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.     

On the construction of normal wildly ramified extensions over Q p , (p ≠ 2)

Let p be an odd prime number, and let Q _p be the field of rational p-adic numbers.The aim of this work is the determination of the standard form of an Eisenstein polynomial defining a normal wildly ramified extension of Q _p . We prove first the equivalence between normality and cyclicity, give some essential normality conditions for the general case (degree p ⁿ ), then we solve the problem completely for the case (degree p ²) also, we obtain that the normality depends on seven congruences modulo p ^m between the coefficients of the considered polynomial with just m = 2 or 3. Note that the case (degree p) was solved by Öystein Ore (see Math. Annalen 102 (1930), 283–304). Also examples are given.     

Hausdorff operators defined by measures on <Emphasis Type="Italic">p</Emphasis>-adic linear spaces and their norms

For Hausdorff operator defined by a measure on p-adic linear space Q _p ⁿ we give the exact values for its norms in power type Morrey space,BMO(Q _p ⁿ ) and BLO(Q _p ⁿ ). Also we prove the sharp two-sided estimate for its norm in Herz space. These results generalize some previous results of the author.     

Lucas sequences and quadratic orders

We consider the Lucas sequences (U _n ) _{n ≥ 0} defined by U ₀ = 0, U ₁ = 1, and U _n =  PU _n–1 – QU _n–2 for non-zero integral parameters P, Q such that Δ = P ² – 4Q is not a square. We use the arithmetic of the quadratic order with discriminant Δ to investigate the zeros and the period length of the sequence (U _n ) _{n ≥ 0} modulo a positive integer d coprime to Q. For a prime p not dividing Q, we give precise formulas for p-powers, we determine the p-adic value of U _n , and we connect the results with class number relations for quadratic orders.     

Computational Complexity over thep-adic Numbers

A model of computation over thep-adic numbers, for odd primesp, is defined following the approach of Blum, Shub, and Smale. This model employs branching on the property of being a square inQ_pand unit height. The feasibility of systems of quadratic polynomials is shown to be NP-complete in this model. A polynomial time algorithm is given for feasibility of a single quadratic polynomial overQ_p.     

The Artin conjecture for three diagonal cubic forms

Let p be a prime, p ≠ 3or 7. Then any three diagonal cubic forms over the p-adics in at least 28 variables possess a common nontrivial p-adic zero. This verifies the Artin conjecture for three diagonal cubic forms over all p-adic fields with the possible exception of $\text{Q}$₃ and $\text{Q}$₁.     

p-Adic dynamical systems of Chebyshev polynomials

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 ? _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 ? _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 ? _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.     

Wavelet bases in the Lebesgue spaces on the field of p-adic numbers

In this paper we prove that p-adic wavelets form an unconditional basis in the space L ^r (? _p ⁿ ) and give the characterization of the space L ^r (? _p ⁿ ) in terms of Fourier coefficients of p-adic wavelets.Moreover, the Greedy bases in the Lebesgue spaces on the field of p-adic numbers are also established.     

On the Classification of <Emphasis Type="Italic">p</Emphasis>-Adic UHF Banach Algebras

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e _ij ^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.     

Abelian group matrices over the p-adic and rational integers

Let G be a finite abelian group of order n. Let Z and Q denote the rational integers and rationals, respectively. A group matrix for G over Z (or Q) is an n-square matrix of the form Σ_{g ∈ G}a_gP(g), where a_g ∈ Z (or Q) and P is the regular representation of G so that P(g) is an n-square permutation matrix and P(gh) = P(g)P(h) for all g, h ∈ G. It is known that if M is an arbitrary positive definite unimodular matrix over Z then there exists a matrix A over Q such that M = A^τA, where τ denotes transposition. This paper proves that the exact analogue of this theorem holds if one demands that M and A be group matrices for G over Z and Q, respectively. Furthermore, if M is a group matrix for G over the p-adic integers then necessary and sufficient conditions are given for the existence of a group matrix A for G over the p-adic numbers such that M = A^τA.     

