Rigged Hilbert Space of Free Coherent States and p-Adic Numbers

We investigate the rigged Hilbert space of free coherent states. We prove that this rigged Hilbert space is isomorphic to the space of generalized functions over a p-adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show that the considered example realizes the isomorphism between the noncommutative line and the p-adic disk.     

Central units in p-adic group rings

If p is a natural prime and G any p-group then an exact sequence is constructed to describe the group of units in the center of the p-adic group ring _p [G].     

Remarks on the Transfer Factors for Unipotent Orbital Integrals in p-adic Classical Split Groups

This article studies unipotent orbital integrals on symplectic and orthogonal groups from the point of view of endoscopy. It begins by partitioning stable unipotent classes into packets and goes on to propose a transfer of these packets. It then discusses (in rough form) the associated transfer factors. Some supporting calculations in split odd orthogonal groups are given.     

A remark on congruences for coefficients of Faber polynomials

Faber polynomials appear when weight zero Hecke operators act on the modular j-invariant. They are polynomials in j with rational integral coefficents. Using the theory of p-adic modular forms we establish some congruences and divisibilities for these coefficients. 2000 Mathematics Subject Classification Primary—11F33 Supported by NSF grant DMS-0501225.     

p-Adic Automorphic Functions for the Unitary Group in Three Variables

Let G be a special unitary group in three variables over a local field and, a co-compact subgroup of G. We construct, -automorphic functions on the rigid analytic space of points in the projective plane stable under the action of tori in G.This research was partially supported by an ARC grant.2000 Mathematics Subject Classification: 11F55, 11F85, 20G25, 22E35     

Sobolev Space and Dirichlet Space Associated with Symmetric Markov Process on a Local Field

The space ℱ_r,p, which was designed so as to play similar roles to the ordinary Sobolev space W^r,p(ℝⁿ), introduced as a cornerstone for analyzing nonlinear potential theoretic features of the state space with a measure-symmetric transition probability semi-group. The aim of this article is revealing a sufficient condition for the counterpart of the Sobolev space to coincide with domain of some Dirichlet form on a local field and discussing some other features of those counterparts on the non-Archimedean metric space. For example, we will see a sufficient condition for the space ℱ_r,2 to be viewed as domain of some Dirichlet form and microscopic property such as polarity of singleton will be investigated. Mathematics Subject Classifications (2000) 31C45, 11D88 , 46E35, 60J25.     

Gaussian Estimates for Random Walks on Some Unimodular p-adic Groups

It is shown in this paper that the transition kernels corresponding to simple random walks on certain unimodular solvable p-adic groups admit upper Gaussian estimates.      

Recurrence and Transience Criteria for Symmetric Hunt Processes

In this paper, we will discuss recurrence, transience and other potential theoretic aspects based on symmetric regular Dirichlet space. We will first deal with Dirichlet space with the strong local property and give a recurrence criterion in terms of exhaustion function. This criterion shows that recurrence automatically provides us with an exhaustion function which is usable to verify a Liouville property on subharmonic functions. Secondly, a recurrence criterion and a transience criterion for a Nonlocal Dirichlet space will be presented. Those criteria can be applied to Albeverio–Karwowski"s random walks on p-adic number field. Lastly, we will prove the assertions which cover other potential theoretic aspect of p-adic number field such as Liouville property on harmonic functions.     

Weakly infinitely divisible measures on some locally compact Abelian groups

On the torus group, on the group of p-adic integers, and on the p-adic solenoid, we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of ℝ. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid.     

Homogeneity of Certain Invariant Distributions on the Lie Algebra of p-adic GLn

Let F be a non-Archimedean local field with ring of integers R and prime ideal . Suppose T is a GL _n (F)-invariant distribution on =M _n (F), the Lie algebra of GL _n (F). If T has support in the set of topologically nilpotent elements, then the restriction of T to the set of functions which are compactly supported and invariant under M _n ( ) may be expressed as a linear combination of nilpotent orbital integrals restricted to the same set of functions.     

Formal Flows on the Non-Archimedean Open Unit Disk

This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series (t,x) satisfying the conditions (0,x)=x and (F(s,t),x)=(s,(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows with any specified divisor of fixed points, where a point of the open unit disk is a fixed point of order n if (x–) ⁿ |((t,x)–x). Furthermore, if is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow , an element of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of (x) is equal to (,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.     

Stochastic Integrals and Stochastic Differential Equations over the Field of p-Adic Numbers

We construct, for various classes of p-adic-valued functions, stochastic integrals with respect to the Poisson random measure. This leads to the construction of Markov processes over the field of p-adic numbers by means of stochastic differential equations.     

关于调和数的一个结论

设n是大于1的正常数,并且设n=pα11p2α2…ptαt,其中pi为素数,i=1,2,…,t,ω(n)表示n的不同素因子的个数,即ω(n)=t.若n的所有因子的倒数和为整数,即0≤∑ij≤αjj=1,2,…,t1p1i1pi22…ptit为整数,称n是调和数.证明了和调和数相关的一个结论.     

Irreducibility of the Igusa tower

We shall give a simple （basically） the Igusa tower over Shimura varieties of PEL purely in characteristic p proof of the irreducibility of type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert-Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.     

On the Prime Radical of PI-Representable Groups

The notion of PI-representable groups is introduced; these are subgroups of invertible elements of a PI-algebra over a field. It is shown that a PI-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated PI-representable group is solvable. The class of PI-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.     

<Emphasis Type="Italic">p</Emphasis>-Adic Pseudodifferential Operators and Analytic Continuation of Replica Matrices

We consider the procedure for analytic continuation of the replica matrix. We formulate a particular form of this procedure in which the analytic continuation is defined by a sequence of maps. Using this definition, we construct a solution that breaks the Parisi replica symmetry and find the corresponding p-adic pseudodifferential operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 336–341, August, 2005     

Infinitely divisible probabilities on linearp-adic groups

In this paper we extend the work of Shah, on the structure of infinitely divisible probabilities onp-adic linear groups, to give a classification for all such probabilities.     

On the Fontaine–Mazur Conjecture for CM-Fields

Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.     

Milnor K-Groups and Zero-Cycles on Products of Curves over p-Adic Fields

We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH ₀(X)/m for X a product of curves over a p-adic field.     

On the divisibility of the discriminant of an integral polynomial by prime powers

The discriminant of an integral polynomial is one of its main characteristics. It influences the distribution of its roots, the structure of the finite extension of the rational field generated by the polynomial's roots. In the paper, we show that, for any given prime power p ^b , there exists an irreducible polynomial with discriminant being a multiple of p ^b .