On unramified Galois extensions constructed using Galois representations

 For a field k, We denote the maximal abelian extension of k by k _ab and (K _ab ^r−1 _ab by k _ab ^r . In this paper, unramified Galois extensions over k _ab ^r are constructed using Galois representations of arbitrary dimension with larger coefficient rings. Received: 31 August 2001 / Revised version: 22 March 2002 Mathematics Subject Classification (2000): 11R21     

Small maximal pro-p Galois groups

For an odd primep we classify the pro-p groups of rank ≤4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of orderp.     

Small maximal pro-p Galois groups

For an odd prime p we classify the pro-p groups of rank ≤ 4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of order p. Received: 2 September 1997     

Deformations of locally abelian Galois representations and unramified extensions

We study the deformation theory of Galois representations whose restriction to every decomposition subgroup is abelian. As an application, we construct unramified non-solvable extensions over the field obtained by adjoining all p-power roots of unity to the field of rational numbers.     

On pro-p analogues of limit groups via extensions of centralizers

We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call ${\mathcal{L}}$ this new class of pro-p groups. We show that the pro-p groups of ${\mathcal{L}}$ have finite cohomological dimension, type FP _?? and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion free nilpotent) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-p group in the class ${\mathcal{L}}$ is either free pro-p or abelian.     

On differential Galois groups of strongly normal extensions

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in , we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.     

On tame pro-p Galois groups over basic {\mathbb{Z}_p}-extensions

For a prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z_p}$ -extension of the rational number field. In this paper, we give a family of S for which the Galois group is a metacyclic pro-p group with an application to Greenberg’s conjecture.     

On cubic Galois field extensions

We study Morton's characterization of cubic Galois extensions F/K by a generic polynomial depending on a single parameter s∈K. We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F. For K=Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.     

Galois module structure of unramified covers

We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).     

Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.     

Galois 2-extensions unramified outside 2

We classify quadratic, biquadratic and degree 4 cyclic 2-rational number fields. We also classify those quadratic number fields which are not 2-rational, but have a degree 2 extension, which is Galois over Q and is 2-rational. In this case we explicitly describe the Galois group of their maximal pro-2 extension unramified outside 2 and infinity using a result of Herfort-Ribes-Zalesskii on virtually free pro-p groups.     

Virtually free pro-p groups

We prove that in the category of pro-p groups any finitely generated group G with a free open subgroup splits either as an amalgamated free product or as an HNN-extension over a finite p-group. From this result we deduce that such a pro-p group is the pro-p completion of a fundamental group of a finite graph of finite p-groups.     

On some pro-p groups from infinite-dimensional Lie theory

We study some pro- \(p\) -groups arising from infinite-dimensional Lie theory. The starting point is incomplete Kac–Moody groups over finite fields. There are various completion procedures always providing locally pro- \(p\) groups. We show topological finite generation for their pro- \(p\) Sylow subgroups in most cases, whatever the (algebraic, geometric or representation-theoretic) completion. This implies abstract simplicity for complete Kac–Moody groups and provides identifications of the pro- \(p\) groups obtained from the same incomplete group. We also discuss the question of (non-)linearity of these pro- \(p\) groups.     

Subgroup properties of pro-p extensions of centralizers

We prove that a finitely generated pro- $p$ group acting on a pro- $p$ tree $T$ with procyclic edge stabilizers is the fundamental pro- $p$ group of a finite graph of pro- $p$ groups with vertex groups being stabilizers of certain vertices of $T$ and edge groups (when non-trivial) being stabilizers of certain edges of $T$ , in the following two situations: (1) the action is $n$ -acylindrical, i.e., any non-identity element fixes not more than $n$ edges; (2) the group $G$ is generated by its vertex stabilizers. This theorem is applied to obtain several results about pro- $p$ groups from the class $\mathcal L $ defined and studied in Kochloukova and Zalesskii (Math Z 267:109–128, 2011) as pro- $p$ analogues of limit groups. We prove that every pro- $p$ group $G$ from the class $\mathcal L $ is the fundamental pro- $p$ group of a finite graph of pro- $p$ groups with infinite procyclic or trivial edge groups and finitely generated vertex groups; moreover, all non-abelian vertex groups are from the class $\mathcal L $ of lower level than $G$ with respect to the natural hierarchy. This allows us to give an affirmative answer to questions 9.1 and 9.3 in Kochloukova and Zalesskii (Math Z 267:109–128, 2011). Namely, we prove that a group $G$ from the class $\mathcal L $ has Euler–Poincaré characteristic zero if and only if it is abelian, and if every abelian pro- $p$ subgroup of $G$ is procyclic and $G$ itself is not procyclic, then $\mathrm{def}(G)\ge 2$ . Moreover, we prove that $G$ satisfies the Greenberg–Stallings property and any finitely generated non-abelian subgroup of $G$ has finite index in its commensurator.