Prime form and sigma function

In this article, we study some cyclic $(r,s)$ curves $X$ given by $y^r =x^s + λ_{1} x^{s-1} +...+ λ_{s-1} x + λ_s$. We give an expression for the prime form $\cE(P,Q)$, where $(P, Q \in X)$, 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 $(r,s) = (3,4)$, $$ \cE(P,Q) =\frac{σ_{\natural_{r}}(u - v)}{\sqrt{du_1}\sqrt{d v_1}}, $$ where $\natural_r$ is a certain 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 $w: X \to \CC^g$, where $g$ is the genus of $X$.

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