To a polynomial f over a p{adic eld K and a character of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s; f;), which is a meromorphic function on C. Several theorems and conjectures relate the poles of Z(s; f;) to the monodromy of f; the so{called holomorphy conjecture states roughly that if the order of does not divide the order of any eigenvalue of...

Consider the Weil sumWF,d(u) = ∑ x∈F ψ(x +ux), where F is a finite field of characteristic p, ψ is the canonical additive character of F , d is coprime to |F |, and u ∈ F . We say that WF,d(u) is threevalued when it assumes precisely three distinct values as u runs through F : this is the minimum number of distinct values in the nondegenerate case, and three-valuedWF,d are rare and desirable. W...

Abstract When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas rings p -adic power series, as $\mathbb Q_{p}\langle X\rangle $ , Z_{p}\langle related series over more general valuation their fraction fields. obtain definable, uniform for radicality, and, on...

1 Introduction This paper is an extended version of the talk given by Miguel Olalla at the International Conference on Valuation Theory in El Escorial in July 2011. Its purpose is to provide an introduction to our joint paper [7] without grinding through all of its technical details. We refer the reader to [7] for details and proofs; only a few proofs are given in the present paper. Notation. T...

Wavelet analysis as a p–adic spectral analysis Abstract New orthonormal basis of eigenfunctions for the Vladimirov operator of p–adic fractional derivation is constructed. The map of p–adic numbers onto real numbers (p–adic change of variable) is considered. p–Adic change of variable maps the Haar measure on p–adic numbers onto the Lebesgue measure on the positive semiline. p–Adic change of var...

Our principal interest in this paper is to study higher order degenerate Hermite-Bernoulli polynomials arising from multivariate $p$-adic invariant integrals on $mathbb{Z}_p$. We give interesting identities and properties of these polynomials that are derived using the generating functions and $p$-adic integral equations. Several familiar and new results are shown to follow as special cases. So...

1. Iwasawa functions, p-adic measures, and distributions 1 1.1. Some preliminaries 1 1.2. Iwasawa functions and the Iwasawa algebra 3 1.3. Iwasawa functions and interpolation 4 1.4. p-adic distributions and p-adic measures 8 2. The p-adic L-functions of Dirichlet characters 10 2.1. Preliminaries 10 2.2. Iwasawa’s construction 12 3. The p-adic L-functions of modular forms 15 3.1. Preliminaries 1...

We show that translations and dilations of a p–adic wavelet coincides (up to the multiplication by some root of one) with a vector from the known basis of discrete p–adic wavelets. In this sense the continuous p–adic wavelet transform coincides with the discrete p–adic wavelet transform. The p–adic multiresolution approximation is introduced and relation with the real multiresolution approximat...

Let p be prime, N be a positive integer prime to p, and k be an integer. Let Pk(t) be the characteristic series for Atkin’s U operator as an endomorphism of p-adic overconvergent modular forms of tame level N and weight k. Motivated by conjectures of Gouvêa and Mazur, we strengthen a congruence in [W] between coefficients of Pk and Pk′ for k ′ p-adically close to k. For p − 1 | 12, N = 1, k = 0...

— Let h : X → Y be a separated morphism of adic spaces of finite type over a non-archimedean field k with Y affinoid and of dimension 6 1, let L be a locally closed constructible subset of X and let g : (X, L)→ Y be the morphism of pseudo-adic spaces induced by h. Let A be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of k and let ...