A course in p adic analysis, alain m robert

The absolute value makes it possible to study the convergenceoiformal power series, thus providing another unifying concept for analysis.. 3.5 Closed Subgroups of the Additive Group of p

Graduate Texts in Mathematics 198

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University of Michigan Ann ArOOr, MI 48109 USA

K.A Ribet Mathematics Department University of California

at Berkeley Berke1ey, CA 94720-3840 USA

Mathematics Subject Classification (2000) : 11-01, IIE95, IISxx

Library of Congress Cataloging-in-Publication Data

Robert, Alain.

A course in p-adic analysis1Alain M Robert

p cm - (Graduate texts in mathematics ; 198)

Includes bibliographical references and index.

ISBN 978-1-4419-3150-4 ISBN 978-1-4757-3254-2 (eBook)

DOI 10.1007/978-1-4757-3254-2

I p-adicanalysis I Title 11 Series.

QA241, R597 2000

Printed on acid-free paper.

© 2000 Springer Science+Business Media New York

Originally published by Springer-Verlag New York, Inc in 2000.

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Kurt Hensel (1861-1941) discovered or invented the p-adic numbers' around theend of the nineteenth century In spite of their being already one hundred yearsold, these numbers are still today enveloped in an aura of mystery within thescientific community Although they have penetrated several mathematical fields,number theory, algebraic geometry, algebraic topology, analysis, , they haveyet to reveal their full potential in physics, for example Several books on p-adic

analysis have recently appeared:

F.Q.Gouvea: p-adic Numbers (elementary approach);

A Escassut: Analytic Elements in p-adic Analysis, (research level)

(see the references at the end of the book), and we hope that this course willcontribute to clearing away the remaining suspicion surrounding them This book

is a self-contained presentation of basic p-adic analysis with some arithmeticalapplications

* * *

Our guide is the analogy with classical analysis In spite of what one may think,these analogies indeed abound Even if striking differences immediately appearbetween the real field and the p-adic fields, a better understanding reveals strong

common features We try to stress these sirnilarities and insist on calculus with the

p-adics,letting the mean value theorem play an important role An obvious reasonfor links between real/complex analysis and p-adic analysis is the existence of

IThe letterpstands for a fixed prime (chosen in the list 2, 3, 5, 7,11, ) except when explicitly stated otherwise.

vi Preface

an absolute value in both contexts.? But if the absolute value is Archimedean in

reallcomplex analysis,

ifx =f:.0, for anyythere is an integernsuch thatInxl > lyl,

it is non-Archimedean in the second context, narnely, it satisfies

Inxl = I;X +x -+: +x,1 ::: Ix\.

ntenns

In particular, Inl ::: 1 for all integers n , This implies that for any r ::: 0 the subset

of elements satisfying lxi::: r is an additive subgroup, even a subring if r =1.

For such an absolute value, there is (except in a trivial case) exactly one prime

p such that Ipl < 1.3Intuitively, this absolute value plays the role of an order

of magnitude Ifx has magnitude greater than 1, one cannot reach it from 0 bytaking a finite number ofunit steps (one cannot walk or drive to another galaxy!).Furthermore,Ipl< 1 implies thatIpn I~ 0, and the p-adic theory provides a linkbetween characteristic 0 and characteristicp

The absolute value makes it possible to study the convergenceoiformal power series, thus providing another unifying concept for analysis This explains theimportant role played by formal power series They appear early and thereafterrepeatedly in this book, and knowing from experience the feelings that they inspire

in our students, Itryto approach them cautiously, as if to tarne them

* * *

Here is a short summary of the contents

Chapter I: Construction of the basic p-adic setsZp, QpandSp,

Chapters II and III: Algebra, construction of Cp andQp,

Chapters IV, V, and VI: Function theory,

Chapter VII: Arithmetic applications

I have tried to keep these four parts relatively independent and indicate by anasterisk in the table of contents the sections that may be skipped in a first reading

I assume that the readers, (advanced) graduate students, theoretical physicists, andmathematicians, are familiar with calculus, point set topology (especially metricspaces, normed spaces), and algebra (linear algebra, ring and field theory) Thefirst five chapters of the book are based solelyon these topics

The first part can be used for an introductory course: Several definitions of thebasic sets of p-adic numbers are given The reader can choose a favorite approach!Generalities on topological algebra are also grouped there

2Both Newton's method for the determination of real roots off = 0 and Hensel's lemma in the p-adic context are applications of the existence of fixed points for contracting maps in a complete metric space.

3Since the prime pis uniquely determined, this absolute value is also denoted by1.lp However, since we use it systematically, and hardly ever consider the Archimedean absolute value, we simply write1.1

Preface viiThe second- more algebraic - part starts with a basic discussion of ultrametricspaces (Section 11.1) and ends (Section lIlA) with a discussion of fundamentalinequalities and roots of unity (not needed before the study of the logarithm inSection VA) In between, the main objective is the construction of a complete andalgebraically closed field C p , which plays a role similar to the complex field C

of classical analysis The reader who is willing to take for granted that the p-adic

absolute value has a unique extension I.IKto every finite algebraic extension K

of Qp can skip the rest of Chapter II: If K and K ' are two such extensions, the

restrictions ofI.IKand I.IK'to K nK' agree This proves that there is a uniqueextension of the p-adic absolute value ofQp to the algebraic closureQ'; ofQp.

Moreover, ifa E Aut(K /Qp), thenx t-+ IxO"IK is an absolute value extendingthe p-adic one, hence this absolute value coincides with I.IK'This shows that

o is isometrie.Ifone is willing to believe that the completion Qi =C p is also

algebraieally closed,most of Chapter III may be skipped as weIl

In the third part, functions of a p-adic variable are examined In Chapter IV,continuous functions (and, in particular, locally constant ones) / :Zp -+ Cparesystematically studied, and the theory culminates in van Hamme's generalization

of Mahler's theory Many results conceming functions of a p -adic variable are tended from similar results conceming polynomials For this reason, the algebra ofpolynomials plays a central role, and we treat the systems of polynomials - umbralcalculus - in a systematic way Then differentiability is approached (Chapter V):Strict differentiability plays the main role This chapter owes much to the presenta-

ex-tion by W.H Schikhof: Ultrametrie Caleulus, an Introdueex-tion to p-adic Analysis.

