Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

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 the Boussinesq equation and a new addition formula.

Keywords

• Abelian function
• Kleinian sigma function
• Jacobi inversion problem
• cyclic trigonal curve