### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 1655-1672 |

Journal | Central European Journal of Mathematics |

Volume | 10 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2012 |

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### Keywords

- generalised elliptic functions
- equivariance
- sigma functions

### Cite this

*Central European Journal of Mathematics*,

*10*(5), 1655-1672. https://doi.org/10.2478/s11533-012-0083-x

**Generalised elliptic functions.** / England, Matthew; Athone, Chris.

Research output: Contribution to journal › Article

*Central European Journal of Mathematics*, vol. 10, no. 5, pp. 1655-1672. https://doi.org/10.2478/s11533-012-0083-x

}

TY - JOUR

T1 - Generalised elliptic functions

AU - England, Matthew

AU - Athone, Chris

PY - 2012

Y1 - 2012

N2 - 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.

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.

KW - generalised elliptic functions

KW - equivariance

KW - sigma functions

UR - http://www.scopus.com/inward/record.url?scp=84864498436&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1111.0777

UR - http://dx.doi.org/10.2478/s11533-012-0083-x

U2 - 10.2478/s11533-012-0083-x

DO - 10.2478/s11533-012-0083-x

M3 - Article

VL - 10

SP - 1655

EP - 1672

JO - Central European Journal of Mathematics

JF - Central European Journal of Mathematics

SN - 1895-1074

IS - 5

ER -