Specific References for the Text

1.2A. Cuoco: Visualizing the p-adic integers, Amer. Math. Monthly 98 (1991), pp. 355-364.A. ROBERr. Euclidean Models of p-adic Spaces , Proc. of the 4th Int. Conf. (Nijmegen),

M. Dekker (1997), pp. 349-361.

1.3.1N. BOURBAKI: Topologie Generale, Hermann (Paris 1974), Chap. IX §3.

1.6.8EK. SCHMIDT: Mehrfa ch Perfekte Körper, Mathematische Annalen 108 (1933), pp. 1-25

(cf. Th . V, p. 7) .

I Exercise 1These numbers were apparently already considered by Gergonne and Lucas in the nineteenth

century and called congruent numbers:R. CUCULIERE: Jeux Mathematiques, in Pour la Science (Juin 1986), pp. 10-15.

11.1.2N. BOURBAKI: Topologie Generale, Hermann (Paris 1974), Chap. IX §3.

11.4.6A. ROBERT:A Good Basisfor Computing with Complex Numbers, Elemente Math. 49 (1994),

pp. 111-117.It was pointed out to me by H. Brunotte (Düsseldorf) that the surjectivity proof given in

this article has a gap. One can deduce it fromI. KATAI: Number systems in imaginary quadratic fields, Ann. Univ. Sei . Budapest, Sect.

Comp. 1994, pp . 91-93, MR#95k: 11134.

II.AN. BOURBAKJ: Algebre commutative, Hermann (Pari s 1964) , Chap. VI §9.A. WEIL: Basic Number Theory, Springer Verlag (3d ed. 1974), Chap. 1.

420 Specific References for the Text

m.I.6M. KRASNER: Nombre des extensions d'un degre donne d'un corps p-adique, in: Les ten­

dances geometriques en algebre et theorie des nornbres, Colloques Internat. du CNRSNÜ 143 (1966), pp. 143-169.

m .2.4For spherical completeness, cf. Chapter 4 of AC.M. VAN Roou: Non-Archimedean Func­

tional Analysis

m.2.5B. DIARRA: Ultraproduits ultrametriques de corps values, Ann. Sc. de I'Univ. de Clermont

H, Serie Math. 22 (1984), pp. 1-37.

m.3.3S. LANG: Algebra, Addison-Wesley (2nd ed. 1984), p.412 and 420, or (3d ed. corrected

1994), p. 347 and p. 482.

m Exercise 7Trees have always played an important role in p-adic analysis. They were revived recently

by Berkovitch, Escassut, Mainetti, and others in the context of circular filters, fromwhich our version for balls is derived (Exercise 2 in Chapter II).

IV.I.3J.F. ADAMS: On the groups J(X)-llI, Topology,vo1.3 (1965), pp. 193-222.

IV.I.5L. VAN HAMME: Three generalizations ofMahler's expansion . . . , in p-adic Analysis, Proc.

Trento 1989, ed. by F. Baldassari et al., Springer Verlag (1990), Lect. Notes #1454,pp. 356-361.

Iv.4.6 For compact operators and compactoidsJ.-P. SERRE: Endomorphismes completement continus des espaces de Banach p-adiques,

IHES n012 (1962), pp. 69-85 .AC.M. VAN Roou: Non-Archimedean FunctionalAnalysis p.133.

IV.5.5L. VAN HAMME: Continuous operators which commute with translations . . . , Proc. of the 1st

Int. Conf. (Laredo), M. Dekker (1992), pp. 75-88.G.-C. ROTA, D. KAHANER, A ODLYZKO: Finite Operator Calculus , J. of Math. Anal. and

Appl., vol.42, NÜ 3, (1973) pp. 684-760 .S.M. ROMAN, G.-C. ROTA: The Umbral Calculus, Adv. in Math., vol.27, nb. 2 (1978) pp. 95­

188.

V.3A ROBERf: A note on the numerators ofthe Bemoulli numbers, Expositiones Math. 9 (1991)

pp. 189-191.

V.4.5M.-C. SARMANT(-DURlx): Prolongement de lafonction exponentielle en dehors de son cercle

de convergence, C.R. Acad. Sc. Paris (A), 269 (1969), pp. 123-125.

VI.1.4For an example ofentire function, bounded on Qp see the book by W.Schikhof, Ultrametric

Calculus, p. 126.

Speeifie Referenees for the Text 421

VI.2.2S. LANG: Algebra, Addison-Wesley2nd ed. p. 215, Th. 11.2 (or 3d ed. p. 208, Th. 9.2).

VI.2.3 Part (c) is already inL. SCHNlRELMANN: Sur les fonctions dans les corps normes . . . , Bull. Aead. Sei. USSR

Math., vol. 5/6 (1938) pp.487-497.

VI.2.6D. HUSEMÖLLER: Elliptic Curves, Springer-Verlag, GTM 111 (1987) ISBN: 0-387-96371-5

(3-540-96371-5 Berlin),l.H. SILVERMAN: Advanced Topics in The Arithmetic of Elliptic Curves, Springer-Verlag,

GTM 151 ISBN: 0-387-94325-0 (3-540-94325-0 Berlin).

VI.3For reallcomplex analysis, my favorite isW. RUDIN: Real and Complex Analysis, 2nd.ed., MeGraw-Hill, New-York(1974) ISBN:

0-07-054233-3.[Runge p.288 ; Mittag-Lerner p.291; Hadamard three-eircle p.281; Picard for essential

singularities p. 227.]

VI.3.6B. GuENNEBAUD:Sur une notion de spectre pour les algebres normees ultrametriques (These,

Univ.Poitiers 1973).G. GARANDEL: Les semi-normes multiplicatives sur les algebres d 'elements analytiques ... ,

Indag. Math. 37 n° 4 (1975) pp. 327-341.

VI.4.7M. ABRAMOWITZ, I. SlEGUN: Handbook ofMathematical Functions, Dover (1972) ISBN:

486-61272-4 (Chap. 24 p. 824).

VI.4.8E. MarzKIN: La decomposiiion d'un element analytique en facteurs singuliers,

Ann. Inst. Fourier 27, fase.l (1977), pp. 67-82.

VII.1.6A. ROBERT, M. ZUBER: The Kazandzidis Supercongruences. A simple Proofand an Applica­

tion. Rend. Sem. Mat. Univ, Padova, vol. 94 (1995), pp. 235-243 .

Vn.2.5B.GROSS, N. KOBLm: Gauss sums and the p-adic r -function, Annals ofMath. 109 (1979),

pp. 569-581 .R.F. COLEMAN: The Gross-Koblitz formula, pp. 21-52 in Galois representations and arith­

metic algebraic geometry, Papers from a Symposium held at Kyoto Univ. (1985) andTokyo Univ. (1986), ed. by Y. Ihara, Adv. Studies in Pure Maths.12, North-Holland(1987), ISBN 0-444-70315-2.

S. LANG: Cydotomic Fields , Springer-Verlag, GTM 59 (1978),- . - : Cydotomic Fields ll, GTM 69 (1980).

VII.2.6A simple proof of the Gross-Koblitz formula appears in A.M. Robert: The Gross-Koblitz

Formula Revisited, Rend. Sem. Mat. Univ. Padova (to appear in 2001).

Vn.3.2Proposition 4 appears repeatedly in various formsN. BOURBAKI: Algebre commutative, Chap. VI, lemme 1, p. 157and Chap. IX, lemme 1, p. 2.

422 Specific References for the Text

VII.3.3M. HAZEWINKEL : Formal Groups and Applications, Academic Press, New-York (1978),

573 p. (contains over 500 references!).

Vll.3.4L. VAN HAMME: The p-adic Moment Problem, pp. 151-163 in p-adic Functional Analysis

(Santiago-Chile 1992), Editorial Univ. de Santiago.

VII.3.5T. HONDA: Two Congruence Properties 01Legendre Polynomials, Osaka Journal of Math .,

vol.13 (1976) , pp. 131-133.A. ROBERT: Polynomes de Legendre mod p, in Univ. BlaisePascal, Sem. d'analyse 1990-91,

1.6 (pp. 19.01-19.11).- . - : Polynomes de Legendre mod 4,c.a. Acad. Sc. Paris, 1.316, Serie I (1993), pp. 1235-1240.

Vll.3.6M. ZUBER : Proprietes de congruence de certaines familles de polynomes, c.R. Acad. Sc.

Paris, 1.315, Serie I (1992), pp. 869-872.

Bibliography

General

Y. AMICE: Les nombres p-adiqu es, PUF, Collection SUp. "Le mathematicien" 14 (Paris1975).

G. BACIIMANN: Introduction to p-adic numbers and valuation theory, Academic Press,New York (1964).

A. Esc ASSUT: Analytic Elements in p-adic Analysis , World Scient ific (1995) ISBN: 981­02-2234-3.

F. GOUVEA: p-adic Numbers: An Introduction, Springer-Verlag, Universitext (1993 ) ISBN:0-387-56844-1 (3-540-56844-1 Berlin).

N. K OBLITZ: p-adic Numbers, p-adic Analysis and Zeta-Functions , Springer-Verlag, GTM58 (1977 ,1984) ISBN: 0-387-90274-0 (3-540-90274-0 Berlin).

K. MAHLER: p-adic Numbers and their Functions , Cambridge University Press , Cambridgetract 76 (2nd ed. 1981).

A. MONNA: Analyse non-archimedienne, Springer-Verlag, New York (1970) .

A.C.M. VAN RooIJ: Non-Archimedean Functional Analysis, Marcel Dekker, Pure and Appl.Math. 51 (1978).

W.H. SCHIKHOF:Ultrametric Calculus, An Introduction to p-adic Analysis. Cambridge Stud­ies in Adv. Math. 4, Cambridge Univ. Press (1984), ISBN : 0-521-24234-7.

J.-P. SERRE: Cours d 'arithmetique, PUF, Collection Sup. "Le mathematicien" 2 (Paris 1970) .English translation: A Course in Arithmetic, Springer-Verlag (1973).

424 Bibliography

Advanced, More Specialized

S. BOSCH, U. GUNTZER, R. REMMERf: Non-Archimedean Analysis, A Systematic Approach toRigid Analytic Geometry, Springer-Verlag, Grundlehren Nr.261 (1984).

B. DwORK: Lectures on p-adicDifferential Equations, Springer-Verlag, Grundlehren Nr. 253(1982).

B. DwORK, G. GERGITO, F.J. SULLIVAN: An Introduction to Gsfunctions, Ann. ofMath. Studies133, Princeton Univ. Press (1994), ISBN: 0-691-03681-0.

J. FRESNEL, M. VAN DER Pur: Geometrie Analytique Rigide et Applications, Birkhäuser,Progress in Math. 18 (1981).

L. GERRlTZEN, M. VAN DER Pur: Schottky Groups and Mumford Curves , Springer-Verlag,L. N. in Math. 817 (1980).

Proceedings of Congresses in p-adic Functional Analysis

J. BAYOD ET AL. (editors): p-adic Functional Analysis, Proc. of the 1st Int. Conf. (1990,Laredo, Spain) Lect. Notes in pure and appl. math. 137, Marcel Dekker (1992).

N. DE GRANDE-DE KiMPE ET AL. (editors): p-adic Functional Analysis, Proc. of the 2nd Int.Conf. (1992, Santiago, Chile) Editorial Universidad de Santiago (1994).

A. ESCASSur ET AL. (editors): p-adic Functional Analysis, Proc. of the 3d Int . Conf. (June1994, Clermont-Ferrand, France) Annales Math. Blaise Pascal, vol. 2, NÜl (1995).

W.H. SCHIKHOF ETAL. (editors): p-adic Functional Analysis, Proc. ofthe 4th Int. Conf. (June1996, Nijmegen, The Netherlands) Lect. Notes in pure and appl. math. 192, MarcelDekker (1997) ISBN: 0-8247-0038-4.

J. KAKOL ET AL. (editors): p-adic Functional Analysis, Proc. ofthe 5th Int. Conf. (June 1998,Poznan, Poland) Lect. Notes in pure and appl. math. 207, Marcel Dekker (1999) ISBN:0-8247-8254-2.

A.K. KATSARAS ET AL. (editors): p-adic Functional Analysis, Proc. of the 6th Int. Conf. (July2000, Ioannina, Greece) (to appear).

Tables

Numberofquadratic

Field Units Squares Roots of unity extensions

Qz Z~=1+2Zz 1 + 8Zz JLz ={±l} 7index 4inZ~

Qp Z; :> 1 + pZp index 2 JLp-l 3P odd prime index p-l inZ;

Field:> B::;l :> B<1 Residue field Nonzero 1.1 Properties

Qp :> z, :> pZp Fp pz locally compact!Z { ef = dimQp K < 00K:>R:>P=1rR Fq (q = pt) 11rlz = r' locally compact

Q~ :> AD :> MD kD =F~ =Fpoo pQ { algebraically closednot locally compact

c, :> A p:> M p ~ =Fpoo pQ { algebraically closedcomplete

a, :> Ao :> Moko

R>o{ algebraically closed

uncountable spherically complete

426 Tables

Umbral calculus

Deltaoperator

(IV.5)

D = dfdx

T

Basic sequenceof polynomials

(IV.5.2)

umbraloperator

(Pn)n?O

Relatedsequences(IY.6.l)

Appell sequences

Dp; =npn-l

Sheffer sequences

Ss; = nSn-l

(IV.5.5)

( _ x - pn(x + ny»)x+ny n>Otranslation principle

Binomial identity: Pn(x +y) ="(p(x) + p(y»n ,"Appell sequences: Pn(x + y) = " (p(x) + y)n ,"Sheffer sequences: sn(x + y) =" (s(x ) + p(y»n,"

Analytic elements

Formalpower series

power serie sconverging in

[x] « 1

power seriesboundedin

lxi< I

analytic H(Mp )

elements in[x] « I

analytic H(Ap) = Cp{x}elements in

lxi::: 1

polynomials Cp[x]

Sequences

(an)n?o

lim sup lan11/ n ::: 1(r! 2: 1)

(an)n?oboundedsequence

Christol-Robbacondition

(4.6)

an -+ 0(n -+ (0)

an i= 0 forfinitely many n's

Co

C(N)p

Radius of convergence of some exponential series(listed in increasing order)

Tables 427

Ep (Artin-Hasse) E! (Dwork)

f ~ ex+!f L xpi e,,(x-xq) (q = pi)exp ">0~J_ pi

1 7 ldrl rp = Ipl? r» ppq

Basic Principles of UltrametricAnalysis in an Abelian Group

(1) The strongest wins

[x] » lyl =} Ix + yl = IxI.

(2) Equilibrium: All triangles are isoseeles (or equilateral )

a +b+c = 0, [c] « Ibl =} lai = Ibl.

(3) Competitivity

al + a2 + ...+ an = 0 =}

there is i#- j such that lad = lajl = max jakl.

(4) A dream realized

(an)n:::O is a Cauchy sequence~ dta.; an+l) ~ o.(5) Another dream come true (in a complete group)

Ln:::o an converges~ an ~ o.When Ln:::oan converges, Ln:::o lan I may diverge, but

ILanl s sup lanl = max lanln:::O

and the infinite version of (3) is valid.(6) Stationarity ofthe absolute value

an ~ a #- 0 =} there is N with lanl = lal/or n 2: N.

Conventions, Notation, Terminology

We use the abbreviations ,iff " if and only if," := "equal by definition," == nontriv ial equality.• is the "end of proof" (or "absence of proof") sign .In a statement: (I), (ii), . . . always denote equivalent properties.In the table of contents, an asterisk * before a section indicates that it will not be used later

and may be ornitted in a first reading .

Set Theory

P(E) power set of E : Set of subsets of E ; 0 : Empty set.A C B means "x E A ==> x E B" hence: AcE {:=:} A E P(E).

(certain authors denote this inclusion by ~).

When AC B CE, B - A = B \ Adenotes the complement of A in B,E - A = ACis the complement of a subset ACE.

A subset of E having only one element is a singleton set: x E E ==> {x} E P(E).U:Disjoint union symbol, partition of a set.EI : Set of families (or functions) I ~ E.E(l) : Set of familie s I ~ E having components equal to the base point

of E (the neutral element in a group G, the 0 in a ring A . . . )except f or finitely many indices.

Let f : E ~ F, x t-+ fex) be a map. Thenfis injective when x =1= y ==> fex) =1= f(y) , namely f is one-to-one,or equivalently when fex) = f(y) ==> x = y,fis surj ective when f(E) = F (namely f is onto),f is bijective when it is one-to-one and onto.

432 Conventions, Notation, Tenninology

The characteristic function of a subset ACE is the function

{I if XE A,

gl(x) = glA(X) = 0if x!f. A .

Fundamental Sets of Numbers

N = {O, 1,2, . . . , n , ... } C Z c Q c R c C, N* = {I, 2, .. . , n, . ..}= N>o.When pE {2, 3, 5, 7,11 , . ..} is a prime, Fp = ZlpZ.p In means p divides n, ptn means p does not divide n,

pV 11 n means that p" is the highest power of p dividing n,R>o = {x ER : x > O}, R:;:o = {x ER: x ~ O} , [a, b) : interval a ::::: x < b.Z(p) = {alb: a E Z, b ~ 1, b prime to p} C Q,Z[I/p]={apv:aEZ, VEZ}CQ.When a > 0 and S C R, aS = las : SES} C R >o, e.g., pZ C pQ C R>o.[x] E Z integral part of x ER: [x] ::::: x < [x] + 1.(x) fractional part ofx ER: x = [x] + (x) .gcd: Greatest common divisor ; lern: Least common multiple.äij : Kronecker symbol (= 1 ifi = i, = 0 otherwise).

Groups, Rings and Modules

A x: Multiplicative group of units (i.e., invertible elements) in a ring A.A[X]: Polynomial ring in one indetenninate X and coefficients in the ring A,

a monic polynomial fis a polynomial having leading coefficient 1:X" + an_IXn-1 + ...+ ao if deg f = n.

A[[X]]: Formal power series ring.A{X}: Restricted power series over a valued ring A

(Chapter V: Power series with coefficients - 0) A[X] C A{X} C A[[X]] .An integral domain is a commutative ring A =1= {O} having no zero divisor.K = Frac A: Fraction field of an integral domain A. In particular,

K(X) = Frac A[X] : Rational fractions,K((X» = Frac A[[X]] (:J K(X»: Formal Laurent series ring .

A [1/q ]: Partial fraction ring corresponding to denominators in {l, q , q2, .. .},where q is not a zero divisor in the ring A.

If G is an abelian group , then {g E G : gn = e for some integer n ~ I}is the torsion subgroup of G: In particular,Jl(A) denotes the group of roots of unity in a commutative ring A,

Jl = Jl(C X) = Jlpoo x Jl(p) , where

Jlp oo : pth-power roots ofunity (p-Sylow subgroup of Jl),

Jl(p) : Roots of unity having order prime to p,Jln(A) = {x E A : x n = I}: nth roots of unity in the ring A.

A pair ofhomomorphisms A ~ B ~ Cis exact when f(A) = ker g.A short exact sequence (SES) is an exact pair with

f injective and g surjective ; hence Cis a quotient of B by f(A) ~ A,

written 0 _ A ~ B ~ C - 0 for additive groups(replace 0 by 1 for multiplicative groups).

Conventions , Notation, Tenninology 433

Fields, Extensions

Characteristic of a field K : Either 0 or the prime p such that p . IK = 0 E K,in which case the prime field Fp is contained in K.

For each prime p, the group F; is cyclic; when the prime p is odd, the squares in F; make

up a subgroup of index two , kernel of the Legendre symbol (~) = ± 1.

In a field (or a ring) of characteristic p we have (x + y)P =x P + y" ,KU: Aigebraic closure of a field K ; when K = KU is algebraically closed of characteristic

0, J-Ln(K) is cyclic and isomorphie to Z/nZ.pi (K) = K U {oo} denotes the projective line over the field K .

Topology, Metric Spaces

The closure of a subset A C X (X being a topological space) is denoted by A.A Hausdorffspace is a topological space X in which for every pair of distinct points, it is

possible to find disjoint neighborhoods ofthese points: Equivalently, the diagonal ßx isclosed in the product X x X.

The diameter of a sub set A C X with respect to ametrie d is

diam(A) = 8(X) = SUPx,yEA d(x, y) ~ 00.

We say that A is bounded when diam(A) < 00.

The distance ofa point x E X to a subset AC Xis d(x, A) = infuEA d(x, a),

d(x , A) =0 <===> XE A.

The balls in ametrie space (X, d) are denoted by

B:::r(a) = B:::r(a; X) = {x EX : d(x , a) ~ r} : closed (dressed) ball,

B<r(a) = B<r(a; X) = {x EX: d(x , a) < r} : open (stripped) ball.

For a ball with center a equal to the base point (the neutral element in a group, the 0 elementin a ring), the notation will be just B:::r , B<r .

The sphere of radius r > 0 and center a in the metric space (X, d) is

Sr(a) = {x EX : d(x , a) = r} = B:::r(a) - B<r(a).

Ametrie space is separable if it has a countable dense subset.C(X ; K): Space of continuous functions X -+ K , or simply C(X) when K is understood;

Cb(X ; K): Subspace consisting ofthe bounded continuous functions (when K is a valuedfield). The sup norm of a bounded function is

IIfll = IIfllx = sup If(x)1 Cf E Cb(X ; K».XEX

Index

A

absolute value 11.1.3- over Q 11.2.1

algebraic variety 1.6.1Amice-Fresnel theorem VL4.4analyt ic element VI.4 .2Appell sequence IV.6.1Actin-Hasse exponential VII .2.1

B

Baire space 111.1.4balanced subset ILA.6balls (stripped and dressed) 11.1.1Banach space (ultrametric -) IV.4.1basic system of polynomials IV.S.2Bell (numbers and polynomials) IY.6.3Bell-Carlitz polynomials (IV, exercise 19)Bernoulli (numbers and polynomials) V.5.4Beukers proposition VII.3.4binomial identities IY.S.2

- polynomial IY.I .I

c

Cantor set 1.2.2carry (operations in basi s p) 1.1.2Chebyshev polynomials (V, exercise 7)Christol-Robba theorem VL4.6Clausen-von Staudt theorem V.5.Sclopen set 11.1.1commutant (bicommutant) IV.5.3

composition operator IV.5.3continuity of roots of equations III.I .Scontinuous retraction LA.6convexity (and duality) VI.1.4covering of circle LA.Icritical radius VI. 1.4cyclotomic polynomial (- units) 11.4.2

D

delta operator IV.5.Idiagonal (in a Cartesian product) 1.3.3Dieudonne-Dwork criterion VII .2.2, VII.3.3differential quotient (higher order-) V.2.4divisible group 111.4.1dominant (monomial) VI.1.4dressed ball 1I.1.1duality (convexity theory) VI.l .6Dwork series VII .2.3

E

Eisenstein (irreducibility criterion)11.4.2

- polynomial 11.4.2entire function VI.2 .3enveloping ball Bv VL4.lequivalent absolute values 11.1.7

- norms 11.3.1 , III.3.2Euclidean model 1.2.5extension of absolute values

- existence 11.3.4- uniqueness 11.3.3

436 Index

F K

Fibonacci numbers (IV, exercise 20)filters III.A.lfiniteness (extensions of Qp of given degree)

III.1.6formal power series 1.4.8, IV.5, Vl.lfractal subset 1.2.3fractional part (x) 1.504fundamental inequalities IIIA .3

G

gamma function r p (Morita) VII .1.IGaussian 2-adic numbers IIA.5Gauss multiplication formula for r p

VII.1.3-norm V.2.1

generalized absolute value 11.2.2- ball V1.3.1- over Q II.2A- Taylor expansion IV.5.2

Gould polynomials (IV, exercise 21)granulation, type of - V.l.2growth modulus VI.1A, V1.3.3

H

Haar measure II.A.lHahn-Banach theorem (p-adic) IVA.7Hadamard formula (radius of convergence)

VI.1.2- three-circle theorem VI.2.6

Hazewinkel maps VII.3 .2- theorem VII.3.3

Hensel's lemma 1.6.4, II.1.5hexagonal field (3-adic numbers) 11.4.6Honda sequence (and congruences)

VII.3.2homothety (= dilatation) 1.5.6, V1.3.1

I

IFS (iterated function system) 1.2.5indecomposable compact space I.A.6indefinite sum IV.1.5Ingleton theorem IVA.7index with respect to a hole VI.3 .5infinite product VI.2.3infraconnected set V1.4.1injective Z-module IIIA.lp-integer 1.504integral part (p-adic) [xl 1.504inverse system (= projective system) 1.4.2involution a 1.1.2irreducibility criterion (Eisenstein) 1104.2isolated singularity V1.2.6Iwasawa logarithm Log VA.5

Kazandzidis congruences VII.1 .6Krasner's lemma III.l.5, III.3.2

L

Legendre polynomials VII.3.5- relation for r p VII.1.2- quadratic residue symbol 1.6.6

length of an expansion in basis p IV.3.2- of a word 1.204

linear fractional transformation V1.3.1Liouville theorem Vl.lALipschitz function V.1.5locally analytic function V1.4.7locally constant function IV.3.1local ring 11.1.4Lucas sequence VII.3A

M

Mahler series IV.2.3- theorem IV.2A

maximum principle VI.2 .5, V1.2.6mean value theorem V.3.2, V.3Ametric, p -adic - 1.2.1Mittag-Leffler theorem VI.3A, VIA.5Möbius function p.(n) VII.2.1module of an automorphism II.A.lMonna-Fleischer theorem IVA.5Motzkin factorization VI.3.5, VIA.8multiplicative norm VI.I.4, V1.3.6

N

Newton algorithm 1.604- approximation method 1.6.3- polygon VI.1 .6

normal basis (ultrametric Banach space)IVA.2

o

order vp = ord , 1.1.4,1.5 .1order of composition operator IV.5.3order offormal power series IV.5.3, VI. 1.1

p

p-adic integer I.1.1, - number 1.5.1- metric 1.2.1

Perrin sequence VII.3 .4Picard theorem (essential singularity) V1.2.6Pochhammer symbol (x)n IV.1.1principal ideal domain 1.1.6

- part at a pole VI.3 .2projective limit (inverse system) 104.2

Q

quadratie residue symbol (Legendre)1.6.6

R

radius of eonvergenceVI.I .2- (exp and log) VA.I

ramification index 1104.1reduetion mod p 1.1.5

- of ultrametrieBanach space IV.4.3regular radius VI.IArepresentationtheorem IVAAresidue degree 11.4.1

- field 11.104restrietedfactorialVII.1.1

- formal power series V.2.1Rodriguesfonnula VII.35Rolle's theorem VI.2Aroots ofunity in C 1.504

- in Cp IIIA.2Runge theorem V1.4.2

s

saturatedset 1.3.3Sehnirelman's theorem V1.2.3self-similaritydimension 1.2.3Sheffer polynomials,- sequences IV.6.1Sierpirisky gasket1.2.5solenoids, LA.Ispherieallycomplete metrie space III.2Astereographie projeetion 1.A.6Stirling numbers (l st and 2nd kind)

V1.4.7Strassmantheorem V1.2.1striet differentiability V.1.1strippedball 11.1.1support (ofa family) IVA.I

- differentiability V.1.1

T

tarneramifieation 1104.1Tate homomorphism t» 1.504

Index 437

Teichmüllereharaeter IIIAAtopologicalfield1.3.7

- group 1.3.1- ring 1.3.6

totally diseonneeted1.2.1- ramified(extension)11.4.1

transition map (inversesystem) 1.4.2translationprincipleIV.5.5type of a granulationV.I.2

u

ultrafilterIII.A.2ultrametrieabsolute value 11.1.3

- Banachspace IVA- distance, - space 11.1.1- field 11.1 .3- group II.1.2

ultraproduct111.2.2unifonnly equivalentmetries1.2.1unit (p-) 1.504universalfield Qp 111.2.2universalpropertyof inverse limits1.4.2unramified extension1104.1

maximal- 11.404, III.1.2

v

valuationofn! V.3.1- polygon VI.1.6- subring 11.104

valuedfield1.3.7van der Put sequence IV.3.2

- theorem IV.3.3van Hamme theorem IV.5AVolkenbom integral V.5.1

w

wild ramifieation 1104.1WilsoneongruenceVlI.1.1

z

Zuber theoremVlI.3.6

Graduate Texts in Mathematics(continued from page ii)

72 STILLWELL. Classical Topology and 104 DUBROVIN/FoMENKOINOVIKOV. ModemCombinatorial Group Theory. 2nd ed. Geometry-Methods and Applications.

73 HUNGERFORD. Algebra. Part 11.74 DAVENPORT. Multiplicative Number 105 LANG. SL2(R).

Theory. 2nd ed. 106 SILVERMAN. The Arithmetic ofElliptic75 HOCHSCHILD. Basic Theory of Algebraic Curves ,

Groups and Lie Algebras. 107 OLVER. Applications ofLie Groups to76 IITAKA. Algebraic Geometry. Differential Equations. 2nd ed.77 HECKE. Lectures on the Theory of 108 RANGE. Holomorphic Functions and

Algebraic Numbers. Integral Representations in Several78 BURRJS/SANKAPPANAVAR. A Course in Complex Variables.

Universal Algebra. 109 LEHTO. Univalent Functions and79 WALTERS. An Introduction to Ergodie Teichmüller Spaces.

Theory. 110 LANG. Algebraic Number Theory.80 ROBINSON. A Course in the Theory of I11 HUSEMÖLLER. Elliptic Curves.

Groups. 2nd ed. 112 LANG. Elliptic Functions.81 FORSTER. Lectures on Riemann Surfaces. 113 KARATZAS/SHREVE. Brownian Motion and82 BOTT/Tu. Differential Forms in Algebraic Stochastic Calculus. 2nd ed.

Topology. 114 KOBLITZ. A Course in Number Theory and83 WASHINGTON. Introduction to Cyclotomic Cryptography. 2nd ed.

Fields. 2nd ed. 115 BERGERIGOSTIAUX. Differential Geometry:84 IRELANDlROSEN. A Classical Introduction Manifolds, Curves, and Surfaces.

to Modem Number Theory. 2nd ed. 116 KELLEy/SRINlVASAN. Measure and Integral.85 EDWARDS. Fourier Series. VoL 11. 2nd ed. VoLl.86 VAN LINT. Introduction to Coding Theory. 117 SERRE. Algebraic Groups and Class Fields.

2nd ed. 118 PEDERSEN. Analysis Now.87 BROWN. Cohomology ofGroups. 119 ROTMAN. An Introduction to Algebraic88 PIERCE. Associative Algebras. Topology.89 LANG. Introduction to Algebraic and 120 ZIEMER. Weakly Differentiable Functions:

Abelian Functions. 2nd ed. Sobolev Spaces and Functions of Bounded90 BRONDSTED. An Introduction to Convex Variation.

Polytopes. 121 LANG. Cyclotomic Fields land II.91 BEARDON. On the Geometry ofDiscrete Combined 2nd ed.

Groups. 122 REMMERT. Theory ofComplex Functions.92 DIESTEL. Sequences and Series in Banach Readings in Mathematics

Spaces. 123 EBBINGHAUslHERMES et aL Numbers.93 DUBROVIN/FOMENKOlNovIKOV. Modem Readings in Mathematics

Geometry-Methods and Applications. 124 DUBROVIN/FOMENKOlNovIKOV. ModemPart I. 2nd ed. Geometry-Methods and Applications.

94 WARNER. Foundations ofDifferentiable Part III.Manifolds and Lie Groups. 125 BERENSTEIN/GAY. Complex Variables: An

95 SHIRYAEV. Probability. 2nd ed. Introduction.96 CONWAY. A Course in Functional 126 BOREL. Linear Aigebraic Groups. 2nd ed.

Analysis. 2nd ed. 127 MASSEY. A Basic Course in Aigebraic97 KOBLITZ. Introduction to Elliptic Curves Topology.

and Modular Forms. 2nd ed. 128 RAUCH. Partial Differential Equations.98 BRÖCKERITOM DIECK. Representations of 129 FULTONIHARRIs. Representation Theory: A

Compact Lie Groups. First Course.99 GROVEIBENSON. Finite Reflection Groups. Readings in Mathematics

2nd ed. 130 DODSONIPOSTON. Tensor Geometry.100 BERG/CHRJSTENSENIRESSEL. Harmonie 131 LAM. A First Course in Noncommutative

Analysis on Semigroups: Theory of Rings.Positive Definite and Related Functions. 132 BEARDON. Iteration ofRational Functions.

101 EDWARDS. Galois Theory. 133 HARRJS. Aigebraic Geometry: A First102 VARADARAJAN. Lie Groups, Lie Algebras Course.

and Their Representations. 134 ROMAN. Coding and Information Theory.103 LANG. Complex Analysis. 3rd ed. 135 ROMAN. Advanced Linear Algebra.

136 AOKINsIWEINTRAUB. Algebra: An 167 MORANDI. Field and Galois Theory.Approach via Module Theory. 168 EwALD. Combinatorial Convexity and

137 AXLERIBOURDONIRAMEY. Harmonie Algebraic Geometry.Function Theory. 169 BHATIA. Matrix Analysis.

138 COHEN. A Course in Computational 170 BREDON. SheafTheory. 2nd ed.Algebraic Number Theory. 171 PETERSEN. Riemannian Geometry.

139 BREOON. Topology and Geometry. 172 REMMERT. Classical Topics in Complex140 AUBIN. Optima and Equilibria. An Function Theory.

Introduction to Nonlinear Analysis. 173 DIESTEL. Graph Theory. 2nd ed.141 BECKERIWEISPFENNINGIKREOEL. Gröbner 174 BRIDGES. Foundations ofReal and

Bases. A Computational Approach to Abstract Analysis.Commutative Algebra. 175 LiCKORISH.An Introduction to Knot

142 LANG. Real and Functional Analysis. Theory.3rd ed. 176 LEE.Riemannian Manifolds.

143 DOOB. Measure Theory. 177 NEWMAN. Analytic Number Theory.144 DENNISIFARB. Noncommutative 178 CLARKEILEDYAEV/STERNIWOLENSKI.

Algebra. Nonsmooth Analysis and Control145 VICK. Homology Theory. An Theory.

Introduction to Algebraic Topology. 179 DOUGLAS. Banach Algebra Techniques in2nded. Operator Theory. 2nd ed.

146 BRIDGES. Computability: A 180 SRIVASTAVA. A Course on Borel Sets.Mathematical Sketchbook. 181 KRESS. Numerical Analysis.

147 ROSENBERG. Aigebraic K-Theory 182 WALTER. Ordinary Differentialand Its Applications. Equations.

148 ROTMAN. An Introduction to the 183 MEGGINSON. An Introduction to BanachTheory ofGroups. 4th ed. Space Theory.

149 RATCLIFFE. Foundations of 184 BOLLOBAS. Modem Graph Theory.Hyperbolic Manifolds. 185 COXILITTLElO'SHEA. Using Algebraic

150 EISENBUD. Commutative Algebra Geometry.with a View Toward Algebraic 186 RAMAKRISHNANIVALENZA.FourierGeometry. Analysis on Number Fields.

151 SILVERMAN. Advanced Topics in 187 liARRISIMORRISON. Modu1i ofCurves.the Arithmetic ofElliptic Curves. 188 GOLDBLATT. Lectures on the Hyperreals:

152 ZIEGLER. Lectures on Polytopes. An Introduction to Nonstandard Analysis.153 FULTON. Algebraic Topology: A 189 LAM. Lectures on Modules and Rings.

First Course. 190 ESMONDEIMURTY. Problems in Algebraic154 BROWNIPEARCY. An Introduction to Number Theory.

Analysis. 191 LANG. Fundamentals ofDifferential155 KASSEL. Quantum Groups. Geometry.156 KECHRIS. Classical Descriptive Set 192 HIRSCHILACOMBE. Elements ofFunctional

Theory. Analysis.157 MALLIAVIN. Integration and 193 COHEN. Advanced Topics in

Probability. Computalional Number Theory.158 ROMAN. Field Theory. 194 ENGELINAGEL. One-Parameter Semigroups159 CONWAY. Functions ofOne for Linear Evolution Equations.

Complex Variable 11. 195 NATHANSON. Elementary Methods in160 LANG. Differential and Riemannian Number Theory.

Manifolds. 196 OSBORNE. Basic Homological Algebra.161 BORWEIN/ERDELVI. Polynomials and 197 EISENBuol HARRIs. The Geometry of

Polynomial Inequalities. Schemes.162 ALPERIN/BELL.Groups and 198 ROBERT. A Course in p-adic Analysis.

Representations. 199 HEDENMALMlKoRENBLUMIZHU. Theory163 DIXON/MORTIMER. Permutation ofBergman Spaces.

Groups. 200 BAO/CHERNISHEN. An Introduction to164 NATHANSON. Additive Number Theory: Riemann-Finsler Geometry.

The Classical Bases. 201 HINDRY/SILVERMAN. Diophantine165 NATHANSON. Additive Number Theory: Geometry: An Introduction.

Inverse Problems and the Geometry of 202 LEE. Introduction to TopologicalSumsets. Manifolds.

166 SHARPE. Differential Geometry: Cartan'sGeneralization ofKlein's ErlangenProgram.