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1.
LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G ; that ifH is epimorphic inG then the action ofH onG/G is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.  相似文献   

2.
Let ℚ ab denote the maximal abelian extension of the rationals ℚ, and let ℚabnil denote the maximal nilpotent extension of ℚ ab . We prove that for every primep, the free pro-p group on countably many generators is realizable as the Galois group of a regular extension of ℚabnil(t). We also prove that ℚabnil is not PAC (pseudo-algebraically closed). This paper was inspired by the author's participation in a special year on the arithmetic of fields at the Institute for Advanced Studies at the Hebrew University of Jerusalem. I would like to express my appreciation to the Institute for its hospitality, and to the organizers, especially Moshe Jarden. Partially supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund-Japan Technion Society Research Fund.  相似文献   

3.
LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL 1 of ℚ(T) with Galois groupG. LetL be the subfield ofL 1 fixed byH. We make the hypothesis thatL 1 admits a quadratic extensionL 2 which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL 1 then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL 2. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL 1 admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).  相似文献   

4.
Let K2 be the Milnor functor and let Фn (x)∈ Q[X] be the n-th cyclotomic polynomial. Let Gn(Q) denote a subset consisting of elements of the form {a, Фn(a)}, where a ∈ Q^* and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn(Q) is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn(Q) is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h(Q(ζp)) is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.  相似文献   

5.
Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ p , s(x)=s(h) is determined by the semisimple part h of xG. For every finite extension K of ℚ p , the scale functions of G and H:=G(K) are related by s H G =s G [ K :ℚ p ]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998  相似文献   

6.
LetG be an algebraic group over a fieldk. We callg εG(k) real ifg is conjugate tog −1 inG(k). In this paper we study reality for groups of typeG 2 over fields of characteristic different from 2. LetG be such a group overk. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element inG(k) is real if and only if it is a product of two involutions inG(k). Every unipotent element inG(k) is a product of two involutions inG(k). We discuss reality forG 2 over special fields and construct examples to show that reality fails for semisimple elements inG 2 over ℚ and ℚp. We show that semisimple elements are real forG 2 overk withcd(k) ≤ 1. We conclude with examples of nonreal elements inG 2 overk finite, with characteristick not 2 or 3, which are not semisimple or unipotent.  相似文献   

7.
We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μn represented by then th roots of unity 1, ζ, ζ2, …, ζn-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μn in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.  相似文献   

8.
The author proves a conjecture of the author: IfG is a semisimple real algebraic defined over ℚ, Γ is an arithmetic subgroup (with respect to the given ℚ-structure) andA is a diagonalizable subgroup admitting a divergent trajectory inG/Γ, then dimA≤ rank G.  相似文献   

9.
Let G/P be a homogenous space with G a compact connected Lie group and P a connected subgroup of G of equal rank. As the rational cohomology ring of G/P is concentrated in even dimensions, for an integer k we can define the Adams map of type k to be l k : H*(G/P, ℚ) → H*(G/P, ℚ), l k (u) = k i u, uH 2i (G/P, ℚ). We show that if k is prime to the order of the Weyl group of G, then l k can be induced by a self map of G/P. We also obtain results which imply the condition that k is prime to the order of the Weyl group of G is necessary.  相似文献   

10.
LetG be a semisimple algebraic ℚ-group, let Γ be an arithmetic subgroup ofG, and letT be an ℝ-split torus inG. We prove that if there is a divergentT -orbit in Γ\G , and ℚ-rankG≤2, then dimT≤ℚ-rankG. This provides a partial answer to a question of G. Tomanov and B. Weiss.  相似文献   

11.
In this paper we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ pr(G), is the minimum cardinality of a paired-dominating set of G. The graph G is paired-domination vertex critical if for every vertex v of G that is not adjacent to a vertex of degree one, γ pr(Gv) < γ pr(G). We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least 3/2 (γ pr(G) − 2). For γ pr(G) ⩽ 8, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph. The first author was supported in part by the South African National Research Foundation and the University of KwaZulu-Natal, the second author was supported by the Natural Sciences and Engineering Research Council of Canada.  相似文献   

12.
We prove that the number of elliptic curves E/ℚ with conductorN isO(N 1/2+ε). More generally, we prove that the number of elliptic curves E/ℚ with good reduction outsideS isO(M 1/2+ε), whereM is the product of the primes inS. Assuming various standard conjectures, we show that this bound can be improved toO(M c/loglogM ). Research partially supported by NSF DMS-9424642.  相似文献   

13.
Some equivariant compactifications of the quotients PGL r n +1/PGL r are constructed. Each one is decomposed into locally closed strata which are smooth, are indexed by the entire convex pavings of the simplex of dimension n and admit a modular interpretation deduced from that of the Grassmann varieties. Together, they form a simplicial scheme which “compactifies” the classifying simplicial scheme of PGL r consisting of all the quotients PGL r n +1/PGL r , n≥0.
Oblatum 8-IV-1998 & 8-X-1998 / Published online: 28 January 1999  相似文献   

14.
We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G n,p without destroying Hamiltonicity. We show that if pK ln n/n then there exists a constant α > 0 such that whp GH is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n. Research supported in part by NSF grant CCR-0200945. Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 526/05 from the Israel Science Foundation.  相似文献   

15.
We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p 2 and p 2 + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p 2 + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p 2, and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p 3 are exceptional.  相似文献   

16.
We prove a version of the L 2-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ⊂λ G ⊂ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K 0(C * r (G))→ℝ takes values in λ G . The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [15]. Oblatum 10-IV-2001 & 18-X-2001?Published online: 15 April 2002  相似文献   

17.
The group Aff(ℚ) of affine transformations with rational coefficients acts naturally not only on the real line ℝ, but also on the p-adic fields ℚp. The aim of this note is to show that all these actions are necessary and sufficient to represent bounded μ-harmonic functions for a probability measure μ on Aff(ℚ) that is supported by a finitely generated subgroup, that is, to describe the Poisson boundary. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.  相似文献   

18.
Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H2cb(G, Lp(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X). Partially supported by Sonderforschungsbereich 611.  相似文献   

19.
Let S 1 and S 2 be two Shimura curves over ℚ attached to rational indefinite quaternion algebras B 1 ℚ and B 1 ℚ with maximal orders B 1 and B 2 respectively. We consider an irreducible closed algebraic curve C in the product (S 1×S 2) such that C(ℂ) ∩ (S 1×S 2)(ℂ) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields, that C is of Hodge type. Received: 3 January 2000 / Revised version: 2 October 2000  相似文献   

20.
Two graphs G1 and G2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer proved that if Δ (G1) Δ (G2) < 0.5n, then G1 and G2 pack. In this note, we study an Ore-type analogue of the Sauer–Spencer Theorem. Let θ(G) = max{d(u) + d(v): uvE(G)}. We show that if θ(G1)Δ(G2) < n, then G1 and G2 pack. We also characterize the pairs (G1,G2) of n-vertex graphs satisfying θ(G1)Δ(G2) = n that do not pack. This work was supported in part by NSF grant DMS-0400498. The work of the first author was also partly supported by grant 05-01-00816 of the Russian Foundation for Basic Research.  相似文献   

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