It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type $^2A_2$ , $2^B_2$ and $^2G_2$ defined over the finite field $F_n$ all have the maximum number of $F_n$ -rational points allowed by the Weil “explicit formulas”, and that these curves are $F_{q^2}$ -maximal curves over infinitely many algebraic extensions $F_{q^2}$ of $F_n$ . Serre showed that an $F_{q^2}$ -rational curve which is $F_{q^2}$-covered by an $F_{q^2}$-maximal curve is also $F_{q^2}$ -maximal. This has posed the problem of the existence of $F_{q^2}$-maximal curves other than the Deligne–Lusztig curves and their $F_{q^2}$-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 $F_{q^2}$-maximal curve X that is not $F_{q^2}$-covered by any $F_{q^2}$ -maximal Deligne–Lusztig curve. Interestingly, X has a very large $F_{q^2}$-automorphism group with respect to its genus.

A new family of maximal curves over a finite field

GIULIETTI, Massimo;
2009

Abstract

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type $^2A_2$ , $2^B_2$ and $^2G_2$ defined over the finite field $F_n$ all have the maximum number of $F_n$ -rational points allowed by the Weil “explicit formulas”, and that these curves are $F_{q^2}$ -maximal curves over infinitely many algebraic extensions $F_{q^2}$ of $F_n$ . Serre showed that an $F_{q^2}$ -rational curve which is $F_{q^2}$-covered by an $F_{q^2}$-maximal curve is also $F_{q^2}$ -maximal. This has posed the problem of the existence of $F_{q^2}$-maximal curves other than the Deligne–Lusztig curves and their $F_{q^2}$-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 $F_{q^2}$-maximal curve X that is not $F_{q^2}$-covered by any $F_{q^2}$ -maximal Deligne–Lusztig curve. Interestingly, X has a very large $F_{q^2}$-automorphism group with respect to its genus.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11391/37547
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