Abelian functions for cyclic trigonal curves of genus 4

We discuss the theory of generalized Weierstrass σσ and ℘℘-functions defined on a trigonal curve of genus 4, following earlier work on the genus 3 case. The specific example of the “purely trigonal” (or “cyclic trigonal”) curve y3=x5+λ4x4+λ3x3+λ2x2+λ1x+λ0y3=x5+λ4x4+λ3x3+λ2x2+λ1x+λ0 is discussed in detail, including a list of some of the associated partial differential equations satisfied by the ℘℘-functions, and the derivation of addition formulae.