An Introduction to G-Functions. (AM-133), Volume 133 9781400882540

Table of contents :
CONTENTS
PREFACE
INTRODUCTION
LIST OF SYMBOLS
CHAPTER I Valued Fields
1. Valuations
2. Complete Valued Fields
3. Normed Vector Spaces
4. Hensel’s Lemma
5. Extensions of Valuations
6. Newton Polygons
7. The y–intercept Method
8. Ramification Theory
9. Totally Ramified Extensions
CHAPTER II Zeta Functions
1. Logarithms
2. Newton Polygons for Power Series
3. Newton Polygons for Laurent Series
4. The Binomial and Exponential Series
5. Dieudonné’s Theorem
6. Analytic Representation of Additive Characters
7. Meromorphy of the Zeta Function of a Variety
8. Condition for Rationality
9. Rationality of the Zeta Function
Appendix to Chapter II
CHAPTER III Differential Equations
1. Differential Equations in Characteristic p
2. Nilpotent Differential Operators. Katz-Honda Theorem
3. Differential Systems
4. The Theorem of the Cyclic Vector
5. The Generic Disk. Radius of Convergence
6. Global Nilpotence. Katz’s Theorem
7. Regular Singularities. Fuchs’ Theorem
8. Formal Fuchsian Theory
CHAPTER IV Effective Bounds. Ordinary Disks
1. p–adic Analytic Functions
2. Effective Bounds. The Dwork–Robba Theorem
3. Effective Bounds for Systems
4. Analytic Elements
5. Some Transfer Theorems
6. Logarithms
7. The Binomial Series
8. The Hypergeometric Function of Euler and Gauss
CHAPTER V Effective Bounds. Singular Disks
1. The Dwork–Frobenius Theorem
2. Effective Bounds for Solutions in a Singular Disk: the Case of Nilpotent Monodromy. The Christol–Dwork Theorem: Outline of the Proof
3. Proof of Step V
4. Proof of Step IV. The Shearing Transformation
5. Proof of Step III. Removing Apparent Singularities
6. The Operators (ϕ) and ψ
7. Proof of Step I. Construction of Frobenius
8. Proof of Step II. Effective Form of the Cyclic Vector
9. Effective Bounds. The Case of Unipotent Monodromy
CHAPTER VI Transfer Theorems into Disks with One Singularity
1. The Type of a Number
2. Transfer into Disks with One Singularity: a First Estimate
3. The Theorem of Transfer of Radii of Convergence
CHAPTER VII Differential Equations of Arithmetic Type
1. The Height
2. The Theorem of Bombieri–André
3. Transfer Theorems for Differential Equations of Arithmetic Type
4. Size of Local Solution Bounded by its Global Inverse Radius
5. Generic Global Inverse Radius Bounded by the Global Inverse Radius of a Local Solution Matrix
CHAPTER VIII G–Series. The Theorem of Chudnovsky
1. Definition of G–Series. Statement of Chudnovsky’s Theorem
2. Preparatory Results
3. Siegel’s Lemma
4. Conclusion of the Proof of Chudnovsky’s Theorem
Appendix to Chapter VIII
APPENDIX I Convergence Polygon for Differential Equations
APPENDIX II Archimedean Estimates
APPENDIX III Cauchy’s Theorem
BIBLIOGRAPHY
INDEX

Citation preview

Annals of Mathematics Studies

Number 133

An Introduction toG-Functions

by

Bernard Dwork, Giovanni Gerotto, and Francis]. Sullivan

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1994

Copyright© 1994 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America 2 4 6 8 10 9 7 5 3 1 Library of Congress Cataloging-in-Publication Data Dwork, Bernard M. An introduction to G-functions I by Bernard Dwork, Giovanni Gerotto, Francis]. Sullivan. p. em. -(Annals of mathematics studies ; no. 13 3) On t.p. "G" is italicized. Includes bibliographical references and index. ISBN 0-691-03675-6- ISBN 0-691-03681-0 (pbk.) l. H-functions. 2. p-adic analysis. I. Gerotto, Giovanni. ll. Sullivan, Francis]., 194 3- . ill. Title. N. Series. QA242.5.D96 1994 515'.55-dc20 94-2414

The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed

Dedicated to the memory of Yvette Amice (1936-1993)

CONTENTS

xi

PREFACE INTRODUCTION LIST OF SYMBOLS CHAPTER I

1. 2. 3. 4. 5. 6. 7. 8. 9.

Xlll

XlX

Valued Fields

Valuations Complete Valued Fields Normed Vector Spaces Hensel's Lemma Extensions of Valuations Newton Polygons The y-intercept Method Ramification Theory Totally Ramified Extensions

CHAPTER II

1. 2. 3. 4.

10 17 24

28 30 33

Zeta Functions

1. Logarithms 2. Newton Polygons for Power Series 3. Newton Polygons for Laurent Series 4. The Binomial and Exponential Series 5. Dieudonn~?s Theorem 6. Analytic Representation of Additive Characters 7. Meromorphy of the Zeta Function of a Variety 8. Condition for Rationality 9. Rationality of the Zeta Function Appendix to Chapter II CHAPTER III

3 6 8

38 41 46 49 53

56 61 71

74 76

Differential Equations

Differential Equations in Characteristic p Nilpotent Differential Operators. Katz-Honda Theorem Differential Systems The Theorem of the Cyclic Vector Vll

77 81 86 89

viii

CONTENTS 5. 6. 7. 8.

CHAPTER

The Generic Disk. Radius of Convergence Global Nilpotence. Katz's Theorem Regular Singularities. Fuchs' Theorem Formal Fuchsian Theory

IV

Effective Bounds. Ordinary Disks

1. p -adic Analytic Functions

2. 3. 4. 5. 6. 7. 8. CHAPTER

Effective Bounds. The Dwork-Robba Theorem Effective Bounds for Systems Analytic Elements Some Transfer Theorems Logarithms The Binomial Series The Hypergeometric Function of Euler and Gauss

V

VI

155

159 168 169 170 173 176 180 189

Transfer Theorems into Disks with One Singularity

1. The Type of a Number 2. Transfer into Disks with One Singularity: a First Estimate 3. The Theorem of Transfer of Radii of Convergence CHAPTER VII

114 119 126 128 133 138 140 150

Effective Bounds. Singular Disks

1. The Dwork-Frobenius Theorem 2. Effective Bounds for Solutions in a Singular Disk: the Case of Nilpotent Monodromy. The Christol-Dwork Theorem: Outline of the Proof 3. Proof of Step V 4. Proof of Step IV. The Shearing Transformation 5. Proof of Step III. Removing Apparent Singularities 6. The Operators ¢ and 1/J 7. Proof of Step I. Construction of Frobenius 8. Proof of Step II. Effective Form of the Cyclic Vector 9. Effective Bounds. The Case of Unipotent Monodromy CHAPTER

92 98 100 102

199 203 212

Differential Equations of Arithmetic Type

1. The Height 2. The Theorem of Bombieri-Andre 3. Transfer Theorems for Differential Equations of Arithmetic Type

222 226 234

CONTENTS 4. Size of Local Solution Bounded by its Global Inverse Radius 5. Generic Global Inverse Radius Bounded by the Global Inverse Radius of a Local Solution Matrix CHAPTER

VIII

IX

243 254

G-Series. The Theorem of Chudnovsky

1. Definition of G-Series.

Statement of Chudnovsky's Theorem 2. Preparatory Results 3. Siegel's Lemma 4. Conclusion of the Proof of Chudnovsky's Theorem Appendix to Chapter VIII

263 267 284 289 300

Convergence Polygon for Differential Equations

301

APPENDIX II

Archimedean Estimates

307

APPENDIX III

Cauchy's Theorem

310

APPENDIX I

BIBLIOGRAPHY

317

INDEX

321

PREFACE

This book is based upon a course given with the collaboration of G. Gerotto and F. J. Sullivan during the academic year 1992/93 at the University of Padova. The plan was to present completely the p-adic foundations of the basic properties of G-functions, assuming only a basic knowledge of algebra. In particular we give a complete proof of the main result: the analytic continuation of a G-series is again a G-series. Algebraic number theory enters in an essential way only in the use of Cebotarev's density theorem in the proof of Katz's Theorem (Theorem 111.6.1). The language of algebraic number theory is used starting with Chapter VII, but only the most basic facts are used: normalization of valuations, product formula, relation between local and global degrees.

B. DWORK

Xl

INTRODUCTION

What is a G-function? Modifying the standard terminology, we say that 00

z

= LAjXj

E K[[X]]

j=O

is a G-series at the origin (defined over an algebraic number field K), if a) There exists a non-zero L E K(X)[djdX] such that Lz = 0; b) For each imbedding of K into C, the series z has a non-zero radius of convergence; c) There exists a sequence of positive integers { c8 } such that

s~p ( ~ log

C8 )

< oo

and such that Aj c8 E 0 K, the ring of integers of K, for all j

:S s.

In this definition we may replace K by qalg, the algebraic closure of Q since each element of qalg(X) is necessarily defined over an algebraic number field, and this property of L then forces the coefficients Aj to all lie in a finite extension of Q. Condition (b) is needed only if the origin is a singular point of L. If ( E qalg then we may use the term "G-series at (" without ambiguity. It is easy to check that the G-series at the origin form a ring (see Proposition VIII.2.1 (ii)). Easy examples of G-series are the expansions at 1 of log X and of X"' for a E Q. A celebrated theorem of Eisenstein (see Dwork-van der Poorten [1]) assures us that any element of qalg[[X]] algebraic over Q(X) must be a G-series. We say that an irreducible element of qalg(X)[d/dX] is an irreducible G-operator if it has a solution which is a G-series at some point, say, the origin. It follows from the remarkable theorem of the Chudnovskys' (see Chapter VIII) and the work of Andre, Christo!, Baldassarri and Chiarellotto that if L is an irreducible G--operator then for Xlll

xiv

INTRODUCTION

( E Qalg each formal solution y E Qalg[[X - (]] of Ly = 0 is again a Gseries. (By Theorem VIII.l.5, L satisfies the Galockin condition, hence, by Theorem VII.2.1, L satisfies the Bombieri condition, so by Theorem VII.3.3 it satisfies the local Bombieri condition at ( and the assertion then follows from Theorem VII.4.2). We extend the notion of "G-series at(" to include the case in which ( E C is transcendental. For such a ( we say that y E Qalg(()[[X- (]] is a G-series if a) There exists a non-zero L E Qalg(X)[d/dX] such that Ly 0; b) No archimedean condition; c) This condition is the same as previously with CJ K replaced by

=

CJ(

= {>. E Qalg(() IIAiv,gauss ~ 1 for

all finite valuations v of Qalg}.

Here Hv,gauss denotes the v-adic Gauss norm relative to ( (for the definition see Section 1.4). It follows from the Galockin condition that if ( is transcendental over the field Q and if y E Qalg(()[[X- (]]is a solution of Ly = 0 where L is an irreducible G-operator then y is a G-series at ( in the above sense. We can show by an easy specialization argument that conversely if L is an irreducible element of Qalg(X)[d/dX] which has a solution which is a G-series at a transcendental point ( E C then L is indeed an irreducible G-operator. Finally we extend the notion of G-series (at each fixed ( E C) so as to make the set of such series a C-vector space. With these definitions we may conclude that the analytic continuation of a G-series at x = 0 to x = ( is again a G-series provided we exclude a finite number of singular points. We may define a G-function to be a multivalued, locally analytic function on C\S (with cardS < =) which at some point is represented by a G-series. Equivalently, a G-function is the solution of an irreducible G-operator. We may eliminate the emphasis on irreducible operators. We say that L E Qalg(X)[d/dX] is a G-operator if all solutions at some non singular point are G-series. Then all solutions at any non-singular point are G-series. If L is a G-operator then all its singularities are regular and the exponents are rational. Thus if the origin is a (possibly) singular point, there exists a basis (Yl, ... , Yn) for the solutions at X = 0 of the form (yl, .. · , Yn) = (z1, ... , Zn)XA (*) where z 1 , ... , Zn lie in K[[X]] and A E Mn ( Qalg) with rational eigenvalues. In fact if L is a G-operator then z 1 , ... , Zn are G-series and the converse is also valid.

INTRODUCTION

XV

So much for the definition of G-functions. How can we determine whether an operator is a G-operator? Since the singularities must be regular we may exclude ex, la(X) and hypergeometric functions kFm with k < m + 1. Since the exponents must be rational we may exclude X"' and 2 F 1 (a, b, c, X) if a ft. Q or if a, b, or c is irrational. There is no general method for verifying that a given operator L is a G--operator. Three methods known to us are: 1. If L 1 and L2 are G-operators then so is the composition L1 o L2. 2. A sufficient condition is that in the representation ( *) of the basis of solutions, the eigenvalues of A be rational and the series z 1 , ... , Zn satisfy the local Bombieri condition, i.e., it is sufficient that

where Rv is the common radius of convergence of all the valuation v.

Zi

for each finite

Remark. In all known cases of G-operators, Rv = 1 for almost all v. In general it is very difficult to compute Rv directly. For example, the second order differential equation satisfied by 2F 1

(a).(b)s 8 (a, b, c, X ) = ~ ~ () 1 X

•=0

c

.s.

is a G-operator if a, b, c E Q because this solution and a second independent one are known so explicitly (Section IV.8) that we can check that Rp = 1 for almost all p and Rp > 0 for all p. We do not know how to verify directly that the same is valid for the power series solution at a non-singular point, say at x = 2. 3. If L comes from geometry then it is known that a strong Frobenius operator exists for almost all v and hence by a classical argument Rv = 1 for almost all v. This is the most important general example of G-operators and it is believed that no other examples exist. This is the situation in which L is associated to a differential module M having a filtration of submodules whose factor modules are isomorphic, as differential modules, to factor modules of the relative cohomology, in some dimension, of a family of varieties defined over K(X). A very general example of such G-modules is associated with the construction given in Section 7 of Chapter II. Let f be an arbitrary element of K(X)[Y1, ... , Yn], put g

=Yo!

INTRODUCTION

XVI

and let R be the sub ring of K(X)[Y0 , Y1 , ... , Yn] generated by the monomials in g. Define operators E; on R by

E; = Y;fJ I fJY; ' for i = 0, 1, ... , n, and set

g; = E;g' Let

D; = E;

+ g;.

n

W=R(LD;R. i=O

Then W is a finite dimensional K(X)-vector space which is a differential module via the action of the operator ux = 8/fJX + fJgjfJX. This construction can be generalized, replacing each D; by D;,a = D; +a; where (ao, ... , an) E Qn+l.

The object of these lectures is to provide an introduction to the theory of a-functions at the level of a first year graduate course. It is our hope that the reader will acquire all the tools needed to study applications to number theory, in particular to the theory of irrationality of special values of functions. We have been strongly influenced by Bombieri [1) and by Andre [1). It is our hope that the present volume will make their work more accessible. Our work is divided into three parts. The first two chapters are devoted to the elementary theory of p-adic fields and functions. The previous paragraph provides a motivation for the inclusion of the proof of the rationality of the zeta function. Chapters III to VI, the second part of this volume, are devoted to the theory of ordinary linear differential equations over finite fields and p -adic fields. The main results here are the notion of nilpotence, Katz's theorem on global nilpotence, and the effective (p-adic) bounds for ordinary disks and for disks with one regular singularity. The basic problem is that of determining padic radii of convergence and rate of growth of formal solutions at the boundary of the disk of convergence. The third part of the book, Chapters VII and VIII, deals with global problems, i.e., how do radii of convergence vary with valuation and what profit we may draw from knowing the variation. We do not treat the applications to number theory.

INTRODUCTION

xvii

It is our belief that the p-adic theory of differential equations is still in its infancy. We mention two questions.

