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 Local-global principles for Weil–Châtelet divisibility in positive characteristic

2017 ◽  
Vol 163 (2) ◽  
pp. 357-367 ◽  
Author(s):  
BRENDAN CREUTZ ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curve ◽  
Rational Point ◽  
Positive Characteristic ◽  
Number Fields ◽  
Global Field ◽  
Global Fields ◽  
Characteristic 2 ◽  
Global Principles ◽  
Global Principle

AbstractWe extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

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.

scholarly journals The Farey density of norm subgroups of global fields (II)

1977 ◽  
Vol 18 (1) ◽  
pp. 57-67
Author(s):  
S. D. Cohen ◽  
R. W. K. Odoni
Keyword(s):  
Algebraic Number ◽  
Function Fields ◽  
Rational Functions ◽  
Number Fields ◽  
Ground Field ◽  
Algebraic Number Fields ◽  
Global Fields ◽  
Finite Constant

In this paper we shall derive for function fields in one variable over finite constant fields results analogous to [1], where algebraic number fields were considered. The ground field P will be the set of all rational functions in a given transcendent X, with coefficients in k = GF(q), q = pr, p a prime; thus P = k(X).

scholarly journals ON LOGARITHMIC DERIVATIVES OF ZETA FUNCTIONS IN FAMILIES OF GLOBAL FIELDS

2011 ◽  
Vol 07 (08) ◽  
pp. 2139-2156 ◽  
Author(s):  
PHILIPPE LEBACQUE ◽  
ALEXEY ZYKIN
Keyword(s):  
Error Term ◽  
Function Fields ◽  
Zeta Functions ◽  
Number Fields ◽  
Global Fields ◽  
And Function ◽  
Derivatives Of ◽  
Logarithmic Derivatives ◽  
Siegel Theorem

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer–Siegel theorem both for number fields and function fields.

scholarly journals Non-abelian Cohen–Lenstra heuristics over function fields

2017 ◽  
Vol 153 (7) ◽  
pp. 1372-1390 ◽  
Author(s):  
Nigel Boston ◽  
Melanie Matchett Wood
Keyword(s):  
Function Fields ◽  
Quadratic Function ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Quadratic Extensions ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Unique Distribution ◽  
Quadratic Function Fields

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$extensions of imaginary quadratic number fields for$p$an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of$\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$extensions, as$q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

scholarly journals Extensions of number fields defined by cohomology groups

1983 ◽  
Vol 92 ◽  
pp. 179-186 ◽  
Author(s):  
Hans Opolka
Keyword(s):  
Natural Number ◽  
Galois Group ◽  
Cohomology Group ◽  
Algebraic Closure ◽  
Number Fields ◽  
Cohomology Groups ◽  
Absolute Galois Group ◽  
The Absolute

Letkbe a field of characteristic 0, letbe an algebraic closure ofkand denote byGk= G(/k) the absolute Galois group ofk. Suppose that for some natural numbern≥ 3 the cohomology groupHn(Gk) Z) is trivial.

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.

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