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A GENERALIZATION OF BRING’S CURVE IN ANY CHARACTERISTIC

GABOR KORCHMAROS, STEFANO LIA, AND MARCO TIMPANELLA

Abstract. Let p ≥ 7 be a prime, and m ≥ 5 an integer. A natural generalization of Bring’s curve valid overany field K of zero characteristic or positive characteristic p, is the algebraic variety V of PG(m−1,K) which isthe complete intersection of the projective algebraic hypersurfaces of homogeneous equations xk

1+· · ·+xk

m =0 with 1 ≤ k ≤ m− 2. In positive characteristic, we also assume m ≤ p− 1. Up to a change of coordinatesin PG(m− 1,K), we show that V is a projective, absolutely irreducible, non-singular curve of PG(m− 2,K)

with degree (m − 2)!, genus g = 1

4((m − 2)(m − 3) − 4)(m − 2)! + 1, and tame automorphism group G

isomorphic to Symm. We compute the genera of the quotient curves of V with respect to the stabilizers ofone or more coordinates under the action of G. In positive characteristic, the two extremal cases, m = 5 andm = p− 1 are investigated further. For m = 5, we show that there exist infinitely many primes p such thatV is Fp2 -maximal curve of genus 4. The smallest such primes are 29, 59, 149, 239, 839. For m = p − 1 we

prove that V has as many as (p− 2)! points over Fp and has no further points over Fp2 . We also point out a

connection with previous work of Redei about the famous Minkowski conjecture proven by Hajos (1941), aswell as with a more recent result of Rodrıguez Villegas, Voloch and Zagier (2001) on plane curves attainingthe Stohr-Voloch bound, and the regular sequence problem for systems of diagonal equations introduced byConca, Krattenthaler and Watanabe (2009).

Keywords: algebraic curves, function fields, positive characteristic, automorphism groups.

Subject classifications: 14H37, 14H05.

1. Introduction

Bring’s curve is well known from classical geometry as being the curve with the largest automorphismgroup among all genus 4 complex curves; see [3]. Its canonical representation in the complex 4-space is thecomplete intersection of three algebraic hypersurfaces of equation xk

1+· · ·+xk5 = 0 with 1 ≤ k ≤ 3. A natural

generalization of Bring’s curve valid over any field K of zero characteristic, or positive characteristic p ≥ 7,is the algebraic variety V of PG(m − 1,K) which is the complete intersection of the projective algebraichypersurfaces of homogeneous equations

(1)

X1 +X2 + . . .+Xm = 0;X2

1 +X22 + . . .+X2

m = 0;· · · · · ·· · · · · ·Xm−2

1 +Xm−22 + . . .+Xm−2

m = 0;

where m ≥ 5. From now on we assume m ≤ p − 1 when K has characteristic p. The first equation in (1)implies that V is contained in the hyperplane of homogeneous equation X1 +X2 + . . .+Xm = 0. Up to achange of coordinates in PG(m− 1,K), we show that V is a projective, absolutely irreducible, non-singularcurve of degree (m− 2)! embedded in PG(m− 2,K); see Theorem 4.16.

The symmetric group Symm has a natural action on the coordinates (X1 : . . . : Xm) of PG(m − 1,K).Therefore, the automorphism group Aut(V ) of V has a subgroup G isomorphic to Symm. It seems plausiblethat G = Aut(V ), and this is proven to be true if Aut(V ) is tame, and so in particular in zero characteristic;

1

http://arxiv.org/abs/2112.10886v2

2 G. KORCHMAROS, S. LIA, AND M. TIMPANELLA

see Theorem 5.6. The non-tame case, i.e. if V has an automorphism of order p fixing a point of V , remainsto be worked out. In Section 7 we investigate the quotient curves of V with respect to subgroups of G. Themost interesting cases for the choice of a subgroup are the stabilizers of one or more coordinates. Since twosuch subgroups are conjugate in G if they fix the same number of coordinates, it is enough to consider thestabilizer Gd+1,...,m of the coordinates Xd+1, . . . , Xm in Aut(V ) with 1 ≤ d ≤ m− 1. A careful analysis ofthe actions of these stabilizers on the points of V , carried out in Section 7, together with the Hurwitz genusformula, allows to compute the genus of the quotient curve Vd = V/Gd+1,...,m; see Proposition 7.3. Thequotient curve Vd turns out to be rational for d = m− 2, otherwise its genus g is quite large; see (16).

The projection of V from the m−d−1 dimensional projective subspace Σ of equationsX1 = 0, . . . , Xd = 0is also a useful tool for the study of V by virtue of the fact that Σ∩ V = ∅. Let Vd be the curve obtained byprojecting V from the vertex Σ to a projective subspace Σ′ of dimension d− 1 disjoint from Σ. Comparisonof Vd with the quotient curve Vd shows that they are actually isomorphic; see Proposition 7.1. Therefore,the function field extension K(V ) : K(Vd) is Galois, that is, the vertex Σ can be viewed as a Galois subspace(also called higher dimensional Galois-point) for V . This shows that there are many higher dimensionalGalois-subspaces for V , which is a somewhat rare phenomena. For results and open problems on Galois-points; see the recent papers [1, 10, 11, 13, 18]. In Section 7.1 the case d = m − 3 is investigated moreclosely. An explicit equation of Vm−3 is given in Theorem 7.4, which shows that if K is either an algebraicclosure of the rational field Q, or of the prime field Fp, then Vm−3 coincides with the plane curve of degreem− 2 investigated by Rodrıguez Villegas, Voloch and Zagier [31], who pointed out that this curve has manypoints; in particular, for m = p− 1, it is a non-singular plane curve attaining the Stohr-Voloch bound.

From an algebraic number theory point of view, (1) is a particular system of diagonal equations. Proposi-tion 6.1 shows that every solution of (1) also satisfies further diagonal equations. On the other hand, Lemmas5.2 and 5.3 imply that 1, 2, . . . ,m is associated to a regular sequence of symmetric polynomials, i.e. thesystem of diagonal equations of xk

1 + · · · + xkm = 0 with k ranging over 1, 2, . . . ,m has only the trivial

solution (0, 0, . . . , 0). This gives a different proof for Proposition 2.9 of the paper of Conca, Krattenthalerand Watanabe [5] where the authors relied on the interpretation of the q-analogue of the binomial coefficientas a Hilbert function. In the same paper [5], the general notion of a regular sequence associated to A ⊂ N∗

was introduced as any system of diagonal equations xk1 + · · · + xk

m = 0 with k ∈ A which has only thetrivial solution (0, 0, . . . , 0). [5, Lemma 2.8] shows (over the complex field) a simple necessary condition onA to be associated to a regular sequence, namely that m! divides the product of the integers in A. In thiscontext, the major issue to find easily expressible sufficient conditions turned out be surprisingly difficult.For the smallest case m = 3, the conjecture is that the above necessary condition is also sufficient, that is,A = a, b, c is associated to a regular sequence whenever abc ≡ 0 (mod 6). Evidences for this conjecturewere given in [5] and later in [4]. For further developments related to regular sequences, see [8, 9, 12, 21, 22].

In positive characteristic, an algebraic closure of the prime field Fp with p ≥ 7 is chosen for K and twoextremal cases are investigated further, namely m = 5 and m = p− 1. In the former case, V can be viewedas the characteristic p version of the Bring curve, and for infinitely many primes p we prove that V is anFp2 -maximal curve, that is the number of points of V defined over Fp2 attains the Hasse-Weil upper boundp2 + 1 + 2gp = p2 + 1 + 8p. We also show that the smallest such primes are 29, 59, 149, 239, 839. Our resultgives a new contribution to the study of Fp2-maximal curves of small genera, initiated by Serre in 1985 andcontinued until nowadays; see [2, 15, 16, 27]. For m = p − 1, V is never Fp2 -maximal instead. In fact, thenumber of points of V does not increase passing from Fp to Fp2 so that V has as many as (p − 2)! points

defined over Fp2 whereas its genus is equal to 14 (((p − 3)(p − 4) − 4)(p − 3)!) + 1; see Lemmas 3.2, 4.14,

4.15, and Theorem 9.3. A more general question for m = p− 1 is to determine the length of the set V (Fpi)consisting of all points of V in PG(p− 2,Fpi). For i = 1, Lemma 3.2 shows V (Fp) = (p− 2)!. Unfortunately,the elementary argument used in the proof of Lemma 3.2, seems far away from being sufficient to tacklethe general case. Even in the second particular case i = 2, settled in Theorem 9.3, the proof relies on theStohr-Voloch bound and on the structure of the G-orbits on the points of V determined in Section 5. On

A GENERALIZATION OF BRING’S CURVE IN ANY CHARACTERISTIC 3

the other hand, Theorem 9.6 yields |V (Fpd)| > |V (Fp)| where d stands for the smallest integer such that

(p − 2) | (pd − 1). The proof uses methods from Galois theory. It remains unsolved the problem whether|V (Fpi)| > |V (Fp)| holds for some intermediate value i.

An important question on the geometry of curves is to compute the possible intersection multiplicitiesI(P, V ∩ Π) between V and hyperplanes Π through a point P generally chosen in V , also called orders. Inthe classical case (zero characteristic), the orders are precisely the non negative integers smaller than thedimension of the space where the curve is embedded, but this may fail in positive characteristic. If this occursthen the curve is called non-classical. For plane curves, non-classicality means that all non-singular pointsof the curve are flexes. In our case, Lemma 9.8 shows the existence of a hyperplane π such that I(P, V ∩Π)is at least p. Since V is embedded in PG(p− 3, p), this implies that V is non-classical; see Theorem 9.9. InSection 9.3, the intersection multiplicities I(P, V ∩Π) for P ∈ V (Fp) are investigated. By Proposition 9.12,1, 2, 3 are such intersection multiplicities. This together with Lemma 9.8 yield that 1, 2, 3, p are orders of V .

Another concept of non-classicality, due to Stohr and Voloch [28], arose from their studies on the maximumnumber Npi = Npi(g, r, d) of points of an irreducible curve of a given genus g, embedded in PG(r,Fpi) asa curve of degree d can have. The Stohr-Voloch upper bound on Npi = Npi(g, r, d) is known to be a deepresult; in particular it may be used to give a proof for the Hasse-Weil upper bound. Also, the Stohr-Volochbound shows that the curves with large Npi = Npi(g, r, d) have the property that the osculating hyperplaneto the curve at a generically chosen point on it also passes through the Frobenius image of the point. Acurve with this purely geometric property can only exist in positive characteristic, and if this occurs thecurve is called Frobenius non-classical. Apart from the trivial case pi = 2, Frobenius non-classical curves arealso non-classical. Theorem 9.10 shows that V is Frobenius non-classical. However this does not hold truefor the quotient curves of V with respect to the stabilizer of the coordinates. For instance, for m = p− 1,Vm−3 has degree p− 3 < p and hence it is a classical (and Frobenius classical) plane curve.

Over a finite field, the number of solutions of a system of diagonal equations has been the subject of manypapers where the authors mainly rely on character sums and the distributions of their values. In the presentpaper we are not moving in this direction. We point out a connection between (1) over Fp and Redei’swork related with the famous Minkowski conjecture, originally proven by Hajos in 1941. Redei proved thatMinkowski conjecture holds if the following claim is true: if an elementary abelian group of order p2 isfactored as the product of two sets of length p, both containing the identity element, then at least one of thefactors is a subgroup; see [26]. Redei and later on Wang, Panico and Szabo [32] showed this claim is true ifeach solution [ξ1, . . . , ξp] over Fp of the system of diagonal equations

(2)

X1 +X2 + . . .+Xp = 0;X2

1 +X22 + . . .+X2

p = 0;· · · · · ·· · · · · ·

X(p−1)/21 +X

(p−1)/22 + . . .+X

(p−1)/2p = 0.

has either equal components or [ξ1, . . . , ξp] is a permutation of the elements of Fp. That such particularp-tuples are exactly the solutions over Fp of system (2) was first shown by Redei himself in [26]. Theabove quoted paper by Wang, Panico and Szabo also contains a proof. In Section 9.6, we give a geometricinterpretation of their results in terms of the variety V .

2. Basic definitions and notation

2.1. Automorphism groups. In this paper, p ≥ 7 stands for a prime and K for an algebraically closedfield of characteristic zero or p. Also, we let m ≥ 5, and we assume m ≤ p− 1 when K has characteristic p.Henceforth, Vm denotes the m-dimensional K-vector space with basis X1, X2, . . . , Xm, and PG(m− 1,K)the (m− 1)-dimensional projective space over K arising from Vm.


