<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators

In this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in is constructed. We also study the connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators. A criterion for a multidimensional p-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models. The text was submitted by the authors in English.     

Representation Results for Equivariant Rigid Analytic Functions

Given a prime number p and the Galois orbit O(x) of an element x of ℂ _p , the topological completion of the algebraic closure of the field of p-adic numbers, we are interested in the representation results for equivariant rigid analytic functions defined on ℙ¹(ℂ _p ) \ O(x) with values in ℂ _p that vanishes at ∞.     

Analytic normal basis theorem

Let p be a prime number, ℚ _p the field of p-adic numbers, and a fixed algebraic closure of ℚ _p . We provide an analytic version of the normal basis theorem which holds for normal extensions of intermediate fields ℚ _p ⊆ K ⊆ L ⊆ .      

A note on symbolic and ordinary powers of homogeneous ideals

In this note we are interested in the graded modulesM _k=I^(k)/I^k and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x₀,…,x_n] of ℙⁿ,I ^(k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙⁿ.

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Sunto In questa nota siamo interessati ai moduli graduatiM _k=I(k)/I^k e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x₀,…,x_n] di ℙⁿ,I ^(k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙⁿ.

     

Equivariant vector bundles on Drinfeld’s upper half space

Let be Drinfeld’s upper half space over a finite extension K of ℚ _p . We construct for every GL _d+1-equivariant vector bundle on ℙ ^d _K , a GL _d+1(K)-equivariant filtration by closed subspaces on the K-Fréchet . This gives rise by duality to a filtration by locally analytic GL _d+1(K)-representations on the strong dual . The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum [ST1] and that of the structure sheaf by Pohlkamp [P].     

Weighted integral formulas on manifolds

We present a method of finding weighted Koppelman formulas for (p,q)-forms on n-dimensional complex manifolds X which admit a vector bundle of rank n over X×X, such that the diagonal of X×X has a defining section. We apply the method to ℙ ⁿ and find weighted Koppelman formulas for (p,q)-forms with values in a line bundle over ℙ ⁿ . As an application, we look at the cohomology groups of (p,q)-forms over ℙ ⁿ with values in various line bundles, and find explicit solutions to the -equation in some of the trivial groups. We also look at cohomology groups of (0,q)-forms over ℙ ⁿ ×ℙ ^m with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.     

p-Adic orthogonal wavelet bases

We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L ²(ℚ _p ). The text was submitted by the authors in English.     

Lower Bound for the Poles of Igusa’s p-adic Zeta Functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n >1 variables and let χ be a character of . Let M _i (u) be the number of solutions of f = u in (R/P ⁱ ) ⁿ for and. These numbers are related with Igusa’s p-adic zeta function Z _f,χ(s) of f. We explain the connection between the M _i (u) and the smallest real part of a pole of Z _f,χ(s). We also prove that M _i (u) is divisible by , where the corners indicate that we have to round up. This will imply our main result: Z _f,χ(s) has no poles with real part less than − n/2. We will also consider arbitrary K-analytic functions f.     

A relation between <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions and the Tamagawa number conjecture for Hecke characters

We prove that the submodule in K-theory which gives the exact value of the L-function by the Beilinson regulator map at non-critical values for Hecke characters of imaginary quadratic fields K with cl (K) = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soulé regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsens conjecture, an upper bound for in terms of the valuation of these p-adic L-functions, where V_p denotes the p-adic realization of a Hecke motive.Received: 4 June 2003     

On Convergence and Convolutions of Random Signed Measures

Let μ _n be a sequence of random finite signed measures on the locally compact group G equal to either or ℝ ^d . We give weak conditions on the sequence μ _n and on functions K such that the convolution product μ _n *K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces or L ^p (G). Examples for sequences μ _n covered are the empirical process (possibly arising from dependent data) and also random signed measures where is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.      

On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

Irrationality of some p-adic L-values

We give a proof of the irrationality of p-adic zeta-values ξp（κ） for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function.     

A divergent Khintchine theorem in the real,complex, and <Emphasis Type="Italic">p</Emphasis>-adic fields

In this paper, we show that if the sum ∑_r=1^∞ Ψ(r) diverges, then the set of points (x, z, w) ∈ ℝ × ℂ × ℚ_p satisfying the inequalities , and for infinitely many integer polynomials P has full measure. With a special choice of parameters v _i and λ _i , i = 1, 2, 3, we can obtain all the theorems in the metric theory of transcendental numbers which were known in the real, complex, or p-adic fields separately.     

Typical sheaves of generalized CM-modules

Let M be a generalized Cohen-Macaulay module over a noetherian local ring (R,m). Fix a standard system x₁, …, x_d∈m with respect to M and let . We construct a coherent Cohen-Macaulay sheafK over the projective space ℙ _R/I ^d-1 whose cohomological Hilbert functions depend only on the lengths of the local cohomology modules H _m ⁱ (M), (i=0, …, d−1).     

On the smallest poles of Igusa's p-adic zeta functions

Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kⁿ. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n=2 and less than −1 for n=3 which occur as the real part of a pole.     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

Büchi’s problem for ultrametric meromorphic functions inside an open disk

Let K be a complete ultrametric algebraically closed field of characteristic zero such as ℂ _p . Büchi’s problem was solved for p-adic meromorphic functions in the whole field K. Here we show similar conclusions for meromorphic functions in an open disk that are not quotients of bounded analytic functions. The main method is the secondMain Theorem for p-adic meromorphic functions inside a disk, a specific p-adic theorem.