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Abelian functions associated with a cyclic tetragonal curve of genus six

This page is intended to contain a repository of results for Abelian functions associated with the cyclotomic tetrogonal curve of genus six.
This is the curve C, defined by f(x,y)=0 where

f(x,y) = y4 - ( x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0 )

We define the higher genus sigma function that is associated with this curve. We then use this to define sets of Abelian functions, including generalisations of the Weierstrass p-function, and Bakers Q-functions. The following paper has been recently published by J Phys A and contains full details of the theory:

M.England and J. C. Eilbeck Abelian functions associated with a cyclic tetragonal curve of genus six, J Phys A: Math. Theor. 42 (2009) 095210
The final version of the paper can be found here.

This web page doubles as a full Appendix to this paper. Links to the full sets of equations described in the paper can be found below.

The sigma function expansion

We defined a set of weights for the variables and constants within the theory. This allowed us to calculate a power series expansion of the sigma function, by partitioning it into polynomials whose terms have the same weight ratio. We calulate the expansion as

σ(u) = C15 + C19 + C23 + ... + C15+4n + ...
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 15-k.

We have calculated sigma up to and including C59. These polynomials are given in the text files below as funtions of {v1,v2,v3,v4,v5,v6} and the curve constants.

Click on the following file names for the text file containing that polynomial.

Relations from the Kleinian formula expansion

In Section 5 of the paper we gave Theorem 5.1, and described how we had expanded this result in terms of a local parameter. We then described how we could obtain an infinite set of equations between z,w and the p-functions. We gave the first three in equations (5.1)-(5.3) and mentioned that we had calculated the first 14. These can be found in the following text file, labelled pp1-pp14, with an obvious notation used: zw_eqs.txt.

We used resultants to remove w from pairs of these equations. We then manipulated them to give equations of degree 5 in z. Such equations must be identically zero. The first two such equations were labelled (T1) and (T2) and can be found in the following text files:

Sets of differential equation satisfied by the Abelian functions

We have derived several sets of equations which are satisfied by the Abelian functions associated with C. For the precise definitionf of these functions, and the derivation of these relations, please see the paper cited above.

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This page is maintained by Matthew England, Heriot Watt University.
last updated: 30th january 2009