Hausdorff operator of special kind on <Emphasis Type="Italic">p</Emphasis>-adic field and <Emphasis Type="Italic">BMO</Emphasis>-type spaces

For Hausdorff operator with generating function having support in the unit ball of p-adic field ℚ _p we give sufficient and necessary conditions of its boundedness in BMO-type spaces: BLO(ℚ _p ⁿ ), Q _r ^α,q (ℚ _p ⁿ ) and BMO _r ^α,q (ℚ _p ⁿ ). Some embedding relations between these spaces and Besov spaces are established.     

Period relations and p-adic measures

Let (π,σ) be a pair of cuspidal automorphic representations of GL ⁿ × GL ^{n −1} over the adele ring of ℚ having non-vanishing cohomology with constant coefficients. The p-adic distribution interpolating the critical values of the twisted corresponding Rankin–Selberg convolution is shown to be p-adically bounded thus leading to an associated p-adic L-function. Received: 18 October 2000 / Revised version: 27 July 2001     

<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

Some Elements of Finite Order in K2Q

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.     

Path integrals for quadratic lagrangians on <Emphasis Type="Italic">p</Emphasis>-adic and adelic spaces

Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of the number fields ℝ ↔ ℚ _p and ℚ ↔ ℚ _p , p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.     

On compression of Bruhat-Tits buildings

Consider an affine Bruhat-Tits building Lat _n of type A_n−1 and the complex distance in Lat _n, i.e., the complete system of invariants of a pair of vertices of the building. An element of the Nazarov semigroup is a lattice in the duplicated p-adic space ℚ _p ⁿ ⊕ ℚ _p ⁿ . We investigate the behavior of the complex distance with respect to the natural action of the Nazarov semigroup on the building. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 163–170.     

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of ℚ _p and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of p-adic geometry, to encode a dendrogram X in a p-adic field K means to fix a set S of K-rational punctures on the p-adic projective line ℙ₁. To ℙ₁ \ S is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers X, a method first used by F. Kato in 1999 in the classification of discrete subgroups of PGL₂(K). Next, we show how the p-adic moduli space of ℙ¹ with n punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on ℙ¹. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a p-adic algebraic curve with totally degenerate reduction modulo p. Finally, we indicate some of our results in the study of general discrete actions on ℙ¹, and their relation to p-adic Hurwitz spaces. The text was submitted by the author in English.     

On the iwasawa algebra associated to a normal element of ℂ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

Given a prime number p and the Galois orbit O(x) of a normal element x of ℂ _p , the topological completion of the algebraic closure of the field of p-adic numbers, we study the Iwasawa algebra of O(x) with scalars drawn from ℚ _p and relate it with ℚ _p -distributions and functionals.     

p-Adic Haar Multiresolution Analysis and Pseudo-Differential Operators

The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p ⁻¹. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ℒ²(ℚ₂) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ²(ℚ₂ ⁿ ) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).      

2-Adic wavelet bases

It is shown that a “p-adic plane wave” f(t + ω ₁ x ₁ + ... + ω _n x _n ), (t, x ₁, ..., x _n ) ∈ ℚ _p ^{n + 1}, where f is a Bruhat-Schwartz complex-valued test function and max_1≤j≤n |ω _j | _p = 1, satisfies, for any f, a certain homogeneous pseudodifferential equation, an analog of the classical wave equation. A theory of the Cauchy problem for this equation is developed.     

Scale functions on $p$-adic Lie groups

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 x∈G. 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     

On the volume of a spherical octahedron with symmetries

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 ℚ_p. 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.     

p-Adic orthogonal wavelet bases

We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L ²(ℚ _p ). The text was submitted by the authors in English.     

A derivative on the field of <Emphasis Type="Italic">p</Emphasis>-adic numbers

Using the fact that the multiplicative group of units in a p-adic field is a compact Vilenkin group, we introduce a new pointwise derivative on ℚ _p ^* and investigate basic properties of this derivative.     

Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

<Emphasis Type="Italic">p</Emphasis>-Adic nonorthogonal wavelet bases

A method for constructing MRA-based p-adic wavelet systems that form Riesz bases in L ²(ℚ _p ) is developed. The method is implemented for an infinite family of MRAs.     

On graded quotient modules of mapping class groups of surfaces

Let Γ _{g, n} be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n<0,g≥1,n≥0). The Torelli subgroup of Γ _{g, n} has a natural weight filtration {Γ_{g, n}(m)} _m≥1. Each graded quotient gr ^m Γ _{g, n} ⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S _n naturally acts. In this paper, we have determined the Sp(2g, ℚ)×S _n module structure of gr ^m Γ _{g, n} ⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr ^m Γ _{g, n} ⊗ ℚ by constructing explicitly elements in these modules.     

Cyclotomic and simplicial matroids

We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μ_n represented by then ^th roots of unity 1, ζ, ζ², …, ζ^n-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 μ_n in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.     

Free pro-p groups as galois groups over ℚ(p)(t)

Letp be a prime and let ℚ(p) denote the maximalp-extension of ℚ. We prove that for every primep, the free pro-p group on countably many generators is realizable as a regular extension of ℚ(p)(t). As a consequence, if ℚ _nil denotes the maximal nilpotent extension of ℚ, then every finite nilpotent group is realizable as a regular extension of ℚ _nil (t).     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999