P-adic Functions Research Articles

Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

It is known that a p-adic, locally Lipschitz continuous semi-algebraic function, is piecewise Lipschitz continuous, where finitely many pieces suffice and the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1 (again, with finitely many pieces). We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parameterized versions and in the subanalytic setting.

Shilov boundary and p-adic injective analytic functions

Let K be an algebraically closed field complete with respect to a dense ultrametric absolute value |.|. Let D be an infraconnected affinoid subset of K and let H(D) be the Banach algebra of analytic elements on D. Let f ∈ H(D) be injective in D and let f* be the mapping defined on the multiplicative spectrum of H(D) that identifies with the set of circular filters on D. We show that f* is injective and maps bijectively the Shilov boundary of H(D) onto this of H(f(D)). Thanks to this property we give a new proof of the equality \(\left| {f(x) - f(y)} \right| = \left| {x - y} \right|\sqrt {\left| {f'(x)f'(y)} \right|} \).

Operational calculus on the ring of p-adic integers

We develop a one-dimensional Mikusinski type Operational Calculus on the ring of p-adic integers ℤ p . The Calculus utilizes the space C(ℤ p ) of continuous functions with values in the field of p-adic numbers ℚ p . C(ℤ p ) is equipped with a convolution, which turns it into an integral domain. A hyperfunction is by definition an element of a fraction field of C(ℤ p ). We refer to this fraction field as the p-adic Mikusinski field, prove its separability and incompleteness. Hyperfunctions corresponding to operations of shift, taking difference, indefinite summation, differentiation and integration are considered. A generalization of the p-adic exponential function a x is constructed for all a∈ℚ p . Also a variant of the Mikusinski field is considered that allows to treat continuous functions and p-adic measures simultaneously.

Van der Put basis and p-adic dynamics

We consider a function g: Zp → Zp and its the van der Put series. Then we get a criteria of Haar’s measure preserving compatible p-adic functions which, actually, need not be uniformly differentiable modulo p. This is used to study ergodicity of p-adic dynamical systems [2, 16].

Integrable functions for the Bernoulli measures of rank 1

In this paper, following the p-adic integration theory worked out by A. F. Monna and T. A. Springer [4, 5] and generalized by A. C. M. van Rooij and W. H. Schikhof [6, 7] for the spaces which are not σ-compacts, we study the class of integrable p-adic functions with respect to Bernoulli measures of rank 1. Among these measures, we characterize those which are invertible and we give their inverse in the form of series.

Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree T (O p ) of balls in O p . In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree T (O p ) and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.

The p-adic quantum plane algebras and quantum Weyl algebra

Let q be a principal unit of the ring of valuation of a complete valued field K, extension of the field of p-adic numbers. Generalizing Mahler basis, K. Conrad has constructed orthonormal basis, depending on q, of the space of continuous functions on the ring of p-adic integers with values in K. Attached to q there are two models of the quantum plane and a model of the quantum Weyl algebra, as algebras of bounded linear operators on the space of p-adic continuous functions. For q not a root of unit, interesting orthonormal (orthogonal) families of these algebras are exhibited and providing p-adic completion of quantum plane and quantum Weyl algebras.

Applications of patching to quadratic forms and central simple algebras

This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over Henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh (in Preprint arXiv:0708.3128 , 2007) on the u-invariant of p-adic function fields, p≠2. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C p . When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).

Rectilinearization of semi-algebraic p-adic sets and Denef's rationality of Poincaré series

In [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105–114], it is shown that a p-adic semi-algebraic set can be partitioned in such a way that each part is semi-algebraically isomorphic to a Cartesian product ∏ i = 1 l R ( k ) where the sets R ( k ) are very basic subsets of Q p . It is suggested in [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105–114] that this result can be adapted to become useful to p-adic integration theory, by controlling the Jacobians of the occurring isomorphisms. In this paper we show that the isomorphisms can be chosen in such a way that the valuations of their Jacobians equal the valuations of products of coordinate functions, hence obtaining a kind of explicit p-adic resolution of singularities for semi-algebraic p-adic functions. We do this by restricting the used isomorphisms to a few specific types of functions, and by controlling the order in which they appear. This leads to an alternative proof of the rationality of the Poincaré series associated to the p-adic points on a variety, as proven by Denef in [J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984) 1–23].

The Cauchy problem for a class of pseudodifferential equations over p-adic field

In this paper, some classes much more general than the one in [N.M. Chuong, Yu.V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (Eds.), Harmonic, Wavelet and p-Adic Analysis, World Scientific, Singapore, 2007] of Cauchy problems for an interesting class of pseudodifferential equations over p-adic fields are studied. The used functions belong to mixed classes of real and p-adic functions. Even for p-adic partial differential equations such problems in such function spaces have not been discussed yet. The established mathematical foundation requires very complicated and very difficult proofs. Days after days, these equations occur increasingly in mathematical physics, quantum mechanics. Explicit solutions of such problems are very needed for specialists on applied mathematics, physics, and engineering.

