The cohomological Brauer group

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

In this chapter, we investigate infinite Galois extensions and prove an analog of the fundamental theorem of Galois theory for infinite extensions. The key idea is to put a topology on the Galois group of an infinite dimensional Galois extension and then use this topology to determine which subgroups of the Galois group arise as Galois groups of intermediate extensions. We also give a number of constructions of infinite Galois extensions, constructions that arise in quadratic form theory, number theory, and Galois cohomology, among other places.

A profinite group G is said to have cohomological dimension at most n if Hr (G, A) is zero for all r > n and all discrete Z-torsion G-modules A. The groups of co omological dimension at most 1 are just the projective groups. Among other important groups which have finite cohomological dimension are the torsion-free profinite groups of finite rank and certain Galois groups Gal (K/k), where K is either a separable closure of k or the subfield of a separable closure generated by all finite Galois extensions of p-power degree.

A characterization of number fields by p-closed extensions

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

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.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

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.

In this paper we prove that, for any Galois finite field extension $F/K$ on which a separated group of operators $\Gamma$ is acting, there is an isomorphism between the group of equivariant isomorphism classes of finite dimensional central simple $K$-algebras endowed with a $\Gamma$-action and containing $F$ as an equivariant strictly maximal subfield and the second equivariant cohomology group of the Galois group of the extension.

We show that there exist infinitely many quartic number fields K with large class numbers such that K∕Q is a Galois extension whose Galois group is isomorphic to a given finite group. Cho and Kim proved that there are infinitely many totally real cyclic extensions over Q of degree 4 with large class numbers. We consider all the other cases of quartic Galois extensions.

A class of soluble groups

A celebrated theorem of Artin and Schreier [AS] characterizes the fields K whose absolute Galois group GK is isomorphic to that of R as the real closed fields. In the present paper we consider the analogous problem for nonarchimedean local fields of positive characteristic F = Fpn((t)). We show that a field K with absolute Galois group isomorphic to GF possesses a Henselian valuation v such that: (1) the value group Γ of v satisfies Γ/l ∼= Z/l for all prime numbers l = p; (2) the residue field K of v has characteristic p; (3) the maximal prime to p Galois group GK(p) of K is Ẑ/Zp; (4) if charK = 0 then Γ = pΓ and K is perfect. For every positive integer r we construct such fields K of characteristic p with Γ/p ∼= (Z/p). Likewise we construct examples with Γ ∼= Z, GK ∼= Ẑ and K imperfect. The similar problem for p-adic fields was answered by Koenigsmann [Kn] and the first named author [E1] (for p = 2), extending earlier results by Neukirch [N2] and Pop [P1]: the fields K such that GK ∼= GF for some finite extension F of Qp are precisely the p-adically closed fields in the sense of [PR].

Let X be a subgroup of a group Y. The \\emphinterval (X, Y) is the set of subgroups of Y that contain X including X and Y. By local class field theory the interval (N K/k K*, k*) contains a finite number of norm groups for any finite extension K of a 𝔭-adic number field k. In the present work we investigate the number of norm groups in the interval (N K/k K*, k*) for a given finite extension K/k of algebraic number fields. We prove that if K/k is an extension of prime degree, or of degree n such that the normal closure of K over k has the Galois group isomorphic to A n or S n , then the interval (N K/k K*, k*) contains only the obvious two norm groups. Also, the interval (N K/k K*, k*) contains a finite number of norm groups for any Galois extension of degree 4, and there are extensions with Galois groups order 8 for which the corresponding intervals contain a finite number of norm groups. The main theorem in our earlier work states that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois extension of even degree that is not a 2-extension, the so-called 2𝔫-\\emphextensions. In the present work we generalize the main theorem to non-Galois 2𝔫-extensions K/k, and determine some subintervals of (N K/k K*, k*) that contain infinitely many norm groups. We then use this theorem to prove that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois 2-extension K/k with the Galois group that either contains an element of order 8 or contains the quaternion group Q8 of order 8, or Q8 is a homomorphic image of the Galois group.

In a 2016 paper by Alan Reid, Martin Bridson and the author, it was shown using the theory of profinite groups that if $\Gamma$ is a finitely-generated Fuchsian group and $\Sigma$ is a lattice in a connected Lie group, such that $\Gamma$ and $\Sigma$ have exactly the same finite quotients, then $\Gamma$ is isomorphic to $\Sigma$. As a consequence, two triangle groups $\Delta(r,s,t)$ and $\Delta(u,v,w)$ have the same finite quotients if and only if $(u,v,w)$ is a permutation of $(r,s,t)$. A direct proof of this property of triangle groups was given in the final section of that paper, with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. Unfortunately, part of the latter direct proof was flawed. In this paper two new direct proofs are given, one being a corrected version using the same approach as before (involving direct products of small quotients), and the other being a shorter one that uses the same preliminary observations as in the earlier version but then takes a different direction (involving further use of the `Macbeath trick').

Fix a number field F⊂ℂ, an abelian variety A∕F and let GA be the Mumford–Tate group of A∕ℂ. After replacing F by finite extension one can assume that, for every prime number l, the action of the absolute Galois group ΓF= Gal(F∕F) on the etale cohomology group Het1(AF,ℚl) factors through a morphism ρl:ΓF→GA(ℚl). Let v be a valuation of F and write ΓFv for the absolute Galois group of the completion Fv. For every l with v(l)=0, the restriction of ρl to ΓFv defines a representation ′WFv→GA∕ℚl of the Weil–Deligne group.It is conjectured that, for every l, this representation of ′WFv is defined over ℚ as a representation with values in GA and that the system above, for variable l, forms a compatible system of representations of ′WFv with values in GA. A somewhat weaker version of this conjecture is proved for the valuations of F, where A has semistable reduction and for which ρl(Frv) is neat.