Irreducible characters with bounded root Artin conductor

按 Artin-Schreier 导子计数阿贝尔扩张

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。

一个动机谱的二次 Artin 导子

The quadratic Artin conductor of a motivic spectrum

本文为光滑正常概型上对偶化的动机谱定义了二次 Artin 导子，证明了关联其二次欧拉示性数、秩与导子的公式，并以此得出经典 Grothendieck-Ogg-Shafarevich 公式的二次加细。

通过Artin导数计数有理数上的代数tori

Counting algebraic tori over $\mathbb{Q}$ by Artin conductor

本文通过Artin导数研究有理数上代数tori的计数问题，提出渐进行为猜想并证明其与Malle猜想的关系，同时给出二维tori的上界估计。

拟分圆域的不可约表示与Artin L函数

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。

Artin L-函数与D4四次函数域的非消失性

Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor

研究D4四次函数域的Artin L-函数低频零点，证明按判别式排序时至少77%的L-函数在中心点非消失，扩展了Durlanik在有理数域上的结果。

Conductors of l-adic representations

We give a new formula for the Artin conductor of an $\ell$-adic representation of the Weil group of a local field of residue characteristic $p\neq\ell$.

三维可解Artin表示在一个素数上的分支

Three-Dimensional Solvable Artin Representations Ramified at One Prime

利用$\PGL_3(\C)$的有限不可约子群的分类方法，对$\Q$的不可约三维Artin表示的核的不动点的可能性进行了分类.这使我们能够用给定的阿廷导体来约束这样的表示的数量。

具有低于给定界限的导体的原始射线类特征数的上界

Upper bounds for the number of primitive ray class characters with conductor below a given bound

本文给出了与Artin的原根猜想有关的某些和的上界，这些和也可用于射线类特征标的计数.

Twist-minimal trace formulas and the Selberg eigenvalue conjecture

We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.

Swan导子为1的局部伽罗瓦表示

Local Galois representations of Swan conductor one

本研究借助有限域仿射直线上Artin-Schreier层的平展上同调构造Swan导子为1的复局部伽罗瓦表示，给出单超尖点表示局部Langlands对应的显式描述，并探讨该类伽罗瓦表示在Artin-Schreier簇平展上同调中的更自然实现。

Distribution of Artin-Schreier-Witt extensions

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.

Generic Newton Slopes for Artin-Schreier-Witt Tower in two variables

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.

椭圆曲线的Artin扭L函数的非零性

Non-vanishing of Artin-twisted L-functions of Elliptic Curves

设E是椭圆曲线，ρ是Artin表示，都定义在有理数上.设p是E有好约化的素数。证明了存在一个无限集个Dirichlet特征标χ，且只在p处分歧，使得当β位于临界带的一个指定区域内时，Artin扭曲L-值L（E，ρχ，β）不为零（假定这些L函数的拓扑延拓和函数方程）.本文的新贡献是我们可以选择我们的特征标只在一个素数处分歧，这可以划分ρ的导体。

Newton slopes for twisted Artin--Schreier--Witt Towers

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曲线的p秩分层

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。

Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjecture

The concept of Artin transfer pattern $((\ker(T_{K,N_i}))_i,(\mathrm{Cl}_p(N_i))_i)$ for homogeneous multiplets $(N_1,\ldots,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer homomorphisms$T_{K,N_i}:\,\mathrm{Cl}_p(K)\to\mathrm{Cl}_p(N_i)$ is generalized for heterogeneous multiplets of ramified extensions. By application to quadratic subfields K of dihedral fields N of degree 2p with an odd prime p, a conjecture of Scholz concerning the index of subfield units, $(U_N:U_0)$, for ramified extensions N/K with conductor f>1 is verified computation…

Newton slopes for Artin-Schreier-Witt towers

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…

Independence of Artin L-functions

Let $K/\mathbb Q$ be a finite Galois extension. Let $χ_1,\ldots,χ_r$ be $r\geq 1$ distinct characters of the Galois group with the associated Artin L-functions $L(s,χ_1),\ldots, L(s,χ_r)$. Let $m\geq 0$. We prove that the derivatives $L^{(k)}(s,χ_j)$, $1\leq j\leq r$, $0\leq k\leq m$, are linearly independent over the field of meromorphic functions of order $<1$. From this it follows that the L-functions corresponding to the irreducible characters are algebraically independent over the field of meromorphic functions of order $<1$.