The group G = Symm, viewed as the symmetric group on X1, . . . , Xm, has a natural faithful repre-sentation in Vm, where g ∈ G takes the vector v =

∑mi=1 λiXi to the vector v′ =

∑mi=1 λig(Xi). With

v′ = g(v), g becomes a linear transformation of Vm and G is viewed as a subgroup of GL(m,K). Since nonon-trivial element of G preserves each 1-dimensional subspace of Vm, G acts faithfully on PG(m − 1,K)and G can also be regarded as a subgroup of the projective linear group PGL(m,K) of PG(m − 1,K). For1 ≤ i < j ≤ m, the transposition σ = (XiXj), as a linear transformation of Vm, is associated with the(0, 1)-matrix (gl,n)l,n whose 1 entries are gl,n with l = n and gij , gji. As a projectivity of PG(m − 1,K),σ is the involutory homology whose center is the point C = (0 : · · · : 1 : · · · : −1 : · · · : 0), where 1 and−1 are in i-th and j-th positions, respectively, and whose axis is the hyperplane Π of equation Xi = Xj .Therefore, the fixed points of σ are those of Π and C, whereas the fixed hyperplanes of σ are those throughC and Π. Since any two transpositions with no common fixed point commute, the center of either lies onthe axis of the other. The m-cycle σ = (XmXm−1 . . . X1), as a linear transformation of Vm, is associatedwith the (0, 1)-matrix (gl,n)l,n whose 1 entries are gl,l+1 for l = 1, . . . ,m− 1 and gm,1. As a projectivity, σfixes the point Pω = (ω : ω2 : · · · : ωm = 1), where ω is any element whose order divides m. Similarly, the(m−1)-cycle σ = (Xm−1Xm−2 · · ·X1), as a linear transformation of Vm, is associated with the (0, 1)-matrix(gl,n)l,n whose 1 entries are gl,l+1 for l = 1, . . . ,m− 2 and gm−1,1, gm,m. As a projectivity, σ fixes the pointPε = (ε : ε2 : · · · : εm−1 = 1 : 0) for any element ε of K whose order divides m− 1.

2.2. Projections. We also recall how the projection from a linear subspace Σ of dimension d to a disjointlinear subspace Σ′ is performed.

If C is an irreducible, non-singular algebraic curve embedded in PG(m− 1,K), which is disjoint from Σ,then the projection determines a regular mapping π : C → PG(m− 2− d,K). The geometric interpretationof π is straightforward. Take any (m − 2 − d)-dimensional subspace Σ′ of PG(m − 1,K) disjoint from Σ.Then through any point of P ∈ C there is a unique (d+ 1)-dimensional subspace ΛP containing Σ and thissubspace meets Σ′ in a unique point, the image π(P ) of P projected from Σ. Moreover, C is projected intoan irreducible, possibly singular, curve F embedded in Σ′ ∼= PG(m− 2− d,K).In the particular case where Σ and Σ′ are the subspaces defined as the intersections of the hyperplanes ofequations Xd+2, . . . , Xm = 0 and X1, . . . , Xd+1 = 0, respectively, then the projection from the vertex Σ toΣ′ maps a point P = (x1 : . . . : xm) ∈ PG(m− 1,K) \ Σ to the point P ′ = (0 : . . . : 0 : xd+2 : . . . : xm).Let H be the subgroup of PGL(m − 1,K) which preserves C and fixes each d + 1-dimensional subspacethrough Σ. The points of the quotient curve C = C/H can be viewed as H-orbits of the points of C.Therefore every point P ∈ C is identified by a unique H-orbit, say HP . Moreover, such an H-orbit HP

is contained in a unique (d + 1)-dimensional subspace through Σ which also meets C in a unique point.Therefore, notation ΛP for that subspace passing through the point P ∈ C is meaningful. Thus the sequenceP = HP → ΛP → π(P ) is well defined. Actually it is a surjective homomorphism from C to F . It is bijectiveif and only if |H | = |ΛP ∩ C| for all but finitely many points P ∈ C. In this case Σ is called a d-dimensionalouter Galois subspace, and for d = 0 an outer Galois-point; see [18]. If N is a subgroup of the normalizer ofH in the K-automorphism group of C then the quotient group N/H is a subgroup of the K-automorphismgroup of C. The genera g(C) and g(C) of the curves C and C are linked by the Hurwitz genus formula. Inparticular, if H is tame, that is, the characteristic of K is either zero, or equal p and in the latter case p isprime to the order ℓ = |H | of H then

(3) 2g(C)− 2 = |H |(2g(C)− 2) +m∑

i=1

(ℓ− ℓi)

where ℓ1, . . . , ℓr denotes the lengths of the short-orbits of H on C.The equations in (1) define a (possibly reducible and singular) projective variety V of PG(m − 1,K) so

that the points of V are the nontrivial solutions of (1) up to a non-zero scalar. Actually, V is contained inthe hyperplane of PG(m − 1,K) of equation X1 + X2 + . . .Xm = 0. Therefore, V is a projective variety


embedded in PG(m − 2,K). Moreover, G = Symm preserves V and no nontrivial element of G fixes Vpointwise. Therefore, G is a subgroup of the K-automorphism group of V .

2.3. Bezout’s theorem. The higher dimensional generalization of Bezout’s theorem about the number ofcommon points of m − 1 hypersurfaces H1, . . . ,Hm−1 of PG(m − 1,Fq) states that either that number isinfinite or does not exceed the product deg(H1) · . . . · deg(Hm−1). For a discussion on Bezout’s theorem andits generalization; see [30].

2.4. Background on algebraic curves. We report some background from [14, Chapter 7]; see also [28].Let Γ be a projective, absolutely irreducible, not necessarily non-singular curve, embedded in a projectivespace PG(r,K). For a non-singular model X of Γ, let K(X ) = K(Γ) denote the function field of X . Thereexists a bijection between the points of X , and the places of K(X ) and the branches of Γ. For any point P ∈ Γ(more precisely, for any branch of Γ centered at P ) the different intersection multiplicities of hyperplaneswith the curve at P are considered. There is only a finite number of these intersection multiplicities, thenumber being equal to r+1. There is a unique hyperplane, called osculating hyperplane with the maximumintersection multiplicity. The hyperplanes cut out on Γ a simple, fixed point free, not-necessarily completelinear series Σ of dimension r and degree n, where n is the degree of the curve Γ. An integer j is a (Σ, P )-order if there is a hyperplane H such that I(P,H ∩ Γ) = j. Notice that, if P is a singular point, then Pis intended as a branch of Γ centered at P . In the case that Σ is the canonical series, it follows from theRiemann–Roch theorem that j is a (Σ, P )-order if and only if j+1 is a Weierstrass gap. For any non-negativeinteger i, consider the set of all hyperplanes H of PG(r,K) for which the intersection number is at least i.Such hyperplanes correspond to the points of a subspace Πi in the dual space of PG(r,K). Then we havethe decreasing chain

PG(r,K) = Π0 ⊃ Π1 ⊃ Π2 ⊃ · · · .

An integer j is a (Σ, P )-order if and only if Πj is not equal to the subsequent space in the chain. In

this case Πj+1 has codimension 1 in Πj . Since deg Σ = n, we have that Πi is empty as soon as i > n.The number of (Σ, P )-orders is exactly r + 1; they are j0(P ), j1(P ), . . . , jr(P ) in increasing order, and(j0(P ), j1(P ), . . . , jr(P )) is the order-sequence of Γ at P .

Here j0(P ) = 0, and j1(P ) = 1 if and only if the branch P is linear (in particular when P is a non-singularpoint).

Consider the intersection Πi of hyperplanes H of PG(r,K), for which

I(P,H ∩ γ) ≥ ji+1.

Then the flag Π0 ⊂ Π1 ⊂ · · · ⊂ Πr−1 ⊂ PG(r,K) can be viewed as the algebraic analogue of the Frenetframe in differential geometry. Notice that Π0 is just P , the centre of the branch γ, and Π1 is the tangentline to the branch γ at P . Furthermore, Πr−1 is the osculating hyperplane at P .

The order-sequence is the same for all but finitely many points of Γ, each such exceptional point is calleda Σ-Weierstrass point of Γ. The order-sequence at a generally chosen point of Γ is the order sequence of Γand denoted by (ε0, ε1, . . . , εr). Here ji(P ) is at least εi for 0 ≤ i ≤ r at any point of Γ.

Now let Γ be defined over Fℓ and viewed as a curve over the algebraic closure K = Fℓ. Stohr and Voloch[28] (see also [14, Chapter 8]) introduced a divisor with support containing all points of P ∈ Γ for whichthe osculating hyperplane contains the Frobenius image Φ(P ) of P . Since every Fℓ-rational point has thisproperty an upper bound on the number of Fℓ-rational points is obtained, unless all (but a finitely many)osculating hyperplanes have that property. In such an exceptional case, the curve is called Frobenius non-classical. Curves with many rational points are often Frobenius non classical (and in particular, non-classicalfor q 6= 2). Actually, Stohr and Voloch were able to give a bound on the number of Fℓ-rational points for any(Frobenius classical, or non-classical) curve Γ. There exists a sequence of increasing non-negative integersν0, . . . , νr−1 with ν0 ≥ 0 such that det(W

ν0,...,νr−1

ζ (x0, . . . , xr)) 6= 0. In fact, if the νi are chosen minimally


in lexicographical order, then there exists an integer I with 0 < I ≤ r such that

νi =

εi for i < I,εi+1 for i ≥ I.

The Stohr-Voloch divisor of Γ is

S = div(Wν0,...,νr−1

ζ (x0, . . . , xr)) + (ν1 + · · ·+ νr−1) div(dζ) + (q + r)E,

where E and eP are defined as before for the ramification divisor. The sequence (ν0, ν1, . . . , νr−1) is theFrobenius order-sequence, and Γ is Frobenius-classical if νi = i for 0 ≤ i ≤ r. Also, degS = (ν1 + · · · +νr−1)(2g− 2) + (q + r)n, where n = deg(Γ). From this the Stohr-Voloch bound follows:

|Γ(Fℓ)| ≤1

r

(

(ν1 + · · ·+ νr−1)(2g− 2) + (ℓ+ r)n)

,

which is a deep result on the number of Fℓ-rational points of Fℓ-rational curves. For instance, the Hasse-Weilupper bound was re-proven in [28]; see also [14, Chapter 9].

3. Some particular solutions of system (1) of diagonal equations

Lemma 3.1. Let ε ∈ K be a (m− 1)-th primitive root of unity. Then (ε, ε2, . . . , εm−1 = 1, 0) is a solutionof system (1).

Proof. Take an integer i such that 1 ≤ i ≤ m− 2, and let θ = εi. Then θ is a (m− 1)-th root of unity (nonnecessarily primitive). Furthermore, θ 6= 1 as ε is a primitive (m− 1)-th root of unity. The i-th equation in

(1) is satisfied by (ε, ε2, . . . , εm−1, 0) if and only if∑m−1

k=1 θk = 0. This sum equals (θm − 1)/(θ− 1)− 1, andhence it is zero.

The same argument also proves the following result.

Lemma 3.2. Let ω ∈ K be a m-th primitive root of unity. Then (ω, ω2, . . . , ωm = 1) is a solution of system(1).

Lemma 3.3. System (1) has no nontrivial solution (x1, x2, . . . , xm−1, xm) for xm−1 = xm = 0.

Proof. For a solution x = (x1, x2, . . . , xm−1, xm) of system (1), let y1, . . . , yk be the pairwise distinct non-zero coordinates of x. For 1 ≤ j ≤ k, the multiplicity of yj is defined to be the number nj of the coordinatesof x which are equal to yj . Since xm−1 = xm = 0 we have k ≤ m− 2. The first k equations of (1) read

(4)

n1y1 + n2y2 + . . .+ nkyk = 0;n1y

21 + n2y

22 + . . .+ nky

2k = 0;

· · · · · ·· · · · · ·n1y

k1 + n2y

k2 + . . .+ nky

kk = 0.

Then (y1, y2, . . . , yk) is a nontrivial solution of the linear system

(5)

n1X1 + n2X2 + . . .+ nkXk = 0;n1y1X1 + n2y2X2 + . . .+ nkykXk = 0;· · · · · ·· · · · · ·n1y

k−11 X1 + n2y

k−12 X2 + . . .+ nky

k−1k Xk = 0.

whose determinant is equal to the product of∏k

i=1 ni by the k × k Vandermonde determinant ∆ =∏

1≤i<j≤k(yi − yj). Since either p = 0 or, if p > 0 then ni < p holds, it turns out that ∆ = 0 andhence yi = yj for some i 6= j. But this contradicts the definition of y1, . . . , yk.

Lemma 3.4. No nontrivial solution (x1, x2, . . . , xm−1, xm) of system (1) has either


(i) four coordinates xi1 , xi2 , xi3 , xi4 such that xi1 = xi2 and xi3 = xi4 with pairwise distinct indicesi1, i2, i3, i4, or

(ii) three coordinates xi1 , xi2 , xi3 such that xi1 = xi2 = xi3 with pairwise distinct indices i1, i2, i3, or(iii) three coordinates xi1 , xi2 , xi3 such that xi1 = xi2 and xi3 = 0 with two distinct indices i1, i2.

