On a characterization of certain maximal curves

Miriam Abdón

Dep. Matemática, PUC-Rio Marquês de S. Vicente 225 22453-900, Rio de Janeiro, RJ E-mail: miriam@mat.puc-rio.br

and

Arnaldo Garcia

IMPA, Estrada Dona Castorina 110, 22.460-320, Rio de Janeiro, Brazil E-mail: garcia@impa.br

Abstract

For a 𝔽_{q^{2}}-maximal curve X of genus (q/n − 1)q/2, where q/n is a nongap at some point, we show that X is 𝔽_{q^{2}}-isomorphic to the nonsingular model of a curve given by P (y) = A(x), where P is an additive separable polynomial of degree q/n and where A is a polynomial of degree q + 1, under the further hypothesis that a certain field extension is Galois. In the particular case n = p = char(𝔽_{q^{2}}) we were able to characterize such maximal curves without the assumption that a certain extension is Galois.

Key words: Finite field, Maximal curve, Artin-Schreier extension, Additive polynomial

1 2000 Math. Subj. Class.: Primary 11G20, Secondary 14G05,14G10 2 The first author was supported by FAPERJ and both authors by PRONEX-CNPq #662408/1996-3 (Brazil)

Preprint submitted to Elsevier Science 1 August 2002

1 Introduction

A projective geometrically irreducible nonsingular algebraic curve X defined over 𝔽_{q^{2}}, the finite field with q^{2} elements, is called 𝔽_{q^{2}}-maximal if the number of its 𝔽_{q^{2}}-rational points attains the Hasse-Weil upper bound; i.e.,

#X (𝔽_{q^{2}}) = q^{2} + 1 + 2qg ,

where g is the genus of the curve.

Maximal curves became important in Coding Theory after Goppa’s paper [8], and they have been intensively studied (see for example [13], [7], [6], [2]).

In [6], it is studied the following type of maximal curves with the following results:

Let X be a maximal curve defined over 𝔽_{q^{2}} and suppose there exists a rational point P_{0} ∈ X (𝔽_{q^{2}}) such that nm = q + 1, where m is a nongap at P_{0} i.e., there exists y ∈ 𝔽_{q^{2}}(X) such that div_{∞}(y) = mP_{0}. Then they conclude in [6] that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(y) is a Galois cyclic extension of degree m and moreover they show that the curve X is 𝔽_{q^{2}}-isomorphic to the nonsingular model of the curve given by the plane affine equation:

y_{1}^{q} + y_{1} = x_{1}^{m}.

In the case when mn = q, i.e., when the nongap m divides q, they conjecture that if the genus g of X is equal to (m−1)q/2 then the curve is 𝔽_{q^{2}}-isomorphic to a curve given by

P (z) = x^{q+1},

where P (z) is a 𝔽_{p}-linear (or additive) separable polynomial of degree m, with p being the characteristic of 𝔽_{q^{2}} (see Remark in [6], p. 46).

Here we prove a weak form of this conjecture (see Theorem 4.10). More precisely, let X be a maximal curve over 𝔽_{q^{2}} of genus g = (m − 1)q/2, where m is a nongap at some point P_{0} ∈ X that divides q. Then we show that the point P_{0} is rational over 𝔽_{q^{2}} and under the further hypothesis that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is a Galois extension, where x is a function such that div_{∞}(x) = mP_{0}, we prove that the curve is 𝔽_{q^{2}}-isomorphic to a curve given by

P (z) = A(x),  (1)

where P (z) ∈ 𝔽_{q^{2}}[z] is an additive separable polynomial of degree m and where A(x) ∈ 𝔽_{q^{2}}[x] is a polynomial of degree (q + 1).

In the particular case m = q/p we show in Theorem 5.11 here, without the assumption that 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is Galois, that the curve X is 𝔽_{q^{2}}-isomorphic to the nonsingular model of the curve given by (with q = p^{t}):

∑_{i=1}^{t} z^{p^{t−i}} = c · x^{q+1}, where c^{q−1} + 1 = 0.  (2)

Similar results were obtained before in [2] for p = 2 and in [3] for p = 3. Hence our Theorem 5.11 here generalizes results from [2] and [3] to any characteristic. Moreover, the maximal curve over 𝔽_{q^{2}} given by Equation (2) is also classified in [5] under the hypothesis that it is Galois covered by the Hermitian curve. Here we classify it under a hypothesis about nongaps at a rational point.

While the proof of Theorem 4.10 is relatively simple (just uses Hilbert’s higher ramification groups), the proof of Theorem 5.11 here is very involved (and uses Stöhr-Voloch Theory, see [17], applied to a linear system associated to a maximal curve and properties of higher Hasse’s differential operators). The reason for this is that we do not assume in Theorem 5.11 the strong hypothesis that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is a Galois extension.

The uniqueness obtained in Theorem 5.11 (see Remark 5.1 here) does not hold in general, i.e., it does not hold in general when m is a proper divisor of q such that m < q/p. In Sections 3 and 4 (see Theorem 3.2 and Theorem 4.13) we give several examples illustrating this nonuniqueness.

2 Maximal Curves

Throughout we use the following notations:

• X is a projective, geometrically irreducible, nonsingular curve over 𝔽_{q^{2}}.

• k(X) is the field of k-rational functions on X, where k = 𝔽_{q^{2}} or \overline{𝔽_{q^{2}}}.

• For x ∈ k(X) we denote by div(x) the divisor of the function x, and by div_{∞}(x) its pole divisor.

• H(P) stands for the Weierstrass nongap semigroup at the point P ∈ X and m_{i}(P) = m_{i} stands for the i-th nongap at P; we have 0 = m_{0} < m_{1} < ⋯ < m_{g} = 2g.

• If k denotes an algebraic closure of 𝔽_{q^{2}} and D is a divisor on X, then L(D) = {f ∈ k(X) | div(f) + D ≥ 0} and we denote by ℓ(D) its dimension as a vector space over k.

The key property of a 𝔽_{q^{2}}-maximal curve X is the existence of a base-point-free linear system D := |(q + 1)P_{0}| where P_{0} ∈ X (𝔽_{q^{2}}), and this linear system carries useful information about the curve (see [6]).

In fact if X is a maximal curve over 𝔽_{q^{2}}, then we have that the Frobenius map F r (relative to 𝔽_{q^{2}}) acts as multiplication by (−q) on the Jacobian variety J of the maximal curve X (see [13], Lemma 1).

Proposition 2.1 (see [6] and [13]). For a maximal curve X over 𝔽_{q^{2}} and P_{0} ∈ X (𝔽_{q^{2}}), it holds that

F r(P) + qP ∼ (q + 1)P_{0}, for each point P on X.

Therefore it follows immediately the following:

Proposition 2.2 ([13], Lemma 1). Let X be a maximal curve over 𝔽_{q^{2}} and let P_{0}, P_{1} be two rational points, then

(q + 1)P_{1} ∼ (q + 1)P_{0}.

Let us then consider the linear system D = g_{q+1}^{n+1} := |(q +1)P_{0}|; i.e., the integer n is defined by the equality ℓ((q + 1)P_{0}) = n + 2.

Proposition 2.2 above says that D is an invariant of the curve, in particular its projective dimension (n + 1) is independent of P_{0} ∈ X (𝔽_{q^{2}}). Moreover we have that q + 1 and q are nongaps at each rational point (see [6]) and so the linear system D is base-point-free.

Applying the techniques in [17] to the linear system D, we have the following results (with the notations as in [6]):

Theorem 2.3 ( [6], Thm. 1.4). For a maximal curve X over 𝔽_{q^{2}}, the D-orders and the Frobenius-orders satisfy:

(i) ϵ_{n+1} = ν_{n} = q. (ii) j_{n+1}(P) = q + 1 if P ∈ X (𝔽_{q^{2}}), and j_{n+1}(P) = q otherwise; in particular,

all rational points over 𝔽_{q^{2}} are D-Weierstrass points of X. (iii) j_{1}(P) = 1 for all points P ∈ X; in particular, ϵ_{1} = 1. (iv) If n ≥ 2, then ν_{1} = ϵ_{1} = 1.

The next proposition gives relations between the D-orders and the nongaps at points of the maximal curve X.

