Ring Of P-adic Integers Research Articles

On some local properties of sequences of big Galois representations

In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field F with coefficients in a domain, which admits a finite monomorphism from a power series ring over a p-adic integer ring, the set of places of F where some of the representations ramifies has density zero. Using this, we extend a result of Das–Rajan to such convergent sequences. We also establish a strong multiplicity one theorem for big Galois representations.

The Set of p-Adic Continuous Functions not Satisfying the Luzin (N) Property

In this short note, we will study the existence of a vector space of continuous functions f:{mathbb {Z}}_prightarrow mathbb Q_p, where {mathbb {Z}}_p and {mathbb {Q}}_p are, respectively, the ring of p-adic integers and the field of p-adic numbers, such that each nonzero function does not satisfy the Luzin (N) property and the dimension of the vector space is the continuum.

Representation of the universe as dendrogramic hologram empowered with relational interpretation

This is a brief review on the basics of recently established Dendrogramic Holographic theory (DH-theory). This is the special model of the event-universe based on the clustering transformation of experimental data into dendrogram, a finite tree which branches encoding the events. These event-branches are coupled via the hierarchic interrelation determined by the dendrogram. Such relational universe differs from the universe with space-time mathematically described by the real numbers. Dendrogram is endowed with the common root ultrametric. Finite dendrograms correspond to the epistemic level of description; in the limit we obtain an infinite tree providing the ontic description. In the simplest model, the tree is homogeneous, p-adic tree. It can be endowed with the algebraic structure of the ring of p-adic integers. Hence, DH-theory is a part (but very special) of p-adic theoretical physics. In this paper we discuss the foundations of DH-theory and its applications to quantum-classical interrelation including the novel interpretation of the violations of the CHSH-inequality, to general relativity, and to emergence of quantum mechanics from the event-picture of the universe. Since both quantum theory and general relativity can be emergent from DH-theory, creation of the latter can be viewed as a step towards unification of these two fundamental physical theories.

On the differential transcendentality of the Morita p-adic gamma function

In p-adic analysis one can find an analog of the classical gamma function defined on the ring of p-adic integers. In 1975, Morita defined the p-adic gamma function Gamma _p by a suitable modification of the function n mapsto n!. In this note we prove that for any given prime number p the Morita p-adic gamma function Gamma _p is differentially transcendental over {mathbb {C}}_p(X). The main result is an analog of the classical Hölder’s theorem, which states that Euler’s gamma function Gamma does not satisfy any algebraic differential equation whose coefficients are rational functions.

Generalizations of results of Friedman and Washington on cokernels of random p-adic matrices

Let p be prime and X be a Haar-random n×n matrix over Zp, the ring of p-adic integers. Let P1(t),…,Pl(t)∈Zp[t] be monic polynomials of degree at most 2 whose images modulo p are distinct and irreducible in Fp[t], where Fp denotes the finite field of p elements. For each j, let Gj be a finite module over Zp[t]/(Pj(t)). We show that as n goes to infinity, the probabilities that cok(Pj(X))≃Gj are independent, and each probability can be described in terms of a Cohen–Lenstra distribution. We also show that for any fixed n, the probability that cok(Pj(X))≃Gj for each j is a constant multiple of the probability that cok(Pj(X¯))≃Gj/pGj for each j, where X¯ is an n×n uniformly random matrix over Fp. These results generalize work of Friedman and Washington and prove new cases of a conjecture of Cheong and Huang.

A LeVeque-type inequality on the ring of p-adic integers

We derive an inequality on the discrepancy of sequences on the ring of [Formula: see text]-adic integers [Formula: see text] using techniques from Fourier analysis. This is a [Formula: see text]-adic analogue of the classical LeVeque inequality on the circle group [Formula: see text]. Some applications of the inequality are also given.

Implicit linear difference equations over a non-Archi-medean ring

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering p-adic integers ~ the completion of the ring of integers with respect to the non-Archimedean p-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of p-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of p-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.

On Certain Probabilistic Properties of Polynomials over the Ring of p-adic Integers

In this article, we study several probabilistic properties of polynomials defined over the ring of p-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials being strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of p-adic integers from polynomials defined over the integers modulo powers of p.

Dynamic structures of p-adic monomials for small primes p

We study the dynamic structures of the monomial xm over the ring of p-adic integers for every positive integer m and for primes p=2,3 and 5. The dynamic structures are described by investigating minimal decompositions which consist of periodic points, minimal subsystems and attracting basins.

