On additive polynomials and certain maximal curves

Abstract

a b s t r a c tWe show that a maximal curve over F q 2 given by an equationhas degree m prime to the characteristic p, is such that all roots of A(X) belong to F q 2 . In the particular case where F(Y) = Y m , we show that the degree m is a divisor of q + 1.

New maximal curves as ray class fields over Deligne-Lusztig curves

“…In particular, if r > 0 then either r = 1 and e P = k/3 for a single P , or r ∈ {1, 2} and e P ≥ k/2 for all P . But either of these cases gives a contradiction in (5).…”

“…Since X rcf is maximal over F q d , so is C mk . But a theorem of Garcia and Tazafolian [5] states that a curve of the form u r = x q − x may be maximal over F q d only for r dividing q d/2 + 1. Thus k divides (q d/2 + 1)/m, as desired.…”

Certain maximal curves and Cartier operators

“…In fact, if q is a square, there exist several curves that attain the above upper bound (see [4], [5], [14] and [23]). We say a curve is maximal (resp.…”

“…Since C is maximal over F q 2 , one can show easily that the additive polynomial A(x) := x q + µx has a nonzero root β ∈ F * q 2 . In fact, more is true: it follows from [5,Theorem 4.3] that all roots of A(x) belong to F q 2 .…”

Weierstrass points on Kummer extensions

“…A natural question that arises is whether this result can be extended to a larger family of curves of type y m = f (x). This question was somewhat addressed in subsequent papers by Garcia, Tafazolian and Torres ( [5], [22], [23], [24]), where the same result was proved when 1 − x m is replaced by some other polynomials f (x) ∈ F q 2 [x]. In section 4, we present a general class of polynomials f (x) ∈ F q 2 [x] for which the F q 2 -maximality of y m = f (x) implies m|(q + 1).…”