Abelian functions for cyclic trigonal curves of genus 4

摘要

We discuss the theory of generalized Weierstrass sigma and p-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) + lambda(4)x(4) + lambda(3)x(3) + lambda(2)x(2) + lambda(1)x + lambda(0) is discussed in detail, including a list of some of the associated partial differential equations satisfied by the p-functions, and the derivation of addition formulae.