Proposition 2.4 ([6], Prop. 1.5). Let X be a maximal curve over 𝔽_{q^{2}}, then:

(i) For each point P on X, we have ℓ(qP) = n+1; i.e., we have the following behavior for the first nongaps at P:

0 < m_{1}(P) < ⋯ < m_{n}(P) ≤ q < m_{n+1}(P).

(ii) If P is not rational over 𝔽_{q^{2}}, the numbers below are D-orders at P:

0 ≤ q − m_{n}(P) < ⋯ < q − m_{1}(P) < q.

(iii) If P is rational over 𝔽_{q^{2}}, the numbers below are exactly all the (D, P)- orders:

0 < q + 1 − m_{n}(P) < ⋯ < q + 1 − m_{1}(P) < q + 1.

In particular, if j is a D-order at a rational point P, then q + 1 − j is a Weierstrass nongap at P.

(iv) If P ∈ X (𝔽_{q^{4}})\X (𝔽_{q^{2}}), then q − 1 is a nongap at P. If P ∉ X (𝔽_{q^{4}}), then

q is a nongap at P. If P ∈ X (𝔽_{q^{2}}), then q and q + 1 are nongaps at P.

(v) Let P be a non-Weierstrass point of X (for the canonical morphism) and suppose that n ≥ 2, then we have (for the nongaps at P) that m_{n−1}(P) = q − 1 and m_{n}(P) = q.

Analogously, for the linear system

2D = |2(q + 1)P_{0}|, where P_{0} ∈ X (𝔽_{q^{2}}),

we can prove the following:

Proposition 2.5 For each P ∈ X we have that ℓ((2q + 1)P) = N or N + 1, where N + 1 is the projective dimension of the linear system 2D. Moreover we have that ℓ((2q + 1)P) = N + 1 if and only if 2q + 1 is a nongap at P. In particular, ℓ((2q + 1)P) = N + 1 for each P ∈ X (𝔽_{q^{2}}).

Proof. Consider the following strict inclusions of vector spaces:

L(2qP) ⊂ L(2qP + F r(P)) ⊂ L(2qP + 2F r(P)).

Since ℓ(2qP + 2F r(P)) = N + 2, it follows that ℓ(2qP + F r(P)) = N + 1 and that ℓ(2qP) = N.

Observe that L(2qP) ⊆ L((2q + 1)P) and hence we obtain that

ℓ((2q + 1)P) = \begin{cases}N & if 2q + 1 is a gap at P \\ N + 1 & if 2q + 1 is a nongap at P . \end{cases}  (3)

■

Next we show two interesting properties of maximal hyperelliptic curves over 𝔽_{q^{2}}; i.e., 𝔽_{q^{2}}-maximal curves with a rational function x ∈ 𝔽_{q^{2}}(X) such that [𝔽_{q^{2}}(X) : 𝔽_{q^{2}}(x)] = 2.

Proposition 2.6 If X is a hyperelliptic maximal curve over 𝔽_{q^{2}} of genus g, then g ≤ q/2.

Proof. As X is hyperelliptic we know that its function field is an extension of degree 2 of the rational function field over 𝔽_{q^{2}}. Using that X is maximal, we obtain the inequality

1 + q^{2} + 2gq ≤ 2(q^{2} + 1).

The proposition now follows immediately from the inequality above. ■

Proposition 2.7 Let X be a maximal curve over 𝔽_{q^{2}} and assume that there exists a point P on X such that q − 1 is a (D, P)-order, then the curve X is hyperelliptic.

Proof. If P ∈ X (𝔽_{q^{2}}), then Proposition 2.4 (iii) shows that (q +1)−(q −1) = 2 is a nongap at P. Hence X is hyperelliptic in this case. If P ∉ X (𝔽_{q^{2}}) and using that q − 1 is a (D, P)-order, we have

(q − 1)P + D ∼ qP + F r(P) and hence D ∼ P + F r(P),

where D is an effective divisor of degree two such that both P and F r(P) do not belong to Supp(D). Hence the curve X is also hyperelliptic in this case. ■

We consider now a hyperelliptic maximal curve X over 𝔽_{q^{2}}, with q even. Such a curve X can be given by an equation of the following type:

z^{2} + z = f(x) ∈ 𝔽_{q^{2}}(x).

We may assume that all the pole-orders of f(x) are odd numbers ( see [15], Lemma III.7.7). If α and β are poles of f(x), then the function (x−α)/(x−β) = t shows that 2P_{α} ∼ 2P_{β}, where P_{α} (resp. P_{β}) is the unique zero of (x − α) (resp. of (x − β)). Then, since we are working in characteristic two, we have that

qP_{α} ∼ qP_{β}.

Using that X is maximal over 𝔽_{q^{2}}, we obtain that

qP_{α} + F r(P_{α}) ∼ qP_{β} + F r(P_{β}),

and then we conclude that P_{α} = P_{β}.

This shows that f(x) has an unique pole and sending this pole to x = ∞, we can assume that f(x) ∈ 𝔽_{q^{2}}[x] is a polynomial of odd degree m.

The genus g of the curve X is then equal to g = (m − 1)/2, and from Proposition 2.6, we get

m ≤ q + 1.

The following example shows a hyperelliptic maximal curve with m = q + 1.

Example 2.8 In characteristic two, the curve given by z^{2} + z = x^{q+1} is a maximal curve over 𝔽_{q^{2}} since it is covered by the Hermitian curve given by y^{q} + y = x^{q+1}; in fact just take

z = y^{q/2} + y^{q/4} + … + y^{2} + y.

One can generalize the example above to odd characteristic as follows: let q = p^{t} with t an odd natural number, then the curve given by z^{p} + z = x^{q+1} is a maximal curve over 𝔽_{q^{2}} since it is covered by the Hermitian curve given by y^{q} + y = x^{q+1}; in fact just take

z = ∑_{i=0}^{t−1}(−1)^{i}y^{p^{i}}.

3 Nonisomorphic Maximal Curves (in char. 2)

Our aim here is to construct nonisomorphic maximal curves of the same genus. These examples will be of the following type

P (z) = x^{q+1}, where P (z) is an additive polynomial.  (4)

All the examples here are defined over a finite field 𝔽_{q^{2}} of characteristic two (see [1]).

Let q = 2^{t} with t ≥ 3 and let H_{t} denote the Hermitian curve over 𝔽_{q^{2}} given by:

y^{q} + y = x^{q+1}.

For each 2 ≤ r < t and each \underline{a} := (a_{1}, …, a_{r−1}) ∈ 𝔽_{2}^{r−1}, we consider the subcover X of H_{t} whose function field satisfies

𝔽_{q^{2}}(X) = 𝔽_{q^{2}}(x, z) with z := y^{2^{r}} + a_{1}y^{2^{r−1}} + … + a_{r−1}y^{2} + y.

We will denote this subcover X by X (r, \underline{a}).

Given a pair (r, \underline{a}) as before, there are some special values of t such that the curve X (r, \underline{a}) can be given by an affine equation of the form

P (z) = x^{q+1}, with P (z) ∈ 𝔽_{2}[z]

an additive separable polynomial of degree 2^{t−r}.

(5)

Those special values of t are of the form t ≡ 0 (mod K) where K depends on the fixed pair (r, \underline{a}). The following tables give the values of K for all possibilities with 2 ≤ r ≤ 5.

r  \underline{a}  z  K
2  (1)  y^{4} + y^{2} + y  3
(0)  y^{4} + y  4
3 (1,1)  y^{8} + y^{4} + y^{2} + y  4
(1,0)  y^{8} + y^{4} + y  7
(0,1)  y^{8} + y^{2} + y  7
(0,0)  y^{8} + y  3
4 (1,1,1) y^{16} + y^{8} + y^{4} + y^{2} + y  5
(1,0,1)  y^{16} + y^{8} + y^{2} + y  6
(1,0,0)  y^{16} + y^{8} + y  15
(1,1,0)  y^{16} + y^{8} + y^{4} + y  7
(0,1,1)  y^{16} + y^{4} + y^{2} + y  7
(0,1,0)  y^{16} + y^{4} + y  6
(0,0,1)  y^{16} + y^{2} + y  6
(0,0,0)  y^{16} + y  4

r  \underline{a}  z  K
5 (1,1,1,1) y^{32} + y^{16} + y^{8} + y^{4} + y^{2} + y  6
(1,1,1,0)  y^{32} + y^{16} + y^{8} + y^{4} + y  31
(0,1,1,1)  y^{32} + y^{8} + y^{4} + y^{2} + y  31
(0,0,1,1)  y^{32} + y^{4} + y^{2} + y  14
(0,1,0,1)  y^{32} + y^{8} + y^{2} + y  15
(0,1,1,0)  y^{32} + y^{8} + y^{4} + y  12
(0,1,0,0)  y^{32} + y^{8} + y  31
(1,1,0,1)  y^{32} + y^{16} + y^{8} + y^{2} + y  31
(1,1,0,0)  y^{32} + y^{16} + y^{8} + y  14
(1,0,0,1)  y^{32} + y^{16} + y^{2} + y  8
(1,0,0,0)  y^{32} + y^{16} + y  21
(1,0,1,0)  y^{32} + y^{16} + y^{4} + y  15
(1,0,1,1)  y^{32} + y^{16} + y^{4} + y^{2} + y  31
(0,0,1,0)  y^{32} + y^{4} + y  31
(0,0,0,1)  y^{32} + y^{2} + y  21
(0,0,0,0)  y^{32} + y  5

For example for q = 2^{7} and r = 3 with (a_{1}, a_{2}) = (1, 0), we have that

z = y^{8} + y^{4} + y.