1. (Modified) Robba Conjecture Let G E Mn(Q(X)) and let p be a fixed prime. We will use only the p-adic valuation. Suppose that all the singularities of D- Gin D(O, 1-) lie in D(O, R) and that for each a with R < Ia I < 1, the solution matrix at a of D-G converges in D(a, lal-). Then (conjecturally) there exists a solution matrix V X A where A is a constant matrix and V is analytic on the annulus. 2. Accessory parameter problem Consider the general nth order scalar differential operator, L, with m + 1 singularities, all regular, and a fixed set of exponents all in Q n Zp. Taking three of the singularities to be 0, 1, oo, the parameters consist of the m- 2 unspecified singularities and the (n- 1)[n(m- 1)- 2]/2 accessory parameters. It is known (Schmitt [1]) that the variety of nilpotence for the mod p -reduction is a complete intersection of dimension n- 2. We ask for the set of parameters for which the solutions of Ly = 0 at a generic point converge in a disk of radius unity.

LIST OF SYMBOLS Throughout the text C, R, Q, F q, Z and N denote the fields of complex, real and rational numbers, the finite field with q elements, the ring of rational integers and the set of natural numbers, respectively. If K is a field then K[X], K[[X]], K(X) and K((X)) are respectively the polynomials, formal power series, rational functions, and (finitely negative) formal Laurent series in X with coefficients from K. The algebraic closure of the field K is denoted by Kalg. We use Gi(n,A), Mn,m(A) and Mn(A) to denote respectively the general linear group of order n, the set of n x m matrices and the ring of n x n matrices over the commutative ring A. If M E Mn(A) then t M (or Mt), det M, Tr M and adj M are the transpose, determinant, trace and adjoint of M respectively. The symbol In denotes the n X n identity matrix. If n; K is a finite extension of fields then Nn/K and Trn;K are the norm and trace maps respectively. If 0/ K is a Galois extension, 9(0/ K) denotes the Galois group of 0/ K. If S is a set #S denotes the cardinality of S, and if u is a permutation, sgn(u) is the sign of u. If A and B are sets, we denote by BA the set of maps from A into B. As usual the symbols EB and 1\ denote direct sums and wedge products respectively, and A(M) denotes the exterior algebra of the module M. The submodule generated by the subset X of the module M will be indicated by (X). In discussion of differential modules over fields containing K(X) the symbol Dis used for d/dX while 6 indicates X d/dX. If X is a metric space with distance dist, for any a E X and p > 0 we will denote the open and closed disks with center a and radius p by D(a, p-) and D(a, p+) respectively, that is we set

D(a,p-) D(a,p+)

= {x E XI dist(a,x) < p} = {x E XI dist(a,x):::; p}.

We occasionally use the simpler notation D(a,p) for D(a,p+). xix

LIST OF SYMBOLS

XX

117, 118 117, 118

Ao, Aa,p A~, A~,p

117, 118 117, 118

Bo, Ba,p B~, B~,p

5 K{{X}}

J{

den

dv ~

~(t)

56

255

La,b,c

150

Lt a L2 78 Lp 98 £2(X) 125 log(1 +X) 38

224 222 86 68

log X log+ a

E 128 Ea, Eo 128 E~ 129 e = e(fl/ K) exp 50 E(X) 55

F(a,b,c;X) f = J(fl/ K)

J¢

Jc!>q

Frob

gA(X) ga(X) Q(b)

19

150 19

104 140 58

284

5 24

P, Pf, Poo p, PK 5 PE 155 1/Jq 173 7r 38

Qp 105

h(o:) 223 h(q),h(q 0 , ... ,qN) 284 h j ( 0:), hco ( 0:) 224 h(s, v) 227, 263, 292 h(s, v) 267, 292 ht q, ht(q 0 , ... , qN) 284

x

Ox ord

255

222

'1/JA,B, '1/JA 107 '1/Jq 65, 173

86

Gs

102 223

v(G), v'(G) 272 v(P)

165 213 32

G[H],b> G[H] 5 Gx

129

~~:(G)

24

CP

x,xa

ht

67 78

h¢(t) H[D]

R R' RN

7

81, 159 190 288 206 Ra(Y), ra(Y) 271 254 R(Y), r(Y) 254 R(C), r(C) 226 Rv 257 Rv(C) 241 Rv(Y), rv(Y) R(s) 272

Rm

LIST OF SYMBOLS

3 R+ 257 p(C/X) p(G) 226 234 p(YD where D(v,c) ={wE Vlllw- vii< c}. We will also make use of the following notation: V. = D(O, c+) = { v E VI llvll ::; € }. The definitions of Cauchy sequence, completeness, etc. are analogous to those given in Section 2. Let (V, 11-11) and (W, 11-11) be normed spaces over K, and let 8 : V--+ W be a K-linear map. We define 11811 = sup ll8(v)ll E (0, oo] llvll:9

Proposition 3.1.

The linear map (} is continuous if and only if

Proof. LetT E GK (the valuation group of K) and let a E K be such that lal = r. Suppose that 11811 = c < oo. If llvll ::; T then lla- 1 vll ::; 1

VALUED FIELDS

9

so that IIB(a- 1v)ll::::; c and IIB(v)ll::::; cr. Hence we have established that B(Vr) ~ Wcr, for every r E GK, so B is continuous since GK contains elements arbitrarily close to 0. Conversely, let B be continuous. Then for every c:' > 0 there exists c: > 0 such that B(Ve) ~ W.-'· Fix a E K, a :f. 0 such that lal ::::; €. If v E vl we have llavll::::; € and then IIB(av)ll::::; € 1 , so that IIB(v)ll::::; € 1 /lal. Hence, if we put c = €1 /Ia I if follows that IIBII ::::; c. Q.E.D. Now let's suppose that V has finite dimension n over K. Let w1, ... ,wn be a base of V over K. Define, for i = 1, 2, ... , n, the K-linear form

A;: V---+ K v 1---+ A;(v) = x; n

where the x; are defined by v =

L x;w;. Now we can put i=l

llvll' = sup (IA;(v)l). l~i~n

It is easily checked that 11-11' is indeed a norm on V (note that this definition depends on the choice of basis for V). If the field K is complete, then obviously V is also complete with respect to this norm. We say that two norms 11-11 1 and 11-112 on V are equivalent if there exist positive constants cl and c2 such that

and for every v E V.

Theorem 3.2. Let K be a complete field and V = (V, 11-11) a finite dimensional normed space over K. Then the norm 11-11 and the norm 11-11' defined above are equivalent. As a consequence, V is complete. Proof. We proceed by induction on n, the dimension of V. If n = 1, the result is clear since there exists C > 0 such that llvll = Cllvll' for every v E V. We now suppose that n > 1 and the result true for n

dimensions less than n. If v =

L .>.; (v) w;, then i=l

llvll::::; sup IIA;(v)w;ll =sup IA;(v)lll w;ll i

i

:S: supi.A;(v)l·supllw;ll = C1llvll', i

i

CHAPTER I

10

where

cl

= supJJw;JJ. i

On the other hand, we have llvll' =sup IA;(v)l ~ llvll· sup IIA;JI. i

i

Therefore it suffices to prove that A; is continuous for every i. To fix ideas, we prove that A1 is continuous. Suppose the contrary. Then there exists a sequence ( vCil) such that vCi) __,. 0 but A1 ( vCil) f> 0, so we can find a subsequence (which we still call ( vCil)) such that vCj) __,. 0 but A1 ( vCil) 2:: € > 0. Let

so that uCi) = Wl +X~j)w2+. ·+X~j)Wn for suitable x~j)' ... 'x~j) E K. Since uCi) __,. 0 it follows that .lim

J--+00

(x~j)W2 +

···+

X~)Wn)

=

-Wl.

If V' is the subspace spanned by w2, ... , Wn, then V' is complete by the induction hypothesis, so that w1 E V' which gives a contradiction.

Q.E.D. 4.

Hensel's Lemma.

In this section we shall assume K to be a complete valued field. The ring of polynomials in one variable K[X] is an infinite dimensional K-vector space and the set { 1, X, X 2, · · · } forms a basis. We now define a norm on the K-vector space K[X], called the Gauss norm by setting

IL a;Xil 1

=

s~p Ja;J.

gauss

Note that K[X] is not complete with respect to the Gauss norm. Note also that the definition of the Gauss norm does not require that K be complete. The following equality is known as Gauss's lemma: If/, g E K[X], then

VALUED FIELDS

11

Let Wn = {! E K[X]j degf :S n- 1}. Clearly the dimension of Wn over K is n, so that it is a complete K-space. Let g, h be fixed elements of K[X] and let deg g = n and deg h = m. Consider the linear map

B: Wn EB Wm---+ Wn+m

(e, TJ) In terms of the basis {1, X, X 2 , · ·

· ,

( 4.1)

g,., +he

f---+

xs-l }, for Ws, s = 1, 2, ... , one has

det(matrix(B)) = ±R(h,g), where R(h, g) is the Sylvester resultant of h and g.

Theorem 4.1 (Hensel's Lemma). Let K be a complete field with valuation 1-1 and valuation ring 0. Let f,g, hE O[X] with

Let 0
0. Let

v=

h-hEWm

1-1·

We obviously can assume

{e E Wn I lei::; ciR(g, h)l},

W = {TJ E Wm I ITJI

:S

ciR(g, h)l}

and define the norm

(4.2)

ll(e, TJ)II =sup{ lei, ITJI} on V EB W. Hypothesis (b) gives the map cP : V

EB W

---+

(e, ,.,)

f---+

Wn+m gh-

J-

e'T]

CHAPTER I

12

so that we can consider the map A= 8- 1 ¢. We claim that (4.3)

A(V EB W) ~ V EB W. Indeed, from hypothesis (a) it follows that If- gh- ~111 :::; sup{lf- uhl, I~7JI}

:::; sup{ciR(g, hW, c2 IR(g, h)l 2 }

= ciR(g, hW

Since g, h E O[X] the matrix of 8 (with respect to the previously described basis) has elements in 0 so that 1 1 llo- II :::; IR(g, h)l

and therefore

118- 1 (!- gh- ~7J)II:::; ciR(g, h)l which, by ( 4.2), proves (4.3). We now prove that A is a contractive map. Let V Ef:) W. Since E < 1, this follows from

(~,

7J), ((, 7J 1 ) E

IIA((~, 7]))- A((e', 7J'))II = II8- 1 ((7J'- ~7J)II

= ll8- 1 (e'(7J' -11) + 11(e'- ~))II = ll8- 1 (e'(7J' -11)) + 8- 1 (7J(e'- ~))II ::S: sup IR(:, h)l {ie'II7J'- 7JI, I7JIIe':::; E sup{ I7J'

- 7]1,

~I}

le' -~I}·

Now, from the Fixed Point Theorem it follows that there exists an element (~, 7J) E V Ef:) W such that A((~, 7J)) = (~, 7J), that is f- gh -~1] = 8((~, 7])) = 91J + h~. Finally, take g = g +~'and h = h + 7J. Q.E.D. Note that the role played by the resultant in the preceding discussion comes from the fact that ±R(g, h) is the Jacobian determinant of the map (~, 7J) ---+ - f + (g + ~)(h + 7J), i.e. Hensel's lemma is a special case of a multivariable implicit function theorem. Before proceeding we make some remarks on Taylor's formula for polynomials with coefficients in a field of arbitrary characteristic. Let

VALUED FIELDS

13

f(X) E K[X], where K is a field (not necessarily valued) and let a E K. As in the classical case, when K has characteristic 0, one can write f(X)

= f(a) + (Df)(a)(X- a)+ 21!(D 2 f)(a)(X- a) 2 + · · · 1

+ 1n. (D" f)(a)(X- a)" where n is the degree of/, and D = d~. If the characteristic of K is p > 0 , then n! = 0 for n 2: p so that the formula, as it stands, loses its meaning. However, we observe that form 2: 0 if m

>s (4.4)

otherwise. Furthermore, (m) = ~ m(m- 1) · · · (m- s + 1) has a well defined s s. image in K, and if K is a field with valuation then that image lies in 0.

Hence we can take formula (4.4) as the definition of the symbol ~ D'. s. With this interpretation we are free to apply Taylor's formula in the general case. We shall also write

In particular, if K is a valued field and f(X) E O[X] then we have

(:,v• f)

(X) E O[X].

We can now give an application of Hensel's lemma which allows one to "refine" an approximate root a of a polynomial over a complete field to a true root a of the given polynomial. Theorem 4.2 (Newton's Lemma). Let K be a complete field with valuation H and valuation ring 0. Let f E O[X] and a E 0. Suppose that lf(a)l:::; cl/'(aW with 0 < c < 1. Then there is an

aE 0

such that

!(a)=

o

and

Ia- til :::; cl/'(a)l.

CHAPTER I

14

Proof.

We apply Hensel's lemma to the given polynomial f and taking

g(X) =X- a, h(X)

= f(X)-

f(a)

= J'(a) +

f'i!a) (X- a)+ ... +

X- a

f(n~fa) (X- at-1 '

f,

g and h being viewed as polynomials in the new variable Y = X - a. We have g, hE O[Y] and clegg= 1, degh = n-1. Since f- gh = f(a), condition (b) in Theorem 4.1 is obviously satisfied and so we need only verify that lf(a)l :::; c:lf'(a)i2 is equivalent to condition (a) of Theorem 4.1, where we take the Gauss norm with respect to Y. To prove this equivalence, it suffices to observe that

0

0

:! !" (a)

1

R(g, h)=± det 0 Thus, by Theorem 4.1,

f'(a)

1

=±!'(a).

_.!._,(~)(a) n!

f = (X- a)h(X) for some a E 0. Q.E.D.

As an explicit application of Newton's lemma we can consider K = Qs = the completion of Q with respect to the valuation 1-1 = l-Is, and f(X) = X 2 + 1. Now /(2) = 5 so that 1/(2)1 = 1/5 < 1, and /'(2) = 4 so that 1/'(2)1 = 1 and therefore the hypotheses of Newton's lemma are satisfied. Thus we obtain a E Qs with a 2 = -1 and Ia- 21 :::; 151. Indeed we can find an explicit formula for a. Since we can write -4 = 1 - 5 we obtain

2. (-1)1/2 = (-4)1/2 = (1- 5)1/2 =

~ c~2}-5)·.

(4.5)

We recall that the binomial coefficient is defined for a E Q and s a non-negative integer by

(:) = {

~a(a-1)

(a->+1)

ifs:;i:O, if s

= 0.

15

VALUED FIELDS

We wish to prove that the series in (4.5) is convergent. To that end, it suffices to prove that map

Ic~ 2) I : :; 1.

Observe that, for fixed s :;::: 0, the

Q-+Q

a~(:) is continuous (with respect the 1-1 5 -topology), it being of polynomial type. Now let (aN) be a sequence of positive integers converging in Q5 to 1/2 (for example, since

1

1

1

= 1+ 2L

5n). Then

L 00

- = 1 - - = 1+ 2- = 1+ 2 5n 2 2 1-5 n=O N

we can take aN

n=O

as desired. Finally from (4.5) it follows that

(-1)1/2 = ~. 2

f:•=0 (1/2) (-5Y. s

The same reasoning we used to prove that

c~ 2 )

E Z 5 allows one

to conclude more generally that

Proposition 4.3.

Let p be a rational prime. If a E Zp then (:) E Zp

for all s:;::: 0. We now offer a weaker version of Hensel's lemma. If a E 0, f(X) E CJ[X] we will denote by a, ](X) the image of a, finK, K[X]; respectively (K = Ofp is the residue field). Furthermore, we recall that two polynomials g, h, with coefficients in a field, are relatively prime if and only if R(g, h)::/= 0. In fact, we have R(g, h)= 0 if and only if ker 0 is different from 0, where 0 is the map defined in (4.1). This is equivalent to saying that the greatest common divisor (g, h) ::/= 1. Indeed, if t;h + TJY = 0 then hI Y"' and since deg h > deg 71 this division is possible only if (g, h)= r with deg r > 0.

CHAPTER I

16

Theorem 4.4. Let J( be a complete field with valuation valuation ring 0. Let f, g, hE O(X] with

degg

= n,

degh

= m,

degf

1-1

and

= m + n.

Suppose that

If- ghlgauss < 1

and

f-

gh E Wn+m

and suppose g, hare relatively prime. Then there exist g, that

h E O[X] such

f = gh,

I[J- glgauss < Proof.

lh- hlgauss < 1.

