Abelian functions associated with a cyclic tetragonal curve of genus six

Abstract

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve $y^4=x^5+{\lambda }_4{x}^4+{\lambda }_3{x}^3+{\lambda }_2{x}^2+{\lambda }_{1}x+{\lambda }_0$. We construct Abelian functions using the multivariate $\sigma$-function associated to the curve, generalising the theory of the Weierstrass $\wp$-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi Inversion Problem, a power series expansion for $σ(\bu)$ and a new addition formula.

31 pages. Version 2 with corrected typos, updated references, and improved structure of the paper

References in corpus (2)

• Abelian Functions for Cyclic Trigonal Curves of Genus Four
• Identities for hyperelliptic P-functions of genus one, two and three in covariant form

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• Klein's Fundamental 2-Form of Second Kind for the $C_{ab}$ Curves
• On the equivariant algebraic Jacobian for curves of genus two
• Abelian function fields on Jacobian varieties
• Building Abelian Functions with Generalised Baker-Hirota Operators
• Some New Addition Formulae for Weierstrass Elliptic Functions