Generalised Elliptic Functions

TL;DR

This paper explores generalized elliptic functions related to algebraic curves, presenting two approaches—analytic via sigma-functions and algebraic via curve modification—to understand their properties and applications in mathematical physics.

Contribution

It introduces two novel methods for defining and analyzing Abelian functions associated with algebraic curves, extending existing theories to higher genus and non-hyperelliptic cases.

Findings

Derived power series expansions of Abelian functions.

Extended the sigma-function approach to non-hyperelliptic curves.

Developed an algebraic framework using representation theory for hyperelliptic curves.

Abstract

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach discussed sees the functions defined as logarithmic derivatives of the sigma-function, a modified Riemann theta-function. We can make use of known properties of the sigma function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given. The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future.