Pseudodifferential <Emphasis Type="Italic">p</Emphasis>-adic vector fields and pseudodifferentiation of a composite <Emphasis Type="Italic">p</Emphasis>-adic function

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree T (O _p ) of balls in O _p . In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree T (O _p ) and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.     

Some aspects of <Emphasis Type="Italic">m</Emphasis>-adic analysis and its applications to <Emphasis Type="Italic">m</Emphasis>-adic stochastic processes

In this paper we consider a generalization of analysis on p-adic numbers field to the m case of m-adic numbers ring. The basic statements, theorems and formulas of p-adic analysis can be used for the case of m-adic analysis without changing. We discuss basic properties of m-adic numbers and consider some properties of m-adic integration and m-adic Fourier analysis. The class of infinitely divisible m-adic distributions and the class of m-adic stochastic Levi processes were introduced. The special class of m-adic CTRW process and fractional-time m-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time m-adic random walk.     

Hausdorff dimension in a family of self-similar groups

For every prime p and every monic polynomial f, invertible over p, we define a group G _{p, f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group . We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a characterization of finitely constrained groups in terms of the branching property, and as a corollary conclude that all defined groups are finitely constrained. In addition, we show that all infinite, finitely constrained groups of p-adic automorphisms have positive and rational Hausdorff dimension and we provide a general formula for Hausdorff dimension of finitely constrained groups. Further “finiteness” properties are also discussed (amenability, torsion and intermediate growth). Partially supported by NSF grant DMS-0600975.     

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.     

Behaviour of Hensel perturbations of p-adic monomial dynamical systems

In the framework of non-Archimedean (p-adic) analysis we study cyclic behaviour of polynomial discrete dynamical systems (iterations of polynomial maps). One of the main tools of our investigation is Hensel's lemma (a p-adic analogue of Newton's method). Our considerations will lead to formulas for the number cycles of a specific length and for the total number of cycles. We will also study the distribution of cycles in the different p-adic fields.     

Arrangements with one bounded component andp-adic integrals

In this article we study arrangementsA, such that ℝ ⁿ \A has exactly one bounded component. We obtain a result about their structure which gives us a method to construct all combinatorially different such arrangements in a given dimension. (A complete list for dimensions 1,2,3 and 4 is included). Furthermore we associate ap-adic integral to each such arrangement and proof that this integral can be written as a product ofp-adic beta functions. This is analogous to results of Varchenko and Loeser for integrals over ℝ and character sums over finite fields respectively.     

<Emphasis Type="Italic">p</Emphasis>-adic renormalization group solutions and the euclidean renormalization group conjectures

We discuss algebraic similarity of the Wilson’s renormalization groups in the Euclidean and p-adic spaces. Automodel Hamiltonians have identical form in both cases in the framework of perturbation theory. Fermionic p-adic model has exact renormalization group solution which generates a list of non-trivial conjectures for the Euclidean case.     

Structure,classification, and conformal symmetry of elementary particles over non-archimedean space-time

It is well known that at distances shorter than Planck length, no length measurements are possible. The Volovich hypothesis asserts that at sub-Planckian distances and times, spacetime itself has a non-Archimedean geometry. We discuss the structure of elementary particles, their classification, and their conformal symmetry under this hypothesis. Specifically, we investigate the projective representations of the p-adic Poincaré and Galilean groups, using a new variant of the Mackey machine for projective unitary representations of semidirect products of locally compact and second countable (lcsc) groups. We construct the conformal spacetime over p-adic fields and discuss the imbedding of the p-adic Poincaré group into the p-adic conformal group. Finally, we show that the massive and the so called eventually masssive particles of the Poincaré group do not have conformal symmetry. The whole picture bears a close resemblance to what happens over the field of real numbers, but with some significant variations.     

Syntomic regulators and special values of p-adic L-functions

In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields F and odd prime numbers p, which generalize Leopoldt's p-adic class number formula, and express special values of p-adic L-functions in terms of orders of K-groups and higher p-adic regulators. The approach uses syntomic regulator maps, which are the p-adic equivalent of the Beilinson regulator maps. They can be compared with étale regulators via the Fontaine-Messing map, and computations of Bloch-Kato in the case that p is unramified in F lead to results about generalized Coates-Wiles homomorphisms and cyclotomic characters. Oblatum 14-V-96 & 9-X-97     

p-adic multiple zeta values I.

Our main aim in this paper is to give a foundation of the theory of p-adic multiple zeta values. We introduce (one variable) p-adic multiple polylogarithms by Colemans p-adic iterated integration theory. We define p-adic multiple zeta values to be special values of p-adic multiple polylogarithms. We consider the (formal) p-adic KZ equation and introduce the p-adic Drinfeld associator by using certain two fundamental solutions of the p-adic KZ equation. We show that our p-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p-adic KZ equation and our p-adic multiple zeta values appear as coefficients of the p-adic Drinfeld associator. We show various properties of p-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the p-adic case, via the p-adic KZ equation.     

Self-similar p-adic fractal strings and their complex dimensions

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. The text was submitted by the authors in English.     

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 p-adic meromorphy of the function field height zeta function

In this brief note, we will investigate the number of points of bounded height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the p-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is p-adic meromorphic.     

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of Cp. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).     

Digraph representations of rational functions over the <Emphasis Type="Italic">p</Emphasis>-adic numbers

In this paper, we construct a digraph structure on p-adic dynamical systems defined by rational functions. We study the conditions under which the functions are measure-preserving, invertible and isometric, ergodic, and minimal on invariant subsets, by means of graph theoretic properties.     

Kirillov's Orbit Method for p-Groups and Pro-p Groups

In this text, we study Kirillov's orbit method in the context of Lazard's p-saturable groups when p is an odd prime. Using this approach we prove that the orbit method works in the following cases: torsion free p-adic analytic pro-p groups of dimension smaller than p, pro-p Sylow subgroups of classical groups over ? _p of small dimension and for certain families of finite p-groups.     

A p-adic analogue of Siegel's theorem on sums of squares

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded.     

Fourier series, measures and divided power series in the theory of function fields

Much of classical number theory is based on Fourier series. Such series play a vital role in the study of characteristic-0 zeta-functions: In the complex theory one has theta-series and Tate's thesis. In the p-adic theory one has Mahler's theorem on binomial coefficients which is used to déscribe the ring of p-adic measures. In this paper, we discuss a version of binomial coefficients for function fields due to L. Carlitz. We will show how these functions arise naturally out of gamma functions for function fields. We will also use some work of C. Wagner to establish that the ring of -adic measures is canonically isomorphic to the ring of divided power-series. The computation of these power-series in specific instances is now an important problem in the theory. Finally, we show the existence of many Fourier transforms in the -adic theory. The explicit computation of these would also be very interesting.Partially supported by NSF grant DMS-8521678. Current address: Department of Mathematics, UMBC, MD 21228, U.S.A.Dedicated to L. Carlitz     

Automata finiteness criterion in terms of van der Put series of automata functions

In the paper we develop the p-adic theory of discrete automata. Every automaton \mathfrakA\mathfrak{A} (transducer) whose input/output alphabets consist of p symbols can be associated to a continuous (in fact, 1-Lipschitz) map from p-adic integers to p-adic integers, the automaton function f_\mathfrakA f_\mathfrak{A} . The p-adic theory (in particular, the p-adic ergodic theory) turned out to be very efficient in a study of properties of automata expressed via properties of automata functions. In the paper we prove a criterion for finiteness of the number of states of automaton in terms of van der Put series of the automaton function. The criterion displays connections between p-adic analysis and the theory of automata sequences.