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

Abstract

In this paper, we study configurations of three rational points on the Hermitian curve over 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.