The sigma function for trigonal cyclic curves

TL;DR

This paper explicitly calculates the sigma function for cyclic trigonal curves with a specified point, providing key algebraic and geometric insights into their structure and solutions to related inversion problems.

Contribution

It completes the explicit calculation of the sigma function for cyclic trigonal curves at a point, including Riemann constants, basis construction, and vanishing order analysis.

Findings

Explicit sigma function for cyclic trigonal curves derived

Riemann constant and basis of cohomology constructed

Order of vanishing and Jacobi inversion solutions provided

Abstract

A recent generalization of the "Kleinian sigma function" involves the choice of a point $P$ of a Riemann surface $X$, namely a "pointed curve" $(X, P)$. This paper concludes our explicit calculation of the sigma function for curves cyclic trigonal at $P$. We exhibit the Riemann constant for a Weierstrass semigroup at $P$ with minimal set of generators $\{3, 2r+s,2s+r\}$, $r<s$, equivalently, non-symmetric, we construct a basis of $H^1(X, \mathbb{C})$ and a fundamental 2-differential on $X\times X$, we give the order of vanishing for sigma on Wirtinger strata of the Jacobian of $X$, and a solution to the Jacobi inversion problem.