Elliptic normal curves of even degree and theta functions

TL;DR

This paper investigates the embedding of elliptic curves of even degree into projective space using theta functions, extending classical results known for odd degrees and analyzing the associated quadratic equations.

Contribution

It introduces a new immersion method for even degree elliptic curves via theta functions and studies their quadratic relations, filling a gap in classical theory.

Findings

Defined an immersion of elliptic curves of even degree using theta functions

Analyzed the quadratic equations satisfied by these theta functions

Extended classical results from odd to even degree cases

Abstract

An elliptic curve may be immersed in ${\mathbf{P}}^{N-1}$ as a degree $N$ curve using level $N$ structure. In the case where $N$ is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the case of even $N$ in some detail. In particular, over the complex number field, we define an immersion using suitably chosen theta functions, and study the quadratic equations satisfied by them.