A new family of maximal curves over a finite field
By Massimo Giulietti, Gábor Korchmáros. 2007
136 citations recorded by the source corpus.
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 = n3 with n = pr > 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 n3(n3 + 1)(n2 − 1)(n2 − n + 1). Interestingly, $${\mathcal {X}}$$ has a very large $${\mathbb {F}_{q^2}}$$-automorphism group with respect to its genus $${g = \frac{1}{2}\,(n^3 + 1)(n^2 - 2) + 1}$$.
Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Information Systems
View Paper: https://www.semanticscholar.org/paper/ed8e1f772e662777b427544563e45bd78cd69cc5
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Citations in the corpus
- Handbook of Finite Fields (2013, background): “44 Example [19] Over the field Fq with q = r, consider the function field F = Fq(x, y, z) which is defined by the equations x + x = y and y · x r2 − x xr + x = z r3+1 r+1 .”
- A generalization of the Giulietti-Korchmaros maximal curve (2010, background): “Very recently, Giulietti and Korchmáros introduced a new example of a maximal curve and they showed that their curve is not covered by the Hermitian curve (see [6]).”