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Deriving bases for Abelian functions


This page is a repository of results related to the following paper: Journal website.
Preprint: arXiv:1103.0468.


Abstract: We present a new method to explicitly define Abelian functions on algebraic curves, for the purpose of finding bases for the relevant vector spaces of such functions. We demonstrate the procedure with the functions associated to 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.


This paper makes reference to large sets of relations, which for reasons of brevity have not been included in the paper. As such links to these relations are provided here. We also provide text files of some of the results in the paper for use by anyone who is interested.


Abelian functions with prescribed pole orders

The paper derives several new classes of Abelian functions whose maximum pole orders are known for a general (n,s)-curve. They are all contained in the following text files, organised by pole order and number of indices. Note that there are an infinite number of functions with poles of order 2 given by the m-index Q-functions. Hence here we begin with poles of order 3.


The cyclic (3,5)-curve

The paper considers the cyclic trigonal curve of genus four, which can be defined by f(x,y)=0 where

f(x,y) = y3 - ( x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0 ).

The sigma-function expansion

Throughout the paper we refer to a power series expansion for the sigma function associated to the curve. This was constructed by partitioning it into polynomials whose terms have the same weight ratio. The expansion was given as

σ(u) = SW3,5 + C11 + C14 + ... + C8+3n + ...
where Ck is a finite polynomial composed of products of monomials in ui of total weight k, multiplied by monomials in λj of total weight 8-k. These polynomials are given in the text files below as functions of {v1,v2,v3,v4} and the curve constants lambda. Click on the file names for the text file containing that polynomial.

Relations between the Abelian functions

In Section 4 of the paper we derived three complete sets of differential equations for the functions associated with the (3,5)-curve.

The new addition formula

Theorem 5.2 presented a new addition formula for the cyclic (3,5)-curve. It was constructed using the polynomials in Appendix C which we make available in a text file here.


Maple code to evaluate products of series expansions

Many results were derived in Maple using the sigma function expansion. This required evaluating products of expansions to gain a new expansion valid up to a certain weight in lambda. The following code was written for this purpose. The code is specified to the (3,5)-case but it is trivial to edit it for any (n,s)-case.


This page is maintained by Matthew England.
last updated: 7th Sep 2011