Daniele Bartoli, Luciane Quoos, Zülfükar Saygı, Emrah Sercan Yılmaz
In this work we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$.
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We provide first slope estimates of the Newton polygon of generalized Artin-Schreier curves, which are proved using the action of Frobenius and Verschiebung on cohomology. We provide a number of applications such as an improved Hasse-Weil bound for this class of curves.
The arrangement of all Galois lines for the Artin-Schreier-Mumford curve in the projective 3-space is described. It may be surprising that there exist infinitely many Galois lines intersecting this curve.
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.
Artin-Schreier Root Stacks and lifts of group actions
本文证明，当连通仿射代数群G无非平凡特征时，其在光滑射影簇上的作用可提升至相关的Artin-Schreier根叠，并证明了这类根叠上某自伴可逆层的G-线性化存在性。
Projective normality of Artin-Schreier curves
在本文中，我们研究了某些Artin-Schreier曲线$Y_f$的投影正规性，该曲线$Y_f$由方程$y^q+y=f(x)$定义在特征$p$的域$\F$上，$q$是$p$的幂，而$f\in \F[x]$是$m$次$x$中的多项式，其中有$(m,p)=1$。许多$Y_f$曲线是奇异的，因此，准确地说，在这里，我们研究适当的投影模型的正规化的投影正规性。
Cartier Duals of Two-Dimensional Kummer--Artin--Schreier--Witt Type Kernels
本文研究二维Kummer--Artin--Schreier--Witt型同态核的Cartier对偶，在幂零条件下给出其用Witt向量表示的显式描述，并利用Sekiguchi--Suwa方法和形式完备化分析完成证明。
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 consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined over $\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of $\mathbb F_{p^n}$-rational points on $C_k^{(p,a)}$ for all $n$, and uses a result we proved elsewhere about the number of rational points on supersingular curves.
Wildly Ramified Actions and Surfaces of General Type Arising from Artin-Schreier Curves
该研究分析特定Artin–Schreier曲线乘积的对角商，得出其光滑模型多为一般型曲面且陈斜率渐近趋1，同时明确了典范模型的奇点类型与嵌入性质，丰富了正特征下一般型曲面的构造与性质研究。
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.
Dependent Artin-Schreier Defect Extensions and Strong Monomialization
本文肯定地回答了F.库尔曼。我们证明了Cutkosky和Piltant的强单项式反例中的第一个Artin-Schreier亏损扩张是一个依赖扩张。我们的主要工具是使用生成估值序列。
We give an example of an extension of two dimensional regular local rings in a tower of two independent defect Artin-Schreier extensions for which strong local monomialization does not hold.
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。
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。
An elementary proof of a theorem of Montanucci and Zini on the automorphism group of generalized Aritn-Schreier-Mumford curves is presented, with the argument of Korchmaros and Montanucci for Artin-Schreier-Mumford curves being improved. Although the characteristic of a ground field is assumed to be odd in the article of Montanucci and Zini, the proof in the present article is applicable to the case of characteristic two also. As an application of the theorem of Montanucci and Zini, the arrangement of Galois points or Galois lines for the generalized Artin-Schreier-Mumford curve is determined.
Ramification of surfaces: Artin-Schreier extensions
设$A$是特征为$p>0$的正则2维局部环，$L/K$是其分式域的度$p$的循环扩张，使得相应的分支因子是正规交叉.对于每个高度为1的$\gp\in\Spec A$，使得$A/\gp$是正则的，考虑剩余域在$\gp$处的扩展的分支跳跃$h_\gp$。本文证明了$h_\gp$关于适当喷流空间的Zariski拓扑的半连续性。讨论了$h_\gp$关于由$\gp$定义的素因子与分支因子的交重数的渐近性.
Artin-Schreier curves of prescribed p-rank without extra automorphisms
该研究针对奇素数p，构造了无额外自同构的指定p-秩的Artin-Schreier曲线，确定了自同构群仅为Cp的精确数值范围，通过Valentini-Madan定理及相关界完成例外族排除。
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。
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。
Local stability of Frobenius and Artin--Schreier base-change: a comparison
针对正特征下光滑曲线上的对族，研究其沿Artin–Schreier与Frobenius覆盖基变换的局部稳定性，借助分歧数据与纯不可分扩张分析偏差，应用所得结果恢复Hu和Zong的相关结论。
The Geometry of the Artin-Schreier-Mumford Curves over an Algebraically Closed Field
对于素数$p$的幂$q$，亏格$g=(q-1)^2$的Artin-Schreier-Mumford曲线$ASM(q)$是在特征$p$的域$\mathbb{K}$上定义仿射方程$(X^q+X)(Y^q+Y)=c,\, c\neq 0,$的不可约平面曲线的非奇异模型$\mathcal{X}$。Artin-Schreier-Mumford曲线是从定义在非阿基米德赋值域上的代数曲线的研究中得知的，因为对于$|c|<1$，它们是具有大的$2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$阶可解自同构群的曲线，远离零特征有效的Hurwitz界$84(g-1)$。本文讨论了$\mathbb{K}$是特征$p$的代数闭域的情形。证明了$\mathbb{K}$的所有自同构的群$Aut(\mathcal{X})$都有$2q^2(q-1)$阶，并且是半直积$Q\rtimes D_{q-1}$，其中$Q$是$q^2$阶的初等阿贝尔群，$D_{q-1}$是$2(q-1)$阶的二面体群.对于特殊情况$q=p$，这个结果被Valentini和Madan证明。此外，我们还证明了$ASM(q)$在三维射影空间$PG(3,\mathbb{K})$中存在一个非奇异模型$\mathcal{Y}$，该模型在有限域$\mathbb{F}_{q^2}$上既不是经典的，也不是Frobenius经典的。
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.
