Finite Galois Extension Research Articles

Galois averages

In this paper, we introduce a notion of “Galois average” which allows us to give a suitable answer to the question: how can one extend a finite Galois extension E / F by a prime degree extension N / E to get a Galois extension N / F ? Here, N / E is not necessarily a Kummer extension.

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L)G]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(Ax/K)] whereAx is the twist ofA by the non-trivial characterχ ofG.

Galois cohomology of Witt vectors of algebraic integers

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p\gt 0$ . Let $L/K$ be a finite Galois extension with Galois group $G\,{=}\,G_{L/K}$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. By the normal basis theorem, $L$ is always a projective $K[G]$ -module. But the ring of integers $\mathcal{O}_L$ is a projective $\mathcal{O}_K[G]$ -module if and only $L/K$ is tamely ramified. For wildly ramified extensions, the structure of $\mathcal{O}_L$ as an $\mathcal{O}_K[G]$ -module is very complicated and quite far from understood. We propose that the ring of Witt vectors $W\s(\mathcal{O}_L)$ is a more well-behaved object. Indeed, we show that for a large class of extensions $L/K$ , the pro-abelian group $H^1(G,W\s(\mathcal{O}_L))$ is zero. We conjecture that this is true in general.

Torsion of abelian varieties over large algebraic fields

We prove: Let A be an abelian variety over a number field K. Then K has a finite Galois extension L such that for almost all σ∈Gal( L) there are infinitely many prime numbers l with A l( K ̃ (σ))≠0 . Here K̃ denotes the algebraic closure of K and K ̃ (σ) the fixed field in K̃ of σ. The expression “almost all σ” means “all but a set of σ of Haar measure 0”.

Euclidean Rings of Algebraic Integers

AbstractLet K be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

On the values of equivariant zeta functions of curves over finite fields

Let K/k be a finite Galois extension of global function fields of characteristic p . Let C_K denote the smooth projective curve that has function field K and set G:= \operatorname{Gal}(K/k) . We conjecture a formula for the leading term in the Taylor expansion at zero of the G -equivariant truncated Artin L -functions of K/k in terms of the Weil-étale cohomology of \mathbb G_m on the corresponding open subschemes of C_K . We then prove the \ell -primary component of this conjecture for all primes \ell for which either \ell \not= p or the relative algebraic K -group K_0(\mathbb z _\ell [G],\mathbb Q_\ell) is torsion-free. In the remainder of the manuscript we show that this result has the following consequences for K/k : if p \nmid |G| , then refined versions of all of Chinburg's ' \Omega -Conjectures' in Galois module theory are valid; if the torsion subgroup of K^\times is a cohomologically-trivial G -module, then Gross's conjectural 'refined class number formula' is valid; if K/k satisfies a certain natural class-field theoretical condition, then Tate's recent refinement of Gross's conjecture is valid.

Equivariant Epsilon Constants, Discriminants and Étale Cohomology

Let $L/K$ be a finite Galois extension of number fields. We formulate and study a conjectural equality between an element of the relative algebraic K-group $K_0(\mathbb{Z}[\mathrm{Gal}(L/K)], \mathbb{R})$ which is constructed from the equivariant Artin epsilon constant of $L/K$ and a sum of structural invariants associated to $L/K$. The precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the Tamagawa Number Conjecture of Bloch and Kato (as formulated by Flach and the second-named author) should be compatible with the functional equation of the associated L-function. We show that, more concretely, our conjecture also suggests a completely systematic refinement of the central approach and results of classical Galois module theory. In particular, the evidence for our conjecture that we present here already strongly refines many of the main results of Galois module theory.

Capitulation and Transfer Triples

Let K/k be a finite unramified Galois extension of number fields with Galois group G. This determines two homomorphisms from the ideal class group Clk of k: the capitulation map Clk → CK and the Artin map Ck↠Gab onto the abelianization Gab of G. We call (ker (capitulation), Clk, Artin) the capitulation triple of K/k. Artin's transition to group theory shows that any triple (X, Y, ζ) which arises in this way satisfies the group-theoretic property of being a transfer triple for G, defined as follows: there exist a group extension A ↣ H ↠ G with A finite abelian and an isomorphism η : Y → ∼ H a b such that η(X) is the kernel of the transfer homomorphism Hab → A, and ζ is the composite of η with Hab → Gab. When G is abelian, we show that a triple (X, Y, ζ) is a transfer triple for G if and only if |G| X = 0 and |G| divides |X|. Whether all transfer triples for G can be realized arithmetically remains an unsolved problem. 2000 Mathematics Subject Classification 11R33, 20J99.

On Poincaré sums for local fields

Let $K/k$ be a finite Galois extension of local fields. To each class $\gamma = [c]$ in $H^1(\operatorname{Gal}(K/k), U_K)$, $U_K$ being the group of units of $K$, we associate an index $i_\gamma(K/k) = (M_c : P_c)$ after the model of Poincaré series and study its relation to the ramification theory of $K/k$.