The value t = 7 is special since we have P (z) = x^{q+1}, with

P (z) = z^{16} + z^{8} + z^{4} + z.

From now on, we will only consider the special values of t such that the curve X (r, \underline{a}) can be given as in Equation (5). The genus g_{t,r} of the curve X (r, \underline{a}) satisfies:

g_{t,r} = (2^{t−r} − 1)2^{t−1}.

From the assumption that t is special, one deduces that all the 2^{r} roots of

y^{2^{r}} + a_{1}y^{2^{r−1}} + ⋯ a_{r−1}y^{2} + y = 0  (6)

are elements belonging to the finite field 𝔽_{q} where q = 2^{t}. In fact if y = α is a root, then z = 0 and P (z) = 0. Hence x = 0 and α^{q} + α = 0. Since p = 2 we conclude that α ∈ 𝔽_{q}. For each element α ∈ 𝔽_{q} which is a root of Equation (6), one can consider the automorphism σ_{α} of the Hermitian curve H_{t} given by:

σ_{α}(x) = x and σ_{α}(y) = (y + α).

One checks easily that σ_{α}(z) = z and hence we have:

Proposition 3.1 The field extension 𝔽_{q^{2}}(x, y)|𝔽_{q^{2}}(x, z) is a Galois extension with Galois group G isomorphic to (ℤ/2ℤ)^{r}. The group G consists exactly of the automorphisms σ_{α} above; i.e.,

G = {σ_{α}; α is a root of Equation (6)}.

Fixing r with r ≥ 2, we have 2^{r−1} distinct possibilities for \underline{a} ∈ 𝔽_{2}^{r−1}. For each choice of \underline{a} ∈ 𝔽_{2}^{r−1} there exists K = K(\underline{a}) ∈ ℕ such that if t ≡ 0 (mod K), then t is special for the choice (r, \underline{a}). Hence there is an integer K_{r} ∈ ℕ such that if t ≡ 0 (mod K_{r}), then t is special for all choices of \underline{a} ∈ 𝔽_{2}^{r−1}. In fact just take K_{r} as

K_{r} = lcm\{K(\underline{a}) ; \underline{a} ∈ 𝔽_{2}^{r−1}\},

where lcm stands for the least common multiple. For example we have from the table above that K_{4} = 4 × 3 × 5 × 7 = 420.

Then for a fixed value of r with r ≥ 2 and for values of t satisfying t ≡ 0 (mod K_{r}), we have 2^{r−1} maximal curves X (r, \underline{a}), with \underline{a} ∈ 𝔽_{2}^{r−1}, all having the same genus g_{t,r} where

g_{t,r} = (2^{t−r} − 1)2^{t−1}.

We can now state:

Theorem 3.2 Let r ≥ 2 be an integer and let q = 2^{t} where t > r satisfy t ≡ 0 (mod K_{r}). If \underline{a} and \underline{b} are distinct elements of 𝔽_{2}^{r−1}, then the 𝔽_{q^{2}}-maximal curves X (r, \underline{a}) and X (r, \underline{b}) are not isomorphic over an algebraic closure k of the finite field 𝔽_{q}.

The proof of Theorem 3.2 will be given in the next section (see Theorem 4.13 below).

Since for each fixed r there exist 2^{r−1} distinct possible choices of elements \underline{a} ∈ 𝔽_{2}^{r−1}, we obtain the following corollary:

Corollary 3.3 Given r ∈ ℕ and taking q = 2^{t} with t ≡ 0 mod(K_{r}) as before, there exist 2^{r−1} maximal curves over 𝔽_{q^{2}} with the same genus g_{t,r} that are not isomorphic to each other. They are Galois covered by the Hermitian curve with the same Galois group G ≃ (ℤ/2ℤ)^{r}.

Proof. Follows directly from Theorem 3.2 and Proposition 3.1. ■

Remark 3.4 Examples of nonisomorphic maximal curves over 𝔽_{q^{2}} with the same genus appeared already in [4]. They showed that (for q ≡ 3 (mod 4)):

y^{q} + y = x^{(q+1)/4} and u^{(q+1)/2} + v^{(q+1)/2} = 1

are equations of nonisomorphic maximal curves over the finite field 𝔽_{q^{2}} with genus satisfying 8g = (q − 1)(q − 3). Both curves above are Galois covered by the 𝔽_{q^{2}}-Hermitian curve with a Galois group G satisfying:

a) G ≃ ℤ/4ℤ in the case of y^{q} + y = x^{(q+1)/4}.

b) G ≃ ℤ/2ℤ × ℤ/2ℤ in the case of u^{(q+1)/2} + v^{(q+1)/2} = 1.

Note that the examples of nonisomorphic maximal curves contained in Corollary 3.3 above, all have the same group G ≃ (ℤ/2ℤ)^{r}.

4 The general case (arbitrary characteristic)

Hypothesis 4.1 In this section we consider the more general situation of a maximal curve X over 𝔽_{q^{2}} of genus g = (m − 1)q/2, with m a proper divisor of q, such that there exists a point P_{0} on X having m as a Weierstrass nongap at P_{0}. We define n ≥ 2 by mn = q.

We start with a particular example:

Example 4.2 The curves X (r, \underline{a}) considered in the previous section, with q = 2^{t} and with t special for the choice of (r, \underline{a}), are examples of such maximal curves over 𝔽_{q^{2}} with m = 2^{t−r} and n = 2^{r}. The point P_{0} is the unique pole of the function x on the curve X (r, \underline{a}). We also have in the case X = X (r, \underline{a}) that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is a Galois extension of degree m = 2^{t−r}, as follows from Equation (5) (see Further Hypothesis 4.8 below). We note that the roots of the polynomial P (z) ∈ 𝔽_{2}[z] appearing in Equation (5) are elements of the finite field 𝔽_{q} with q = 2^{t}.

We have used the same letter n to denote either the number n_{1} such that ℓ((q + 1)P_{0}) = n_{1} + 2 or the number n_{2} such that m · n_{2} = q. We will see shortly after the proof of Claim 4.3 that (under the hypothesis above) we have the equality n_{1} = n_{2}.

We now turn back to curves X in the more general situation (see Hypothesis 4.1 above):

Claim 4.3 The point P_{0} on the curve X as above is a rational point over 𝔽_{q^{2}}.

Proof. Suppose that P_{0} ∉ X (𝔽_{q^{2}}) and let x ∈ k(X) with k = 𝔽̄_{q^{2}} be a function such that div_{∞}(x) = mP_{0}.

We define the integer e ≥ 1 by e = v_{F r(P_{0})}(x − x(F r(P_{0}))), where v_{P} denotes the valuation associated to the point P.

We can write

div(x−x(F r(P_{0}))) = eF r(P_{0})+D−mP_{0}, where D ≥ 0 and P_{0}, F r(P_{0}) ∉ SuppD.

For j = 1, 2, …, n = q/m we have:

div((x − x(F r(P_{0})))^{j}) = jeF r(P_{0}) + jD − jmP_{0}.

