Hui June Zhu
In this short paper we prove that the following two 1-dimensional families of Artin-Schreier curves are supersingular: y^7 - y = x^5 + c.x^2 over F_7 y^5 - y = x^7 + c.x over F_5 (for some parameter c). Our method is developed upon the p-adic Dwork's method.
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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
In this paper, we proved that the F_p completioin of fundamental group of an F_p shceme can be computed by the bar complex of Artien-Schreier differential graded algebra. Artin-Schreier differential graded algebra is obtained by a relation diagram of identity map and frobenius map by Kats' theorem.
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 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。
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…
We study a moduli space AS_g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification of AS_g by p-rank into strata AS_{g,s} of Artin-Schreier curves of genus g with p-rank exactly s. We enumerate the irreducible components of AS_{g,s} and find their dimensions. As an application, when p=2, we prove that every irreducible component of the moduli space of hyperelliptic k-curves with genus g and 2-rank s has dimension g-1+s. We also determine all pairs (p,g) for which AS_g is irreducible. Finally, we study deformations of Artin-Sc…
For Artin-Schreier curve y^q -y = f(x) defined over a finite field F_q of q elements, we show that the Weil bound for the number of the rational points over extension fields of F_q can often be greatly improved, essentially removing an extra factor of size about the square root of q in the error term.
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。
In this paper we define the $p$-density of a finite subset $D\subset\ma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$.
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 …
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…
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。
Let $k$ be an algebraically closed field of characteristic $p > 0$. We consider the problem of lifting $p$-cyclic covers of $\Proj_k$ as $p$-cyclic covers of the projective line over some DVR under the condition that the wild monodromy is maximal. We answer positively the question for covers birational to $w^ p-w=tR(t)$ for some additive polynomial $R(t)$.
In this note, we prove a criteria for supersingularity when the variety has a large automorphism group and a perfect bilinear pairing. This criteria unifies and extends many known results on the supersingularity of curves and varieties and in particular, applies to a large family of Artin-Schreier curves.
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。
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.
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.
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.
We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$, with total Galois group $\mathbb Z_p$. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are inde…
Let $R$ be a complete discrete valuation ring of equal characteristic $p>0$. Given a $\mathbb{Z}/p$-Galois cover of a formal disc over $R$, one can derive from it a semi-stable model for which the specializations of branch points are distinct and lie in the smooth locus of the special fiber. The description leads to a combinatorial object which resembles a classical Hurwitz tree in mixed characteristic, which we will give the same name. The existence of a Hurwitz tree is necessary for the existence of a $\mathbb{Z}/p$-cover whose branching data fit into that tree. We show that the conditions i…
Let f(x) = x^d + a_{d-1}x^{d-1} + ... + a_0 be a polynomial of degree d in Q[x]. For every prime number p coprime to d and f(x) in (Z_p \cap Q)[x], let X/F_p be the Artin-Schreier curve defined by the affine equation y^p - y = f(x) mod p. Let NP_1(X/F_p) be the first slope of the Newton polygon of X/F_p. We prove that there is a Zariski dense subset U in the space A^d of degree-d monic polynomials over Q such that for all f(x) in U, the limit of NP_1(X/F_p) is equal to 1/d as p goes to infinity. This is a ``first slope version'' of a conjecture of Wan. Let X/F_pbar be an Artin-Schreier curve d…
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.
We show how to construct non-isotrivial families of supersingular K3 surfaces over rational curves using a relative form of the Artin-Tate isomorphism and twisted analogues of Bridgeland's results on moduli spaces of stable sheaves on elliptic surfaces. As a consequence, we show that every point of Artin invariant 10 in the Ogus space of marked supersingular K3 surfaces lies on infinitely many pairwise distinct rational curves canonically associated to elliptic structures on the underlying K3 surface.
We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the $n$-level correlations of eigenvalues of random unitary matrices as well as a new proof of a formula due to M. Diaconis and P. Shahshahani expressing averages of trace products over the unitary matrix ensemble. Our method uses the zero statistics of Artin-Schreier L-functions and a deep equidistribution result due to N. Katz and 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.
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p>0$ and $B \subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$-number of a $\mathbf{Z}/p \mathbf{Z}$-cover of $X$ with branch locus $B$. For odd primes $p$, in most cases it is not known if this lower bound is realized. In this note, when $X$ is ordinary, we use formal patching to reduce that question to a computational question about $a$-numbers of $\mathbf{Z}/p\mathbf{Z}$-covers of the affine line. As an application, when $p…
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.
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.
We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of $[-π,π)$ has a standard Gaussian distribution (when properly normalized).
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…
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.
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…
Let $q$ be a power of a prime number $p$. Let $k=\mathbb{F}_{q}(t)$ be the rational function field with constant field $\mathbb{F}_{q}$. Let $K=k(α)$ be an Artin-Schreier extension of $k$. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of $K$ . Using these results we study the $p$-part of the ideal class group of the integral closure of $\mathbb{F}_{q}[t]$ in $K$. And we also give an analogy of R$\acute{e}$dei-Reichardt's formulae for $K$.
We study the distribution of the number of points and of the zeroes of the zeta function in different $p$-rank strata of Artin-Schreier covers over $\F_q$ when $q$ is fixed and the genus goes to infinity. The $p$-rank strata considered include the ordinary family, the whole family, and the family of curves with $p$-rank equal to $p-1.$ While the zeta zeroes always approach the standard Gaussian distribution, the number of points over $\F_q$ has a distribution that varies with the specific family.
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.
Effective Artin-Schreier-Witt theory for curves
本文提出一种算法，计算特征 $p>0$ 代数闭域上光滑射影曲线的所有 $\mathbb{Z}/p^n\mathbb{Z}$-étale Galois 覆盖，给出有限域上的复杂度与 SageMath 实现，并应用于计算局部常系数层的上同调复形。
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。
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…
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。
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。
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。
$p$-torsion for unramified Artin--Schreier covers of curves
本文研究特征$p$的完美域上曲线的$\mathbb{Z}/p\mathbb{Z}$-非分歧Galois覆盖中，上下曲线雅可比簇的$p$-挠子群概形，分析上层雅可比$p$-挠的Galois模结构及其对该结构（如Ekedahl–Oort型）的限制。
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。
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 …
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 …
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。
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