On The Curve Y N = X M + X Over Finite Fields

We characterize certain maximal curves over finite fields defined by equations of type y n = x m + x. Moreover, we show that a maximal curve over Fq2 defined by the affine equation y n = f(x), where f(x)∈Fq2[x] is separable of degree coprime to n, is such that n is a divisor of q + 1 if and only if f(x) has a root in Fq2. In this case, all the roots of f(x) belong to Fq2; cf. Theorems 1.2 and 4.3 in Garcia and Tafazolian (2008) [9]. © 2014 Elsevier Inc.1455166Aguglia, A., Korchmáros, G., Torres, F., Plane maximal curves (2001) Acta Arith., 98, pp. 165-179Duursma, I., Mak, K.-H., On maximal curves which are not Galois subcovers of the Hermitian curve (2012) Bull. Braz. Math. Soc. (N.S.), 43, pp. 453-465Fuhrmann, R., Garcia, A., Torres, F., On maximal curves (1997) J. Number Theory, 67, pp. 29-51Garcia, A., Kawakita, M.K., Miura, S., On certain subcovers of the Hermitian curve (2006) Comm. Algebra, 34, pp. 973-982Garcia, A., Özbudak, F., Some maximal function fields and additive polynomials (2007) Comm. Algebra, 35, pp. 1553-1566Garcia, A., Stichtenoth, H., A maximal curve which is not a Galois subcover of the Hermitian curve (2006) Bull. Braz. Math. Soc. (N.S.), 37, pp. 139-152Garcia, A., Stichtenoth, H., Xing, C.P., On subfields of Hermitian function fields (2000) Compos. Math., 120, pp. 137-170Garcia, A., Tafazolian, S., Certain maximal curves and Cartier operators (2008) Acta Arith., 135, pp. 199-218Garcia, A., Tafazolian, S., On additive polynomials and certain maximal curves (2008) J. Pure Appl. Algebra, 212, pp. 2513-2521Giulietti, M., Hirschfeld, J.W.P., Korchmáros, G., Torres, F., Curves covered by the Hermitian curve (2006) Finite Fields Appl., 12, pp. 539-564Giulietti, M., Korchmáros, G., A new family of maximal curves over a finite field (2009) Math. Ann., 343, pp. 229-245Hirschfeld, J.W.P., Korchmáros, G., Torres, F., (2008) Algebraic Curves over a Finite Field, , Princeton Univ. PressKazemifard, A., Naghipour, A.R., Tafazolian, S., A note on superspecial and maximal curves (2013) Bull. Iranian Math. Soc., 39, pp. 405-413Kazemifard, A., Tafazolian, S., A note on some Picard curves over finite fields, preprint. Lachaud, G., Sommes d'Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis (1987) C. R. Acad. Sci. Paris, 305, pp. 729-732Rück, H.-G., Stichtenoth, H., A characterization of Hermitian function fields over finite fields (1994) J. Reine Angew. Math., 457, pp. 185-188Stichtenoth, H., Algebraic Function Fields and Codes (2009) Grad. Texts in Math., 254. , Springer-VerlagTafazolian, S., A note on certain maximal hyperelliptic curves (2012) Finite Fields Appl., 18, pp. 1013-1016Tafazolian, S., Torres, F., On maximal curves of Fermat type (2013) Adv. Geom., 13, pp. 613-617Tate, J., Endomorphisms of Abelian Varieties over Finite Fields (1996) Invent. Math., 2, pp. 134-144Wolfmann, J., The number of points on certain algebraic curves over finite fields (1989) Comm. Algebra, 17, pp. 2055-206