A polynomial <Emphasis Type="Italic">p</Emphasis>-adic dynamical system

We completely describe the Siegel discs and attractors for the p-adic dynamical system f(x) = x ²ⁿ⁺¹ + axⁿ⁺¹ on the space of complex p-adic numbers.     

Representation Results for Equivariant Rigid Analytic Functions

Given a prime number p and the Galois orbit O(x) of an element x of ℂ _p , the topological completion of the algebraic closure of the field of p-adic numbers, we are interested in the representation results for equivariant rigid analytic functions defined on ℙ¹(ℂ _p ) \ O(x) with values in ℂ _p that vanishes at ∞.     

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).      

Irrationality of some p-adic L-values

We give a proof of the irrationality of p-adic zeta-values ξp（κ） for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function.     

On p-adic quaternionic Eisenstein series

We show that certain p-adic Eisenstein series for quaternionic modular groups of degree 2 become “real” modular forms of level p in almost all cases. To prove this, we introduce a U(p) type operator. We also show that there exists a p-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of p-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).     

p-adic Siegel–Eisenstein series of degree two

We prove an explicit formula for Fourier coefficients of Siegel–Eisenstein series of degree two with a primitive character of any conductor. Moreover, we prove that there exists the p-adic analytic family which consists of Siegel–Eisenstein series of degree two and a certain p-adic limit of Siegel–Eisenstein series of degree two is actually a Siegel–Eisenstein series of degree two.     

Families of Siegel modular forms, <Emphasis Type="Italic">L</Emphasis>-functions and modularity lifting conjectures

For a prime p and a positive integer g, by making use of certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus g and associated p-adic L-functions. Describing L-functions attached to Siegel modular forms and their analytic properties from the point of view of motivic L-functions studied by Deligne and Yoshida, we discuss critical values of the L-functions and p-adic interpolation problems. In particular, we formulate a general conjecture on the existence of the modularity lifting from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.     

On zeta functions and families of Siegel modular forms

Let p be a prime, and let G = \textS\textp_g( \mathbbZ ) \Gamma = {\text{S}}{{\text{p}}_g}\left( \mathbb{Z} \right) be the Siegel modular group of genus g. This paper is concerned with p-adic families of zeta functions and L-functions of Siegel modular forms; the latter are described in terms of motivic L-functions attached to Sp _g ; their analytic properties are given. Critical values for the spinor L-functions are discussed in relation to p-adic constructions. Rankin’s lemma of higher genus is established. A general conjecture on a lifting of modular forms from GSp_2m   × GSp_2m to GSp_4m (of genus g = 4 m) is formulated. Constructions of p-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions.     

On rational Drinfeld associators

We prove an estimate on denominators of rational Drinfeld associators. To obtain this result, we prove the corresponding estimate for the p-adic associators stable under the action of suitable elements of Gal([`(\mathbbQ)]/\mathbbQ){{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})} . As an application, we settle in the positive Duflo’s question on the Kashiwara–Vergne factorizations of the Jacobson element J _p (x, y) = (x + y) ^p − x ^p − y ^p in the free Lie algebra over a field of characteristic p. Another application is a new estimate on denominators of the Kontsevich knot invariant.     

Some congruences for Saito-Kurokawa lifts

We give two examples of congruences between Saito-Kurokawa lifts. Moreover, we prove that a certain p-adic limit of Siegel-Eisenstein series of level one becomes a Siegel modular form of level p and trivial character.     

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.     

On the functional equation <Emphasis Type="Italic">Af</Emphasis><Superscript>2</Superscript> + <Emphasis Type="Italic">Bg</Emphasis><Superscript>2</Superscript> = 1 on the field of complex <Emphasis Type="Italic">p</Emphasis>-adic numbers

For a fixed prime p, let C _p denote the complex p-adic numbers. For polynomials A, B ε C p [x] we consider decompositions A (x) f ² (x) + B (x) g ² (x) = 1 of entire functions f, g on C p and try to improve an impossibility result due to A. Boutabaa concerning transcendental f, g. We also provide a new proof of a p-adic diophantic statement due to D. N. Clark, which is an important ingredient of Boutabaa’s method.     

Upper triangular operators and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

For a newform f for Γ₀(N) of even weight k supersingular at a prime p ≥ 5, by using infinite dimensional p-adic analysis, we prove that the p-adic L-function L _p (f,α; χ) has finite order of vanishing at any character of the form [(c)\tilde] _s ( x ) = x^s\tilde \chi _s \left( x \right) = x^s. In particular, under the natural embedding of ℤ _p in the group of ℂ* _p -valued continuous characters of ℤ* _p , the order of vanishing at any point is finite.     

On a class of rational p-adic dynamical systems

In this paper we investigate the behavior of trajectories of one class of rational p-adic dynamical systems in complex p-adic field Cp. We studied Siegel disks and attractors of such dynamical systems. We found the basin of the attractor of the system. It is proved that such dynamical systems are not ergodic on a unit sphere with respect to the Haar measure.     

On the number of equivalence classes of attracting dynamical systems

We study discrete dynamical systems of the kind h(x) = x + g(x), where g(x) is amonic irreducible polynomial with coefficients in the ring of integers of a p-adic field K. The dynamical systems of this kind, having attracting fixed points, can in a natural way be divided into equivalence classes, and we investigate whether something can be said about the number of those equivalence classes, for a certain degree of the polynomial g(x). The text was submitted by the authors in English.     

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.     

Onp-adic functions preserving Haar measure

Let {a _n } _n ^=0/ be a uniformly distributed sequence ofp-adic integers. In the present paper we study continuous functions close to differentiable ones (with respect to thep-adic metric); for these functions, either the sequence {f(a _n )} _n ^=0/ is uniformly distributed over the ring ofp-adic integers or, for all sufficiently largek, the sequences {f _k (_k(a_n))} _n ^=0/ are uniformly distributed over the residue class ring modp ^k , where _k is the canonical epimorphism of the ring ofp-adic integers to the residue class ring modp ^k andf _k is the function induced byf on the residue class ring modp ^k (i.e.,f k _(x) =f( _k (x)) (modp ^k )). For instance, these functions can be used to construct generators of pseudorandom numbers.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 935–950, June, 1998.In conclusion, the author wishes to express his deep gratitude to V. S. Anashin for permanent attention to this research and for support.     

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.     

Congruences between Siegel modular forms II

Using a p-adic monodromy theorem on the affine ordinary locus in the minimally compactified moduli scheme modulo powers of a prime p of abelian varieties, we extend Katz?s results on congruence and p-adic properties of elliptic modular forms to Siegel modular forms of higher degree.     

<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.