A construction of classifying spaces for p-adic group actions

One major open problem in geometric topology is the Hilbert-Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor's construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions.     

Badly approximablep-adic integers

It is known that thep-adic integers that are badly approximable by rationals form a null set with respect to Haar measure. We define a [0,1]-valued dimension function on thep-adic integers analogous to Hausdorff dimension inR and show that with respect to this function the dimension of the set of badly approximablep-adic integers is 1.     

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

On periodic p-adic distributions

In this paper we introduce a notion of periodic p-adic distribution defined on ? _p — the set of p-adic integers. This periodicity depends on a partition of ? _p . For several concrete partitions we describe corresponding periodic p-adic distributions. Moreover, we construct a periodic p-adic measure.     

On recent results of ergodic property for p-adic dynamical systems

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.     

On ergodicity of p-adic dynamical systems for arbitrary prime p

In this paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of 1-Lipschitz p-adic functions that are defined on (and valuated in) the space ? _p of p-adic integers for any prime p. The conditions are stated in terms of coordinate representations of p-adic functions.     

A note on p-adic L-functions

An elementary proof is given for the existence of the Kubota-Leopoldt p-adic L-functions. Also, an explicit formula is obtained for these functions, and a relationship between the values of the p-adic and classical L-functions at positive integers is discussed.     

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.     

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.     

A Dirichlet Space on Ends of Tree and Dirichlet Forms with a Nodewise Orthogonal Property

Stochastic processes on the field of p-adic numbers have been studied for more than two decades. The theory of Dirichlet space is applied in important parts of the studies. In some recent observations, the hierarchical structure is handled by tree structure and the structure gives an overview for measure symmetric stochastic processes on the field of p-adic numbers and ends of a tree. An orthogonal property described in terms of Dirichlet form for eigenfunctions associated with distinct nodes is involved coherently in the existing formalisms based on Dirichlet space theory. The objective of the present article is paying a close attention to some characteristic property implied by the orthogonal property and presenting a wider formalism in accordance with the characteristic property.     

On the representation of integers by p-adic diagonal forms

Let p be an odd prime, let d be a positive integer such that (d,p?1)=1, let r denote the p-adic valuation of d and let m=1+3+3²+…+3^r. It is shown that for every p-adic integer n the equation Σ_i=1^mX_i^d=n has a nontrivial p-adic solution. It is also shown that for all p-adic units a₁, a₂, a₃, a₄ and all p-adic integers n the equation Σ_i=1⁴a_iX_i^p=n has a nontrivial p-adic solution. A corollary to each of these results is that every p-adic integer is a sum of four pth powers of p-adic integers.     

p-Adic dynamics and angle doubling

We consider the squaring map over the p-adic numbers for an odd prime p, and study its symbolic dynamics on the unit circle in ? _p , the p-adic integers. When the map is restricted to the set of squares, we show an equivalence to angle doubling (mod 1) for rational angles. For primes p ≡ 3 (mod 4), this map may be represented as a unitary permutation matrix of the type used in quantum phase estimation.     

Simultaneous approximation problems of <Emphasis Type="Italic">p</Emphasis>-adic numbers and <Emphasis Type="Italic">p</Emphasis>-adic knapsack cryptosystems - Alice in <Emphasis Type="Italic">p</Emphasis>-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l _∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.     

On Carlitz's q-Bernoulli numbers

Properties of q-extensions of Bernoulli numbers and polynomials which generalize those satisfied by B_k and B_k(x) are used to construct q-extensions of p-adic measures and define a q-extension of p-adic Dirichlet L-series.     

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).     

Extensions of the representation modules of a prime order group

For the ring R of integers of a ramified extension of the field of p-adic numbers and a cyclic group G of prime order p we study the extensions of the additive groups of R-representations modules of G by the group G.     

Solvability of cubic equations in p-ADIC integers (p > 3)

We give a criterion for the existence of solutions to an equation of the form x ³ + ax = b, where a, b ∈ ? _p , in p-adic integers for p > 3. Moreover, in the case when the equation x ³ + ax = b is solvable, we give necessary and sufficient recurrent conditions on a p-adic number x ∈ ?* _p under which x is a solution to the equation.     

On p-adic Diamond–Euler Log Gamma functions

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In this paper, using the fermionic p  -adic integral on _Zp${\mathbb{Z}}_p$, we define the corresponding p-adic Log Gamma functions, so-called p-adic Diamond–Euler Log Gamma functions. We then prove several fundamental results for these p-adic Log Gamma functions, including the Laurent series expansion, the distribution formula, the functional equation and the reflection formula. We express the derivative of p-adic Euler L  -functions at s=0$s=0$ and the special values of p-adic Euler L-functions at positive integers as linear combinations of p-adic Diamond–Euler Log Gamma functions. Finally, using the p-adic Diamond–Euler Log Gamma functions, we obtain the formula for the derivative of the p  -adic Hurwitz-type Euler zeta function at s=0$s=0$, then we show that the p-adic Hurwitz-type Euler zeta functions will appear in the studying for a special case of p  -adic analogue of the (S,T)$(S,T)$-version of the abelian rank one Stark conjecture.

Video

For a video summary of this paper, please click here or visit http://youtu.be/DW77g3aPcFU.     

A family of p-adic twisted interpolation functions associated with the modified Bernoulli numbers

The main purpose of this paper is to construct a family of modified p-adic twisted functions, which interpolate the modified twisted q-Bernoulli polynomials and the generalized twisted q-Bernoulli numbers at negative integers. We also give some applications and examples related to these functions and numbers.     

p-adic affine dynamical systems and applications

We consider p-adic affine dynamical systems on the ring ${\mathbb{Z}}_p$ of all p-adic integers, and we find a necessary and sufficient condition for such a system to be minimal. The minimality is equivalent to the transitivity, the ergodicity of the Haar measure, the unique ergodicity, and the strict ergodicity. When the condition is not satisfied, we prove that the system can be decomposed into strict ergodic subsystems. One of our applications is the study of the divisibility, by a power of prime number, of the sequence of integers $a^n?b$ with positive integers $a,b$ and n. To cite this article: A.-H. Fan et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).