In Chapter VI, a previous acquaintance with complex analysis is desirable, sincethe purpose is to give the p-adic analogues of the classical theorems linked to thenames ofWeierstrass, Liouville, Picard, Hadamard, Mittag-Leffler, among others

In the last part (Chapter VII), some familiarity with the classical gamma functionwill enable the reader to perceive the similarities between the classical and theP:

adic contexts Here, a means of unifying many arithmetic congruences in a generaltheory is supplied For example, the Wilson congruence is both generalized andembedded in analytical properties of the p-adic gamma function and in integralityproperties of the Artin-Hasse power series I explain several applications of p -adicanalysis to arithmetic congruences

* * *

Let me now indicate one point that deserves more justifieation The study of metric

spaces has developed around the classical examples of subsets ofRn(we makepictures on a sheet of paper or on the blackboard, both models of R2 ),and a famous

treatise in differential geometry even starts with "The nieest example0/ametrie spaee is Euclidean n-spaceRn." This point of view is so widely shared that onemay be 100to think that ultrametrie spaces are not genuine metric spaces! Thus thecommonly used notation for metric spaces has grown on the paradigmatic model

of subsets of Euclidean spaces For example, the "closed ball" of radius rand

center a - defined by d(x, a) :s: r- is often denoted by B(a ;r) or Br(a) This

notation comforts the belief that it is the closure of the "open ball" having the same

viii Preface

radius and center If the specialists have no trouble with the usual terrninology andnotation (and may defend it on historical grounds), our students lose no opportunity

to insist on its rnisleading meaning In an ultrametric space all balls of positive

radius (whether defined by d(x, a) ::::: r or by d(x, a) < r) are both open and

closed They are dopen sets Also note that in an ultrametric space, any point of

a ball is a center of this ball The systematic appearance of totally disconnectedspaces in the context of fractals also calls for a renewed view of metric spaces Ipropose using a more suggestive notation,

B<r(a)={x : d(x, a) < r}, B~r(a)={x : d(x, a) ::::: r}

which has at least the advantage of clarity In this way I can keep the notation

A strictly for the closure of a subset A of a topological space X The algebraicclosure of a field Kis denoted byK" ,

* * *

Finally, let methank all the people who helped me during the preparation of thisbook, read prelirninary versions, or corrected mistakes I would like to mentionespecially the anonymous referee who noted many mistakes in my first draft,suggested invaluable improvements and exercises; W.H Schikhof, who helped

me to correct many inaccuracies; and A Gertsch Hamadene, who proofread thewhole manuscript I also received encouragement and help from many friends andcollaborators Among them, it is a pleasure for me to thank

D Barsky, G Christol, B Diarra, A Escassut, S Guillod-Griener,

A Junod, V Schüreh,C.Vonlanthen,M Zuber

My wife, Ann, also checked my English and removed many errors

Cross-references are given by number: (11.3.4) refers to Section (3.4) of Chapter

11 Within Chapter 11 we ornit the mention of the chapter, and we simply refer

to (3.4) Within a section, lemmas, propositions, and theorems are individuallynumbered only if several of the same type appear I have not attempted to trackhistorical priorities and attach names to some results only for convenience Generalassumptions are repeated at the head of chapters (or sections) where they are inforce

Figures 1.2.5a, 1.2.5c, 1.2.5d, and 1.2.6 are reproduced here (some with rninormodifications) with written perrnission from Marcel Dekker They first appeared in

my contribution to the Proceedings ofthe 4th InternationaL Conference on p-adic FunctionaL AnaLysis(listed in the References)

Alain M RobertNeuchätel, Switzerland, July 1999

1.2 Addition of p-adic Integers

1.3 The Ring of p-adic Integers

1.4 The Order of a p-adic Integer

1.5 Reduction mod p

1.6 The Ring of p-adic Integers is a Principal Ideal Domain

2 The Compact SpaceZp

3.2 Closed Subgroups of Topological Groups

3.3 Quotients of Topological Groups

3.4 Closed Subgroups of the Additive Real Line

3.5 Closed Subgroups of the Additive Group of p-adic Integers

3.7 Topological Fields, Valued Fields

v11

12345677891112161717192022232425

4.4 Projective Limits of Topological Spaces

4.5 Projective Limits of Topological Groups

4.6 Projective Limits of Topological Rings

4.7 Back to the p-adic Integers

*4.8 Formal Power Series and p-adic Integers

5 The Field Qp of p-adic Numbers

5.2 Ultrametric Structure on Qp

*5.3 Characterization ofRational Numbers Among

p-adic Ones

5.4 Fractional and Integral Parts of p-adic Numbers

5.5 Additive Structure of Qp and Zp

6.4 The Newtonian Algorithm

6.5 First Application: Invertible Elements in Zp

6.6 Second Application: Square Roots in Qp

6.7 Third Application: nth Roots of Unity in Zp

Table: Units, Squares, Roots ofUnity

*6.8 Fourth Application: Field Automorphisms of Qp

Appendix to Chapter I: The p-adic Solenoid

*A.l Definition and First Properties

*A.2 Torsion of the Solenoid

*A.3 Embeddings of R and Qp in the Solenoid.

