Weierstrass semigroup and automorphism group of the curves Xn,r

Finite Fields and Their Applications 36 (2015) 121–132

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Finite Fields and Their Applications www.elsevier.com/locate/ffa

Weierstrass semigroup and automorphism group of the curves Xn,r H. Borges a , A. Sepúlveda b , G. Tizziotti b,∗ a

Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, São Paulo, 13566-590, Brazil b Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, Minas Gerais, 38408-100, Brazil

a r t i c l e

i n f o

Article history: Received 26 November 2014 Received in revised form 26 March 2015 Accepted 26 July 2015 Available online xxxx Communicated by Michael Tsfasman

a b s t r a c t In this paper, we determine the Weierstrass semigroup H(P∞ ) and the full automorphism group of a certain family of curves Xn,r , which was recently introduced by Borges and Conceição. © 2015 Elsevier Inc. All rights reserved.

MSC: 11G20 14H55 14H37 Keywords: Algebraic curves Weierstrass semigroup Automorphism group

* Corresponding author. E-mail addresses: [email protected] (H. Borges), [email protected] (A. Sepúlveda), [email protected] (G. Tizziotti). http://dx.doi.org/10.1016/j.ffa.2015.07.004 1071-5797/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction Since the algebraic geometric (AG) codes were introduced by V.D. Goppa [5,6], the study of several topics in the theory of algebraic curves has been intensified. A particular case is the investigation of Weierstrass semigroups H(P ) at points P of an algebraic curve. It is well known that the structure of H(P ) provides both local and global information regarding this curve. The knowledge of such structure can prove quite useful, for instance, in improving the lower bound on the minimum distance of AG codes arising from the curve (see [2,4,7,16]). The group of automorphisms of a curve is another topic that plays a prominent role in several aspects of the theory (see [10,11,17]). To be concise, it should be noted that many curves have a characterization based on the structure and the action of their automorphism groups. It is worth mentioning that AG codes can also be studied in connection with these groups (see [18,19]). Applications of the automorphism groups in the systematics of encoding and decoding can be found in [12] and [15], respectively. In [1], Borges and Conceição used minimal value set polynomials to introduce a large family of curves that generalize the Hermitian curve. An example is the following curve over Fqn : H:

Tn (y) = Tn (xq

r

+1

n

(mod xq − x)

)

where, for a symbol z, Tn (z) := z q

n−1

+ zq

n−2

+ ... + z,

and r = r(n) ≥ n/2 is the smallest positive integer such that gcd(n, r) = 1, i.e.,

r=

⎧ 1, ⎪ ⎪ ⎨ n/2 + 1, n/2 + 2, ⎪ ⎪ ⎩ (n + 1)/2,

if if if if

n=2 n ≡ 0 (mod 4) n ≥ 6, n ≡ 2 (mod 4) n is odd .

(1)

Borges and Conceição have shown that the curve H has a somewhat large ratio #H(Fqn )/g(H), where H(Fqn ) is the set of Fqn -rationals points, and g(H) is the genus of H. They also computed the Weierstrass semigroup H(P∞ ), where P∞ = (0 : 1 : 0) is the only singular point on the projective closure of H. In particular, it was proved that H is an example of a so-called Castle curve, which is a special type of curves potentially suitable for construction of AG codes with good parameters (see [14]). In this paper, we further investigate the family of curves introduced in [1]. More precisely, within this family, we consider a set of curves Xn,r (which includes the curve H)

