On Galois cohomology of semisimple groups over local and global fields of positive characteristic

We treat a case that was omitted from consideration in our article [2] in Math Zeit, 2007.     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic, III

We extend some well-known results on Galois cohomology in its relation with weak approximation for connected linear algebraic groups over number fields to the case of global fields of positive characteristic. Some applications are considered.     

Galois cohomology of the classical groups over imperfect fields

Elaborating on techniques of Bayer-Fluckiger and Parimala, we prove the following strong version of Serre’s Conjecture II for classical groups: let G be a simply connected absolutely simple group of outer type A_n or of type B_n, C_n or D_n (non trialitarian) defined over an arbitrary field F. If the separable dimension of F is at most 2 for every torsion prime of G, then every G-torsor is trivial.     

On Galois cohomology and realizability of 2-groups as Galois groups II

In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2 ⁿ , n ≥ 4, having a cyclic subgroup of order 2 ⁿ⁻², over fields containing a primitive 2 ⁿ⁻³th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups.     

On some Hasse principles for homogeneous spaces of algebraic groups over global fields of positive characteristic

We extend to global function fields some Hasse principles for homogeneous spaces of connected linear algebraic groups proved earlier by several authors in the case of number fields. We also give some applications.     

Galois cohomology of the classical groups over fields of cohomological dimension≦2

Oblatum 8-XII-1994 7-III-1995     

On the admissibility of finite groups over global fields of finite characteristic

Let k be a global field of characteristic p. A finite group G is called k-admissible if there exists a division algebra finite dimensional and central over k which is a crossed product for G. Let G be a finite group with normal Sylow p-subgroup P. If the factor group G/P is k-admissible, then G is k-admissible. A necessary condition is given for a group to be k-admissible: if a finite group G is k-admissible, then every Sylow l-subgroup of G for l ≠ p is metacyclic with some additional restriction. Then it is proved that a metacyclic group G generated by x and y is k-admissible if some relation between x and y is satisfied.     

Cycles on curves over global fields of positive characteristic

Let be a global field of positive characteristic, and let be a smooth projective curve. We study the zero-dimensional cycle group and the one-dimensional cycle group , addressing the conjecture that is torsion and is finitely generated. The main idea is to use Abhyankar's Theorem on resolution of singularities to relate the study of these cycle groups to that of the -groups of a certain smooth projective surface over a finite field.

     

The cohomology of period domains for reductive groups over local fields

We compute the étale cohomology of period domains over local fields in the case of a basic isocrystal for quasi-split reductive groups. Period domains, which have been introduced by Rapoport and Zink [RZ], are open admissible rigid-analytic subsets of generalized flag varieties. They parametrize weakly admissible filtrations of a given isocrystal with additional structure of a reductive group. Mathematics Subject Classification (2000) 14L05, 14F30, 14L24     

On Galois cohomology and realizability of 2-groups as Galois groups

In this paper we develop some new theoretical criteria for the realizability of p-groups as Galois groups over arbitrary fields. We provide necessary and sufficient conditions for the realizability of 14 of the 22 non-abelian 2-groups having a cyclic subgroup of index 4 that are not direct products of groups.     

Frobenius Galois groups over quadratic fields

There exists a quadratic fieldQ(√D) over which every Frobenius group is realizable as a Galois group.     

Some Galois groups over number fields

Some conditions are stated which imply that certain finite groups are Galois groups over some number fields and related fields.     

Galois groups over complete valued fields

We propose an elementary algebraic approach to the patching of Galois groups. We prove that every finite group is regularly realizable over the field of rational functions in one variable over a complete discrete valued field. Partially supported by NSF grant DMS 9306479.     

Galois groups of intersections of local fields

LetK be a denumerable Hilbertian field with separable algebraic closure and Galois group , letw ₁,...w _n be absolute values on . Then for almost allσ ∈ G _K ⁿ (in the sense of Haar measure) there are no relations between the decomposition groups G _K (ω ₁ σ ₁),...,G _K (w _n σ _n ) of the absolute valuesw ₁ σ ₁,...,w _n σ _n i.e. the subgroup of G _K generated by these groups is the free product of these groups.     

Computation of Galois groups over function fields

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.

     

On the Galois cohomology of ideal class groups

We use étale cohomology to prove some explicit results on the Galois cohomology of ideal class groups. Received: 3 May 2007     

On the characterization of local fields by their absolute Galois groups

For finite field extensions of the field of Henselian p-adic rational numbers necessary and sufficient conditions are given which state that the fields have isomorphic absolute Galois groups; it is thereby supposed that a p-th root of unity (a 4-th when p = 2) belongs to the fields. Also examples are discussed.