Summing up (qP_{0} + F rP_{0}) to the divisors above we see that the numbers 0, 1, e + 1, 2e + 1, …, ne + 1 are all the (n + 2) distinct (D, F r(P_{0}))-orders. Since P_{0} ∉ X (𝔽_{q^{2}}) we must have that ne + 1 = q, which is a contradiction since n divides q and n ≥ 2. ■

From Claim 4.3 and the assumption g = (m − 1)q/2, we see that the Weierstrass nongap semigroup H(P_{0}) at P_{0} is generated by m and q + 1; i.e., we have H(P_{0}) = ⟨m, q + 1⟩. This is so because q + 1 is a nongap at any rational point. It follows from Proposition 1.10 in [6] that the assumption on the genus is equivalent to ℓ((q + 1)P_{0}) = n + 2.

Consider now x, z ∈ 𝔽_{q^{2}}(X) such that:

div_{∞}(x) = mP_{0} and div_{∞}(z) = (q + 1)P_{0}.

Clearly we have that 𝔽_{q^{2}}(X) = 𝔽_{q^{2}}(x, z).

The morphism Φ associated to the linear system D = |(q + 1)P_{0}| is given by:

Φ = (1 : x : x^{2} : ⋯: x^{n} : z).

If P ≠ P_{0} the morphism Φ can also be given as Φ = (1 : u : u^{2} : ⋯: u^{n} : z) where u = x−x(P). Then it follows immediately that v_{P} (u), 2v_{P} (u), …, nv_{P} (u) are (D, P)-orders.

Claim 4.4 v_{P} (u) = 1 for each point P on the curve with P ≠ P_{0}.

Proof. Suppose that v_{P} (u) > 1 for some point P ≠ P_{0}, then 0, 1, v_{P} (u), 2v_{P} (u), …, nv_{P} (u) are all the (D, P)-orders. We must have that

nv_{P} (u) = \begin{cases}q + 1 & if P ∈ X (𝔽_{q^{2}}) \\ q & if P ∉ X (𝔽_{q^{2}}). \end{cases}

From the fact that n ≥ 2 and n divides q, it follows that P is not a rational point if v_{P} (u) > 1 . Then we must have that nv_{P} (u) = q or equivalently that v_{P} (u) = m. It then follows that

div(u) = mP − mP_{0} and hence qP ∼ qP_{0}.

Using that the curve X is maximal over 𝔽_{q^{2}} and so that qP +F rP ∼ (q+1)P_{0}, we conclude from qP ∼ qP_{0} that P = P_{0}, a contradiction. ■

Remark 4.5 Claim 4.4 implies that the place corresponding to the point P_{0} over the pole of x is the only ramified place in the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x). Moreover, using that X is 𝔽_{q^{2}}-maximal and hence that #X (𝔽_{q^{2}}) = mq^{2} + 1, we have that all the other rational places of 𝔽_{q^{2}}(x) split completely in 𝔽_{q^{2}}(X) and that x = ∞ is totally ramified.

Proposition 4.6 With the hypothesis on the curve X above we have:

(i) The D-orders on the curve X are 0, 1, 2, …, n, q.

(ii) For P ∉ X (𝔽_{q^{2}}), the (D, P)-orders are 0, 1, 2, …, n, q.

(iii) For P ∈ X (𝔽_{q^{2}}) \ {P_{0}}, the (D, P)-orders are 0, 1, 2, …, n, q + 1.

(iv) The (D, P_{0})-orders on X are 0, 1, m + 1, 2m + 1, …, q + 1 = nm + 1.

Proof. The proofs of items (i)-(iii) follow directly from Claim 4.4. The proof of item (iv) follows from Proposition 2.4 and the fact that the Weierstrass nongap semigroup at P_{0} is H(P_{0}) = ⟨m, q + 1⟩. ■

The D-ramification divisor R contains in its support (with multiplicities) exactly the D-Weierstrass points of the curve (see [17]).

Corollary 4.7 Let R be the ramification divisor on the curve X associated to the linear system D, then Supp(R) = X (𝔽_{q^{2}}).

Further Hypothesis 4.8 From now on we suppose that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is a Galois extension of degree m, where x is a function on X satisfying div_{∞}(x) = mP_{0}.

Since the place corresponding to P_{0} is the only ramified place in this extension, we must have:

2g − 2 = m(−2) + d(P_{0}),

where d(P_{0}) is the different exponent at P_{0}. Then d(P_{0}) = (m − 1)(q + 2).

As the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is Galois with a Galois group H of order m, we can consider the i-th ramification group H_{i} at P_{0} over x = ∞.

We remind the reader that H_{i} is defined as (with O_{P_{0}} the local ring at P_{0}):

H_{i} = {σ ∈ H | v_{P_{0}}(σ(z̃) − z̃) ≥ i + 1, ∀z̃ ∈ O_{P_{0}}}.

The Hilbert different formula states that

d(P_{0}) = ∑_{i=0}^{∞}(|H_{i}| − 1),

and then we have that

(m − 1)(q + 2) = ∑_{i=0}^{∞}(|H_{i}| − 1).

Claim 4.9 |H_{q+2}| = 1.

Proof. Let σ ∈ H and let z ∈ 𝔽_{q^{2}}(X) with div_{∞}(z) = (q + 1)P_{0}. Then σ(P_{0}) = P_{0} and σ(z) = cz + ∑_{i=0}^{n} b_{i}x^{i}, where c ≠ 0 and c, b_{i} ∈ 𝔽_{q^{2}}. We also have σ(x) = x.

Consider u = x^{n}/z. This element u is a local parameter at P_{0}, and we have that σ ∈ H_{q+2} if and only if v_{P_{0}}(σ(u) − u) ≥ q + 3.

Since

σ(u) − u = x^{n}/σ(z) − x^{n}/z = x^{n}(z − σ(z))/zσ(z),

we have

v_{P_{0}}(σ(u) − u) = −nm + 2(q + 1) + v_{P_{0}}(σ(z) − z)

and hence

v_{P_{0}}(σ(u) − u) = q + 2 + v_{P_{0}}((c − 1)z + ∑_{i=0}^{n} b_{i}x^{i}).  (7)

Then v_{P_{0}}(σ(u) − u) ≥ q + 3 if and only if v_{P_{0}}((c − 1)z + ∑_{i=0}^{n} b_{i}x^{i}) ≥ 1.

But v_{P_{0}}((c − 1)z + ∑_{i=0}^{n} b_{i}x^{i}) ≤ 0 if σ is not the identity. ■

Theorem 4.10 Let X be a maximal curve over 𝔽_{q^{2}} of genus g = (m − 1)q/2, where m is a nongap at a point P_{0} ∈ X such that nm = q, and suppose that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is a Galois extension, where x ∈ 𝔽_{q^{2}}(X) is a function such that div_{∞}(x) = mP_{0}. Then the curve X is 𝔽_{q^{2}}-isomorphic to a curve given by

P (z) = A(x),

where P (z) ∈ 𝔽_{q^{2}}[z] is an additive separable polynomial of degree m and A(x) ∈ 𝔽_{q^{2}}[x] is a polynomial of degree q + 1.

Proof. As |H_{i}| ≤ m, ∀i, and since (m − 1)(q + 2) = ∑_{i=0}^{q+1}(|H_{i}| − 1), as follows from Claim 4.9 above, we have that

|H_{i}| = m, ∀i = 0, 1, … q + 1.

Then, it follows from ([14], Chapter IV) that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is elementary abelian and the theorem follows immediately from this fact. ■

Remark 4.11 The hypothesis that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is a Galois extension is equivalent to the hypothesis that the extension F̃|𝔽_{q^{2}}(X) is not ramified, where F̃ is the Galois closure of 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x).

In fact, let g̃ denote the genus of F̃; since F̃|𝔽_{q^{2}}(X) is unramified we have:

2g̃ − 2 = (2g − 2)[F̃ : 𝔽_{q^{2}}(X)].

Since from Remark 3.4 we have that (x−α) splits completely in 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) for each α ∈ 𝔽_{q^{2}}, it follows that

mq^{2}[F̃ : 𝔽_{q^{2}}(X)] ≤ q^{2} + 1 + 2g̃q and then [F̃ : 𝔽_{q^{2}}(X)] ≤ 1.

