Independence of ℓ \ell -adic representations of geometric Galois groups

2018 ◽  
Vol 2018 (736) ◽  
pp. 69-93 ◽  
Author(s):  
Gebhard Böckle ◽  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Field Extension ◽  
Galois Groups ◽  
Closed Field ◽  
Absolute Galois Group ◽  
Arbitrary Characteristic ◽  
Cohomology Modules

AbstractLetkbe an algebraically closed field of arbitrary characteristic, let{K/k}be a finitely generated field extension and letXbe a separated scheme of finite type overK. For each prime{\ell}, the absolute Galois group ofKacts on the{\ell}-adic étale cohomology modules ofX. We prove that this family of representations varying over{\ell}is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure ofKof the kernels of the representations for all{\ell}become linearly disjoint over a finite extension ofK. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

scholarly journals Independence of -adic Galois representations over function fields

2013 ◽  
Vol 149 (7) ◽  
pp. 1091-1107 ◽  
Author(s):  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Galois Representations ◽  
Function Fields ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Prime Numbers ◽  
Positive Answer ◽  
Absolute Galois Group ◽  
The Family

AbstractLet$K$be a finitely generated extension of$\mathbb {Q}$. We consider the family of$\ell $-adic representations ($\ell $varies through the set of all prime numbers) of the absolute Galois group of$K$, attached to$\ell $-adic cohomology of a separated scheme of finite type over$K$. We prove that the fields cut out from the algebraic closure of$K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF

2017 ◽  
Vol 234 ◽  
pp. 46-86
Author(s):  
MOSHE JARDEN ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Galois Group ◽  
Abelian Variety ◽  
Haar Measure ◽  
Rational Point ◽  
Algebraic Closure ◽  
Prime Numbers ◽  
Absolute Galois Group ◽  
Finitely Generated ◽  
Fixed Field ◽  
Almost All

Let$K$be a finitely generated extension of$\mathbb{Q}$, and let$A$be a nonzero abelian variety over$K$. Let$\tilde{K}$be the algebraic closure of$K$, and let$\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$be the absolute Galois group of$K$equipped with its Haar measure. For each$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, let$\tilde{K}(\unicode[STIX]{x1D70E})$be the fixed field of$\unicode[STIX]{x1D70E}$in$\tilde{K}$. We prove that for almost all$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, there exist infinitely many prime numbers$l$such that$A$has a nonzero$\tilde{K}(\unicode[STIX]{x1D70E})$-rational point of order$l$. This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.

scholarly journals VARIÉTÉS DE KISIN STRATIFIÉES ET DÉFORMATIONS POTENTIELLEMENT BARSOTTI-TATE

2016 ◽  
Vol 17 (5) ◽  
pp. 1019-1064 ◽  
Author(s):  
Xavier Caruso ◽  
Agnès David ◽  
Ariane Mézard
Keyword(s):  
Galois Group ◽  
Finite Extension ◽  
Finite Union ◽  
Two Dimensional ◽  
Explicit Computation ◽  
Absolute Galois Group ◽  
Dimensional Representation ◽  
The Absolute

Let $F$ be a unramified finite extension of $\mathbb{Q}_{p}$ and $\overline{\unicode[STIX]{x1D70C}}$ be an irreducible mod $p$ two-dimensional representation of the absolute Galois group of $F$. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil–Kisin modules associated to certain families of potentially Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$. We prove that this variety is a finite union of products of $\mathbb{P}^{1}$. Moreover, it appears as an explicit closed connected subvariety of $(\mathbb{P}^{1})^{[F:\mathbb{Q}_{p}]}$. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$.

Decidable regularly closed fields of algebraic numbers

1985 ◽  
Vol 50 (2) ◽  
pp. 468-475 ◽  
Author(s):  
Lou van den Dries ◽  
Rick L. Smith
Keyword(s):  
Galois Group ◽  
Elementary Theory ◽  
Algebraic Closure ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Integer Polynomials ◽  
Forthcoming Book ◽  
Closed Fields ◽  
Free Profinite Group ◽  
Theoretic Argument

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

scholarly journals One-Relator Maximal Pro-p Galois Groups and the Koszulity Conjectures

2020 ◽  
Author(s):  
Claudio Quadrelli
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Group Algebra ◽  
Galois Groups ◽  
Cohomology Algebra ◽  
Augmentation Ideal ◽  
Graded Algebra ◽  
Absolute Galois Group ◽  
Quadratic Algebras ◽  
Elementary Type

Abstract Let p be a prime number and let ${\mathbb{K}}$ be a field containing a root of 1 of order p. If the absolute Galois group $G_{\mathbb{K}}$ satisfies $\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p)\lt\infty$ and $\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for ${\mathbb{K}}$. Also, under the above hypothesis, we show that the $\mathbb{F}_p$-cohomology algebra of $G_{\mathbb{K}}$ is the quadratic dual of the graded algebra ${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{\mathbb{K}}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s elementary type conjecture.

scholarly journals Galois action on homology of generalized Fermat Curves

2020 ◽  
Vol 71 (4) ◽  
pp. 1377-1417
Author(s):  
Aristides Kontogeorgis ◽  
Panagiotis Paramantzoglou
Keyword(s):  
Fundamental Group ◽  
Galois Group ◽  
Projective Line ◽  
Galois Groups ◽  
Absolute Galois Group ◽  
Burau Representation ◽  
Galois Action ◽  
Unified Study ◽  
Fermat Curves ◽  
Ramified Covers

Abstract The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups H. We provide a unified study of the action of both cover Galois group H and the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

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 BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS

2014 ◽  
Vol 2 ◽  
Author(s):  
TOBY GEE ◽  
MARK KISIN
Keyword(s):  
Galois Group ◽  
Finite Extension ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Serre's Conjecture ◽  
Weight Part ◽  
The Absolute ◽  
Serre’S Conjecture

Abstract We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$ , $K$ a finite extension of $\mathbb{Q}_{p}$ , for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

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