1,

Since g, hE O[X], then R(g, h) E 0, and we can write R(g, h)= R(g, h) .

To conclude it suffices to observe that R(g, h) =f. 0 if and only if IR(g,h)l = 1, so that if (g,h) = 1 then the conditions (a) and (b) of Theorem 4.1 become the hypotheses of the present theorem. Q.E.D. As an application of Theorem 4.4 we prove the following reducibility criterion. Corollary 4.5.

Let

with Ia; I ~ 1 and lan+j I < 1 for 0 ~ i ~ n - 1 and 1 ~ j ~ f(X) is reducible in K[X]. Proof. Then

Take g = xn

f- gh

+ an_ 1 xn-l + · · · +

E Wn+m

and

Moreover g = xn + · · · , h = 1, and (g, h) apply Theorem 4.4. Q.E.D.

m.

Then

a 0 and h = 1 + an+mxm.

If- ghl
1 and so I± 1 =f a1 ±···±ami> 1. Hence there is ani< m with Ia; I > 1 and we may choose the minimum i such that Ia; I 2: laj I for all j. Now we have

_.!_ f(X) = _!_Xm a;

a;

+ ~xm-1 + ... + xm-i + ai+1 xm-i- 1 + a;

By Corollary 4.5 we conclude that tion. Q.E.D.

a;

... +am. a;

f is reducible, which is a contradic-

In the sequel we will denote the valuation (} defined above by the same symbol used for the valuation on K .Then, for every a E 0, we have lal = INnjK(a)I 1/[!1:K] (5.1) where [0 : K] is the degree of the field extension. Let 0 be a finite extension of K. We recall that 0 = On/'Pn and ]{ = OK/'PK· Then since 'Pn n ()K = 'PK we haven 2 K, and so we may define

f

= f(O/ K)

= the relative degree of 1-1 = [0 : K] .

Similarly GK is a subgroup of Gn, so that we can define

e = e(O/K) =the ramification index of

1-1 =

(Gn: GK)

where ( Gn : G K) is the index of the subgroup G K in Gn. We now wish to prove that e and have a more precise result.

f are finite numbers. We actually

Theorem 5.2. Let K be a complete valued field and 0 a finite extension of K. Then ef ~ [0: K]. Proof. Let w 1, · · · , w p, E On be such that their residues w1, · · · , wJ1. in n are linearly independent over [{ (here J.l is a positive integer). Furthermore, let 11"1, · · · , 1r" elements of ox such that the cosets l1r 1IGK, ... , 17rviGK in Gn/GK are distinct. We claim that the elements 11"jWi, for 1 ~ j ~ v and 1 ~ i ~ J.l, are linearly independent over ]{. This will give the desired result since both J.l and v must be bounded by n = [0 : K], so that e and f are finite and ef ~ n.

CHAPTER I

20

To prove the claim, let us prove first that if

B = YtWl

+ · · · + yl-'wl-'

with the y; E K and not all zero, then

Clearly we may assume that IYtl is the maximum of the particular, y 1 "t 0. Now the element

belongs to On and its residue in independent over K. Hence

0

is not 0 since the

IYd, so that, in

w;

are linearly

whence IBI = IYtl as desired. If the 'lrjWi were linearly dependent we would have II

J-1

L1ri L:x;iw; =

j=l

i=l

o

with X;i E K and not all zero. Let us define }j

L 7rj }j II

i=l

=

= I:r=l X;iw;.

Now

sup {17rj }j I} 1$;j$;11

since the l1ri }j I are in distinct cosets by what we have seen above. This gives the desired contradiction. Q.E.D. The inequality ef :=::; [0 : K] of Theorem 5.2 also holds if K is not complete, and in that case one should not expect equality to hold. However even in the complete case (which we consider) the inequality can be strict. We cite without proof the following theorem, called

VALUED FIELDS

21

Ostrowski's Defect Theorem. With the same hypotheses and notation as in Theorem 5.1, there exists an integer 6, called the defect of 0 over K, such that [0 : K] = ef{j .

Moreover, 6 = 1 unless the residue class field K has characteristic p > 0 in which case 6 = p1 for some E N. For a proof see Artin [1], p. 60. There are examples of fields for which the integer I is positive. However, in the particular case which we will now discuss the equality ef = [0 : K] holds. We say that the valuation on K is discrete if G K is a discrete subgroup of the topological group R x. If 1r is an element of K such that l1rl = max{z E GK I z < 1} then GK = (17rl), the multiplicative cyclic group generated by l1rl and PK = 1rOK. If e = (Gn : G K) < oo then Gn is also a cyclic group, so that there exists a II EOn such that IIII' = l1rl and Gn = (IIII), Pn =liOn. Theorem 5.3. Let K be a complete field with respect to a discrete valuation and let 0 be a finite extension of K. Then

ef Proof.

= [0: K].

Let w1 , ... , w J E On be a set of representatives for a basis of

IT over K. We prove that the set { w;IIih9~/,0~j~e-1 is a basis of 0

over K . We observe that for every a E 0 there exists a power 1rt such that 1rta E On since l1rl < 1 and the group Gn is archimedean. Then it suffices to prove that J e-1

On= LLOKw;IIi. i=1 j=O

We first prove that if a E On there exist X1,o, ... , XJ,O E OK and E On such that

0::1

This follows immediately from the fact that we can write

CHAPTER I

22

for suitable X;,o E K. Now we iterate this procedure and write

ae-1

x1,e-1W1

with suitable X;,j E OK and f

ah

+

0

0

0

+ Xj,e-1Wj + Ilae.

EOn. Hence we have

e-1

a= LLXi,jW;IIj

+ Ileae.

i=1 j=O

But rre = 1ru where u EOn satisfies we can write f

a

lui= 1; so if we put {31 = uae

e-1

= L L X;,j,ow;ITi + 1r/31 i=1 j=O

where X;,j,o E OK, and then, iterating, f

e-1

LL

!31

X;,j,1Willj

+ 11"/32

i=1 j=O

J e-1

f3s

L L xi,j,sW;IIj + 1i"f3s+1 i=1 j=O

for suitable X;,j,>. E OK and

We now take the limit for s

because

11" 8 +1 f3s+l --+

/31-' E On. Thus

--+

oo and obtain

0 since f3s+1 E On and the series

EOn,

VALUED FIELDS

is convergent in

OK

since

X;,j,>.7r>.

23

tends to 0 for..\-+ oo.

Q.E.D.

Let I< be a field complete with respect to the valuation H and let f{alg its algebraic closure. From Theorem 5.1 it easily follows that H admits a unique extension to Kalg. However Kalg need not be complete with respect to this valuation. We call its completion n and we will still use H to denote the unique extension of H to n. Theorem 5.4. Let K be a complete field and let Q be the completion of its algebraic closure. Then 0 is algebraically closed. Proof. Assume not. Then there exists a monic irreducible polynomial in O[X] of degree n > 1 such that 1/lgauss :S 1. We begin with the case of a separable polynomial, that is we assume that f'(X) f:. 0 (the inseparable case can occur only in positive characteristic). We can write

f

=II (X- ,8;) n

f(X)

i=1

with ,8; f:. ,8j for i f:. j. We again use to 0(,81, ... ,,Bn)· Let

R;

H to denote its

unique extension

= min{I,B;- ,81·1} . j~i

Let a E Kalg; since f(a) = f};(a-,8;), from la-,8;1 ~ R; for every i we deduce If( a)!~ R1R2 · · · Rn. On the other hand, if Ia- ,8;1 < R; for at least one index i, we claim that lf(a)l

= Ia- ,B;IIf'(,B;)I.

To fix notation assume i = 1. It suffices to show that Ia- ,Bj I = !,81- ,8j I for every j f:. 1, and this follows from the identity a-,Bj = a-,81+,81-,Bj since !,81 - ,8j I ~ R1. Since Katg is dense in n there exists a sequence (g.) of monic elements from Kalg[X] with deg g. = n and lgs !gauss :S 1 such that lim lgs -/!gauss

S-+00

=0 ·

Let a 5 E f{alg be a root of g •. Then we have las I :S 1, and lim lf(a.)l = 0

S-+00

CHAPTER I

24

smce IJ(a,)l = lg,(a,)- f(a,)l :S l9s- /lgauss · We can conclude that there exists an index j with 1 :S j :S n, and a subsequence (a,#) of (a,) such that Ia,#- ,Bj I -+ 0. Hence, the sequence (a,#) converges to ,8j, so that ,8j E 0, which gives a contradiction. Finally, iff{ has characteristic p and f is inseparable then f(X) = g(Xq) with q a power of p and g(t) separable, monic and irreducible. By the case just discussed we see that if a is a root of g then a = limn an with an E f{alg_ But then (a;,/q) is a Cauchy sequence in f{alg which tends to a root of f. Q.E.D. The completion of the algebraic closure of the field Qp of p-adic numbers will be denoted by Cp.

6.

Newton Polygons.

Given a valuation to 1-1 by setting

1-1 on a field J{, we define the order function associated orda = -loglal.

If J{ has characteristic 0, and K is of positive characteristic p then one frequently takes

orda =_log lal . log p The following are the defining properties of the ord function

(01)

ord a= +oo if and only if a= 0,

(02)

ord (ab) = ord a+ ord b,

(03)

ord (a+ b) 2: min{ ord a, ord b} with equality when ord a # ord b.

Now let f E K[X] be of degree n, say f(X) = anXn +an_ 1 xn-t + · · · + a 0 • We define the Newton polygon of f(X) as the convex hull of the set of points

where Y+oo denotes the point at infinity of the positive vertical axis; if = 0, we define (j, ord aj) = Y+oo.

aj

VALUED FIELDS

25

The following is the Newton polygon of the polynomial f(X) 5X 10 +5 2 X 6 +5- 1X 5 + X 4 +5 2 X+ 57 over the field Q5.

Theorem 6.1. Let K be field complete with respect to a valuation, and let f E K[X]. Then to each finite side of the Newton polygon of f there corresponds at least one root of f. The number (counting multiplicities) of roots corresponding to a given side is equal to the length of the projection of that side on the x-axis, and all roots a corresponding to the same side have ord a = -A where A is the slope of the side. If

f>..(X) =

IJ

(X- a)

J()=O ord~=-.\

then f>..(X) E K[X]. Proof. We first prove the last statement. Let f be of degree n and let the roots of f (counted with multiplicities) be a1, ... , an. Let us first assume that K ( a1, ... , an) is a Galois extension of K (this is always the case if K has characteristic zero). As we have seen in the proof of Theorem 5.1, if u is any automorphism of K(a 1, · · · , an) over K then lu( a;) I = Ia; I, that is the values of the roots are invariant under u. If we set R>. ={a I f(a) = 0 and orda =-.A}, we have uR>. = R>. and then f>.. = IlaeR~ (X- a) is invariant for u so that it is in K[X]. If K has positive characteristic p we proceed as follows. We first observe that if f is irreducible, then all roots of f have the same magnitude: this is certainly true if f is also separable, since in that case K ( a1, ... , an) is a Galois extension; otherwise it suffices to observe that f(X) = g(XPv) with g E K[X] irreducible and separable, and v a positive integer. If f is reducible we proceed by induction on n, the case n = 1 being obvious. Let h(X) E K[X] be the minimal polynomial of a1 over K, and put f(X) = g(X)h(X), with g(X) E K[X]. Then, if we define 9>.(X) = (X- a),

IT

g()=O ord o=-~

CHAPTER I

26

then by the inductive hypothesis g,x(X) E K[X] for every A. Let At = -ord at. Then since h(X) is irreducible, /,x 1 (X) = g,x 1 (X)h(X) and f.x(X) = g,x(X) for all A f. At. Therefore, JA(X) E K[X] for all A, as desired. We now prove the first part of the theorem. It is not restrictive to assume, as we do, that a 0 = 1 (multiplying the polynomial f by a non-zero constant or by X has the effect of a translation on the Newton polygon); changing the preceding notation we write

f =

n

II(1 +a; X) = 1 +at X+ a2X 2 + ... + anxn i=t

and suppose

(6.1) Suppose {ordat,··· ,ordan} = {vt,··· ,vl} with Vt < v2 < ... < v1 and let the order v; appear with multiplicity K;, for 1 ::::; i ::::; l (that is, ordat = orda2 = ... = orda,. 1 = Vt, orda,. 1 +t = orda,. 1 +2 = ... = orda,. 1 +,. 2 = v2, ... ). We must prove that the Newton polygon off has l (non-vertical) sides PoPt, PtP2, ... , P1-tP1 where

Pp

p

p

i=t

i=t

= (LK;,LV;K;), ...

This is equivalent to proving that

p

orda,. 1 + ... +~ ord a.; therefore from (6.3) it follows that

L: ord •

ord a 8 = ord (a 1 a 2 ···a,)=

L: 8

a;

i=l

=

v;x:; .

(6.4)

i=l

Q.E.D. Example 6.2.

Let K

= Q 5 and f = X + 1. 2

There are two roots, each of which has order 0. If we make the change of variable X= Y + 2, then f(X) = g(Y) = Y 2 + 4Y + 5. Now the two roots have different orders; one is of order 0 and the other of order 1.

28

7.

CHAPTER I

They-intercept Method.

Let

For .A E R, the line of support, of slope .A, for the Newton polygon of f : Y = .AX + b>-. with b>-. maximal such that each of the points (j, ord ai) lies on or above f. Trivially b>-. = iJ!.f(ordaj- j.A), a continuous function of .A.

f is the line

J

If f3 E n, ord/3 = -.A then ord(ajf3i) ~ ord f(f3) ~ b)... We distinguish two cases

h

for all j and hence

1. General Case. If no side of the polygon is of slope .A then there exists precisely one subscript j such that h = ord aj - .Aj and hence ord f(f3) = h, i.e. n

ord A+ Lord (/3- ai)

= b>-.

.

i=l

But in this case ord f3 and ord ai are distinct for all i and hence

= ord A+ 'L: inf(ord ai, -.A) n

b>-.

i=l

provided .A is not the slope of a side. But both sides of this equation are continuous functions of .A and hence the relation is valid also if .A is the slope of a side.

VALUED FIELDS

29

2. Special case. For ord,B = ->., >.a slope of a side, we cannot hope to determine ord /(,8) in terms of>. alone since ord /(,8) may assume arbitrarily large values. To assist in a geometric presentation of the situation we use a multiplicative notation. Let I,BI = R correspond to ord ,8 = ->.. Let rR = {,BII,BI = R},D(a,R-) = {,BIIa-,81 < R}. Let {a1, ... ,ah} be the zeros off in rR. If ,8

h

ErR\ U D(a;,R-)

then, for

i=l

1::; i::; n,

Ia; - ,81 = sup (Ia; J, I,BI) and hence

IAI IT n

l/(,8)1 =

sup (Ia; I, 1,81)

i=l

i.e. once again n

ord/(,8) = ordA + L:inf(orda;, ->.)=h. i=l

Summarizing: Let ord ,8 = ->.; then ord /(,8) ~ b>. in all cases. Equality holds if ord (,8- a;) = ->. for each root a; with ord a; = ->.. In particular equality holds if the polygon has no side of slope >.. We invite the reader to note that in "ultrametric" geometry a "circumference" of radius R (like r R above) contains open disks ofradius R centered at points of the circumference. Thus, if one wishes to make an analogy with "classical" geometry, one should view the circumference as an annulus rather than a circle.

30

8.

CHAPTER I

Ramification Theory

Let K be a complete field and let 0 be a finite extension of K. We say that 0/ K is an unramified extension if a)

b)

0/ K

is a separable extension,

/(0/K)

= [0: K].

Clearly, by Theorem 5.2, if 0/ K is an unramified extension then e(O/ K) = 1. Conversely, if the valuation of K is discrete, this last condition, by Theorem 5.3, implies that 0/ K is unramified (subject to condition (a)). If f(O/K) = 1 we say that the extension 0/K is totally ramified.