Remark 4.12 We obtain, as a consequence of Equation (7) and the fact that H = H_{q+1}, that for each σ ∈ H there exists an element b_{0} ∈ 𝔽_{q^{2}} such that σ(z) = z + b_{0}. This element b_{0} satisfies P (b_{0}) = 0, where P (z) is the additive polynomial appearing in Theorem 4.10. We note also that all roots of P (z) belong to the finite field 𝔽_{q^{2}} and hence

H = {σ_{b}; σ_{b}(z) = z + b and P (b) = 0}.

Now we will prove Theorem 3.2 in the more general case (see Hypothesis 4.1 and Example 4.2 in the beginning of this section).

Theorem 4.13 Let X_{1} and X_{2} be two maximal curves over 𝔽_{q^{2}} satisfying the hypothesis of Theorem 4.10 with n = p^{r}, r ≥ 2, and q = p^{t}. Suppose that they can be given as below:

X_{1} := (z^{p^{t−r}} + α_{t−r−1}z^{p^{t−r−1}} + ⋯ + α_{1}z^{p} + α_{0}z − A_{1}(x) =: Q_{1}(x, z)),

X_{2} := (w^{p^{t−r}} + β_{t−r−1}w^{p^{t−r−1}} + ⋯ + β_{1}w^{p} + β_{0}w − A_{2}(x̃) =: Q_{2}(x̃, w)),

where

A_{1}(x) = x^{q+1} + ∑_{i=0}^{q} γ_{i}x^{i} and A_{2}(x̃) = x̃^{q+1} + ∑_{i=0}^{q} δ_{i}x̃^{i}, with γ_{i}, δ_{i} ∈ 𝔽_{q^{2}}.

If there exists an index i_{0} ∈ {0, 1, ⋯, t − r − 1} such that α_{i_{0}} ≠ 0 and β_{i_{0}} = 0 (or α_{i_{0}} = 0 and β_{i_{0}} ≠ 0), then the two curves above X_{1} and X_{2} are not isomorphic over an algebraic closure k of the finite field 𝔽_{q^{2}}.

We need the following lemma in our proof of Theorem 4.13.

We will denote by P_{0} the point corresponding to the only place above the pole of x in the Galois extension 𝔽_{q^{2}}(X_{1})|𝔽_{q^{2}}(x) and by Q_{0} the point corresponding to the only place above the pole of x̃ in the Galois extension 𝔽_{q^{2}}(X_{2})|𝔽_{q^{2}}(x̃).

Lemma 4.14 The point P_{0}(resp. the point Q_{0}) is the only point of the curve X_{1} (resp. of the curve X_{2}) whose Weierstrass nongap semigroup is generated by two elements.

Proof. Since we assume in Theorem 4.13 that r ≥ 2, we have that m = m_{1}(P_{0}) is strictly smaller than q/p, where m = p^{t−r} and q = p^{t}. For a point P on X_{1} distinct from P_{0}, we consider two cases:

• For P ∈ X (𝔽_{q^{2}}) it follows from Proposition 4.6 that 0, 1, 2, ⋯, p^{r}, q + 1 are the (D, P)-orders. It then follows from Proposition 2.4 that the numbers

m_{1} = q+1−p^{r}, m_{2} = q+2−p^{r}, m_{3} = q+3−p^{r}, ⋯, m_{n+1} = q+1, with n = p^{r},

are nongaps at P; in particular as m_{2} = m_{1} + 1 and m_{3} = m_{2} + 1, then the Weierstrass nongap semigroup H(P) cannot be generated by two elements.

• For P ∉ X (𝔽_{q^{2}}) it follows from Proposition 4.6 that the (D, P)-orders are 0, 1, 2, ⋯, p^{r}, q. From Proposition 2.4, we then have the following possibilities for the set {m_{1}, m_{2}, m_{3}} of the first three nongaps at P:

\{m_{1}, m_{2}, m_{3}\} = \begin{cases}\{p^{t} − p^{r}, p^{t} − p^{r} + 1, p^{t} − p^{r} + 2\} \\ \{p^{t} − p^{r}, p^{t} − p^{r} + 2, p^{t} − p^{r} + 3\} \\ \{p^{t} − p^{r}, p^{t} − p^{r} + 1, p^{t} − p^{r} + 3\} \\ \{p^{t} − p^{r} + 1, p^{t} − p^{r} + 2, p^{t} − p^{r} + 3\}. \end{cases}

In all the four cases above we have that m_{2} < 2m_{1} and then if the semigroup H(P) could be generated by two elements, we would have that H(P) = ⟨m_{1}, m_{2}⟩ and that m_{3} = 2m_{1}. A direct analysis of each case give us a contradiction. ■

Now we will give the proof of Theorem 4.13:

Proof. [Proof of Theorem 4.13]

We will denote by k an algebraic closure of 𝔽_{q^{2}}. Suppose that there exists an isomorphism ϕ : X_{1} ↦ X_{2}. It follows from the lemma above that ϕ must map the point P_{0} onto Q_{0}.

We recall that m = p^{t−r}, q = p^{t} and n = p^{r}. Also we have:

div_{∞}(x) = mP_{0} and div_{∞}(z) = (p^{t} + 1)P_{0}

and analogously, div_{∞}(x̃) = mQ_{0} and div_{∞}(w) = (p^{t} + 1)Q_{0}.

In particular we obtain that

ϕ(x̃) = bx + a and ϕ(w) = ∑_{i=0}^{n} d_{i}x^{i} + ez,

where a, b, e, d_{i} ∈ k and eb ≠ 0. Substituting x by (x + a/b) one can assume that a = 0. Now we show that one can assume simultaneously that:

ϕ(x̃) = bx and ϕ(w) = ∑_{i=0}^{n} d_{i}x^{i} + z;

i.e., one can assume simultaneously that a = 0 and e = 1. In fact multiplying the equation of definition of X_{1} by e^{p^{t−r}} we get

(ez)^{p^{t−r}} + αt−r−1^{′}(ez)^{p^{t−r−1}} + ⋯ + α1^{′}(ez)^{p} + α0^{′}(ez) = e^{p^{t−r}}.A_{1}(x), (8)

where αj^{′} = α_{j}e^{p^{t−r}−p^{j}}. Note that α_{j} ≠ 0 if and only if αj^{′} ≠ 0. Let d ∈ k be such that d^{q+1} = e^{p^{t−r}} and substitute x by (dx). This makes the right hand side of Equation (8) a monic polynomial in the new variable (dx). Hence ϕ(x̃) = bx = (b/d).(dx), as we wanted. Applying now ϕ with a = 0 and e = 1 to the equation of definition of the curve X_{2}, we obtain

ϕ(Q_{2}(x̃, w)) = ∑_{j=0}^{t−r} β_{t−r−j}ϕ(w)^{p^{t−r−j}} − ϕ(A_{2}(x̃))

= ∑_{j=0}^{t−r} β_{t−r−j}(∑_{i=0}^{n} d_{i}x^{i} + z)^{p^{t−r−j}} − ∑_{i=0}^{q+1} δ_{i}b^{i}x^{i}

= ∑_{j=0}^{t−r} β_{t−r−j} ∑_{i=0}^{n}(d_{i}x^{i})^{p^{t−r−j}} − ∑_{i=0}^{q+1} δ_{i}b^{i}x^{i} + ∑_{j=0}^{t−r} β_{t−r−j}z^{p^{t−r−j}}.

As ϕ(X_{1}) = X_{2}, we obtain that ϕ(Q_{2}(x̃, w)) must be a constant multiple of Q_{1}(x, z),

ϕ(Q_{2}(x̃, w)) = c.Q_{1}(x, z), for some contant c.

Comparing the coefficients of the terms in x^{q+1} we have that c = δ_{q+1}b^{q+1} = b^{q+1} ≠ 0.

We also have

β_{j} = c.α_{j} ∀ j = 0, ⋯, t − r.

Hence if the two curves are isomorphic we have that β_{j} = 0 if and only if α_{j} = 0, and this finishes the proof of Theorem 4.13. ■

5 The particular case m = q/p

In this section we will consider a maximal curve X over 𝔽_{q^{2}} where q = p^{t} with genus g = (q/p − 1)q/2 such that q/p is a nongap at some point P_{0} of the curve. We will prove that the curve X is 𝔽_{q^{2}}-isomorphic to a curve given by the plane equation:

∑_{i=1}^{t} z^{p^{t−i}} = c · x^{p^{t}+1}, with c ∈ 𝔽_{q^{2}}^{∗} such that c^{q} + c = 0.  (9)

This section generalizes some of the results in [2] and [3]. Note that here we do not assume that the extension 𝔽_{q^{2}}(X)|𝔽_{q^{2}}(x) is Galois, for some function x on the curve X with div_{∞}(x) = (q/p) · P_{0}.

