Higher genus Abelian functions associated with cyclic trigonal curves
This page is a repository of results related to the following paper, published by
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications):
- M. England Higher genus Abelian functions associated with cyclic trigonal curves.
SIGMA 6 (2010), 025, 22 pages.
A pdf of the final version of the paper is available here.
Abstract: We develop the theory of Abelian functions associated with cyclic trigonal curves by considering two new cases.
We investigate curves of genus six and seven and consider whether it is the trigonal nature or the genus which dictates certain areas of the theory.
We present solutions to the Jacobi Inversion Problem, sets of relations between the Abelian function, links to Boussinesq equation
and new addition formula.
This paper makes reference to a large number of relations, which for reasons of brevity have not been included in the paper.
As such links to these relations are provided here.
The (3,7)-curve
The paper considers the cyclic trigonal curve of genus six, which can be defined by f(x,y)=0 where
f(x,y) = y3 - ( x7 + λ6x6 + λ5x5 + λ4x4 + λ3x3 + λ2x2
+ λ1x + λ0 ).
Relations derived from the Kleinian formula expansion
In Section 3 of the paper we give the Kleinian formula for the curve 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 the paper
and give the rest in the following text file. They are labelled pp1-pp10, with an obvious notation used:
rhoi.txt
The sigma-function expansion
In Section 4 of the paper we derive 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) = SW3,7 + C19 + C22 + ... + C16+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 16-k.
These polynomials are given in the text files below as functions of
{v1,v2,v3,v4,v5,v6} and the curve constants lambda.
Click on the file names for the text file containing that polynomial.
Relations between the Abelian functions
We have derived several sets of equations which are satisfied by the Abelian functions associated with the curve, as detailed in Section 5 of the paper.
- We have a set of relations that express those 4-index Q-functions not in the basis as a linear combination of basis entries.
The first few were given in Lemma 1 with the set down to weight -37 available in this text file: FiQ.txt
- We have similar expression for the 6-index Q-functions with the set to weight -32 available in this text file: SiQ.txt
The (3,8)-curve
In Section 7 of the paper we repeat some of the calculations for the cyclic trigonal curve of genus seven, which can be defined by f(x,y)=0 where
f(x,y) = y3 - ( x8 + λ7x7 + λ6x6 + λ5x5 + λ4x4 + λ3x3 + λ2x2
+ λ1x + λ0 ).
Relations derived from the Kleinian formula expansion
We again derive relations from the Kleinian formula and give the first eight in the following text file: rhoi.txt
The sigma-function expansion
We also derive the sigma function expansion in this case. This was was given as
σ(u) = SW3,8 + C24 + C27 + ... + C21+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 21-k.
The polynomials are given in the text files below.
Relations between the Abelian functions
We have derived several sets of equations which are satisfied by the Abelian functions associated with the curve, as detailed in Section 7.4 of the paper.
- We have a set of relations that express those 4-index Q-functions not in the basis as a linear combination of basis entries.
The first few were given in the paper with the set down to weight -20 available in this text file: FiQ.txt
The Jacobi inversion problem for curves of even higher genus
In Theorem 1 and Theorem 5 of the paper we present the solution of the Jacobi inversion problem for the (3,7) and (3,8)-curves respectively.
In Appendix B we comment that the same method of solution can be used on trigonal curves of much higher genus without any computational difficulty.
We presented the details for the (3,10) and (3,11) curves in this Appendix.
The explicit solutions have also been derived for the next six trigonal curves. In each case we use the first two equations derived from the
Kleinian expansion, rho[1] and rho[2]. We take the resultant of these, eliminating the variable w, to leave rho[1,2] (which is degree g in z
in each case). The problem can then be solved by finding the roots of this polynomial and the corresponding w from the equation rho[1]=0
(which is degree one in w in each case). The relevent polynomials are given in the text files below.
- The (3,13)-curve, (genus 12).
- The (3,14)-curve, (genus 13).
- The (3,16)-curve, (genus 15).
- The (3,17)-curve, (genus 16).
- The (3,19)-curve, (genus 18).
- The (3,20)-curve, (genus 19).
last updated: 24th March 2010