Theorem 8.1. Let K be a complete field and 0 be a finite extension of K. Suppose that 0/ K is a separable extension. Then 0 contains a unique field extension T of K such that OfT is totally ramified and T / K is unramified. Proof. We remind the reader that 0/ K is a separable extension if and only if 0 = K(a) where Ci is a root of a monic irreducible polynomial g(X) E K[X] (of degree f) such that g'(a) f 0. There exists g(X) E On[X] with degg = f and whose reduction modulo .PK[X] is equal to g. We claim that g is irreducible. Indeed, we have g(O) f 0 and hence the Newton polygon of g consists of one side of slope zero and length f. Thus the roots of g (in an extension field of 0) have valuation one. If g factored in K[X], it would have monic factors in CJK[X] and by reduction mod .PK[X], the polynomial g would be reducible in K[X]. Let aEOn satisfy (a mod .Po)= Ci; then we have g(a) = g(Ci) = 0, g'(a) = g'(a) f 0, so that lg(a)l < 1 and lg'(a)l = 1. Hence by Newton's lemma there exists f3 E 0 such that g(/3) = 0 and 1/3- aj < 1, that is P = Ci. We define T = K(/3). Now we have T = K(/3) 2 K(P) = 0, so that OfT is totally ramified. On the other hand, [T: K] = [K(/3) : K] = deg g = f = [0 : K] = [T : K] so that T / K is unramified. To prove that T is unique with the.se properties, assume that E is another such field. Then E = 0 and so we can find 1 E E such that ('Y mod .PE) = Ci. This implies that jg('Y)I < 1 and jg'('Y)I = 1 since g is a separable polynomial. Repeating the preceding argument, we can assume that 1 is actually a root of g and so it must be equal to f3 since Q.E.D. distinct roots of g have distinct reductions. The field T is called the inertial subfield of 0/ K.

VALUED FIELDS

31

Theorem 8.2. Let K be a complete field. Let 0./ K be a finite Galois extension, and assume IT/ K to be a separable extension. Then IT/ K is a Galois extension and there is a group epimorphism

9(0./ K) ___,. 9(0/ K) whose kernel is (](0./T), where Tis the inertial subfield off2/ K. Therefore there is a natural group isomorphism

(](T/K)

'-

9(0/K).

Proof. We maintain the notations from the proof of Theorem 8 .1. We can write f

g(X)

= Il(X- a;)

with the a; E 0. and a1 =a.

i=l

We recall that the polynomial g(X) is irreducible over K. Since g is separable the a; are distinct elements of IT so that

in IT = K (a). Thus IT = K (a) is the splitting field of a separable polynomial, and so is a Galois extension. If u E 9(0./ K) then uOn ~ On and upn ~ Pn since ju,BI = I,BI for every ,8 E 0.. Let

u : On/Pn ___,. On/Pn =IT

be defined by for every ,8 E On. It is an immediate verification that 7Y is an automorphism of IT/ K and that

9(0./ K) ___,.(](IT/ K) is a group homomorphism. To prove surjectivity, let u* E (](IT/ K) be defined by u*a = a; and let u E 9(0./ K) be defined by ua = a;. It is clear that 7Y = u*. To conclude, observe that by varying i from 1 to f we obtain all the elements of (](IT/ K).

32

CHAPTER I

=

=

If u E Q(OI K) is such that u 1 then ua a so that we have 1. The roots of g being all in distinct classes modulo .Pn,

lua - al
R it is not convergent. The additive version of this criterion is the following: let . . ford an - M = llmln --; n n---+00

CHAPTER II

42

then M E [-oo, +oo] and might be called the additive radius of convergence. If x E n satisfies ord x > M then f( x) is convergent, if ord x < M then f(x) is not convergent. We recall also that the series f(x) converges if and and only if ian llx In --+ 0 (or equivalently, ord an +nord x --+ +oo) as n--+ oo. As in the case of polynomials, the Newton polygon off is defined to be the convex hull of the set { (j, ord aj)} i2~:o U {Y+oo}. If ai = 0 then we set (j,+oo) = Y+oo· There is in any case a first infinite vertical side. There are three possible cases: (i) there is no other side (e.g. EnP-n 2 xn), (ii) there are I+ 1 other sides, I 2:: 0, (iii) there is an infinite set of other sides. In case ( ii), I of the other sides are finite in length, the (I+ 1)-st is of infinite length. Each finite side has a first point and a last point. These are vertices of the polygon. The initial point (h, ord aiJ of the last side is again a vertex, but there need not be any other point (j, ord ai) on the infinite side. We must have ord ai 2:: ord aj, + )..(j- h) for all j > j 1 , and r . f ordaj- ordaj, -).. l!Jllll J-+00

.

.

J - Jl

-

.

In case (iii) the slopes of the sides form a strictly monotonic sequence. An example is given by log(1 +X). At this point it is not difficult to prove the following useful formula for the additive radius of convergence off: M =-sup{)..

I)..

is a slope of the Newton polygon of!}.

As in the classical case, on the "circle of convergence" the series may or may not converge, as we see from the following two examples. •

Let

/(X) = 1 +X+ X 2 + ... + xn + ....

The Newton polygon off is the positive x-axis so the additive radius of convergence is 0, but for x a unit the series f(x) does not converge. •

Let

00

f(X)

= Epn x2". n=O

The Newton polygon of this f is again the positive x-axis, but in this case when ord x = 0 we have convergence. More generally we have the

ZETA FUNCTIONS

43

2: pg(n) xn where g( n) is a 00

same behavior for any series of the type

n=O

sequence satisfying lim g(n) =+=and lim g(n) = 0. n--+oo n-+oo n As promised, we now prove the analogue of Theorem 1.6.1 for power series. In the proof we use the spaces WN introduced in the proof of Hensel's lemma (Theorem 1.4.1).

2: anXn 00

Theorem 2.1.

Let K be a complete field and let f =

E

n=O

K[[X]]. Then to each finite side of the Newton polygon off there correspond I zeros a (counting multiplicities) off where I is the length of the horizontal projection of the side. Moreover, if A is the slope of the side, then ord a= -A. Conversely if a is a root then -ord a is the slope of a (possibly infinite) side. Proof. If ord a = -A and A is not the slope of a side then the slope intercept method shows that ord /(a) is the y-intercept of the line of support of slope A and hence f( a) ::J 0. This proves the converse part of the theorem. As to the direct part of the theorem, it suffices to prove the assertion in the following special situation: the Newton polygon off contains a finite side with extreme vertices (m, 0) and (m + n, 0).

m

m+n

In fact we can easily reduce to this situation as follows: We start from a side of slope A whose horizontal projection has length I. We take a j3 E n with ord j3 = -A. Then the series f(j3X) has a horizontal side of length I. Now the series 8f(j3X), for a suitable 8 ::J 0, has a side on

CHAPTER II

44

the X-axis of length/. So, assume that we are in the special situation. There are two cases to consider. Case 1. The number of sides is infinite, so that we can find an infinite sequence of vertices (N, ord aN) from the Newton polygon of f with N > m + n. Let (N, ord aN) be one of these vertices, and take N

!N(X)

=L

anxn

n=O

. Then in view of Theorem 1.6.1 we can write !N(X) = PNQN where

with lc)N) I < 1 and ld~N) I < 1. The polynomial PN corresponds to the horizontal side, the first factor of QN corresponds to the sides of negative slope, and the second factor to sides of positive slope. Here IN is a unit in OK since 1 = I!Nigauss = IPNigaussiQNigauss = I'YNI· Let (N',ordaN') be another vertex with N' > N. Then fN' = PN'QN' as above. We also have !N'- PN(QN

+ aN'xN'-n) E WN'

,

I

where WN' = {! E K[X] deg f ~ N' - 1 }. The reduction mod PK of QN + aN'xN'-n is the polynomial QN = 'fNXm while PN =

+ ... + b~N). Since b~N) prime. Now let us define

xn

#

Then lfN'- PN(QN

0 we see that p N and Q N are relatively

+ aN'xN'-n)lgauss

~ €N.

Hensel's lemma gives a factorization !N' = PN'QN' with PN', QN' E OK[X] such that degPN' n, degQN' N'- n, that PN'- PN E Wn, QN' - (QN + aN'xN'-n) E WN'-n and that lPN' - PNigauss ~ €N, IQN' - (QN + aN'XN'-n)lgauss ~ €N. From this we can deduce first

=

=

ZETA FUNCTIONS

45

that PN' is monic and its roots are all units. Since the unit roots of !N' are the roots of PN' we obtain PN' = PN' and then QN' = QN'· Since C:N-+ 0 for N-+ oo we have that lPN'- PNigauss-+ 0. Similarly IQN'- (QN

+ aN'xN'-n)lgauss:::::; C:N,

so that IQN'- QNigauss:::::; C:N. We can now take P = lim PN N-+oo

Q

= N-+oo lim QN

with respect to the topology on K[[X]] induced by the Gauss norm. Since the PN are all monic polynomials of degree n, the same holds for P; on the other hand Q is a power series which is surely different from zero since the coefficient of xm is a unit, this being the case for all of the QN. Taking the limit in fN PNQN we find that f PQ. Observe that the Newton polygon of Q is obtained by "removing" the horizontal side from the Newton polygon off, and so Q has the same radius of convergence as f. In particular Q is convergent for lxl = 1. Thus we have found at least n unit roots of f, namely the roots of P (counting multiplicities, if necessary). Now let a:: satisfy lo:l = 1 and f(o:) = 0. We wish to prove that Q(o:) # 0 so that P(o:) = 0. To see that Q(o:) # 0 it suffices to observe that Q ;yxm with 1 lim IN which is a unit.

=

=

=

= N-+oo

m Case 2. The number of sides is finite, so that there is a last side. Let A be the slope of the last side, ( N 0 , ord aNa) the initial point of the last side, and A' the slope of the next to last side. For the points (j, ord ai) lying on or above the Newton polygon, we have ord aj 2:: ord aNa+ .A(j- N 0 ) for all j. We now consider the partial sums fN with N >max{ No, m + n} and, to fix ideas, we may insist that aN # 0 so that degfN = N. The Newton polygon of fN is not, as in Case 1, the truncation of the

CHAPTER II

46

Newton polygon off at N. The Newton polygon of fN can have several sides of slopes > A', but the Newton polygon of fN lies on or above the truncation at N of the Newton polygon of f. Bearing this in mind, we can once again follow the same steps as in case 1 to obtain the polynomial P and the series Q. The Newton polygon of Q lies on or above the Newton polygon off with its horizontal side removed. Then sup {slopes of the Newton polygon of Q} ~A and we can conclude that the radius of convergence of Q is at least -A so that f = PQ, as in case 1. Then the radius of convergence of Q must be equal to -A and the Newton polygon of Q must be equal to the Newton polygon off with its horizontal side removed.

m

N

m+n

As a final remark we observe that if a is a root of P of multiplicity s, then it is of the same multiplicity as a zero off (that is f(a) = f'(a) = · · · = J(•-l)(a) = 0, while J(•)(a) f 0). This follows immediately from Leibniz's formula for the successive derivatives of a product and the fact that the various derived series have better convergence than their primitives. Q.E.D. 3.

Newton Polygons for Laurent Series.

We now wish to extend our considerations on Newton polygons to the case of Laurent series, that is series of the form

L 00

f(X) =

anxn

with

an E [{ .

n=-oo

We can write f(X) = J+(X)

+ f-(1/X)

where

+oo

J+(X)

= LanXn n=O

+oo

and

r(x)

= L:a-nxn. n=l

47

ZETA FUNCTIONS

We say that f(X) converges at x

#

0, x E

n if J+

converges at x and

f- converges at 1/x. This is is equivalent to saying that lim (ord an+ nord x) = +oo. In l-oa

If M+ and M- are the additive radii of convergence of J+ and /respectively, then f converges for ordx in [-oo,-M-) n (M+,+oo]. If M+ < -M- then we have an annulus of convergence, and we will consider only this case. The Newton polygon of a Laurent series has the same definition as in the case of a power series. In order to have an annulus of convergence we assume that .\ 1 = inf {slopes} < .\2 = sup{ slopes} and then we have convergence for -.\ 1 > ord x > -.\ 2 .

We now generalize Theorem 2.1 to the case of Laurent series. We give a sketch of the proof. As in the case of a power series, we first reduce to the case of a finite side on the x-axis. Using the notation introduced above, we can consider the truncated Laurent polynomial N2

!N,,N2

=

L

a.x·

s=-N 1

where N1 and N2 are positive integers with N 1, N2-+ oo. Now we write

where PN,,N,

with

lbol

= 1 and

lb;l::;

= xn + bn-lxn-l + ... + bo 1 fori= 1, 2, ... , n- 1, and

CHAPTER II

48

where and

_ XN1+m SN1,N2-

+ dN1+m-1 XN1+m-1 + . . . + d0,

with lei I < 1, ldd < 1. These conditions ensure that hN1 ,N2 1 = 1, and that the reductions PN1,N2 and QN 1,N2 mod .PK are relatively prime. Now take N{ > N 1 and N2 > N 2 and write, as above, X

N'

1

/N'1'N'2

= PN' N' 1'

2

·QN'1' N'· 2

We then have n) E WN{+N~-n , X N'1 !N{,N~- PN1,N2 ( QN1,N2X N'1- N 1 +aN~ X N'+N' 1 2-

and

where

as N 1 , N 2 --+ oo. Now, the reduction modulo .PK ofthe two polynomials PN1,N2 and QN 1,N2XN{ -N1 +aN~xN{+N~-n are the polynomials PN1,N2 and xN{-N1QN1' N2 respectively, so that they are relatively prime (since b0 I.../.. 0). Thus, Hensel's lemma furnishes us a PN't' N'2 and a QN't' N'2 such that deg PN'1' N'2

= n,

and

so that PN'1' N'2 is monic and has only unit roots. Therefore PN'1' N'2 PN{,N~ and We obtain similar results for the x-N1QN1,N2· Passing to the limit as N 1 and N 2 --+ oo we obtain a polynomial Panda Laurent series Q such that f = PQ, and then one proceeds as in the proof of Theorem 2.1.

ZETA FUNCTIONS

49

Remark 3.1. The restriction to finite sides is actually unnecessary. Suppose ord aj ~ 0 for all j, ord ao 0 ord am and ord aj > 0 both for j < 0 and for j > m. Further suppose iai I -+ 0 as Iii -+ oo. Then

= =

f(X) =

00

L:

j=-oo

ajXi has precisely m unit roots.

The proof is as before except for one point. The Newton polygon of x-N1 QN1 ,N2 coincides with that of /N 1 ,N2 except that the horizontal side is excised. From this we deduce Q(X) = Ls b,X' with Ibm I = 1, lb. I-+ 0 as lsi-+ oo and lb. I< 1 for all s -:f m.

4.

The Binomial and Exponential Series.

Let a E Cp. We define

( 4.1)

where

(~) = 1 and

(a) = a( a n

1) .. · (a - n + 1) . n!

The problem of convergence for general a is rather delicate and will be treated in Section IV.7. However, by Proposition 1.4.3, if a E Zp

then

(:) E Zp ,

so that in this case the series is convergent if Moreover

lxl
0 for n ::=: 1. Now, by (4.6), we have

n-S

ord (pn /n!) =n-ord n! = n- ___ n >0 since Sn

> 0 and n

(1 --p-11-)

p-1

::=: 0 for every positive prime.

It follows that E(x) is defined as a function for note that if we let

xP xP2 fi(x)=x+-+-+ p p2 and

h(Y)

00

lxl
jp2-

for every

j

~

2.

On the other hand, since

Ci we have ord C · J

71'j

= """":T J.

for

= p_j1

1 :::; j :::; p - 1 ,

for

1 :::; j :::; p- 1 ,

and, since

2

mod XP ,

O(X) =: E(11'X) then ord C · > _ j p-1

J-

This proves statement (i). We now claim that 0(1)

# 1.

for

j :::; p 2

-

1.