Remark 5.1 The curves given by Equation (9) are 𝔽_{q^{2}}-isomorphic to each other. In fact if c_{1}, c_{2} ∈ 𝔽_{q^{2}}^{∗} satisfy c_{i}^{q} + c_{i} = 0, then one can take α ∈ 𝔽_{q^{2}} satisfying α^{q+1} = c_{1}/c_{2} and the map ϕ below is an isomorphism

ϕ : X_{1} ↦ X_{2}
(x, z) ↦ (αx, z)

where X_{i} is the curve given by Equation (9) corresponding to c_{i} (for i = 1, 2).

As before let x, z ∈ 𝔽_{q^{2}}(X) be such that

div_{∞}(x) = p^{t−1}P_{0} and div_{∞}(z) = (p^{t} + 1)P_{0}.

Let v = v_{P_{0}} be the valuation at P_{0}, and let D^{(i)} := D_{x}^{(i)} be the i-th Hasse differential operator on 𝔽̄_{q^{2}}(X) with respect to the variable x (see e.g. [17]). We also set D := D^{(1)}.

Claim 5.2 v(Dz) = −q^{2}/p.

Proof. (cf. [2], p. 47). Let u be a local parameter at P_{0}; then

v(Dz) = v(dz/du) − v(dx/du) = −(q + 2) − (2g − 2) = −q^{2}/p,

since by Remark 4.5 the morphism x : X → ℙ^{1}(𝔽̄_{q^{2}}) is totally ramified at P_{0} and unramified outside P_{0}, and since a canonical divisor has degree 2g − 2. ■

Let x and z be as above; they are related to each other by an equation of the type (see e.g. [10])

x^{q+1} + az^{q/p} + ∑_{i=0}^{q/p−1} A_{i}(x)z^{i} = 0 ,  (10)

where a ≠ 0 and the A_{i}(x)’s are polynomials in 𝔽_{q^{2}}[x] such that

deg(A_{i}(x)) ≤ q − pi.

Claim 5.3 (1) If i ≥ 2 is not a power of p, then A_{i}(x) = 0.

(2) A_{p^{j}}(x) ∈ 𝔽_{q^{2}}^{∗} for each j = 0, 1, …, t − 2.

Proof. First we show that A_{i}(x) = 0 if i ≥ 2 and i ≢ 0 (mod p). To do that, we apply the operator D to Equation (10) obtaining

0 = x^{q} + ∑_{i=0}^{q/p−1} z^{i}D(A_{i}(x)) + ( ∑_{i=1}^{q/p−1} A_{i}(x)iz^{i−1})Dz .

Suppose that A_{i}(x) ≠ 0 for some i ≥ 2 and i ≢ 0 (mod p). Then

v ( ∑_{i=1}^{q/p−1} A_{i}(x)iz^{i−1}Dz) = v ( ∑_{i=0}^{q/p−1} z^{i}D(A_{i}(x))) ,

since by Claim 5.2 we have the following inequality:

v ( ∑_{i=1}^{q/p−1} A_{i}(x)iz^{i−1}Dz) < v(x^{q}).

We note by reasoning modulo q/p that the minimum value of v is attained just once in both sums. Then there exist integers 2 ≤ i_{0} ≤ (q/p) − 1 with i_{0} ≢ 0 (mod p) and 1 ≤ j_{0} ≤ (q/p) − 1 such that

v(A_{i_{0}}(x)z^{i_{0}−1}Dz) = v(z^{j_{0}}D(A_{j_{0}}(x))),

and then

q/p(deg(DA_{j_{0}}) − deg(A_{i_{0}})) = (q + 1)(i_{0} − 1 − j_{0}) + q^{2}/p.

Reasoning again modulo q/p we conclude that i_{0} ≡ j_{0}+1 (mod q/p) and hence i_{0} = j_{0} + 1. Therefore if A_{i_{0}}(x) ≠ 0 we get deg(DA_{j_{0}}) = deg(A_{i_{0}}) + q ≥ q, a contradiction. This proves that A_{i}(x) = 0 if i ≥ 2 and i ≢ 0 (mod p). Then Equation (10) is reduced to

x^{q+1} + az^{q/p} + A_{0}(x) + A_{1}(x)z + ∑_{i=1}^{q/p^{2}−1} A_{pi}(x)z^{pi} = 0 .  (11)

Note that if q = p^{2} then the last sum in Equation (11) is nonexisting.

Now we can see that A_{1} ∈ 𝔽_{q^{2}}. Indeed, applying D to Equation (11) we get

x^{q} + DA_{0}(x) + D(A_{1}(x)z) + ∑_{i=1}^{q/p^{2}−1} z^{pi}D(A_{pi}(x)) = 0.  (12)

If u is an uniformizer at P_{0}, we have

v(D(A_{1}(x)z)) = v(d(A_{1}(x)z)/du) − v(dx/du)
= −(q + 2) − q/p deg(A_{1}(x)) − (2g − 2)
= −q^{2}/p − (q/p) · deg(A_{1}(x)).

Suppose that deg(A_{1}(x)) ≥ 1, then v(D(A_{1}(x)z)) is the unique minimum value that the valuation v takes in Equation (12) and this gives us a contradiction. Hence A_{1}(x) ∈ 𝔽_{q^{2}}.

Assuming that q ≥ p^{3} we show now that A_{pi}(x) = 0 for i ≥ 2 and i ≢ 0 (mod p). In order to do that we need to compute v(D^{(p)}z).

Lemma 5.4 v(D^{(p)}z) = −q^{2}.

Proof. From the fact that the morphism (1 : x : ⋯: x^{p} : y) has 𝔽_{q^{2}}-Frobenius orders 0, 1, 2, ⋯, p − 1 and q, we conclude that the following determinant D is identically zero, where

D = det (\begin{matrix}1 & x^{q^{2}} & x^{2q^{2}} & ⋯ & x^{pq^{2}} & z^{q^{2}} \\ 1 & x & x^{2} & ⋯ & x^{p} & z \\ 0 & 1 & Dx^{2} & ⋯ & 0 & Dz \\ 0 & 0 & 1 & ⋯ & 0 & D^{(2)}z \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & D^{(p)}z \end{matrix}) .

After some computations we see that D = 0 gives us an equation as below

z^{q^{2}} − z = ∑_{i=1}^{p} B_{i}(x) · (D^{(i)}z),  (13)

where B_{1}(x) = (x^{q^{2}} − x), B_{p}(x) = (x^{q^{2}} − x)^{p}, and moreover deg(B_{j}(x)) = jq^{2} for each j ∈ ℕ with 1 ≤ j ≤ p.

For each 1 ≤ j ≤ (p − 1) we have that

v(B_{j}(x)D^{(j)}z) > −q^{2}(q + 1),

as follows by applying the operator D^{(j)} to Equation (11) and using that deg(B_{j}(x)) = jq^{2}. We then conclude from Equation (13) that

v((x^{pq^{2}} − x^{p})D^{(p)}z) = −q^{2} · (q + 1),

and this finishes the proof of Lemma 5.4. ■

Returning to the proof of Claim 5.3, we apply D^{(p)} to Equation (11) and we obtain:

\begin{aligned}0 = D^{(p)}A_{0}(x) + A_{1}D^{(p)}z & + ∑_{i=1}^{q/p^{2}−1} z^{pi} · D^{(p)}(A_{pi}(x)) \\ & + ( ∑_{i=1}^{q/p^{2}−1} A_{pi}(x)iz^{p(i−1)})(Dz)^{p}. \end{aligned}  (14)

Since we have from Lemma 5.4 the following inequalities

\begin{cases}v(D^{(p)}z) < v(D^{(p)}A_{0}(x)) and \\ v(D^{(p)}z) < v(∑_{i=1}^{q/p^{2}−1} z^{pi} · D^{(p)}(A_{pi}(x))), \end{cases}

we must have that

−q^{2} = v(D^{(p)}z) = v( ∑_{i=1}^{q/p^{2}−1} A_{pi}(x)iz^{p(i−1)}) − q^{2},

and therefore

v ( ∑_{i=1}^{q/p^{2}−1} A_{pi}(x)iz^{p(i−1)}) = 0.