Proof. We use the argument in the proof of Lemma 3.3. For a non-trivial solution x = (x1, x2, . . . , xm−1, xm)of system (1), let y1, . . . , yk be the pairwise distinct non-zero coordinates of x with multiplicities nj forj = 1, . . . , k. If x is a counterexample to Lemma 3.4 then k ≤ m − 2. But the proof of Lemma 3.3 showsthat this is actually impossible.

Lemma 3.5. Up to a non-zero scalar, system (1) has finitely many solutions (x1, x2, . . . , xm−2, xm−1, 0).

Proof. We again use the idea from the proof of Lemma 3.3. For a non-trivial solution x = (x1, x2, . . . , xm−1, 0)of system (1), let y1, . . . , yk be the pairwise distinct non-zero coordinates of x with multiplicities nj forj = 1, . . . , k. Furthermore, one of them can be assumed to be equal to 1. Up to a relabelling of indices, nk

counts the coordinates equal to 1. Then (y1, . . . , yk−1) is a solution of

(6)

n1X1 + n2X2 + . . .+ nk−1Xk−1 = −nk;n1y1X1 + n2y2X2 + . . .+ nk−1yk−1Xk−1 = −nk;· · · · · ·· · · · · ·n1y

k−21 X1 + n2y

k−22 X2 + . . .+ nk−1y

k−2k−1Xk−1 = −nk.

Assume on the contrary that system (6) has infinitely many solutions. Since nk 6= 0, the determinant of (6)vanishes. On the other hand, up to a non-zero constant, it is a Vandermonde determinant with parameters(y1, . . . , yk−1). Therefore, yi = yj must occur for some i 6= j contradicting the definition of y1, . . . , yk.

4. The variety defined by system (1)

We keep upon with the notation introduced in Section 2. In particular, V stands for the projectivealgebraic variety of PG(m− 1,K) defined by the equations of system (1).

Our goal is to prove that V is an irreducible non-singular curve. This requires some technical lemmasinvolving both the geometry of V and the transpositions of G. We begin by stating and proving them.

Lemma 4.1. V has dimension 1.

Proof. From [24, Corollary 5] applied to r = m− 2 and n = m− 1, we have dim(V ) ≥ 1. On the other hand,we show that there exists a projective subspace of codimension 2 which is disjoint from V . By [24, Corollary4], this will yield dim(V ) ≤ m− 1− (m− 1− 2)− 1 ≤ 1. Lemma 3.3 ensures that a good choice for such asubspace of codimension 2 is the intersection Λ of the hyperplanes of equation Xm = 0 and Xm−1 = 0.

Lemma 4.2. V is non-singular.

Proof. The Jacobian matrix of V is

∇(V ) =∂(f1, . . . , fm−2)

∂(X1, . . . , Xm)=

∂f1∂X1

· · · ∂f1∂Xm

∂f2∂X1

· · · ∂f2∂Xm

· · · · · · · · ·∂fm−2

∂X1· · · ∂fm−2

∂Xm

=

1 · · · 1

2X1 · · · 2Xm

· · · · · · · · ·(m− 2)Xm−3

1 · · · (m− 2)Xm−3m

.

Up to the non-zero factor (m−2)!, the determinants of maximum orderm−2 are Vandermonde determinants.Therefore, for some point P = (x1 : . . . : xm) ∈ V , ∇(V ) evaluated at P has rank less than m− 2 if and onlyif (x1, . . . , xm) has one of the properties (i) and (ii). But Lemma 3.4 rules out these possibilities, and hencethe point P is non-singular.


Lemma 4.3. deg(V ) = (m− 2)!.

Proof. Let Π be the hyperplane with equation Xm−1 = Xm. Since V has dimension 1, Π intersects Vin at least one point P = (x1 : x2 : . . . : xm−2 : xm−1 : xm−1). By Lemma 3.3, xm−1 6= 0 and soP = (x1/xm−1 : . . . : xm−2/xm−1 : 1 : 1). Also, from Lemma 3.4 x1/xm−1, . . . , xm−2/xm−1 are pairwisedistinct. Since any permutation of x1/xm−1, . . . , xm−2/xm−1 gives rise to a new point of V on P , we obtain|Π ∩ V | = (m− 2)! and hence deg(V ) ≥ (m− 2)!. On other hand, the higher dimensional generalization ofBezout’s theorem yields deg(V ) ≤ (m− 2)! whence the claim follows.

Lemma 4.4. Let P = (ξ1 : ξ2 : · · · : ξm) be a point of V , and fix r ≥ 2 indices 1 ≤ j1 < · · · < jr ≤ m. LetR = (x1 : x2 : · · · : xm) be a point of V such that xj1 = ξj1 , . . . , xjr = ξjr . Then there is a permutation ρ on1, . . . ,m with ρ(ji) = ji for i = 1, . . . , r such that xk = ξρ(k) for k = 1, . . . ,m.

Proof. If there is a pair i, ℓ with i 6= ℓ such that ξji = ξjℓ , let Π be the hyperplane of equation Xj1 = Xjℓ .Lemma 3.4 shows that the m − 2 coordinates of P other than ξji and ξjℓ are pairwise distinct. Thereforethe permutations on the coordinates other than Xji and Xjℓ give rise as many as (m− 2)! pairwise distinctpoints of V . By Lemma 4.3 these are all points in V ∩Π. From this the claim follows. If no such pair i, ℓexists, take for Π the hyperplane of equation Π : Xj1 = ξj1Xm. Then the above argument still works, andthe claim holds true.

Lemma 4.5. For a m-th primitive root of unity ω, let Pω = (ω : ω2 : · · · : ωm = 1) be a point of V .Then the stabilizer of Pω in G is a cyclic group of order m, and it acts on X1, . . . , Xm as a m-cycle.Moreover if Oω is the G-orbit of Pω, then |Oω| = (m− 1)!, and if Πω is the hyperplane Xm−1 = ωXm, thenΠω ∩ Oω = V ∩ Πω.

Proof. Let u ∈ PGL(m − 1,K) be given by the (0, 1)-matrix gi,j whose 1 entries are gi,i+1 for i =1, 2, . . . , p− 2, and gm,1. Clearly, u ∈ G and Pω is fixed by u. Also u has order m and acts on X1, . . . , Xmas a m-cycle. Therefore, |GP | ≥ m. We show that equality holds. Let σ be permutation on 1, 2, . . . ,m−1.If Pω = P σ

ω , then ωi = ωσ(i) for any i ≤ m − 1. Thus σ(i) − i ≡ 0 (mod m) whence σ(i) = i follows.Therefore |GP | = m. From this |Oω | = |G|/|GP | = (m− 1)! follows. Since Oω ⊂ Πω and |V ∩Πω| ≤ deg(V )by Lemma 4.2, the last claim follows from Lemma 4.3.

Lemma 4.6. For a (m − 1)-th primitive root of unity ε, let Pε = (ε : ε2 : . . . , εm−1 = 1 : 0). Then thestabilizer of Pε in G is a cyclic group of order m− 1 which acts on X1, . . . , Xm−1 as a cycle.

Proof. Let h ∈ PGL(m−1,K) be given by the 0, 1-matrix gi,j whose 1 entries are gi,i+1 for i = 1, 2, . . . ,m−2, gm−1,1 and gm,m. Clearly h takes Pε to the point Q = (ε2 : ε3 : . . . : 1 : ε : 0). Actually, Q = Pε asthe coordinates of Q are proportional to those of Pε. It remains to show that any g ∈ G fixing Pε is apower of h. Let g : (X1 : X2 : . . . : Xm−1 : 0) → (Y1 : Y2 : . . . : Ym−1 : 0) where Y1Y2 · · ·Ym−1 is apermutation π of X1X2 · · ·Xm−1. Since g fixes Pε, there exists (y1 : y2 : . . . : ym−1) with π(εi) = yi fori = 1, . . . ,m− 1 such that (ε, ε2, . . . , εm−1 = 1) and (y1, y2, . . . , ym−1) are proportional. Then ym−1ε

i = yifor i = 1, . . . ,m − 1. Also, there exists 1 ≤ j ≤ m − 1 such that yj = 1. Therefore ym−1 = ε−j whenceyi = εi−j . Thus π(εi) = εi−j whence Yi = Xi+(m−1−j) where the indices are taken modulo m − 1. Hence

g = hm−1−j which proves the first claim. Since h fixes Xm, G acts on X1, . . . , Xm−1 as a cycle.

Lemma 4.7. Let Π∞ be the hyperplane of equation Xm = 0. Then Π∞ intersects V transversally at each oftheir (m− 2)! common points. Moreover, Oε = V ∩ Π∞ is the orbit of Pǫ under the action of the stabilizerof Π∞ in G.

Proof. Since |G| = m!, the subgroup of G preserving Π∞ has order (m − 1)!. Since Pε ∈ Π∞, Lemma 4.6yields that Π∞ contains at least (m − 1)!/(m − 1) = (m − 2)! pairwise distinct points of V . On the otherhand, |V ∩ Π∞| ≤ deg(V ) by Lemma 4.2. As deg(V ) = (m − 2)! by Lemma 4.3, the first claim follows.Since the stabilizer of Π∞ in G has order (m− 1)!, the first claim together with Lemma 4.6 prove the secondclaim.


Lemma 4.8. Let g ∈ G be a nontrivial element fixing a point of V . If g acts on Vm fixing Xi and Xj theng is a transposition on X1, X2, . . . , Xm. Furthermore, the fixed points of g in V are as many as (m− 2)!and they are all the common points of V with the hyperplane of equation Xl = Xn where Xn = g(Xl), andl 6= n.

Proof. Up to a change of the indices, g fixes Xm and Xm−1. Let P = (x1, . . . , xm−1, xm) ∈ V be a fixedpoint of g. From Lemma 3.3, xm−1 and xm do not vanish simultaneously. Therefore, xm = 1 may beassumed. Furthermore, if Xg(l) = g(Xl) then g(P ) = P yields xg(l) = cxl for some c ∈ K and l = 1, . . . ,m.In particular, since g(Xm) = Xm, we have xm = cxm whence c = 1. As g is nontrivial, the set M = l |l 6= g(l), 1 ≤ l ≤ m − 2 is non empty. By c = 1, l ∈ M yields that xg(l) = xl. By (i) and (ii) of Lemma3.4. this implies |M| ≤ 2. Since also |M| ≥ 2 holds, the claim follows. To show the other claims, observethat the transposition g = (XlXn) is the involutory homology of PG(m − 1,K) associated with the (0, 1)matrix gu,v whose 1 entries are gu,u for u ∈ 1, 2, . . . ,m \ l, n, gl,n and gn,l. In particular, its axis is thehyperplane Π of equation Xl = Xn, and its center is the point C = (0 : 0 : · · · : −1 : · · · : 0 : · · · : 1 : · · · : 0).Moreover, xl = xn as P is fixed by g. From Lemma 3.3, xl = xn = 1 may be assumed. This together with(i) of Lemma 3.4 yield that xu 6= xv for 1 ≤ u < v ≤ m and (u, v) 6= (i, j). Therefore, the images of P underthe action of the stabilizer H of Xl and Xn in G are all pairwise distinct and their number is |H | = (m− 2)!.Since H preserves both V and Π, all the images of P are in V ∩Π. Thus |V ∩ Π| ≥ (m− 2)!. On the otherhand, Lemma 4.3 shows |V ∩Π| ≤ (m− 2)! whence the second part of the claim follows.

Lemma 4.9. V has a unique component of dimension 1.

Proof. From Lemma 4.1, the irreducible components of V have dimension at most 1, and at least one ofthem is an absolutely irreducible curve C. Let P be a common point of C and Π∞. By the second claimof Lemma 4.7, we can assume P = Pε for a (m − 1)-th primitive root of unity ε. Since P is a nonsingularpoint of V , C is the unique irreducible component of V containing P . The last claim in Lemma 4.7 ensuresthat this holds for each point in V ∩ Π∞. In particular, the 1-dimensional irreducible components of V areexactly the images of C under the action of G. Clearly, these absolutely irreducible curves C = C1, C2, . . . , Clhave the same degree k = deg(C), and kl = deg(V ). The hyperplane Π of equation Xm = Xm−1 meetsC nontrivially, and let R ∈ C ∩ Π. Clearly the transposition g = (XmXm−1) in G which fixes each Xi for1 ≤ i ≤ m − 2 and interchanges Xm with Xm−1 fixes R. Furthermore, C contains a point S ∈ V lyingon the hyperplane of equation Πω : Xm−1 = ωXm. By Lemma 4.5, S ∈ Oω and some element u ∈ GS

acts on the basis (X1, X2, . . . , Xm) as a m-cycle. Since S is a nonsingular point, u preserves C. Indeed,assume on the contrary that u takes C to Ci, i 6= 1. Then, since u fixes S, S must be in the intersectionC ∩ Ci, a contradiction with the non-singularity of S. Therefore g and u are automorphisms of C. By a wellknown result, the group generated by the transposition g and the m-cycle u is the whole symmetric groupG = Symm. This shows that G preserves C and hence l = 1.

