Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 025, 22 pages      arXiv:1003.4144
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics

Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

Matthew England
Department of Mathematics, University of Glasgow, UK

Received December 31, 2009, in final form March 19, 2010; Published online March 24, 2010

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.

Key words: Abelian function; Kleinian sigma function; Jacobi inversion problem; cyclic trigonal curve.

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