Triples of rational points on the Hermitian curve and their Weierstrass semigroups

TL;DR

This paper classifies triples of rational points on the Hermitian curve over finite fields based on their Weierstrass semigroups, revealing a connection to divisors of q+1 and providing explicit descriptions.

Contribution

It introduces a classification of triples of rational points on the Hermitian curve by their Weierstrass semigroups and extends the analysis using two-point discrepancies.

Findings

Number of semigroups equals the number of positive divisors of q+1 for q>3

Explicit descriptions of Weierstrass semigroups for each triple

A criterion for arbitrary curves over finite fields based on two-point discrepancies

Abstract

In this paper, we study configurations of three rational points on the Hermitian curve over $\mathbb{F}_{q^2}$ and classify them according to their Weierstrass semigroups. For $q>3$, we show that the number of distinct semigroups of this form is equal to the number of positive divisors of $q+1$ and give an explicit description of the Weierstrass semigroup for each triple of points studied. To do so, we make use of two-point discrepancies and derive a criterion which applies to arbitrary curves over a finite field.