Prime form and sigma function

Original · EN

In this article, we study some cyclic (r,s) curves X given by yʳ =xˢ + λ₁ xˢ⁻¹ +...+ λₛ₋₁ x + λₛ. We give an expression for the prime form (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), (P,Q) =σᵣ(u - v)√du₁√d v₁, where ᵣ is a certain 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 w: X → ᵍ, where g is the genus of X.