However, for reductive padic groups, there is very little known, and there are many open problems in harmonic analysis. 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. In 49, the behavior of a trajectory of a rational p adic dynamical system in a complex p. Mathematically it is natural to work over the padic.
There is a beautiful theory of representations over such and so there is no serious problem. Over the last century, padic numbers and padic analysis have come to playa central role in modern number theory. In press, journal preproof what are journal preproof articles. These notes are essentially the lecture notes for that course. 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. We study the behaviour of the iterates of the chebyshev polynomials of the first kind in p adic fields. Therefore, naturally there arises a need to study the dynamics of these functions in the p adic analysis. Before studying random p adic functions, some basic details of the p adic numbers themselves must be set down.
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. A more computationally oriented point of view, based on padic analysis, can be seen in the. Reactiondiffusion equations on complex networks and. Constructing the complex padic numbers mathematics. Z pconsists of all the p adic numbers whose p adic expansion only contains positive power of p. 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. On the other hand, such p adic dynamical systems appear in the process of studying p adic gibbs measures 11, 16, 17, 26. It took more than 25 years before padic numbers were taken seriously by number theorists. It means that the padic l function should have a zero of multiplicity strictly bigger than the one of the complex.
The first term, i presented several classical results on zeta functions in characteristic p weils calculation of the zeta. In general, a trivial zero should appear when 1 or p. By lg we shall denote the space of continuous complex valued. Z pconsists of all the padic numbers whose padic expansion only contains positive power of p.
The classification of finite field extensions of the. Ii representations of classical groups over p adic fields 43. We shall sometime need to consider a bigger eld than q p, the algebraic. On irreducible representations of compact padic analytic groups. The ring of padic integers zp is the unit disk in the space qp with the norm p. 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. It showcases research results in functional analysis. Theres a lot more where this came from, but we wont need the rest of it just yet. It took more than 25 years before p adic numbers were taken seriously by number theorists. 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. Mathematics subject classification 2010 eisbn 9781461401957 doi 10.
In this presentation we will discuss upto the algebraic closure of qp. The basics of padic analysis and of padic mathematical physics are explained in 47, 48, 9. For example, the characters of the irreducible representations of reductive groups over padic elds are not yet well understood. In this paper a few analogs are developed for the padic case. However, here we wish to relay another, deeper, relation between the cohomology. On the other hand, such padic dynamical systems appear in the process of studying padic gibbs measures 11, 16, 17, 26. 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. It showcases research results in functional analysis over nonarchimedean valued complete fields. The behaviour of the analytic elements on an infraconnected set d in k an algebraically closed complete ultrametric field is mainly explained by the circular filters and the monotonous filters on d, especially the tfilters. In this paper a few analogs are developed for the p adic case. A course in padic analysis american mathematical society. We will call padic digita natural number between 0 and p 1 inclusive.
One consequence of this is that the roots of any polynomial f. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Representations of reductive padic groups under revision. Then there exists a complex manifold y and a proper sur. Then jaj 2 2, jaj 3 3 8, jaj 5 53, jaj p 1 for p 7.
Therefore, naturally there arises a need to study the dynamics of these functions in the padic analysis. The classification of finite field extensions of the field. 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. 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. Applications of p adic analysis have mainly been in number theory, where it has a significant role in diophantine. Presents a systematic treatment of the moduli space of curves from the point of view of \p\adic galois representations. Multiplicative seminorms and berkovich spaces 19 5. Dec 06, 2012 over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. 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. Existence of invariant norms in padic representations of. Over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. Click download or read online button to get introduction to p adic analytic number theory book now. This elementary introduction offers a broad understanding of p adic numbers.
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. Lectures on some aspects of padic analysis school of. Galois representations institute for advanced study. The basics of p adic analysis and of p adic mathematical physics are explained in 47, 48, 9. Introduction to p adic analytic number theory download. Weils theorem on counting points over finite fields via padic integration will essentially.
In both 1 and 2 kinematical variables a, b, c are real. 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 write this conventionally as a i a 2 a 1 a 0 that is, the a iare written from left to right. Reactiondiffusion equations on complex networks and turing patterns, via padic analysis.
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. 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. In section 4, we will continue expanding upon the theory of divisors to prove. Indeed, recently this proof has actually been converted into an algorithm for computing. Complex analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. This latter approach will allow to show that padic integers form a ring, denoted by zp.
But as there are no hilbert spaces we have to work in the category of banach spaces over the. Constructing the complex padic numbers mathematics stack. 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. Before studying random padic functions, some basic details of the padic numbers themselves must be set down. 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 notes are as self contained as possible, covering basic number theory and analytic ideas which would probably be familiar to to more advanced readers. This site is like a library, use search box in the widget to get ebook that you want. Course notes and supplementary material pdf format. 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. The usual meaning taken for p adic analysis is the theory of p adic valued functions on spaces of interest. 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. Pdf padic description of characteristic relaxation in.