Relationship between the prime form and the sigma function for some cyclic (r,s) curves

Abstract

In this article, we study some cyclic (r, s) curves X given byWe give an expression for the prime form E(P, Q), where (P, Q ∈ X), in terms of the sigma function for some such curves, specifically any hyperelliptic curve (r, s) = (2, 2g+1) as well as the cyclic trigonal curve (r, s) = (3, 4),where ♮ r is a certain multi-index of differentials. Here u 1 and v 1 are respectively the first components of u = w(P ) and v = w(Q) which are given by the Abel map w : X → C g , where g is the genus of X. These explicit formulae are useful in applications, for instance to the problem of constructing classes of Schwarz-Christoffel maps to slit domains.