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123

and determine the Weierstrass semigroup H(P∞ ) as well as the curves’ full group of automorphisms. In particular, we arrive at a larger set of Castle curves. This paper is organized as follows. In Section 2, some basics concepts related to Weierstrass semigroups and automorphism groups of curves are recalled, and some properties of the curves Xn,r are presented. In Section 3, we determine the Weierstrass semigroup H(P∞ ) and prove that Xn,r are Castle curves, while providing an alternative, somewhat simpler, plane model for the curves Xn,r . Finally, in Section 4, the full automorphism group of Xn,r is obtained. 2. Preliminaries Hereafter we use the following notation: • Fq is the finite field with q elements. • X is a projective, non-singular, geometrically irreducible algebraic curve of genus g ≥ 2 defined over Fq . • Fq (X ) denotes the field of Fq -rational functions on X . • For a point P ∈ X , the discrete valuation at P is denoted by vP . • For f ∈ Fq (X )\{0}, the pole divisor of f is denoted by div ∞ (f ). • #X (Fq ) denotes the number of Fq -rational points of X . • N0 = {0, 1, 2, . . .} denotes the semigroup of non-negative integers. 2.1. Weierstrass semigroup Here we present the relevant data on Weierstrass semigroups that will be used in Section 3. For a more detailed discussion on the basics of this topic, see [13, Chapter 6]. For a rational point P ∈ X , the Weierstrass semigroup of X at P is defined by H(P ) := {n ∈ N0 : ∃f ∈ Fq (X ) with div ∞ (f ) = nP }, and the set G(P ) = N0 \ H(P ) is called Weierstrass gap set of P . From the Weierstrass Gap Theorem, G(P ) = {α1 , . . . , αg } and 1 = α1 < · · · < αg ≤ 2g − 1. The semigroup H(P ) is called symmetric if αg = 2g − 1. The curve X is called Castle curve if H(P ) = {0 = m1 < m2 < · · ·} is symmetric and #X (Fq ) = m2 q + 1. For more details about Castle curves and their applications in coding theory, see [14]. Definition 2.1. Let (a1 , . . . , am ) be a sequence of positive integers whose greatest common divisor is 1. Set d0 = 0, and define di := gcd(a1 , . . . , ai ) and Ai := { ad1i , . . . , adii } for i = 1, . . . , m. If adii lies in the semigroup generated by Ai for i = 2, . . . , m, then the

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sequence (a1 , . . . , am ) is called telescopic. A semigroup is called telescopic if it is generated by a telescopic sequence. For a semigroup S, the number of gaps and the largest gap of S will be denoted by g(S) and lg (S), respectively. The following result will be a significant factor in determining the semigroup H(P∞ ) of the curves Xn,r . Lemma 2.2. (See [7], Lemma 6.5.) If Sm is the semigroup generated by a telescopic sequence (a1 , . . . , am ), then m  di−1 ( − 1)ai di i=1 • g(Sm ) = dm−1 g(Sm−1 ) + (dm−1 − 1)(am − 1)/2 = (lg (Sm ) + 1)/2,

• lg (Sm ) = dm−1 lg (Sm−1 ) + (dm−1 − 1)am =

where d0 = 0. In particular, telescopic semigroups are symmetric. 2.2. Automorphism group This subsection briefly recalls few facts related to automorphism groups of algebraic curves over finite fields. For a more detailed discussion on this subject, see [8,9], [13, Chapter 11]. Let Aut(X ) be the automorphism group of X and G ⊆ Aut(X ) be a finite subgroup. For a rational point P ∈ X , the stabilizer of P in G, denoted by GP , is the subgroup of G consisting of all elements fixing P . For a non-negative integer i, the i-th ramification (i) group of X at P is denoted by GP and defined by (i)

GP = {α ∈ GP : vP (α(t) − t) ≥ i + 1} , (0)

(1)

where t is a local parameter at P . Here GP = GP and GP is the unique Sylow (1) p-subgroup of GP . Moreover, GP has a cyclic complement H in GP , i.e., (1)

G P = GP  H

(2)

where H is a cyclic group of order coprime to p. The following result will be essential to determine the full group Aut(Xn,r ) of automorphisms of the curves Xn,r . Theorem 2.3. (See [13], Theorem 11.140.) Let X be a curve of genus g ≥ 2 over Fq , where q is a prime power, and let G be an automorphism group of X such that X has an (1) Fq -rational point P satisfying the condition |GP | > 2g + 1. Then one of the following cases occurs:

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125

1) G = GP . 2) X is birationally equivalent to one of the following curves: (i) the Hermitian curve v(Y n + Y − X n+1 ) with n = q t ≥ 2 and g = 12 (n2 − n) (ii) the DLS curve (the Deligne–Lusztig curve arising from the Suzuki group) v(X n0 (X n + X) − (Y n + Y )) with p = 2, q = n, n0 = 2r , r ≥ 1, n = 2n20 and g = n0 (n − 1) 2 (iii) the DLR curve (the Deligne–Lusztig curve arising from the Ree group) v(Y n − (1 + (X n − X)n−1 )Y n + (X n − X)n−1 Y − X n (X n − X)n+3n0 with p = 3, q = n, n0 = 3r , n = 3n20 and g = 32 n0 (n − 1)(n + n0 + 1). Where v(F (X, Y )) is the plane projective curve with affine equation F (X, Y ) = 0. 2.3. The curves Xn,r In this subsection, we define the principal object of this study, namely the curves Xn,r , and recall some of their basic properties. For additional details, see [1]. Fix integers n ≥ 2 and r ∈ { n2 , . . . , n − 1}, with gcd(n, r) = 1. Consider the polynomial   r n fr (x) := Tn x1+q (3) mod (xq − x), where Tn (x) = x + xq + . . . + xq

