Abelian functions for cyclic trigonal curves of genus 4

Abstract We discuss the theory of generalized Weierstrass σ 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 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 ℘ -functions, and the derivation of addition formulae.