On the image of complex conjugation in certain Galois representations
Ana Caraiani, Bao V. Le Hung
我们计算图像的任何选择的复共轭的伽罗瓦表示相关的定期代数尖形自守表示和扭转类的局部对称空间的上同调$GL_n$在一个完全真实的领域$F$。
We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $GL_n$ over a totally real field $F$.
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Eigenvarieties for classical groups and complex conjugations in Galois representations
本文的目的是删除理查德泰勒定理中的不可约性假设描述图像的复杂共轭$p$ -进伽罗瓦表示与正规的，代数的，本质上是自对偶的，尖点自守表示$\GL_{2n+1}$在一个全实数域$F$。我们还将其扩展到$\GL_{2n}/F$的乘法字符是“奇”的表示的情况。我们使用$p$ -进变形参数，更精确地说，我们证明了辛，甚至正交群的本征簇，有“许多”点对应于（准）不可约伽罗瓦表示。最近的工作詹姆斯阿瑟描述的自守谱为这些群体是用来定义这些伽罗瓦表示，也转移自对偶自守表示的一般线性群，这些典型群。
Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations
这篇文章是一系列的三篇文章的第一部分关于兼容系统的辛伽罗瓦表示和应用程序的逆伽罗瓦问题。在第一部分中，我们确定了一个满足某些自然条件的辛群表示在其上的投影可以被定义的最小域。答案只取决于内心的扭曲。我们将其应用到一个兼容系统的辛伽罗瓦表示满足一些温和的假设，并获得精确的信息，他们的投影图像的系统的几乎所有成员，在巨大的残留图像的假设下，我们的意思是，一个辛群的全维的素域包含共轭。最后，我们得到一个应用程序的逆伽罗瓦问题。
The image of the adelic Galois representation of an elliptic curve with complex multiplication
本文针对具有复乘且j-不变量非0或1728的椭圆曲线E/Q，描述并实现了一种高效算法，计算其adelic Galois表示在GL(2, Ź)中的像（共轭意义下）。
Let $F$ be a CM field with totally real subfield $F^+$ and let $π$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of) discrete series. We attach to $π$ a Galois representation $R_π:\mathrm{Gal}(\overline F/ F^+)\to{}^C\mathrm{U}(a,b)(\overline{\mathbf Q}_\ell)$ such that, for any complex conjugation element $c$, $R_π(c)$ is as predicted by the Buzzard--Gee conjecture. As a corollary, we deduce that the Galois representations attached to certain irregular, $C$-algebraic (esse…
Proofs that an arbitrary field has a separable closure are necessarily non-constructive, and separable closures are unique only up to non-canonical isomorphism. This means that the absolute Galois group of a field is defined only up to inner automorphism. Here we construct a profinite algebraic group which is an inner form of the absolute Galois group. Our construction uses no form of the axiom of choice, and the group is defined up to canonical isomorphism. We also show that the Frobenius associated with a prime of a number field unramified in an extension, which is classically defined only u…
Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image
这篇文章是一系列的三篇文章的第二部分关于兼容系统的辛伽罗瓦表示和应用的逆伽罗瓦问题。这一部分讨论了具有巨大余象的辛伽罗瓦表示，我们的意思是，在素域上的全维辛群包含到共轭。一个关键的成分是辛表示的分类，其图像包含一个非平凡的transvection：这些分为三个非常简单的描述类，可约的，诱导的和那些巨大的图像。利用Khare，Larsen和Savin的（n，p）-群的思想，我们给出了系数在有限域上的辛伽罗瓦表示具有巨象的简单条件.最后，我们把这个分类结果与第一部分的主要结果结合起来，得到了它在Galois逆问题中的一个加强应用。
In this paper, we will calculate the number of Galois extensions of local fields with Galois group A_n and S_n.
The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let $ρ_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\t…
The non-existence of certain mod 2 Galois representations of some small quadratic fields
对少数二次域，证明了2次模2不可约的模2连续伽罗瓦表示不存在于2外。
Construction Of Non-Odd Galois Representations With Large Image
该研究构造了满足特定条件的非GL_r-奇性伽罗瓦表示，其像与GL_r(Z_p)可公度，复共轭迹绝对值符合要求，还得到了正交群的类似结果。
A control theorem for the images of Galois actions on certain infinite families of modular forms
在这封信中，我们表明，对于某些无限家庭的模形式的增长水平是可能有一个控制结果的例外素数所附的伽罗瓦表示。作为应用，证明了任意大指数的有限域上线性群的WieseonGalois实现的一个结果的统一形式.
