Hilbertian fields and Galois representations

Bary-Soroker, Lior; Fehm, Arno; Wiese, Gabor

doi:10.1515/crelle-2013-0116

Cited by 6 publications

(24 citation statements)

References 19 publications

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“…The generalized derived series is defined inductively: D 0 (G) = G and D i+1 (G) = D(D i (G)) for i ≥ 0. Plainly, this is a strictly descending series of closed normal subgroups of G (see [BFW,Lemma 2.1]). Moreover, we define the abeliansimple length of G, denoted by l(G), to be the smallest r ∈ N for which D r (G) = {1}.…”

Section: Groups Of Finite Lengthmentioning

confidence: 99%

“…Proof Set N = U G ⊳ o G (the intersection of all the G-conjugates of U ). By [BFW,Proposition 2.9, Lemma 2.2, Lemma 2.7 (2)]:…”

Section: Groups Of Finite Lengthmentioning

confidence: 99%

“…In the following theorem we utilize the twisted wreath product approach to prove the generalization of [BFW,Theorem 5.7] to arbitrary infinite rank, thus treating the case of infinite rank in Theorem 1.1. Our proof is just a translation to group theory (using the Galois correspondence) of the field theoretic argument given in the proof of [BFW,Theorem 3.2].…”

Section: Groups Of Finite Lengthmentioning

confidence: 99%

“…does not have a proper solution. Towards a contradiction, suppose that there is a proper solution λ : F/N 0 → A wr G 0 G. Applying [BFW,Lemma 2.7 (1)] to F → F/L we see that…”

Section: Groups Of Finite Lengthmentioning

confidence: 99%

“…Proof Let δ : F → R be the quotient map, and let H 0 = δ(H). We may assume, without loss of generality, that N = H F (that is, there are no proper closed F -normal subgroups between N and H) for otherwise we change N accordingly (taking it to be the intersection of all the F -conjugates of H), and the length does not grow in view of [BFW,Lemma 2.7 (1)]. Since l(R) < ∞, there is a maximal k ∈ N for which D k (R) is nontrivial.…”

Section: Groups Of Finite Lengthmentioning

confidence: 99%

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Free subgroups of finitely generated free profinite groups

Shusterman

2016

J. London Math. Soc.

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We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated nonabelian free profinite group is contained in an infinitely generated free profinite subgroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for subgroups to be free profinite. As a result of our work, we are able to address a conjecture of Jarden on the Hilbertianity of fields generated by torsion points of abelian varieties. IntroductionOne of the most celebrated theorems of classical group theory is the theorem of Nielsen and Schreier stating that every subgroup of a free group is free. This theorem has attracted much attention, and proofs from rather diverse mathematical disciplines like abstract group theory, algebraic topology, geometric group theory, and Bass-Serre theory have been given (see [DHLS], [FJ, Proposition 17.5.6 In Field Arithmetic, a central task is to extend the Nielsen-Schreier theorem to the realm of profinite groups. Unfortunately, the 2-Sylow subgroup of a free profinite group has only 2-groups as finite images, so it is not a free profinite group. It is therefore apparent that the theorem does not extend to the profinite setting in the strict sense, and much of the work carried out by Bary-Soroker, Binz, Chatzidakis, 1 Fehm, Gildenhuys, Haran, Harbater, Iwasawa, Jarden, Lubotzky, Lim, Melnikov, Neukirch, Ribes, Steinberg, V.D. Dries, Wenzel, Wiese, Zalesskii and others, produced sufficient conditions for the permanence of profinite freeness in closed subgroups.Besides the intrinsic group theoretic interest of exploring a free profinite group by examining the structure of its closed subgroups (these are the projective objects in the category of profinite groups), extensions of the Nielsen-Schreier theorem to profinite groups prove to be of arithmetical importance for two reasons:• Free profinite groups arise as absolute Galois groups of fields, so any information about a closed subgroup, tells us something about the absolute Galois group of a field extension. In particular, the profinite freeness of a closed subgroup provides us with a solution of the inverse Galois problem over its fixed field in a rather strong sense.• Free profinite groups and absolute Galois groups of Hilbertian fields (see Definition 6) exhibit many similar properties, and the techniques used to study these object have a lot in common ( The definition of the class of groups of fixed abelian-simple length appears in [BFW, Section 2], and is given in the section devoted to these groups in our work (Section 5, Definition 5). The reason to consider this class of groups is twofold: First, this class, being defined by 2 restricting the length of certain normal series,...

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Section: Groups Of Finite Lengthmentioning

confidence: 99%

“…Proof Set N = U G ⊳ o G (the intersection of all the G-conjugates of U ). By [BFW,Proposition 2.9, Lemma 2.2, Lemma 2.7 (2)]:…”

Section: Groups Of Finite Lengthmentioning

confidence: 99%

Section: Groups Of Finite Lengthmentioning

confidence: 99%

“…does not have a proper solution. Towards a contradiction, suppose that there is a proper solution λ : F/N 0 → A wr G 0 G. Applying [BFW,Lemma 2.7 (1)] to F → F/L we see that…”

Section: Groups Of Finite Lengthmentioning

confidence: 99%

Section: Groups Of Finite Lengthmentioning

confidence: 99%

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