Higher genus Abelian functions associated with algebraic curves

Abstract

We investigate the theory of Abelian functions with periodicity properties defined from an associated algebraic curve. A thorough summary of the background material is given, including a synopsis of elliptic function theory, generalisations of the Weierstrass σ and 0functions and a literature review. The theory of Abelian functions associated with a tetragonal curve of genus six is considered in detail. Differential equations and addition formula satisfied by the functions are derived and a solution to the Jacobi Inversion Problem is presented. New methods which centre on a series expansion of the σ function are used and discussions on the large computations involved are included. We construct a solution to the KP equation using these functions and outline how a general class of solutions can be generated from a wider class of curves. We proceed to present new approaches used to complete results for the lower genus trigonal curves. We also give some details on the the theory of higher genus trigonal curves before finishing with an application of the theory to the Benney moment equations. A reduction is constructed corresponding to Schwartz-Christoffel maps associated with the tetragonal curve. The mapping function is evaluated explicitly using derivatives of the σ function.