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 $y^4 = x^5 + λ_4x^4 + λ_3x^3 + λ_2x^2 + λ_1x + λ_0$. 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.