Computing the number of certain Galois representations mod $p$
利用$\qu$的模$p$伽罗瓦表示与Serre猜想建立的模$p$模形式之间的联系，对每个素数$p\leq 1999$，计算了$\fbar$上二维向量空间上$\qu$的连续伽罗瓦表示的同构类个数的下界，这些同构类在$p$外不可约、奇且不可分歧.
Finding Galois representations corresponding to certain Hecke eigenclasses
1992年，阿夫纳·阿什和马克·麦康奈尔提出了计算证据，证明了三维伽罗瓦表示和某些算术上同调类之间的联系。对于某些例子，他们无法确定所附的代表性。对于几个Hecke本征类（包括一个灰和麦康奈尔没有找到的伽罗瓦表示），我们发现一个伽罗瓦表示似乎是附加的，并显示出强有力的证据，这种表示的唯一性。我们使用的技术，以找到定义多项式的伽罗瓦表示包括有针对性的亨特搜索，类域理论，和椭圆曲线。
Non-existence of certain Galois representations with a uniform tame inertia weight
本文证明了数域的某些半稳定伽罗瓦表示的不存在性。我们的结论可应用于某些几何问题。例如，我们证明了Rasmussen和Tamagawa猜想的一个特殊情况，与有限的集合的同构类的阿贝尔簇约束素幂挠。
In this article, we use deformation theory of Galois representations valued in the symplectic group of degree four to prove a freeness result for the cohomology of certain quaternionic unitary Shimura variety over the universal deformation ring for certain type of residual representation satisfying a property called rigidity. This result plays an important role in the proof of the arithmetic level raising theorem for the symplectic similitude group of degree four over the field of rational numbers by the author.
The Nonexistence of Certain Representations of the Absolute Galois Group of Quadratic Fields
对于一个二次域K，我们研究了K的绝对伽罗瓦群的连续模p表示，这些绝对群是从p到无穷远的非分歧群。我们证明了对于某些对（K，p），不存在这样的不可约表示.我们还列出了一些存在这种不可约表示的虚二次域。作为一个应用，我们看看椭圆曲线与好约化远离2超过二次域。
In this paper we discuss applications of our earlier work in studying certain Galois groups and splitting fields of rational functions in $\mathbb Q\left(X_0(N)\right)$ using Hilbert's irreducibility theorem and modular forms. We also consider computational aspect of the problem using MAGMA and SAGE.
Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with residue field $\mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $\mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $\mathcal{G}(R)$ with full residual image $\mathcal{G}(\mathbb{F})$ is a conjugate of a group $\mathcal{G}(A)$ for $A\subset R$ a closed subring that is local and has residue field $\mathbb{F}$ …
Local-Global principles for certain images of Galois representations
设$K$是数域，$E/K$是椭圆曲线，其模$\ell$伽罗瓦表示局部有象包含在群$G$中，直到共轭.在$G$是Cartan子群或Cartan子群的正规化子的情况下，我们对全局伽罗瓦表示的可能图像进行分类。当$K = \mathbf{Q}$时，我们在$G$是分裂Cartan和$\ell = 13$的正规化子的情况下，推导出局部-整体原理的一个反例.特别地，在$\mathbf{Q}$上至少有三条椭圆曲线（直到twist），其Galois的mod $13$像局部包含在分裂Cartan的正规化子中，但其全局像不包含在分裂Cartan的正规化子中。
Potential automorphy for certain Galois representations to GL_2n
基于Clozel、Harris、Shepherd-Barron和Taylor的工作，本文证明了在对基域进行适当大的全实扩张后，某些伽罗瓦表示会变得自守。这里的主要创新之处是，结果适用于伽罗瓦表示到GL_{2n}，以前的工作处理的表示GSp_n。主要的技术是考虑的上同调的Dwork超曲面，特别是，在自然的群作用下的不变量，这一上同调的其他部分。
We study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes. These representations are symplectic of dimension $4$. Following a method of Dieulefait, we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is "full" for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases.
