Ring Of P-adic Integers Research Articles

The narrow class groups of the ℤ17- and ℤ19-extensions over the rational field

Let p be either 17 or 19, let ℤp denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p2. We shall prove, with the help of a computer, that the l-class group of the ℤp-extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤp-extension.

The p-adic quantum plane algebras and quantum Weyl algebra

Let q be a principal unit of the ring of valuation of a complete valued field K, extension of the field of p-adic numbers. Generalizing Mahler basis, K. Conrad has constructed orthonormal basis, depending on q, of the space of continuous functions on the ring of p-adic integers with values in K. Attached to q there are two models of the quantum plane and a model of the quantum Weyl algebra, as algebras of bounded linear operators on the space of p-adic continuous functions. For q not a root of unit, interesting orthonormal (orthogonal) families of these algebras are exhibited and providing p-adic completion of quantum plane and quantum Weyl algebras.

Some Properties of Riesz Products on the Ring of p-Adic Integers

Riesz products on the ring of p-adic integers are introduced and studied from the points of view of harmonic analysis and dynamical system relative to the shift transformation. We find necessary and sufficient conditions for a Riesz product to be invariant, quasi-invariant or quasi-Bernoulli. We study the mutual absolute continuity of two Riesz products and the almost everywhere convergence of lacunary series with respect to a Riesz product. We also compute the Hausdorff dimension and the energy dimension and prove a multifractal formalism for a given Riesz product.

Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

AbstractAn integer may be represented by a quadratic form over each ring ofp-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

On Algebraically Compact Λ-Modules

We study the structure and properties of algebraically compact modules over Λ = ℤ p [[T]], the power series ring over the p-adic integer ring.

Liapunov Stability and the ring of P-adic integers

It is well-known that stable Cantor sets are topologically conjugate to adding machines. In this work we show are also conjugate to an algebraic object, the ring of P−adic integers with respect to group tramnslation. This ring is closely related to the field of p-adic numbers; connections and distintions are explored. The inverse limit construction provides a purely dynamical proof of an algebraic result: the classification of adding machines, or P−adic integers, up to group isomorphism.

P-adic Superspaces and Frobenius

The notion of a p-adic superspace is introduced and used to give a transparent construction of the Frobenius map on p-adic cohomology of a smooth projective variety over \({\mathbb{Z}_p}\) (the ring of p-adic integers), as well as an alternative construction of the crystalline cohomology of a smooth projective variety over \({\mathbb{F}_p}\) (finite field with p elements).

A uniformly distributed sequence on the ring of p-adic integers

This article will be started with a fundamental observation on consecutive non-negative integers. Based on the observation, we will demonstrate that the sequence of non-negative integers is regarded as a uniformly distributed sequence in the ring of the p -adic integers. As a result, the sequence provides us with a faster rate of convergence than the one obtained by simply applying law of the iterated logarithm.

Rank Two Integral Representations of the Infinite Dihedral Group

We give an effective classification of the representations of the infinite dihedral group in GL 2(R) where R is either the valuation ring ℤ(p) or the ring of p-adic integers.

Riesz products on the ring of p-adic integers

We introduce a class of probability measures, called Riesz products, on the ring Z p of p-adic integers. We prove a result on the almost everywhere convergence, with respect to a Riesz product, of some related series and then obtain the Hausdorff dimension of the Riesz product. Other properties of these measures are also discussed, like the mutual absolute continuity, the quasi-invariance with respect to the shift transformation and the quasi-Bernoulli property. To cite this article: A. Fan, X.Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

ANALYTIC PROPERTIES OF THE q-VOLKENBORN INTEGRAL ON THE RING OF p-ADIC INTEGERS

In this paper, we consider the q-Volkenborn integral of uniformly differentiable functions on the p-adic integer ring. By using this integral, we obtain the generating functions of twisted q-generalized Bernoulli numbers and polynomials. We find some properties of these numbers and polynomials.

On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Q p . Let K / k be a totally ramified elementary abelian Kummer extension of degree p 2 with Galois group G. We determine the isomorphism class of the ring of integers in K as an o G -module under some assumptions. The obtained results imply there exist extensions whose rings are Z p G -isomorphic but not o G -isomorphic, where Z p is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.

On Strongly Clean Matrix and Triangular Matrix Rings

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.

Supergeometry and arithmetic geometry

We define a superspace over a ring R as a functor on a subcategory of the category of supercommutative R-algebras. As an application the notion of a p-adic superspace is introduced and used to give a transparent construction of the Frobenius map on p-adic cohomology of a smooth projective variety over Z p (the ring of p-adic integers).

On Crawley Modules

ABSTRACT This continues recent work in an article by Corner et al. (2004). A particular question was left open: Is it possible to carry over the results concerning the undecidability of torsion-free Crawley groups to modules over the ring of p-adic integers? We will confirm this and also strengthen one of the results in Corner et al. (2004) by replacing the hypothesis of ⋄ by CH. For details see the introduction.

When is the [formula omitted] matrix ring over a commutative local ring strongly clean?

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local rings are strongly clean. It is unknown when a matrix ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005] that for any prime number p, the 2 × 2 matrix ring M 2 ( Z ˆ p ) is strongly clean where Z ˆ p is the ring of p-adic integers, but M 2 ( Z ( p ) ) is not strongly clean where Z ( p ) is the localization of Z at the prime ideal generated by p. Let R be a commutative local ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for M 2 ( R ) to be strongly clean. As consequences, M 2 ( R ) is strongly clean iff M 2 ( R 〚 x 〛 ) is strongly clean iff M 2 ( R [ x ] / ( x n ) ) is strongly clean iff M 2 ( R C 2 ) is strongly clean.

A wavelet theory for local fields and related groups

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = ℚp, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤp, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.

Existence of certain fundamental discriminants and class numbers of real quadratic fields

Let D be the fundamental discriminant of the quadratic field Q( √ D), h(D) its class number, and χD := ( D · ) the usual Kronecker character. Let p be prime, Zp the ring of p-adic integers, and λp(Q( √ D)) the Iwasawa λ-invariant of the cyclotomic Zp-extension of Q( √ D). Let Rp(D) denote the p-adic regulator of Q( √ D), and | · |p denote the usual multiplicative p-adic valuation normalized so that |p|p = 1 p . In [9], by applying Sturm’s theorem on the congruence of modular forms to Cohen’s half integral weight modular forms, Ono proved the following theorem.

One-Dimensional p -Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is the ring of p-adic integers. One of these theorems [2, 3.32] says that each subanalytic subset of Zp is semialgebraic. This is extended here as follows.

Local Zeta Functions on Hermitian Forms and Its Application to Local Densities

We give an explicit description of functional equations satisfied by zeta functions on the space of unramified hermitian forms over a p-adic field. Further, as an application, we give explicit expressions of local densities of integral representations of nondegenerate unramified hermitian matrices with entries in the ring of p-adic integers.