Bring’s curve: old and new

Abstract

Bring’s curve, the unique Riemann surface of genus 4 with automorphism group S₅, has many exceptional properties. We review, give new proofs of, and extend a number of these including giving the complete realisation of the automorphism group for a plane curve model, identifying a new elliptic quotient of the curve and the modular curve X₀ (50), providing a complete description of the orbit decomposition of the theta characteristics, and identifying the unique invariant characteristic with the divisor of the Szegő kernel. In achieving this we have used modern computational tools in Sagemath, Macaulay2, and Maple, for which notebooks demonstrating calculations are provided.

Metadata

Item Type:	Article
Authors/Creators:	Braden, H.W. Disney-Hogg, L. https://orcid.org/0000-0002-6597-2463
Copyright, Publisher and Additional Information:	© The Author(s) 2023. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords:	Bring's curve; Automorphisms; Theta characteristics; Weierstrass points; Isogeny; Computation
Dates:	Accepted: 16 October 2023 Published (online): 18 December 2023 Published: March 2024
Institution:	The University of Leeds
Academic Units:	The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds)
Date Deposited:	21 Apr 2026 10:01
Last Modified:	21 Apr 2026 10:02
Published Version:	https://link.springer.com/article/10.1007/s40879-0...
Status:	Published
Publisher:	Springer Nature
Identification Number:	10.1007/s40879-023-00706-0
Related URLs:	Author Dataset
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:239895