Geometric local $\varepsilon$-factors in higher dimensions

We use former results on geometric local $\varepsilon$-factors over curves in order to prove a factorization result for the determinant of the cohomology of an $\ell$-adic sheaf over an arbitrary proper scheme over a perfect field of positive characteristic $p$ distinct from $\ell$. The local contributions are constructed by iterating vanishing cycle functors as well as certain "refined Artin conductors", the latter being exact additive functors which can be considered as linearized versions of Artin conductors and local $\varepsilon$-factors. We provide several applications of our higher dime…

A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes

Let $F/k$ be a cyclic extension of number fields of prime degree. Let $ρ$ be an irreducible $2$-dimensional representation of Artin type of the absolute Galois group of $F$, and $π$ a cuspidal automorphic representation of GL$_2(\mathbb A_F)$, such that the $L$-functions $L(s,ρ_v)$ and $L(s,π_v)$ agree at all (but finitely many of) the places $v$ of degree one over $k$. We prove in this case that we have the global identity $L(s,ρ)=L(s,π)$, with $ρ_v \leftrightarrow π_v$ being given by the local Langlands correspondence at all $v$. In particular, $π$ is tempered and $L(s,ρ)$ is entire.

An explicit conductor formula for ${\rm GL}_n \times {\rm GL}_1$

We prove an explicit formula for the conductor of an irreducible, admissible representation of ${\rm GL}_n(F)$ twisted by a character of $F^{\times}$ where the field $F$ is local and non-archimedean. As a consequence, we quantify the number of character twists of such a representation of fixed conductor.

弱几乎单项群与 Artin 猜想

Weak almost monomial groups and Artin's conjecture

本文定义弱几乎单项群并证明其封闭性，进而在 Galois 群属于该类时，用 Artin L-函数幺半群的阶乘性及单零点条件刻画 Artin 猜想。

Multiplicities in Selmer groups and root numbers for Artin twists

Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the t…

On the formal degree conjecture for simple supercuspidal representations

We prove the formal degree conjecture for simple supercuspidal representations of symplectic groups and quasi-split even special orthogonal groups over a p-adic field, under the assumption that p is odd. The essential part is to compute the Swan conductor of the exterior square of an irreducible local Galois representation with Swan conductor 1. It is carried out by passing to the equal characteristic local field and using the theory of Kloosterman sheaves.

The second moment of Dirichlet twists of a $\textrm{GL}_{4}$ automorphic $L$-function

In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.

忠实阿廷归纳与切博塔廖夫密度定理

Faithful Artin induction and the Chebotarev density theorem

本研究证明有限群G的忠实不可约特征标张成的向量空间由其内单项式特征标生成，由此推出固定数域F的几乎所有G-扩张满足强有效Chebotarev密度定理，并将其用于推导数域族类群的相关结论。

$p$-adic estimates of abelian Artin $L$-functions on curves

The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $ρ:π_1^{et}(V) \to \mathbb{C}^\times$. In this article, we prove a lower bound on the Newton polygon of the $L$-function $L(ρ,s)$. The estimate depends on monodromy invariants of $ρ$: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions this lower bound agrees with the irregular Hodge filt…

Chebotarev密度定理中的矩：一般类函数

Moments in the Chebotarev density theorem: general class functions

本文研究Chebotarev密度定理误差项高阶矩的下界，针对一般类函数，利用显式公式中的正性和Artin L-函数零点组合分析，结合分歧与伽罗瓦理论信息，证明矩至少为高斯型。

关于Character出现在其伴随Weil表示中的扭转计数

On Counting Twists of a Character Appearing in its Associated Weil Representation

考虑GL（2，$F$）的不可约容许表示$π$，其对GL（2，$F)^+$）的限制分解为两个不可约表示$π_+ + π_-$的和。如果$π=r_θ$，GL（2，$F$）的Weil表示，附加到$K^*$的字符$θ$，该字符不通过从$K$到$F$的范数映射进行因子分解，则$χ\in \hat{K^*}$与$(χ>. θ^{-1})|_{F^{*}}=ω_{K/F}$在${r_θ}_+$中出现当且仅当$ε(θχ^{-1},ψ_0)=ε(\bar θχ^{-1},ψ_0)=1$和在${r_θ}_-$中出现当且仅当两个因子都是-1。但是给定一个指挥家$n$，我们能准确地说有多少这样的$χ$会出现在$π$中吗？在这项工作中，我们计算在每个给定的导体$n$这样的字符的数量。

奇剩余特征下超椭圆曲线的导子-判别式不等式

Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

本文证明了在特征不为2的奇剩余特征下，离散赋值域上超椭圆曲线的导子与判别式之间的不等式，将Artin导子的负值与特殊纤维不可约分支数量由判别式赋值上界控制，推广了前人工作。

Large values of newforms on GL(2) with highly ramified central character

We give a lower bound for the sup-norm of an $L^2$-normalized newform in an irreducible, unitary, cuspidal representation $π$ of $GL_2$ over a number field. When the central character of $π$ is sufficiently ramified, this bound improves upon the trivial bound by a positive power of $N$ where $N$ is the norm of the conductor of $π$. This generalizes a result of Templier, who dealt with the special case when the conductor of the central character equals the conductor of the representation. We also make a conjecture about the true size of the sup-norm in the $N$-aspect that takes into account thi…

相反导子新形式的Rankin--Selberg积分

Rankin--Selberg integrals of opposite conductor--one newforms

本文显式计算了相反导子新形式的Rankin--Selberg积分，证明其中心值非零，并由此验证了Disegni--Zhang猜想的一个情形。

关于特征 $p>0$ 下 Hasse-Weil-Artin $L$-函数的 Birch 和 Swinnerton-Dyer 型猜想

On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin $L$-functions in characteristic $p>0$

本文在特征 $p>0$ 的整体函数域上，对阿贝尔簇与不可约伽罗瓦表示建立 Hasse-Weil-Artin $L$-函数首项系数的 BSD 型公式，并利用等变向量丛的 Riemann-Roch 定理处理 $p$-部分。

导子-判别式不等式I：缓分歧循环覆盖

Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I

本文证明了离散赋值域上Z/n-覆盖的P^1的导子-判别式不等式，推广了超椭圆曲线情形，并加强了Kohls关于Jacobian导子指数的结果。

关于小判别域的大次伽罗瓦表示的存在性

On the existence of large degree Galois representations for fields of small discriminant

设$L/K$是数域的Galois扩张.证明了${\rm Gal}(L/K)$的不可约复表示的最大次数的两个下界，其中最大次数的下界是以Artin猜想和广义黎曼假设为条件的.当$[K : \mathbb{Q}]$很小时，并且$L$有小的根判别式时，我们的界是非平凡的，可以概括为这样的域不能是“太阿贝尔的”。"

Conjectures, consequences, and numerical experiments for p-adic Artin L-functions

We conjecture that the p-adic L-function of a non-trivial irreducible even Artin character over a totally real field is non-zero at all non-zero integers. This implies that a conjecture formulated by Coates and Lichtenbaum at negative integers extends in a suitable way to all positive integers. We also state a conjecture that for certain characters the Iwasawa series underlying the p-adic L-series have no multiple roots except for those corresponding to the zero at s=0 of the p-adic L-function. We provide some theoretical evidence for our first conjecture, and prove both conjectures by means o…

李型有限群的低秩群的特征值与导数

Character values and conductors of low-rank groups of Lie type

研究低秩李型有限群特征值的导数性质，证明某些秩1群中导数在单个群元处实现，并在某些准单情况下证明特征值域由单个值生成。

关于多重诱导特征

On multiply induced characters

该研究探讨能同时由多个不同子群诱导复不可约特征的有限群结构，揭示其作为数域伽罗瓦扩张的伽罗瓦群时，对应多个子域共享同一Artin L-函数的结论。

局部伽罗瓦表示与商奇点的质量公式 I：计数函数的比较

Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions

本文研究局部伽罗瓦表示中 Artin 导子与野 Deligne-Mumford 栈 motivic 积分权的关系，并借此证明 McKay 对应的一种形式，将 Bhargava 质量公式与点的 Hilbert 概型联系起来。