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 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 with the curve, generalizing the theory of the Weierstrass P-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 sigma(u) and a new addition formula.