription. See below.
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 𝒳(p)2 containing the integral points 𝒳() 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 𝒳(p)2 𝒳(). 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.

PDF Access Denied

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.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords
quadratic Chabauty, p-adic heights, integral points on hyperbolic curves
Mathematical Subject Classification 2010
Primary: 11D45
Secondary: 11G50, 11Y50, 14H52
Milestones
Received: 12 April 2019
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