n−1

, and define the curve Xn,r by affine equation Tn (y) = fr (x).

(4)

It is easy to check that the polynomial fr satisfies fr (a) ∈ Fq for all a ∈ Fqn , and that if n > 2, then fr (x) can written as fr (x) =

n−r−1 

r

i

(x1+q )q +

i=0

r−1 

(x1+q

n−r

i

)q .

(5)

i=0

Proposition 2.4. (See [1], Proposition 3.1.) The following hold: 1) The curve Xn,r has degree d = q n−1 + q r−1 , genus g = q r (q n−1 − 1)/2 and N = q 2n−1 + 1 Fqn -rational points. 2) In the projective closure of Xn,r , the point P∞ = (0 : 1 : 0) is the only singular point. 3. The Weierstrass semigroup H(P∞ ) In this section, we consider the point P∞ = (0 : 1 : 0) ∈ Xn,r and determine its Weierstrass semigroup H(P∞ ). Let Fn,r = Fqn (x, y) be the function field of the curve Xn,r over Fqn , and let vP∞ be the discrete valuation at P∞ .

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126

Remark 3.1. 1) By equation (4), the functions x, y ∈ Fn,r satisfy y q − y = (xq − x)fr (x) = (xq − x)(xq + xq n

n

n

r

n−r

).

(6)

2) The notation z0 := y q

n−r

− xq

n−r

+1

∈ Fn,r

will be used in what follows. Lemma 3.2. For the functions x, y, z ∈ Fn,r , where z := z0q

2r−n

− xq

r

+1

+ xq

2r−n

−1

y,

we have that 1) div ∞ (x) = q n−1 P∞ 2) div ∞ (y) = (q n−1 + q r−1 )P∞ 3) div ∞ (z) = (q 2r−1 + q n−r−1 )P∞ . Proof. The proof of assertions 1 and 2 follows similarly to [1, Lemma 3.6]. To compute n vP∞ (z), we use the function z q . Making use of equation (6), we have n

z q = z0q = yq

2r

− xq

n+r

n+r

− xq

+q n

n+r

n

+q

+ xq 2r

r

2r

−q n

q n +q r

= (x

−q n

− xq

= ((xq − x)(xq + xq + xq

2r

n−r

n+r

n

+q

n

+ xq

) + y)q − xq

n

+x

2r

r

r

((xq − x)(xq + xq q n +q n−r

r

((xq − x)(xq + xq

−x

q r +1

n−r

−x

−q

n

n+r

n−r

) + y)

n

r

((xq − x)(xq + xq

+q

2r

− xq

n+r

+q

n−r

) + y)

n

) + y)

q n−r +1

r

+ y)q − xq

n+r

+q 2r

− xq

n+r

+q n

+ xq −q (xq +q + xq +q − xq +1 − xq +1 + y)   n+r  n+r n 2r  r n r r n+r  n+r n    +q 2r +q 2r xq + xq +q −  xq+q − xq +q + y q −  xq − xq +q = 2r  r 2r n−r 2r n r 2r n n−r 2r n  + xq+q + xq +q − xq −q +q +1 − xq −q +q +1 + xq −q y. 2r

n

n

r

n

Thus z q = xq +q − xq −q And triangle inequality gives us n

2r

n−r

2r

n

n

n−r

+q r +1

vP∞ (z q ) = vP∞ (xq

2r

r

− xq

+q n−r

2r

n−r

−q n +q n−r +1

+ xq

2r

−q n

) = −q n−1 (q 2r + q n−r ).

Therefore, div ∞ (z) = (q 2r−1 + q n−r−1 )P∞ . 2

y − xq

n

+q r

r

+ yq .

