Abelian Functions for Trigonal Curves of Genus Three

Abstract

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations satisfied by the \wp-functions, a proof that the coefficients of the power series expansion of the sigma-function are polynomials of moduli parameters, and the derivation of two addition formulae.

32 pages, no figures. Revised version has the a fuller description of the general (3,4) trigonal curve results, the first version described only the "Purely Trigonal" case

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