Existence of invariant norms in padic representations of. The basics of p adic analysis and of p adic mathematical physics are explained in 47, 48, 9. In this presentation we will discuss upto the algebraic closure of qp. These notes are essentially the lecture notes for that course. A presentation of results in padic banach spaces, spaces over fields with an infinite rank valuation, frechet and locally convex spaces with schauder bases, function spaces, padic harmonic analysis, and related areas. This importance comes from the fact that they afford a natural and powerful language for talking about congruences between integers, and allow the use of methods borrowed from calculus and analysis for studying such problems.
In this paper a few analogs are developed for the p adic case. Z pconsists of all the padic numbers whose padic expansion only contains positive power of p. The first term, i presented several classical results on zeta functions in characteristic p weils calculation of the zeta. Mathematically it is natural to work over the padic.
The p adic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. In 49, the behavior of a trajectory of a rational p adic dynamical system in a complex p. In this paper a few analogs are developed for the padic case. On irreducible representations of compact padic analytic. However, here we wish to relay another, deeper, relation between the cohomology. Historically, this research was inspired by the discovery that the veneziano amplitude of the open bosonic string, which is calculated using an integral over the real numbers, can be generalized to the padic numbers. Such a correspondence has been shown for g gl2 when the. The usual meaning taken for p adic analysis is the theory of p adic valued functions on spaces of interest. The classification of finite field extensions of the. Therefore, naturally there arises a need to study the dynamics of these functions in the p adic analysis. In section 3, we will build the background of elementary padic analysis and the theory of newton polygons, which will be used to bound the number of zeros of particular padic analytic functions whose zeros which lie in pz pcorrespond to the rational points on the curve. This site is like a library, use search box in the widget to get ebook that you want.
In both 1 and 2 kinematical variables a, b, c are real. Indeed, recently this proof has actually been converted into an algorithm for computing. Then there exists a complex manifold y and a proper sur. Pdf padic description of characteristic relaxation in. The ring of padic integers zp is the unit disk in the space qp with the norm p. We write this conventionally as a i a 2 a 1 a 0 that is, the a iare written from left to right. Complex analysis has successfully maintained its place as the standard elementary text on functions of one complex variable.
One of its main achievements is a successful formulation and development of p adic and adelic quantum mechanics, which have complex valued wave functions of p adic and adelic arguments, respectively. However, for reductive padic groups, there is very little known, and there are many open problems in harmonic analysis. This elementary introduction offers a broad understanding of p adic numbers. This book aims to bridge the gap between the approach presented and the classical uniformization of a hyperbolic riemann surface that is studied in undergraduate complex analysis.
One consequence of this is that the roots of any polynomial f. In general, a trivial zero should appear when 1 or p. There is, never theless, need for a new edition, partly because of changes in current mathe matical terminology, partly because of differences in student preparedness and aims. But as there are no hilbert spaces we have to work in the category of banach spaces over the. Z pconsists of all the p adic numbers whose p adic expansion only contains positive power of p.
A little padic functional analysis part 1 of 2 im going to start with a little bit of terminology and notation about nonarchimedean banach spaces trusting that you can. The basics of padic analysis and of padic mathematical physics are explained in 47, 48, 9. There is a beautiful theory of representations over such and so there is no serious problem. This work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the padic analysis.
Constructing the complex padic numbers mathematics. The classification of finite field extensions of the field. Ii representations of classical groups over p adic fields 43. Therefore, naturally there arises a need to study the dynamics of these functions in the padic analysis. Over the last century, padic numbers and padic analysis have come to playa central role in modern number theory. One of its main achievements is a successful formulation and development of padic and adelic quantum mechanics, which have complexvalued wave functions of padic and adelic arguments, respectively. Introduction to p adic analytic number theory download. It means that the padic l function should have a zero of multiplicity strictly bigger than the one of the complex. A more computationally oriented point of view, based on padic analysis, can be seen in the.
On irreducible representations of compact padic analytic groups. Click download or read online button to get introduction to p adic analytic number theory book now. The complex representation theory of these groups has been intensively studied in recent years, starting with mautner, gelfandgraev, bruhat, harishchandra, and then by langlands and others motivated by arithmetic. We all know the fields r and c of real and complex numbers as the completion of the field q of rational numbers and its algebraic closure. The notes are as self contained as possible, covering basic number theory and analytic ideas which would probably be familiar to to more advanced readers. However, for reductive p adic groups, there is very little known, and there are many open problems in harmonic analysis.
This is probably the first book dedicated to this topic. Constructing the complex padic numbers mathematics stack. This work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p adic analysis. Reactiondiffusion equations on complex networks and turing patterns, via padic analysis. Applications of p adic analysis have mainly been in number theory, where it has a significant role in diophantine. Over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. Before studying random padic functions, some basic details of the padic numbers themselves must be set down. Representations of reductive padic groups under revision. We will call padic digita natural number between 0 and p 1 inclusive. Our endgoal in building up this complex analysis background is to understand the jacobian of a curve, x, which is an abelian variety over the. We study the behaviour of the iterates of the chebyshev polynomials of the first kind in p adic fields. By lg we shall denote the space of continuous complex valued. A course in padic analysis american mathematical society.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The purpose of the padic langlands programme is to create a similar correspondence between some of the padic representations of the reductive group, and some of the padic representations of the galois group. The representation theory of padic groups over padic. In press, journal preproof what are journal preproof articles. Theres a lot more where this came from, but we wont need the rest of it just yet.
Lectures on some aspects of padic analysis school of. This latter approach will allow to show that padic integers form a ring, denoted by zp. Weils theorem on counting points over finite fields via padic integration will essentially. Before studying random p adic functions, some basic details of the p adic numbers themselves must be set down.
Course notes and supplementary material pdf format. A presentation of results in p adic banach spaces, spaces over fields with an infinite rank valuation, frechet and locally convex spaces with schauder bases, function spaces, p adic harmonic analysis, and related areas. On the other hand, such padic dynamical systems appear in the process of studying padic gibbs measures 11, 16, 17, 26. Otherwise there would be a rational r between these two numbers and. It took more than 25 years before padic numbers were taken seriously by number theorists.
On the other hand, such p adic dynamical systems appear in the process of studying p adic gibbs measures 11, 16, 17, 26. Dec 06, 2012 over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. Multiplicative seminorms and berkovich spaces 19 5. Galois representations institute for advanced study. It showcases research results in functional analysis over nonarchimedean valued complete fields. It means that the padic l function should have a zero of multiplicity strictly bigger than the one of the complex l function the following work has been done by. Reactiondiffusion equations on complex networks and. For example, the characters of the irreducible representations of reductive groups over p adic elds are not yet well understood. Then jaj 2 2, jaj 3 3 8, jaj 5 53, jaj p 1 for p 7. We shall sometime need to consider a bigger eld than q p, the algebraic.
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