Normal elements of noncommutative Iwasawa algebras over SL3(ℤ_ p )

Abstract Let p be a prime integer and let ℤ p {\mathbb{Z}_{p}} be the ring of p-adic integers. By a purely computational approach we prove that each nonzero normal element of a noncommutative Iwasawa algebra over the special linear group SL 3 ⁢ ( ℤ p ) {\mathrm{SL}_{3}(\mathbb{Z}_{p})} is a unit. This gives a positive answer to an open question in [F. Wei and D. Bian, Erratum: Normal elements of completed group algebras over SL n ⁢ ( ℤ p ) \mathrm{SL}_{n}(\mathbb{Z}_{p}) [mr2747414], Internat. J. Algebra Comput. 23 2013, 1, 215] and makes up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over SL n ⁢ ( ℤ p ) \mathrm{SL}_{n}(\mathbb{Z}_{p}) , Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously.

On the digits of Schneider's p-adic continued fractions

Let Zp be the ring of p-adic integers, λ the Haar measure on pZp and an(x) the n-th digit of Schneider's p-adic continued fractions of x∈pZp. We prove that an(x) are independent and identically distributed with respect to λ. Moreover, we obtain the Hausdorff dimensions of some sets defined by the growth rate of an(x) and the sum of an(x) respectively. Such dimensional results are different from that in the cases of real numbers and formal series.

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a b , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a b through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let R be the Galois ring GR(pk,s) of characteristic pk and cardinality psk. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length n over R, and a p-adic integer ring with gcd(p,n)=1. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length rlm over R, where r,l, and t are three primes with 2∤l, r|(qt−1), lv∥(qt−1) and gcd(rl,q(q−1))=1. Finally, as applications, weight distributions of all irreducible cyclic codes for t=2 and generator polynomials of self-dual cyclic codes of length lm and rlm over R are given.

Periodicity in the p-adic valuation of a polynomial

For a prime p and an integer x, the p-adic valuation of x is denoted by νp(x). For a polynomial Q with integer coefficients, the sequence of valuations νp(Q(n)) is shown to be either periodic or unbounded. The first case corresponds to the situation where Q has no roots in the ring of p-adic integers. In the periodic situation, the period length is determined.

Finite automata and numbers

Abstract We study finite automata representations of numerical rings. Such representations correspond to the class of linear p-adic automata that compute homogeneous linear functions with rational coefficients in the ring of p-adic integers. Finite automata act both as ring elements and as operations. We also study properties of transition diagrams of automata that compute a function f(x)= cx of one variable. In particular we obtain precise values for the number of states of such automata and show that for c > 0 transition diagrams are self-dual (this property generalises self-duality of Boolean functions). We also obtain the criterion for an automaton computing a function f(x)= cx to be a permutation automaton, and fully describe groups that are transition semigroups of such automata.

Ramification filtrations of certain abelian Lie extensions of local fields

Let G⊂xFq〚x〛 (q is a power of the prime p) be a subset of formal power series over a finite field such that it forms a compact abelian p-adic Lie group of dimension d≥1. We establish a necessary and sufficient condition for the APF extension of local field corresponding to (Fq⸨x⸩,G) under the field of norms functor to be an extension of p-adic fields. We then apply this result to study invertible power series over a ring of p-adic integers which commute with a fixed noninvertible power series under composition.

Q-Hardy-Littlewood-Type Maximal Operator with Weight Related to Fermionic p-Adic q-Integral on Z<SUB>p</SUB>

The q-extension of Hardy-littlewood-type maximal operator in accordance with q Volkenborn integral in the p-adic integer ring was recently studied . A generalization of Jang's results was given by Araci and Acikgoz . By the same motivation of their papers, we aim to give the definition of the weighted <i>q</i>-Hardy-littlewood-type maximal operator by means of fermionic p-adic q-invariant distribution on Z<SUB>p</SUB>. Finally, we derive some interesting properties involving this-type maximal operator.

Algebraic Sets in a Finitely Generated 2-Step Solvable Rigid Pro-p-Group

A 2-step solvable pro-p-group G is said to be rigid if it contains a normal series of the form G = G 1 > G 2 > G 3 = 1 such that the factor group A = G/G 2 is torsionfree Abelian, and the subgroup G 2 is also Abelian and is torsion-free as a ℤ p A-module, where ℤ p A is the group algebra of the group A over the ring of p-adic integers. For instance, free metabelian pro-p-groups of rank ≥ 2 are rigid. We give a description of algebraic sets in an arbitrary finitely generated 2-step solvable rigid pro-p-group G, i.e., sets defined by systems of equations in one variable with coefficients in G.

P-adic and Motivic Measure on Artin n-stacks

AbstractWe define a notion of p-adic measure on Artin n-stacks that are of strongly finite type over the ring of p-adic integers. p-adic measure on schemes can be evaluated by counting points on the reduction of the scheme modulo pn. We show that an analogous construction works in the case of Artin stacks as well if we count the points using the counting measure defined by Toën. As a consequence, we obtain the result that the Poincaré and Serre series of such stacks are rational functions, thus extending Denef's result for varieties. Finally, using motivic integration we show that as p varies, the rationality of the Serre series of an Artin stack defined over the integers is uniform with respect to p.

Extended q-Dedekind-type Daehee-Changhee sums associated with extended q-Euler polynomials

In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.