In this paper we study the images of certain families $\{ρ_{π,\ell} \}_\ell$ of $G_2$-valued Galois representations of $\mbox{Gal}(\overline{F}/F)$ associated to $L$-algebraic regular, self-dual, cuspidal automorphic representations $π$ of $\mbox{GL}_7(\mathbb{A}_F)$, where $F$ is a totally real field. In particular, we prove that, under certain automorphic conditions, the images of the residual representations $\overlineρ_{π,\ell}$ are as large as possible for infinitely many primes $\ell$. Moreover, we apply our result to some examples constructed by Chenevier, Renard and Taïbi.
On the dynamical Galois group of certain affine polynomials in positive characteristic
本文在正特征下研究了特定多项式动力学伽罗瓦群的判据，提出了一种在特征2情况下与Stoll判据相关的动态判据方法。
For every prime $p\geq 5$ for which a certain condition on the class group $\text{Cl}(\mathbb{Q}(μ_p))$ is satisfied, we construct a $p$-adic analytic Galois extension of the infinite cyclotomic extension $\mathbb{Q}(μ_{p^{\infty}})$ with some special ramification properties. In greater detail, this extension is unramified at primes above $p$ and tamely ramified above finitely many rational primes and is isomorphic to a finite index subgroup of $\text{SL}_2(\mathbb{Z}_p)$ which contains the principal congruence subgroup. For the primes $107,139,271$ and $379$ such extensions were first constru…
On the potential automorphy of certain odd-dimensional Galois representations
在前一篇文章中，继Harris，Shepherd-Barron和Taylor的工作之后，利用Clozel，Harris和Taylor的提升定理，建立了某些伽罗瓦表示对GL_n（n为偶数）的势自同构.本文将这些结果推广到n=3和n=5，并有条件地推广到所有其它的奇数n.关键的额外工具是必要的结果，使自同构或潜在的自同构对称权力的椭圆曲线，最显着的盖尔伯特，雅凯，金，沙希迪和哈里斯。
Given two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's u…
Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{P}Ω^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PO}_{2m}^\pm(\mathbb{F}_{\ell^s})$ or $\mbox{PGO}^\pm_{2m}(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for infinitely many integers $s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorp…
In this article we study local constancy of the mod $p$ reduction of certain $2$-dimensional crystalline representations of $\mathrm{Gal}\left(\bar{\mathbb{Q}}_p/\mathbb{Q}_p\right)$ using the mod $p$ local Langlands correspondence. We prove local constancy in the weight space by giving an explicit lower bound on the local constancy radius centered around weights going up to $(p-1)^{2} +3$ and the slope fixed in $(0, \ p-1)$ satisfying certain constraints. We establish the lower bound by determining explicitly the mod $p$ reductions at nearby weights and applying a local constancy result of Be…
Let $p$ be an odd prime and $q$ a power of $p$. We examine the deformation theory of reducible and indecomposable Galois representations $\barρ:G_{\mathbb{Q}}\rightarrow \text{GSp}_{2n}(\mathbb{F}_q)$ that are unramified outside a finite set of primes $S$ and whose image lies in a Borel subgroup. We show that under some additional hypotheses, such representations have geometric lifts to the Witt vectors $\text{W}(\mathbb{F}_q)$. The main theorem extends that of Hamblen and Ramakrishna in which the $n=1$ case was treated.
In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending these to the maximal totally real field of the reflex field. We prove the existence of such descent for many Shimura varieties whose associated adjoint group has certain factors of type A or D. This includes a large family of Shimura varieties of abelian type. Our considerations and constructions are carried out purely at the level of Shimura data and group theory.