From now on, the irreducible non-singular curve C stands for the unique 1-dimensional component of V .As G preserves C and G ≤ PGL(m,K), G is a K-automorphism group of C. Since every hyperplane meetsC, the final claim of Lemma 4.5 shows that some point of Oω (and hence Pω) is in C. Therefore, the G-orbitΩω of Pω has length |G|/m = (m − 1)!. Similarly, Lemmas 4.6 and 4.7 yield that Pε ∈ C, and hence theG-orbit Ωε of Pε has length |G|/(m− 1) = m(m− 2)!. A third short G-orbit Ωθ arises from transpositions,as Lemma 4.8 shows that every transposition of G fixes a point of C.

Lemma 4.10. The hyperplane of equation X1+X2+ . . .+Xm = 0 is the unique hyperplane of PG(m−1,K)which contains C.

Proof. We have Pω ∈ C. Since G preserves C, this yields that Q = (ω : 1 : ω2 : · · · : ωm−1) is in C, as well.Assume that C is contained in a hyperplane Π of equation α1X1 + α2X2 + . . . + αmXm = 0 with αi ∈ K.Then α1 +ωα2 +ω2α3 + . . .+ ωm−1αm−1 = 0 and ωα1 +α2 +ω2α3 + . . .+ωm−1αm−1 = 0 whence α1 = α2

by subtraction. Similar argument shows that any two consecutive coefficient in the equation of Π are equal.


This yields α1 = α2 = . . . = αm−1, that is Π has equation X1 +X2 + . . .+Xm = 0. On the other hand, aswe have already pointed out in Section 2, the first equation in System (1) shows that hyperplane of equationX1 +X2 + . . .+Xm = 0 contains C.

Lemma 4.11. Every transposition of G fixes exactly (m− 2)! points of C.

Proof. We have already pointed out in the proof of Lemma 4.9 that the transposition g = (Xm−1Xm) fixesa point R ∈ C ∩ Π where Π is the hyperplane of equation Xm = Xm−1. By Lemma 3.3 and 3.4, we canassume R = (x1 : . . . : xm−2 : 1 : 1) and that x1, . . . , xm−2 are pairwise distinct and non zero. Thereforeany point whose first m− 2 coordinates are a permutation on x1, . . . , xm−2 is also fixed by g. Thus g hasat least (m − 2)! fixed points. On the other hand, if P ∈ C is not on Π then the last two coordinates of Pare different and hence P is not fixed by g. Therefore, the claim holds for g. Since the transpositions arepairwise conjugate in G, the claim holds true for every transposition in G.

Lemma 4.12. Let P be a point of C which is fixed by a transposition g ∈ G. Then the tangent ℓ to C at Pis the line joining P with the center of g.

Proof. W.l.o.g. we can assume g = (Xm−1Xm). Then g is an homology with center C = (0 : 0 : · · · : 0 : −1 :1) and axis the (pointwise fixed) hyperplane Π of equation Xm = Xm−1. Also, P = (x1 : · · · : xm−2 : 1 : 1).The tangent line ℓ is the intersection of the tangent hyperplanes in P of the hypersurfaces of equation fi = 0of system (1), i = 1, . . . ,m− 2, and hence its equation is given by

(7)

X1 +X2 + . . .+Xm−2 +Xm−1 +Xm = 0;x1X1 + x2X2 + . . .+ xm−2Xm−2 +Xm−1 +Xm = 0;· · · · · ·· · · · · ·xm−2−11 X1 + xm−2−1

2 X2 + . . .+ xm−2−1m−2 Xm−2 +Xm−1 +Xm = 0.

Since the coordinates of C satisfy system (7), the claim follows.

Lemma 4.13. Let H be the stabilizer of Xi and Xj in G. Then the quotient curve C of C with respect to His rational.

Proof. Up to a reordering of the coordinates, we can assume (i, j) = (m − 1,m). The hyperplanes Πλ,µ ofequations λXm−1 + µXm = 0 form the pencil through the intersection Σ of the hyperplanes Xm−1 = 0 andXm = 0. Σ is a subspace of PG(m,F) of codimension 2, and it is disjoint from C by Lemma 3.3. Furthermore,H preserves each Πλ,µ. Take any point of P ∈ C whose stabilizer HP in H is trivial. Then the H-orbit ∆of P has length (m− 2)! and ∆ is contained in the unique hyperplane Πλ,µ of the pencil which contains P .Since deg(C) = (m − 2)!, ∆ coincides with the intersection of C with Πλ,µ. If the stabilizer HP of P ∈ Cin H is nontrivial, from Lemma 4.8 and the proof of Lemma 4.11, then |HP | = 2 and the only nontrivialelement in HP is a transposition h. Indeed if h′ is a nontrivial element of HP , from Lemma 4.8 it is atransposition (XiXj), and from the proof of Lemma 4.11, xi = xj and xm−1 = xm, which is a contradictionwith Lemma 3.4. From Lemma 4.12, the tangent line ℓ to C at P contains the center C of h. The hyperplaneΠ of equation Xm−1 −Xm = 0 is the axis of the transposition g which interchanges Xm and Xm−1. Sinceg and h commute, it follows that C ∈ Π. Therefore, ℓ is contained in Π, and hence I(P, C ∩ Π) ≥ 2, whereI(P, C ∩ Π) is the intersection multiplicity of C and Π at P . From the higher dimensional generalization ofBezout’s theorem, |C ∩Π| ≤ 1

2 (m− 2)!. Thus, |C ∩Π| = 12 (m− 2)!, and, again, ∆ coincides with C ∩Π. This

shows that C is isomorphic to the rational curve which is the projection of C from the vertex Σ.

Lemma 4.14. Let g be the genus of C. Then

(8) 2g− 2 = 12 ((m− 2)(m− 3)− 4)(m− 2)!.


Proof. Let H ∼= Symm−2 be the subgroup of G which fixes both Xm and Xm−1. From Lemma 4.8, if h ∈ Hhas a fixed point in C then h is a transposition. If P is a fixed point of a transposition then, up to a reorderingof the coordinates, P = (1, 1, x3, . . . , xm−1, xm). A point whose coordinates are a permutation of those of Pis another fixed point of the transposition. Among these points, those which are contained in the hyperplaneXm−1 = Xm, fixed by H , are as many as (m − 2)!, and no more, since deg(C) = (m − 2)!. Therefore, thenumber of short orbits of H is equal to the number of choices of two values among x3, . . . , xm. As theseare m− 2 such distinct values, the short orbits are as many as (m− 2)(m− 3). The claim follows from theRiemann-Hurwitz formula.

Lemma 4.15. V is irreducible, that is, V = C.

Proof. Suppose on the contrary the existence of a point Q = (q1 : · · · : qm−1 : qm) of V which is not in C.Since (1 : 0 : 0 : · · · : 0) is not a point of V , both qm = 1 and qm−1 = λ 6= 0 may be assumed. Then Q iscontained in the hyperplane Π of equation Xm−1 = λXm. Choose a point P ∈ C lying on Π. The stabilizerH of Xm−1 and Xm in G preserves Π. Since H also preserves C, the H-orbit ∆ of P is contained in Π. AsP ∈ Π, this together with Lemma 4.3 yield |∆| < |H |, that is, HP is nontrivial. As in the proof of Lemma4.13, from Lemma 4.8 and Lemma 4.11 we obtain |HP | = 2. Therefore, |∆| = 1

2 (m− 2)!. Thus, the higherdimensional generalization of Bezout’s theorem yields that the length of the H-orbit of Q is also less than(m − 2)!, that is, HQ is non-trivial. This is a contradiction, since Lemma 4.8 together with Lemma 4.11,yields that all the fixed points of any transposition of G are on C.

Lemmas 4.2 and 4.15 have the following corollary.

Theorem 4.16. V is an irreducible non-singular curve of PG(m− 2,K).

5. The automorphism group of V

As we have already pointed out, G has at least three short orbits on V , named Ωω, Ωε and Ωθ. We provethat they are the only short G-orbits.

Lemma 5.1. The short G-orbits on V are exactly Ωω, Ωε and Ωθ, and they have lengths (m−1)!, m(m−2)!and m!/2.

Proof. Since |Ωω| = (m − 1)!, |Ωε| = m(m − 2)! and Ωθ are short G-orbits, and |Ωθ| ≤12m!, the Hurwitz

genus formula (3) applied to the G yields

2g− 2 ≥ −2m! + (m!− (m− 1)!) + (m!−m(m− 2)!) + (m!− 12m!).

Comparison with (8) shows that equality holds. Therefore, no further short G-orbits on V exists, and|Ωθ| = 1

2m!, that is, the stabilizer of a point on V which is fixed by a transposition contains no morenontrivial element of G.

Lemma 5.2. The short G-orbit Ωω consists of the common points of V and the hypersurface Σm−1 ofequation

(9) Xm−11 +Xm−1

2 + . . .+Xm−1m = 0.

Proof. We observe first that no point of the G-orbit Ωε is in Σm−1. In fact, if P = (ξ1 : . . . : ξm−1 : 0) ∈ Ωε

then ξm−1j = 1 and hence

∑m−1j=1 ξm−1

j = m − 1 6= 0, thus P 6∈ Σm−1. On the other hand, it is readilyseen that P ∈ Σm−1 for any P ∈ Ωω. From the higher dimensional generalization of Bezout’s theorem,|V ∩Σm−1| ≤ (m−1)!. Furthermore, since G also preserves Σm−1, the intersection V ∩Σm−1 is G-invariant,as well. Therefore, Lemma 5.1 yields V ∩ Σm−1 = Ωω.

A similar argument can be used to prove the following lemmas.


Lemma 5.3. The short G-orbit Ωε consists of the common points of V and the hypersurface Σm of equation

(10) Xm1 +Xm

2 + . . .+Xmm = 0.

Lemma 5.4. The short G-orbit Ωθ consists of the common points of V and the hypersurface Σm(m−1)/2 ofequation

(11) Xm(m−1)/21 +X

m(m−1)/22 + . . .+Xm(m−1)/2

m = 0.

Lemma 5.5. Let Pω be the point of V given in Lemma 4.5. Then the number of fixed points on V of theinvolution in the stabilizer of Pω in G is

(12) 2m/2 (m/2)!

m,

if m is even, and

(13) 2(m−1)/2 ((m− 1)/2)!

m,

if m is odd.

Proof. We will perform the proof for m even. The same approach can be applied for m being odd. Let u bethe (unique) involution in G which fixes Pω. Then u acts on (X1, . . . , Xm) as the involutory permutation(X1X(m+2)/2)(X2X(m+4)/2) · · · (Xm/2Xm). Therefore, the centralizer CG(u) of u in G has order 2m/2(m/2)!,and hence u has as many as

k =m!

2m/2(m/2)!

conjugate in G. We show that if v is conjugate of u in G and u 6= v then u and v has no common fixedpoint. Assume on the contrary the existence of a point Q ∈ V such that u(Q) = v(Q). Then u and v aretwo distinct involutions in the stabilizer GQ of Q in G. On the other hand, since either p = 0 or p > 0 andp ∤ |G|, GQ is cyclic, and hence it contains at most one involution; a contradiction. Therefore, each pointin the G-orbit Ωω is the fixed point of exactly one involution which is conjugate to u in G. If Nu countsthe fixed points of u in Ωω, this yields that |Ωω| = kNu. Therefore, the number of fixed points u in Ωω

equals (12). It remains to show that u has no further fixed points on V . By Lemma 4.6 the stabilizer of anypoint Q ∈ Ωε in G has odd order and hence contains no involution. By Lemma 5.1, the 1-point stabilizerof the remaining short orbit has order 2 and its non-trivial element is a transposition. Since u is not atransposition, the claim follows.

Theorem 5.6. If K has zero characteristic, or it has characteristic p and the K-automorphism group of Vis tame, then G is the K-automorphism group of V .

Proof. By way of a contradiction assume that G = Symm is a proper subgroup of the K-automorphismgroup Γ of V . Then two cases arise, according as G is a normal subgroup of Γ or is not.

In the former case, assume that the centralizer CΓ(G) of G in Γ is trivial. Then for any γ ∈ Γ themap g 7→ γ−1gγ is a non trivial automorphism of G. Hence Γ is isomorphic to a subgroup of Aut(G).However, if m 6= 6, then Aut(G) ∼= G and hence G = Γ, a contradiction. In the remaining case, m = 6,then G ∼= PγL(2, 9) and Aut(G) ∼= PΓL(2, 9). Therefore, [Γ : G] = 2. Lemma 5.1 shows that G has aunique orbit of length (m− 1)! = 120, namely Ωω. Since G is a normal subgroup of Γ, this yields that Γ alsopreserves Ωω. Hence, the stabilizer of Pω ∈ Ωω of Γ has order 12. This yields that Γ has a cyclic subgroupof order 12, but this is impossible since PΓL(2, 9) has two subgroups of order 12, up to conjugation, butneither is cyclic.

