Galois Cohomology Groups Research Articles

On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL1(A). For a field extension K of F, we study the first Galois Cohomology group H 1(K,SL1(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.

On K2 of varieties over number fields

AbstractLet k be a number field and let X be a smooth, projective and geometrically integral k-variety. We show that, if the geometric Néron-Severi group of X is torsion-free, then the Galois cohomology group is finite. Previously this group was only known to have a finite exponent. We also obtain a vanishing theorem for this group, showing in particular that it is trivial if X belongs to a certain class of abelian varieties with complex multiplication. The interest in the above cohomology group stems from its connection to the torsion subgroup of the Chow group CH2(X) of codimension 2 cycles on X. In the last section of the paper we record certain results on curves which must be familiar to all specialists in this area but which we have not formerly seen in print.

Iwasawa theory and the Eisenstein ideal

We verify, for each odd prime p<1000, a conjecture of W. G. McCallum and R. T. Sharifi on the surjectivity of pairings on p-units constructed out of the cup product on the first Galois cohomology group of the maximal unramified outside p extension of Q(μp) with μp-coefficients. In the course of the proof, we relate several Iwasawa-theoretic and Hida-theoretic objects. In particular, we construct a canonical isomorphism between an Eisenstein ideal modulo its square and the second graded piece in an augmentation filtration of a classical Iwasawa module over an abelian pro-p Kummer extension of the cyclotomic Zp-extension of an abelian field. This Kummer extension arises from the Galois representation on an inverse limit of ordinary parts of first cohomology groups of modular curves which was considered by M. Ohta in order to give another proof of the Iwasawa main conjecture in the spirit of that of B. Mazur and A. Wiles. In turn, we relate the Iwasawa module over the Kummer extension to the quotient of the tensor product of the classical cyclotomic Iwasawa module and the Galois group of the Kummer extension by the image of a certain reciprocity map that is constructed out of an inverse limit of cup products up the cyclotomic tower. We give an application to the structure of the Selmer groups of Ohta's modular representation taken modulo the Eisenstein ideal

COHOMOLOGY GROUPS OF RADICAL EXTENSIONS

If k is a subfield of Q(em) then the cohomology group H2(k (en)/k) is isomorphic to H(k(en′ )/k) with gcd(m, n ′) = 1. This enables us to reduce a cyclotomic k-algebra over k(en) to the one over k(en′ ). A radical extension in projective Schur algebra theory is regarded as an analog of cyclotomic extension in Schur algebra theory. We will study a reduction of cohomology group of radical extension and show that a Galois cohomology group of a radical extension is isomorphic to that of a certain subextension of radical extension. We then draw a cohomological characterization of radical group.

The group SK 2 of a biquaternion algebra

M. Rost has proved the existence of an exact sequence relating the group SK 1 —kernel of the reduced norm for K 1 —of a biquaternion algebra D whose center is a field F with Galois cohomology groups of F . In this paper, we relate the group SK 2 —kernel of the reduced norm for K 2 —of D with Galois cohomology of F through an exact sequence.

Vanishing of some cohomology groups and bounds for the Shafarevich–Tate groups of elliptic curves

Let E be an elliptic curve over Q and ℓ be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ℓ . Under certain assumptions, we prove the vanishing of the Galois cohomology group H 1 ( Gal ( K ( E [ ℓ i ] ) / K ) , E [ ℓ i ] ) for all i ⩾ 1 . When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ℓ -primary part of Shafarevich–Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton–Dyer conjecture.

