Artin--Schreier and Cyclotomic Extensions
Julio Cesar Salas--Torres, Martha Rzedowski--Calderón, Gabriel Villa--Salvador
本文证明了同余有理函数域的任意Artin-Schreier扩张都包含在明确规定的分圆函数域和常量域扩张的合成中。
In this paper we prove that any Artin--Schreier extension of a congruence rational function field is contained in the composite of a cyclotomic function field and a constant field extension that are explicitly prescribed.
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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.
We study a tower of function fields of Artin-Schreier type over a finite field with $2^s$ elements. The study of the asymptotic behavior of this tower was left as an open problem by Beelen, García and Stichtenoth in $2006$. We prove that this tower is asymptotically good for $s$ even and asymptotically bad for $s$ odd.
The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin-Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case.
Artin-Schreier L-functions and Random Unitary Matrices
本文给出了一个由Z引起的恒等式的新的推导。Rudnick和P. Sarnak关于随机酉矩阵特征值的$n$级相关性的结果以及M. Diaconis和P. Shahshahani表示在酉矩阵系综上的迹积的平均值。我们的方法利用了Artin-Schreier L函数的零统计量和N. Katz和P. Sarnak。
We study the local statistics of zeros of $L$-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier $L$-functions: the ordinary, polynomial (the $p$-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.
For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin-Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A ramification at a closed point of X is understood by the invariant r_x defined by Kato [2]. The main theme of this paper is to give a simple formula to compute r_x' defined in [4], which is equal to r_x for good Artin-Schreier extension. We also prove Kato's conjecture for upper bound of r_x.
The aim of this paper is twofold: on one hand we study the invariants of traces of quadratic forms over a finite field of characteristic two. On the other hand, we give results about the zeta functions of certain curves studied by van der Geer and ven der Vlugt.
Distribution of zeta zeroes of Artin--Schreier curves
研究了当$q$固定且亏格趋于无穷大时，$\mathbb{F}_q$上投影线的Artin-Schreier覆盖族的zeta函数零点的分布.我们认为全球和介观制度，证明当属无穷大，在规定的非平凡子区间的$[-π,π)$的角度的零的数量有一个标准的高斯分布（当适当规范化）。
The corresponding Hasse-Weil bound was a major breakthrough in history of mathematics. It has found many applications in mathematics, coding theory and theoretical computer science. In general, the Hasse-Weil bound is tight and cannot be improved. However, the Hasse-Weil bound is no longer tight when it is applied to some specific classes of curves. One of the examples where the Hasse-Weil bound is not tight is the family of Artin-Schreier curves. Due to various applications of Artin-Schreier curves to coding, cryptography and theoretical computer science, researchers have made great effort to…
Distribution of Artin-Schreier extensions
这篇文章手头包含确切的渐近公式的分布导体的基本阿贝尔p-扩展的全球功能领域的特征p.作为一个后果的分布判别式，这导致了一个确切的渐近公式简单的循环扩张和一个有趣的下界非循环基本阿贝尔扩展。
We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-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…
The genus fields of Artin-Schreier extensions
设$q$是素数的幂$p$。设$k=\mathbb{F}_{q}(t)$是具有常数域$\mathbb{F}_{q}$的有理函数域。令$K=k(α)$是$k$的Artin-Schreier扩张。本文明确地描述了$K$的模糊理想类和亏格域。利用这些结果研究了$K$中$\mathbb{F}_{q}[t]$的积分闭包的理想类群的$p$部分.并给出了R $\acute{e}$ dei-Reichardt公式对$K$的类比.
Statistics for ordinary Artin-Schreier covers and other $p$-rank strata
研究了当$q$固定且亏格趋于无穷大时，$\F_q$上Artin-Schreier覆盖的不同$p$秩层中zeta函数的点数和零点的分布.考虑的$p$秩层包括普通族、整个族和$p$秩等于$p-1.$的曲线族。虽然zeta零点总是接近标准高斯分布，但$\F_q$上的点数具有随特定族而变化的分布。
Classification of quadratic forms over finite fields with maximal and minimal Artin-Schreier curves
本文研究了有限域上与 Artin-Schreier 曲线相关的二次型，利用多项式系数定义的矩阵对其进行刻画，并将结果应用于显式求得极大和极小 Artin-Schreier 曲线。
The article at hand contains exact asymptotic formulas for the distribution of conductors of abelian p-extensions of global function fields of characteristic p. These yield a new conjecture for the distribution of discriminants fueled by an interesting lower bound.
