Bases normales,unités et conjecture faible de leopoldt

Letk be a number field,p an odd prime,R _k the ring ofp-integers ofk. We use Iwasawa theory to study theZ _p -moduleG(R _k ,Z _p ) (resp.NB (R _k ,Z _p )) ofclasses ofZ _p -extensions (resp.Z _p -extensions having a normal basis overR _k ) ofR _k . The rank ofG(G _k ,Z _p ) (resp.NB(R _k ,Z _p )) is related to Leopoldt's conjecture (resp. weak Leopoldt's conjecture) fork andp.      

Normal series and character values in p-standard table algebras

We investigate the character values and structures of p-standard table algebras (A,B) with o(B) = p^N. If N≤3, then B has a complete normal series. If for every χ∈Irr(B), χ has at most p distinct classes of character values, and if either B has a complete normal series or p = 2, then B is an elementary abelian p-group.     

CirculantGH(p 2; Z p ) exist for all primesp

The only known circulant ordinary Hadamard matrix is developed from the initial row-1, 1, 1, 1. Letp be a prime, and letZ _p denote the cyclic group of orderp. In this paper, we construct circulantGH(p ²;Z _p ) for all primesp. Whenp is odd, this result also extends the earlier result that there exist circulantGH(p;Z _p ) for all odd primesp. Other families ofGH-matrices which are developed modulo a group are discussed.     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

The Central Limit Theorem on SO0(p, q)/SO(p)×SO(q)

We obtain a central limit theorem for the space SO ₀(p, q)/SO(p)×SO(q). To achieve this, we derive a Taylor expansion of the spherical function on the group SO ₀(p, q).     

Summability Kernels for <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Multipliers

In this article we study the problem of extending Fourier Multipliers on L ^p (T) to those on L ^p (R) by taking convolution with a kernel, called a summability kernel. We characterize the space of such kernels for the cases p = 1 and p = 2. For other values of p we give a necessary condition for a function to be a summability kernel. For the case p = 1, we present properties of measures which are transferred from M(T) to M(R) by summability kernels. Furthermore it is shown that every l _p sequence can be extended to some L ^q (R) multipliers for certain values of p and q.     

Fitting ideals and the Gorenstein property

Let p be a prime number and G be a finite commutative group such that p ² does not divide the order of G. In this note we prove that for every finite module M over the group ring Z _p [G], the inequality #M  ￡  #Z_p[G]/Fit_Zp[G](M){\#M\,\leq\,\#{\bf Z}_{p}[G]/{{\rm Fit}}_{{\bf Z}_{p}[G]}(M)} holds. Here, Fit_Zp[G](M){\rm Fit}_{{\bf Z}_{p}[G]}(M) is the Z _p [G]-Fitting ideal of M.     

Definable one dimensional structures in o-minimal theories

Let N be a structure definable in an o-minimal structure M and p ∈ S _N (N), a complete N-1-type. If dim _M (p) = 1, then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N-definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to a generic type of an N-definable real closed field.     

The eigen functions of the complex projective space

In the complexn-dimensional projective spaceCP ⁿ , let λ _p (=4p(p+n)) be the eigen value of the Laplace-Beltrami operator andH _p be the space of all eigen functions of eigen value λ _p . The reproducing kernelh _p (z, w) ofH _p is constructed explicitly in this paper, and a system of complete orthogohal functions ofH _p is constructed fromh _p (z,w)(p=1,2, …). Partially supported by NSF of China     

Higher Power Residue Codes

For certain primeslandp, and characters χ:F*_p→F*_l², we construct codesWof lengthp+ 1 overF_l²which are linear overF_l, but not overF_l², and which are invariant under a monomial action of the group SL(2,p). We consider the cases of cubic and quartic characters in detail and use theWto construct linear codes overF_lin these cases.     

On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.     

Farrell cohomology of GL(n,Z)

The Tate-Farrell cohomology of GL(n,Z) with coefficients inZ/p is computed forp an odd prime andp−1 ≦n ≦ 2p−3. Its size depends on the Galois structure of the class group of the cyclotomic fieldQ(p√1) and is shown to be quite large in general. Research partially supported by NSF Grant No. DMS-8701758.     

