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arXiv · 0806.2377

Abelian functions associated with a cyclic tetragonal curve of genus six

M. England·J. C. Eilbeck
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Abstract

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve $y^4 = x^5 + λ_4x^4 + λ_3x^3 + λ_2x^2 + λ_1x + λ_0$. We construct Abelian functions using the multivariate $σ$-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.

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M. England, J. C. Eilbeck. 2008-11-03. Abelian functions associated with a cyclic tetragonal curve of genus six. https://doi.org/10.1088/1751-8113%2F42%2F9%2F095210

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