A note on l-class groups of certain algebraic number fields

The structure of ideal class groups of number fields is investigated in the following three cases: (i) Abelian extensions of number fields whose Galois groups are of type (p, p); (ii) non-Galois extensions $\text{Q(p}\text{d}{}_0\text{3}\text{,p}\text{d}{}_1\text{3}\text{)}$ of degree p² over Q; (iii) dihedral extensions of degree 2^{n + 1} over Q. It is shown that it is possible to obtain class number relations by group-theoretic methods. Subgroups of ideal class groups whose orders are prime to the extension degree are considered.     

On the lipschitz property of a class of mappings

Open discrete annular Q-mappings with respect to the p-modulus in ? ⁿ , n ≥ 2, are considered in this paper. It is established that such mappings are finite Lipschitz for n ? 1 < p < n if the integral mean value of the function Q(x) over all infinitesimal balls B(x ₀, ?) is finite everywhere.     

Weak and strong estimates for rough Hausdorff type operator defined on <Emphasis Type="Italic">p</Emphasis>-adic linear space

For rough Hausdorff type operator defined on p-adic linear space Q _p ⁿ and its commutator with symbol from Lipschitz space, we give sufficient conditions of their boundedness from one Lorentz space into another.     

Identifying variable points on a smooth curve

LetX be a compact Riemann surface,n ≥ 2 an integer andx = [x ₁, …,x _n ] an unorderedn-tuple of not necessarily distinct points onX. Byf _x :X →Y _x we denote the normalization which identifies thex ₁, …,x _n and maps them to the only and universal singularity of a complex curveY _x . Thenf _x depends holomorphically onx and is uniquely determined by this parameter. In this context we consider the fine moduli spaceQ _X of all complex-analytic quotients ofX and construct a morphismS ⁿ (X) →Q _X such that each and everyf _x corresponds to the image of the pointx on then-fold symmetric powerS ⁿ (X). For everyn ≥ 2 the mappingS ⁿ (X) →Q _X is a closed embedding; the points of its image have embedding dimensionn(n ? 1) inQ _X . HenceS ²(X) is a smooth connected component ofQ _X . On the other hand, a deformation argument yields thatS ⁿ (X) is part of the singular locus of the complex spaceQ _X provided thatn ≥ 3.     

The image of Colmez’s Montreal functor

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL₂(Q _{?? p} ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL₂(Q _{?? p} ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.     

On the density of odd integers of the form (p − 1)2−n and related questions

It is shown that odd integers k such that k · 2ⁿ + 1 is prime for some positive integer n have a positive lower density. More generally, for any primes p₁, …, p_r, the integers k such that k is relatively prime to each of p₁,…, p_r, and such that k · p₁^n₁p₂^n₂ … p_r^n_r + 1 is prime for some n₁,…, n_r, also have a positive lower density.     

p-Adic Schrödinger-type operator with point interactions

A p-adic Schrödinger-type operator Dα+VY is studied. Dα (α>0) is the operator of fractional differentiation and (bij∈C) is a singular potential containing the Dirac delta functions δx concentrated on a set of points Y={x₁,…,xn} of the field of p-adic numbers Qp. It is shown that such a problem is well posed for α>1/2 and the singular perturbation VY is form-bounded for α>1. In the latter case, the spectral analysis of η-self-adjoint operator realizations of Dα+VY in L₂(Qp) is carried out.     

p-adic valuations and k-regular sequences

A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x)∈Q_p[x], we consider the sequence , where v_p is the p-adic valuation. We show that this sequence is p-regular if and only if f(x) factors into a product of polynomials, one of which has no roots in Z_p, the other which factors into linear polynomials over Q. This answers a question of Allouche and Shallit.     

Pointwise consistency of the hermite series density estimate

The Hermite series estimate of a density f?L_p, p > 1, convergessin the mean square to f (x) for almost all x? |R, ifN (n) → ∞ and N (n) / n² → ) as n → ∞, where N is the number of the Hermite functions in the estimate while n is the number of observations. Moreover, the mean square and weak consistency are equivalent. For m times differentiable densities, the mean squares convergence rate is O(n^?(2m?1)/2m). Results for complete convergence are also given.