Generalised elliptic functions

England, M. and Athorne, C. (2012) Generalised elliptic functions. Central European Journal of Mathematics, 10(5), pp. 1655-1672. (doi: 10.2478/s11533-012-0083-x)

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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.<p></p> 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.<p></p> 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.<p></p>

Item Type:Articles
Additional Information:Paper originally presented at Finite Dimensional Integrable Systems in Geometry and Mathematical Physics 2011 (FDIS2011), Jena, Germany, 26-29 July 2011. Accepted for publication 03 April, 2012.
Glasgow Author(s) Enlighten ID:England, Dr Matthew and Athorne, Dr Chris
Authors: England, M., and Athorne, C.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Central European Journal of Mathematics
Publisher:SP Versita
ISSN (Online):1644-3616
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