双特征的Beltrami方程和拟正则映射

设Ω为Ｒｎ上的一个区域，ｎ２，对于具有双特征矩阵Ｇ（ｘ），Ｈ（ｘ）∈Ｃｋ，α（Ω，Ｒｎ），ｋ１，０＜α＜１的Ｂｅｌｔｒａｍｉ方程（１．４），建立了在Ｓｏｂｏｌｅｖ空间Ｗ１，ｎｌｏｃ（Ω，Ｒｎ）上广义解的正则性：ｆ（ｘ）∈Ｃｋ＋１，δｌｏｃ（Ω），对某一δ：０＜δ＜１．     

Quelques proprietes des espaces homogenes spheriques

Let G be a connected, reductive, algebraic group on an algebraically closed field k of characteristic zero. Let H be aspherical subgroup of G, i.e. H is a closed subgroup of G such that every Borel subgroup of G operates on G/H with an open orbit.It is shown that for a spherical subgroup H, the homogeneous space G/H is a deformation of a homogeneous space G/H₀, where H₀ contains a maximal unipotent subgroup of G (such a H₀ is spherical). It is also shown that every Borel subgroup of G has a finite number of orbits in G/H.     

Projective Extensions of Fields

Every field K admits proper projective extensions, that is,Galois extensions where the Galois group is a non-trivial projectivegroup, unless K is separably closed or K is a pythagorean formallyreal field without cyclic extensions of odd degree. As a consequence,it turns out that almost all absolute Galois groups decomposeas proper semidirect products. We show that each local field has a unique maximal projectiveextension, and that the same holds for each global field ofpositive characteristic. In characteristic 0, we prove thatLeopoldt's conjecture for all totally real number fields isequivalent to the statement that, for all totally real numberfields, all projective extensions are cyclotomic. So the realizabilityof any non-procyclic projective group as Galois group over Qproduces counterexamples to the Leopoldt conjecture.     

The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z_p. We prove the existence of a canonical Ore set S^* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S^*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.     

The absolute Galois groups of finite extensions of $${\mathbb{R}}(t)$$

Let R be a real closed field and L be a finite extension of R(t). We prove that Gal(L) ≅ Gal(R(t)) if L is formally real and Gal(L) is the free profinite group of rank card (R) if L is not formally real. Received: 3 April 2007     

Arithmetic of quasi-cyclotomic fields

We call a quadratic extension of a cyclotomic field a quasi-cyclotomic field if it is non-abelian Galois over the rational number field. In this paper, we study the arithmetic of any quasi-cyclotomic field, including to determine the ring of integers of it, the decomposition nature of prime numbers in it, and the structure of the Galois group of it over the rational number field. We also describe explicitly all real quasi-cyclotomic fields, namely, the maximal real subfields of quasi-cyclotomic fields which are Galois over the rational number field. It gives a series of totally real fields and CM fields which are non-abelian Galois over the rational number field.     

数域中tame 核的p-秩

Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K₂O_E if the group K₂O_E is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K₂O_E, where E/F is not necessarily abelian. In particular, in the case of G=C_n, D_n, A₄, we easily get some results on the divisibility of the p-rank of the tame kernel K₂O_E by the character table. Let E/Q be a normal extension with Galois group D_l, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K₂O_F, where f is the smallest positive integer such that p^f≡±1(mod l).     

About Knop's Action of the Weyl Group on the Set of Orbits of a Spherical Subgroup in the Flag Manifold

Let G be a connected reductive algebraic group defined on an algebraically closed field k of characteristic different from 2. Let B denote the flag variety of G. Let H be a spherical subgroup of G. F. Knop defined an action of the Weyl group W of G on the finite set of the H-orbits in B. Here, we define an invariant, namely the type, separating the orbits of W.     

BASES NORMALES AUTODUALES ET GROUPES UNITAIRES EN CARACT ÉRISTIQUE 2

Let k be a field of characteristic 2, and let L/k be a finite Galois extension, with Galois group G. We show the equivalence of the following two properties: (?) The group G is generated by elements of order 2 and by elements of odd order. (??) There exists x ∈ L such that Tr(x) = 1 and Tr(x.g(x)) = 0 for every g ∈ G, g = 1.     

Relative spinor class fields: A counterexample

It is known that classes of indefinite quadratic forms in a genus are classified by the Galois group of a spinor class field [4]. Hsia has proved the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. Spinor class fields can also be used to classify conjugacy classes of maximal orders in a central simple algebra. In [1] we left open the issue of whether for every fixed given non-maximal order in a central simple division algebra there exists a representation field L with the property that embeds into a given maximal order if and only if the corresponding element of the Galois group is trivial on L. In this work we give a negative answer to this question for central simple division algebras of dimension ≥ 3². The case of non-division algebras is also treated by replacing the phrase embeds into by is contained in a conjugate of. As a byproduct of the techniques used in this paper we compute the representation field of an Eichler order in a quaternion algebra. Received: 8 April 2008     

Witt rings and the quaternion group

Let F be a formally real field which admits no quaternionic Galois extension. The structure of the Witt ring and the maximal pro-2 Galois group of F are investigated. Received: 3 July 1997 / Revised version: 2 February 1998     

On the galois module structure of semisimple holomorphic differentials

LetL/K be a finite Galoisp-extension of algebraic function fields of one variable over an algebraically closed fieldk of characteristicp, with Galois groupG=Gal(L/K). The space Ώ _L ^s (0) of semisimple holomorphic differentials ofL is thek-vector space of holomorphic differentials which are fixed by the Cartier operator. We obtain the isomorphism classes and multiplicities of the summands in a Krull-Schmidt decomposition of thek[G]-module Ώ _L ^s (0) into a direct sum of indecomposablek[G]-modules. Partially supported by CONACyT, project No. 25063-E.     

On a maximal torus in subgroups of a general linear group

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), G_o=Aut(k K) and fix in K a basis ω₁,...,ω_n over k. In this basis, to any automorphism group of kK there corresponds a matrix group, which is denoted by the same symbol. Let G′≤G., In this paper, the conditions under which G′⊎G_o is a maximal torus in G′ are studied. The calculation of N_G′(G′⊎G_o) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎G_o is a maximal torus in G′. Bibligraphy: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995. pp. 15–22.     

Abelsche Galoiserweiterungen von R[X]

Our main result states that for a commutative ring R and a finite abelian group G the following conditions are equivalent: (a) Gal(R,G)=Gal (R[X],G), i.e. every commutative Galois extension of R[X]with Galois group G is extended from R. (b) The order of G is a non-zero-divisor in R/Nil(R). The proof uses lifting properties of Galois extensions over Hensel pairs and a Milnor-type patching theorem.