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

Abstract

In this paper, we study some cyclic (r, s) curves X given byyr=xs+λ1xs−1+⋯+λs−1x+λs.?>We give an expression for the prime form , 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),E(P,Q)=σ♮r(u−v)du1dv1,?>where ♮r is a certain multi-index of differentials. Here u1 and v1 are respectively the first components of u = w(P) and v = w(Q) which are given by the Abel map , 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.