*A.4 The Solenoid as a Quotient

*A.5 Closed Subgroups of the Solenoid

*A.6 Topological Properties of the Solenoid

Exercises for Chapter I

2 Finite Extensions of the Field of p-adic Numbers

1 Ultrametric Spaces

Table: Properties of Ultrametric Distances

1.2 Ultrametric Principles in Abelian Groups

Table: Basic Principles of Ultrametric Analysis

1.3 Absolute Values on Fields

1.4 Ultrametric Fields : The Representation Theorem

1.5 General Form of Hense!'s Lemma

26262828303132333436363739404344454546464749 49

51

5353545555565760616369 69 697373777779 80

Contents1.6 Characterization of Ultrametric Absolute Values

1.7 Equivalent Absolute Values

2 Absolute Values on the Field Q

*2.1 Ultrametric Absolute Values onQ

*2.2 Generalized Absolute Values

*2.3 Ultrametric Among Generalized Absolute Values

*2.4 Generalized Absolute Values on the Rational Field

3 Finite-Dimensional Vector Spaces

3.3 Uniqueness ofExtension of Absolute Values

3.4 Existence of Extension of Absolute Values

3.5 Locally Compact Ultrametric Fields

4 Structure of p-adic Fields

4.1 Degree and Residue Degree

4.2 Totally Ramified Extensions

4.3 Roots of Unity and Unramified Extensions

*4.5 Example 1: The Field of Gaussian 2-adic Numbers

*4.6 Example 2: The Hexagonal Field of 3-adic Numbers

*4.7 Example 3: A Composite ofTotally Ramified Extensions

Appendix to Chapter11:Classification of Locally Compact Fields

*A.1 Haar Measures

*A.2 Continuity of the Modulus

*A.3 Closed Balls are Compact

*A.4 The Modulus is a Strict Homomorphism

*A.5 Classification

*A.6 Finite-Dimensional Topological Vector Spaces

*A.7 Locally Compact Vector Spaces Revisited

*A.8 Final Comments on Regularity of Haar Measures

Exercises for Chapter11

3 Construction of Universal p-adic Fields

1.1 Extension of the Absolute Value

1.2 Maximal Unramified Subextension

1.3 Ramified Extensions

1.4 The Algebraic Closure~is not Complete

*1.6 A Finiteness Result

*1.7 Structure ofTotally and Tamely Rarnified Extensions

2 Definition of a Universal p-adic Field

2.1 More Results on Ultrametric Fields

2.2 Construction of a Universal FieldQp

2.3 The FieldQpis Aigebraically Closed

xi8283858586888890 90939495969797101104107111112114115115116116118118

119

121122123127127127128129129130132133134134137138

xii Contents

3.2 Finite-Dimensional Vector Spaces over a Complete

3.3 The Completion is Algebraically Closed 143

*3.4 The Field C, is not Spherical1y Complete 143

*3.5 The Field C, is Isomorphie to the Complex Field C 144Table: Notation 145

4 Multiplicative Structure ofC p • • • • • • • • • • • • • • 1464.1 Choiee ofRepresentatives for the Absolute Value 146

4.3 Fundamental Inequalities 1484.4 Splitting by Roots of Unity of Order Prime top 1504.5 Divisibility ofthe Group ofUnits Congruent to 1 151Appendix to Chapter III: Filters and Ultrafilters 152

1 Functions of an Integer Variable 1601.1 Integer-Valued Functions on the Natural Integers 1601.2 Integer-Valued Polynomial Functions 1631.3 Periodie Functions Taking Va1ues in a Field

of Characteristicp 1641.4 Convolution of Functions of an Integer Variable 1661.5 Indefinite Sum ofFunctions of an Integer Variable 167

2 Continuous Functions onZp 1702.1 Review of Some Classieal Results 170

2.3 Mahler Series 1722.4 The Mahler Theorem 1732.5 Convolution ofContinuous Functions onZp 175

3 Locally Constant Functions onZ p 178

4.5 The Monna-Fleischer Theorem

*4.7 The p-adic Hahn-Banach Theorem

5.2 The Basic System of Polynomials of a Delta Operator .5.3 Composition Operators

5.4 The van Ramme Theorem

5.5 The Translation Principle

Table: Umbral Calculus

6 Generating Functions

6.1 Sheffer Sequences

6.2 Generating Functions

6.3 The Bell Polynomials

Exercises for Chapter IV

*1.4 Limited Expansions of the Second Order

1.5 Differentiability of Mahler Series

1.6 Strict Differentiability of Mahler Series

2 Restricted Formal Power Series

2.1 A Completion of the Polynomial Algebra

2.2 Numerical Evaluation of Products

2.3 Equicontinuity of Restricted Formal Power Series

2.4 Differentiability of Power Series

2.5 Vector-Valued Restricted Series

3.1 The p-adic Valuation of a Factorial

3.2 First Form of the Theorem

3.3 Application to Classical Estimates

3.4 Second Form ofthe Theorem

4 The Exponentiel and Logarithm

4.1 Convergence of the Defining Series

4.2 Properties of the Exponential and Logarithm

4.3 Derivative of the Exponential and Logarithm

4.4 Continuation of the Exponential

4.5 Continuation of the Logarithm

5.1 Definition via Riemann Sums

5.2 Computation via Mahler Series

xiii

190192194195195197198201204

207 207

207209211212

217

217

217221222224226232233233235236238

240

241241242245247248

249

251251252257258259263263265

xiv Contents

5.4 Relation to Bemoulli Numbers

5.5 Sums of Powers

5.6 Bemoulli Polynomials as an Appell System Exercises for Chapter V

6 Analytic Functions and Elements 1 Power Series

1.1 Formal Power Series

1.2 Convergent Power Series

1.3 Formal Substitutions

1.4 The Growth Modulus

1.5 Substitution of Convergent Power Series

1.6 The valuation Polygon and its Dual

1.7 Laurent Series

2 Zeros of Power Series 2.1 Finiteness of Zeros on Spheres

2.2 Existence of Zeros

2.3 Entire Functions 2.4 Rolle's Theorem

2.5 The Maximum Principle 2.6 Extension to Laurent Series 3 Rational Functions 3.1 Linear Fractional Transformations 3.2 Rational Functions 3.3 The Growth Modulus for Rational Functions *3.4 Rational Mittag-Leffler Decompositions *3.5 Rational Motzkin Factorizations *3.6 Multiplicative Norms onK(X) 4 Analytic Elements

*4.1 Enveloping Balls and Infraconnected Sets *4.2 Analytic Elements *4.3 Back to the Tate Algebra *4.4 The Amice-Fresnel Theorem *4.5 Thep-adicMittag-Leffler Theorem *4.6 The Christol-Robba Theorem Table: Analytic Elements

*4.7 Analyticity ofMahler Series *4.8 The Motzkin Theorem Exercises for Chapter VI

