Abelian Functions for Cyclic Trigonal Curves of Genus Four

Abstract

We discuss the theory of generalized Weierstrass $\sigma$ and $\wp$ 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 $y^3=x^5+{\lambda }_4{x}^4+{\lambda }_3{x}^3+{\lambda }_2{x}^2+{\lambda }_{1}x+{\lambda }_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the $\wp$ functions, and the derivation of an addition formulae.

23 pages

References in corpus (1)

• On Algebraic Expressions of Sigma Functions for (n,s) Curves