Abelian Functions for Cyclic Trigonal Curves of Genus Four

Abstract

We discuss the theory of generalized Weierstrass $σ$ and $\wp$ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the "purely trigonal" (or "cyclic trigonal") curve $y^3=x^5+λ_4 x^4 +λ_3 x^3+λ_2 x^2 +λ_1 x+λ_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the $\wp$ functions, and the derivation of an addition formulae.