Prime form and sigma function

Abstract

In this article, we study some cyclic $(r,s)$ curves $X$ given by $y^r=x^s+{\lambda }_1{x}^{s-1}+...+{\lambda }_{s-1}x+{\lambda }_s$. 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 ${♮}_r$ 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 $X$.

References in corpus (1)

• On Algebraic Expressions of Sigma Functions for (n,s) Curves