Abelian functions associated with a cyclic tetragonal curve of genus six

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y⁴ = x ⁵ + ?₄x⁴ + ?₃x ³ + ?₂x² + ?₁x + ?₀. We construct Abelian functions using the multivariate s-function associated with the curve, generalizing the theory of the Weierstrass -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 s(u) and a new addition formula. © 2009 IOP Publishing Ltd.