Otherwise, let CΓ(G) be a nontrivial subgroup disjoint from G. Furthermore, since CΓ(G) is contained inthe normalizer of G in Γ, CΓ(G) induces a permutation group on the set of the short orbits of G. By Lemma


5.1, there are three such orbits and they have pairwise different lengths. Therefore, this permutation groupis trivial, that is, CΓ(G) preserves each of the three short orbits of G. Also, G = CΓ(G) ∼= CΓ(G)G/G is anontrivial K-automorphism group of the quotient curve V = V/G which G fixes three points of V . However,this is impossible as V is rational by Lemma 4.13.

In the case where G is not normal in Γ, take any conjugate G = γ−1Gγ of G with γ ∈ Γ, and considertheir intersection N = G ∩ G. Assume first N = Altm. If this occurs for every γ ∈ Γ, then N is a normalsubgroup of Γ. As shown before, this is impossible. Now take G such that N 6= Altm. Then |N | ≤ (m− 1)!;

see [6, Theorem 5.2B]. Therefore, | < G, G > | ≥ |G|2/(m− 1)! = m ·m!, whence |Γ| ≥ m ·m!. A comparisonwith (8) gives

|Γ|

g− 1=

4m2(m− 1)

(m− 2)(m− 3)− 4

whence |Γ| > 84(g− 1) for m = 5, 6 and m > 15 whereas 40(g− 1) < |Γ| ≤ 84(g− 1) for the remaining casesof m.

Assume that Γ is tame. Then Γ has exactly three short orbits, and the classical Hurwitz bound yields|Γ| ≤ 84(g− 1). So we are left with 7 ≤ m ≤ 15. From Hurwitz’s proof given in [25] or [14, Theorem 11.56],it follows that |Γ| > 40(g− 1) is only possible when V has two points with stabilizers of Γ of order 2 and 3,respectively. But Lemma 5.1 shows that the three nontrivial point stabilizers of Γ have order m,m− 1 and2. This completes the proof.

6. More equations

Proposition 6.1. Every solution of (1) also satisfies the following equations.

(14)

Xm+21 +Xm+2

2 + . . .+Xm+2m = 0,

Xm+31 +Xm+3

2 + . . .+Xm+3m = 0,

· · · · · ·· · · · · ·X2m−3

1 +X2m−32 + . . .+X2m−3

m = 0.

Proof. Let P = (a1, . . . , am) be any point of V . If P ∈ Ωω then ai = am+1i and hence am+1+l

i = a1+li for any

positive integer l. Therefore, the claim holds for every P ∈ Ωω.Let Hj be the (irreducible) hypersurface of equation Xm+1+j

1 +Xm+1+j2 + . . .+Xm+1+j

m = 0. Then Ωω

is contained in Hj . We prove that Hj contains V as far as j ≤ m − 4. Assume on the contrary that thisdoes not occur for some j. Then Lemma 4.3 together with Bezout’s theorem applied to the intersection ofHj with V yield

(15)∑

Q∈H∩V

I(Q,H ∩ V ) = (m+ 1 + j)(m− 2)!.

We show that Hj contains some points of V other than those in Ωω. As G also preserves Hj , the intersectionnumber I(Q,Hj ∩V ) is invariant when Q ranges over an G-orbit. For a point Q ∈ Ωω, let λ = I(P,Hj ∩V ).Then

∑

Q∈ΩωI(Q,Hj ∩ V ) = λ(m − 1)! whence λ(m − 1) ≤ m + 1 + j. For λ ≥ 2, this would yield

2(m−1) ≤ m+1+j, that is, j > m−4, a contradiction. Therefore λ = 1. Since (m+1+j)(m−2)! > (m−1)!,the claim follows.

Let R be a point R ∈ H∩V not in Ωω. From Lemma 5.1, the G-orbit of R has length at least m(m− 2)!.Then Hj and V have at least ((m − 1) +m)(m− 2)! = (2m− 1)(m− 2)! common points. Since j ≤ m− 3implies 2m− 1 > m+ 1 + j, this contradicts (15).

Lemma 6.2. The group Gm,m−1,m−2 acts on Ωθ with λ1 short orbits and λ2 long orbits where

λ1 = (m− 2)(m− 3)(m− 4), λ2 = 3(m− 2)2.


Proof. Fix a point P ∈ Ωθ. As in the proof of Lemma 4.11, assume that P = (x1, . . . , xm) where xi = xj = tfor some 1 ≤ i < j ≤ m, t ∈ K and xk 6= xl whenever (k, l) 6= (i, j). So, the coordinates (x1, . . . , xm)of P are m − 1 different values from K. The points of Ωθ are those whose coordinates are a permutationof x1, . . . , xm, that is Q ∈ Ωθ if and only if Q = (xσ(1), . . . , xσ(m)) for a permutation σ ∈ G. Two pointsare in the same short Gm,m−1,m−2-orbit if and only if they share the last three coordinates, hence a shortGm,m−1,m−2-orbit arises every time we fix an ordered triple (xa, xb, xc), with xa, xb, xc 6= t. This can bedone in (m− 2)(m− 3)(m− 4) different ways. A long Gm,m−1,m−2-orbit arises every time we fix an orderedtriple (xa, xb, xc), with either one or two of the values xa, xb, xc being equal to t. This can be done in3(m − 2)(m − 3) and 3(m − 2) different ways respectively. Therefore λ1 = (m − 2)(m − 3)(m − 4) andλ2 = 3(m− 2)2.

7. Quotient curves of V

We have already determined the quotient curve of V with respect to the subgroup of G which fixes twogiven coordinates Xi and Xj ; see Lemma 4.13. In this section, we consider the more general case where thesubgroup Hl of G fixes l ≥ 3 coordinates. Let d = m− 1− l. W.l.o.g. these coordinates are assumed to beXd+2, . . . , Xm. The hyperplanes Πi : Xi = 0 with i = d+2, . . . ,m meet in a d-dimensional subspace Σ whichis disjoint from V by Proposition 3.3. Clearly, Hl preserves Σ. Furthermore, the hyperplanes Πi : Xi = 0with i = 1, . . . , d + 1 meet in a (m − d − 2)-dimensional subspace Σ′ disjoint from Σ. Clearly, Hl fixes Σ′

pointwise. Projecting V from Σ on Σ′ produces a curve V of Σ′ whose degree is equal to (m− 2)!/(m− l)!.

Proposition 7.1. Let Hl be the stabilizer of (Xj1 , . . . , Xjl) in G. If l ≥ 2 then the quotient curve V/Hl isisomorphic to V .

Proof. Take a point P ∈ V such that no nontrivial element of Hl fixes P . Then the Hl-orbit Ω of P haslength (m− l)! = (d+1)!. On the other hand, Lemma 4.4 shows that Σ and P generate a (d+1)-dimensional

subspace Σ that cuts out on V a set of (d+1)! points. Since Ω is contained in Σ, it turns out that Ω = V ∩Σ′

whence the claim follows.

Remark 7.2. Proposition 7.1 shows that Σ is an outer Galois subspace of V .

Proposition 7.3. Let g be the genus of the quotient curve V/Hl where Hl is the stabilizer of (Xj1 , . . . , Xjl).If l ≥ 2 then

(16) 2g− 2 =(m− 2)(m− 3)− 4− (m− l)(m− 1− l)

2(m− l)!(m− 2)!

Proof. The number of transpositions in Hl is equal to 12 (m − l)(m − 1 − l). By Lemma 4.11 each such

transposition has as many as (m − 2)! fixed points. From Lemmas 4.5, 4.6, and 5.1, no nontrivial elementof Hl other than its transpositions fixes a point of V . Therefore, the claim follows from the Hurwitz genusformula (3).

7.1. Quotient curve of V by the 3-coordinate stabilizer of G. In this section H = Gm,m−1,m−2 isthe stabilizer of Xm, Xm−1, Xm−2 in G, and X is the quotient curve of V with respect to H . Then X is anirreducible plane curve of degreem−2 whose genus equals 1

2 (m2−7m+12) by (16). Hence X is non-singular.

The following proposition shows that X coincides with the curve introduced and investigated in [31].

Theorem 7.4. X has homogeneous equation Gm−2(x, y, z) = 0 where

(17) Gm−2(x, y, z) =∑

i,j,k≥0,i+j+k=m−2

xiyjzk.


Proof. Since |H | = (m− 3)!, Lemma 4.5 yields that X contains as many as (m− 1)(m− 2) points (α : β : 1)

with αm = βm = 1 but α, β 6= 1. Let X be the plane curve with homogeneous equation Gm−2(x, y, z) = 0.By [31, Equation (2)]

(18) Gm−2(x, y, z) =1

x− y

(

xm − zm

x− z−

ym − zm

y − z

)

.

This shows that X also contains each point (α : β : 1) with αm = βm = 1 but α, β 6= 1. Therefore, X and Xhave at least (m− 1)(m− 2) pairwise distinct points. On the other hand, since X and X both have degree

m− 2, Bezout’s theorem applied to X and X yields that if X and X were distinct then they could share atmost (m− 2)2 points. Therefore, X = X .

Theorem 7.4 shows that K(X ) = K(x, y) with Gm−2(x, y, 1) = 0. From Lemma 4.13, the quotient curveof V with respect to the stabilizer of Xm, Xm−1 is rational. Therefore its function field is K(x).

Theorem 7.5. The Galois closure M of K(X )|K(x) is K(V ), with Galois group isomorphic to Symm−2.

Proof. SinceK(V )|K(x) is a Galois extension, M may be assumed to be a subfield ofK(V ). ThusK(V )|M is aGalois extension, as well. Galois theory yields that Gal(K(V )|M) is a normal subgroup of Gal(K(V )|K(x)).Since Gal(K(V )|K(x)) ≃ Symm−2, for m ≥ 7 and m = 5, the unique non-trivial normal subgroup ofGal(K(V )|K(x)) is Altm−2. On the other hand, as K(X ) is the fixed field of Gm,m−1,m−2

∼= Symm−3,K(X ) ⊆ M 6= K(V ) would imply that Altm−2 is isomorphic to a subgroup of Symm−3 which is impossibleby |Symm−3| < |Altm−2|. If m = 6, a further case arises, namely Gal(K(V )|M) is normal in Alt4 and oforder 4. However, this is again impossible since 4 ∤ |Sym3| = 6. Thus M = K(V ) and Gal(K(X )|K(x)) isisomorphic to the subgroup of G fixing x. Since this subgroup is isomorphic to the stabilizer of Xm, Xm−1

in G, we have Gal(K(X )|K(x)) ∼= Symm−2.

In the subsequent sections we further investigate the two extremal cases in positive characteristic, namelym = 5 and m = p − 1. Accordingly, from now on, K stands for the algebraic closure of the field Fp wherep ≥ 7 is a prime. Then, V is defined over Fp and viewed as a curve defined over K. The group G is alsodefined over Fp, and it preserves the set X (Fpi) of points of V defined over Fpi for every i ≥ 1.

8. The case of positive characteristic; m = 5

In this section m = 5, that is, V is the p-characteristic analog of the Bring curve. Therefore, V hasgenus 4 and is embedded in PG(4,K) as the complete intersection of an hyperplane, a quadratic and a cubicsurface, both non-singular. The (homogeneous) function field K(V ) is F = K(x1, x2, x3, x4, x5) with

(19)

x1 + x2 + x3 + x4 + x5 = 0;x21 + x2

2 + x23 + x2

4 + x25 = 0;

x31 + x3

2 + x33 + x3

4 + x35 = 0.

Our goal is to show that V is an Fp2 -maximal curve, that is the number of points of V defined over Fp2 attainsthe Hasse-Weil upper bond p2+1+2gp = p2+1+8p for infinitely many values of p. The essential tool for theproof is the Jacobian variety JV associated with V , with the following characterization of maximal curvesdue to Tate [29, Theorem 2(d)] and explicitly pointed out by Lachaud [19, Proposition 5]: a curve definedover Fp of genus g is an Fp2 -maximal curve if and only if its Jacobian is Fp2-isogenous to the g-th powerof an Fp2 -maximal elliptic curve. To determine JV we use the following Kani-Rosen theorem [17, TheoremB] about the decomposition of the Jacobian variety of an algebraic curve with respect to an automorphismgroup G equipped by a partition, that is, the group G has a family of subgroups H1, . . . , Ht such thatG = H1 ∪ · · · ∪Ht and Hi ∩Hj = 1 for 1 ≤ i < j ≤ t.


Theorem 8.1 (Kani-Rosen). Let G be a finite automorphism group of an algebraic curve X . If G is equippedby a partition, then the following isogeny relation holds:

(20) J t−1X × J

|G|X/G ∼ Jh1

X/H1× · · · × Jht

X/Ht,

where H1, . . . , Ht are the components of the partition and hi = |Hi| for i = 1, . . . t.

