Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves
Matthew England
TL;DR
This paper advances the theory of Abelian functions for cyclic trigonal curves by exploring genus six and seven cases, providing solutions to classical problems, and establishing links to integrable systems.
Contribution
It introduces new cases of cyclic trigonal curves, analyzes their Abelian functions, and derives novel addition formulas and relations to the Boussinesq equation.
Findings
Solutions to the Jacobi inversion problem for new curve cases
New addition formulas for Abelian functions
Connections between Abelian functions and the Boussinesq equation
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.
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