The group of automorphisms of the Heisenberg curve

TL;DR

This paper investigates the automorphism group of the Heisenberg curve, an algebraic curve linked to the Fermat curve, providing explicit equations and analyzing its symmetries.

Contribution

It offers a detailed study of the automorphism group of the Heisenberg curve and presents an explicit equation for the specific case of $C_3$, advancing understanding of its structure.

Findings

Automorphism group of the Heisenberg curve characterized.

Explicit equation for the curve $C_3$ provided.

Connections established between the Heisenberg and Fermat curves.

Abstract

The Heisenberg curve is defined to be the curve corresponding to an extension of the projective line by the Heisenberg group modulo $n$, ramified above three points. This curve is related to the Fermat curve and its group of automorphisms is studied. Also we give an explicit equation for the curve $C_3$.