If there exists i ≥ 2 and i ≢ 0 (mod p) such that A_{pi}(x) ≠ 0, then the last equality above never holds. Moreover we obtain that A_{p} ∈ 𝔽_{q^{2}} and A_{p} ≠ 0. We have shown that Equation (10) can be reduced to

x^{q+1} + az^{q/p} + A_{0}(x) + A_{1}z + A_{p}z^{p} + ∑_{i=1}^{q/p^{3}−1} A_{p^{2}i}(x)z^{p^{2}i} = 0 .  (15)

Note that if q = p^{3} then the last sum in Equation (15) is nonexisting. Applying D^{(p^{2})} to Equation (15) and using that D^{(p^{2})}z = 0 since p < p^{2} < q, as follows from the fact that 0, 1, …, p and q are the generic D-orders, we get

D^{(p^{2})}(A_{0}(x)) + A_{p}(D^{(p)}z)^{p} + ∑_{i=1}^{q/p^{3}−1} z^{p^{2}i} · D^{(p^{2})}(A_{p^{2}i}(x))

+ [ ∑_{i=1}^{q/p^{3}−1} i · z^{(i−1)p^{2}} · (A_{p^{2}i}(x))] · (Dz)^{p^{2}} = 0.

As before we conclude that

v ( ∑_{i=1}^{q/p^{3}−1} i · z^{(i−1)p^{2}} · (A_{p^{2}i}(x))) = 0.

This shows that A_{p^{2}i} = 0 for i ≥ 2 and i ≢ 0 (mod p), and also that A_{p^{2}} ∈ 𝔽_{q^{2}}^{∗}. Continuing with these arguments we conclude the proof of Claim 5.3. ■ Thus Equation (10) becomes

az^{q/p} + x^{q+1} + A_{0}(x) + ∑_{i=0}^{t−2} A_{p^{i}} · z^{p^{i}} = 0, with each A_{p^{i}} ∈ 𝔽_{q^{2}}^{∗}.  (16)

We rewrite Equation (16) as

0 = x^{q+1} + A_{0}(x) + ∑_{i=1}^{t} a_{i} · z^{p^{t−i}} ,  (17)

where deg(A_{0}(x)) ≤ q and a_{i} ∈ 𝔽_{q^{2}}^{∗}.

We can assume that degA_{0}(x) < q. In fact, writing A_{0}(x) = ∑_{i=0}^{q} b_{i} · x^{i} and making the substitution z ↦ z̃ = (z + αx^{p}) where α satisfies α^{q/p} = −b_{q}/a_{1}, then

a_{1}(z̃)^{q/p} = a_{1}(z + αx^{p})^{q/p} = a_{1}z^{q/p} + a_{1}α^{q/p}.x^{q} = a_{1}z^{q/p} − b_{q}x^{q}.

Claim 5.5 After a change of variables, we can suppose that the polynomial A_{0}(x) has the form

A_{0}(x) = b_{0} + ∑_{i=0}^{t−1} b_{p^{i}} · x^{p^{i}}.

Proof. We write A_{0}(x) = ∑_{i=0}^{q−1} b_{i} · x^{i}. Since the D-orders of (1 : x : x^{2} : ⋯: x^{p} : z) are 0, 1, ⋯, p and q we obtain that D^{(i)}z = 0, ∀i = p + 1, ⋯, q − 1. Then, applying D^{(i)} to Equation (17) for each i ≥ p + 1 such that p ∤ i, we obtain that b_{i} = 0.

For j > p such that j ≠ kp^{s} with 1 ≤ k ≤ p − 1 and s ≥ 1, we write j = c_{i}p^{i} + c_{i+1}p^{i+1} + ⋯ + c_{i+l}p^{i+l} with c_{i} ≠ 0 and c_{i+l} ≠ 0, then

0 = D^{(j)} (∑_{k=1}^{t} a_{k} · z^{p^{t−k}}) = ∑_{k=t−i}^{t} a_{k} · (D^{(j/p^{t−k})}z)^{p^{t−k}},

since D^{(j/p^{t−k})}z = 0 for k = t − i, ⋯, t. From this fact it follows that b_{j} = 0. We have therefore shown that

A_{0}(x) = ∑_{i=0}^{p−1} b_{i} · x^{i} + ∑_{i=1}^{t−1} b_{p^{t−i}} · x^{p^{t−i}}  (18)

+ ∑_{i=1}^{t−1} b_{2p^{t−i}} · x^{2p^{t−i}} + ⋯ + ∑_{i=1}^{t−1} b_{(p−1)p^{t−i}} · x^{(p−1)p^{t−i}}.

This finishes the proof of Claim 5.5 if p = 2. So we assume now that p ≥ 3.

Applying D, D^{(2)}, and D^{(p)} to Equation (17) with A_{0}(x) as given in Equation (18), we obtain:

−a_{t} · Dz = x^{q} + ∑_{i=1}^{p−1} ib_{i} · x^{i−1}.  (19)

−a_{t} · D^{(2)}z = ∑_{j=2}^{p−1} \binom{j}{2}b_{j} · x^{j−2}, if char ≠ 2.  (20)

−a_{t} · D^{(p)}z = ∑_{k=1}^{p−1} k(b_{kp} − \frac{a_{t−1}}{a_{t}^{p}} b_{k}^{p})x^{(k−1)p} − \frac{a_{t−1}}{a_{t}^{p}} · x^{qp}.  (21)

Now, for i = 1, ⋯, t − 1 we compute D^{(2p^{i})} on Equation (18) and using the expressions for Dz and D^{(p)}z obtained above, we obtain that

Lemma 5.6 b_{kp^{i}} = a_{t−i}(\frac{b_{k}}{a_{t}})^{p^{i}} for all k = 2, ⋯, p − 1 and all i = 1, ⋯, t − 1.

Proof. In fact, D^{(2p^{i})} applied to Equation (17) gives us (since we have D^{(j)}z = 0, for each value of j ∈ ℕ with p + 1 ≤ j ≤ q − 1):

D^{(2p^{i})} (∑_{k=1}^{t} a_{k}z^{p^{t−k}}) = −D^{(2p^{i})}(A_{0}(x)).

Hence a_{t−i}(D^{(2)}z)^{p^{i}} = − ∑_{j=2}^{p−1} \binom{j}{2}b_{jp^{i}} · x^{(j−2)p^{i}}, and therefore

\frac{a_{t−i}}{a_{t}^{p^{i}}} . ∑_{j=2}^{p−1} \binom{j}{2}b_{j}^{p^{i}} · x^{(j−2)p^{i}} = ∑_{j=2}^{p−1} \binom{j}{2}b_{jp^{i}} · x^{(j−2)p^{i}}.

Then Lemma 5.6 follows directly from the equation above by comparing coefficients. ■

Returning to the proof of Claim 5.5, Lemma 5.6 above allows us to make the following reduction on the defining equation of the curve

− ∑_{i=1}^{t} a_{i} · z_{1}^{p^{t−i}} = x^{q+1} + ∑_{i=1}^{t−1} b_{p^{i}} · x^{p^{i}} + b_{0} + b_{1}x.  (22)

In fact, Equation (22) is obtained from Equation (17) and Equation (18) by making the following substitution:

z ↦ z_{1} = (z + \frac{1}{a_{t}} . ∑_{k=2}^{p−1} b_{k} · x^{k}) .

This finishes the proof of Claim 5.5. ■

Claim 5.7 Changing variables, we can assume that the plane equation of the maximal curve X has the form:

∑_{i=1}^{t} a_{i} · z^{p^{t−i}} = x^{q+1} + ∑_{i=1}^{t−1} b_{p^{i}} · x^{p^{i}}.  (23)

Proof. We want to show that we can assume b_{0} = b_{1} = 0 in Equation (22). To make b_{1} = 0 we perform the substitution

z_{1} ↦ z_{2} = −(z_{1} + \frac{b_{1}}{a_{t}} · x).

