Quadratic functions and maximal Artin–Schreier curves

Abstract

For an odd prime p and an even integer n with gcd(n, p) = 1, we consider quadratic functions from F p n to F p of codimension k. For various values of k, we obtain classes of quadratic functions giving rise to maximal and minimal Artin-Schreier curves over F p n . We completely classify all maximal and minimal curves obtained from quadratic functions of codimension 2 and coefficients in the prime field F p .

Explicit maximal and minimal curves of Artin–Schreier type from quadratic forms

“…If we take all the coefficients s i of S(x) in F q m we obtain the following explicit classifications in Corollaries 1, 2 and 3. These results include the maximal curves obtained in [2] as a very special subcase. Also note that in [2] only the case q = p (prime case) is considered under the condition that gcd(p, n) = gcd(p, 2m) = 1.…”

“…These results include the maximal curves obtained in [2] as a very special subcase. Also note that in [2] only the case q = p (prime case) is considered under the condition that gcd(p, n) = gcd(p, 2m) = 1. Here we have no such condition.…”

“…For q = 2 t a full classification of quadratic forms from F q k to F q of codimension 2 is provided in the following cases: all the coefficients are from F 2 or at least three are in F 4 ; as an application maximal and minimal curves are obtained, see [4,5,12,13,14]. Latter on some results on quadratic functions and maximal Artin-Schreier curves over finite fields having odd characteristic are presented in [1] and [2]. In [15] by using some techniques developed in [3] a Conjecture presented in [2] is proved and explicit classes of maximal and minimal Artin-Schreier type curves over finite fields having odd characteristics are presented.…”

“…Latter on some results on quadratic functions and maximal Artin-Schreier curves over finite fields having odd characteristic are presented in [1] and [2]. In [15] by using some techniques developed in [3] a Conjecture presented in [2] is proved and explicit classes of maximal and minimal Artin-Schreier type curves over finite fields having odd characteristics are presented. Throughout this paper by a curve we mean a smooth geometrically irreducible and projective curve over a finite field of odd characteristic.…”

Zeta Functions of a Class of Artin–Schreier Curves with Many Automorphisms

“…We start with a discussion of the case h = 0, and then turn our attention to R(X) = X p h . For more results along the same lines we refer to [3] and [1]. In [4] it is shown that all curves C R that are maximal over the field F p 2n are quotients of the Hermite curve H p n with affine equation y p n − y = x p n +1 .…”