P-adic Sets Research Articles

Analytic Tate Spaces and Reciprocity Laws

We consider a functional analytic variant of the notion of Tate space, namely the category of those topological vector spaces which have a direct sum decomposition where one summand is nuclear Frechet space and the other is the dual of a nuclear Frechet. We show that, both in the complex and in the p-adic setting, one can use this formalism to define symbols for analytic functions which satisfy Weil-type reciprocity laws.

Cell decomposition for semibounded p-adic sets

We study a reduct \({\mathcal{L}_*}\) of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the \({\mathcal{L}_*}\) -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, \({\mathcal{L}_*}\)) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multiplication can only be defined on bounded sets, and we consider the existence of definable Skolem functions.

The trace of Frobenius of elliptic curves and thep-adic gamma function

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p >3, that the trace of Frobenius of any elliptic curve over Fp, whose j -invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over Fp when p 1 .mod 12/. 1. Introduction and statement of results Let Fp denote the finite field with p, a prime, elements. Consider E=Q an elliptic curve with an integral model of discriminantA.E/. We denote Ep the reduction of E modulo p. We note that Ep is nonsingular, and hence an elliptic curve over Fp, if and only if p› A.E/, in which case we say p is a prime of good reduction. Regardless, we define

On the structure of p-adic subanalytic functions and sets

We prove a p-adic version of the Lion–Rolin preparation theorem. As a consequence, we obtain a cell decomposition theorem, which can be viewed as an extension of Denef’s cell decomposition theorem to the p-adic analytic case. Using these theorems, we give shorter and more explicit proofs of some of the results by Denef and van den Dries on p-adic subanalytic sets.

EXTENDING GAUSSIAN HYPERGEOMETRIC SERIES TO THE p-ADIC SETTING

We define a function which extends Gaussian hypergeometric series to the p-adic setting. This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes. We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series. These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas.

Fields with analytic structure

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o -minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principal, previously studied, analytic structures on Henselian valued fields, as well as new ones. The b -minimality is shown, as well as other properties useful for motivic integration on valued fields. The paper is reminiscent of [Denef, van den Dries, p-adic and real subanalytic sets . Ann. of Math. (2) 128 (1988) 79–138], of [Cohen, Paul J. Decision procedures for real and p-adic fields . Comm. Pure Appl. Math. 22 (1969)131–151], and of [Fresnel, van der Put, Rigid analytic geometry and its applications . Progress in Mathematics, 218 Birkhäuser (2004)], and unifies work by van den Dries, Haskell, Macintyre, Macpherson, Marker, Robinson, and the authors.

Quincunx multiresolution analysis for L 2(ℚ 2 2 )

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.

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].

P-adic sets of range uniqueness

We study sets of range uniqueness (SRU’s) in a complete, ultrametric, algebraically closed fieldK for analytic elements. We find monotonic distances sequences which appear to be SRU’s completely different from those known in ©. On the other hand, most of open closed sets cannot be SRU’s.

Multivariate Igusa theory: decay rates of exponential sums

We obtain general estimates for exponential integrals of the form Ef(y)=∫ℤpnψ(∑j=1ryjfj(x))|dx|⁠, where the fj are restricted power series over ℚp, yj ε ℚp and ψ is a nontrivial additive character on ℚp. We prove that if (f1,…,fr) is a dominant map, then |Ef(y)|<c|y|α for some c > 0 and α < 0, uniform in y, where |y|=max(|yi|)i⁠. In fact, we obtain similar estimates for a much bigger class of exponential integrals. To prove these estimates we introduce a new method to study exponential sums, namely, we use the theory of p-adic subanalytic sets and p-adic integration techniques based on p-adic cell decomposition. We compare our results with some elementarily obtained explicit bounds for Ef with fj polynomials.

Classification of semi-algebraic p-adic sets up to semi-algebraic bijection

We prove that two infinite p-adic semi-algebraic sets are isomorphic (i.e. there exists a semi-algebraic bijection between them) if and only if they have the same dimension.

One-Dimensional p -Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is the ring of p-adic integers. One of these theorems [2, 3.32] says that each subanalytic subset of Zp is semialgebraic. This is extended here as follows.

J. Denef and L. van den Dries. p-Adic and real subanalytic sets. Annals of mathematics, ser. 2 vol. 128 (1988), pp. 79–138.

J. Denef and L. van den Dries. p-Adic and real subanalytic sets. Annals of mathematics, ser. 2 vol. 128 (1988), pp. 79–138.

Flatness and smooth points of p-adic subanalytic sets

We give a new proof of the subanalyticity of the regular locus of a p-adic subanalytic set, replacing use of an approximation theorem by a more natural argument based on the flatness of certain homomorphisms given by Taylor expansions of strictly convergent power series at a non-standard point of Z m p .

Analytic cell decomposition and the closure of p-adic semianalytic sets

The p-adic semianalytic sets are defined, locally, as boolean combinations of sets of the form over the p-adic fields ℚp, where f is an analytic function. A well-know example due to Osgood showed the projection of a semianalytic set need not be a semianalytic set. We call those sets that are, locally, the projections of p-adic semianalytic sets p-adic subanalytic sets. The theory of p-adic subanalytic sets was presented by Denef and Van den Dries in [5]. The basic tools are the quantifier elimination techniques together with the ultrametric Weierstrass Preparation Theorem. Simultaneously with their developments of the p-adic subanalytic sets, they established some basic properties of p-adic semianalytic sets.In this paper, we prove that the closure of any p-adic semianalytic set is also a semianalytic set. The analogous property for real semianalytic sets was proved in [12] and that for rigid semianalytic sets, informed by the referee, has been proved recently by a quite different method in [14] (cf. [9]). The keys to the proof are a separation lemma (Lemma 2) and an analytic cell decomposition theorem (Theorem 2) which is an analytic version of Denef's cell decomposition theorem (see [3, 4]; A total different form of anayltic cells appeared in [13]). The analytic cell decomposition theorem allows us to partition certain kinds of basic subsets into analytic cells that possess the closure property (see §1 for the definition).

P-adic and Real Subanalytic Sets

p-adic and Real Subanalytic Sets

P-adic semi-algebraic sets and cell decomposition.

p-adic semi-algebraic sets and cell decomposition.