In fact, the Kani-Rosen theorem applies to G as G ∼= Sym5 and Sym5∼= PGL(2, 5) is equipped with a

partition whose components form three conjugacy classes of lengths 15, 6, 10, namely those consisting of allcyclic subgroups of order 4, 5 and 6, respectively. Since Aut(V ) acts on the set of five coordinates X1, . . . X5,representatives of the conjugacy classes are: C4 = 〈(X1, X2, X3, X4)〉, C5 = 〈(X1, X2, X3, X4, X5)〉, andC6 = 〈(X1, X2, X3)(X4, X5)〉, respectively. In our case, since V/G is rational, Theorem 8.1 reads

(21) J30V ∼ J60

V/C4× J30

V/C5× J60

V/C6

Therefore, V is an Fp2-maximal curves if each JV/Ciwith i = 4, 5, 6 is either rational, or a Fp2 -maximal

elliptic curve. We show first that several quotient curves of V are rational.

Proposition 8.2. Each of the following quotient curves is a rational curve:

• V/C5

• V/G8, where G8 is a (dihedral) subgroup of G of order 8;• V/G24, where G24 is a subgroup (isomorphic to Sym4) of G of order 24;• V/G12, where G12 is a subgroup of G of order 12;• V/G20, where G20 is a subgroup of G of order 20.

Proof. We apply the Riemann-Hurwitz formula to each of the groups C5, G8, G24, and G12. For i ∈5, 8, 12, 24, let gi denote the genus of the corresponding quotient curve. From Theorem 5.1, G actson V with exactly 3 short orbits, namely Ωω, Ωǫ and Ωθ, of length 24, 30 and 60 respectively. For C5 theRiemann-Hurwitz formula reads

6 = 10(g5 − 1) +∑

(5− |oi|);

with oi running over the set of short orbits of C5. Since C5 has at least four fixed points on Ωω, it followsg5 = 0. Also, since a group of order 20 contains a subgroup of order 5, this implies g20 = 0. For G8 theRiemann-Hurwitz formula reads

6 = 16(g8 − 1) +∑

(8− |oi|);

with oi running over the set of short orbits of G8. This implies g8 ≤ 1. If equality holds then G8 has aunique short orbit of length 2. On the other hand, since neither 30 nor 60 is divisible by 8, G8 has a shortorbit in both Ωǫ and Ωθ, a contradiction which implies g8 = 0. Finally, since Sym4 contains a subgroup oforder 8, g24 = 0 also holds. For G12, the Riemann-Hurwitz formula reads

6 = 24(g12 − 1) +∑

(12− |oi|);

with oi running over the set of short orbits of G12. This implies g12 ≤ 1. If equality holds then G12 has aunique short orbit of length 12, contained in Ωǫ. From the orbit-stabilizer theorem follows that G12 containsan involution fixing a point in Ωǫ. This implies that the involution must be the product of two transposition,but it can be checked that the group G12 does not contains any such element. It follows g12 = 0.

Next we show for every p ≥ 7 that JV/Ciwith i = 4, 6 is an elliptic curve, and that these two elliptic

curves are pairwise isogenous over Fp2 .The above defined C4 can be viewed as an automorphism group of F generated by the automorphism

(x1, x2, x3, x4, x5) 7→ (x2, x3, x4, x1, x5). A Magma aided computation shows that each of following elements


b, c, d,∈ F is left invariant by C4:

b = −48x2x23x

44x

25 − 48x2

3x54x

25 − 48x2x

64x

25 − 48x7

4x25 − 24x2x

23x

34x

35 − 24x2x3x

44x

35 − 48x2

3x44x

35−

48x2x54x

35 − 24x3x

54x

35 − 72x6

4x35 − 36x2x

23x

24x

45 − 12x2x3x

34x

45 − 48x2

3x34x

45 − 84x2x

44x

45 − 24x3x

44x

45−

108x54x

45 − 208/9x2x

23x4x

55 − 28/9x2x3x

24x

55 − 370/9x2

3x24x

55 − 668/9x2x

34x

55 − 82/9x3x

34x

55 − 1046/9x4

4x55−

16/3x2x23x

65 − 104/9x2x3x4x

65 − 152/9x2

3x4x65 − 44x2x

24x

65 − 118/9x3x

24x

65 − 730/9x3

4x65 + 64/27x2x3x

75−

8/3x23x

75 − 712/27x2x4x

75 − 92/27x3x4x

75 − 1306/27x2

4x75 − 2128/243x2x

85 + 32/27x3x

85 − 5332/243x4x

85−

1064/243x95,

c = −67/3x2x23x

24x

45 + 1/3x2x3x

34x

45 + 68/3x2

3x34x

45 − 67/3x2x

44x

45 + 67/3x5

4x45 − 11x2x3x

24x

55 + 1/6x2

3x24x

55−

11x2x34x

55 + 23/2x3x

34x

55 + 67/6x4

4x55 − 68/9x2x

23x

65 + 2x2x3x4x

65 + 86/9x2

3x4x65 − 217/9x2x

24x

65+

5/18x3x24x

65 + 439/18x3

4x65 + 272/81x2x3x

75 + 578/81x2

3x75 − 790/81x2x4x

75 − 97/81x3x4x

75 + 107/6x2

4x75−

116/81x2x85 + 632/81x3x

85 + 217/81x4x

85 + 554/81x9

5,

d = 72x2x23x

54x5 + 72x2x

74x5 + 54x2x

23x

44x

25 + 36x2x3x

54x

25 + 18x2

3x54x

25 + 90x2x

64x

25 + 18x7

4x25 + 63x2x

23x

34x

35+

27x2x3x44x

35 + 36x2

3x44x

35 + 144x2x

54x

35 + 9x3x

54x

35 + 45x6

4x35 + 178/3x2x

23x

24x

45 + 9x2x3x

34x

45 + 16x2

3x34x

45+

154x2x44x

45 + 55/3x3x

44x

45 + 142/3x5

4x45 + 50/3x2x

23x4x

55 + 24x2x3x

24x

55 + 29x2

3x24x

55 + 298/3x2x

34x

55+

8/3x3x34x

55 + 209/3x4

4x55 + 52/9x2x

23x

65 − 2/9x2x3x4x

65 + 110/9x2

3x4x65 + 613/9x2x

24x

65 + 74/9x3x

24x

65+

139/3x34x

65 − 208/81x2x3x

75 + 532/81x2

3x75 + 2260/81x2x4x

75 + 226/27x3x4x

75 + 2738/81x2

4x75 + 4x2x

85+

208/81x3x85 + 1460/81x4x

85 + 676/81x9

5.

Moreover, let b1 = b/c, d1 = d/c. From the above computation,

(22) 135b31 − 360b21 + 240b1 + 256 + 256d21 = 0.

Let F1 be the subfield of F generated by b1, d1. Then F1 is elliptic and the linear map (b1, d1) 7→ (x, y) withx = −256/135b1, y = −65536/18225d1, gives an equation

y2 = x3 + 2113−45−1x2 + 2203−85−2x− 2323−125−4

for F1. Furthermore, F1 is the fixed field FC4 of C4. Indeed, since both b1 and d1 are fixed by C4, F1 ⊆ FC4

holds. On the other hand, if equality does not hold then Galois theory yields that F1 is the fixed field of asubgroup N of G strictly containing C4. Since C4 is cyclic and G ∼= Sym5, this yields that the order of N iseither 8, 12, 20 or 24. But then V/N is rational by Proposition 8.2. Therefore, F1 = FC4 .

The quotient curve V/C6 can be investigated in a similar way. The subfield FC6 of F is generated byA,B with

(23)5585034240000A4+ 23225726880000A3B + 27897294510000A2B2+7952734845000AB3+ 1056082140000B4+ 13606338560000A2B+28775567360000AB2+ 6849136640000B3+ 11767644160000B2 = 0.

The birational map

(A,B) 7→ (2193−65−2AB + 2203−65−2B2, 23237 · 3−125−4A2+230313 · 3−125−4AB + 229149 · 3−125−4B2+2383−125−4B,−A2 − 4AB − 4B2),

is a birational isomorphism over Fp to the elliptic curve E2 with equation

(24) y22 = x32 − 2203−85−271x2

2 − 2433−165−441x2 − 26423 · 3−245−6.

Since, by Proposition 8.2, the function field of the quotient curve of any subgroup of G properly containingC6 is rational, it follows that K(A,B) = K(x2, y2) is the function field of V/C6.

Furthermore, the elliptic curves E1 and E2 are isogenous over Fp via the isogeny

(x, y) 7→ ((2103−4x3 + 220 · 17 · 3−85−2x2 + 23031 · 3−12 · 5−3x+ 24011 · 3−165−4)/(x2 − 2113−45−1x+ 2203−85−2), (2153−6x3y − 2253−95−1x2y − 23513 · 3−145−2xy−24553 · 3−185−4y)/(x3 − 2103−35−1x2 + 2203−75−2x− 2303−125−3)).


Observe that, since p ≥ 7 the denominators do not vanish.Summing up, the following theorem holds.

Theorem 8.3. For p ≥ 7, the Jacobian variety JV of V has a decomposition over Fp of the form JV ∼ E4

where E = E1 is the elliptic curve of Weierstrass equation

y2 = x3 + 2113−45−1x2 + 2203−85−2x− 2323−125−4.

Since E is maximal over Fp2 for infinitely many primes [7], Theorem 8.3 has the following corollary.

Corollary 8.4. V is a Fp2-maximal curve for infinitely many primes p.

Remark 8.5. The primes p ≤ 10000 such that V is Fp2-maximal are

p = 29, 59, 149, 239, 269, 839, 1439, 1559, 2789, 2909, 4079, 4799, 5519, 6959, 8069, 8819, 9479, 9749.

9. The case of positive characteristic: m = p− 1

In this section we assume m = p− 1.

9.1. Further results on V (Fpi).

Theorem 9.1. System (1) mod p has as many as (p− 1)! solutions.

Proof. One can count the solutions of (1) modulo p up to a non-zero constant factor by computing thenumber of points of V over Fp. Since any primitive (p− 1)-th roots of unity in K is in Fp, Lemma 4.5 yields|V (Fp)| = (p− 2)!, and the claim follows.

From the proof of Theorem 9.1, V (Fp) is the G-orbit Oω defined in Lemma 4.5. Furthermore, Lemmas4.6 and 4.7 have the following corollary.

Lemma 9.2. The G-orbit Oε is contained in V (Fpi) but not in V (Fpj ) for j < i, where Fpi is the smallestsubfield of K containing a (p− 2)-th primitive root ε of unity.

We are in a position to prove the following theorem.

Theorem 9.3. The curve V has no proper Fp2-rational point, that is V (Fp2) = V (Fp).

Proof. By way of a contradiction, |V (Fp2)| > |V (Fp)| is assumed. Then, since G takes Fp2-rational pointsto Fp2 -rational points, there exists a G-orbit Ω entirely contained in V (Fp2) \ V (Fp).

Assume first that Ω is a long orbit, that is |Ω| = (p− 1)!, and let H be the stabilizer of Xp−1, Xp−2, Xp−3

in G. Then H partitions Ω into (p − 1)(p − 2)(p − 3) long H-orbits. Since each H-orbit corresponds to apoint of X = V/H , it follows that

|X (Fp2 ) \ X (Fp)| ≥ (p− 1)(p− 2)(p− 3).

On the other hand, the Stohr-Voloch bound [28], (see also [14, Theorem 8.41]) applied to X gives

2|X (Fp2)| ≤ (2g(X )− 2) + (p2 + 2)(p− 3) = (p− 6)(p− 3) + (p2 + 2)(p− 3) = (p− 3)(p2 + p− 4).

Therefore, 2(p− 1)(p− 2)(p− 3) ≤ (p− 3)(p2 + p− 4). But then p < 7, a contradiction.Assume now that Ω is a short orbit. From Lemma 5.1, the only possibility is that Ω = Ωθ for some

transposition θ ∈ G. Since each H-orbit corresponds to a point of X , Lemma 6.2 implies

|X (Fp2) \ X (Fp)| ≥ λ1 + λ2 = (p− 3)(p2 − 6p+ 11).

This time the Stohr-Voloch bound gives

2 · (p− 3)(p2 − 6p+ 11) ≤ (p− 3)(p2 + p− 4),

which is only possible for p = 7. However, a Magma aided computation rules out this possibility.


From Lemmas 4.5 and 4.6, Ωω = V (Fp) and Ωε ⊂ V (Fpj ) respectively, where Fpj is the smallest subfieldof K containing a primitive (p− 2)-th root of unity. We prove an analog claim for Ωθ. As pointed out in theproof of Lemma 4.11, Ωθ has a point P whose last two coordinates are equal 1.