Potential automorphy of certain non self-dual 3-dimensional Galois representations
本文利用潜在自守性与自守提升的最新进展，证明了 van Geemen–Top 曲面族所对应的三维伽罗瓦表示对所有非零整数参数 z 均潜在自守，从而得到其 L 函数的解析延拓与函数方程。
Modularity of certain mod $p^n$ Galois representations
本文针对满足奇且绝对不可约剩余表示的二维模 $p^n$ 伽罗瓦表示，在合适假设下通过形变理论构造特征0提升，结合模性提升结果证明其模性，并给出非分歧情形下对应模形式权的应用。
Determining class groups as Galois modules up to equivalence for some nonabelian extensions
本文将负类群的伽罗瓦模结构研究从阿贝尔CM扩张推广到亚循环非阿贝尔扩张，并在作者早先引入的模等价关系下完整刻画了可作为负类群出现的伽罗瓦模。
A positive density of elliptic curves are diophantine stable in certain Galois extensions
本文证明，对于循环p-扩张L/Q（p=3或5），存在正密度的椭圆曲线E（按高度排序）在L中具有丢番图稳定性。
Invariance Galoisienne des zéros centraux de fonctions L
研究证明了L函数在1/2处的消去性质及其epsilon因子的伽罗瓦不变性，通过构造离散自动形式表示和处理intertwining运算符的全纯性。
On intersective polynomials with non-solvable Galois group
研究具有不可解伽罗瓦群（如射影线性群、仿射线性群、仿射辛群等）的相交多项式的存在性，给出相关理论结果，并构造了部分非可解群的显式相交多项式族。
Modularity of p-adic Galois representations via p-adic approximations
我们给出了一个直接的方法来恢复Wiles和Tayor关于Q的绝对伽罗瓦群的某些二维p-adic表示的模性的一些结果。
Congruence covers of triangular modular curves and their Galois groups
本文计算了某些双曲三角形群的同余覆盖的伽罗瓦群。作为这种计算的结果，推导出相应曲线的属。
Galois groups of certain even octic polynomials
本文研究了某些偶八次多项式的伽罗瓦群，通过算术条件确定其因式分解模式，并完全分类了六个可能的伽罗瓦群。
In the 80's Aschbacher classified the maximal subgroups of almost all of the finite almost simple classical groups. Essentially, this classification divide these subgroups into two types. The first of these consist roughly of subgroups that preserve some kind of geometric structure, so they are commonly called subgroups of geometric type. In this paper we will prove the existence of infinitely many compatible systems $\{ ρ_\ell \}_\ell$ of $n$-dimensional Galois representations associated to regular algebraic, essentially self-dual, cuspidal automorphic representations of $\mbox{GL}_n(\mathbb{…
Galois Groups Over Nonrigid Fields
令$F$是具有特征$\neq 2$的字段。我们证明了$F$是非刚性场当且仅当某些小的2-群作为伽罗瓦群出现在$F$上.这些结果为$F$上的伽罗瓦群提供了新的“自动可实现性”结果.证明了当$F$为非刚性域时，$F$的两个特殊亚交换2-扩张的不等式.利用已知的结果之间的连接刚性和存在的某些估值，我们得到伽罗瓦理论的标准，这些估值的存在。
Automorphic Galois representations and the inverse Galois problem
解决Q上的逆伽罗瓦问题的策略包括利用附加到某些自守形式的伽罗瓦表示的知识。更确切地说，如果仔细选择这样的形式，它们提供了满足某些期望性质的伽罗瓦表示的兼容系统，例如反映在系统成员的图像上的性质。在这篇文章中，我们调查使用这种策略获得的一些结果。
Deformations of Galois representations arising from degenerate extensions
Nigel Boston和Barry Mazur已经证明了如何确定有理数的某些S_3-扩张的自然子空间，他们称之为“一般的”。我们推广他们的一些结果到另一类扩张，称为“退化”。
In this article we study the Galois group of field generated by division points of special class of formal group laws and prove an equivalent condition for the group to be abelian. Further, we explore relations between the endomorphism ring of a formal group and the Galois group of field generated by division points.
Galois descents of certain multiple polylogarithms
该研究证明了K. C. Au关于多重对数和割圆级下降的猜想，还确认了Broadhurst关于偶权荣誉多重zeta值的猜想。
The spin of prime ideals and level-raising of even Galois representations
本文通过扩展素理想旋量概念，证明短特征和猜想与偶Galois表示提升的素数密度关系。
Gerth's heuristics for a family of quadratic extensions of certain Galois number fields
本文推广了Fouvry和Kluners于2007年证明的Gerth猜想，针对某些Galois数域的一族无限二次扩张，获得了类群相关量平均值的下界。
A level raising result for modular Galois representations modulo prime powers
在这项工作中，我们提供了一个$\mod λ^n$模块化伽罗瓦表示的水平提升定理。当满足$p$上的某个局部条件时，它允许人们看到这样一个伽罗瓦表示，它是$N$级的模，权重为2，平凡Neutralypus是$Np$级的模，对于一个与$N$互质的素数$p$。它是Ribet关于$\mod \ell$伽罗瓦表示的一个结果的推广.
On Galois realizations of special linear groups
本文通过研究中间卷积构造的平展层的行列式，给出在有理数域上将特殊线性群正则实现为伽罗瓦群的方法。
Remarks on methods of Fontaine and Faltings
本文证明了绝对伽罗瓦群的某些不可约二维表示的不存在性。这样的结果是预测塞尔的猜想，我们使用Fontaine的方法来验证这些预测在少数情况下。我们还利用Faltings方法研究了伽罗瓦表示的有限性问题。关键词：塞尔猜想，方丹-马祖尔图，伽罗瓦表示。
The Artin Conjecture for some S_5-extensions
我们建立了Artin猜想的一些新的例子。我们的结果适用于伽罗瓦表示$\Q$与图像$S_5$满足一定的局部假设，其中最重要的是，复共轭共轭是共轭的$(12)(34)$。事实上，我们证明了朗兰兹提出的更强有力的主张，即这些表示是自守的。对于4维和6维的不可约表示，我们的结果来自已知的2维情况下的阿廷猜想（证明佐佐木）以及函子性质的Asai转移证明Ramakrishnan。对于维数为5的不可约表示，我们遇到了从与伽罗瓦表示相容的二次扩张下降自守形式的问题。这个问题部分地通过用$S_5$的某个中心扩展的四维表示来解决。在这种情况下，我们的模块性结果是偶然的，在临界带的实直线上的某个戴德金ζ函数的非零。一个结果布克表明，一个可以（原则上）明确验证这种非零，并与布克的帮助下，我们给出了一个例子，验证阿丁猜想的代表来自（伽罗瓦闭包）的五次领域$K$的最小判别式（1609）。
We realize Frobenius conjugacy classes in Galois groups of certain $q$-polynomials over $\mathbb{F}_q(t)$ using specific degree 1 ideals. We combine this with methods from elementary linear algebra and group theory to realize transvections in some linear Galois groups. This enables the Galois group to be identified as a known classical group in several reasonably general cases.
Let $ρ$ be a two-dimensional even Galois representation which is induced from a character $χ$ of odd order of the absolute Galois group of a real quadratic field. After imposing some additional conditions on $χ$, we attach $ρ$ to a Hecke eigenclass in the cohomology of ${\rm GL}(2,\mathbb Z)$ with coefficients in a certain infinite-dimensional vector space over a field of characteristic not equal to 2.
Galois invariants of K_1-groups of Iwasawa algebras
本文研究了系数在整数环的某些子环上的群代数K_1的Galois下降，它是Q_p的一个代数闭包的完备化.
Framed Deformation of Galois Representation
研究了二维伽罗瓦表示的框架变形，其剩余表示限制于分解群是纯量，并在一定情况下建立了模提升定理.然后，我们证明了一个家庭版本的结果，并使用它来确定结构的变形环上的特征零域。这些结果可应用于p-adic L-函数例外零点的研究。
Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of K_n, which is the Galois extension field of K fixed by the stabilizer of T/p^n T. By applying this result, we prove an asymptotic inequality which describes an explicit lower bound of the class numbers along a tower K(A[p^\infty])/K for a given abelian variety A with certain conditions in term…
Galois groups of Mori trinomials and hyperelliptic curves with big monodromy
本文计算了S. S. S引入的有理数域上一类多项式的伽罗瓦群。Mori和研究相应的超椭圆雅可比算子的单值性。
Arboreal Galois groups of rational maps with nonreal Julia sets
本文研究具有非实 Julia 集的有理映射的树状 Galois 群，证明相关映射及与椭圆曲线倍乘映射关联的 Lattès 映射具有非交换树状 Galois 群，并探讨实系数多项式何时具有实 Julia 集。
We prove that the arboreal Galois representations attached to certain unicritical polynomials have finite index in an infinite wreath product of cyclic groups, and we prove surjectivity for some small degree examples, including a new family of quadratic polynomials. To do this, we use a combination of local techniques including the Chabauty-Coleman method and the Mordell-Weil sieve.
On irreducibility of certain low dimensional automorphic Galois representations
本文研究了与GL_n(A_Q)的7或8维正则代数本质自对偶尖点自守表示相关的Galois表示的不可约性，并在特定李类型与Hodge-Tate权重条件下证明了其除有限多个外均不可约。
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