Some New Addition Formulae for Weierstrass Elliptic Functions

TL;DR

This paper introduces new addition formulae for Weierstrass elliptic functions associated with general elliptic curves, expanding the understanding of their structure and explicit formulas for multiple variables.

Contribution

It provides the first explicit addition formulae for Weierstrass functions in multiple variables for general elliptic curves, generalizing previous special cases.

Findings

Derived explicit 2- and 3-variable addition formulae

Established the structure of n-variable addition formulae

Connected new formulae to higher genus curve generalizations

Abstract

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.