Lemma 9.4. Let P = (ξ1, ξ2, . . . , ξp−3, 1, 1) be a point of Ωθ. Then ξp−3 ∈ Fpj for some 1 < j ≤ p − 3.Furthermore, if ξp−3 6∈ Fpj with j < p− 3 then, up to a permutation of the indices 1, 2, . . . , p− 4,

xj = xpj

p−3 j = 1, 2, . . . p− 4.

Proof. Since both V and Ωθ are defined over Fp, Lemma 5.1 shows that the Frobenius map Φ takes P to

the point P (p) ∈ Ωθ where P (p) = (ξp1 , ξp2 , . . . , ξ

pp−3, 1, 1). Also, Φ

i takes P to the point

P (pi) = (ξpi

1 , ξpi

2 , . . . , ξpi

p−3, 1, 1)

of Ωθ. To prove the first claim, assume on the contrary ξp−3 6∈ Fpj for j ≤ p− 3. Then ξp−3, ξpp−3, . . . , ξ

pp−3

p−3

are pairwise distinct. On the other hand, from Lemma 4.4, ξpi

p−3 ∈ ξ1, ξ2, . . . , ξp−3 for any i ≥ 0; acontradiction. To prove the second claim, we may assume that ξp−3 ∈ Fpp−3 . Then the previous argument

shows that ξp−3, ξpp−3, . . . , ξ

pp−3

p−3 = ξ1, ξ2, . . . , ξp−3 whence the claim follows.

Theorem 7.4 together with Lemma 9.4 have the following corollary.

Proposition 9.5. Let Fpj be the subfield of K which is the splitting field of the polynomial f(X) = Xp−3 +2Xp−4 + 3Xp−5 + . . .+ (p− 3)X + p− 2. Then Ωθ ⊂ V (Fpj ) but Ωθ * V (Fpi) for i < j.

The proof of the following theorem relies on Proposition 9.5.

Theorem 9.6. Let d be the smallest positive integer such that (p− 2) | (pd − 1). Then Ωθ,Ωǫ ⊂ V (Fpd) butΩθ,Ωǫ * V (Fpi) for i < d.

Proof. By definition, Fpd is the smallest extension of Fp containing a primitive (p− 2)-th root of unity. So,the claim holds for Ωǫ. To complete the proof, consider the polynomial

g(X) =Xp−2 − 1

X − 1= Xp−3 +Xp−4 + . . .+X + 1,

whose splitting field is Fpd . We prove that g(1−X) = f(X). Indeed

g(1−X) =

p−3∑

i=0

i∑

j=0

(

i

j

)

(−X)j,

and for i ∈ 0, . . . , p− 3 the coefficient of X i in g(1−X) is

(−1)ip−3∑

k=i

(

k

i

)

= (−1)ip−3−i∑

k=0

(

i+ k

i

)

= (−1)i(

p− 2

p− 3− i

)

= (−1)i(

p− 2

i+ 1

)

.

Thus,

g(1−X) =

p−3∑

i=0

(

p− 2

i+ 1

)

(−1)iX i =

p−3∑

i=0

(p− 2− i)X i = f(X).

Therefore the splitting field of f(X) coincides with the splitting field of g(X) and Proposition 9.5 yields theclaim.


9.2. Non-classicality and Frobenius non-classicality of V . For m = p − 1, Proposition 6.1 has thefollowing corollary.

Proposition 9.7. If K has characteristic p then V is contained in the Hermitian variety Hp−3 which is theintersection of the hyperplane Π of equation X1 +X2 + . . .+Xp−1 = 0 with the Hermitian variety Hp−2 of

equation Xp+11 + . . .+Xp+1

p−1 = 0.

Lemma 9.8. For a point P ∈ V , let ΠP be the tangent hyperplane to Hp−3 at P . Then I(P, V ∩ ΠP ) ≥ p.

Proof. Clearly, ΠP is the intersection of the hyperplane Π with the tangent hyperplane αP to the Hermitianvariety Hp−2 at P . Since V is contained in Π, I(P, V ∩ ΠP ) = I(P, V ∩ αP ) holds. Hence, it is enough toshow that I(P, V ∩ αP ) is at least p.

Let P = (ξ1 : · · · : ξp−2 : ξp−1). Up to a change of coordinates, ξp−1 = 1 may be assumed. In the affine

space AG(p−2,K) with infinite hyperplane Xp−1 = 0, Hp−2 has equation Xp+11 +Xp+1

2 + . . .+Xp+1p−2 +1 = 0.

For every i = 1, . . . , p − 2, let xi(t) = ξi + ρi(t) with xi(t), ρi(t) ∈ K[[t]] and ord(ρi(t)) ≥ 1 be a primitiverepresentation of the unique branch of V centered at P . By Proposition 9.7, V is contained in H. Therefore,x1(t)

p+1 + x2(t)p+1 + . . .+ xp−2(t)

p+1 + 1 vanishes in K[[t]]. From this

ξp1(ξ1 + ρ1(t)) + ξp2 (ξ2 + ρ2(t)) + . . .+ ξpp−2(ξp−2 + ρp−2(t)) + 1 = tpv(t), v(t) ∈ K[[t]], ord(v(t)) ≥ 0.

Since ΠP has equation ξp1X1 + ξp2X2 + . . .+ ξpp−2Xp−2 + 1 = 0 in AG(p− 2,K), the claim follows.

Since the dimension of PG(p− 2,K)) is smaller than p, Lemma 9.8 has the following corollary.

Theorem 9.9. V is a non-classical curve.

Theorem 9.10. V is a Frobenius non-classical curve.

Proof. Assume on the contrary that V is Frobenius classical. Then 0, 1, . . . , p− 4 are orders at a genericallychosen point of V . Lemma 9.8 yields that the last order, εp−3, is equal to p. Therefore, Lemma 9.8 also yieldsthat the osculating tangent hyperplane to V at P coincides with the tangent hyperplane Π to Hermitianvariety Hp. Since Π passes through the Frobenius image of P , it follows that V is Frobenius non-classical,a contradiction.

9.3. Some results on the orders of V at Fp-rational points. Since G is transitive on V (Fp), the ordersof V are the same at every point P ∈ V (Fp). From Lemma 3.2, such a point is P = (1 : ηp−2 : ηp−3 : · · · : η)for a primitive element η of Fp. From Lemma 4.5, the stabilizer of P in G is a cyclic group of order p− 1generated by the projectivity σ associated with the matrix

(25) Mσ =

0 1 0 · · · 00 0 1 · · · 0

. . ....

. . .

0 0 0 · · · 0 11 0 0 · · · 0 0

.

The eigenvalues of Mσ are λi = ηi for i = 0, . . . , p − 2. Moreover, then eigenvectors of Mσ are wi =(1, ηi, η2i, . . . , η(p−2)i). Thus the point of PG(p−2,Fp) represented bywi is in V if and only if g.c.d.(i, p−1) =1. Therefore, σ has as many as ϕ(p− 1) fixed points on V .


Let ηi = ηi for i = 0, . . . , p−2, and consider the change of the projective frame from (X1 : X2 : . . . : Xp−1)to (Y1 : Y2 : . . . : Yp−1) defined by

(26)

X1 = Y1 + Y2 + · · ·+ Yp−1;

X2 = Y1 + η1Y2 + η21Y3 · · ·+ ηp−21 Yp−1;

X3 = Y1 + η2Y2 + · · ·+ ηp−22 Yp−1;

...

Xp−1 = Y1 + ηp−2Y2 + · · ·+ ηp−2p−2Yp−1.

With this transformation, P is taken to the fundamental point O = (0 : 0 : · · · : 0 : 1). Let R be the matrixwhose rows are the vectors wi, i = 1, . . . , p− 1. Then R−1MσR is the diagonal matrix

D =

η 0 · · · 00 η2 0 · · · 0...

. . .

0 · · · 0 η2u

,

where 2u = p − 1, and hence it is the matrix associated to σ in the projective frame (Y1 : Y2 : . . . : Yp−1).Replacing Xi by (26) in (1), we obtain the equations of V in the projective frame (Y1 : Y2 : . . . : Yp−1). Now,we explicitly write down the equations defining V in the projective frame (Y1 : Y2 : . . . : Yp−1). From the

first equation in (1), we obtain Y1 = 0. In fact,∑p−1

i=1 Xi = (p− 1)Y1 + Y2

∑p−2j=0 η

j1 + · · ·+ Yp−1

∑p−2j=0 η

jp−2,

and for i = 1 . . . , p− 2 we have∑p−2

j=0 ηji = (ηp−1

i − 1)/(ηi − 1) = 0. Therefore, V is a projective variety of

PG(p− 3,Fp) with projective frame (Y2 : Y3 : · · · : Yp−1).Since O is off the hyperplane of homogenous equation Yp−1 = 0, a branch representation of the unique

branch γ of V centered at O has as components y2 = y2(t), . . . , yp−2 = yp−2(t), yp−1 = 1. Here yi(t) ∈ Fp[[t]].Furthermore, we can assume ord(yp−2(t)) = 1, since O is a simple point of V and the tangent hyperplaneto V at O does not contain the line of homogeneous equation Y2 = 0, . . . , Yp−3 = 0. Therefore,

(27)

y2(t) = α2,1t+ α2,2t2 + . . .

y3(t) = α3,1t+ α3,2t2 + . . .

...

yp−3(t) = αp−3,1t+ αp−3,2t2 + . . .

yp−2(t) = t

yp−1(t) = 1.

Since the projectivity σ preserves V and fixes O, it also preserves γ. Moreover, σ is associated to the abovediagonal matrix D. This yields that an equivalent branch representation is given by

(28)

y1(t) = 0

y2(t) = η2(α2,1t+ α2,2t2 + . . . )

y3(t) = η3(α3,1t+ α3,2t2 + . . . )

...

yp−3(t) = ηp−3(αp−3,1t+ αp−3,2t2 + . . . )

yp−2(t) = ηp−2t = η−1t

yp−1(t) = 1.


Replacing t by η−1t, the equations of (28) become

(29)

y1(t) = 0

y2(t) = η2(ηα2,1t+ η2α2,2t2 + . . . )

y3(t) = η3(ηα3,1t+ η2α3,2t2 + . . . )

...

yp−3(t) = ηp−3(ηαp−3,1t+ η2αp−3,2t2 + . . . )

yp−2(t) = t.

Therefore (27) and (29) are the same branch representation of γ, whence

η3α2,1 = α2,1;...

η2+iα2,i = α2,i;...

η2u−1α2u−1,1 = α2u−1,1;...

η2u−2+iα2u−2,i = α2u−2,i;...

From this α2,i = 0 for i < 2u− 2. More generally, αk,i = 0 for 2 ≤ k ≤ p− 3 and i < p− 1− k. Therefore,ord(yk(t)) ≥ p− 1− k for 2 ≤ k ≤ p− 3. Moreover, the only coefficients αk,i 6= 0 are among those verifyingi+ k ≡ 0 (mod p− 1), where 2 ≤ k ≤ p− 3 and i ≥ 1. Thus, the branch representation is

(30)

y2(t) = α2,p−3tp−3 + α2,2p−4t

2p−4 + · · ·+ α2,w(p−1)−2tw(p−1)−2 + . . .

y3(t) = α3,p−4tp−4 + α3,4u−3t

4u−3 + · · ·+ α3,w(p−1)−3tw(p−1)−3 + . . .

...

yp−4(t) = αp−4,3t3 + αp−4,p+2t

p+2 + · · ·+ αp−4,w(p−1)+3tw(p−1)+3 + . . .

yp−3(t) = αp−3,2t2 + αp−3,p+1t

p+1 + · · ·+ αp−3,w(p−1)+2tw(p−1)+2 + . . .

yp−2(t) = t

yp−1(t) = 1.

We now show how to compute the remaining orders.Recall that, in the new coordinates, the k-th equation in (1) reads

(31)

p−1∑

j=1

( p−2∑

i=2

ηi−1j−1Yi + ηp−2

j−1Yp−1

)k

,

and that by the multinomial theorem( p−2∑

i=2

si + v

)k

=∑

k2+···+kp−2+l=k

(

k

k2, . . . , kp−2, l

) p−2∏

w=2

skww · vl.

Replacing Yi by yi(t) =∑∞

s=1 αi,(p−1)s−it(p−1)s−i in the k-th equation, it is obtained

p−1∑

j=1

( p−2∑

i=2

ηi−1j−1

(

∞∑

s=1

α(p−1)s−i,it(p−1)s−i

)

+ ηp−2j−1

)k

= 0.(32)


First, ord(yp−3(t)) = 2, and, more precisely, αp−3,2 = 2 6= 0. This can be observed by looking at thequadratic term in t in the (p− 3)-th equation, after the substitution Yi = yi(t).

By Proposition 6.1, k ∈ 1, 2, . . . , p− 3 ∪ p+ 1, p+ 2, . . . , 2p− 5.In order to compute the orders of yi(t), with i < p − 1 − 2, let k ∈ p + 1, p + 2, . . . , 2p − 5, and write

k = 2p− 2− k, k ∈ 3, . . . , p− 3.

