The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines (original) (raw)

On Weierstrass points of Hurwitz curves

Journal of Algebra, 2006

Let X be a compact Riemannn surface, or curve for short. Let g ≥ 2 be its genus, and G its automorphism group. Then |G| ≤ 84(g − 1), the well known Hurwitz bound. Curves attaining this bound are called Hurwitz curves and the corresponding groups are called Hurwitz groups. Recall that a finite group is a Hurwitz group if and only if it is generated by three elements of orders 2, 3 and 7 whose product is 1. The only Hurwitz curves of genus g < 14 are the famous Klein curve (of genus 3) and the MacBeath curve [Mb]of genus 7, see [Co]. Their automorphism groups G = L 2 (7) respectively G = L 2 (8) act transitively on their Weierstrass points. In this paper we prove that there are no other Hurwitz curves with this property, except possibly the Hurwitz curves of genus 14.

The automorphism groups of a family of maximal curves

Journal of Algebra, 2012

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C3 which is maximal over F q 6 and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves Cn, indexed by an odd integer n ≥ 3, such that Cn is maximal over F q 2n. In this paper, we determine the automorphism group Aut(Cn) when n > 3; in contrast with the case n = 3, it fixes the point at infinity on Cn. The proof requires a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point.

The supersingularity of Hurwitz curves

Involve, a Journal of Mathematics, 2019

We study when Hurwitz curves are supersingular. Specifically, we show that the curve H n,ℓ : X n Y ℓ + Y n Z ℓ + Z n X ℓ = 0, with n and ℓ relatively prime, is supersingular over the finite field Fp if and only if there exists an integer i such that p i ≡ −1 mod (n 2 −nℓ+ℓ 2). If this holds, we prove that it is also true that the curve is maximal over F p 2i. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.

Jacobian varieties of Hurwitz curves with automorphism group PSL(2,q)

Involve, a Journal of Mathematics, 2016

The size of the automorphism group of a compact Riemann surface of genus g > 1 is bounded by 84(g − 1). Curves with automorphism group of size equal to this bound are called Hurwitz curves. In many cases the automorphism group of these curves is the projective special linear group PSL(2, q). We present a decomposition of the Jacobian varieties for all curves of this type and prove that no such Jacobian variety is simple. where e i are certain idempotents in End(JX) ⊗ ‫ޚ‬ ‫.ޑ‬ More details about this decomposition may be found in [Paulhus 2008]. It is important to note here that this decomposition may not be the finest possible decomposition. Some of the abelian variety factors e i (JX) could decompose further. Decomposable Jacobian varieties have applications to rank and torsion questions in number theory [

Automorphism groups of algebraic curves with p-rank zero

Journal of the London Mathematical Society, 2010

In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g 4 automorphisms. It has been observed on many occasions that the most anomalous examples invariably have zero p-rank. In this paper, the K-automorphism group Aut(X ) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that if the curve has genus g ≥ 2 and |Aut(X )| > 24g 2 , then Aut(X ) has a fixed point on X , apart from few exceptions. In the exceptional cases the possibilities for Aut(X ) and g are determined.

Automorphisms of the Generalized Fermat curves

The automorphism group of the generalized Fermat Fk,nF_{k,n}Fk,n curves is studied. We use tools from the theory of complete projective intersections in order to prove that every automorphism of the curve can be extended to an automorphism of the ambient projective space. In particular if k−1k-1k−1 is not a power of the characteristic, then a conjecture of of Y. Fuertes, G. Gonz\'alez-Diez, R. Hidalgo, M. Leyton is proved.

Plane curves with a large linear automorphism group in characteristic ppp

2022

Let GGG be a subgroup of the three dimensional projective group mathrmPGL(3,q)\mathrm{PGL}(3,q)mathrmPGL(3,q) defined over a finite field mathbbFq\mathbb{F}_qmathbbFq of order qqq, viewed as a subgroup of mathrmPGL(3,K)\mathrm{PGL}(3,K)mathrmPGL(3,K) where KKK is an algebraic closure of mathbbFq\mathbb{F}_qmathbbFq. For the seven nonsporadic, maximal subgroups GGG of mathrmPGL(3,q)\mathrm{PGL}(3,q)mathrmPGL(3,q), we investigate the (projective, irreducible) plane curves defined over KKK that are left invariant by GGG. For each, we compute the minimum degree d(G)d(G)d(G) of GGG-invariant curves, provide a classification of all GGG-invariant curves of degree d(G)d(G)d(G), and determine the first gap varepsilon(G)\varepsilon(G)varepsilon(G) in the spectrum of the degrees of all GGG-invariant curves. We show that the curves of degree d(G)d(G)d(G) belong to a pencil depending on GGG, unless they are uniquely determined by GGG. We also point out that GGG-invariant curves of degree d(G)d(G)d(G) have particular geometric features such as Frobenius nonclassicality and an unusual variation of the number of mathbbFqi\mathbb{F}_{q^i}mathbbFqi-rational points. F...

Algebraic curves with many automorphisms

Advances in Mathematics, 2019

Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g ≥ 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 element-wise. It is known that if |Aut(X)| ≥ 8g 3 then the prank (equivalently, the Hasse-Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f (g) such that whenever |Aut(X)| ≥ f (g) then X has zero prank. For even g we prove that f (g) ≤ 900g 2. The odd genus case appears to be much more difficult although, for any genus g ≥ 2, if Aut(X) has a solvable subgroup G such that |G| > 252g 2 then X has zero prank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from Group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz-Gabber covers.