Nicole Lemire

Detecting pro-p-groups that are not absolute Galois groups

Abstract Let p be a prime. It is a fundamental problem to classify the absolute Galois groups GF of fields F containing a primitive pth root of unity ξ p . In this paper we present several constraints on such GF , using restrictions on the cohomology of index p normal subgroups from N. Lemire, J. Mináč, and J. Swallow, Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics, J. reine angew. Math. 613 (2007), 147–173. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these GF . In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of GF . Finally, in section 3 we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.

Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics

Abstract Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F. Using the Bloch-Kato Conjecture we determine the structure of the cohomology group Hn (U, 𝔽 p ) as an 𝔽 p [GF/U]-module for all . Previously this structure was known only for n = 1, and until recently the structure even of H 1(U, 𝔽 p ) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. For the case when the maximal pro-p-quotient T of GF is finitely generated, we apply these results to study the partial Euler-Poincaré characteristics of χ n(N) of open subgroups N of T. We show in particular that the nth partial Euler-Poincaré characteristic χ n(N) is determined by only χ n(T) and the conorm in Hn (T, 𝔽 p ).

Demuškin groups, Galois modules, and the Elementary Type Conjecture

Abstract Let p be a prime and F ( p ) the maximal p-extension of a field F containing a primitive pth root of unity. We give a new characterization of Demuskin groups among Galois groups Gal ( F ( p ) / F ) when p = 2 , and, assuming the Elementary Type Conjecture, when p > 2 as well. This characterization is in terms of the structure, as Galois modules, of the Galois cohomology of index p subgroups of Gal ( F ( p ) / F ) .

Cohomological Dimension and Schreier's Formula in Galois Cohomology

Let p be a prime and F a field containing a primitive p-th root of unity. Then for n \in {mathbb N}, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is at most n if and only if the corestriction maps Hn(H,{mathbb F}p) \to Hn(G,{mathbb F}p) are surjective for all open subgroups H of index p. Using this result, we generalize Schreiers formula for dim{mathbb F}p H1(H,{mathbb F}p) to dim{mathbb F}p Hn(H, {mathbb F}p).

Hilbert 90 for Galois Cohomology

We determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n.

Equivariant Euler characteristics of discriminants of reflection groups

Abstract Let G be a finite, complex reflection group acting on a complex vector space V, and δ its disciminant polynomial. The fibres of δ admit commuting actions of G and a cyclic group. The virtual G × Cm character given by the Euler characteristics of a fibre is a refinement of the zeta function of the geometric monodromy, calculated in [8]. We show that this virtual character is unchanged by replacing δ by a slightly more general class of polynomials. We compute it explicitly, by studying the poset of normalizers of centralizers of regular elements in G, and the subspace arrangement given by the proper eigenspaces of elements of G. As a consequence we also compute orbifold Euler characteristics and find some new ‘case-free’ information about the discriminant.