Sigma functions for a space curve (3, 4, 5) type with an appendix by J. Komeda

Details Sigma functions for a space curve (3, 4, 5) type with an appendix by J. Komeda Sigma functions for a space curve (3, 4, 5) type with an appendix by J. Komeda

2011-12-18

arxiv.org/abs/1112.4137v1

Physics

Mathematical Physics

p.22

Scientific paper

In this article, a generalized Kleinian sigma function for an affine $(3,4,5)$ space curve of genus 2 was constructed as the simplest example of sigma function for an affine space curves, by applying the construction method of sigma function for $(r,s)$ plane curve provided by Eilbeck Enolskii and Leykin (SIDEIII, CRM Proc. Lecture Notes, 25 2000) to the space curve. By defining the fundamental differential of the second kind over it, the Legendre relations was obtained as the symplectic structure over it. It was showed that with the abelian map to $\CC^2$, the symplectic structure determines the sigma function. Using the sigma function, the Jacobi inversion formulae for the curve are obtained. It means that the generalization of the sigma functions for the affine plane curves to ones for the space curves is basically possible. An interesting relation between a semigroup generated by $(6,13,14,15,16)$ and Norton number associated with Monster group is also mentioned with an Appendix by Komeda.

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