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.

中文翻译：

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Hermitian 曲线上的三重有理点及其 Weierstrass 半群

在本文中，我们研究了 $\mathbb{F}_{q^2}$ 上 Hermitian 曲线上三个有理点的配置，并根据它们的 Weierstrass 半群对它们进行分类。对于 $q>3$，我们证明这种形式的不同半群的数量等于 $q+1$ 的正除数的数量，并给出了所研究的每个三元组的 Weierstrass 半群的明确描述。为此，我们利用两点差异并推导出适用于有限域上任意曲线的标准。