Distribution of Artin–Schreier extensions

The article at hand contains exact asymptotic formulas for the distribution of conductors of elementary abelian p-extensions of global function fields of characteristic p. As a consequence for the distribution of discriminants, this leads to an exact asymptotic formula for simple cyclic extensions and an interesting lower bound for noncyclic elementary abelian extensions.     

Artin–Schreier extensions of the rational function field

In this paper we study the arithmetic of Artin–Schreier extensions of $\mathbb {F}_{q}(T)$ . We determine the integral closure of $\mathbb {F}_{q}[T]$ in Artin–Schreier extension of $\mathbb {F}_{q}(T)$ . We also investigate the average values of the $L$ -functions of orders of Artin–Schreier extensions and study the average values of ideal class numbers when $p=3$ in detail.     

On the distribution of zeroes of Artin–Schreier L-functions

We study the distribution of the zeroes of the L-functions of curves in the Artin–Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a random unitary matrix model.     

Improvements of the Weil bound for Artin–Schreier curves

For the Artin–Schreier curve y ^q ? y = f(x) defined over a finite field \({{\mathbb F}_q}\) of q elements, the celebrated Weil bound for the number of \({{\mathbb F}_{q^r}}\)-rational points can be sharp, especially in super-singular cases and when r is divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz’s work on ?-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra \({\sqrt{q}}\) factor in the error term.     

Galois lines for the Artin–Schreier–Mumford curve

The arrangement of all Galois lines for the Artin–Schreier–Mumford curve in the projective 3-space is described. Surprisingly, there exist infinitely many Galois lines intersecting this curve.     

Lifting Artin–Schreier covers with maximal wild monodromy

Let k be an algebraically closed field of characteristic p > 0. We consider the problem of lifting p-cyclic covers of ${\mathbb{P}^{1}_k}$ as p-cyclic covers C of the projective line over some discrete valuation field K under the condition that the wild monodromy is maximal. We answer positively the problem for covers birationally given by w ^p ?w = t R(t) for any additive polynomial R(t). One gives further informations about the ramification filtration of the monodromy extension and in the case when p = 2, one computes the conductor exponent f (Jac(C)/K) and the Swan conductor sw(Jac(C)/K).     

Multisequences with large linear and k-error linear complexity from a tower of Artin–Schreier extensions of function fields

In this paper, we construct multisequences with both large (joint) linear complexity and k-error (joint) linear complexity from a tower of Artin–Schreier extensions of function fields. Moreover, these sequences can be explicitly constructed.     

On loxodromic actions of Artin–Tits groups

Artin–Tits groups act on a certain delta-hyperbolic complex, called the “additional length complex”. For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that “most” elements of Artin–Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin–Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin–Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.     

Deformations of Koszul Artin–Schelter Gorenstein algebras

We compute the Nakayama automorphism of a Poincaré–Birkhoff–Witt (PBW)-deformation of a Koszul Artin–Schelter (AS) Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi–Yau algebra to be Calabi–Yau. The relations between the Calabi–Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul AS–Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi–Yau algebra is still Calabi–Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this article is elementary and based on linear algebra. The results obtained in this article will be applied in a subsequent paper (He et al., Skew polynomial algebras with coefficients in AS regular algebras, preprint, 2011).     

Gröbner–Shirshov Bases for Schreier Extensions of Groups

In this article, by using the Gröbner–Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.