Algebraic independence of -adic numbers

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ℚ p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ℚ p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T p in the field ℚ p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.

Let K be a field of degree n over Q p , the field of rational p-adic numbers, say with residue degree f, ramification index e and differential exponent d. Let O be the ring of integers of K and P its unique prime ideal. The trace and norm maps for K/Q p are denoted Tr and N, respectively. Fix q=p r , a power of a prime p, and let η be a numerical character defined modulo q and of order o(η). The character η extends to the ring of p-adic integers ℤ p in the natural way; namely η(u)=η(u ˜), where u ˜ denotes the residue class of u modulo q, and similarly for the root of unity ζ q u =exp(2πiu ˜/q). Fix a positive integer γ≥re-d for which N(1+P γ )⊆1+qℤ p so that the (twisted) Kloosterman sumsR(η,z)=∑α∈(O/Pγ)*η(Nα)ζqTrα+z/Nα(z∈Z/qZ*)are well-defined.Saliè explicitly determined R(η,z) in the classical case n=1 (so K=Q p ) for q=p r with r>1 and o(η)=1 or 2. Here I generalize Saliè’s result for the general case n>1 for characters η with o(η)|p-1 (also o(η)=2 when p=2), and for all γ≥re-d>1 but for a few small exceptional values r. My evaluation relies on the author’s recent explicit determination of Gauss sums for prime powers in p-adic fields and exponential sums of the form ∑χ(x) ax ζ q bx .

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

Let p be a prime number, a2 the completion of the algebraic closure of the field of rational p-adic numbers and let A be the residue class field of Q. The field A is the algebraic closure of its prime subfield and is of characteristic p. If T* is the set of all roots of unity in a2 of order prime to p then the restriction of the residue class map to T* is a multiplicative isomorphism of T* onto the multiplicative group of R. The elements of T T* U {O} form the Teichmiiller representatives of A in Q2 and for each x C A the representative of x in Q will be understood to be the element of T in the class x. The non-archimedean valuation of 0 will be denoted by the ordinal function, abbreviated , and normalized by the condition ord p = 1.

The p-adic numbers were first considered by Hensel in the 19th century. He observed that the primes play an analogous role in the integers as linear polynomials do in \({\mathbb C}[X]\). The Laurent expansion of a rational function led him to consider the p-adic expansion of a rational number. In this chapter, for a fixed prime p, we will construct block codes over the rings \({\mathbb Z}/p^h{\mathbb Z}\) simultaneously, by constructing codes over the p-adic numbers and then considering the coordinates modulo ph. These codes will be linear over the ring but when mapped to codes over \({\mathbb Z}/p{\mathbb Z}\) will result in codes which are not equivalent to linear codes. We start with a brief introduction to p-adic numbers, which will cover enough background for our purposes. The classical cyclic codes, that we constructed in Chapter 5, lift to cyclic codes over the p-adic numbers. In the case of the cyclic Hamming code, this lift extends to a code over \({\mathbb Z}/4{\mathbb Z}\) which, when mapped to a binary code, gives a non-linear code with a set of parameters for which no linear code exists.

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction. In the real case, it is well known that there exists an intimate relation between poles of complex powers of polynomial functions and the roots of the b-functions (cf. [B]). Many examples suggest that a similar relation between them would exist also in the p-adic case. Recently the case of polynomials over a p-adic field in two variables has been closely examined (cf. F. Loeser [L], W. Veys [V]). In this paper we shall deal with the case of relative invariants of prehomogeneous vector spaces over a p-adic field. Let us explain our problem more precisely. Let K be a p-adic field, i.e., a finite algebraic extension of Qp. We denote by ?)K the maximal compact subring of K, by rrT)K the ideal of nonunits of )K, and put q = #(DKI/'XC)K). We denote by | IKthe absolute value on K normalized as ITWK = q-. Let fQ(KX) be the set of quasi-characters of KX. Then the identity component fl(KX)O of fl(KX) consists of w,(t) = Itls for all s in C and fl(Kx)o CX under ws q-S. Let (G, p, V) be a prehomogeneous vector space (abbrev. P.V.) defined over K, where we assume that

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, e.g., in 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 thesis we restrict our study to 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.In this thesis, we present a description of 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p.

Using the fact that the multiplicative group of units in a p-adic field is a compact Vilenkin group, we introduce a new pointwise derivative on ℚp* and investigate basic properties of this derivative.

Nondegenerate σ-additive measures with ranges in ℝ and ℚq (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚq, where ℚq is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory.

In this work we develop the concept of biological referents to analyze the origin of complexity of biological systems. The concept, as we demonstrate, can be formalized by classes of the objects which constitute hierarchic scale-free patterning at different levels of biological complexity. By this reason, ultrametric relationships between these classes are assumed to be relevant to the referent representation. To explore this idea, we realize particular formalization of the referent concept including construction of objects and classes, construction of ultrametric space of classes, and description of the ultrametric space by a field of p-adic numbers. We discuss how a notation “evolution” can be introduced through ultrametric formalism. An example of ultrametric evolutionary equations is presented. Finally, we demonstrate that different aspects of the origin and evolution of the Biosphere (such as macroevolution and development) being verified in the frame of the referent concept acquire new contours and interpretations.

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.

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

Dynamical systems with discrete time (discrete systems) are widely used in various applications (e.g., see the monographs [1–4]). Most often, the parameters of such systems are chosen from the field R of real numbers or C of complex numbers. However, if one deals with the so-called sequential machines [5] used in construction of nodes of discrete objects for communication and control, finite fields, in particular, the field GF (p) of residues modulo a prime number p [6], are chosen as the parameter spaces. Further, a number of problems in contemporary theoretical physics require that techniques for studying equations of various types over the field of p-adic numbers be developed [7]. Thus, numerous problems in applied and general mathematics are stated in terms of discrete systems over specific fields; moreover, the range of such fields is fairly broad. Therefore, it seems worthwhile to study discrete equations over arbitrary fields so as to simulate well-known models using a single mathematical object and to investigate them with the help of unified approaches. In this paper, we prove stabilization criteria for linear discrete nonstationary systems whose parameters may belong to an arbitrary field.

Feynman-Kac formulas for heat conduction equations with Vladimirov’s operator (acting here as a Laplace operator) are proved; it is assumed that unknown functions are determined on the product of the real axis and a space over the field of p-adic numbers and take real or complex values. Similar formulas may be obtained for Schrodinger type equations. Such equations may be useful in construction both mathematical models of processes whose scales are characterized by the Planck length and time and mathematical models describing phenomenology in chemistry, mechanics of continua, and also in psychology.