Since 00

0(1) = 1 + 71' +

L Cj '

j=2

it suffices to prove that ord Cj > - 1 - if j If j ~ p 2 we have

p-1

.p- 1

2 p-

~ 2. 1

This is clear for j < p 2 .

ordC·J> J - - > p - - =p-1 p2p2

CHAPTER II

60

which is

> _!._1 if p 2:: 3. In the

pwhat happens for j

so that

= 4.

case of p

Since, in this case,

= 2 we have only to check 11"

= -2, we have

c4 = -42 + -42 . 2 + -16 = 4 + 2 . -43 4!

and ord c4 2:: 2 > 1. Similarly to what we have seen for the Artin-Hasse exponential function, although O(x) converges for Jxl ::::; 1 + €, it is not true that O(x) = exp(1r(x- xP)) when Jxl 2:: 1. However, this last equality does hold for Jxl < 1 since for such x the series

is convergent (see Lemma IV.7.1). If Jxl = 1, it is possible to have Jx - xP I = 1 and for such x the series is not convergent. Therefore we have logO(x) = 1r(x- xP) for JxJ < 1. On the other hand JO(x) -11 < 1 ifordx >

_P~ 1 , so that logO(X) p

is

> - p ~ 1 . It follows that log 0( x) = p with € > 0; from this equality we obtain

a series which converges for ord x

1r(x - xP) for JxJ < 1 + €, log0(1) = 0 so that 0(1) is a primitive p 1-th root of unity for some I> 0. Since J0(1) -11 = J1rl we can conclude that I= 1. This proves statement (ii) Now let x E Cp be such that xP' = x. We prove that the equality in (iii) holds. We define the series 0

which converges for JtJ

< 1 + €.

If JtJ

< 1 we have

61

ZETA FUNCTIONS

so that

H(t) = exp { 1r [(tx)- (tx)P

= exp ( 1r(t -

+ (txP)- (txP)P + ... -

tP )(X + xP

+ · · · + xP•-

_ ( ( P))x+xP+··+xP -exp1rt-t

s-1

1

(txP'- 1 )P)}

))

.

The last equality follows from the fact that x + xP being the trace over Qp of x, is an element of Zp. Hence

and, since both series converge for Q.E.D. specialized at T = 1.

It I < 1 + t:,

+ ... + xP'-

1

,

this equality holds when

·

We now generalize the preceding construction. Let q = p1• We define

fh(X)

= exp (1r(X- Xq)) = B(X)B(XP) ... 9(XP

Since B(XP') E g ((p- 1)/p2pi) for all i, we have We can now define a non-trivial additive character

£11

E

1

-

1

).

g ((p- 1)/pq).

Xs: Fq•---+ CP

by setting

(6.9) where

7.

z

= Teich z.

Meromorphy of the Zeta Function of a Variety.

Let V be an algebraic set defined over F q, with q = p 1• For s E N, we denote by V. = V(F q•) the set of points of V rational over F q• and let N. = # V,. We define the zeta function of V to be ((V,t)

= exp

00

(

t•)

~N·-;

.

(7.1)

CHAPTER II

62

We will consider the special case of

V = {x E F~oo

I f(x)

= 0,

X1X2 · · · Xn

"# 0} ,

where f E F q[X 1, ... , Xn] and F q"" denotes the algebraic closure of F q. The algebraic set V is an open set of the affine hypersurface W = {x E F~oo I f(x) = 0}, precisely the set of the points of W which do not belong to the union of the coordinate hyperplanes. Clearly V. = { x E Fq• I f(x) = 0, X1X2 · · ·Xn "# 0}. We will show that ((V, t) E Q(t), i.e., it is a rational function. From this, using a combinatorial argument, one can obtain the same result for W and then for an arbitrary algebraic set over F q, see Dwork [1]. We begin with a simple observation. As a consequence of Galois theory we have Proposition 7.1.

With the notation given above we have ((V, t) E Z[[t]] .

Proof.

Let

be the Frobenius automorphism of F q"" /F q. The automorphism u acts naturally on V: if x = (x 1 , ... ,xn) is a point of V then ux = xq = (xi, ... , x~) is again a point of V. The orbits of this action are finite sets and give a partition of V. Let x = (x1, ... , Xn) be a point of V and let I be the length of the orbit { x, xq, ... , xq 1 _ , } of x. Then Fq(x1,··· ,xn) = Fql and x E V8 if and only if ll s. If we let D1 = #of orbits of length I, then, N8

= #V. = LID1. II•

Now

= Loo 1=1

D1 ( -log(l - t 1))

oo ( 1 ) D1 = log fi -1-t1 1=1

,

ZETA FUNCTIONS

so that

((V, t)

= IT 00

(

1 1 _ tl

63

)D' = 1 + a1t + a2t 2 + · · ·

1=1

Q.E.D.

where the a; are positive integers.

Remark 7.2. Since N, :=; q•n = #F~. the series ((V, t) is dominated (in the sense of Cauchy, cf. Appendix III) by the series exp

00

( nt)") "'"' _q__

(

~ s=1

S

= exp ( log 1 - 1qnt ) = 1 - 1qnt = "'"' qnj ti ~ 00

so that a· ]

for every j

,

j=O

2:: 0 where

aj

-< qnj

is as in the proof of the preceding proposition.

Our aim is now to find a useful formula for the numbers N 8 • In order to do so we note first that if Xs is the additive character defined in (6.9), we have

"'"'

Xs (xof(x1, x2, ... , Xn))

~

xoE F q•

={

if J(x1, ... ,xn) ::/= 0 if J(x1, ... ,xn) = 0.

0 q

8

Thus

q"N. = .roEF qs (.r}l ...

,.rn)E{F:s)n

(7.2)

= (q"- 1)n + Now we suppose that

f has degree d and set

Xof(X1,··· ,Xn)

= LauXu

(7.3)

u

where au E F q and the u = (uo, u1, ... , un) are multi-indices with u 0 = 1 and d 2:: U1 + ... + Un. Moreover we set xu = x~o Xf 1 .•• X~n. If 1 we then have x = (xo, ... , xn) E (F:.

r+

CHAPTER II

64

If we now put au = Teich au and

x = (xo, Xt, ... , Xn) =Teich x = (Teich xo, Teich Xt, ... , Teich Xn) then from (7.3) and (6.9) we obtain

x.(xof(x))

=II (ol(auxu)Ol(a~xqu)

0

0

0

Ol(af-' xq'-'u))

u

q

stnce au = au. We define 0

v=(vo,··· ,vn) dv 0 ~v 1

'7-f),

Since 81 E 9 ( whenever v1 moment's reflection, that ord Av 2: where

+···vn

+ · · · + Vn

:::;

dvo we find, after a

(7.4)

Vol\:

p-1

~~:=--.

pq

Therefore F(X) and also F(X)F(Xq) · · · F(Xq'-') converge for

with a suitable positive c:. Then Xs (xof(x))

= F(x)F(xq) ... F(xq

,_,

)

and finally from (7 .2) it follows that

q• N.

= (q"- lt +

L

F(x)F(xq) · · · F(xq'-').

(7.5)

xq• - 1 =1

Before proceeding we need some algebraic considerations. Let T be a field of characteristic 0, and consider the T-vector space W =

ZETA FUNCTIONS

65

T[X 0 ,X 1 , ... ,Xn]· For HEW we indicate by the same letter H the multiplication by H in W: H :W ____. W ~

f---+

~H

;

obviously H is aT-linear map. Let

1/Jq:

w----> w

be the unique T-linear map such that if qJv otherwise, where v = (vo, v1, ... , vn) E Nn+l and q I v means q I v; for all i. A more intrinsic definition of 1/Jq is the following: if~ E W

Let R be positive real number and define (7.6) uo::SR du 0 ~u 1

+···+un

Now we fix a positive integer N and suppose that H E W_N. The composed map 1/Jq o H

is an endomorphism of the T-vector space WR_ if R is sufficiently large. In fact if we assume R > Nj(q- 1) then we have (R + N)jq < R and W(R+N)Jq ~ WR_. We denote by Tr ¢; the trace of a vector-space endomorphism ¢;.

66

CHAPTER II

Lemma 7.3.

In the preceding notations we have

(q-1t+ 1 Tr(1/JqoH)Iw~=

L

H(x).

x•- 1 =1

Proof. Since both members are linear in H we can assume H to be a monomial. Let H = X" with u = (uo, u 1, ... , Un) such that uo :S R, duo~ u1 + · · · + un; then we have

.,.~ 1 x" =

{

0 (q- 1)n+l

if (q- 1))'u otherwise

since x 1---+ x" is a non-trivial multiplicative character if (q - 1))' u. On the other hand, for fixed u we have that ( 1/Jq o X")Xv = 1/!qXu+v is zero or a monomial for every v. To calculate Tr ( 1/!q o X") we need only the v 's such that (1/Jq oX" )Xv = xv, that is such that u + v = qv. It follows that 0 if (q- 1))'u Tr(1/Jq oX")= { 1 otherwise, as desired.

Q.E.D.

From this Lemma it follows in particular that Tr ( 1/!q o H) I w~ does not depend on R for large R. (More generally it is possible to prove that the characteristic polynomial of (1/!q o H)1 w~ does not depend on R.) Now, lets be a positive integer; since H(X)H(Xq) · · · H(Xq"- 1 ) is an element of W~(l+q+··+q•- 1 ) and (R+ N(1 + q + ... + q•- 1 )) jq•
N/(q-1), the map 1/Jq• oH(X)H(Xq) · · · H(Xq"- 1 ) is an endomorphism of W~ for R sufficiently large independently of s. Since as is easily verified, we have

1/!q2 o H(X)H(Xq) = 1/!q o 1/!q o H(Xq) o H(X) = 1/!q o H(X) o 1/!q o H(X)

= (1/!q o H(X)) 2 and, inductively, for s > 2

ZETA FUNCTIONS

67

Then, from Lemma 7.3, it follows that

(q 8 -l)n+lTr (lfq o H(X))f w;,

=

L

H(x)H(xq) · · · H(xq'-') (7.7)

xqs-1=1

and again Tr (!fq o H(X))f w;, is independent of R. We define

Let v 0 '5N +···vn

dv 0 ~v 1

Then, since the series F(X)F(Xq) · · · F(Xq'-') converges for JxJ < l+c:, by (7.7) we obtain

so that

where the limit is taken "coefficient by coefficient" (i.e., identifying Cp[[t]] with C"J: endowed with the product topology). Now, for any h(t) E Cp[[t]], we define

h(t) Consider

= h(qt).

CHAPTER II

68

then we have exp

(~ t; (q'- 1)Tr (~q o FN )') = hh:\~t! = ht- (t)

exp

(~ t; (q" -1t+ 1 Tr(~q oFN)') = h}t- 1)n+\t).

and

1

On the other hand, we recall the following elementary fact: Proposition 7.4. Iff is an endomorphism of a finite dimensional vector space over a field then exp

t• (!') ) = (det(J- tf))?; -;Tr 00

(

1 ,

where I is the identity map. Proof.

Consider the Jordan canonical form for f.

Q.E.D.

Returning to our situation, we have 00 t• ) =(det(I-t(~qoFN)))- 1 , hN(t)=exp ( ?;-;Tr(~qoFN)'

so that

Z(t) = Ll(t)-(4>-W+l =

II Ll(qit)(-l)n-i(ntl)

n+1

(7.10)

i:O

where

Ll(t)

= N-+oo lim det(I- t( ~q o FN)) .

(7.11)

Our next step will be to prove that Ll(t) is a p-adic entire function, that is an everywhere convergent power series. From this result and (7.10) it will follow that Z(t) is a p-adic meromorphic function, that is a quotient of two p-adic entire functions. Now, from (7.7) and (7.9) it follows that

((V,qt)

t')

N = exp ( ~q"~

ZETA FUNCTIONS

69

Moreover, since 00

exp

(

t6) (exp (~-;t•)) 00

~(q 8 - 1t-; =

we obtain ((V, qt)

= (1 -

1

(¢- )"

1 - ( 1- t)C¢-1)" '

1

t)C¢-1)" Z(t) '

so that also ((V, t) will be a p-adic meromorphic function. We now prove that Ll(t) is a p-adic entire function. We set

LlN(t)

= det(l- t(tf; o FN)).

We then have

il(t) Un

= N-+oo lim LlN(t) .

Let L 6 be the number of points u E Nn+ 1 such that du 0 ;::: u 1 + · · ·+ and consider the polygon with vertices at (£ 0 , 0), (L 0 +L 1 , (q-1)x:L!), L;, (q- 1)x: iL;), ....

·.. , o::=o

L:!=o

We claim that the Newton polygon of Ll(t) lies on or above the polygon just constructed. It suffices to establish this for each LlN(t). We consider the endomorphism tPq 0 FN of the space w~f(q-1) defined in (7.6). We take a basis for W~f(q- 1 ) of the form {av 0 Xv}dvo~v 1 +···v,. for a suitable choice of av 0 E Cp. We then have

v

w

CHAPTER II

70

2: (qwo -

Since by (7.4) we have ord Aqw-u

uo)~~:,

if we put

we obtain ord (Aqw-u ll'uo)

2: (qwo- uo)~~: + II:Uo - II:Wo = (q- 1)wo~~:.

ll'wo

Thus the matrix B of 1/;q o FN has entries Bu,w satisfying ordBu,w

2: (q-

1)wo~~:.

We now consider the coefficient {31 oft 1 in det(J -tB). Let Lo+ ... +Lv:::; l :::; Lo + ... + Lv+l· The coefficient f31 is the sum of all products of the form (sgn u) Bu(ll,u< .. 0 and all its zeros z satisfy lzl > R then 1/g(t) converges in the disk ltl :5 R.

Proof. We may assume without loss of generality that B 0 = 1. Let p. be the first slope of the Newton polygon of g. Then we have p~-' > R and ordBj 2': p.j. Thus g E 9(p.) (cf. (6.8)), whence 1/g E 9(p.). Q.E.D.

CHAPTER III

DIFFERENTIAL EQUATIONS 1.

Differential Equations in Characteristic p.

We now turn our attention to differential operators over arbitrary fields. Let H be a differential field, that is a field endowed with a map D:H----.H

satisfying D(x+y)=Dx+Dy D(xy)

(1.1)

= xDy + (Dx)y

for all x, y E H. (Such a map is called a derivation of H.) We assume that D is non-trivial, that is D # 0. Let H 0 = ker(D, H) be the kernel of D in H. One checks easily that H 0 is a subfield of H. If u 1 , u 2 , ••. , Un are elements of H we define the Wronskian matrix W( u 1 , u 2 , ... , un) of Ut, u2, ... , Un with respect to D by

W(ut, u2, ... , un) =

(

.. )

u, Du1

U2 Du 2

Dun

Dn~lul

Dn-lu2

Dn~lun

(1.2)

and the Wronskian (determinant) of u 1 , u 2 , ... , Un by w(ut, u2, ... , un)

= det W(ut, u2, ... , un).

(1.3)

Lemma 1.1. The elements Ut, u2, ... , Un E H are linearly dependent over Ho if and only if w( Ut, u2, ... , un) = 0. Proof. If the u's are linearly dependent then there exist elements a1, a2, ... , an E Ho, not all zero, such that

77

78

CHAPTER III

so that on applying iterates of D we obtain

which proves w(u1, ... ,un) = 0. Conversely, assume w(u1, ... ,un) = 0 and w(u1, ... ,un-d # 0 (otherwise we proceed inductively), so that the columns of the Wronskian W(u1, ... , un) are linearly dependent over H while the first n- 1 are independent over H. Then there exist z1, ... , Zn-1 E H such that

Applying D to both sides of this identity we obtain

(1.5)

Now removing the last row in (1.5) and comparing with (1.4), from which the first row has been removed, we obtain W(u1, ... , Un-1)Dz = 0 whence z E Hon- 1 in virtue of the invertibility of W(u1, ... , Un-d· Q.E.D. We define the ring H[D] of (ordinary) differential operators on H by We shall denote the product of £ 1 and £ 2 in H[D] by £ 1 o £ 2 or simply (with due attention to the meaning) £ 1 £ 2 . The elements of H[D] are

DIFFERENTIAL EQUATIONS

79

to be thought of as non-commutative polynomials with the usual rules of multiplication, except that, for a E H,

Doa=D(a)+aoD. Let L = A 0 Dn + A 1 Dn- 1 +···+An E H[D], then if Ao =j:. 0 we say that L has order n, and we write n = ord L. Clearly ord (L1 o L2) = ord L 1 + ord L 2. One easily verifies that the ring H[D] is a left and right euclidean ring with respect to the function ord, so that, in particular, each (right or left) ideal is a principal ideal. Let ( H', D') be another differential field. We say that H' is a differential extension of H if H' 2 H and D'IH = D. In this case, for simplicity, we will write D for D'. We observe that for each differential extension E of H, any element L E H[D] can be interpreted as an endomorphism of E.

=

Lemma 1.2. Let L A 0 Dn+A 1 nn- 1 +· ··+An E H[D], with Ao =j:. 0. Then ker(L, H) is an Ha-vector space and

dimH 0 ker(L, H)::=; ord L.

Proof. s

Assume that A 0

= 1, . . . , n + 1 we have

= 1.

Let u 1 , ...

, Un+ 1

E ker( L, H). Then for

so that w( u 1 , ... , Un+l) = 0, whence { u 1 , ... , Un+ 1 } are linearly dependent over H 0 . Thus we conclude that dimH 0 ker(L, H) ::=; n. Q.E.D. Example 1.3.

d Let H = Fp(X) where X is an indeterminate, let D = dX, and assume p ;::: 3. If L = D 2 then 1, X E ker(L, H) so dimHo ker(L, H)= 2 = ord L. If L = DP+l then obviously 1,X, ... ,XP- 1 E ker(L,H); it follows that dimH 0 ker(L, H)= p < ordL, since H 0 = Fp(XP) and the elements 1, X, ... , XP- 1 constitute a basis of H over H 0 •

a)

d b) H as above and I) = X dX . Then o2 is an operator of order 2 whose kernel has dimension 1.

We say that L E H[D] is trivial on H if dimH 0 ker(L, H)= ord L.

CHAPTER III

80

Lemma 1.4. If 1 = 1 1 o · · · o 1m with 1; E H[D] and 1 is trivial on H, then 11, ... , 1m are also trivial on H.

Proof.

It suffices to handle the case m = 2. Then

ord 1 = dimHo ker(1, H)~ dimH 0 ker(11, H)+ dimH0 ker(12, H) ~

ord 11

+ ord 12 = ord 1

.

Thus all the inequalities must be equalities.

Q.E.D.

The converse of this Lemma is not true in general, as Example 1.3 (b) shows. Lemma 1.5. Let 1 = Dn + A 1Dn- 1+···+An E H[D] with n 2: 1 and let B E H. Then there exists a differential extension n of H in which the equation 1y=B

has a non-zero solution. Proof. Let n = H(zo, Z1, Zn-1) be a transcendental extension of H of degree of transcendence n over H. We can extend the derivation D to n as follows 0

Dz; Dzn-1 Then 1z0 =B.

0

0,

= Zi+1 (0 ~ i ~ n - 2) = -A1Zn-1 - A2Zn-2- · · · -

Anzo + B .

Q.E.D.

Corollary 1.6. Let 1 E H[D]. Then there exists an extension E of H such that 1 is trivial on E.

=

=

Proof. We proceed by induction on n ord 1. If n 1 the assertion is evident. Assume n > 1. Let n be an extension of H with 0 # u En such that 1u

= 0.

If we divide 1 by D- Du in il[D] (on the right) we

u

obtain 1

= 1 1 o ( D - ~u) + 1 2

where 11, 12 E il[D] and 12 is an element of n since its order must be 0. Applying both sides of this identity to u we obtain that 1 2 = 0. Hence in il[D] we have

DIFFERENTIAL EQUATIONS

81

Since ord LI = n - 1 there exists an extension S1' 2 S1 such that LI is trivial on Sl 1 ; thus there exist VI, ... , Vn-I E Sl' such that VI, ... , Vn-I lie in ker(LI,S1') and w(vi, ... ,vn-d :f 0. Now, by Lemma 1.5, we can find an extension E of S1' and non zero elements ui, ... , Un-I in S1' such that for Since Lui= Lio ( D-

~u) Ui

i

= 1, . . . , n -

(1.6)

1.

= LIVi = 0 we have that u, ui, ... , Un-I

are elements of ker(L,E). We claim that u,ui,··· ,un-I are linearly independent over Eo = ker(D, E). Let Ti = u;ju; then from (1.6) it follows that Dri

Dui = ----;;+ uiD ( ~1 ) = ~1

( Vi+ -;;ui Du )

+ uiD

( 1) ~

Vi =-:;-; .

It suffices to prove that 1, TI, ... , Tn-I are independent over Ea. Now

we have VI Vn-I) w(1,ri,··· ,Tn-d=w(Dri,··· ,Drn-I ) =w ( -:;;, ... ,-uBut VI, ... , Vn-I are linearly independent over (S1') 0 so that they are also independent over Eo and from this it follows that ~' ... , Vn-I are u u independent over Eo. Q.E.D. 2.

Nilpotent Differential Operators. Katz-Honda Theorem.

Let F = K(X) be the field of rational functions in the indeterminate X over the field ]{ of characteristic p :f 0; let D = djdX; we write R = F[D]. Let L E R. We say that L is nilpotent if L splits in R into trivial factors, that is if there exist LI, ... , Lm E R, trivial over F, such that L = L I o · · · o Lm. Theorem 2.1. Let L E R be of order n. Then i) if L is nilpotent then DPn E RL, that is there exists A E R such that DPn = AL; ii) if DPJJ E RL for some p,, then L is nilpotent. Proof.

i) Let us assume L to be trivial on F. Then DP = AL+B'

CHAPTER III

82

with A, B E R, ord B < ord L = n. Since dimF0 ker(L, F) = n and Bv = 0 for every v E ker(L, F), we must have, by Lemma 1.2, B = 0, so that DP E RL. Let L be non-trivial. We proceed by induction on n = ord L. Since Lis nilpotent, L = £1 o £2 with n; = ord L; < n fori= 1, 2 and £ 1 and £ 2 are nilpotent so by induction there exist B 1, B 2 E R such that

Therefore

B2B1L = B2(B1Ll)L2 = B2Dpn, £2 = Dpn, B2L2 = Dpn, DPn2 = Dp(n1+n2) = Dpn, since, as is easily verified,

DPoL=LoDP

for every

L ER .

ii) Assume DPI-' E RL, ord L > 0, so that Jl. 2: 0; moreover, assume Jl. to be minimal with this property. Since in R every ideal is principal, there exists £ 1 E R such that

RDP

+ RL =

R£1 .

If 1 E RDP + RL then DP(I-'- 1) E RDP~-' + RL s; RL and this contradicts the minimality of Jl.· Therefore ord £ 1 > 0 and L = A 1£ 1 for a suitable A 1 E R. Since DP E R£ 1 and DP is trivial on F, we have, by Lemma 1.4, that £1 is trivial on F. Let E be an extension ofF on which L is trivial and let u E ker(L, E). Then we have DP~-'u = 0 and 0 = L 1 DP~-'u = DP~-'(L 1 u), so that DPI-' annihilates £ 1 (ker(L, E)) which by the proof of Lemma 1.4 is precisely ker(A 1 , E). The dimension of this last vector space over E 0 is ord A 1 . Let

with A1, A2, B E R and ord B < ord A1. Then B annihilates ker(A 1, E) and thus we have B = 0 and DPI-' E RA 1. Since ord £ 1 > 0, we have ord A 1 < n, so that, by induction, A 1 is nilpotent. Therefore, we have L = A1 £1 with A 1 nilpotent and £ 1 trivial on F and we can conclude that L is nilpotent, as desired. Q.E.D. Theorem 2.1 establishes the equivalence of the following conditions for the operator L to be nilpotent:

DIFFERENTIAL EQUATIONS

83

L is the product of trivial factors, E RL for a suitable positive integer p.

D~'

There is yet another characterization of nilpotence. To establish it, let n be an extension of :F = K(X) such that DP' is trivial over n and let n. = ker(DP', n) . We observe that n. is not merely a vector space over no, but is in fact a differential field. Indeed by Leibniz's formula, bearing in mind that, for 0

< i < p•, the binomial coefficients

have DP' (yz)

(p;)

are all divisible by p, we

= yDP' z + zDP' y.

Hence if yz :f 0 and if two of the three elements y, z, and yz lie in n. then so does the third. Let E be any extension ofn and put E. = ker(DP', E), which is again a field. Since DP' is trivial on n it is also trivial on n. and on E. 2 n •. Moreover we have [Es : Eo] = (n. : no] = p• and so Es = n.Eo.

This shows that the choice of the extension on which DP' is trivial is not important. Proposition 2.2. Let L E R be of order n and assume p• 2: pn. Then L is nilpotent if and only if L is trivial over n,. Proof. If L is nilpotent then DPn E RL by Theorem 2.1, so that DP' E RL and this implies that L is trivial over n •. Conversely, let DP' = AL + B with A, B E R, and ord B < n. Since DP' and L both annihilate ker( L, n,) we have that ker( B, n,) 2 ker( L, n,). On the other hand, since L is trivial on n. we have dimno ker(L, n.) = n, so that B = 0 and L is nilpotent. Q.E.D. Let{;= X d/dX so that :F[D] = :F[o] =Rand let

(Observe that after multiplying by a suitable power of X any element of order n in R, monic with respect to D, can be written in this form.)

CHAPTER III

84

If we take the Laurent expansion ofthe Aj at 0, that is if we consider the Aj as elements of K((X)), and observe that

for every non-negative integer q and all s E Z, then it is easily seen that there exists an integer m such that for each s E Z LX"=

xm [ 1 (s + 1)x•+l + ...

=

=

and that ¢>; E K[s] and deg ¢>o = n. If IAi 111 ,gauss :::; 1 for j = 1, 2, ... , n then the coefficients of ¢> 0 are bounded by 1 in the valuation p, and so the indicial polynomial of Lp is the reduction mod p of ¢> 0 (s). Since by Theorem 2.3 all roots of this reduction are in F P, and this holds for almost all p, the exponents of L are rational. Q.E.D. Remark 6.2.

If

eE K(X) has a representation in K((X)) e=

2:: 00

ejxi

i=-m

e

then lelsauss = supi lei 1provided that has no pole in the punctured disk D(O,l-)\{0}. The proof may be reduced to the case in which = f(X)fg(X) where g(X) = 1 + alX + ... + alX 1 with lai I :::; 1 for j = 1, ... 'l, so that IYigauss = 1 and lelgauss = 1/lgauss· From

e

e= f(X) L( -alX- ... 00

alX 1)"

s=O

we see that

lei I :::; Ill gauss.

Conversely, from

f(X)

= g(X)

L 00

eixi

i=-m

we deduce that

1/lgauss :::; supi

lei I, which completes the proof.

100

CHAPTER III

Remark 6.3. In statement (ii) of Theorem 6.1 it would suffice to have nilpotence of Lp for all primes of K dividing a set of primes of Q of density strictly greater than 1/2 since Cebotarev Theorem is valid for such sets of primes. By way of illustration, let K = Q(/5); then /5 (mod lJ) E Fp if and only if 5 is a quadratic residue modulo p for p f:. 5; such primes have density 1/2 and in fact /5 is not rational. This explains the necessity of a stronger condition on the set of primes in the second assertion. 7.

Regular Singularities. Fuchs' Theorem.

Consider the differential system

d~ fi = G(X) fi,

(7.1)

where G(X) E Mn(F) and F is the field of complex meromorphic functions in an open disk V centered at the origin. We can assume that the radius of V is so small that zero is the only singularity of G(X). Let zo E V' = V\ {0}. Then the vector space of solutions of (7.1) in a neighborhood of z 0 has dimension n, by Cauchy's theorem (cf. Appendix III, Poole [1), p. 5). Let Fzo be the ring of complex analytic functions at z0 and let U E Gf. (n, Fz 0 ) be a matrix solution to (7.1) at zo, that is let

dU dX

= G(X)U.

We recall that, having fixed a closed circuit 1 C V' starting at z 0 and wrapping once counterclockwise around zero, analytic continuation of an element f E Fzo along 1 produces in general a new element T f E :Fzo. We call the map T the monodromy map (relative to 1). As an example consider any determination of log X; then TlogX =log X+ 27ri. One easily verifies that analytic continuation commutes with differentiation. Now the matrix TU is a new matrix solution of (7.1) since TG =G. Indeed, that det(TU) f:. 0 follows from the fact that TdetU = det(TU) and the fact that det U satisfies the following differential equation d dX det U = (TrG)( det U) .

Thus there exists a matrix C E Gf. ( n, C) such that

TU =UC.

101

DIFFERENTIAL EQUATIONS

If, for A E Mn(C) we define XA by

xA = exp(AlogX) = L oo

(AlogX)"

s=O

I

s.

then we have TXA

= exp(AlogX + 21riA) = XA exp (27riA).

If we take A such that exp (21riA)

= C then

so that W = U x-A is "uniform", that is, W is analytic in V'. There are now two possibilities: either zero is an essential singularity of W or W has at worst a pole at zero. Classically if the latter occurs the singularity is called regular and otherwise irregular. By the theorem of the cyclic vector, system (7.1) is equivalent to a scalar equation Ly = 0 where

with the A; E :F. In analogy with the definition given in Section 2 we say that L has a regular singularity at zero if the A; are analytic functions at zero. The following Theorem shows that there is no inconsistency in the two uses of the word "regular". Theorem 7.1 (Fuchs). The differential system (7.1) has a regular singularity at zero if and only if one (and so any) associated scalar equation is regular at zero.

If the singularity is irregular it is very difficult to calculate the matrices A and W since it is difficult to compute the map T. On the contrary, if the singularity is regular there is an algebraic method for computing A which avoids analytic continuation. This is particularly important for our aims since we shall be interested in the p-adic case where analytic continuation is not available (if two disks overlap then one is contained in the other by the strong triangle inequality).

102

8.

CHAPTER III

Formal Fuchsian Theory.

In order to give an algebraic treatment of the Fuchsian theory we first require a definition of the matrix X A encountered in its classical incarnation in the preceding Section. Let K be an algebraically closed field of characteristic zero, and let :F be a differential extension field of K((X)) containing the algebraic closure of K((X)) and containing a solution, which we will call log X, of the equation 6y = 1, where as usual 6 = X d/ dX. For every positive integer n we fix a root, which we call X 1 fn, of the equation zn =X in such a way that (X 1fmn)n = X 1fm, for all m, n > 0. Moreover, for all m, n > 0, we define

Clearly, for a E Q we have 6Xa = aX a. We now wish to extend the definition of xa to all a E K. To this end, for each non-zero class a of (K\Q)/Z we choose a representative a and we enlarge :F so as to contain a non-zero solution in :F, which we call xa, for the equation 6y = ay. Clearly any other non-zero solution in :F of this equation is obtained by multiplying xa by a non-zero constant in :F, that is an element of the subfield :F0 = ker(6,:F). We also assume that that is we take -a to be the representative of just shown. Finally, we define

-a,

and choose x-a as

for all m E Z. In this way xa is defined for every a E K. Moreover the two preceding equalities are easily verified for any a E K and each mEZ. Warning. Let a and j3 E K. Then there exists a non-zero constant Ca,{3 E /{ such that xa x/3 Ca,{3xa+f3 However, the constant Ca,/3 need not be equal to 1, if a, j3 fl. Q. In particular for each a E /{ and n E Z there exists a non-zero c E :F0 , which depends on a and n, such that (xat cxna

=

0

=

Observe that the symbol (xnt for n

0

> 1 has not

been defined.

103

DIFFERENTIAL EQUATIONS

Now let

~

E Mn(K) be the diagonal matrix

Then we define the matrix X~ E Mn(:F) to be the diagonal matrix

(8.1) On the other hand, if N E Mn(K) is a nilpotent matrix then we define XN =

oo

Ni

j=O

J.

L:-. (logX)j =exp(NlogX). 1

(8.2)

Let ~' N as above and assume ~N = N ~. Then, from (~i- ~j )Nij = 0, it follows immediately that X~ N = N X~ so that x~xN

= xNx~.

