Certain maximal curves and Cartier operators

Weierstrass points on Kummer extensions

“…For an F q 2 -maximal curve of type y m = f (x), which conditions on the polynomial f (x) ∈ F q 2 [x] will assure that m divides q + 1? Substantial progress towards an answer of Question 1 has been made by Garcia, Tafazolian and Torres in a sequence of papers [4,5,23,24]. In particular, they proved that m must divide q + 1 in the following cases.…”

“…Our first application of Theorem 3.5 is in the theory of maximal curves over finite fields. In 2008, using Cartier Operators, Garcia and Tafazolian [4] proved that if the Fermat curve y m = 1 − x m is maximal over F q 2 , then m must be a divisor of q + 1. The converse is a well known result.…”

The a-numbers of Fermat and Hurwitz curves

“…The Cartier operator is related to the number X (F q ) of points of X over a finite field F q , as the following theorem shows; see [6].…”

A new tower with good $p$-rank meeting Zink’s bound

“…We refer to [10] for more information on the properties of the Cartier operator. For us it will be more convenient to use powers of the Cartier operator.…”

“…In this article, we will introduce a new tower F/F p 3e satisfying the same reducible recursive equation as the one used in [3] (see Equation (10)). However, the defining equation of F/F p 3e is coming from a different factor; namely the unique factor of degree q 2 − 1 (see Equation (11)), where q = p e .…”