Prime form and sigma function

TL;DR

This paper derives an explicit expression for the prime form on certain cyclic algebraic curves using sigma functions, extending known formulas to hyperelliptic and trigonal cases.

Contribution

It provides a new formula for the prime form in terms of sigma functions for hyperelliptic and cyclic trigonal curves, generalizing previous results.

Findings

Explicit prime form expression for hyperelliptic curves

Explicit prime form expression for cyclic trigonal curves

Connection between prime form and sigma functions for these curves

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 $(r,s) = (3,4)$, $$ \cE(P,Q) =\frac{\sigma_{\natural_{r}}(u - v)}{\sqrt{du_1}\sqrt{d v_1}}, $$ 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 $X$.