# Deriving bases for Abelian functions

This page is a repository of results related to the following paper:
• M. England Deriving bases for Abelian functions. Comput. Methods Funct. Theory 11:2 (2011).
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.
• Bfunction.txt. Poles or order 3 and 5 indices.
• Tfunction.txt. Poles or order 3 and 6 indices.
• Sfunction.txt. Poles or order 3 and 6 indices.
• Pfunction.txt. Poles or order 3 and 7 indices.
• I5P4a.txt. Poles or order 4 and 5 indices.
• I6P4.txt. Poles or order 4 and 6 indices.
• Ffunction.txt. Poles or order 4 and 6 indices. (May be constructed from functions above).
• I7P4.txt. Poles or order 4 and 7 indices.
• I8P4.txt. Poles or order 4 and 8 indices.
• Gfunction.txt. Poles or order 4 and 8 indices. (One of the function above which was has been used elsewhere).

## 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.

• Theorem 4.1 presented a set of relations that express all the 4-index P-functions as a degree 2 polynomial in the fundamental basis entries. They were printed in Appendix B and are available in a text file here.
• Theorem 4.2 presented an expression for each of the 210 products of pairs of 3-index functions as a degree 3 polynomial in the fundamental basis entries. They are available in a text file here. The first few were printed in the paper while a full print out may be viewed in this Maple File.
• Theorem 4.3 derived a set of 65 relations bilinear in the 2 and 3-index functions, from which all others can be constructed. They are available in a text file here. This text file also contains the relations in the notation of B-functions, which is how they were derived. The first few relations were printed in the paper while a full print out may be viewed in this Maple File.