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

#### 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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