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.
Bibliographical noteCopyright © and Moral Rights are retained by the author(s) and/ or other copyright
owners. A copy can be downloaded for personal non-commercial research or study,
without prior permission or charge. This item cannot be reproduced or quoted extensively
from without first obtaining permission in writing from the copyright holder(s). The
content must not be changed in any way or sold commercially in any format or medium
without the formal permission of the copyright holders.
- Abelian functions
- Addition formula
- Kleinian sigma-functions
- Trigonal curves
- Weier-strass functions
ASJC Scopus subject areas
- Applied Mathematics
- Computational Theory and Mathematics