H. Borges et al. / Finite Fields and Their Applications 36 (2015) 121–132

127

Remark 3.3. Note that, if gcd(n, r) = 1, then there exist positive integers α and β such that (n − r)α − βn = 1. Proposition 3.4. Let α and β be positive integers such that (n − r)α − βn = 1, and consider the following functions in Fn,r :

w :=

α−1 

z0 q

(n−r)i

−

β 

i=0

(xq

n−r

+1

+ xq

n

+q n−r

q n(β−i)+1

)

(7)

i=1

and t := xq

2r−n+1

−q

w + z q + xq

2r−n+1

−q 2r−n −q+1

z.

(8)

Then div ∞ (w) = (q n + q n−r )P∞ , and div ∞ (t) = (q 2r − q n + q r + 1)P∞ . n

Proof. We begin by computing wq − w. Note that β 

(x

q n−r +1

+x

q n +q n−r

)

q n(β−i)+1 q

n

−

i=1

β 

(xq

n−r

+1

+ xq

n

+q n−r

q n(β−i)+1

)

i=1

= (xq

n−r

+1

+ xq

n

+q

n−r

)q

nβ+1

− (xq

n−r

+1

+ xq

n

+q n−r q

) .

Since α−1 

n

(z0q − z0 )q

(n−r)i

= (xq

n−r

− xq

+1

n

+q n−r

) − (xq

n−r

− xq

+1

n

+q n−r q nβ+1

)

,

i=0

it follows that n

w q = w + xq

n−r

+1

− xq

n

+q n−r

+ xq

n−r+1

+q

n

− xq

n+1

n+1

+q n−r+1

.

(9)

n−r+1

Therefore, by triangle inequality, vP∞ (wq ) = vP∞ (xq +q ). Thus q n vP∞ (w) = n+1 n−r+1 n−1 n n−r −(q +q )q , and then div ∞ (w) = (q + q )P∞ . Now to find div ∞ (t), we see that t q = xq

2r+1

−q n+1

wq + z q

= xq

2r+1

−q n+1

(w + xq

n

+ (x

n

q 2r +q n−r

+ xq

2r+1

− xq

n

+q

−x

n+1

n−r

r

+ yq )

+1

2r+1

− xq

q 2r −q n +q r +1

−q 2r −q n+1 +q n

r

+ xq

(xq

2r

n

−q 2r −q n+1 +q n q n zr

+q n−r

−x

+q n−r

+ xq

n−r+1

+q

q 2r −q n +q n−r +1

− xq

2r

− xq

+x

−q n +q r +1

n+1

+q n−r+1

q 2r −q n

− xq

2r

y−x

)

q n +q r

−q n +q n−r +1

r

+ y q )q

+ xq

2r

−q n

y

H. Borges et al. / Finite Fields and Their Applications 36 (2015) 121–132

128

((( ((( 2r+1 2r+1 (( (( +q n−r +1 +q n +q n−r (n+1 (n+1 w +( xq((−q −( xq((−q ( (( (( 2r+1 n−r+1 2r+1 ( 2r+1 ( n−r+1 2r+1 (( +q( +q ( +q n−r+1 −q n+1 +q r+1 +q (( (n+1 +( xq((−q −xq (( +( xq((+q − xq ( ( (( 2r+1 n−r+1 2r+1 n+1 r+1 r+1 +q( +q −q n+1 q (( (n+1 −( xq((−q + xq y − xq +q + yq ( ((( 2r+1 n ( 2r+1 2r+1 (( (( +q( +q n−r −q n+1 +q r +1 +q n−r +1 (n+1 (n+1 +( xq((−q − xq −( xq((−q

= xq

2r+1

+ xq

−q n+1

2r+1

−q n+1

y − xq

2r+1

−q 2r −q n+1 +q n +q n +q r

+ xq

2r+1

−q 2r −q n+1 +q n q r

y .

Then, n

t q = xq

2r+1

−q n+1

w − xq

2r+1

−q n+1 +q r+1 +q

− xq

2r+1

−q n+1 +q r +1

+ xq

2r+1

−q 2r −q n+1 +q n q r

+ xq

2r+1

−q n+1

+ xq

2r+1

y − xq

−q n+1 q

2r+1

y − xq

n+1

+q r+1

+ yq

r+1

−q 2r −q n+1 +q n +q n +q r

y . n

Using triangle inequality, we have vP∞ (tq ) = vP∞ (xq follows that div ∞ (t) = (q 2r − q n + q r + 1)P∞ . 2

2r+1

−q n+1 +q r+1 +q

) from which it

Proposition 3.5. The curve Xn,r has a plane model given by yq

n−1

+ · · · + y q + y = xq

n−r

+1

− xq

n

+q n−r

.