Then Equation (22) is transformed into

∑_{i=1}^{t} a_{i} · z_{2}^{p^{t−i}} = x^{q+1} + ∑_{i=1}^{t−1} b_{p^{i}} · x^{p^{i}} + b_{0}.  (24)

Since the equation for the curve X has the property that for each x = α ∈ 𝔽_{q^{2}} we have m = q/p rational points over 𝔽_{q^{2}} with first coordinate equal to x = α, taking in particular the value x = 0 in Equation (24) we have

∑_{i=1}^{t} a_{i} · β^{p^{t−i}} = b_{0}, with β ∈ 𝔽_{q^{2}}.

Now we change variable z_{2} ↦ z = (z_{2} − β) to conclude the proof of Claim 5.7. ■

Lemma 5.8 For each 2 ≤ i ≤ t − 1 we have:

b_{p^{i}} = a_{t−i+1} · (b_{p}/a_{t})^{p^{i−1}} and a_{t−i} = a_{t−i+1} · (a_{t−1}/a_{t})^{p^{i−1}}.

Proof. Applying D and D^{(p)} to Equation (23) we get:

a_{t−i}(Dz) = x^{q} and b_{p} = a_{t−1} · (Dz)^{p} + a_{t} · (D^{(p)}z).

For each i with 2 ≤ i ≤ t − 1, we apply D^{(p^{i})} to Equation (23) and using that D^{(j)}z = 0 for p < j < q, we get

a_{t−i}(Dz)^{p^{i}} + a_{t−i+1}(D^{(p)}z)^{p^{i−1}} = b_{p^{i}}.

We then have b_{p^{i}} − a_{t−i+1} · (\frac{b_{p}}{a_{t}} )^{p^{i−1}} = [a_{t−i} − a_{t−i+1} · (\frac{a_{t−1}}{a_{t}} )^{p^{i−1}}] · (Dz)^{p^{i}}.

Lemma 5.8 now follows from the equality above. ■

Lemma 5.9 In Equation (23) we must have:

a) b_{p} = 0.

b) (a_{t}^{p+1}/a_{1}^{pq}) + a_{t−1} = 0. Proof. From Equation (23) we see that

D^{(j)}z = 0, for each 2 ≤ j < p.

Equation (13) has then the form

z^{q^{2}} − z = (x^{q^{2}} − x) · (Dz) + (x^{pq^{2}} − x^{p}) · (D^{(p)}z),  (25)

where Dz = \frac{x^{q}}{a_{t}} and D^{(p)}z = \frac{b_{p}}{a_{t}} − \frac{a_{t−1}}{a_{t}} · (\frac{x^{q}}{a_{t}})^{p}.

From Eq. (23) we have z^{q^{2}} = (\frac{1}{a_{1}} )^{pq} · (x^{q+1} + ∑_{i=1}^{t−1} b_{p^{i}} · x^{p^{i}} − ∑_{i=2}^{t} a_{i} · z^{p^{t−i}})^{pq} . The highest pole-order at P_{0} in the right hand side of the equation above is equal to (q^{3} + q^{2}) and comes from (x^{q+1})^{pq}/a_{1}^{pq}. The same order of pole on the right hand side of Equation (25) comes from the term −(a_{t−1}/a_{t}) · (x^{q}/a_{t})^{p} · x^{pq^{2}} = −(a_{t−1}/a_{t}^{p+1}) · (x^{q+1})^{pq}. This proves item b) in Lemma 5.9. Examining again Equation (25), we see that the second highest pole-order at P_{0} is equal to q^{3} and it comes from a unique term, namely it comes from the term (b_{p}/a_{t}) · x^{pq^{2}} on the right hand side of Equation (25). This proves item a) of Lemma 5.9. ■

From Lemmas 5.8 and 5.9, we have that Equation (23) now reads as below

∑_{i=1}^{t} a_{i} · z^{p^{t−i}} = x^{q+1},  (26)

where the coefficients a_{1}, a_{2}, ⋯, a_{t} satisfy:

• (a_{t}/a_{1}^{q})^{p} + a_{t−1}/a_{t} = 0.

• a_{t−i} = a_{t−i+1} · (a_{t−1}/a_{t})^{p^{i−1}} for i = 2, 3, ⋯, t − 1.

Lemma 5.10 With the curve X given by Equation (26) we have:

a) a_{i} = a_{t−1} · (a_{i+1}/a_{t})^{p} for each 1 ≤ i < t − 1.

b) a_{t}^{q} + (a_{t−1}/a_{t}^{p}) · a_{1}^{p} = 0.

c) a_{t−1}/a_{t}^{p} belongs to 𝔽_{q}.

Proof. We have z^{q^{2}} − z = (x^{q^{2}} − x) · (Dz) + (x^{pq^{2}} − x^{p}) · (D^{(p)}z),

where Dz = x^{q}/a_{t} and D^{(p)}z = \frac{b_{p}}{a_{t}} − \frac{a_{t−1}}{a_{t}} · (\frac{x^{q}}{a_{t}} )^{p} .

Hence

z^{q^{2}} − z = \frac{1}{a_{t}} ((x^{q+1})^{q} − x^{q+1}) − \frac{a_{t−1}}{a_{t}} ((x^{q+1})^{pq} − (x^{q+1})^{p}).  (27)

In the equality above we substitute x^{q+1} = ∑_{i=1}^{t} a_{i} · z^{p^{t−i}}, obtaining in this way that the rigth side of Equation (27) is a polynomial in the variable z. Lemma 5.10 now follows from Equation (27) by comparison of coefficients of the polynomials in the variable z. ■

We can now state our main result:

Theorem 5.11 Let q = p^{t} with t ≥ 2 and let m = q/p. There exists a unique maximal curve X over 𝔽_{q^{2}} with genus g = (m−1)q/2 having a point such that m is a nongap at this point. This curve X can be given by the following plane equation:

∑_{i=1}^{t} z^{p^{t−i}} = c · x^{q+1}, with c^{q} + c = 0 and c ≠ 0.  (28)

Proof. Let α := a_{t−1}/a_{t}^{p}. We know from Lemma 5.10 that:

a_{t−i} = α^{(p^{i}−1)/(p−1)} · a_{t}^{p^{i}}, for each 1 ≤ i ≤ t − 1.  (*)

Using the substitution z ↦ z_{3} = a_{t} · z we get from Equation (26):

∑_{i=1}^{t} ã_{i} · z_{3}^{p^{t−i}} = x^{q+1}, with ã_{i} = α^{N_{i}} and N_{i} = \frac{p^{t−i} − 1}{p − 1}.  (29)

We will need the following lemma:

Lemma 5.12 The element α is a (p − 1)-power in the finite field 𝔽_{q^{2}}. More precisely, we have

α^{(q−1)/(p−1)} = −1.

Proof. In fact it follows from Equation (∗) above with i = t − 1 that

α^{(q/p−1)/(p−1)} = \frac{a_{1}}{a_{t}^{q/p}} .

Item b) in Lemma 5.10 gives:

(\frac{a_{1}}{a_{t}^{q/p}} )^{p} · α = −1.

Lemma 5.12 follows from the two preceeding equalities. ■

Returning to the proof of Theorem 5.11, let now c ∈ 𝔽_{q^{2}}^{∗} be such that

c^{(p−1)} = α = \frac{a_{t−1}}{a_{t}^{p}} ∈ 𝔽_{q}.

We can then rewrite Equation (29) as

∑_{i=1}^{t}(cz_{3})^{p^{t−i}} = c · x^{q+1},

and moreover c^{q} + c = 0 as follows from Lemma 5.12.

Substituting z = cz_{3} we finish the proof of Theorem 5.11 (see also Remark 5.1 here). ■

Remark 5.13 One can show that if the equation ∑_{i=1}^{t} z^{p^{t−i}} = c · x^{q+1}, with c ∈ 𝔽_{q^{2}}^{∗}, defines a maximal curve over 𝔽_{q^{2}} with q = p^{t}, then we must have that c^{q} + c = 0.

Remark 5.14 It follows from Theorem 5.11 that the unique maximal curve X over 𝔽_{q^{2}} satisfying the hypothesis in this theorem is Galois covered by the Hermitian curve over 𝔽_{q^{2}} (the covering degree being equal to the characteristic p). In fact the function z = cy − c^{p}y^{p}, with c^{q} + c = 0, inside the function field of the Hermitian curve over 𝔽_{q^{2}} with defining equation as below

y^{q} + y = x^{q+1},

is such that the functions x and z above satisfy Equation (28) in Theorem 5.11. Compare with Theorem 2.1 of [5].

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