On the spectrum for the genera of maximal curves over small fields

Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper ,we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7\leq q\leq 16$. In particular, by using a result in Kudo and Harashita(2016) paper, the set $\mathbf{M}(7^2)$ is completely determined.


Introduction
Let X be a (projective, nonsingular, geometrically irreducible, algebraic) curve of genus g defined over a finite field K = F of order . The following inequality is the so-called Hasse-Weil bound on the size N of the set X (K) of K-rational points of X : In Coding Theory, Cryptography, or Finite Geometry one is often interested in curves with "many points", namely those with N as big as possible. In this paper, we work out over fields of square order, = q 2 , and deal with so-called maximal curves over K; that is to say, those curves attaining the upper bound in (1), namely (2) N = q 2 + 1 + 2g · q .
Key words and phrases: Finite field, Hasse-Weil bound, Stöhr-Voloch theory, maximal curve. The first author was partially supported by FAPESP, grant 2013/00564-1. The second author was in part supported by a grant from IPM (No. 93140117). The third author was partially supported by CNPq (Grant 308326/2014-8).
The subject matter of this note is in fact concerning the spectrum for the genera of maximal curves over K, M(q 2 ) := {g ∈ N 0 : there is a maximal curve of genus g over K} .
In Section 2 we subsume basic facts on a maximal curve X being the key property the existence of a very ample linear series D on X equipped with a nice property, namely (5). We have that Castelnuovo's genus bound (6) and Halphen's theorem imply a nontrivial restriction on the genus g of X , stated in (8) (see [21]); in particular, g ≤ q(q − 1)/2 is the well-known Ihara's bound [19]. Let r be the dimension of D. Then r ≥ 2 by (5), and the condition r = 2 is equivalent to g = q(q−1)/2, or equivalent to X being K-isomorphic to the Hermitian curve y q+1 = x q + x [30,9,20]. Under certain conditions, we have a similar result for r = 3 in Corollary 1 and Proposition 1. In fact, in Section 3 we bound g via Stöhr-Voloch theory [27] applied to D being the main results the aforementioned Proposition and its Corollary 2. Finally, in Section 4 we apply all these results toward the computation of M(q 2 ) for q = 7, 8, 9, 11, 13, 16. In fact, here we improve [11,Sect. 6] and, in particular, we can compute M(7 2 ) (see Corollary 3) by using Corollary 2 and a result of Kudo and Harashita [22] which asserts that there is no maximal curve of genus 4 over F 49 .
We recall that the approach in this paper is quite different from Danisman and Ozdemir [4], where in particular the set M(7 2 ) is missing.
Conventions. P s is the s-dimensional projective space defined over the algebraic closure of the base field.

Basic facts on maximal curves
Throughout, let X be a maximal curve of genus g over the field K = F q 2 of order q 2 . Let Φ : X → X be the Frobenius morphism relative to K (in particular, the set of fixed points of Φ coincides with X (K)). For a fixed point P 0 ∈ X (K), let j : X → J , P → [P − P 0 ] be the embedding of X into its Jacobian variety J . Then, in a natural way, Φ induces a morphismΦ : J → J such that Now from (2) the numerator of the Zeta Function of X is given by the polynomial As a matter of fact, sinceΦ is semisimple and the representation of endomorphisms of J on the Tate module is faithful, from (4) it follows that This suggests to study the Frobenius linear series on X , namely the complete linear series D := |(q + 1)P 0 | which is in fact a K-invariant of X by (5); see [8], [16,Ch. 10] for further information. Moreover, D is a very ample linear series in the following sense. Let r be the dimension of D, which we refer to as the Frobenius dimension of X , and π : X → P r be a morphism related to D; we noticed above that r ≥ 2 by (5). Then π is an embedding [20,Thm. 2.5]. In particular, Castelnuovo's genus bound applied to π(X ) gives the following constrain involving the genus g, r and q (see [16,Cor. 10.25]): Since F (r) ≤ F (2) = q(q − 1)/2, as r ≥ 2, then g ≤ q(q − 1)/2 which is a well-known fact on maximal curves over K due to Ihara [19]. In addition, F (r) ≤ F (3) = (q − 1) 2 /4 for r ≥ 3, so that the genus g of a maximal curve over K satisfies the following condition (see [9]) As a matter of fact, the following holds true.
Lemma 1 (( [25,9,20])). Let X be a maximal curve over K of genus g with Frobenius dimension r. The following sentences are equivalent: Corollary 1. Let X be a maximal curve over K of genus g and Frobenius dimension r. Suppose that Then r = 3.
It is known that g = F (3) if and only if X is is uniquely determined by plane models of type: y (q+1)/2 = x q + x if q is odd, and y q+1 = x q/2 + . . . + x otherwise; see [8,2,21].
Let us consider next an improvement on (7). Suppose that Therefore Halphen's theorem implies that X is contained in a quadric surface and so g = F (3) ; see [21]. In particular, (7) improves to The following important remark is commonly attributed to J.P. Serre.

Remark 2.
Any curve (nontrivially) K-covered by a maximal curve over K is also maximal over K. In particular, any subcover over K of the Hermitian curve is so; see e.g. [11,3].
Remark 3. We point out that there are maximal curves over K that cannot be K-covered by the Hermitian curve H; see [12,28]. We observe that the examples occurring in these papers are all defined over fields of order q 2 = 6 with > 2.
We also point out that there are maximal curves over K that cannot be Galois covered by the Hermitian curve; see [10,5,28,15,14,24,13].

The set M(q 2 )
In this section we investigate the spectrum M(q 2 ) for the genera of maximal curves defined in (3). By using Remark 2 this set has already been computed for q ≤ 5 [11, Sect Let F (r) be the function in (6). Next we complement Corollary 1. Let X be a maximal curve of genus g over K with Frobenius dimension r.
Proof. As in the proof of the above proposition, we can assume that the Frobenius dimension of X equals 3. Now the hypothesis on g is equivalent to (2g − 2) > (q + 1)(q − 4)/3; thus and the result follows from Proposition 1.