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 y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0. We construct Abelian functions using the multivariate σ-function associated to the curve, generalising 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 σ() and a new addition formula.