Holes in [formula omitted

Let F be an arbitrary field of characteristic ≠2. We write W ( F ) for the Witt ring of F, consisting of the isomorphism classes of all anisotropic quadratic forms over F. For any element x ∈ W ( F ) , its dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted by I ( F ) . The filtration of the ring W ( F ) by the powers I ( F ) n of this ideal plays a fundamental role in the algebraic theory of quadratic forms. The Milnor conjectures, recently proved by Voevodsky and Orlov–Vishik–Voevodsky, describe the successive quotients I ( F ) n / I ( F ) n + 1 of this filtration, identifying them with Galois cohomology groups and with the Milnor K-groups modulo 2 of the field F. In the present article we give a complete answer to a different longstanding question concerning I ( F ) n , asking about the possible values of dim x for x ∈ I ( F ) n . More precisely, for any n ⩾ 1 , we prove that (*) dim I n = { 2 n + 1 − 2 i | i ∈ [ 1 , n + 1 ] } ∪ ( 2 Z ∩ [ 2 n + 1 , + ∞ ) ) , where dim I n is the set of all dim x for all x ∈ I ( F ) n and all F. Previously available partial informations on dim I n include the classical Arason–Pfister theorem (saying that ( 0 , 2 n ) ∩ dim I n = ∅ ) as well as a recent Vishik's theorem on ( 2 n , 2 n + 2 n − 1 ) ∩ dim I n = ∅ (the case n = 3 is due to Pfister, n = 4 to Hoffmann). The proof of ( * ) is based on computations in Chow groups of powers of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.

Galois cohomology in degree 3 of function fields of curves over number fields

Let k be a field of characteristic not equal to 2. For n≥1, let H n(k, Z /2) denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements α 1,⋯,α n∈H 2(k, Z /2) , there exist a,b 1,⋯,b n∈k ∗ such that α i =( a)∪( b i ), where for any λ∈k ∗ , ( λ) denotes the image of k ∗ in H 1(k, Z /2) . In this paper we prove a higher dimensional analogue of the Tate's lemma.

A cup product in the Galois cohomology of number fields

Let $K$ be a number field containing the group $μ_n$ of $n$th roots of unity, and let $S$ be a set of primes of $K$ including all those dividing $n$ and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal $S$-ramified extension of $K$ with coefficients in $μ_n$, which yields a pairing on a subgroup of K x containing the $S$-units. In this general situation, we determine a formula for the cup product of two elements that pair trivially at all local places. Our primary focus is the case in which K = ℚ ( μ p ) for $n=p$, an odd prime, and $S$ consists of the unique prime above $p$ in $K$. We describe a formula for this cup product in the case that one element is a pth root of unity. We explain a conjectural calculation of the restriction of the cup product to $p$-units for all $p≤10,000$ and conjecture its surjectivity for all $p$ satisfying Vandiver's conjecture. We prove this for the smallest irregular prime $p=37$ via a computation related to the Galois module structure of $p$-units in the unramified extension of $K$ of degree $p$. We describe a number of applications: to a product map in $K$-theory, to the structure of $S$-class groups in Kummer extensions of $K$, to relations in the Galois group of the maximal pro-$p$ extension of ℚ ( μ p ) unramified outside $p$, to relations in the graded $ℤ_p$-Lie algebra associated to the representation of the absolute Galois group of $\mathbb{Q}$ in the outer automorphism group of the pro-$p$ fundamental group of P 1 ( ℚ ¯ ) − { 0 , 1 , ∞ } , and to Greenberg's pseudonullity conjecture.

Le groupe SK2 d'une algèbre de biquaternions

In this Note, I will sketch the proof that a sequence relating the group SK 2 (kernel of the reduced norm) of a biquaternion algebra over a field F and the Galois cohomology group H 5(F, Z /2) is exact. The main steps of the proof contain computations in spectral sequences for motivic cohomology and in the topological filtration of an Albert quadric. To cite this article: B. Calmès, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Deflation and Residuation for Class Formation

In the present paper, recalling briefly a deflation map and a residuation map in the cohomology theory of finite groups, we shall see that a certain pair of isomorphisms of Tate's main theorem for class formations establishes a correspondence of the associated residuation map to the associated deflation map (Theorem 1). This result, supplementary to the results of Artin and Tate, will further provide us with a commutative diagram for Galois cohomology groups, which just generalizes a fundamental commutative diagram for number knots (Theorem 2).

Torseurs sous une variété abélienne sur R (( t ))

Let V be an henselian discrete valuation ring with real closed residue field and let k be its quotient ring; we denote by k+ and k− the two real closures of k. Consider a k-abelian variety A. We compute the Galois-cohomology group H1(k,A) in terms of the reduction of the dual variety of A and of the semi-algebraic topology of A(k+) and A(k−). The tools we need are Ogg's results concerning valuation rings with algebraically closed residue field, Hochschild–Serre spectral sequence and Scheiderer's local-global principles. At the end we study more precisely the case of an elliptic curve.

Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Nonexcellence

A field extension L / F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L / F is called universally excellent if L ⋅ E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.

The structure of Selmer groups.

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Lambda-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Lambda-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

Congruences between modular forms: raising the level and dropping Euler factors.

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida's result accounts for congruences in terms of the value of an L-function, and Ribet's result is related to the behavior of the period that appears there. Wiles' theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at "nonminimal level" is obtained from one at "minimal level" by dropping Euler factors from the L-function.

Remarks on Galois cohomology and definability

In this paper we develop some basic features of Galois cohomology, specifically the connection between first Galois cohomology groups and principal homogeneous spaces, in a model-theoretic context. “Descent theory” also fits into our approach.The model theory involved is elementary, and the reader is referred to [2]. It should be said that we make crucial use of Meq in our analysis. The reader is also referred to Poizat's seminal paper “Une theorie de Galois imaginaire” ([6]). Although our results do not depend on Poizat's work, it is in his paper that the model-theoretic context is suggested for a generalised treatment of Galois theory.Nothing in this paper is particularly deep. We are concerned mainly with translating between the Galois cohomological language and the language of definable sets and definable families of definable sets. We will introduce (in a suitable context) the notion of a definable cocycle (from an automorphism group to a definable group G). The (classical) situation of profinite and continuous cocycles will be a special case. Kolchin's theory of constrained cohomology will be another special case, and our results yield a substantially simpler proof of his Theorem 5 from Chapter VII of [4]. In any case model-theorists will see that definable cocycles correspond to objects with which they are already quite familiar—commuting families of definable bijections.

A Finiteness Theorem in the Galois Cohomology of Algebraic Number Fields

In this note we show that if $k$ is an algebraic number field with algebraic closure $\overline k$ and $M$ is a finitely generated, free ${{\mathbf {Z}}_l}$-module with continuous $\operatorname {Gal} (\overline k /k)$-action, then the continuous Galois cohomology group ${H^1}(k, M)$ is a finitely generated ${{\mathbf {Z}}_l}$-module under certain conditions on $M$ (see Theorem 1 below). Also, we present a simpler construction of a mapping due to S. Bloch which relates torsion algebraic cycles and étale cohomology.

Derivations of graded rings and Clifford systems

As a counterpart to the commutative Galois theory of groups acting on rings one may consider Lie algebras acting on rings as Lie algebras of derivations. A peculiar situation where techniques from both areas meet and blend arises when studying derivations of graded rings and Clifford systems. Some results for strongly graded rings might benefit from a rephrasing in terms of Hopf algebra actions, the Hopf algebras being smash products of dual group algebras and enveloping algebras, but we did not go into these matters here for simplicity’s sake. In the first part of the paper we provide general results concerning R,derivations of a graded ring of type G, R = eJntC; R, (where e is the neutral clement of G) into weakly cancellative R-bimodules M which need not be graded. In case A4 is graded as a left R-module a useful criterion for an R,derivation D: R -+ M to be a graded (left) R,-morphism may be given. We then investigate when an R,.-derivation is a sum of such “graded” derivation, cg., we establish that for a finitely generated abelian group G, every R,-derivation of R into a weakly cancellative graded R-bimodule has a decomposition as a sum of graded R,-derivations. In the second section we prove a result relating the R,-derivations to some Galois cohomology group for G and as a consequcncc of this, it follows that for a strongly graded ring Hochschild cohomology (whenever defined) coincides with some Galois cohomology of the grading group. Along the way we provide an interesting extension of results about the Miyashita auto-morphisms of strongly graded rings. Our approach leads to some applications in the crossed product theory for Azumaya algebras or even central simple algebras, presenting a new point of view on crossed product structures which seems to deserve further development. In the light of these applications and also to the benefit of general applicability, the consideration of Clifford systems which are not graded rings is important. For example, every algebra generated by invertible elements may be viewed 485 0021-8693/86 $3.00

On the Galois cohomology groups of ${C_K} / {D_K}$

On the Galois cohomology groups of ${C_K} / {D_K}$

On some Galois cohomology groups of a local field and its application to the maximal p-extension

On some Galois cohomology groups of a local field and its application to the maximal p-extension