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