Remark 9.11. For any fixed p, the coefficients αp−1−k,k, for k ∈ 3, . . . , p− 3, can be computed by takinginto account the following constraints.

The expansion of (32) is of the form

p−1∑

j=1

∑

k2+···+kp−2+l

(

k

k2, . . . , kp−2, l

)

η(2p−2)lj−i

p−2∏

i=2

η(i−1)ki

j−1 t(p−1−i)ki .

Here, ki ≤ k for terms of degree k in t. Moreover, since k > p and(

k

k1, k2, . . . , kp−1−1, l

)

=(k)!

k1!k2! · · · k2u−1!l!;

we see that l ≥ p is a necessary condition in order to have a non-zero multinomial coefficient. Further,l+

∑p−2i=1 ki = k, and each term of degree k corresponds to a choice of the indices such that

∑

(p−1−i)ki = k.Moreover, since for u 6= 0 (mod p− 1),

2u∑

j=1

(

ηj−1u

)

=

2u−1∑

j=0

ηju =η2uu − 1

ηu − 1= 0,

every non-zero term must satisfy∑

(i − 1)ki ≡ l (mod p− 1).To illustrate Remark 9.11, we show that αp−1−3,3 is non-zero, whereas α2,p−1−2 = 0.

• Let k = 3, k = 2p− 5. In the k-th equation, the terms of degree k in t are precisely the three termscorresponding to the choices s = 1, i = p − 1 − 1, ki = 3 and l = 2p − 5 − 3 (for j 6= i it will bekj = 0), or s = 1, i = p− 1− 1, ki = 1 and l = 2p− 5− 1(for j 6= i, kj = 0), and s = 1, i1 = p− 1− 1,ki1 = 1, i2 = p− 1− 2, ki2 = 1, i2 = 1 and l = 2p− 5− 2(for j 6= i, kj = 0).

(

2p− 5

k2, . . . , k2u−1, l

)

=(2p− 5)!

k2! · · · k2u−1!l!

Therefore, the term of degree 3 in the (2p− 5)-th equation is[

(

2p− 5

1, 1, 0, . . . , 0, 2p− 5− 2

)

αp−1−2,2+

(

2p− 5

3, 0, 0 . . . , 0, 2p− 5− 3

)

+

(

2p− 5

0, 0, 1, 0 . . . , 0, 2p− 5− 1

)

αp−1−3,3

]

t3 =

=

[

αp−1−2,2 · (−5) · (−6) + (−1)(−5)(−7)− 5αp−1−3,3

]

t3

Since αp−3,3 = 2, it follows αp−1−3,3 = 5 6= 0.

• Let k = p− 3, k = (p+ 1). Observe that, in this case, the factors in the multinomial theorem are of

the form(

p+1k1,k2,...,k2u−1,l

)

with k1 + k2 + · · ·+ k2u−1 + l = p+ 1. Also, the only non-zero coefficients

are those with ki = p or ki = p+ 1 for some i. Moreover, for u 6= 0 (mod p− 1),

2u∑

j=1

(

ηj−1u

)

=

2u−1∑

j=0

ηju =η2uu − 1

ηu − 1= 0.


Therefore, for k = p + 1, the possibly non-vanishing terms in equation (32), are those obtained for

i+ i ≡ 2 (mod 2u), taking ηi−1j−1yi(t) with multiplicity p and ηi−1

j−1yi(t) with multiplicity 1 or taking

ηi−1j−1yi(t) with multiplicity 1 and ηi−1

j−1yi(t) with multiplicity p, and those obtained for i = m+1 and

ηi−1j−1y2(t) with multiplicity p+ 1.

Therefore, the term of degree 2u− 2 in t, can only be obtained by taking η2−1j−1y2(t) with multiplicity

1 and η2u−1j−1 with multiplicity p, namely it is −α2,2u−2t

2u−2, and therefore α2,2u−2 = 0. As a

consequence, ord(y2(t)) ≥ 2p− 4.

Proposition 9.12. Each of the integers 1, 2, 3 are intersection multiplicities I(P, V ∩ π) at any Fp-rationalpoint. Furthermore, the last order is at least 2p− 4.

9.4. Case p = 7. From Section 9.3, y1(t) = 0, y5(t) = t, and ord(y4(t)) = 2, with α4,2 = 2, that isy4(t) = 2t2 + α4,4t

4 + · · · . The second equation reads

−y33 + y2y3y4 + 4y22y5 − y35 + y4y5 + 4y3 = 0;

whence α3,3 = 5, that is y3(t) = 5t3 + α3,6t6 + · · · . The eighth equation reads

y72 + y2 + y73y5 + y3y75 + y84 = 0.

Therefore it must be ord(y2(t)) = 10.Thus, the order sequence of the curve V at the origin is (0, 1, 2, 3, 10).

9.5. Case p = 11. With the support of MAGMA the first terms of the branch expansions of the yi can becomputed.

y1(t) = 0;

y2(t) = 7t18 + · · · ;

y3(t) = 5t7 + 6t17 + · · · ;

y4(t) = 3t6 + 2t16 + · · · ;

y5(t) = 9t5 + t15 + · · · ;

y6(t) = 3t4 + 0t14 + · · · ;

y7(t) = 5t3 + 6t13 + · · · ;

y8(t) = 2t2 + 4t12 + · · · ;

y9(t) = t.

This shows that the order sequence of V at the origin is (0, 1, 2, 3, 4, 5, 6, 7, 18).

9.6. Connection with Redei’s work on the Minkowski conjecture. From [32, Theorem 2.1] and [20,Lemma 6.1], every solution (ξ1, . . . , ξp) with ξi ∈ Fp also satisfies the diagonal equationsX

k1 +Xk

2 +. . .+Xkp =

0 for k = 12 (p+ 1), . . . p− 3.

Now, let W be the algebraic variety of PG(p− 1,K) associated with the system

(33)

X1 +X2 + . . .+Xp = 0;X2

1 +X22 + . . .+X2

p = 0;· · · · · ·· · · · · ·

Xp−31 +Xp−3

2 + . . .+Xp−3p = 0.

of diagonal equations. Clearly E = (1 : 1 : · · · : 1) is a point of W . Furthermore, any ℓ line through E andanother point P ∈ W is entirely contained in W . In fact, if P = (a1 : · · · : ap) then the points on ℓ distinct


from E are Q = (λ+ a1 : · · · : λ+ ap) with λ ∈ K, and a straightforward computation, (λ+ a1, . . . , λ+ ap) isalso a solution of (33). Since the hyperplane Π of equation Xp = 0 does not contain E, the algebraic varietycut out on W by Π is associated with (1) for m = p− 1. Since this variety is V , Theorem 9.1 and Lemma3.2 show that the points of W over Fp are E together with the points (ξ1 : · · · : ξp) such that [ξ1, . . . , ξp] isa permutation of the elements of Fp. This gives a geometric interpretation for the result of Redei [26] andof Wang, Panico and Szabo [26] reported in the Introduction.

Acknowledgements

This research was partially supported by the Italian National Group for Algebraic and Geometric Struc-tures and their Applications (GNSAGA - INdAM). The research of M. Timpanella was funded by the IrishResearch Council, grant n. GOIPD/2021/93.

References

[1] R. Auffarth, S. Rahausen, Galois subspaces for the rational normal curve, Comm. Algebra 49 (2021), 3635-3644.[2] D. Bartoli, M. Giulietti, M.Q. Kawakita, M. Montanucci, New examples of maximal curves with low genus, Finite Fields

Appl. 68 (2020), article 101744.[3] H.W. Braden, T.P. Northover, Timothy P. Bring’s curve: its period matrix and the vector of Riemann constants, SIGMA

Symmetry Integrability Geom. Methods Appl. 8 (2012), Paper 065, 20 pp.[4] Ri-Xiang Chen, On two classes of regular sequences, J. Commut. Algebra 8 (2016), 29-42.[5] A. Conca, C. Krattenthaler, J. Watanabe, Regular sequences of symmetric polynomials, Rend. Semin. Mat. Univ. Padova

121 (2009), 179-199.[6] J.D. Dixon, B. Mortimer, Permutation Groups, Springer, 1996.[7] N.D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89, 561–567

(1987).[8] R. Dvornicich, U. Zannier, Newton functions generating symmetric fields and irreducibility of Schur polynomials, Adv.

Math. 222 (2009), 1982-2003.[9] R. Froberg, B. Shapiro, On Vandermonde varieties, Math. Scand. 119 (2016), 73-91.

[10] S. Fukasawa, Galois lines for the Artin-Schreier-Mumford curve, Finite Fields Appl. 75 (2021), Paper No. 101894, 10 pg.[11] S. Fukasawa, K. Higashine, Galois lines for the Giulietti-Korchmaros curve, Finite Fields Appl. 57 (2019), 268-275.[12] F. Galetto, A.V. Geramita, D.L. Wehlau, Degrees of regular sequences with a symmetric group action, Canad. J. Math.

71 (2019), 557-578.[13] K. Higashine, A criterion for the existence of a plane model with two inner Galois points for algebraic curves, Hiroshima

Math. J. 51 (2021), 163-176.[14] J.W.P. Hirschfeld, G. Korchmaros and F. Torres, Algebraic Curves Over a Finite Field, Princeton Univ. Press, Princeton

and Oxford, 2008.[15] M. Giulietti, M.Q. Kawakita, S. Lia, M. Montanucci, An Fp2 -maximal Wiman’s sextic and its automorphisms, Adv. Geom.

arXiv:1805.06317.[16] M. Giulietti, G. Korchmaros, M. Montanucci, Maximal Curves over Finite Fields, Past, Present and Future, in Proc. of

the international conference Curves over finite fields, Past, Present and Future dedicated to J.P. Serre, pg.30, 2021.[17] E. Kani, M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann. 284 (1989) 307–327.[18] G. Korchmaros, S. Lia, M. Timpanella, Curves with more than one inner Galois point, J. Algebra, 566 (2021), 374-404.[19] G. Lachaud, Sommes d’ Eisenstein et nombre de points de certaines courbes algebriques sur les corps finis, C.R. Acad.

Sci. Paris, Ser. I 305 (1987) 729-732.[20] R. Lidl -H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1983[21] N. Kumar, I. Martino, Regular sequences of power sums and complete symmetric polynomials, Matematiche (Catania)

bf67 (2012), 103-117.[22] H. Melanova, B. Sturmfels, R. Winter, Recovery from Power Sums, arXiv:2106.13981v1 [math.AG] 26 Jun 2021.[23] A. Seidenberg, Elements of the theory of algebraic curves, Addison-Wesley Publishing Co., Reading, Mass.-London-Don

Mills, Ont. 1968[24] I. R. Shafaverich, Basic Algebraic Geometry 1: Varieties in Projective Space, Springer, Heidelberg, third edition, 2013.

[25] H. Stichtenoth, Uber die Automorphismengruppe eines algebraischen Funktionenkorpers von Primzahl- charakteristik. I.Eine Abschatzung der Ordnung der Automorphismengruppe, Arch. Math. 24 (1973), 527–544.

[26] L. Redei, Lacunary Polynomials over Finite Fields, North-Holland Publishing Co., Amsterdam-London; American ElsevierPublishing Co., Inc., New York, 1973.

http://arxiv.org/abs/1805.06317

http://arxiv.org/abs/2106.13981


[27] J.P. Serre, Rational points on curves over finite fields, handwritten notes of Serre’s Harward course 1985 by F. Gouvea,published in Documents Mathematiques (Paris) 18. Societe Mathematique de France, Paris, 2020 with contributions of E.Howe, J. Oesterle and C. Ritzenthaler.

[28] K.O. Stohr, J.F. Voloch, Weierstrass points and curves over finite fields, Proc. Lond. Math. Soc. 52 (1986), 1-19.[29] J. Tate, Endomorphisms of Abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.[30] W. Vogel, Lectures on results on Bezout’s Theorem. Lecture Notes, Tata Institute of Fundamental Research of Bombay.

(Notes by D. P. Patil). Berlin-Heidelberg-New York, Springer. 1984 I, No. 74.[31] F. Rodrıguez Villegas, F. Voloch, D. Zagier, Constructions of plane curves with many points, Acta Arith. 99, 85–96 (2001).[32] C. Ward, V.D. Panico, S. Szabo, Elementary proofs of two results of Redei, J. Algebra Comput. Appl. 2 (2012), 1-6.

Gabor Korchmaros, Stefano Lia, Dipartimento di Matematica, Informatica ed Economia Universita degli Studi dellaBasilicata, Contrada Macchia Romana, 85100, Potenza, Italy.

Email address: [email protected],[email protected]

Marco Timpanella, School of Mathematics and Statistics, University College Dublin, Belfield, Ireland.Email address: [email protected]