Meromorphic functions of uniqueness

Let E be an algebraically closed field of characteristic 0 which is either C or a complete ultrametric field K. We consider the composition of meromorphic functions h ○ f where h is meromorphic in all E and f is meromorphic either in E or in an open disk of K. We then look for a condition on h in order that if 2 similar functions f , g satisfy h ○ f ( a m ) = h ○ g ( a m ) where ( a m ) is a bounded sequence satisfying certain condition, this implies f = g . Particularly we generalize to meromorphic functions previous results on polynomials of uniqueness. The condition on h involves the zeros ( c n ) of h ′ and the values h ( c n ) but is weaker than this introduced by H. Fujimoto (injectivity on the set of zeros of h ′ ). The main tool is the Nevanlinna Theory but also involves some specific p-adic properties and basic affine properties. Results concerning p-adic entire functions only suppose a property involving 2 zeros of h ′ . Polynomials of uniqueness for entire functions are characterized. Every polynomial P of prime degree n ⩾ 3 is a polynomial of uniqueness for p-adic entire functions, except if is of the form A ( x − a ) n + B . A polynomial P such that P ′ has exactly two distinct zeros is a polynomial of uniqueness for meromorphic functions in K if and only if both zeros have a multiplicity order greater than 1. Results on p-adic functions have applications to rational functions in any field of characteristic 0.

On the U-Invariant of P-Adic Function Fields#

ABSTRACT We show that a quadratic form defined over the rational function field ℚ(x 1 , …, x n ) of dimension at least 4.2 n + 1 is isotropic over all fields ℚ p (x 1 , …, x n ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown.

SOME REMARKS ON A q-ANALOGUE OF BERNOULLI NUMBERS

Using the p-adic q-integral due to T. Kim[4], we define a number B＊<TEX>$_{n}$</TEX>(q) and a polynomial B＊<TEX>$_{n}$</TEX>(q) which are p-adic q-analogue of the ordinary Bernoulli number and Bernoulli polynomial, respectively. We investigate some properties of these. Also, we give slightly different construction of Tsumura＇s p-adic function <TEX>$\ell$</TEX><TEX>$_{p}$</TEX>(u, s, <TEX>$\chi$</TEX>) [14] using the p-adic q-integral in [4].n [4].

P-Adic Automorphic Functions for the Unitary Group in Three Variables

Let G be a special unitary group in three variables over a local field and, a co-compact subgroup of G. We construct, -automorphic functions on the rigid analytic space of points in the projective plane stable under the action of tori in G.

Urs, Ursim, and Non-Urs for p-adic Functions and Polynomials

LetWbe an algebraically closed field of characteristic zero, and letKbe an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. LetQnbe the polynomialxn−xn−1+kfor any constantk≠0, (n−1)n−1/nn. LetTn(k) be the set ofndistinct zeros ofQn. For everyn⩾9, we show thatTn(k) is ann-point unique range set (ignoring multiplicities) for bothW[x] and the set A(K) of entire functions inK. However, for everyn>0, we also show thatTn(k) is not a unique range set (counting or ignoring multiplicities) forW(x) and therefore, is also not a unique range set for the set of p-adic meromorphic functions (this was also separately found by Chung-Chun Yang and Xin-Hou Hua). In the same way, we show that there exist no bi-urs for p-adic meromorphic functions of the form ({a,b,c}, {ω}). Moreover, for everyn⩾5, we show that the only linear fractional functions preserving a setTn(k) is the identity, something which was asked (in particular) in Boutabaa and Escassut, “On Uniqueness of p-adic Meromorphic Functions,”Proc. Amer. Math. Soc.(1988).

On the reduction modulo p of an absolutely irreducible polynomial f (x, y)

Let/e fєℤ[x, y] be an absolutely irreducible polynomial. A classical result by Ostrowski states that the reduction modulo p of f remains absolutely irreducible for all large prime numbers p. Here we give a new sufficient condition on p for the conclusion to hold. The result, which holds for polynomials defined over arbitrary discrete valuation rings, also implies equality of the genera of the curves defined by f and its reduction. The method of proof stems from Hensel’s principle and analytic continuation of p-adic analytic functions, following Dwork and Robba.

A Remark on the Satake Transform and Base Change Map

In Ref. [1] Mautner established an identity in a finite form between a p-adic spherical function on GL(2) and its Satake transform. This identity was generalized by the author to GL(3) in Ref. [2]. On the other hand, the author proved in Ref. [3] an identity in a similar finite form expressing the Whittaker transform of a p-adic spheri-

Some new bases for p-adic continuous functions

Some new bases for p-adic continuous functions

A note on some expansions of p-adic functions

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by $(ϕₘ)_{m∈ ℕ₀}$. The system $(ϕₘ)_{m∈ ℕ₀}$ is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski ha