Abelian Functions for Trigonal Curves of Degree Four and Determinantal Formulae in Purely Trigonal Case

In the theory of elliptic functions, there are two kinds of determinantal formulae of Frobenius-Stickelberger [6] and of Kiepert [7], both of which connect the function σ(u) with ℘(u) and its (higher) derivatives through an determinantal expression. These formulae were naturally generalized to hyperelliptic functions by the papers [11], [12], and [13]. Avoiding generality, we restrict the story only for the simplest purely trigonal curve y = x + · · · , where the right hand side is a monic biquadratic polynomial of x. Our main results Theorem 5.3 and Corollary 6.2 are quite natural generalization of those determinantal formulae for such the curves. For more general purely trigonal curve, or for any purely d-gonal curve (d = 4, 5, · · · ), the author would like to publish in other papers, including formulae of Cantor-type (see [13]). In the case of hyperelliptic functions, we considered only the hyperellptic curves ramified at infinity in [13]. The theta divisor of the Jacobian variety of such a curve is symmetric with respect to the origin of the Jacobian variety. Each of purely trigonal curves considered in this paper is also completely ramified at infinity and it is acted by the third roots of unity. Hence, the theta divisor of the Jacobian variety of a purely trigonal curve has third order symmetry with respect to the origin. Similar symmetry is also possessed by any purely d-gonal curve. On the other hand, any curve that is not purely d-gonal does not have symmetry with respect to the origin at all. Since this fact is very serious, the author do not imagine whether generalizations of Frobenius-Stickelberger type and of Kiepert type exist or not. The first two sections are devoted to describe general theory of trigonal curves of genus three. The rest sections are restricted to purely trigonal case.