Bring's curve: Old and New

Abstract

Bring's curve, the unique Riemann surface of genus-4 with automorphism group $S_5$, 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 $X_0(50)$, providing a complete description of the orbit decomposition of the theta characteristics, and identifying the unique invariant characteristic with the divisor of the Szëgo kernel. In achieving this we have used modern computational tools in Sagemath, Macaulay2, and Maple, for which notebooks demonstrating calculations are provided.

Figures (5)

• Figure 1: Geometric realisations.
• Figure 2: Hyperbolic tilings.
• Figure 3: Quotient structure of Bring's curve.
• Figure 4: Subgroup structre of $S_5$ corresponding to the quotients of Bring's curve.
• Figure 5: Quotient structure including marked points. The points $\textcolor{red}{\times}$ correspond to the Weierstrass points $[1: 1: \alpha : \beta: \gamma]$ (and permutations of $\alpha, \beta, \gamma$), the points $\textcolor{blue}{\otimes}$ to the vertices $[1: -1: 0: \pm i : \mp i]$ and $[1: -1: \pm i : 0 : \mp i]$, the points $\textcolor{blue}{\ocircle}$ to the vertices $[1: -1: \pm i : \mp i : 0]$, and the points $\textcolor{blue}{\square}$ correspond to the vertices $[1: \pm i : \mp i : -1 : 0]$ and $[1: \pm i : -1 : \mp : 0]$.

Theorems & Definitions (60)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4: Braden2012
• Lemma 2.5
• Corollary 2.6
• Lemma 2.7: Braden2012
• Lemma 2.8
• Proposition 2.9
• proof