Abelian Functions for Cyclic Trigonal Curves of Genus Four

We discuss the theory of generalized Weierstrass σ and ℘ 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 = x + λ4x 4 + λ3x 3 + λ2x 2 + 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 an addition formulae.