Jan Minac, Andrew Schultz, John Swallow
We establish automatic realizations of Galois groups among groups M\rtimes G, where G is a cyclic group of order p^n for a prime p and M is a quotient of the group ring Fp[G].
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This is a report of a talk given at the Oberwolfach workshop on "cohomology of finite groups: Interactions and applications" which was held during July 25th - July 31st, 2010. It is an announcement of some of the results (with motivation) and their applications from the paper "Quotients of absolute Galois groups which determine the entire Galois cohomology"; see arXiv:0905.1364.
Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F)]-modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.
Galois module structure of cyclic extensions of local Fields of characteristic zero
本文描述了特征零局部域循环扩张的精确Galois模结构及通常作为其直接因子的参数化空间Wa,b,m,n。
Locally cyclic extensions with Galois group $GL_2(p)$
本研究利用椭圆曲线的伽罗瓦表示，构造了所有分解群均为循环群、伽罗瓦群为$GL_2(p)$的有理数域伽罗瓦扩张，首次实现对所有素数p的这类构造。
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.
In the mid-1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F.
We observe that some basic but fundamental constructions in Galois theory can be used to obtain some interesting restrictions on the structure of Galois groups of maximal $p$-extensions of fields containing a primitive $p$th root of unity. This is an extension of some significant ideas of Demushkin, Labute and Serre from local fields to all fields containing a primitive $p$th root of unity. Our techniques use certain natural simple Galois extensions together with some considerations in Galois cohomology and Massey products.
Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to sufficient conditions for the ring of integers $O_L$ to be free of rank 1 over its associated order $A_0$, and to stricter conditions which imply that $A_0$ is a Hopf order in the group ring $K[C_{p^n}]$.
In this paper we concentrate on the relations between the structure of small Galois groups, arithmetic of fields, Bloch-Kato conjecture, and Galois groups of maximal pro-$p$-quotients of absolute Galois groups.
Construction of a cyclic $p$-extension of number fields whose unit group has prescribed Galois module structure
本文证明对任意有限循环p-群G，每个无Z_p-挠的有限生成Z_p[G]-模都可在相差自由直和项的意义下实现为某数域G-扩张的单位群，从而刻画了单位群的伽罗瓦模结构。
Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $\text{Gal}(K/F)$. For fixed values of $m$ and $n$, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provid…
We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first classify all maximal p-elementary abelian-by-order p quotients of such G_F. In the case p>2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. We then derive analog…
Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the number of elliptic curves over ${\bf Q}$ with at least $i$ pairs of Galois-stable cyclic subgroups of order 4, and height at most $X$. In this article we show that $N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$. We also show, as $X\to \infty$, that $N_2(X)=c_{2,1}X^{1/6}+o(X^{1/12})$, the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann ze…
In cyclic, degree 8 extensions of algebraic number fields $N/K$, ambiguous ideals in N are canonical $\mathbb{Z}[C_8]$-modules. Their $\mathbb{Z}[C_8]$-structure is determined here. It is described in terms of indecomposable modules and determined by ramification invariants. Although infinitely many indecomposable $\mathbb{Z}[C_8]$-modules are available (classification by Yakovlev), only 23 appear.
We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first classify all maximal p-elementary abelian-by-order p quotients of such G_F. In the case p>2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. We then derive analog…
Recently, much work has been done to investigate Galois module structure of local field extensions, particularly through the use of Galois scaffolds. Given a totally ramified $p$-extension of local fields $L/K$, a Galois Scaffold gives us a $K$-basis for $K[G]$ whose effect on the valuation of elements of $L$ is easy to determine. In 2013, N.P. Byott and G.G. Elder gave sufficient conditions for the existence of Galois scaffolds for cyclic extensions of degree $p^2$ in characteristic $p$. We take their work and adapt it to cyclic extensions of degree $p^2$ in characteristic $0$.
Let p>2 be prime, and let n,m be positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root of unity, we show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p^{n+1} extension E'/F, we give a more refined F_p[Gal(E/F)]-decomposition of H^m(G_E,mu_p).
Let p be a prime and suppose that K/F is a cyclic extension of degree p^n with group G. Let J be the F_pG-module K^*/K^{*p} of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p^i+1 for some 0<=i<n. We show that for all primes p and all 0<=i<n, there exists a field extension K/F with a summand of dimension p^i+1.
Galois module structure of algebraic integers of cyclic cubic fields
研究通过根式通用循环三次多项式确定三次循环域中整数环的伽罗瓦模结构，并显式给出其生成元。
Wild Galois representations: elliptic curves with wild cyclic reduction
本文完成了剩余特征为2或3且曲线在分歧次数可被p整除的扩张上获得良好约化的所有剩余wild情形下椭圆曲线Galois表示的显式计算。
For prime power $q=p^d$ and a field $F$ containing a root of unity of order $q$ we show that the Galois cohomology ring $H^*(G_F,\dbZ/q)$ is determined by a quotient $G_F^{[3]}$ of the absolute Galois group $G_F$ related to its descending $q$-central sequence. Conversely, we show that $G_F^{[3]}$ is determined by the lower cohomology of $G_F$. This is used to give new examples of pro-$p$ groups which do not occur as absolute Galois groups of fields.
An infinite family of pairs of distinct quartic Galois CM-fields with the same discriminant and regulator
本文构造了无限多个具有相同判别式和调节数的虚二元四次域和虚循环四次域的配对，并给出了具有相同判别式、调节数和类数的示例。
We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.
We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realiza…
For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.
Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1 + r_2 - 1$ where $r_1$ and $r_2$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in a finite cover of this torus with respect to a natural partial ordering induced by Hilbert's Theorem 90.
Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete charac…
The classifying element for quotients of Fermat curves
该研究针对射影直线上在0、1、∞三点分歧的循环伽罗瓦覆叠C，在温和分歧条件下，借助适配绝对伽罗瓦群作用研究的基，确定了其基本群下中心序列的分次李代数结构。
A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper, the authors observed that in a particular example, two total masses coming from two different weightings are dual to each other. We discuss the problem how general such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincaré duality for stringy invariants. We also exhibit several examples.
In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois groups have been regarded as amongst the most mysterious objects in number theory. Very little has hitherto been discovered regarding them despite their importance in studying p-adic Galois representations unramified at p. The conjectures of Fontaine-Mazur say that they should have no p-adic analytic quotients (equivalently, the images of the Galois representa…
We show that finite Galois extensions with cyclic Galois group are radical.
We present a simple method for the construction of polynomials with cyclic Galois groups, hoping to encourage a reader with some background in algebra to make computations of his/her own.
This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic mo…
We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{\mathfrak{p}}$ having residue field with $q= p^f$ elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when $q> n^2+n$. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed $n$, almost all Galois groups are cyclic in the limit $q \to \infty$. In particular, we show that the Galois groups are cyclic with p…
We prove two "large images" results for the Galois representations attached to a degree $d$ Q-curve $E$ over a quadratic field $K$: if $K$ is arbitrary, we prove maximality of the image for every prime $p >13$ not dividing $d$, provided that $d$ is divisible by $q$ (but $d \neq q$) with $q=2$ or 3 or 5 or 7 or 13. If $K$ is real we prove maximality of the image for every odd prime $p$ not dividing $d D$, where $D = \disc(K)$, provided that $E$ is a semistable Q-curve. In both cases we make the (standard) assumptions that $E$ does not have potentially good reduction at all primes $p \nmid 6$ an…
On The Cyclicity of Algebraic Lattices
本文研究代数格的循环性，通过Minkowski嵌入在伽罗瓦数域中建立循环格的充要条件，并给出理想格循环性的判定条件。
Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations themselves were introduced in a previous paper. In the present paper we deal with the solutions of these equations as well as with the differential Galois groups attached to the solutions.
We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.
Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensions
本文给出Tate上同调中杯积的非齐次上链公式，并用于计算分裂于循环扩张的局部域上环面的一阶上同调类的显式上循环。
On cubic polynomials with the cyclic Galois group
本文研究具有循环伽罗瓦群的三次多项式，通过构造二次多项式对根的循环置换引入耦合、相邻类及超类概念，并给出超类的完整描述。
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 tamely ramified infinite Galois extensions
本文通过引入稳定惯性生成pro-p群的概念，刻画并构造了数域最大驯顺分歧扩张的伽罗瓦群的有限生成pro-p商群，并给出了可实现与不可实现商群的分类结果。
Local Minkowski units in non-abelian extensions with cyclic Sylow $p$-subgroups
本文针对伽罗瓦群为非p群与循环p群直积的伽罗瓦扩张，建立了p处局部Minkowski单位的存在准则，应用该准则构造了满足条件的非阿贝尔扩张，分析了相关Iwasawa模结构并推广了研究范围。
Information about the absolute Galois group $G_K$ of a number field $K$ is encoded in how it acts on the étale fundamental group $π$ of a curve $X$ defined over $K$. In the case that $K=\mathbb{Q}(ζ_n)$ is the cyclotomic field and $X$ is the Fermat curve of degree $n \geq 3$, Anderson determined the action of $G_K$ on the étale homology with coefficients in $\mathbb{Z}/n \mathbb{Z}$.The étale homology is the first quotient in the lower central series of the étale fundamental group.In this paper, we determine the structure of the graded Lie algebra for $π$. As a consequence, this determines the…
In this paper, we study the Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We will show that under appropriate congruence condition, if the dual Selmer group of one satisfies the $\M_H(G)$-property, so will the other. In the event that the $\M_H(G)$-property holds, and assuming certain further hypothesis on the decomposition of primes in the $p$-adic Lie extension, we compare the ranks of the $π$-free quotient of the two dual Selmer groups. We then apply our results to compare the characteristic elements attached to the Selmer groups. We also s…
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。
We investigate the Galois cohomology of finitely generated maximal pro-$p$ quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with $p$-Zassenhaus filtration on the group. This paper also contains a survey of Koszul propert…
Let $φ_p(z)=(z-1)^p+2-ζ_p$, where $ζ_p\in\bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate $φ_p^n(z)$ has Galois group $[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.
Stabilization on ideal class groups in potential cyclic towers
本文研究数域的势循环$p$塔结构，将岩泽理论中Fukuda的循环$p$塔$p$类群稳定性结果、以及$\u2124_p$扩张的岩泽类数公式，分别推广至势循环$p$塔和势$\u2124_p$扩张情形。
Hopf--Galois structures of cyclic type on parallel extensions of prime power degree
本文研究了当扩展L/K的度数为素数幂且类型为循环时，平行扩展是否也具有相同Hopf-Galois结构的问题，采用群论方法并结合Greither-Pareigis和Byott的工作予以解决。
We obtain strong information on the asymptotic behaviour of the counting function for nilpotent Galois extensions with bounded discriminant of arbitrary number fields. This extends previous investigations for the case of abelian groups. In particular, the result confirms a conjecture by the second author on this function for arbitrary groups in the nilpotent case. We further prove compatibility of the conjecture with taking wreath products with the cyclic group of order 2 and give examples in degree up to 8.
$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $Γ$ are groups of order $p^{2} q$ with non-is…
We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.
We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the $L$-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients.
Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring $H^*(G_F,\mathbb{Z}/q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E$ of $F$ such that $\mathrm{Gal}(E/F)$ is isomorphic to $\{1\}$, $\mathbb{Z}/p$ or to the nonabelian g…
A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formu…
Mod p Monodromy of Cyclic Covers of the Projective Line
本文证明了对于模p系数的项目线循环覆盖的monodromy的大monodromy定理，扩展了Achter和Pries关于二阶和三阶循环覆盖的结果，采用积分上同调的证明方法。
Motivic Galois theory for one-loop Feynman integrals in momentum space
本文提出了一种动机构建一阶费曼积分在动量空间中的理论，通过关联通用动量学的动机局部系统来处理带有切口的图，并计算了这些系统的重量分量，这些分量是与最大切口商图相关的二次Artin动机的塔扭曲。
We examine in detail the stable reduction of Galois covers of the projective line over a complete discrete valuation field of mixed characteristic (0, p), where G has a cyclic p-Sylow subgroup of order p^n. If G is further assumed to be p-solvable (i.e., G has no nonabelian simple composition factors with order divisible by p), we obtain the following consequence: Suppose f: Y --> P^1 is a three-point G-Galois cover defined over the complex numbers. Then the nth higher ramification groups above p for the upper numbering of the (Galois closure of the) extension K/Q vanish, where K is the field …
In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group $G$ is an almost simple group in which the socle $A$ has prime index $p$. Each Hopf-Galois structure is associated to a group $N$ of the same order as $G$. We give necessary criteria on these $N$ in terms of their group-theoretic properties, and determine the number of Hopf-Galois structures associated to $A\times C_p$, where $C_p$ is the cyclic group of order $p$.
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