Zeta functions of a class of Artin-Schreier curves with many automorphisms
本文描述了一类Artin-Schreier曲线，将Van der Geer和Van der Vlugt的结果推广到奇特征。这些曲线的自同构群包含一个大的超特殊群作为子群。这个子群的精确知识使得有可能在子群中所有自同构的定义域上计算该类中曲线的zeta函数。因此，我们得到了新的极大曲线的例子。
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。
Valuations of exponential sums and Artin-Schreier curves
令$p$表示奇素数。本文研究了特征为$p$的有限域上一元多项式的可加指数和的$p$ -整除性，以及确定某些Artin-Schreier曲线族的Newton多边形，即特征为$p$的射影线的$p$ -循环覆盖的问题.首先给出了固定次数多项式指数和的$p$ -整除性的一个下界。然后证明了定义在特征$p$的有限域上的Artin-Schreier曲线当其亏格$g$对于某些$1\leq i\leq p-1$和$n\geq 1$具有$(p-1)\left(i(p^n-1)-1\right)/2$形式使得$n(p-1)>2$时不可能是超奇异的.我们还确定了固定亏格的$p$ -秩$0$ Artin-Schreier曲线族的一般牛顿多边形的第一个顶点，以及相关的Hasse多项式。
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。
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.
On the existence of a morphism between certain Artin-Schreier curves
本文研究了Artin-Schreier曲线之间映射的存在性，证明在特定条件下，映射的存在性等价于指数关系。
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。
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.
Newton slopes for Artin-Schreier-Witt towers
我们固定一个有限域上的首一多项式$f(x) \in \mathbb F_q[x]$，并考虑由$f(x)$定义的Artin-Schreier-Witt塔;这是一个曲线塔$\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$，具有全伽罗瓦群$\mathbb Z_p$。我们研究这座曲线塔的zeta函数的牛顿斜率。这减少到研究的牛顿斜率的L-功能相关的字符的伽罗瓦群这座塔。证明了当特征标的导体足够大时，L-函数的牛顿斜率形成与特征标的导体无关的等差数列.作为推论，我们得到了与Artin-Schreier-Witt塔相关的本征曲线的斜率的行为的结果，类似于Buzzard和Kilford的结果。
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…
Slope estimates of Artin-Schreier curves
设f（x）= x^d + a_{d-1} x^{d-1} +. + a_0是Q[x]中的d次多项式。对任意素数p与d和f（x）互质，（Z_p \cap Q）[x]，设X/F_p是由仿射方程y^p-y = f（x）mod p定义的Artin-Schreier曲线，NP_1（X/F_p）是X/F_p的牛顿多边形的第一斜率，我们证明了Q上d次首一多项式空间A^d中存在一个Zebraki稠密子集U，使得对所有f（x）在U中，当p趋于无穷大时，NP_1（X/F_p）的极限等于1/d。这是万的一个猜想的“第一个斜率版本”。设X/F_pbar是由仿射方程y^p-y = F（x）定义的Artin-Schreier曲线，其中F（x）= x^d + A_{d-1} x^{d-1} +. + A_0。证明了当p>d>1时，NP_1（X/F_pbar）>= ceiling（（p-1）/d）/（p-1）.当p>2d>3时，给出了等式成立的一个充分条件.
Artin-Schreier geproci configurations in projective spaces of arbitrary dimension
研究人员引入𝔽_N-Artin-Schreier构型，刻画ℙ³中该构型为(q,N)-geproci的条件（q≤N），并通过提升构造得到任意n≥3时ℙⁿ中的geproci集。
In this paper, we will interest in finding the number of zeros of the quadratic forms over finite fields. We will apply the tool for finding the number of rational points of supersingular curves in [6]. We will give some more tools for Artin-Schreier curves related with quadratic form. We will especially interest in the (supersingular) Artin-Schreier curve $y^q-y=x^{q^b+1}-x^{q^a+1}$.
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…
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。
Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$
本文在特征2中构造亏格q+1的普通双椭圆曲线，其自同构群达到极值4q，实现Giulietti-Korchmáros分类中的情形(ib)，并给出该情形的完全刻画与参数化。
$p$-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 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.