We study a mass formula for Artin--Schreier curves of genus $g$ defined over a finite field $k$ of characteristic $p$. For an odd prime $p$ and for small $g$, we determine the number of $k$-isomorphism classes of Artin-Schreier curves of genus $g$, weighted by the order of the centralizer of the Artin-Schreier involution in the automorphism group. This extends earlier results by several authors in characteristic $p=2$. Keywords: Artin-Schreier curve, finite field, automorphism, Mass formula, moduli space, arithmetic statistics
We prove that for any generic polynomial $f$ in two variables of degree $(d_1,d_2)$ over the rationals, for $p$ large enough the Newton slopes of the character power series $C_f^*(χ_m,s)$ of $f$ at $p$ is independent of the choice of the character $χ_m$ (of conductor $p^m$); and the Newton slopes of the $L$-function $L_f^*(χ_m,s)$ of $f$ at $p$ is in weighted arithmetic progression.
Powers of the Cartier operator on Artin-Schreier covers
本文将层论方法推广至 Cartier 算子的任意幂，给出了 Artin-Schreier 覆盖核维数的界限，并对其 Ekedahl-Oort 类型提出了新限制。
Irreducible representations and Artin L-functions of quasi-cyclotomic fields
本文确定了准素分圆域的所有不可约表示。该方法可用于确定任何拟分圆域的不可约表示。我们还计算了一类拟分圆域的Artin L函数。
In this paper, we study a certain Artin--Schreier family of elliptic curves over the function field $\mathbb{F}_q(t)$. We prove an asymptotic estimate on the size of the special value of their $L$-function in terms of the degree of their conductor; loosely speaking, we show that the special values are "asymptotically as large as possible". We also provide an explicit expression for the $L$-function of the elliptic curves in the family. The proof of the main result uses this expression and a detailed study of the distribution of some character sums related to Kloosterman sums. Via the BSD conje…
The Artin-Schreier DGA and the F_p fundamental group of an F_p scheme
本文证明了F_p类的基本群的F_p完备化可由Artien-Schreier微分分次代数的杆复形计算。利用Kats定理，通过恒等映射与Frobenius映射的关系图，得到了Artin-Schreier微分分次代数.
Counting abelian extensions by Artin-Schreier conductor
本文通过定义 Artin-Schreier 导子，对特征 p 整体函数域的有限阿贝尔 p-群 étale 扩张进行计数，并证明其生成函数为有理函数，从而得到精确计数结果。
In this article, we find a lower bound for the order of the coset x+b in the Artin-Schreier extension $\mathbb F_q[x]/(x^p-x-a)$, where $b\in\mathbb F_q$ that satisfies a generic special condition.
On the number of rational points of Artin-Schreier curves and hypersurfaces
本文利用二次型与置换矩阵方法，精确计算了有限域上两类Artin-Schreier曲线和超曲面的有理点个数，并证明Weil界仅在参数λ的迹为零时达到。
Computing Invariants of Artin-Schreier Curves
本文提出算法框架，用于计算 Artin-Schreier 曲线的不变量生成器，并为低 genus 曲线提供了显式不变量。
We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto ω(a)$ be the Teichmüller lift of $\mathbb F_q$, and let $χ:\mathbb{Z}\to \mathbb C_p^\times$ be a finite character of $\mathbb Z_p$. The $L$-function associated to the polynomial $f$ and the so-called twisted character $ω^u\times χ$ is denoted by $L_f(ω^u,χ,s)$. We prove that, when the conductor of the character is large enough, the $p$-adic Newton slopes of this $L$-function form arithmetic progressions.
Artin-Schreier-Witt lifts of purely inseparable extensions
本文研究正特征离散赋值域剩余域纯不可分扩张的循环提升问题，证明任意有限纯不可分模扩张的循环提升及任意有限次Artin-Schreier-Witt提升存在，给出显式构造，回答了相关文献问题并证明野分歧指数无上界，还给出应用及Artin-Schreier-Witt扩张的结构定理。
We prove a conjecture of Johansen, Helleseth and Kholosha concerning equality of exponential sums related to the cross-correlation of $m$-sequences. In the proof we show that certain Artin-Schreier curves have the same number of points over finite fields. This has a consequence regarding the L-polynomials of these curves.