Moreover it is clear from the definitions that (X~)- 1

(XN)-1

= x-~ = x-N .

For a general A E Mn (K) let P E G£ (n, K) be such that A = + N)P- 1 with~ diagonal and N nilpotent satisfying ~N = N ~. Then we define P(~

(8.3) We now check that this gives a well defined matrix in the sense that if we choose some other representation, say A = P'(~' + N')P'- 1 then we obtain the same formula for XA as an element of Mn(:F). To this end, it suffices to remark that P N p- 1 is uniquely defined by A so that PNP- 1 = P'N'P'- 1. On the other hand, by definition, XPNr' =

t

(PN~-1)j (logX)i

j=O

= p

(

J.

Ni (logX)l·) p-1 L -., oo

j=O J

= PXN p-1.

CHAPTER III

104

Likewise we obtain, xP'N'P'-l = P' xN' P'- 1 and therefore pxN p- 1 = P' xN' P'- 1 . So there remains only to check that if Pt:..P- 1 = P' D.' P'- 1 then PXt:. p- 1 = P' xt:.' P'- 1 . Replacing P by (P')- 1 P we have reduced to proving that if PD.= D.'P then PXt:. = xt:.'p. To see this, let P = ( P;j). Then PD. = D.' P means P;j D.i = D.~ P;j for all i, j so that P;ixt:.; = xt:.:pii as desired. It is not difficult to verify that for every A E Mn(K) the matrix XA is invertible in Mn(F); more precisely

Moreover,

xA

is a matrix solution of the system b- A, that is

Remark 8.1. In the sequel whenever we refer to XA in a neighborhood of zero we will mean the object introduced above. For any non-zero a in an extension field of K we shall also use the symbol (X/a)A (but when a = 1 we again write simply X A rather that (X /1 )A) to denote the normalized matrix solution of bY = AY at x = a. In this case

where

( ~) = In

and for any positive integer s 1 A- In) .. · (A- (s- 1)In) ( A8 ) = -,A( s.

( cf. Section IV. 7). Before proceeding we introduce some useful notation. Let (F, b) be any differential field and consider the system by= Gy

with

G E Mn (F)

DIFFERENTIAL EQUATIONS

105

If we set z = Hy with HE Gf(n,F) and we put

(8.4) then we obtain 6z = G[H]Z.

Note that if H 1 , H 2 E Gi (n, F) then

(8.5) In the following lemma we describe what are classically referred to as shearing transformations.

Lemma 8.2. Let G E Mn(K[[X]]) and let a be an eigenvalue ofG(O). Then there exists HE Gi (n, K[X, x- 1]) such that G[H] E Mn(K[[X]]) and such that the eigenvalues ofG[HJ(O) coincide including multiplicities with those of G(O), except that the eigenvalue a is replaced by a+ 1. A similar statement holds with a replaced by a - 1. Proof.

Assume G(O) to be upper triangular, and assume further

G(O) = (

~ ~)

where BE M(n-m)xm(K), DE Mm(K), J E Mn-m(K) and both J and D are upper triangular matrices

with

a;

f.

a for all i. Let

If we write

CHAPTER III

106

then

0) (i-In-m L)

G[H)-_ (Xln-m 0

0

0

+ (X ln-m

lm0 )

0

_ (In-m 0

-

0) 0

Since G 3 (0) = 0 we have

+

( G1 Ga

( G1 :_k-Ga

ln-m

G2 ) ( :_kG4 0

L)

XG2) G4 .

x- 1 G 3 E K[[X)) and

G[Hj(O)

= ( J + :n-m ~)

Thus the matrix G[Hj(O) has o:+l as an eigenvalue with the same multiplicity as that of o: for G(O), while the eigenvalues o: 1 , ... , O:m remain unchanged. If G(O) is not upper triangular, we find a matrix C E G£ (n, I0

II Gs,j lla,p:::; p-•+i {s, n- l}p. Application 2.3. skian

Let u 1 , ...

, Un

E Ao and assume that the Wron-

w(u1, ... ,un)

is never zero in D(O, 1-). If we write

then

L = Dn

where the A; E

A~.

+ A1Dn-l +···+An

Let

ui

U1

U=

Then U

(

...

