# Generalised elliptic functions

Open Mathematics (2012)

• Volume: 10, Issue: 5, page 1655-1672
• ISSN: 2391-5455

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## Abstract

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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 Weierstraß ℘-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 σ-function, a modified Riemann θ-function. We can make use of known properties of the σ-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. We consider how the two approaches may be combined, giving the explicit mappings for the genus 3 hyperelliptic theory. We consider the problem of generating bases of the functions and how these decompose when viewed in the equivariant form.

## How to cite

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Matthew England, and Chris Athorne. "Generalised elliptic functions." Open Mathematics 10.5 (2012): 1655-1672. <http://eudml.org/doc/269465>.

@article{MatthewEngland2012,
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 Weierstraß ℘-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 σ-function, a modified Riemann θ-function. We can make use of known properties of the σ-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. We consider how the two approaches may be combined, giving the explicit mappings for the genus 3 hyperelliptic theory. We consider the problem of generating bases of the functions and how these decompose when viewed in the equivariant form.},
author = {Matthew England, Chris Athorne},
journal = {Open Mathematics},
keywords = {Generalised elliptic functions; Sigma functions; Equivariance; generalised elliptic functions; sigma functions; equivariance},
language = {eng},
number = {5},
pages = {1655-1672},
title = {Generalised elliptic functions},
url = {http://eudml.org/doc/269465},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Matthew England
AU - Chris Athorne
TI - Generalised elliptic functions
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1655
EP - 1672
AB - 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 Weierstraß ℘-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 σ-function, a modified Riemann θ-function. We can make use of known properties of the σ-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. We consider how the two approaches may be combined, giving the explicit mappings for the genus 3 hyperelliptic theory. We consider the problem of generating bases of the functions and how these decompose when viewed in the equivariant form.
LA - eng
KW - Generalised elliptic functions; Sigma functions; Equivariance; generalised elliptic functions; sigma functions; equivariance
UR - http://eudml.org/doc/269465
ER -

## References

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