Klein's Curve

TL;DR

This paper derives a new explicit expression for Klein's curve period matrix by choosing a homology basis aligned with its symmetries, connecting algebraic and hyperbolic models.

Contribution

It introduces a homology basis adapted to Klein's curve symmetries, leading to a novel period matrix expression linked to hyperbolic geometry.

Findings

New explicit period matrix expression for Klein's curve

Connection established between algebraic and hyperbolic models

Enhancement of understanding of Klein's curve symmetries

Abstract

By giving an homology basis well adapted to the symmetries of Klein's curve, presented as a plane curve, we derive a new expression for its period matrix. This is explicitly related to the hyperbolic model and results of Rauch and Lewittes.