P-Adically Projective Groups as Absolute Galois Groups (original) (raw)
Related papers
The Absolute Galois Group of the Field of Totally S-Adic Numbers
Nagoya Mathematical Journal, 2009
For a finite set S of primes of a number field K and for σ1,…, σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1,…,σe in Ktot,S by Ktot,S(σ). We prove that foralmost all σ ∈ Gal(K)e the absolute Galois group of Ktot,S(σ) is the free product of and a free product of local factors over S.
Functiones et Approximatio Commentarii Mathematici, 2012
For a finite set S of local primes of a countable Hilbertian field K and for σ 1 ,. .. , σe ∈ Gal(K) we denote the field of totally S-adic numbers by K tot,S , the fixed field of σ 1 ,. .. , σe in K tot,S by K tot,S (σ), and the maximal Galois extension of K in K tot,S (σ) by K tot,S [σ]. We prove that for almost all σ ∈ Gal(K) e the absolute Galois group of K tot,S [σ] is isomorphic to the free product ofFω and a free product of local factors over S.
3 on the Structure of the Galois Group of the Abelian Closure of a Number Field
2016
Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain some non-trivial information about the structure of the profinite group A, for every number field, by application of results published in our book on class field theory. Résumé.-A partir d'un article de Athanasios Angelakis et Peter Stevenhagen sur la détermination de corps quadratiques imaginaires ayant le même groupe de Galois Abélien absolu A, nous étudions cette propriété pour les corps de nombres quelconques. Nous montrons qu'une telle propriété n'est probablement pas facilement généralisable, en dehors des corps quadratiques imaginaires, en raison d'obstructions p-adiques provenant des unités globales. En se restreignant aux p-sous-groupes de Sylow de A, nous montrons que l'étude correspondante est liée à une généralisation de la notion classique de corps p-rationnels. Cependant, nous obtenons des informations non triviales sur la structure du groupe profini A, pour tout corps de nombres, par application de résultats publiés dans notre livre sur la théorie du corps de classes.
On the structure of the Galois group of the Abelian closure of a number field
Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain (Thms. 2.8, 3.1) non-trivial information about the structure of the profinite group A, for every number field, by application of results of our book on class field theory.
Approximating absolute Galois groups
Journal of Pure and Applied Algebra, 2022
In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.
On the descending central sequence of absolute Galois groups
arXiv (Cornell University), 2008
Let p be an odd prime number and F a field containing a primitive pth root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group G F of F. Namely, the third subgroup G (3) F in the descending p-central sequence of G F is the intersection of all open normal subgroups N such that G F /N is 1, Z/p 2 , or the extra-special group M p 3 of order p 3 and exponent p 2. Determining the profinite groups which are realizable as absolute Galois groups of fields is a major open problem in Galois theory. Our Main Theorem appears to be simple yet powerful restriction on the possible structure of such groups, and on their quotients G F /G
2007
On Galois representations of local fields with imperfect residue fields By Kazuma MORITA * Let K be a complete discrete valuation field of characteristic 0 with residue field k of characteristic p>0 such that [k : k^{p}]=p^{e}<+\infty. Let V be a p-adic representation of the absolute Galois group G_{ K} =\mat hrm{ G} \mat hrm{ a} 1(\overl i ne{ K} /K) where we fix an algebraic closure \ov er l i ne{ K} of K. When the residue field k is perfect (i.e. e=0), Berger has proved a conjecture of Fontaine (Conjecture 1.1. below) which claims that, if V is a de Rham representation of G_{K}, V becomes a potentially semi-stable representation of G_{K} (See Theorem 1.2.) Here, we generalize this result to the case when the residue field k is not necessarily perfect. For this, we prove some results on p-adic representations in the imperfect residue field case (see Theorem 1.3.) which are obtained by using the recent theory of p-adic differential modules and deduce this generalization of the result of Berger as a corollary. (See Theorem 1.4.) In this survey article, we first state the results in Section 1. In Section 2, we review the property of the p-adic periods ring B_ { \ ma t h r m{ d } \ ma t h r m{ R} }. Then, in Section 3 and Section 4, we give a sketch of the proof of Theorem 1.3, §1. Results Let K, k, G_{K} and V be as above. Fontaine, Hyodo, Kato and Tsuzuki define the p-adic periods rings (associated to K) which are equipped with the continuous action of G_{K} .