1,720,987 research outputs found
The complete list of genera of quotients of the Fqjavax.xml.bind.JAXBElement@2b4afbfb-maximal Hermitian curve for q ≡ 1 (mod 4)
No full textLet Fqjavax.xml.bind.JAXBElement@47e531f1 be the finite field with q2 elements. Most of the known Fqjavax.xml.bind.JAXBElement@1831ea17-maximal curves arise as quotient curves of the Fqjavax.xml.bind.JAXBElement@74ff9869-maximal Hermitian curve Hq. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of Hq, but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of Hq for any q≡1(mod4)
Generalized Artin–Mumford curves over finite fields
Full text linkLet Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq ̄ be a subfield of Fq. From any two q ̄-linearized polynomials L1,L2∈F ̅q[T] of degree q, we construct an ordinary curve X(Ljavax.xml.bind.JAXBElement@3de21171,Ljavax.xml.bind.JAXBElement@44e73174) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(Ljavax.xml.bind.JAXBElement@265fda03,Ljavax.xml.bind.JAXBElement@5ee20ea3) over the algebraic closure F ̅q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q ̄−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(Ljavax.xml.bind.JAXBElement@4b2f1fff,Ljavax.xml.bind.JAXBElement@2ddc4e9)) over F ̅q; for L1=L2 there exists an extra involution and Aut(X(Ljavax.xml.bind.JAXBElement@2daa9e77,Ljavax.xml.bind.JAXBElement@1c89ae0d))=Σ⋊Δ with a dihedral group Δ of order 2(q ̄−1) containing Γ. Two different choices of the pair L1,L2 may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F ̅q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(Ljavax.xml.bind.JAXBElement@1301e61e,Ljavax.xml.bind.JAXBElement@61aac551) for some separable q ̄-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1]
On the spectrum of genera of quotients of the Hermitian curve
Full text linkWe investigate the genera of quotient curves Hq∕G of the Fq2 -maximal Hermitian curve Hq, where G is contained in the maximal subgroup Mq ≤ Aut (Hq) fixing a pole-polar pair (P,l) with respect to the unitary polarity associated with Hq. To this aim, a geometric and group-theoretical description of Mq is given. The genera of some other quotients Hq∕G with G≰Mq are also computed. In this way we obtain new values in the spectrum of genera of Fq2 -maximal curves. The complete list of genera g>1 of quotients of Hq is given for q≤29, as well as the genera g of quotients of Hq with g>q2q+30/32 for any q. As a direct application, we exhibit examples of Fq2 -maximal curves which are not Galois covered by Hq when q is not a cube. Finally, a plane model for Hq∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1
Some Ree and Suzuki curves are not Galois covered by the Hermitian curve
Full text linkThe Deligne–Lusztig curves associated to the algebraic groups of type A22, B22, and G22 are classical examples of maximal curves over finite fields. The Hermitian curve Hq is maximal over Fqjavax.xml.bind.JAXBElement@77709645, for any prime power q, the Suzuki curve Sq is maximal over Fqjavax.xml.bind.JAXBElement@99c946, for q=22h+1, h≥1, and the Ree curve Rq is maximal over Fqjavax.xml.bind.JAXBElement@706d12f8, for q=32h+1, h≥0. In this paper we show that S8 is not Galois covered by H64. We also prove an unpublished result due to Rains and Zieve stating that R3 is not Galois covered by H27. Furthermore, we determine the spectrum of genera of Galois subcovers of H27, and we point out that some Galois subcovers of R3 are not Galois subcovers of H27
Ordinary algebraic curves with many automorphisms in positive characteristic
No full textLet X be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus g (X) >= 2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all automorphisms of X which fix K elementwise. For any solvable subgroup G of Aut(X) we prove that vertical bar G vertical bar cp(2) for some positive constant c
A class of linear sets in PG(1,q5)
Full text linkMaximum scattered linear sets in PG(1,qn) have been completely classified for n≤4, see Csajbók and Zanella (2018) [9]; Lavrauw and Van de Voorde (2010) [10]. Here a wide class of linear sets in PG(1,q5) is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in PG(1,q5) are exhibited
On the classification problem for the genera of quotients of the Hermitian curve
Full text linkIn this article, we characterize the genera of those quotient curves H q /G of the F q 2-maximal Hermitian curve H q for which either G is contained in the maximal subgroup M 1 of (H q ) fixing a self-polar triangle, or q is even and G is contained in the maximal subgroup M 2 of (H q ) fixing a pole-polar pair (P,l) with respect to the unitary polarity associated to H q (F q 2) In this way, several new values for the genus of a maximal curve over a finite field are obtained. Our results leave just two open cases to provide the complete list of genera of Galois subcovers of the Hermitian curve; namely, the open cases in [4] when G fixes a point P∈H q (F q 2) and q is even, and the open cases in [33] when G≤M 2 and q is odd
On plane curves given by separated polynomials and their automorphisms
Full text linkLet be a plane curve defined over the algebraic closure K of a finite prime field p by a separated polynomial, that is : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of when m 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve : X(qr - 1)/(q-1) = Yqr-1 + Yqr-2 + ... + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms
Large odd prime power order automorphism groups of algebraic curves in any characteristic
No full textLet X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p ≥ 0, and let Aut(X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by Zomorrodian for Riemann surfaces is |G| ≤ 9(g − 1) where the extremal case can only be obtained for d = 3 and g ≥ 10. We prove Zomorrodian’s result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X/Z for a central subgroup Z of Aut(X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X, we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups
AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves
Full text linkWe investigate several types of linear codes constructed from two families of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup at an Fq-rational point is shown to be symmetric
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