On invariants of Artin-Schreier curves
本文通过建立标准形式、确定同构并计算不变量重构系统，完成了p>2时亏格3和4的Artin-Schreier曲线的完整分类，拓展了其不变量理论。
Moments of Artin-Schreier L-functions
本研究在有限域大小固定、亏格趋于无穷的条件下，计算了$\n\mathbb{F}_q$\n上Artin–Schreier覆盖多项式族对应$L$-函数的多阶矩与绝对值二阶矩，得到精确公式并验证其酉对称类型。
In local class field theory, the Schmid-Witt symbol encodes interesting data about the ramification theory of $p$-extensi-ons of $K$ and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid-Witt symbol of Artin-Schreier extensions of local fields. Later, his formula was generalized to Artin-Schreier-Witt extensions, but still over a local field. In this paper we generalize Schmid's formula to compute the Artin-Schreier-Witt-Parshin symbol for Artin-Schreier-Witt extensions of two-dimensional local …
We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton …
Some families of supersingular Artin-Schreier curves in characteristic > 2
本文证明了以下两个一维Artin-Schreier曲线族是超奇异的：F_7上的y^7 - y = x^5 + c.x^2; F_5上的y^5 - y = x^7 + c.x（对某个参数c）。我们的方法是在p-adic Dwork方法的基础上发展起来的。
Given a prime $p$ and $q$ a power of $p$, we study the statistics of $p$-ranks of Artin--Schreier covers of given genus defined over $\mathbb{F}_q$, in the large $q$-limit. We refer to this problem as the geometric problem. We also study an arithmetic variation of this problem, and consider Artin--Schreier covers defined over $\mathbb{F}_p$, letting $p$ go to infinity. Distribution of $p$-ranks has been previously studied for Artin--Schreier covers over a fixed finite field as the genus is allowed to go to infinity. The method requires that we count isomorphism classes of covers that are unram…
The p-rank stratification of Artin-Schreier curves
研究了特征p的代数闭域k上亏格为g的Artin-Schreier曲线的模空间AS_g，研究了AS_g的p秩分层为亏格为g的p秩恰为s的Artin-Schreier曲线的层AS_{g,s}.我们列举了AS_{g,s}的不可约成分，并找到它们的尺寸。作为应用，当p=2时，我们证明了亏格为g，秩为2的超椭圆k-曲线的模空间的每一个不可约分支的维数为g-1+s.我们还确定了所有的对（p，g），对于它们AS_g是不可约的.最后，我们研究了具有不同p-秩的Artin-Schreier曲线的变形。关键词：Artin-Schreier，超椭圆，曲线，模，p-秩。
Improvements of The Weil Bound For Artin-Schreier Curves
对于定义在q个元素的有限域F_q上的Artin-Schreier曲线y^q -y = f（x），我们证明了F_q的扩域上有理点点的Weil界通常可以大大提高，本质上消除了误差项中关于q的平方根大小的额外因子。
Codes with Hierarchical Locality on Artin-Schreier Surfaces
本文利用形如 $y^p-y=f(x,z)$ 的 Artin-Schreier 曲面的几何结构构造层级局部性码，给出其结构、恢复算法与参数，并通过计算有理点提供了显式层级参数与更优的最小距离界。
Twisted Artin-Schreier Theory I: $(C_p \rtimes C_{p-1})$-extensions of local function fields
该研究针对特征$p$的域，用阿廷-施赖埃尔理论的扭曲变体参数化$(C_p\rtimes C_{p-1})$-扩张，并对有限域上的局部函数域，给出了给定判别式的此类扩张的计数精确公式。
On the Weil descent of Artin-Schreier algebraic function fields over finite fields
本文研究了有限域上广义 Artin-Schreier 代数函数域扩张的 Weil 下降问题，给出了一般性结果，并完全处理了双循环情形，提供了小素数下的显式示例。
On the Galois-module structure of polydifferentials of Artin-Schreier-Mumford curves, modular and integral representation theory
本文利用调和上闭链理论研究了正特征域上 Mumford 曲线多重微分的 Galois 模结构，并显式计算了 Artin-Schreier-Mumford 曲线全纯多重微分的结构。
p-Density, exponential sums and Artin-Schreier curves
本文定义了有限子集$D\subset\ma{N}^r$的$p$ -密度，并证明了它给出了特征$p$的有限域上指数和的$p$ -adic赋值的一个好的下界.我们还提供了一个应用程序：当为$r=1$时，$p$ -密度是与具有$D$中的指数的多项式相关联的Artin-Schreier曲线族的通用牛顿多边形的第一斜率。
Let $π: Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $π$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the …
We study the canonical model of the Artin-Schreier-Kummer-Witt flat family of curves over a ring of mixed characteristic. We first prove the relative version of a classical theorem by Petri, then use the model proposed by Bertin-Mézard to construct an explicit generating set for the relative canonical ideal. As a byproduct, we obtain a combinatorial criterion for a set to generate the canonical ideal, applicable to any curve satisfying the assumptions of Petri's theorem.
This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.