Orbits of euclidean frames under discrete linear groups

We obtain certain sufficient conditions for the orbit of a (euclidean)p-frame over a vector spaceV,p<dimV, under the action of a discrete subgroup of GL(V), to be dense in the corresponding orbit of a Lie subgroup of GL(V). Using the result we classify thep-frames whose orbits under SL (n,Z) are dense in the space ofp-frames and deduce, in turn, a classification of dense orbits of certain horospherical flows. A similar result is obtained for Sp (2n,Z) forp≦n.     

Estimates for Hyperbolic Equations of Space Dimension 3

We discuss the problem of boundedness fromL^p(Rⁿ) toL^p′(Rⁿ) (1/p+1/p′=1, 1?p?2) of operators of the typeM=F⁻¹e^i?(ξ)a(ξ) F, which is related to the study of hyperbolic equations with constant coefficients. The boundedness is dependent on a geometrical property ofΣ=?⁻¹(1), and its dependence has been exactly determined in the casesn=2, 1?p?2 andn?3,p=1, 2 (M. Sugimoto,Math. Z.215(1994), 519–531;222(1996), 521–531). This paper is devoted to the unsolved case 1<p<2, and a strange phenomenon is exhibited in the simplest casen=3.     

On the Q-rational cuspidal subgroup and the component group ofJ 0(p r )

Forp≥3 a prime, we compute theQ-rational cuspidal subgroupC(p ^r ) of the JacobianJ ₀(p ^r ) of the modular curveX ₀(p ^r ). This result is then applied to determine the component group Φ _p ^r of the Néron model ofJ ₀(p ^r ) overZ _p . This extends results of Lorenzini [7]. We also study the action of the Atkin-Lehner involution on thep-primary part ofC(p ^r ), as well as the effect of degeneracy maps on the component groups.     

Lacunary Series in Weighted Hyperholomorphic B p,q (G) Spaces

The goal of this article is two-fold. First, we consider a class of hyperholomorphic functions, the so called B ^{p, q} (G) space in ?³. Then, we use the B ^{p, q} (G) space to characterize the hyperholomorphic α-Bloch space. Second, we obtain characterizations of the weighted hyperholomorphic B ^{p, q} (G)-functions by the coefficients of certain lacunary series expansions in Clifford Analysis.     

Finite Groups with a Disconnected p -Regular Conjugacy Class Graph

Abstract

Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ _p (G) whose vertices are the non-central p-regular conjugacy classes of G and connect two vertices by an edge if their cardinalities have a common prime divisor. In this note we study the structure and arithmetical properties of the p-regular class sizes in p-solvable groups G having Γ _p (G) disconnected.     

Alcune classi di gruppi abeliani misti di rango 1

Riassunto Si definisce una classe diZ _p-moduliQ _β, simili aip-gruppiP _β studiati daE. A. Walker, che danno una caratterizzazione delle sequenze esatte bilanciate diZ _p-moduli e degliZ _p-moduli bilanciati proiettivi. Si definiscono poi dei gruppi misti di rango 1,R _h, che sono un'estensione al caso globale deiQ _β e che generalizzano i gruppi razionali. Per gliR _h vale un teorema analogo a quello di classificazione coi tipi di Baer ed essi danno inoltre una caratterizzazione delle sequenze esatte bilanciate di gruppi e dei gruppi bilanciati proiettivi.

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Summary We define a class ofZ _p-modulesQ _β, like thep-groupsP _β studied byE. A. Walker, that give a characterization of balanced exact sequence ofZ _p-modules and balanced projectiveZ _p-modules. We introduce also some mixed groups of rank one,R _h, which are an extension to the global case ofQ _β and which generalize rational groups. Moreover, forR _h, there is a theorem, which is analogous to Baer's theorem of classification by types. GroupsR _h give too a characterization of balanced exact sequence of groups and of balanced projective groups.

     

Paley Type Inequalities for Several Parameter Vilenkin Systems

The aim of this paper is to prove Paley type inequalities for two-parameter Vilenkin system. Our main result is the following estimate:

Representation ofL p-norms and isometric embedding inL p-spaces

For fixed 1≦p<∞ theL _p-semi-norms onR ⁿ are identified with positive linear functionals on the closed linear subspace ofC(R ⁿ ) spanned by the functions |<ξ, ·>| ^p , ξ∈R ⁿ . For every positive linear functional σ, on that space, the function Φ_σ:R ⁿ →R given by Φ_σ is anL _p-semi-norm and the mapping σ→Φ_σ is 1-1 and onto. The closed linear span of |<ξ, ·>| ^p , ξ∈R ⁿ is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in anyL _p unlessp=2. Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley.