Bring's Curve: its Period Matrix and the Vector of Riemann Constants

TL;DR

This paper analyzes Bring's curve, a genus 4 Riemann surface with maximal symmetry, by deriving its period matrix and Riemann constants using an algebraic model, extending previous hyperbolic studies.

Contribution

It introduces an algebraic approach to study Bring's curve, recovering and extending prior hyperbolic model results, including the period matrix and Riemann constants.

Findings

Explicit period matrix of Bring's curve obtained.

Vector of Riemann constants identified.

Extension of previous hyperbolic model results.

Abstract

Bring's curve is the genus 4 Riemann surface with automorphism group of maximal size, $S_5$. Riera and Rodr\'iguez have provided the most detailed study of the curve thus far via a hyperbolic model. We will recover and extend their results via an algebraic model based on a sextic curve given by both Hulek and Craig and implicit in work of Ramanujan. In particular we recover their period matrix; further, the vector of Riemann constants will be identified.