Weierstrass points on Kummer extensions

Abstract

For Kummer extensions y m = f (x), we discuss conditions for an integer be a Weierstrass gap at a place P . In the case of totally ramified places, the conditions will be necessary and sufficient. As a consequence, we extend independent results of several authors.

Pure Weierstrass gaps from a quotient of the Hermitian curve

“…where A := n k=1 i k . We claim that j 1 = m. Suppose that this is false, so j 1 = m and (n − 1) + (q − n) > A + q by (16). Thus A < −1 giving a contradiction as A 0 by definition.…”

“…In [15], Matthews generalized the results of [7], [14] by determining the Weierstrass semigroup of arbitrary rational places on the quotient of the Hermitian curve defined by the equation y m = x q + x over F q 2 where q is a prime power and m > 2 is a divisor of q + 1. For general Kummer extensions, the authors in [16], [9] recently described the Weierstrass semigroups and gaps at one place and two places. Bartoli, Quoos and Zini [17] gave a criterion to find pure gaps at many places and presented families of pure gaps.…”

Pure gaps on curves with many rational places

“…The majority of maximal curves and curves with many rational places has a plane model of Kummer-type. For those curves with affine equation given by y m = f (x) λ where m ≥ 2, λ ≥ 1 and f (x) is a separable polynomial over F q , general results on gaps and pure gaps can be found in [1,5,18,29]. Applications to codes on particular curves such as the Giulietti-Korchmáros curve, the Garcia-Güneri-Stichtenoth curve, and quotients of the Hermitian curve can be found in [19,28,31].…”

“…Applications to codes on particular curves such as the Giulietti-Korchmáros curve, the Garcia-Güneri-Stichtenoth curve, and quotients of the Hermitian curve can be found in [19,28,31]. In [1] the authors use a decomposition of certain Riemann-Roch vector spaces due to Maharaj (Theorem 2.2) to describe gaps at one place arithmetically. The same idea has been used to investigate gaps and pure gaps at several places [2,5,18,19,[29][30][31].…”

On Weierstrass semigroup at m points on curves of the form f(y)=g(x)

“…. , P a+1 be a + 1 distinct rational points such that Examples of such curves can be found in [1], [10], [12] and [13].…”