Deriving bases for Abelian functions

TL;DR

This paper introduces a new method for explicitly constructing bases of Abelian functions related to algebraic curves, enabling systematic derivation of addition formulas and differential equations, demonstrated on a genus four trigonal curve.

Contribution

It presents a novel approach to define Abelian function bases for algebraic curves, facilitating the derivation of addition formulas and differential equations, with explicit results for a genus four trigonal curve.

Findings

New 3-term 2-variable addition formulae

Complete set of differential equations for the functions

Generalization of Weierstrass identities

Abstract

We present a new method to explicitly define Abelian functions associated with algebraic curves, for the purpose of finding bases for the relevant vector spaces of such functions. We demonstrate the procedure with the functions associated with a trigonal curve of genus four. The main motivation for the construction of such bases is that it allows systematic methods for the derivation of the addition formulae and differential equations satisfied by the functions. We present a new 3-term 2-variable addition formulae and a complete set of differential equations to generalise the classic Weierstrass identities for the case of the trigonal curve of genus four.