The Deligne-lusztig Curve Associated to the Suzuki Group

We give a characterization of the Deligne-Lusztig curve associated to the Suzuki group Sz(q) based on the genus and the number of Fq-rational points of the curve. §0. Throughout this paper by a curve we mean a projective, geometrically irreducible, and non-singular algebraic curve defined over the finite field Fq with q elements. Let Nq(g) denote the maximum number of Fq-rational points that a curve of genus g can have. The number Nq(g) is bounded from above by the Hasse-Weil bound. A curve X of genus g is said to be optimal over Fq, if its number of Fq-rational points #X(Fq) is equal to Nq(g). Optimal curves are very useful for applications to coding theory via Goppa construction [Go]. Besides maximal curves over Fq, that is, curves whose number of Fq-rational points attains the Hasse-Weil bound and some curves of small genus, the only known examples of optimal curves are the Deligne-Lusztig curves associated to the Suzuki group Sz(q) and to the Ree group R(q) [De-Lu, §11], [Han]. Arithmetical and geometrical properties of maximal curves were studied in [R-Sti], [I], [Geer-Vl] (see the references therein), [FTo] and [FGTo]. The other mentioned optimal curves were studied in [Han-Sti], [Han], [Pe] and [Han-Pe]. Hansen and Pedersen [Han-Pe, Thm. 1] stated the uniqueness, up to Fq-isomorphism, of the curve corresponding to R(q) based on the genus, the number of Fq-rational points, and the group of Fq-automorphisms of the curve. They observed a similar result for the curve corresponding to Sz(q) (cf. [Han-Pe, p.100]) as a consequence of its uniqueness up to F̄q-isomorphism (cf. [He]). Hence, by [Han-Sti], the curve under study in this paper is Fq-isomorphic to the plane curve given by y − y = x0(x − x), where q0 = 2 s and q = 2q0. Its genus is q0(q − 1) and its number of Fq-rational points is q + 1 (loc. cit.). The paper was partially written while the author was visiting the University of Essen supported by a grant from the Graduiertenkolleg “Theoretische und experimentelle Methoden der Reinen Mathematik”. The results of this paper were announced in [To].