Artin-Schreier-Witt Theory for Stacky Curves
本文利用高阶分歧数据和 Artin-Schreier-Witt 理论扩展了正特征下 stacky 曲线的分类，引入了 Artin-Schreier-Witt 根栈作为新技术工具，并应用野 Riemann-Hurwitz 定理计算了部分野 stacky 曲线的典范环。
Let $p$ denote an odd prime. In this paper, we are concerned with the $p$-divisibility of additive exponential sums associated to one variable polynomials over a finite field of characteristic $p$, and with (the very close question of) determining the Newton polygons of some families of Artin-Schreier curves, i.e. $p$-cyclic coverings of the projective line in characteristic $p$. We first give a lower bound on the $p$-divisibility of exponential sums associated to polynomials of fixed degree. Then we show that an Artin-Schreier curve defined over a finite field of characteristic $p$ cannot be …
We study the moduli space $\mathcal{AS}_{g}$ of Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of positive characteristic $p$. The moduli space is partitioned by irreducible strata, where 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 $\mathcal{AS}_{g}$ is connected for all possible $g$. When $p>3$, it turns out that $\mathcal{AS}_{g}$ is connected for sufficiently large value of …
In this (mostly historical) note we show how a unified Kummer-Artin-Schreier sequence from [W], [SOS] may be recovered from the relativistic velocity addition law.
We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin-Schreier root stack, allowing us to incorporate this ramification data directly into the stack. As an application, we compute dimensions of Riemann-Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms in this set…
Families of Artin-Schreier curves with Cartier-Manin matrix of constant rank
设k是特征p > 0的代数闭域。对于k（x）中的某个f（x），每个Artin-Schreier k-曲线X都有一个形式为y^p - y = f（x）的方程，使得p不能整除f（x）极点阶数的最小公倍数L。在p与1 mod L全等的条件下，朱文龙证明了X的L-函数的牛顿多边形由f（x）的Hodge多边形决定。特别地，牛顿多边形仅取决于f（x）的极点的阶数，而不取决于极点的位置或f（x）的系数。本文证明了关于X的Jacobian的p-挠群方案的a-数的一个类似结果，给出了具有常数a-数的Jacobian族的第一个非平凡例子。等价地，我们考虑X的正则1-形式层上的半线性Cartier算子，并给出了曲线族的第一个非平凡例子，其Cartier-Manin矩阵具有常秩。
Kummer-Artin-Schreier-Witt Theory
本文提出通过库默-阿廷-施雷尔-维特理论，将正特征中的阿廷-施雷尔-维特同态显式提升到零特征，并探讨其在不分离扩张和斯万展开式研究中的应用。
Double Artin-Schreier extensions of rational function fields with many lifted automorphisms
本文通过双Artin-Schreier扩展构造新的普通函数域，并确定其大自同构群。
Lifting Artin-Schreier covers with maximal wild monodromy
设$k$是特征$p > 0$的代数闭域.在野单值最大的条件下，我们考虑将$\Proj_k$的$p$-循环覆盖提升为某些DVR上投影线的$p$ -循环覆盖的问题。对某些可加多项式$R(t)$，我们肯定地回答了复盖双有理到$w^ p-w=tR(t)$的问题.
Quadratic fields, Artin-Schreier extensions, and Bell numbers
本文证明了联系二次域类数与 Artin-Schreier 多项式根的单项式迹的模 p 同余式，并借助 Bell 数公式推广了 Barsky–Benzaghou 迹公式。
Let $\mathbb F_q$ 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 $y^q - y= x \cdot F(x) - λ,$ where $F(x)$ is a $\mathbb F_q$-linearized polynomial and $λ\in \mathbb F_q$. Our results provide a characterization of the number of affine rational points of this curve in the extension $\mathbb F_{q^r}$ of $\mathbb F_q$, for $\gcd(q,r)=1$. In the case $F(x) = x^{q^i}-x$ we give a complete description of the number of affine rational points in terms of Legendre symbols and …
An Infinite Family of Artin-Schreier Curves with Minimal a-number
本文利用形式粘合法构造任意特征下 a-数达到 Booher-Cais 分歧下界的 Artin-Schreier 曲线无限族，证明该下界的最优性。
Minimal $a$-numbers of Artin--Schreier covers of ordinary curves
本文研究Artin-Schreier覆盖的普通曲线的最小a数，通过计算特定多项式定义的曲线的a数，验证了Booher和Cais给出的下界是紧致的。
Artin-Schreier-Witt extensions and ramification breaks
本研究针对特征p>0的局部域上的Artin-Schreier-Witt循环扩张，采用直接方法证明：既约Witt向量对应的全分歧扩张的上分歧断点计算公式，在剩余域为任意完满域时仍成立，推广了有限域情形下的已有结果。
Artin-Schreier quandles of involutions in absolute Galois groups
本研究提出与绝对伽罗瓦群相关的新域不变量Artin-Schreier quandle，明确其在形式实数域中的自由生成结构，给出洛朗级数域下的计算方式，证实一般情形下该结构存在关系，细化了域的实谱研究。
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is a finite, possibly empty, set of points. The Newton polygon of a degree $p$ Galois cover of $X$ with branch locus $B$ depends on the ramification invariants of the cover. When $X$ is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal.
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