CROSSED PRODUCT ORDERS OVER VALUATION RINGS II: TAMELY RAMIFIED CROSSED PRODUCT ALGEBRAS

Let V be a commutative valuation domain of arbitrary K rull-dimension, with quotient field F, let K be a finite Galois extension of F with group G, and let S be the integral closure of V in K. Suppose that one has a 2-cocycle on G that takes values in the group of units of S. Then one can form the crossed product of G over S, S * G, which is a V-order in the central simple F-algebra K * G. If S * G is assumed to be a Dubrovin valuation ring of K * G, then the main result of this paper is that, given a suitable definition of tameness for central simple algebras, K * G is tamely ramified and defectless over F if and only if K is tamely ramified and defectless over F. The residue structure of S * G is also considered in the paper, as well as its behaviour upon passage to Henselization.

The weight-two motivic complex of equicharacteristic local fields

Let L/ K be a finite Galois extension of local fields in positive characteristic with group G. The weight-two motivic complex Γ(2, L) defines an element of Ext Z[G] 2(K 2(L),K 3(L).) . We show, after inverting the prime 2, that cup-product with this 2-extension induces an isomorphism on Tate cohomology. In fact we show that this isomorphism coincides with cup-product by the K 2/ K 3 local fundamental class previously constructed.

Frattini closed groups and adequate extensions of global fields

LetL be a finite Galois extension of a global fieldF. It is shown that if the Galois groupG=Gal(L/F) satisfies a certain condition, thenL is a maximal commutative subfield of someF-division algebra if and only if the intermediate field corresponding to the Frattini subgroup ofG is also a maximal commutative subfield of someF-division algebra. In particular this condition holds ifG is a supersolvable group.

Toward equivariant Iwasawa theory

Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G ∞ , X ∞ the Galois groups of K ∞ /k and M ∞ /K ∞ , respectively, where K ∞ is the cyclotomic l-extension of K and M ∞ the maximal abelian S-ramified l-extension of K ∞ with S a sufficiently large finite set of primes of k. We introduce a new K-theoretic variant of the Iwasawa ℤ[[G ∞ ]]-module X ∞ and, for K/k abelian, formulate a conjecture, which is the main conjecture of classical Iwasawa theory when ll[K : k]. We prove this new conjecture when Iwasawa's μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l.

Properties of forking in ω-free pseudo-algebraically closed fields

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data:• The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field).• The degree of imperfection of K.• The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K.They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

Elasticity Properties Preserved in the Normset

Let T⊆R be the rings of integers in a number field and a finite Galois extension field. We study relations between the elasticity ρ(R) of the monoid of nonzero elements of R and the elasticity ρ(S) of the monoid S of norms to T of those elements. We show ρ(R)⩾ρ(S) and that equality holds if the norms of irreducible elements of R are irreducible in S, which is true, in particular, if either ρ(R)<2 or ρ(S)=1.

Finite Arithmetic Subgroups of GLn, II

We continue our investigation on the conjecture of Y. Kitaoka that if a finite subgroup G of GLn(OK) is invariant under the action ofGal(K/Q) then it is contained in GLn(Kab). Here OK is the ring of integers in a finite Galois extension K of Q and Kab is the maximal abelian subextension of K. We give a very precise description of a hypothetical counterexample of minimal order for minimal possible n. Using it we prove the conjecture for n=3 and give a new, simplified proof for n=2.

Splitting fields of G-varieties

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is trivial. If $L/K$ is a (finite) Galois extension then $\Gal(L/K)$ is called a splitting group of $X$. We prove a lower bound on the size of a splitting field of $X$ in terms of fixed points of nontoral abelian subgroups of $G$. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.

CROSSED PRODUCT ORDERS OVER VALUATION RINGS

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.

Equivariant Tamagawa numbers, fitting ideals and Iwasawa theory

Let L/K be a finite Galois extension of number fields of group G. In [4] the second named author used complexes arising from etale cohomology of the constant sheaf ℤ to define a canonical element TΩ(L/K) of the relative algebraic K-group K 0(ℤ[G],ℝ). It was shown that the Stark and Strong Stark Conjectures for L/K can be reinterpreted in terms of TΩ(L/K), and that the Equivariant Tamagawa Number Conjecture for the ℚ[G]-equivariant motive h 0(Spec L) is equivalent to the vanishing of TΩ(L/K). In this paper we give a natural description of TΩ(L/K) in terms of finite G-modules and also, when G is Abelian, in terms of (first) Fitting ideals. By combining this description with techniques of Iwasawa theory we prove that TΩ(L/ℚ) vanishes for an interesting class of Abelian extensions L/ℚ.

Equivariant Tamagawa numbers and Galois module theory I

Let L/K be a finite Galois extension of number fields. We use complexes arising from the etale cohomology of Z on open subschemes of Spec OL to define a canonical element of the relative algebraic K-group K0Z[Gal(L/K)], R. We establish some basic properties of this element, and then use it to reinterpret and refine conjectures of Stark, of Chinburg and of Gruenberg, Ritter and Weiss. Our results precisely explain the connection between these conjectures and the seminal work of Bloch and Kato concerning Tamagawa numbers. This provides significant new insight into these important conjectures and also allows one to use powerful techniques from arithmetic algebraic geometry to obtain new evidence in their favour.