7 Special Functions, Congruences 1 The Gamma Functionrp 1.1 Definition

1.2 Basic Properties

266 269 272 275 276 280 280 280 283 286 290 294 297 303 305 305 307 313 315 317 318 321 321 323 326 330 333 337 339 339 342 344 347 348 350 354 354 357 359 366 366 367 368

1.3 The Gauss Multiplication Fonnula

1.4 The Mahler Expansion

1.5 The Power Series Expansion of logrp

*1.6 The Kazandzidis Congruences

2.1 Definition and Basic Properties

2.2 Integrality of the Artin-Hasse Exponential

2.3 The Dieudonne-Dwork Criterion

2.4 The Dwork Exponential

*2.5 Gauss Sums

*2.6 The Gross-Koblitz Fonnula

3 The Hazewinkel Theorem and Honda Congruences

3.1 Additive Version of the Dieudonne-Dwork Quotient

3.4 Applications to Classical Sequences

3.6 Applications to Appell Systems of Polynomials

Exercises for Chapter VII

Specific References for the Text

Bibliography

Tables

Basic Principles of Ultrametric Analysis

Conventions, Notation, Terminology

Index

xv

371374375380382385386388391393397401403403404408410411412414

419 423 425 429 431 435

p-adic Numbers

The letter p will denote a fixed prime.

The aim of this ehapter is the eonstruetion of the eompaet topologieal ring Zp

of p-adic integers and of its quotient field Qp , the loeally eompaet field of p-adic numbers This gives us an opportunity to develop a few eoneepts in topological algebra ,namely the struetures mixing algebra and topology in a eoherent way.Two tools play an essential role from the start:

• the p-adie absolute valueI.I p = I.Ior its additive version, the p -adie valuation

ord,= v p ,

• reduetion mod p

1 The Ring Zp of p-adic Integers

We start by a down-to-earth definition of p-adie integers: Other equivalent sentations for them appear below, in (4.7) and (4.8)

pre-1.1 Definition

A p-adie integer is aformal seriesL i:::Oa, pi with integral coefficients ai satisfying

o~ai ~ P-I

With this definition, a p -adie integer a = Li >Oai pi ean be identified with the

sequenee (ai)i:::O of its eoeffieients, and the set-of p-adie integers eoineides with

2 1.p-adic Numbers

the Cartesian product

x =X p = n{O, 1, ,p-l} ={O, 1, ,p- I}N

expansions in base p of natural integers produce p-adic integers (ending with zero

coefficients: Finite series are special series), and we obtain a canonical embedding

of the set of natural integersN = {O, 1, 2, } into X

From the definition, we immediately infer that the set0/p-adic integers is not countable.Indeed, if we take any sequence of p-adic integers, say

b= Lbi / ,

i:;:O

,

we can define a p-adic integer x = L i:;:Ox i p iby choosing

thus constructing a p-adic integer different froma, b, c, This shows that thesequencea , b, c, does not exhaust the set of p-adic integers A mapping from

the set of natural integers N to the set of p-adic integers is never surjective

1.2 Addition of p-adic 1ntegers

Let us define the sum of two p-adic integersa and b by the following procedure The first component of the sum is ao+bo if this is less than or equal to p - 1, or

ao+bo - p otherwise In the second case, we add a carry to the component of

pand proceed by addition of the next components In this way we obtain aseriesfor the sum that has components in the desired range More succinctIy, we can say

that addition is defined componentwise, using the system0/carries to keep them

in the range {O, 1, , P - l}

An example will show how to proceed Let

a= 1= 1+ °p+ °p2+ ,

b= (p - 1)+(p- 1)p+(p - 1)p2+

The suma+b has a first component 0, since 1+(p - 1)= p. But we have toremember that a carry has to be taken into account for the next component Hencethis next component is also 0, and another carry has to be accounted for in thenext p1ace, etc EventuaIly, we find that all components vanish, and the result is

1.The Ring Zpofp-adicIntegers 3

1+b= 0, namelybis an additive inverse of the integera= 1 (in the set of p-adieintegers), and for this reason writtenb= -1.More generally, if

so that a+b+1 =O This is best summarized bya+a(a)+1 =0 or even

a(a)+1= -a.In partieular, all natural integers have an additive inverse in theset of p-adie integers It is now obvious that the set X of p-adie integers with thepreeedingly defined addition is an abelian group The embedding of the monoid

N in X extends to an injeetive homomorphism Z +X Negative integers havethe form-m- 1= a(m)with all but finitely many eomponents equal top - 1.Considering that the rational integers are p -adie integers, from now on we shalldenote by Zpthe group of p-adie integers (Another natural reason for this notationwill appear in (3.6).) The mappinga :Zp +Zpobviously satisfiesa 2 = a 0a =

id and is therefore aninvolutionon the set of p-adie integers Whenpis odd, thisinvolution has a fixed point, namely the elementa=L i?:O~pi E Zp.

1.3 The Ring 01 p-adic 1ntegers

Let us define the produet of two p-adie integers by multiplying their expansionseomponentwise, using the system of earries to keep these eomponents in the desiredrange {O, 1, ,p - 1}

This multiplieation is defined in sueh a way that it extends the usual tion of natural integers (written in basep) The usual algorithm is simply pursuedindefinitely Again, a eouple of examples will explain the procedure We havefound that -1 = L(P - l)pi.Now we write

4 1.p-adicNumbers

but it is not so easy to deduce the coefficients of-mfrom this relation Together

with addition and multiplication, Zp is a commutative ring When p is odd, the

fixed element under the involutiona is

a = L p - 1 pi =P- 1 L l =p - 1 _1_ = _~ ,

but 2 is not an invertible element ofZz ,-t ~ Zz, and the involutiona = ozhas

no fixed point in Zz.

1.4 The Order ofa p-adic Integer

Let a=Li>O ail be a p-adic integer /fa =f.0, there is afirst index v =v(a) 2:0

such that av-=f. O.This index is the p -adic order v =v(a)= ordp(a), and we get amap

v= ord, :Zp - {O}-+ N

This terminology comes from a formal analogy between the ring of p-adic integersand the ring ofholomorphic functions ofa complex variable zE C.If1is a nonzeroholomorphic function in a neighborhood of a pointa E C, we can write its Taylorseries near this point

I(z) = L an(z - a)n, (am=f.0, [z - al < s)

n~m

The indexmof the first nonzero coefficient is by definition the order (of vanishing)

of1at a: this order is 0 if I(a)=f.0 and is positive if1vanishes at a.

Proposition The ring Zp01p-adic integers is an integral domain.

PRoOF The commutative ring Zp is not {O}, and we have to show that it has no

zero divisor Let therefore a = Li>Oa, pi =f. 0, b = L i>Ob, pi =f. 0, and define

v = v(a), w = v(b) Then av is theflrstnonzero coefficient ofa, 0< av < p, and

similarlyb wis the first nonzero coefficient ofb.In particular,pdivides neithera v

nor bw and consequently does not divide their product avbw either By definition

of multiplication, the first nonzero coefficient of the product ab is the coefficient cv+w of pV+w , and this coefficient is defined by

o< cv+w< p , cv+w==avb w (mod p).

Corollary of proof The order v : Zp - {O} -+ N satisfies

v(ab)= v(a)+v(b), v(a+b)2:min(v(a), v(b»

ifa, b, and a+b are not zero.

•

•

1 The Ring Zp of p-adic Integers 5

It is convenient to extend the definition of the order by v(O) = 00 so thatthe preceding relations are satisfied without restriction onZp ,with the naturalconventions concerning the symbol00.The p-adic order is then a mapping Zp -+

Nu{oo} having the two above-listed properties

Since the quotient is a field, the kernel pZp ofeis a maximal ideal of the ring

Zp. A comment about the notation used here has to be made in order to avoid aparadoxical view of the situation: Far from being ptimes bigger thanZp, the set

pZpis a subgroup of index pinZp(just aspZis a subgroup of indexpin Z)

Proposition. The groupZ;ofinvertible elements in the ring Zp consists ofthe p-adic integers oforderzero,namely

Z; = {Laipi:ao;l: O}

i~O

PROOF Ifa p-adic integerais invertible, so must be its reduction e(a)in Fp.Thisproves the indusion Z; c rLi~Oaipi : ao;l: O} Conversely, we have to showthat any p-adic integeraof orderv(a)= 0 is invertible In this case the reduction

e(a )E Fp is not zero, and hence is invertible in this field Choose 0 < bo < P

withaobo ==1 mod pand writeaobo= 1+kp Hence, ifwe writea= ao+ptx,

then

a bo= 1+kp+pabo= 1+pK

for some p-adic integerK. It suffices to show that the p-adic integer 1+K Pisinvertible, since we can then write

a bo(l+Kp)-I =1, a-I =bo(l+Kp)-I.

In other words, it is enough to treat the caseao =1, a =1+K p.Let us observethat we can take

with integers Ci E {O, 1, ,p - 1} This possibility is assured if we apply therules for carries suitably Such a procedure is cumbersome to detail any further, and

6 1 p-adic Numbers

another, equivalent, definition ofthe ringZp will be given in (4.7) below, making

Corollary 1 The ringZpo/p-adic integers has a unique maximal ideal, namely

The statement ofthe preceding corollary corresponds to a partitionZp = Z; II

pZp (a disjoint union) In fact, one has a partition

Zp - {O}= Ulz; (disjoint union ofpkZ; =v-I(k».

1.6 The Ring 0/ p-adic Integers is a Principal Ideal Domain

The principal ideals ofthe ringZp,

2 The Cornpact SpaceZp 7

2 The Compact Space z;

2.1 Product Topology on Zp

The Cartesian product spaces

X p= TI{O,I,2, ,p-l}={O,I,2, ,p-I}N

i :::O

will now be considered as topological spaces, with respect to the product topologyofthe finite discrete sets {O, 1,2, ,p- I} These basic spaces will be studiedpresently, and we shall give natural models for them (they are homeomorphic for

all p) By the Tychonoff theorem, Xp is compact It is also totally disconnected:

The connected components are points

Let us recall that the discrete topology can be defined by a metric

Although all metrics on a compact metrizable space are uniformly equivalent, they

are not all equally interesting! For example, we favor metrics that give a faithful

image of the coset structure of Zp: For each integer k EN,all cosets of pkZp in

Zp should be isometrie (and in partieular have the same diameter)

The p-adic metric is the first mentioned above Unless speeified otherwise, we

use it and introduee the notation

1d(x,0)= r: if x =I0(v = ordp(x»,

(absolute values will be studied systematieally in Chapter 11) We reeover the

p-adie metrie from this absolute value by d(x, y) = Ix - yl. With this metrie,

multiplication by p in Zp is a contracting map

d(px , py)= ~d(x , y)

and henee is continuous

8 1.p-adic Numbers

2.2 The Cantor Set

In point set topology the Cantor set plays an important role Let us recall itsconstruction From the unit interval Co= I = [0,1] one deletes the open middlethird There remains a compact set

Deleting again the open middle third of each of the remaining intervals, we obtain

a smaller compact set

Iterating the process, we get a decreasing sequence of nested compact subsets of

the unit interval By definition, the Cantor set C is the interseetion0/all C n •

The Cantor set

It is a nonempty compact subset of the unit interval I = [0, 1] The Cantordiagonal process (see 1.1) also shows that this compact set is not countable.lfwe

temporarily adopt a system of numeration in base 3 - hence with digits 0, 1, and

2 - the removal of the first middle third amounts to deleting numbers having firstdigit equal to 1 (keeping first digits°and 2) Removing the second, smaller, middleintervals amounts to removing numbers having second digit equal to 1, and so on.Finally, we see that the Cantor set C consists precisely of the numbers° :sa :s 1

that admit an expansion in base 3:

with digitsCi;= °or 2 We obtain these expansions by doubling the elements ofarbitrary binary sequences This leads to considering the bijection

The definition of the product topology shows that this mapping is continuous, andhence is a homeomorphism, since the spaces in question are compact

2 The Compact Spacez, 9

Binary sequences can also be considered as representing expansions in base 2

of elements in the unit interval This leads to a surjective mapping

Zz +[0, I]

This map is surjective and continuous but is not injective: The numbers Li>i2iand2 i EZzhave the same image in [0, 1], as is immediately seen (in the decimalsystem, a decimal expansion having only 9's after place j can be replaced by adecimal expansion with a single I in place j).In fact, Cardlp-!(t) ::::2 for any

2/9 1/31/9

Gluing the extremities of the Cantor set

2.3 Linear Models ojZp

We choose areal numberb> 1 and use it as numeration base in the unit interval[0, I] In other words, we try to write real numbers in this interval in the form

ao/b+at/bz+ with integral digits°::::a, < b.More precisely, fix the prime

p and consider the maps Vt = Vtb (= Vtb,p) : Zp +[0, 1] defined by the infiniteseries in R

For p= 2 andb =3 we find thattJ= 2, and we recover the special case studied

in the preceding section, where Vrfumished a homeomorphismZ2-+ C C [0, 1]

In general, Vr = Vrbwill be injective ifthe p-adic integers

Theorem The maps Vrb (=Vrb,p) : Zp -+ [0,1]definedfor b» 1by

are continuous When b > p, v»is injective and defines a homeomorphism of

Zp onto its image Vrb(Zp) When b = p, we get a surjective mapVrp which is

a set is "defined" by means of a dilatation producing a union of copies of translates

ofA If we denote by E(A) an intuitive - not formally defined - notion of extent

of A and if ),.A is a union of m translates of A, this self-similarity dimension d

satisfies

mE(A)=E()"A) =),.d E(A),

2 The Compact Spacez; 11

and hence d log A= log m and d = log m / log A In our case, take A= b so

that m= p and the self-similarity dimension of A = 1/fb(Zp)in [0, 1] C R islogp / log b < 1 In this way we obtain a continuous family of fractal models of

increasing dimension for b'\i p degenerating in the limit to a connected intervaI

It may be useful to look at symmetrie models obtained by replacing the digits

a, E{O,1,2, ,p- I} by symmetrie ones in {-P~l,•• • , P~l}.Define

A centered linear model of Z3

2.4 Free Monoids and Balls ojZp

LetB<r(a)denote theballdefined byd(x , a)= Ix- al< r inZp. It is cIear that

this ball does not change if we replace its radius r by the smallest powerr:that isgreaterthan orequal tor.lfthep -adicexpansionofa isaO+aIP+' +anpn+ =

Sn+pn+la,the ball does not change either ifwe replace its center bySn 'This ball

is fully deterrnined by the sequence of digits (of variable length giving the radius)

ao, al, ,an,and we associate to it the word

aOal" ·anE Mp

in the free monoid generated by S= {O, I, ,p - I}

Conversely, to each (finite) word in the elements of S - sayaoal an - weassociate the ball of centera = ao +ai p + +anpn and radiusr = »<Weget in this way a bijective map between M pand the set of balls ofZp:Observethat aballB~r(a)defined byd(x, a) ~ r is the same as a ballB<r,(a)for some

r' > r ,

12 1.p-adicNumbers

The monoidM phas several matrix representations

For example, whenn= 2, we can take

S 1-+ r,= (~ ~) (s ES= {O, 1, ,p- I})

Indeed,

Ta Tb= (~ ~) (~ ~) = ~2 a~ bP) ,

and more generally,

Observe that in this representation thelength of a word corresponds to the order of

the deterrninant of the matrix.In terms of balls, the radius appears as the absolutevalue ofthe deterrninant, whereas a center ofthe ball is read in the upper right-handcorner of the matrix With the preceding notation

Sn)1

Euclidean models of the ring of p-adic integers will be obtained in the next section

by means of injective representations

SinceM pis free, such representations are completely deterrnined by the imagesofthe generators, namely byp matricesMo, , Mp-l

2 The Compact Spacez, 13

For large enough values ofb, the image F = Fv.b = Wv.bZpwill also be adisjoint

union of self-similar images In this way we get a construction of spatial models

W(Zp) by iteration (similar to the construction of the Cantor set as an intersection

so that the imageF of W is also contained in the convex hull of E: F C f = Ko.

Moreover, by choice of the constantf},

Eventually, this leads to a representation of the fractal F as the intersection of

a decreasing sequence of compact sets K n • Several pictures will illustrate thisconstruction

(2.5.1) Take, for example, p = 3,V = R3with canonical basiseo, e), ez,and

v(k) = ek.Then the corresponding vector maps W :Z3 -+R3are given by

in the planex +y+ z= 1 Since the components ofthe imagesW(a) are positive,the image of the map W is contained in the unit simplex of R3(convex span of the

Models of Z3: Sierpirisky gasket

(2.5.2)Takenowp =5,V =R2,andthemapvdefinedbyv(O)=(0,0), v(1) =

(1,0), v(2) = (0,1), v(3) = (-1,0), v(4) = (0, -1).With a suitably chosennormalization constant {}, the components of an image q,(a) = (x,y)will satisfy-1 :5x+Y :51and-1 :5x - Y :5 I The image ofq,is a union of the similar

subsets q,(k+5Zs) (0:5k:54) Observe that q,(5Zs)= b-Iq,(Zs) and that these subsets are disjoint when b > 3 In this case, the image is a fractal of self-similarity

dimension log 5/ log b In the limit case b=3 the image is connected

2 The Compact SpaceZp 15

(2.5.3)It is interesting to refine the preceding construction by addition of an

extra component Take p = 5 as before butV = R3with v' of the form

v'(k) = (v(k), h k ) ER3,

ho= 0, h]= h 3 = -h 2 = -h 4 = h > 0

The corresponding vector maps Whave images in a tetrahedron bounded by anupper edge parallel to the x-axis and a lower edge parallel to the y-axis (hence

two horizontal edges: Choosing h suitably, we get a regular tetrahedron) These

edges give linear models of Z2, and the vertical projection on the horizontal plane(obtained by omitting the third component) is the previous construction But now,the vector maps Ware already injective forb> 2,and in the limit caseb= 2theimage is a well-known connected fractal, parametrized by Zs As in (2.2), thesevector mappings fumish commutative diagrams

i r / g

<I> = W2: Zs -+ <I>(Zs)

Model ofZsas space fractal

(2.5.4)Takep = 7, v: {O, 1,2, , 6}-+R3given byv(O)= °and

2.6 An Exotic Example

There is an interesting example connecting different primes We can add formally(i.e., componentwise) two 2-adic numbers and consider this sum in Z3' We thusobtain a continuous map

+ +

Z3

-!-C+C

n[0,2] Recall that the left vertical map is given by

and hence the diagonal composite is

( "LJa;2, LJb;2; " ;) t-+ 2"LJ a,3i'+!' +b;

Consequently, this composite has an image equal to the whole interval [0,2].Hence addition C x C ~ [0, 2] is also surjective A good way of viewing thesituation is to make a picture of the subset C x C in the unit square ofR2andconsider addition (x , y) t-+ (x+y ,0) as a projection on the x-axis The image ofthe totally disconnected set C x C is the whole interval [0, 2]

18 1.p-adicNumbers

IfGis a topologieal group, the inverse map x 1 + x-I is eontinuous (fix x = e

in the eontinuous map(x, y) 1 +xy-I) and henee a homeomorphism of order 2

of G The translationsx 1 +ax (resp.x 1 +xa)are also homeomorphisms (e.g.,the inverse ofx 1 +ax is x 1 +a-Ix).A subgroup of a topologieal group is atopologieal group for the indueed topology

Examples (1) With addition,Zp is a topologieal group We have indeed

for all n ::: O In other words , using the p-adie metrie (2.1), we have

proving the eontinuity ofthe map(x , y) 1 + X - Yat any point(a, b).

(2) With respeet to multiplieation, Z; is a topologieal group There is a mental system of neighborhoods of its neutral element 1 eonsisting of subgroups:

funda-1+pZp ::J1+p 2Zp ::J • ::J1+pnZp::J •

eonsists of subgroups: If er,ßEZp, we see that(l+ v"ß)-I = 1+pnß'for some

ß'EZp (as in (1.5», and henee

a = 1+pner, b = 1+pnß ===}ab- 1=(l+pner)(l+pnß')= 1+pny

for some y E Zp Consequently,

a' Ea(1 +pnZp), b' Eb(1+pnZ p)==}a'b,-I Eab-I(1 +pnZp) (n::: 1),

and(x, y) 1 +xy-Iis eontinuous As seen in (1.5) ,1+pZp is a subgroup ofindex

p- 1 in Z; It is also open by definition (2.1) With respeet to multiplieation, allsubgroups 1+pnz p (n :::1) are topologieal groups

(3) The realline R is an additive topologieal group

Ifa topologieal group has one eompaet neighborhood of one point, then it is aloeally eompaet spaee.Ifa topologieal group is metrizable, then it is a Hausdorff

spaee and has a eountable fundamental system of neighborhoods of the

neu-tral element Conversely, one ean show that these eonditions are sujjicient for

metrizability.'

Let G be a metrizable topologieal group Then there exists ametrie d on G that

defines the topology of G and is invariant under left translations:

d(gx, gy) = d(x, y)

1 Specific references for the text are listed at the end of the book.

3.2 Closed Subgroups ofTopological Groups

As already observed, a subgroup of a topological group is automatically a gical group for the induced topology

topolo-Lemma Let G be a topological group, Ha subgroup of G.

(a) The closure Hof His a subgroup ofG

(b) Gis Hausdorffprecisely when its neutral element is closed.

PRoOF (a) Let sp : G x G~ G denote the continuous map (x, y) 1-+ xy-I Since

His a subgroup, we have ifJ(H x H) eH and hence

ifJ(Hx H) = ifJ(Hx H) C ifJ(Hx H) C H.

This proves that His a subgroup

(b) Let us recall that a topological space X is Hausdorff precisely when thediagonal D.xis closed in the product space X x X In any Hausdorff space thepoints are closed, and thus

G Hausdorff~ {e} closed

~ D.G= ifJ-1(e)closed in G x G

~ G Hausdorff

-Proposition Let H be a subgroup of a topological group G If H contains

a neighborhood of the neutral element in G, then H is both open and closed

inG

PROOF LetVbe a neighborhood of the neutral element of G contained inH.Then

for each h EH , h V is a neighborhood of h in G contained in H This proves

that H is a neighborhood of all of its elements, and hence is open in G Considernow the cosets g H of H in G Since translations are homeomorphisms of G,these cosets are open in G Any union of such cosets is also open But H is the

complement of the union of all cosets g H f. H Hence H is closed. _

20 1 p-adic Numbers

Examples The subgroups pnz p (n ~ 0) are open and closed subgroups of theadditive groupZp,The subgroups 1+pnz p (n ~ 1) are open and closed subgroupsofthe multiplicative group 1+pZp

Let us recall that a subspace Y of a topological space X is called locally closed

(in X) when each pointy E Y has an open neighborhood V in X such that YnV

is closed inV.When this is so, the union of all such open neighborhoods of points

of Y is an open set U in which Y is closed This shows that the locally closedsubsets of X are the intersections Un F of an open set U and a closed set F

of X In fact, Yis locally closed in X precisely when Yis open in its closureY.

Locally compact subsets of a Hausdorff space are locally closed (a compact subset

is closed in a Hausdorff space) With this concept, the preceding proposition admitsthe following important generalization

Theorem LetGbe a topological group and H a locally closed subgroup Then His closed.

PRooF If H is locally closed in G, then H is open in its closure H But this closure

is also a topological subgroup of G Hence (by the preceding proposition) H is

closed in H (hence H= H) and also closed in G by transitivity of this notion _Altematively, we could replace G by H, thus reducing the general case toH

locally closed and dense in G This case is particularly simple, since all cosetsg H

must meet H: gEH for allgE G, namely H = G

Corollary 1 Let H be a locally compact subgroup0/a Hausdorjftopological

Corollary 2 Letr be a discrete subgroup0/a Hausdorjftopological groupG

The completion Gof G is also a topological group.IfG is locally compact, itmust be closed in its completion, and we have obtained the following corollary.Corollary 3 A locally compact metrizable group is complete. -

3.3 Quotients ojTopological Groups

As the following statement shows, the use of closed subgroups is weIl suited forconstructing Hausdorff quotients Let us recall that ifHis a subgroup of a group

G, then Gj H is the set of cosetsg H (g E G) The group G acts by left translations

on this set WhenHis anormal subgroup of G, this quotient is a group Let now

G be a topological group and

JT:G-+GjH

3 Topological Algebra 21

denote the canonical projection By definition of the quotient topology, the opensetsU' C GIH are the subsets such thatU= 1T-1(U')is open inG Now, ifUisany open set inG,then

1T-1(1TU) = U H = UUh

hEH

is open, and this proves that1T Uis open inGIH Hence the canonical projection

1T :G ~ GI H is a continuous and open map By complementarity, we also seethat the closed sets ofGI H are the images of the closed sets of the form F =F H

(i.e., F = 1T-1 (F ' ) for some complement F' of an open set U' C GIH). It is

convenient to say that a subset A C Gissaturated(with respect to the quotientmap 1T) when A= A H, so that the closed sets ofGI H are the images of thesaturated closed sets of G (but1T is not a closed map in general)

Proposition Let H be a subgroup0/a topological groupG.Then the quotient

GI H (equipped with the quotient topology) is Hausdorffprecisely when H is closed.

PRüOE Let1T : G ~ GIH denote the canonical projection (continuous by nition ofthe quotient topology) Ifthe quotientGI HisHausdorff, then its pointsare closed andH =1T- 1(e)is also closed Assume conversely that H is closed in

defi-G The definition of the quotient topology shows that the canonical projection1T

is an open mapping We infer that

R= R · (H x H).

3.4 Closed Subgroups ojthe Additive Real Line

Let us review a few well-known results conceming the classical realline, viewed

as an additive topological group At first sight, the differences withZpare striking,

but a closer look will reveal formal similarities, for example when compact and discreteare interchanged

Proposition 1 The discrete subgroups0/Rare the subgroups

aZ (0 S aER)

PRooF Let H #-{O} be a nontrivial discrete subgroup, hence closed by (3 2)

Consider any nonzero h in H, so that 0 < IhI(= ±h) EH The intersection H () [0, IhIlis compact and discrete, hence finite, and there is a smallest positive element

a EH Obviously, Z aCH In fact, this inclusion is an equality Indeed, if we

take anybE Handassume (without loss of generality) b> 0, we can write

b= ma+r (m EN, 0 Sr< a) (take for m the integral part of bja) Since r = b - ma E H and 0 Sr< a,

we must haver =0 by construction This proves b=ma E Z a, and hence the

Corollary The quotient0/ R by a nontrivial discrete subgroup H #- {O} is

Proposition 2 Any nondiscrete subgroup0/R is dense.

PRooF Let HeR be a nondiscrete subgroup Then there exists a sequence of

distinct elements hnE H with hn-+h EH Hence Sn= Ihn - hIE H and Sn-+O.Since H is an additive subgroup, we must also have Z Sn C H (for all n ::: 0),

(the inverse isomorphism is the logarithm to the basep)we deduce parallel resultsfor the cIosed (resp discrete) subgroups of the topological group R >o.

Typically, we shall use the fact that the discrete nontrivial subgroups of this

group have the form paZ (a > 0) or, putting ()= p-a, are the subgroups

for some 0 < () < 1

3.5 Closed Subgroups ofthe Additive Group of p-adic Integers

Proposition The closed subgroups ofthe additive group Zp are ideals: They are

PROOF. We first observe that multiplication in Zp is separately continuous, since

Ix'a -xa/= lallx' -x/-+0 (x' -+x).

Since an abelian group is a Z-module, ifH C Zp is a cIosed subgroup, then for

anyheH,

ZH eH==> Zpa C Za C Jj = H.

This proves that a cIosed subgroup is an ideal of Zp (or a Zp-module) Hence the

Corollary 1 The quotient of Zp bya closed subgroup Hi={O} is discrete • Corollary 2 The only discrete subgroup ofthe additive group Zp is the trivial subgroup {O}.

PROOF. Indeed, discrete subgroups are cIosed: We have a complete list of these

(being cIosed in Zp compact, a discrete subgroup is finite hence trivial)

Alterna-tively, if a subgroup Hcontains a nonzero elementh,it contains all multiples ofh,

and hence H :JN h In particular, H :3pnh-+ 0 (n-+ 00).Since the elements

r"hare distinct, H is not discrete •

are continuous.

The second axiom implies in particular that y t-+ - Yis continuous (fixx = -1

in the product) Combined with the first, it shows that

(x, y) t-+ X - Y :A x A -+ A

is continuous and the additive group of A is a topological group A topological ring A is a ring with a topology such that A is an additive topological group and multiplication is continuous on A x A.

IfA is a topological ring, the subgroup Ax of units is not in general a

to-pological group, since x t-+ x-I is not necessarily continuous for the inducedtopology (for an example of this, see the exercises) However, we can consider theembedding

X t-+ (x ,x- I ) : A X -+ A x A,

andgiveA X the initial topology: It is finerthan the topology induced byA Forthis

topology, Ax is a topological group: The continuity of the inverse map, induced bythe symmetry(x, y) t-+ (y, x) ofAx A,is now obvious Still with this topology,the canonical embeddingAx C-)- Ais continuous, but not a homeomorphism ontoits image in general

Proposition With the p-adic metric the ring Zp is a topological ring It is a compact, complete, metrizable space.

PRooF Since we already know that Zpis a topological group (3.1), it is enough tocheck the continuity of multiplication Fixaand binZpand considerx = a+h,

y = b+kinZp Then

[xy - abi =I(a+h)(b+k) - abi =lak +hb - hkl

~ maxf]c],Ibl)(lhl + Ikl)+Ihllkl -+ 0 (Ihl, Ikl -+0).This proves the continuity ofmultiplication at any pointCa,b) EZp x Zp •

Corollary 1. The topological group Zp is a completion ofthe additive group