Some remarks on the correspondence between elliptic curves and four points in the Riemann sphere

TL;DR

This paper explores the relationship between elliptic curves and four-point configurations on the Riemann sphere, providing new proofs of classical normal forms and geometric constructions of cross ratios.

Contribution

It offers an alternative proof that all elliptic curves are isomorphic to the Hesse normal form and clarifies the equivalence of Edwards and Jacobi forms.

Findings

Elliptic curves are isomorphic to the Hesse normal form.

Established the equivalence between Edwards and Jacobi normal forms.

Provided a geometric construction of cross ratios for four-point sets.

Abstract

In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in the Hesse normal form. In this setting, we give an alternative proof of the equivalence between the Edwards and the Jacobi normal forms. Also, we give a geometric construction of the cross ratios for 4-point sets in general position.