Action of the automorphism group on the Jacobian of Klein's quartic curve

TL;DR

This paper investigates the automorphism group action on Klein's quartic curve's Jacobian, revealing that the quotient is a weighted projective space with specific singularities, thus contributing to the understanding of complex crystallographic groups and their quotients.

Contribution

It computes the orbits and stabilizers of the automorphism group action on the Jacobian of Klein's quartic and confirms the quotient's structure as a weighted projective space, addressing a case of a conjecture involving complex reflection groups.

Findings

The quotient $J/G$ is a strongly simply connected variety.

$J/G$ has the same singularities as $ ext{P}(1,2,4,7)$.

The group $ ilde G$ is generated by affine complex reflections.

Abstract

Klein's simple group $H$ of order $168$ is the automorphism group of the plane quartic curve $C$, called Klein quartic. By Torelli Theorem, the full automorphism group $G$ of the Jacobian $J=J(C)$ is the group of order $336$, obtained by adding minus identity to $H$. The quotient variety $J/G$ can be alternatively represented as the quotient $\mathbb C^3/\tilde G$ of the complex $3$-space by the complex crystallographic group $\tilde G$, the extension of $G$ by the period lattice of the Klein quartic. Moreover, it turns out that $\tilde G$ is generated by affine complex reflections. According to a conjecture of Bernstein--Schwarzman, a quotient of $\mathbb C^n$ by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture is known in dimension two and for complexifications of the real crystallographic groups generated by reflections. The case of $\tilde G$ is the first, and in a sense the smallest of the unknown cases. We compute the orbits and the stabilizers of the action of $G$ on $J$ and deduce that $J/G=\mathbf C^3/\tilde G$ is a strongly simply connected variety with the same singularities as the weighted projective space $\mathbb P(1,2,4,7)$.