Relative approximation degrees and the henselian rationality problem over perfect fields
本文研究完美域上的亨塞尔理性问题，完善相对逼近次数理论，推广亨塞尔次数界，分析Artin–Schreier归约，证明特定条件下亨塞尔理性可下传至基域K。
The $L$-functions of Witt coverings
将Dwork，Bengeri和Adolphson-Sperber关于Artin-Schreier覆盖的$L$ -函数的结果推广到Witt覆盖的$L$ -函数.
A motivic derived Fourier transform
针对正特征域上导出Artin stacks的导出向量丛，本文构造了以Artin-Schreier动机为核的有理平展动机傅里叶变换理论，并建立了其核心基本性质，为相关研究提供了重要理论基础。
Maximal subgroups of the Mathieu group $M_{23}$ and symplectic automorphisms of supersingular K3 surfaces
证明了Mathieu群$M_{22}$和$M_{11}$可以作用在特征为11的Artin不变量1的超奇异$K3$曲面上作为辛自同构.更一般地，我们证明了所有的极大子群的Mathieu群$M_{23}$的三个轨道上的24个字母的作用在一个超奇异的$K3$表面与Artin不变1在一个适当的特征。
On a F_{q^2}-maximal curve of genus q(q-3)/6
本文证明了亏格q（q-3）/6的特征3的F_{q^2} -极大曲线要么是q+1次非自反空间曲线，要么是由Artin-Schreier型平面模型在F_{q^2} -同构下唯一确定的
Genus 3 curves with many involutions and application to maximal curves in characteristic 2
设k=F_q是特征为2的有限域.亏格为3的曲线C/k有许多对合，如果k-自同构群允许C_2 \times C_2子群H（如果C是超椭圆的，则不包含超椭圆对合）.则C是三条椭圆曲线的Artin-Schreier覆盖，C的Jacobian与这三条椭圆曲线的乘积k-同构.本文给出了具有多个对合的亏格为3的曲线的显式模型，并计算了椭圆曲线的显式方程。然后，我们刻画了三组（E_1，E_2，E_3）椭圆曲线允许亏格为3的曲线的Artin-Schreier覆盖，并将这个结果应用于极大曲线的构造.当q为非平方数且m= \lfloor 2 sqrt（q）\rfloor = 1，5，7 mod 8时，得到N_q（3）=1+q+3 m.我们还表明，这发生在无限数量的q nonsquare值。
Generalized Artin-Mumford curves over finite fields
设$\mathbb{F}_q$是具有$p>2$素数和$h>1$的$q=p^h$阶有限域，$\mathbb{F}_{\bar{q}}$是$\mathbb{F}_q$的子域.由任意两个$\bar{q}$ -线性化的$q$次多项式$L_1,L_2 \in \overline{\mathbb{F}}_q[T]$，构造出亏格为$(q-1)^2$的普通曲线$\mathcal{X}_{(L_1,L_2)}$，它是射影直线$\mathbb{P}^1$的广义Artin-Schreier覆盖. $\mathcal{X}_{(L_1,L_2)}$在$\mathbb{F}_q$的代数闭包$\overline{\mathbb{F}}_q$上的自同构群包含一个阶为$q^2$的初等交换群$p$ -群$Σ$与一个阶为$\bar{q}-1$的循环群$Γ$的半直积$Σ\rtimes Γ$。我们表明，$L_1 \neq L_2$，$Σ\rtimes Γ$是全自同构群${\rm Aut}(\mathcal{X}_{(L_1,L_2)})$ $\overline{\mathbb{F}}_q$ ; $L_1=L_2$存在一个额外的对合和${\rm Aut}(\mathcal{X}_{(L_1,L_1)})=Σ\rtimes Δ$与二面体群$Δ$的顺序$2(\bar{q}-1)$包含$Γ$。对$\{L_1,L_2\}$的两个不…
Automorphisms of elliptic $K3$ surfaces and Salem numbers of maximal degree
使用椭圆结构，我们表明，任何超奇异K3表面的阿丁不变的$1$在特征$p \not= 5$，$7$，$13$有一个自同构的熵是自然对数的塞勒姆数的程度$22$。
On the number of rational points on curves over finite fields with many automorphisms
使用Weil下降，我们给出了两个家庭的曲线上的有理点的数量的界限在有限域与一个大的阿贝尔群的自同构：Artin-Schreier曲线的形式$y^q-y=f(x)$与$f\in\Fqr[x]$，其上的添加剂组$\Fq$的作用，和Kummer曲线的形式$y^{\frac{q-1}{e}}=f(x)$，其中有一个行动的乘法群$\Fq^\star$。在这两种情况下，当$q$足够大时，我们可以从Weil界中去除$\sqrt{q}$因子。
A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian as large as possible, that is, an $\mathbb{F}_p$ vector space of dimension equal to its genus. In this paper we consider the following question: fix a finite field $\mathbb{F}_q$ and a family $\mathscr{F}$ of curves over $\mathbb{F}_q$. Then, what is the probability that a curve in this family is ordinary? We answer this question when $\mathscr{F}$ is either the Artin-Schreier family in any characteristic or a superelliptic family in characteristic 2.
Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$.
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