The Riemann constant for a non-symmetric Weierstrass semigroup

TL;DR

This paper investigates the Riemann constant for non-symmetric Weierstrass semigroups on compact Riemann surfaces, establishing relations with half periods and semi-canonical divisors, especially for trigonal curves.

Contribution

It extends classical results by analyzing the Riemann constant in the non-symmetric case using semi-canonical divisors and provides explicit identification for trigonal curves.

Findings

Relation between Riemann constant and half periods in non-symmetric case

Identification of semi-canonical divisor for trigonal curves

Remarks on algebraic expression for Jacobi inversion problem

Abstract

The zero divisor of the theta function of a compact Riemann surface $X$ of genus $g$ is the canonical theta divisor of Pic${}^{(g-1)}$ up to translation by the Riemann constant $\Delta$ for a base point $P$ of $X$. The complement of the Weierstrass gaps at the base point $P$ given as a numerical semigroup plays an important role, which is called the Weierstrass semigroup. It is classically known that the Riemann constant $\Delta$ is a half period $\frac{1}{2}\Gamma_\tau$ for the Jacobi variety $\mathcal{J}(X)=\mathbb{C}^g/\Gamma_\tau$ of $X$ if and only if the Weierstrass semigroup at $P$ is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor $D_0$, we show a relation between the Riemann constant $\Delta$ and a half period $\frac{1}{2}\Gamma_\tau$ of the non-symmetric case. We also identify the semi-canonical divisor $D_0$ for trigonal curves, and remark on an algebraic expression for the Jacobi inversion problem using the relation