Abelian functions for cyclic trigonal curves of genus 4

Abstract

Abstract. 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 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 an addition formulae.

Abelian Functions for Trigonal Curves of Genus Three

“…. , the map ι : Sym k (C) → J, 6) and denote its image by W [k] . (W [k] = J for k ≥ 3 by the Abel-Jacobi theorem.)…”

Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

“…In the (4,5)-case every polynomial needed at least two rounds of substitution to reach this stage, with the corresponding equations for z 6 and z 7 far more complicated. In this case we need only substitute once into ρ 1,4 and so the polynomials achieved here are far simpler. They start with…”

“…Note that we can apply equation (15) to the first set of equations in order to derive a generalisation of equation (3), the second differential equation from the elliptic case [10,8,9] and [4], and is discussed further in Remark 2.…”

“…We next define ℘-functions using an analogy of equation (4). Since there is more than one variable we need to be clear which we differentiate with respect to.…”

“…In particular, the two canonical cases of the (3,4)-and (3,5)-curves have been examined in [9] and [4] respectively. Both papers explicitly construct the differentials on the curve, solve the Jacobi inversion problem and obtains sets of differential equations between the ℘-functions.…”