On totally positive units of real holomorphy rings

In this paper we will show that every totally positive unit of the real holomorphy ring of a formally real field is a sum of 2n-th powers of totally positive units for all natural numbersn. Moreover, in the casen=1 we give a bound on the number of summands required in such a representation. This research was supported by the Alexander-von-Humboldt-Foundation and carried out during the author's stay at the Institute for Advanced Studies at the Hebrew University of Jerusalem.     

Fourier series, measures and divided power series in the theory of function fields

Much of classical number theory is based on Fourier series. Such series play a vital role in the study of characteristic-0 zeta-functions: In the complex theory one has theta-series and Tate's thesis. In the p-adic theory one has Mahler's theorem on binomial coefficients which is used to déscribe the ring of p-adic measures. In this paper, we discuss a version of binomial coefficients for function fields due to L. Carlitz. We will show how these functions arise naturally out of gamma functions for function fields. We will also use some work of C. Wagner to establish that the ring of -adic measures is canonically isomorphic to the ring of divided power-series. The computation of these power-series in specific instances is now an important problem in the theory. Finally, we show the existence of many Fourier transforms in the -adic theory. The explicit computation of these would also be very interesting.Partially supported by NSF grant DMS-8521678. Current address: Department of Mathematics, UMBC, MD 21228, U.S.A.Dedicated to L. Carlitz     

On topological extensions of Archimedean and non-Archimedean rings

The usage of the fields of p-adic numbers Q _p , rings of m-adic numbers Q _m and more general ultrametric rings in theoretical physics induced the interest to topological-algebraic studies on topological extensions of rational and real numbers and more generally (commutative and even noncommutative) rings. It is especially interesting to investigate a possibility to proceed to non-Archimedean rings by starting with real numbers. In particular, in this note we present “no-go” theorems (Theorems 3, 4) by which one cannot obtain an ultrametric ring by extending (in a natural way) the ring of real numbers. This puremathematical result has some interest for non-Archimedean physics: to explore ultrametricity one has to give up with the real numbers — to work with rings of e.g. m-adic numbers (where m > 1 is a natural, may be nonprime, number).     

Denseness results in the theory of algebraic fields

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.     

The holomorphy rings of planar ternary rings with rational prime field

By a deep, recent result of Hartmann and Priess-Crampe any ordered planar ternary ring with rational prime field admits a natural order-compatible place into the reals. This allows us to extend the classical machinery about real places and spaces of orderings, the notion of real and relative holomorphy rings, the Kadison-Dubois representation for these rings, and the Brown-Marshall inequality to arbitrary planar ternary rings with rational prime field.     

Towards a general theory of formallyp-adic fields

The work is devoted to develop a general framework for the theory of formallyp-adic fields and to prove in this context some analogues of certain results of the theory of formallyp-adic fields in the sense of Kochen-Roquette. The general theory is illustrated by examples.     

Computable valued fields

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal’cev property, we show that they have computable dimension \(\omega \).     

Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.     

p-Adic wavelets and their applications

The theory of p-adic wavelets is presented. One-dimensional and multidimensional wavelet bases and their relation to the spectral theory of pseudodifferential operators are discussed. For the first time, bases of compactly supported eigenvectors for p-adic pseudodifferential operators were considered by V.S. Vladimirov. In contrast to real wavelets, p-adic wavelets are related to the group representation theory; namely, the frames of p-adic wavelets are the orbits of p-adic transformation groups (systems of coherent states). A p-adic multiresolution analysis is considered and is shown to be a particular case of the construction of a p-adic wavelet frame as an orbit of the action of the affine group.     

A p-adic analogue of Siegel's theorem on sums of squares

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded.     

Application of p-Adic Wavelets to Model Reaction–Diffusion Dynamics in Random Porous Media

Fourier and more generally wavelet analysis over the fields of p-adic numbers are widely used in physics, biology and cognitive science, and recently in geophysics. In this note we present a model of the reaction–diffusion dynamics in random porous media, e.g., flow of fluid (oil, water or emulsion) in a a complex network of pores with known topology. Anomalous diffusion in the model is represented by the system of two equations of reaction–diffusion type, for the part of fluid not bound to solid’s interface (e.g., free oil) and for the part bound to solid’s interface (e.g., solids–bound oil). Our model is based on the p-adic (treelike) representation of pore-networks. We present the system of two p-adic reaction–diffusion equations describing propagation of fluid in networks of pores in random media and find its stationary solutions by using theory of p-adic wavelets. The use of p-adic wavelets (generalizing classical wavelet theory) gives a possibility to find the stationary solution in the analytic form which is typically impossible for anomalous diffusion in the standard representation based on the real numbers.     

On recent results of ergodic property for p-adic dynamical systems

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.     

Completed cohomology of Shimura curves and a p-adic Jacquet–Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet–Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur’s ‘level lowering’ principle.     

Anneaux d'holomorphie et Positivstellensatz archimédien

In this paper we obtain some new Positivstellensatz for rings satisfying some archimedean properties. We also study the properties of the real holomorphy ring introduced recently by Becker and Powers. An application of these results in functional analysis is the resolution of the moment problem for some compact sets in R ⁿ . Re?cu le 20 janviers 1998 / Version revisée: le 2 April 1998     

Iwasawa theory of totally real fields for certain non-commutative p-extensions

In this paper, we will prove the non-commutative Iwasawa main conjecture—formulated by John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha and Otmar Venjakob (2005)—for certain specific non-commutative p-adic Lie extensions of totally real fields by using theory on integral logarithms introduced by Robert Oliver and Laurence R. Taylor, theory on Hilbert modular forms introduced by Pierre Deligne and Kenneth A. Ribet, and so on. Our results give certain generalization of the recent work of Kazuya Kato on the proof of the main conjecture for Galois extensions of Heisenberg type.     

The Bar Construction and Affine Stacks

We discuss the equivalence of two constructions of a unipotent group scheme attached to a differential graded algebra over a ?-algebra. The first construction is using the bar resolution and the second is using Toen's theory of affine stacks. We use this to establish the equivalence of two approaches to the comparison theorem in p-adic Hodge theory for the unipotent fundamental group of varieties defined over p-adic fields.     

Automorphism groups of fields

We consider pairs (K,G) of an infinite field K or a formally real field K and a group G and want to find extension fields F of K with automorphism group G. If K is formally real then we also want F to be formally real and G must be right orderable. Besides showing the existence of the desired extension fields F, we are mainly interested in the question about the smallest possible size of such fields. From some combinatorial tools, like Shelah’s Black Box, we inherit jumps in cardinalities of K and F respectively. For this reason we apply different methods in constructing fields F: We use a recent theorem on realizations of group rings as endomorphism rings in the category of free modules with distinguished submodules. Fortunately this theorem remains valid without cardinal jumps. In our main result (Theorem 1) we will show that for a large class of fields the desired result holds for extension fields of equal cardinality. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag     

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p-adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p-adic type.     

The Darmon-Dasgupta units over genus fields and the Shimura correspondence

We study a special case of the Gross-Stark conjecture (Gross, 1981 [Gr]), namely over genus fields. Based on the same idea we provide evidence of the rationality conjecture of the elliptic units for real quadratic fields over genus fields, which is a refinement of the Gross-Stark conjecture given by Darmon and Dasgupta (2006) [DD]. Then a relationship between these units and the Fourier coefficients of p-adic Eisenstein series of half-integral weight is explained.