On Weierstrass Points and Optimal Curves

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves. This paper continues the study, begun in [FT] and [FGT], of curves over finite fields with many rational points, based on Stöhr-Voloch’s approach [SV] to the Hasse-Weil bound by way of Weierstrass Point Theory and Frobenius orders. Some of the results were announced in [T]. A projective geometrically irreducible non-singular algebraic curve X |Fq of genus g is said to be optimal if #X(Fq) = max{#Y (Fq) : Y |Fq curve of genus g} . Optimal curves occupy a distinguished niche, for example, in coding theory after Goppa’s [Go]. We recall that #X(Fq) is bounded from above by the Hasse-Weil bound, namely q + 2g √ q + 1 . The main goal of this paper is to sharpen and generalize some results in [FGT]. In that paper Garcia and us improved and generalized previous results obtained by RückStichtenoth’s [R-Sti] and Stichtenoth-Xing’s [Sti-X]. We will mainly concerned with the uniqueness (up to Fq-isomorphism) of some optimal curves. Roughly speaking, firstly the Zeta Function of the curve is used to define a linear system on the curve. Then, Stöhr-Voloch’s [SV] is used to obtain the desired property. We distinguish two cases according as q is a square or not. In the first case, say q = l, we look for curves X | Fl2 that attain the Hasse-Weil bound, that is, the so-called maximal curves. These curves were studied in [Sti-X], [Geer-Vl] (see also the references therein), [FT] and [FGT]. X is equipped with the linear system D := |(l + 1)P0|, P0 ∈ X(Fl2) (cf. [FGT, §1], §2 here) and from the application of [SV] to D, Garcia and us deduced properties on the genus and the uniqueness (up to Fl2-isomorphism) of X for some values of genus (cf. [FT], [op. cit.]). The following theorem follows from [FT] and Proposition 2.5; it improves [FGT, Thm. 3.1] and is a typical example of the The paper was partially written while Torres was visiting the University of Essen (supported by the Graduiertenkolleg “Theoretische und experimentelle Methoden der Reinen Mathematik”); ICTP, Trieste -Italy (supported by ICTP) and IMPA, Rio de Janeiro Brazil (supported by Cnpq). 1 2 R. FUHRMANN AND F. TORRES results obtained here. We recall that the biggest genus that X can have is l(l− 1)/2 (cf. Ihara’s [Ih]). Theorem 1. Let X |Fl2 be a maximal curve of genus g and l odd. If g > (l− 1)(l− 2)/4, then 1. X is Fl2-isomorphic to the Hermitian curve y l + y = x so that g = (l− 1)l/2 or 2. X is Fl2-isomorphic to the plane curve y l + y = x so that g = (l− 1)/4 . Theorem 1(1) is valid without restricting the parity of l if g > (l − 1)/4. Indeed, several characterizations of Hermitian curves have already been given, see for example [HSTV] (and the references therein), [R-Sti] and [FT]. See also Theorem 2.4. Furthermore, we show that the morphism associated to D is an embedding. Hence we improve [FGT, Prop. 1.10], that is, we can compute the genus of maximal curves under a hypothesis on non-gaps at Fl2-rational points (see §2.3). Now we discuss the case √ q 6∈ N. Besides some curves of small genus, see for example [Car-He], the only known examples of optimal curves, in this case, are the DeligneLusztig curves associated to the Suzuki group Sz(q) and to the Ree group R(q) [De-Lu, §11], [Han]. They were studied in [Han-Sti], [Han], [P] and [Han-P]. Hansen and Pedersen [Han-P, 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-P, p.100]) as a consequence of its uniqueness up to F̄q-isomorphism (cf. [Henn]). Hence, after [Henn] and [Han-Sti], the curve under study in §3 of this paper is Fq-isomorphic to the plane curve given by y − y = x0(x − x) , where q0 = 2 s and q = 2q0. This curve is equipped with the linear system g 4 q+2q0+1 = |(q + 2q0 + 1)P0|, P0 a Fq-rational point. By applying [SV] to this linear system we prove (see §3) the Theorem 2. Let q0, q be as above, X | Fq a curve of genus g such that: (1) g = q0(q − 1) and (2) #X(Fq) = q + 1. Then X is Fq-isomorphic to the Deligne-Lusztig curve associated to the Suzuki group Sz(q). We remark that a Hermitian curve can be also realized as a Deligne-Lusztig curve associated to a projective special linear group (cf. [Han]). Then its uniqueness (up to Fq) is also a consequence of its uniqueness up to F̄q (cf. [Han-P, p.100]). A. Cossidente brought to our attention a relation between the curve in Theorem 2 and the Suzuki-Tits ovoid. This is described in the Appendix. WEIERSTRASS POINTS AND OPTIMAL CURVES 3 It is our pleasure to thank: A. Garcia, R. Pellikaan and H. Stichtenoth for useful conversations; A. Cossidente for having let us include his observation in the Appendix. In addition, we want to thank Prof. J.F. Voloch for his interest in this work. Convention: Throughout this paper by a curve we mean a projective geometrically irreducible non-singular algebraic curve.