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A new family of maximal curves over a finite field

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Abstract

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type \({^2A_2,\,^2B_2}\) and \({^2G_2}\) defined over the finite field \({\mathbb {F}_n}\) all have the maximum number of \({\mathbb {F}_n}\)-rational points allowed by the Weil “explicit formulas”, and that these curves are \({\mathbb {F}_{q^2}}\)-maximal curves over infinitely many algebraic extensions \({\mathbb {F}_{q^2}}\) of \({\mathbb {F}_n}\). Serre showed that an \({\mathbb {F}_{q^2}}\)-rational curve which is \({\mathbb {F}_{q^2}}\)-covered by an \({\mathbb {F}_{q^2}}\)-maximal curve is also \({\mathbb {F}_{q^2}}\)-maximal. This has posed the problem of the existence of \({\mathbb {F}_{q^2}}\)-maximal curves other than the Deligne–Lusztig curves and their \({\mathbb {F}_{q^2}}\)-subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n 3 with n = p r > 2, p ≥ 2 prime, we give a simple, explicit construction of an \({\mathbb {F}_{q^2}}\)-maximal curve \({\mathcal {X}}\) that is not \({\mathbb {F}_{q^2}}\)-covered by any \({\mathbb {F}_{q^2}}\)-maximal Deligne–Lusztig curve. Furthermore, the \({\mathbb {F}_{q^2}}\)-automorphism group Aut\({(\mathcal {X})}\) has size n 3(n 3 + 1)(n 2 − 1)(n 2 − n + 1). Interestingly, \({\mathcal {X}}\) has a very large \({\mathbb {F}_{q^2}}\)-automorphism group with respect to its genus \({g = \frac{1}{2}\,(n^3 + 1)(n^2 - 2) + 1}\).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli, 1, 06123, Perugia, Italy

    Massimo Giulietti

  2. Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, 85100, Potenza, Italy

    Gábor Korchmáros

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  1. Massimo Giulietti
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  2. Gábor Korchmáros
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Correspondence to Massimo Giulietti.

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Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni, PRIN 2006–2007.

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Giulietti, M., Korchmáros, G. A new family of maximal curves over a finite field. Math. Ann. 343, 229–245 (2009). https://doi.org/10.1007/s00208-008-0270-z

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