# 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.

• We have a set of relations that express the 4-index Q-functions. These were discussed in Lemma 7.2 of the paper.
The first few were given in Appendix B, but the following pdf file contains the complete set: Appendix_Full_4iQ.pdf.
The tex file for the pdf is available here: Appendix_Full_4iQ.tex
• We have a set of relations that express the 4-index p-functions. These were discussed in Corollary 7.3 of the paper.
The following pdf file contains the complete set: Appendix_Full_4iP.pdf.
The tex file for the pdf is available here: Appendix_Full_4iP.tex
We also have these relations in the following text file. These were for use in Maple, and so the notation is different, although still obvious: known_4ip.txt.
• We have a set of relations that express the 6-index Q-functions. These were also discussed in Lemma 7.2 of the paper. We have so far derived relations down to weight -41.
The set is contained in the following text file, with the notation designed for use in Maple: known_6iq.txt.
We can substitute the Q-functions for Kleinian p-functions to give the following set of equations for 6-index p-functions. known_6ip.txt.
• We have derived a set of relations bilinear in the 3-index and 3-index p-functions. These were discussed in Proposition 7.4 of the paper: lin.txt.
• We are starting to derive a set of relations that express the product of two 3-index p-functions: known_q.txt.