Artin-Schreier Curves Research Articles

On complete m-arcs

Let m be a positive integer and q be a prime power. For large finite base fields Fq, we show that any plane curve can be used to produce a complete m-arc as long as some generic explicit geometric conditions on the curve are verified. To show the effectiveness of our theory, we derive complete m-arcs from hyperelliptic curves and from Artin-Schreier curves.

Artin-Schreier curves given by [formula omitted]-linearized polynomials

Let Fq be a finite field with q elements, where q is a power of an odd prime p. In this paper we associate circulant matrices and quadratic forms with the Artin-Schreier curve yq−y=x⋅F(x)−λ, where F(x) is a Fq-linearized polynomial and λ∈Fq. Our results provide a characterization of the number of affine rational points of this curve in the extension Fqr of Fq, for gcd⁡(q,r)=1. In the case F(x)=xqi−x we give a complete description of the number of affine rational points in terms of Legendre symbols and quadratic characters.

Zeta functions of quadratic Artin–Schreier curves in characteristic two

The aim of this paper is twofold: on the one hand, we study the invariants of traces of quadratic forms over a finite field of characteristic $2$. On the other hand, we give results about the zeta functions of certain curves studied by van der Geer and va

Some enumeration results on trace forms of low radical over finite fields of characteristic two

We introduce some enumeration results for quadratic forms written as trace forms with very low dimension of radical for different types of degrees of extension over F2. This work shows a different approach to extend earlier enumeration results, for instance, when the degree of extension is product of two odd primes. We also determine the possible pairs of invariants for some trace forms and use those results to construct some new maximal Artin-Schreier curves with co-dimension 4 and 6 when the degrees of extension are divisible by 8 but not by 16.

The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

We use known characterizations of generalized Paley graphs which are Cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods.

On the existence of curves with prescribed $a$-number

We study the existence of Artin-Schreier curves with large $ a$-number. We also give bounds on the $a$-number of trigonal curves of genus $5$ in small characteristic.

On the projective normality of Artin-Schreier curves

In this paper we study the projective normality of certain Artin-Schreier curves Y_f defined over a field F of characteristic p by the equations y^q+y=f(x), q being a power of p and f in F[x] being a polynomial in x of degree m, with (m,p)=1. Many Y_f curves are singular and so, to be precise, here we study the projective normality of appropriate projective models of their normalization.

Connectedness of the moduli space of Artin-Schreier curves of fixed genus

We study the moduli space ASg of Artin-Schreier curves of genus g over an algebraically closed field k of positive characteristic p. The moduli space is partitioned into strata, which are irreducible. Each stratum parameterizes Artin-Schreier curves whose ramification divisors have the same coefficients. We construct deformations of these curves to study the relations between those strata. As an application, when p=3, we prove that ASg is connected for all g. When p>3, it turns out that ASg is connected for a sufficiently large value of g. In the course of our work, we answer a question of Pries and Zhu about how a combinatorial graph determines the geometry of ASg.

Codes With Hierarchical Locality From Covering Maps of Curves

Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin–Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed–Solomon- and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia–Stichtenoth tower.

On the enumeration of irreducible polynomials over GF(q) with prescribed coefficients

We present an efficient deterministic algorithm which outputs exact expressions in terms of n for the number of monic degree n irreducible polynomials over Fq of characteristic p for which the first l<p coefficients are prescribed, provided that n is coprime to p. Each of these counts is 1n(qn−l+O(qn/2)). The main idea behind the algorithm is to associate to an equivalent problem a set of Artin–Schreier curves defined over Fq whose number of Fqn-rational affine points must be combined. This is accomplished by computing their zeta functions using a p-adic algorithm due to Lauder and Wan. Using the computational algebra system Magma one can, for example, compute the zeta functions of the arising curves for q=5 and l=4 very efficiently, and we detail a proof-of-concept demonstration. Due to the failure of Newton's identities in positive characteristic, the l≥p cases are seemingly harder. Nevertheless, we use an analogous algorithm to compute example curves for q=2 and l≤7, and for q=3 and l=3. Again using Magma, for q=2 we computed the relevant zeta functions for l=4 and l=5, obtaining explicit formulae for these open problems for n odd, as well as for subsets of these problems for all n, while for q=3 we obtained explicit formulae for l=3 and n coprime to 3. We also discuss some of the computational challenges and theoretical questions arising from this approach in the general case and propose some natural open problems.