(10)

Proof. Consider the polynomial in Fqn (x)[Z] given by G(Z) = Z q − Z − xq

n−r

+1

+ xq

n

+q n−r

− xq

n−r+1

+q

+ xq

n+1

+q n−r+1

.

It follows from (9) that G(Tn (w)) = 0. Therefore, since G(Z) may be written as

(Z −

λ∈Fq

(xq

n−r

+1

− xq

n

+q n−r

) − λ), we have that Tn (w) = xq

n−r

+1

− xq

n

+q n−r

+λ

for some λ ∈ Fq . Letting a ∈ Fqn be such that Tn (a) = λ and considering the function w ˜ = w − a, we obtain Tn (w) ˜ = xq

n−r

+1

− xq

n

+q n−r

.

Since [3, Theorem 2.5] implies that (10) is irreducible, the result follows. 2 Theorem 3.6. Let H(P∞ ) be the Weierstrass semigroup at P∞ . Then H(P∞ ) = q n−1 , q n−1 + q r−1 , q n + q n−r , q 2r−1 + q n−r−1 , q 2r − q n + q r + 1. Moreover, H(P∞ ) is a telescopic semigroup and, in particular, symmetric.

H. Borges et al. / Finite Fields and Their Applications 36 (2015) 121–132

129

Proof. Let · · · · · ·

a1 = q n−1 ; a2 = q n−1 + q r−1 ; a3 = q n + q n−r ; a4 = q 2r−1 + q n−r−1 ; a5 = q 2r − q n + q r + 1; and S = a1 , a2 , a3 , a4 , a5 .

By Lemma 3.2 and Proposition 3.4, it follows that S ⊆ H(P∞ ). To prove the equality, we first show that S is telescopic. Using the notations of Definition 2.1, we have · · · · ·

d1 d2 d3 d4 d5

= q n−1 and S1 = 1; = q r−1 and S2 = q n−r , q n−r + 1; = q n−r and S3 = q r−1 , q r−1 + q 2r−n−1 , q r + 1; = q n−r−1 and S4 = q r , q r + q 2r−n , q r+1 + q, q 3r−n + 1; = 1 and S5 = S.

Note that a2 d2 a3 · d3 a4 · d4 a5 · d5 ·

= q n−r + 1 ∈ S1 ; = q r + 1 = (q 2r−n − 1)q n−r + (q n−r + 1) ∈ S2 ; = q 3r−n + 1 = (q 2r−n+1 − q)q r−1 + (q r + 1) ∈ S3 ; = q 2r − q n + q r + 1 = (q r − q n−r − q 2r−n + 1)q r + (q 3r−n + 1) ∈ S4 .

Therefore, S is telescopic and, by Lemma 2.2, symmetric. To complete the proof, we must show that g(S) is equal to g = q r (q n−1 − 1)/2. Using the formula for the genus of a telescopic semigroup given in Lemma 2.2,

5   di−1 g(S5 ) = (lg (S5 ) + 1)/2 = ( − 1)ai + 1 , di i=1 and then g(S) = q r (q n−1 − 1)/2, which completes the proof. 2 Corollary 3.7. The curves Xn,r are Castle curves. 4. Automorphism group of Xn,r Let us consider the q 2n−1 affine points P := (δ, μ) ∈ Xn,r (Fqn ) and all the elements γ ∈ Fqn such that

130

H. Borges et al. / Finite Fields and Their Applications 36 (2015) 121–132



γ q−1 = 1 2 γ q −1 = 1

if n is odd if n is even .

n−r−1 1+qr qi Using that X is given by Tn (y) = fr (x), where fr (x) = (x ) + i=0

r−1 1+qn−r qin,r (x ) , one can easily check that the set G of maps on F , given by n,r i=0 αγ,P : (x, y) −→ (γx + δ, γ 1+q y + (δ q

n−r

r

+ δ q )γx + μ),

(11)

is a subgroup of Aut(Xn,r ), whose elements fix P∞ , i.e., G ⊆ AutP∞ (Xn,r ). Based on the above definition, note that the following subgroups of G: N = {αγ,P ∈ G : γ = 1} and H = {αγ,P ∈ G : P = (0, 0)} ∼ = F∗q2−(n

mod 2)

have order q 2n−1 and q 2−(n mod 2) − 1, respectively. The aim of this section is to prove the following: Theorem 4.1. The group G is the full group of automorphisms of Xn,r . Moreover, N = AutP∞ (Xn,r )(1) and G = AutP∞ (Xn,r ) = N  H. The proof of Theorem 4.1 will follow after some partial results. We begin with the following: Lemma 4.2. AutP∞ (Xn,r ) = G. Proof. As mentioned before, the inclusion G ⊆ AutP∞ (Xn,r ) can be easily verified. Now let α ∈ AutP∞ (Xn,r ) be an automorphism. Note that Lemma 3.2 implies that {1, x} and {1, x, y} are bases for the Riemann–Roch spaces L(q n−1 P∞ ) and L((q n−1 + q r−1 )P∞ ), respectively. Therefore, α(x) ∈ L(q n−1 P∞ ) and α(y) ∈ L((q n−1 + q r−1 )P∞ ), and α(x) = cx + d and α(y) = ay + b1 x + b0 for some a, b0 , b1 , c, d ∈ Fqn with ac = 0. Since α is an automorphism, we must have F (cx + d, ay + b1 x + b0 ) = λF (x, y) for some λ ∈ Fqn and F (x, y) := Tn (y) − fr (x). In particular, • Tn (ay) = λTn (y) • fr (cx + d) − Tn (b1 x + b0 ) = λfr (x).

H. Borges et al. / Finite Fields and Their Applications 36 (2015) 121–132

131

Note that the first equality gives a = λ ∈ Fq \{0}. Moreover, using that fr (x) =

n−r−1 

r

i

(x1+q )q +

i=0

r−1  n−r i (x1+q )q i=0

and comparing coefficients on the second equality yields fr (d) = Tn (b0 ) and r

n−r

(i) cq +1 = λ = cq +1 r n−r (ii) (dq + dq )c = b1 n−r 2n−2r n−r (iii) (d + dq )cq = b1 q . n−r

n

Since λ ∈ Fq \{0}, condition (i) gives cq = λc , and then cq = cλqr = c. That r n−r 2r−n −1 is, c ∈ Fqn . Furthermore, since cq +1 = cq +1 implies cq = 1, we have that gcd(q n −1,q 2r−n −1) c = 1. However, it follows from gcd(n, r) = 1 that gcd(q − 1, q n

2r−n

− 1) =

q−1 q2 − 1

if n is odd if n is even ,

(12)

which proves the assertion about c. In particular, (i) gives a = λ = cq+1 . Using conditions (ii) and (iii), one can easily see that d ∈ Fqn as well. Therefore fr (d) ∈ Fq , and then fr (d) = Tn (b0 ) implies that (d, b0 ) is an Fqn -rational point on Xn,r . This finishes the proof. 2 Lemma 4.3. AutP∞ (Xn,r )(1) = N and AutP∞ (Xn,r ) = N  H. Proof. Since |N | = q 2n−1 and |G| = q 2n−1 (q 2−(n mod 2) − 1), it follows from Subsection 2.2 that N must be the unique Sylow p-subgroup of G = AutP∞ (Xn,r ), i.e., (1) N = AutP∞ (Xn,r ). Also, from (2), it follows that the cyclic group H ∼ = G/N is a complement in G, i.e., G = N  H. 2 Proof of Theorem 4.1. In view of the results in Lemmas 4.2 and 4.3, it suffices to prove r n−1 that Aut(Xn,r ) = AutP∞ (Xn,r ). In fact, first recall that Xn,r has genus g = q (q 2 −1) . Thus, for n > 2, the curve Xn,r cannot be birationally equivalent to any of the curves listed in Theorem 2.3. Therefore, since |AutP∞ (Xn,r )(1) | = q 2n−1 > 2g + 1 = q r (q n−1 − 1) + 1, if follows from Theorem 2.3 that Aut(Xn,r ) = AutP∞ (Xn,r ), as desired. 2 Acknowledgments The authors would like to thank the referees for suggestions that improved the presentation of this work. The author A. Sepúlveda would like to thank FAPEMIG,

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