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.     

On the Iwasawa Theory of p-Adic Lie Extensions

In this paper, the new techniques and results concerning the structure theory of modules over noncommutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k of number fields k 'up to pseudo-isomorphism'. In particular, a close relationship is revealed between the Selmer group of Abelian varieties, the Galois group of the maximal Abelian unramified p-extension of k as well as the Galois group of the maximal Abelian p-extension unramified outside S where S is a certain finite setof places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.     

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.     

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 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.     

Little q-Legendre Polynomials and Irrationality of Certain Lambert Series

Certain q-analogs h _p(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erds (J. Indiana Math. Soc. 12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory 37, 1991, 253–259; Proc. Cambridge Philos. Soc. 112, 1992, 141–146) used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs ln _p (2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math. 20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apéry's proof of irrationality of (2) and (3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Väänänen (Compositio Math. 91, 1994, 175–199) and recently also by Matala-aho and Väänänen (Bull. Australian Math. Soc. 58, 1998, 15–31) (for ln _p (2)). In this paper we show how one can obtain rational approximants for h _p(1) and ln _p (2) (and many other similar quantities) by Padé approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for h _p(1) and a better upper bound as the one given by Matala-aho and Väänänen for ln _p (2).     

<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.     

On irrationality measures of the values of Gauss hypergeometric function

The paper gives irrationality measures for the values of some Gauss hypergeometric functions both in the archimedean andp-adic case. Further, an improvement of general results is obtained in the case of logarithmic function.     

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.     

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.     

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 some 2-adic properties of a recurrence involving stirling numbers

We analyze some 2-adic properties of the sequence defined by the recurrence Z(1) = 1; Z(n) = Σ _k=1 ⁿ⁻¹ S(n, k)Z(k), n ≥ 2, which counts the number of ultradissimilarity relations, i.e., ultrametrics on an n-set. We prove the 2-adic growth property ν ₂(Z(n)) ≥ ⌈log₂ n⌉ −1 and present conjectures on the exact values.     

The approximate identity kernels of product type for the Walsh system

At present there are only a few approximate identity kernels for the Walsh system, for example, the p^N-truncated Dirichlet kernel D_{p^N − 1}(t) = ∑_{j = 0}^{pN − 1} w_j(t) [6]; the Abel-Poisson kernel λ_γ(t) = ∑_{k = 0}^∞ γ^kw_k(t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L¹-norm.     

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.     

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.     

Fonctions L associées aux F-isocristaux surconvergents II: Zéros et pôles unités

Résumé Nous démontrons la conjecture de Katz concernant la méromorphie et la caractérisation des zéros et p?les unités des fonctions L associées aux représentations p-adiques lorsque celles-ci se prolongent sur une compactification du schéma de base. Comme cas particuliers importants, on obtient celui de la fonction zêta d’un schéma quelconque et celui d’une représentation p-adique quelconque sur un schéma propre.

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If X is a smooth variety over a finite field ? _q of characteristic p > 0 and ℱ is a p-adic sheaf associated to a representation of the fundamental group of X, N. Katz conjectures, in his Bourbaki talk 409, that the L function L (X, ℱ, t) has its p-adic unit roots and poles given in terms of p-adic étale cohomology. We prove this conjecture in the case of the structure sheaf ℱ = ℤ _p , that is for the Zeta function, and also more generally when the p-adic sheaf ℱ extends to a smooth sheaf on a compactification of X: as a consequence we get the Unit-Root Zeta function of Dwork and Sperber as an L function. The idea of the proof is to get the p-adic étale cohomology with coefficients and compact support as the fixed points of Frobenius acting on rigid cohomology with compact support. For this purpose, we first build a crystalline Artin–Schreier short exact sequence on the syntomic site of a scheme which is separated of finite type over a perfect field k: this naturally generalizes the work of J.M. Fontaine and W. Messing in the proper smooth case. Then getting rigid cohomology with coefficients as a limit of crystalline cohomologies of variable level we deduce a long exact sequence connecting p-adic étale cohomology (with compact support) to rigid cohomology (with compact support). When X is smooth and affine over an algebraically closed field, the former exact sequence splits into short exact sequences that identify the p-adic étale cohomology with support of X to the part of its rigid cohomology invariant under Frobenius. We can then describe the p-adic unit roots and poles of the Zeta function of X; as a corallary we get the Unit-Root Zeta function of Dwork and Sperber as an L function. In the appendix we show that the characteristic spaces of Frobenius in rigid cohomology commute with isometric extensions of the base, and that isocrystals associated to p-adic sheaves with finite monodromy are overconvergent: we thus obtain a p-adic proof of the rationality of the corresponding L-function.

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Oblatum 8-XII-1994 & 30-IV-1996     

On <Emphasis Type="Italic">k</Emphasis>-pairable graphs from trees

The concept of the k-pairable graphs was introduced by Zhibo Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter p(G), called the pair length of a graph G, as the maximum k such that G is k-pairable and p(G) = 0 if G is not k-pairable for any positive integer k. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved are restricted to be trees. That is, we characterize the trees G with p(G) = 1 and prove that p(G □ H) = p(G) + p(H) when both G and H are trees.     

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.     

K3 et formules de Riemann-Hurwitzp-adiques

The well-known formula of Riemann-Hurwitz gives the change of genuses in ann-fold covering of compact connected Riemann surfaces. In Iwasawa theory, there existp-adic analogues which give the change of certain ^±-invariants in ap-extension ofCM number fields. Using functorial and arithmetical properties ofK ₃, we extend such Riemann-Hurwitzp-adic formulas to non-CM fields, assuming some restrictive hypotheses on the capitulation ofK ₂.