Divisibility of L-polynomials for a family of Artin–Schreier curves

In this paper we consider the curves Ck(p,a):yp−y=xpk+1+ax defined over Fp and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves Ck(p,a). Our proof involves finding an exact formula for the number of Fpn-rational points on Ck(p,a) for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.

A birational embedding with two Galois points for certain Artin–Schreier curves

We show that two curves of Artin–Schreier type have a birational embedding into a projective plane with two Galois points. As a consequence, all curves with large automorphism groups in the classification list by Henn have a birational embedding with two Galois points.

Exact evaluation of second moments associated with some families of curves over a finite field

Let Fq be the finite field with q elements. Given an N-tuple Q∈FqN, we associate with it an affine plane curve CQ over Fq. We consider the distribution of the quantity q−#Cq,Q where #Cq,Q denotes the number of Fq-points of the affine curve CQ, for families of curves parameterized by Q. Exact formulae for first and second moments are obtained in several cases when Q varies over a subset of FqN. Families of Fermat type curves, Hasse–Davenport curves and Artin–Schreier curves are also considered and results are obtained when Q varies along a straight line.

Arithmetic of abelian varieties in Artin-Schreier extensions

We study abelian varieties defined over function fields of curves in positive characteristic p p , focusing on their arithmetic in the system of Artin-Schreier extensions. First, we prove that the L L -function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension g g , exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves.

More on quadratic functions and maximal Artin–Schreier curves

For an odd prime \(p\) and an even integer \(n\) with \(\gcd (n,p) > 1\), we consider quadratic functions from \({\mathbb F}_{p^{n}}\) to \({\mathbb F}_p\) of codimension \(k\). For various values of \(k\), we obtain classes of quadratic functions giving rise to maximal and minimal Artin–Schreier curves over \({\mathbb F}_{p^{n}}\). We completely classify all maximal and minimal curves obtained from quadratic functions of codimension \(2\) and coefficients in the prime field \({\mathbb F}_p\). The results complement our results obtained earlier for the case \(\gcd (n,p) = 1\). The arguments are more involved than for the case \(\gcd (n,p) = 1\).

Quadratic functions and maximal Artin–Schreier curves

For an odd prime p and an even integer n with gcd⁡(n,p)=1, we consider quadratic functions from Fpn to Fp of codimension k. For various values of k, we obtain classes of quadratic functions giving rise to maximal and minimal Artin–Schreier curves over Fpn. We completely classify all maximal and minimal curves obtained from quadratic functions of codimension 2 and coefficients in the prime field Fp.

Complete arcs arising from a generalization of the Hermitian curve

We investigate complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves which is calculated by using exponential sums via Coulter's approach. We also single out some examples of maximal curves.

Families of Artin–Schreier curves with Cartier–Manin matrix of constant rank

Let k be an algebraically closed field of characteristic p>0. Every Artin–Schreier k-curve X has an equation of the form yp−y=f(x) for some f(x)∈k(x) such that p does not divide the least common multiple L of the orders of the poles of f(x). Under the condition that p≡1modL, Zhu proved that the Newton polygon of the L-function of X is determined by the Hodge polygon of f(x). In particular, the Newton polygon depends only on the orders of the poles of f(x) and not on the location of the poles or otherwise on the coefficients of f(x). In this paper, we prove an analogous result about the a-number of the p-torsion group scheme of the Jacobian of X, providing the first non-trivial examples of families of Jacobians with constant a-number. Equivalently, we consider the semi-linear Cartier operator on the sheaf of regular 1-forms of X and provide the first non-trivial examples of families of curves whose Cartier–Manin matrix has constant rank.

Galois points for a plane curve in characteristic two

Let C be an irreducible plane curve. A point P in the projective plane is said to be Galois with respect to C if the function field extension induced by the projection from P is Galois. We denote by δ′(C) the number of Galois points contained in P2∖C. In this article we will present two results with respect to determination of δ′(C) in characteristic two. First we determine δ′(C) for smooth plane curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ′(C). Second we determine δ′(C) for a generalization of the Klein quartic, which is related to an example of Artin–Schreier curves whose automorphism group exceeds the Hurwitz bound. This curve has many Galois points.

Complete arcs arising from a generalization of the Hermitian curve (extended abstract)

Abstract We investigate arcs, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves.