scholarly journals THE DISTRIBUTION OF NUMBER FIELDS WITH WREATH PRODUCTS AS GALOIS GROUPS

2012 ◽  
Vol 08 (03) ◽  
pp. 845-858 ◽  
Author(s):  
JÜRGEN KLÜNERS
Keyword(s):  
Asymptotic Behavior ◽  
Galois Group ◽  
Wreath Product ◽  
Symmetric Groups ◽  
Number Fields ◽  
Wreath Products ◽  
Constant Factor ◽  
Galois Groups ◽  
Mild Conditions ◽  
Counting Functions

Let G be a wreath product of the form C2 ≀ H, where C2 is the cyclic group of order 2. Under mild conditions for H we determine the asymptotic behavior of the counting functions for number fields K/k with Galois group G and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups.

scholarly journals Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
A. Esterov ◽  
L. Lang
Keyword(s):  
Galois Group ◽  
Wreath Product ◽  
Galois Theory ◽  
General System ◽  
Wreath Products ◽  
Galois Groups ◽  
Complex Polynomial ◽  
Polynomial Equations ◽  
Sparse Polynomial ◽  
System Of Polynomial Equations

AbstractWe introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial $$f(x) = c_0 + c_1 x^{d_1} + \cdots + c_k x^{d_k}$$ f ( x ) = c 0 + c 1 x d 1 + ⋯ + c k x d k by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable $$y=x^d$$ y = x d , and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over $$d_k/d$$ d k / d elements and $${\mathbb {Z}}/d{\mathbb {Z}}$$ Z / d Z . We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.

scholarly journals On the minimal ramification problem for ℓ-groups

2010 ◽  
Vol 146 (3) ◽  
pp. 599-606 ◽  
Author(s):  
Hershy Kisilevsky ◽  
Jack Sonn
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Fields ◽  
Symmetric Groups ◽  
Wreath Products ◽  
The Other ◽  
Classical Groups ◽  
Direct Products ◽  
Cyclic Groups ◽  
Other Hand

AbstractLet ℓ be a prime number. It is not known whether every finite ℓ-group of rank n≥1 can be realized as a Galois group over ${\Bbb Q}$ with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite ℓ-groups which contains all the cyclic groups of ℓ-power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow ℓ-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not ℓ. On the other hand, it does not contain all finite ℓ-groups.

MILD PRO-p-GROUPS AS GALOIS GROUPS OVER GLOBAL FIELDS

2009 ◽  
Vol 05 (05) ◽  
pp. 779-795 ◽  
Author(s):  
LANDRY SALLE
Keyword(s):  
Galois Group ◽  
Cohomology Group ◽  
Function Fields ◽  
Number Fields ◽  
Galois Groups ◽  
Global Field ◽  
Global Fields ◽  
Global Principle

This paper is devoted to finding new examples of mild pro-p-groups as Galois groups over global fields, following the work of Labute ([6]). We focus on the Galois group [Formula: see text] of the maximal T-split S-ramified pro-p-extension of a global field k. We first retrieve some facts on presentations of such a group, including a study of the local-global principle for the cohomology group [Formula: see text], then we show separately in the case of function fields and in the case of number fields how it can be used to find some mild pro-p-groups.

scholarly journals ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 273-284 ◽  
Author(s):  
David J. Benson
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Finite Groups ◽  
Cohomology Ring ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Characteristic P ◽  
Residue Classes ◽  
The Difference ◽  
A Finite Group

AbstractLet $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

Totally Real Subfields of p-Adic Fields having the Symmetric Group as Galois Group

1971 ◽  
Vol 14 (3) ◽  
pp. 441-442 ◽  
Author(s):  
Howard Kleiman
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Elementary Proof ◽  
Symmetric Groups ◽  
Rational Numbers ◽  
Galois Groups ◽  
Arbitrary Field ◽  
Totally Real

In this paper, an elementary proof is given of the following proposition:Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.

scholarly journals Galois groups of number fields generated by torsion points of elliptic curves

1986 ◽  
Vol 104 ◽  
pp. 43-53 ◽  
Author(s):  
Kay Wingberg
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Group ◽  
Number Field ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Number Fields ◽  
Galois Groups ◽  
Torsion Points ◽  
Special Case

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).

scholarly journals Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions

2014 ◽  
Vol 10 (08) ◽  
pp. 2045-2095 ◽  
Author(s):  
David Loeffler ◽  
Sarah Livia Zerbes
Keyword(s):  
Galois Group ◽  
Lie Group ◽  
Quadratic Field ◽  
Number Fields ◽  
Galois Groups ◽  
Iwasawa Theory ◽  
Absolute Galois Group ◽  
Abelian Extensions ◽  
Dimension 2 ◽  
Crystalline Representation

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

scholarly journals Constraining Images of Quadratic Arboreal Representations

2020 ◽  
Author(s):  
Andrea Ferraguti ◽  
Carlo Pagano
Keyword(s):  
Galois Group ◽  
Binary Tree ◽  
Number Fields ◽  
Galois Groups ◽  
Finitely Generated ◽  
Quadratic Polynomials ◽  
Global Fields ◽  
Local Class Field Theory ◽  
The One ◽  
Quadratic Case

Abstract In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First, we show that, over global fields, quadratic post-critically finite (PCF) polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes’ conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily PCF, and we combine our results with local class field theory to classify quadratic pairs over ${ {\mathbb{Q}}}$ with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other hand, we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence toward Jones’ finite index conjecture.

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