Un

u~

)

~~~ .. ::: .. ~.~~

= GW(ul,···

,un)

where G = (Gs,j) and W(ul,··· ,un) is the Wronskian matrix. If we denote the adjoint matrix of W(u 1, ... , un) by W then UW=wG, so that wG,,j E Ao for all s 2: 0 and 0 :::; j maximum modulus principle we have

< n-

1. Then, by the

lwG,,j(x)l:::; II wGs,j llo:::; II w llo II Gs,j llo:::; II w llo {s, n- 1}p · Now we observe that since w has no zeros in D( 0, 1-) the Newton polygon of w has no side of negative slope, so that lw(x)l = lw(O)I for all x E D(O, 1-) and so II w llo = lw(O)I. Therefore we have

IGs,j(O)I:::; {s, n- l}p.

EFFECTIVE BOUNDS. ORDINARY DISKS Now let

125

u(X) = LC,X"

be a solution of the equation L = 0, with the initial conditions for

0

~

i

~

n - 1.

From Taylor's formula we have

Since by definition

.!..u 0 00

f(X) = LArXr,

g(X)

=

z=s.x• 00

•=1

r=O

with Ar, B. E 0. Assume that f(X) converges in D(O, Rt) and that g(X) converges in D(O, Rt) and maps D(O, Rt) into D(O, Rt). Then the formal power series h( X) = f(g( X)) has radius of convergence 2: R 2 Straightforward or down-to-earth and if lxl ~ R2 then h(x) = f(g(x)).

Proof.

Let h(X) = I::~= a CnXn. Then Co = A 0 , while for n 2: 1

As g converges in D(O, Rt) and maps D(O, Rt) into D(O, Rt), by Proposition 1.1 it follows that for every s. Therefore

and

Since the right hand side of the inequality tends to zero as n series h(X) converges for lxl ~ R2. Assume lxl ~ R2 and let

L ArXr , N

!N(X) =

r=O

= !N(g(X)) = L Cn,NXn E O[[X]]. 00

hN(X)

n=O

--+

oo, the

CHAPTER IV

142

Since fN(X) is a polynomial, we have hN(x) prove h( x) = f(g( x)) it suffices to prove that

= f(g(x))

lim !N(g(x))

N-+oo

Therefore to

= h(x)

lim hN(x)

N-+oo

and that

= fN(g(x))o

0

The second assertion is clear since lg(x)l S: sup 8 IB.IR2 S: R1 and lxl S: R1o As to the first assertion, we note that

f(X) converges for

ifn

1 then Ya(x) converges if and only if lxl < l1rl/lal. (iii) If a E Zp then Ya(x) converges for lxl < 1. (iv) Ifsup{ord (a- b) lb E Zp} = m + c, with mEN, and c E [0, 1) then Ya(x) converges if and only if 1 1 ordx> pm (p- l ) - tpm - +1

.

(v) If lxl < 1 and x lies in the disk of convergence of Ya then Ya(x) = 1 implies that (1 + x )P' = 1 for some s. (vi) Let orda > l. If(P 1+ 1 = 1 then Ya(( -1) = 1. Conversely if 1

ord (( - 1) ~ P' (p _ 1) 1+1

and Ya((- 1) = 1 then (P = 1. (vii) If Ya converges in D(O, 1-) then a E Zp.

Proof.

(i) We have already noted that as formal power series in X, Y

Ya(X)ga(Y)

= Ya(X + Y + XY)

.

CHAPTER IV

144

Letting G(X, Y) = X+ Y + XY, the assertion follows from Corollary 7.2 (ii). (ii) For lal

> 1 we have

and so convergence is precisely the same as for exp aX. (iii) This follows immediately from Proposition 1.4.3. (iv) Case 1. ord a = m + sEN, we have

€.

Then since ord (a-s) :::; ord a for all

~ s-1 s-1 ord(a(a-1) ... (a-s+1))=L....-(1+[-. ])+c(1+[ m+l]) j=l pJ p while

00

ords! =

L[~] j=l

pJ

Thus by a routine calculation lim

$--+00

~ord s

(a) s

€ 1 --pm+l pm(p _ 1)

This confirms the asserted radius of convergence. To show that there can be no convergence on the boundary we use the above formula to compute for s = p 1+m+l, l E N ord

(a) ( 8

+ s

1 - -€ - ) = -1pm(p- 1) pm+l p- 1

which shows that 9a(x) cannot converge for € 1 ordx = pm(p-1)- pm+l.

Case 2. By hypothesis a ¢ Zp. By compactness of Zp there exists b E Zp such that ord (a- b) = m + €. By case 1 we know the radius of convergence of 9a-b and by (iii) this is strictly smaller than the radii of convergence of 9b and of 9-b· Since 9a = 9a-b9b, 9a-b = 9a9-b we conclude that 9a has precisely the same radius of convergence as 9a-b·

EFFECTIVE BOUNDS. ORDINARY DISKS

145

(v) Let g 0 converge in D(O, R+) and A E D(O, R+). Then

lim (1

S--+00

+ A)P' = 1 .

Letting 1 +As = (1 + A)P', it follows from (i) that if g 0 (A) = 1 then ( As) = 1 and As -+ 0. It follows (e.g. by Proposition 4.2) that {,\.} is a finite set with zero limit and hence As = 0 for s sufficiently large.

g0

(vi) We first explain the converse point. If Ya(( - 1) = 1 then from (v) it follows that (P' = 1 for some s. Let s be minimal; then by Example 1.9.3 1 ord((- 1) = (p _ 1)ps- 1 if s

2: 1 (the

case s = 0 is trivial). Hence s- 1 ~las asserted. For the direct part we consider two cases.

Case_ 1. l = 0. In th1s case ord o:

>

= 1.

0, (P

Hence ord( ( - 1)

1

2: - p-1

and by

(iv) the series Ya converges on D(O, 171"1+). Thus by (i), Ya((- 1)P Thus Ya((- 1) is a ord

(0:) (( s

Thus ord(g 0 pth

1)'

((-

pth

= Ya((P- 1) = Ya(O) = 1.

root of unity but for s

2:

1

2: ord o: + -s- - ord s! 2: ord o: + -1- > -1- . p-1

1) -1) > - 1 - and so g 0 p-1

root of unity, i.e. g 0

((-

1)

p-1

((

p-1

-1) cannot be a primitive

= 1 as desired.

Case 2. l > 1 By hypoth~is o: = p 1j3, ordj3 > 0. Let h1(X) = (1 + X)P' - 1, R = 171"1 1/Pz. Then h1 maps D(O, R+) into D(O, 171"1+) while as formal power series Ya(X) = g13(h1(X)). Thus by Lemma 7.1 (since 9{3 converges on D(O, 171"1+) and ( - 1 E D(O, R+)) and by Case 1, since 1 + h1((- 1) is a pth root of unity, we have Ya((- 1)

= 9{3(hl((- 1)) = 1.

(vii) This follows from (ii) and (iv).

Q.E.D.

CHAPTER IV

146

Remark 7.4. (1) Subject to the hypothesis of Case 1 in the proof of (iv) we may represent fo: as a composition of functions for

ord x

1

1

> pm (p- 1) - -pm ,

in two ways: as exp(a log(1+X)) and as yp(hm(X)) where a= f3pm and hm(X) = (1+X)Pm -1. By these compositions we could obtain alternate demonstrations that Yo: converges on the indicated set. The proof used shows that we have obtained the precise radius of convergence. An alternate procedure would use the formula

1 yp(x) =pm -

'""" L..J

Yo:(Y)

hm(Y)=x

(for a= pm/3, 1!31::; 1) to estimate the radius of convergence of yp from that of Yo:· (2) For fixed x with l1rl < lxl < 1 consider F(a) = (1 + x)o: as a function of a, and suppose that lal 0).

Proposition 7.5. Let A E Mn(K) and let ~1, ... , ~n be the eigenvalues of the matrix A. Then gA(X) converges in D(O, R+), R = 11!'1 1/P', l ~ 0, if and only if

for some

€

>

0. In particular if each ~i E p1+'0K then

(7.2) for all ( such that (P'+' = 1. Moreover, if A is nilpotent equality (7.2) holds for all p-th power roots of unity.

EFFECTIVE BOUNDS. ORDINARY DISKS

149

Corollary 7.6. Let A be as in Proposition 7.5. Then YA(X) converges in D(O, r-) if and only if the eigenvalues of A belong to Zp. The following proposition due to G. Christal gives an example of a new type of transfer theorem. Here we obtain information on the singular disk D(O, r-) from information on the generic disk. Proposition 7.7. Let G E Mn(Eo). Let Ua,t be the solution matrix at the generic point t of the system

dy -

I.a-y

dX- X

such that Ua,t(t) = In. Assume that Ua,t has radius of convergence 1. Then the eigenvalues of the matrix G(O) belong to Zp. Proof.

We define G1

1

= XG

and as usual, for s = 2, 3, ... , we define the matrices Gs by

so that

ds dX• y

= G ,y-

for all solutions yof the system. We know that XG 1 E Mn(Eo) and we now show that indeed x•G. E Mn(Eo) for all s. In fact, we prove by induction that

G (X) = s! (G(O)) _1 • s x•

(-1)

+ x•-1

where

The recursion formula reduces the verification to the following obvious identity

CHAPTER IV

150

By hypothesis

UG,t(X) =

f: G:~t)

(X- t)"

s=O

converges in D( t, 1-) and so by Theorem 2.1 we have

for a suitable constant C. Therefore, we also have

I0 ; I 8·

gauss

::; C{s,n -1}p.

On the other hand

so that the series

converges in D(O, 1-), and therefore by Corollary 7.6 we obtain the desired result. Q.E.D. 8.

The Hypergeometric Function of Euler and Gauss.

Let /{ be as in Section 7. We consider the hypergeometric function defined by

F(a b c· X)-~ (a).(b)s x• ''' - ~ s=O S.'() C s

(8.1) where a, b, c E K, c (a) s is defined by

(a)o

= 1,

fl.

N, and where for any element a E /{ the symbol

(a). =a( a+ 1) ···(a+ s- 1) for

s ~ 1.

The power series (8.1) is a formal solution of the hypergeometric differential equation

(8.2)

La,b,c

=D

2

+

c - (a + b + 1)X ab X(1- X) D- X(1- X) .

151

EFFECTIVE BOUNDS. ORDINARY DISKS

It is well known that La,b,e has only three singular points, namely 0,1 and oo with exponents given by the following table.

0 0 1-c

Exponents of La,b,e 1 0

c-a-b

00

a b

Another formal solution to La,b,e at zero has the form

X 1 -eF(a + 1- c,b+ 1- c,2- c;X) and if c fj. Z then this is independent of the first solution. Kummer gave a list of 24 such formal series solutions; here are four more of them:

F(a, b, a+ b + 1- c; 1- X) , (1- X)e-a-b F(c- a, c- b, c + 1- a- b; 1- X) , x-a F( a, a+ 1 - c, a+ 1 - b; 1/ X) , x-b F(b + 1- c, b, b + 1- a; 1/ X) . We have the following result regarding the domain of convergence of the formal solutions of La,b,e at the generic point. Proposition 8.1. If a, b, and c E Zp then the solutions of La,b,e at the generic point t are convergent in the disk D(t, 1-). Proof. Let (a~c,b/c,clc) E (-N) 3 be a sequence converging to (a,b,c) chosen so that

a) b)

-ak < -c~c, b~c+1-c~c 0. In view of hypothesis (b) we see that Vk is also a polynomial. Now we have uk(O) 1 and vk(O) 0 so that uk and Vk are linearly independent. Let

=

=

where W denotes the Wronskian matrix. Moreover, let uk(X), vk(X) be defined by

Then Uk, Vk are linearly independent solutions of Lak,h,ck normalized by Wk(t) 12 (here wk W(uk. Vk)) and are clearly analytic in the disk D(t, 1-). Therefore, by Proposition 5.4, the same is true for the two solutions u(X) and v(X) of La,b,c defined by

=

=

u(X)

= k-+oo lim uk(X) ,

v(X) = lim vk(X) , k-+oo

where the limits are taken in the topology of uniform convergence on D(t, R-) for each R < 1. Q.E.D. We note that the convergence of the two solutions in the generic disk does not permit one to draw conclusions regarding the convergence of solutions at the points 0, 1 and oo. As an example of this, consider the case a 1 and c b + 1 rt Z with b E Zp. In this situation the exponents are given by the following table.

=

=

Exponents of L1,b,b+l

1 0 0

0 0 -b

00

1 b

Formal Solutions at zero:

F(l b b + 1· X)=~ ( 1).(b). x• = b " - 1-x• ' ' ' LJ s!(b+ 1), LJ b+s s=O

s~O

x-b F(1- b, o, 1- b; X)= x-b . The radius of convergence of the first series at zero can be zero or very small if lb + sl is very small for arbitrarily large values of s, that is if b is what is called a p-adic Liouville number, see Section Vl.l.

EFFECTIVE BOUNDS. ORDINARY DISKS

153

Formal Solutions at oo: _.!._

XF( 1' 1

-

b,

2- . _.!._ - _.!._ ~ b,X)- X~

(_!_)s (1- b)s X (2-b),

1 (1)' 1 :::::X~ X 1-b+s 00

1 (1)b (X1)b F(O,b,b; X)::::: X and so this case is similar to the preceding one. Formal Solutions at 1:

F(1,b,

1; 1- x)::::: ~ (~~·c1- xy::::: ~ (~b)cx -1)'::::: x-b.

Since the two exponents at 1 are both zero, we must expect a solution with a logarithmic term. To find it we first observe that Ll,b,b+ 1 y ::::: 0 is equivalent to 8 o (X- 1) o (8

+ b)y::::: 0,

with

8::::: X

d~.

We further note that if P(8) E K(X)[8] then from Leibniz's formula it follows that

+ a)y. 1) o (8 + b)X-bz::::: 0 we find

P(8)(X"y)::::: X" P(8 Therefore, if y::::: x-bz, from 8 o (X-

(8-b)o(X-I)o8z:::::O so that

(X- 1) o 8z::::: Xb .

If we now put X ::::: 1 + U then we must have d

(I+U)UdUz:::::(l+U)b so that

~z::::: ~ + dU

U

f (b-l)u•-1 •=1

s

CHAPTER IV

154

whence the other solution is

x-b

[log(X- 1) +

e

~ ~ 1) (X~ 1)']

Aside from the factor log(1 - X) both solutions converge in the disk D(1, 1-), and convergence of all solutions in D(t, 1-) could again be established in this case by Proposition 5.2 and Corollary 5.3. Remark 8.2. The reader should check that if L 1 and L 2 are elements of K(X)[D] each with the property that all its solutions at t converge in D(t, 1-) then the same holds for L = L 1 o L 2 . By applying this remark to 1 Ll,b,b+l= X- 1 oc5o(X-1)o(8+b)

we deduce a third proof that all solutions of L 1 ,b,b+l at t converge in the disk D(t, 1-).

CHAPTER V

EFFECTIVE BOUNDS. SINGULAR DISKS In this Chapter J{ will be a fixed complete algebraically closed extension of Qp and 0 will be a sufficiently large extension of J{ so as to contain all our generic points over J{. 1.

The Dwork-Frobenius Theorem.

We recall the E is the field of analytic elements over We denote by .p E the valuation ideal of E that is

.PE =

J{,

cf. Section IV .4.

{e E Ellelgauss < 1}.

We observe that the residue field of E is K(X), the same as that of K(X) with respect to the Gauss norm.

Lemma 1.1. Let G E Mn(Eo) and let Ua,t be the matrix solution at the generic point t of the system

such that Ua,t(t) = In. Assume that Ua,t converges in the disk D(t, where as usuall?rlp-l = IPI· Assume moreover that

IGigauss > If a E J{ satisfies modulo .PE·

Proof.

lai = IGigauss

l1rl-)

1·

then the matrix

a- 1 G

is nilpotent

With G, defined as in (III.5.3) we shall prove by induction that

G, : : (G)' a•

a

155

1

(mod-). a

CHAPTER V

156

Indeed,

Cs+I C d c. - =C. -+a1-dX -a• a•+l a• a

and it suffices to observe that

c.,

I_!}_ dX a•

. E K be such that 1>.1

= 1, ...

, n.

1/" = sup 1 :"0j:"'n(ICjlga~ss) so that

sup (ICjj>.ilgauss)

1:"0j:"'n

=1 ·

Take

and consider G[H]

= H- 1 GH. 0 0

It follows that

1 0

Hence IG[HJigauss

0 1

= l.\1 ·

If ICjlgauss > 1 for some j then 1>.1 > 1 so that by Lemma 1.1 the matrix >.- 1 G[H] is nilpotent modulo PE· On the other hand, the characteristic polynomial of >.- 1 GrHJ is xn- >.- 1 clxn- 1 + ... + (-1)n >.-ncn, which has a root a with lal 1 and this gives a contradiction. Q.E.D.

=

Remark 1.3. For a scalar equation L = Dn + C 1 Dn- 1 + · · · + Cn with Cj E E and ICj Igauss :S 1 for all j we have shown in Section 111.5 that all solutions at the generic point converge in the disk D(t, 171"1-), and for special points one can not generally do better. For example, the equation djdX- 1 has the exponential function as a solution at zero, and this has radius of convergence 111"1. Example 1.4. By a Heun equation we will mean a second order differential equation with four singular points, all regular; we will suppose

CHAPTER V

158

that all exponents belong to Q n Zp. If we assume the singularities to be at the points 0, 1, .A and oo we may take the equation to be in the form D

2

(a

b

c

+ X+ X- 1 +X- .A

)

D

eX+ f

+ X(X-

1)(X- .A) .

It is possible to compute the indicial polynomials at the singularities and

then one sees that the exponents determine the coefficients a, b, c, e but not f which is called, in Klein's terminology, the accessory parameter. If If I ~ sup(1, I.AI) then all the solutions at the generic point converge in the disk D(t, 17TI-), but there may be special values off for which the radius of convergence is larger. Not very much is known about this question. Actually this equation does not "come from geometry" in the sense that there is no integral formula for the solutions, unlike the case of the hypergeometric equation (three singularities) which we have discussed in Section IV.8. The case a= b = c = 1/2, e = -n(n + 1)/4 is known as the Lame equation and is the subject of an extensive literature (cf. Dwork [2]). If .A is algebraic over Q and n E N then there exist 2n + 1 special values of f for which there is a solution which is the square root of an element of Q[.A,J, X] and in this case the radius of convergence at the generic point is 1 for almost all valuations of Q(.A,J). Iff is algebraic over Q but not equal to one of these 2n + 1 special values then the solutions of the equation are given by the exponential of an integral of the third kind on the elliptic curve Y 2 = X(X- 1)(X- .A). In this case it has been shown in Chudnovsky [2] that global nil potence implies finiteness of the monodromy group and so under that hypothesis once again R, = 1 for almost all valuations v of Q(.A, f). For the case n = 0, the special value of f is f = 0. For f # 0 the solutions involve the exponential of an integral of the first kind and hence by the result of Chudnovsky global nilpotence is impossible. An earlier proof based on the Riemann hypothesis for elliptic curves defined over a finite field may be found in Katz [1]. We note that for f not special, 0 for almost all primes, nilpotence implies zero p-curvature, i.e. Gp mod v if v is an extension of p. Thus the result of the Chudnovskys is a special case of Grothendieck's conjecture that Gp 0 mod v for almost all p implies finiteness of monodromy. The main treatment of this conjecture has been given by Katz [2].

=

=

EFFECTIVE BOUNDS. SINGULAR DISKS

2.

159

Effective Bounds for Solutions in a Singular Disk: the Case of Nilpotent Monodromy. The Christol-Dwork Theorem: Outline of the Proof.

Consider a system 6-G

with

G E Eo,

(2.1)

where 6 = XdfdX. The origin is a regular singularity of (2.1) (cf. the end of Section 11!.8). Let A1 , ... , An be the eigenvalues of G(O) and assume that they are prepared ( cf. (11!.8.6)). From Proposition 11!.8.5 we know that there exists a solution matrix YaXG(O) of (2.1) with Ya E Gl(n,I,bcb.

This implies that Q and Q- 1 are analytic at b.

EFFECTIVE BOUNDS. SINGULAR DISKS

169

On the other hand from Lemma 2.4 and since G(l)(O) and F(O) are nilpotent, we obtain that

Q.E.D. We note that the hypothesis that G(l)(O) and F(O) be nilpotent is essential here as is shown by the example

( x1/2 o

0 ) x1/3

c

(x112 o

0 ) x2/3

which can be meromorphic at zero but not constant, with a suitable choice of C.

4.

Proof of Step IV. The Shearing Transformation.

Proposition 4.1. Let 9 E Mn(Eo) and fix an eigenvalue .A of 9(0). Then there exists a unimodular matrix H E Gf (n, K[X, x- 1]) such that ~I[H] E Mn(Eo) and the eigenvalues of 9[HJ(O) are the same as those of9(0) (counting multiplicities), except that the eigenvalue .A is replaced by .A + 1. A similar statement holds for passing from .A to .A - 1. Proof. The shearing transformation is constructed as in the proof of Lemma 111.8.2. The only new point is to prove that we may choose our shearing transformation to be unimodular. But this transformation is obtained by taking a product of a matrix of the type

where 1 is the multiplicity of .A, with a matrix C E Gf (n, K) such that C9(0)C- 1 is upper triangular. The former matrices are obviously unimodular. The following Lemma shows that we may also choose C to be unimodular. Q.E.D.

170

CHAPTER V

Lemma 4.2. Let A E Mn ( K). Then there exists C E G£ (n, Ch) such

that CAC- 1 is upper (resp. lower) triangular. Proof.

Let (recall that K is algebraically closed)

with Ixi I ~ 1 for 2 ~ i ~ n, be an eigenvector of the matrix A corresponding to the eigenvalue >. (if necessary we perform a permutation of the basis which certainly is represented by a matrix in G£ (n, Ch)). If we now put

C1=

(-~'

0)

0

ln-1

-Xn

then

_, e,

cl

=

:

Xn

0 ln-1

l

so that this C1 E Gl(n,OK)· A simple calculation then shows that

and so by induction we obtain the desired result. Q.E.D. In our situation we apply Proposition 4.1 to the matrix g = F[H.HI], and so we see that we can find a unimodular H 3 E G£ (n, K[X, x- 1 ]) such that F[H3 H 2 H,] has prepared eigenvalues. 5.

Proof of Step III. Removing Apparent Singularities.

Let Q E Mn(Eb) (recall that Eb is the quotient field of Eo). Let a E D(O, 1-). If g is analytic at a we say that a is an ordinary point of the

EFFECTIVE BOUNDS. SINGULAR DISKS

171

system D- g where D = dfdX. If a is not ordinary we say that a is a singular point of D - (}. The singular point a is said to be an apparent singularity of D- g if the system has a matrix solution at a belonging to Gf ( n, K ( (X - a))). Finally, an apparent singularity, a, of D - g is said to be a trivial singularity if the system has a matrix solution analytic at a. As an example of a system with a trivial singularity, consider the system of order 2 defined by

This system has a singularity at zero and it has 2 ( 1 x ) 0 2X

as a matrix solution at zero and this is analytic (but not invertible) at zero. Proposition 5.1. Let g E Mn(Eb) and assume that(} is analytic at zero. (i) If the system fJ- g has only apparent singularities in the punctured disk D(O,l-)\{0}, then there exists a polynomial P E K[X] having !Pigauss = 1 such that (}[Pin] is analytic at zero and fJ- (}[Pin] has only trivial singularities in D(O, 1- )\ { 0}. (ii) If the system fJ- g has only trivial singularities in D(O, 1-)\ { 0} then there exists a unimodular matrix HE Gf(n, K(X))nGf(n, K[[X]]) such that g[H] E Mn(Eo).

e

eg

Proof. (i) There exists E Eo such that E Mn(Eo). By Proposition IV.4.2 the number of zeros of in D(O, 1-) is finite. Therefore the number of poles of gin D(O, 1-) is finite. Let S be the set of poles of(} in D(O, 1-) and vb be the order of the pole of Ug,b in b. Put

P(X)

e

=II (X- b)Vb

0

bES

Then !Pigauss = 1 and the matrix (}[Pin] is again analytic at zero. Moreover the system fJ- (}[Pin] now has only trivial singularities in the punctured disk D(O, 1-)\{ 0 }.

172

CHAPTER V

(ii) Let :F be the set of all matrices Q[H] with H unimodular, H E Gf(n,K(X)) n Gf(n,K[[X]]) and such that 6- Q[H] has a solution matrix analytic at b for every 0 # b E D( 0, 1-). Note that each F E :F is analytic at zero. For F E :F let m(F)

=L

orderb (det UF,b) < oo,

b .. O

lbl 1 -

c for all c

> 0.

Q.E.D.

The following result, due to Christo!, was the first in the direction of Theorem 3.1. The final form of Theorem 3.1 is due to Andre, Baldassarri and Chiarellotto. Corollary 3.3. Let G be as in Theorem 3.2, and assume that r(Ua,t) = 1. Then r(Ya) = 1.

We now extend Theorem 3.2 to the case in which the system has trivial singularities in the punctured disk. Corollary 3.4. Let Y E Gf(n, K((X))) n Mn(K[[X]]), A E Mn(K). Let Y X A be a solution matrix at the origin of 6- G where G E Mn ( Eb). We assume that

(i) The differences a i - aj of the eigenvalues a 1 , ... , an of the matrix A all have type one. (ii) The operator 6- G has only trivial singularities in the punctured disk D(O, 1-)\{0}. (iii) The solution matrix Ua,t at the generic point t converges in D(t, R-) with R ~ 1. Then

Proof.

We repeat steps used in the proof of Proposition 2.3.

TRANSFER INTO DISKS WITH ONE SINGULARITY

221

Step 1. By the theorem of the cyclic vector there exists H1 E Gl(n, Eo) such that G[Hl]

=

( .~ ..... ~

.... ·.·.·...

~.)

...................

0 Cn

0 Cn-1

... 1 . . . C1

.

By Theorem 111.8.9, the matrix G[Hd is analytic at the origin. Step 2. Removal of apparent singularities. By Proposition V.5.1 there exists H2 E Gl(n, K(X)) with H 2, H-1 2 analytic at zero such that G[H2 Ht] E Mn(Eo). Step 3. Shearing. By Proposition V.4.1 there exists H 3 E Gl(n, K[X, x- 1]) such that T = G[H3 H2 Hd E Mn(Eo) and the eigenvalues of T(O) are prepared. Let H = H3H2H1. Then YrXT(o) is solution matrix of 6- T with Yr E Gl(n, K[[X]]), Yr(O) =In. A solution matrix at t of 6- Tis given by H Ua,t and hence Ur,t converges in D(t, R-). By Lemma V.2.4 the eigenvalues of T(O) are congruent modulo Z to eigenvalues of A and hence the eigenvalues of T(O) satisfy condition R3". Hence by Theorem 3.2, r 0 (Yr) 2:: Rn 2 • Again by Lemma V.2.4 we find that Yi 1 HY E Gl(n,K(X)) (in fact it lies in Gl(n,K[x,x- 1 ])). It follows that Y is meromorphic in D(O, (Rn 2 ) - ) . Up to this point we have made no use of hypothesis (ii). By that condition, if b E D(O, 1-)\{0} then Y(X)(X/b)A is a solution matrix at b which must be analytic at b. It follows that Y is analytic at b. Q.E.D. With the preceding proof we have actually established the following: Corollary 3.5. Under the same hypotheses as in Corollary 3.4, but

with hypothesis (ii) replaced by the weaker assumption that 6- G has only apparent singularities in the punctured disk, then Ya is meromorphic in the disk D ( 0, (Rn 2 ) - ) .

CHAPTER VII

DIFFERENTIAL EQUATIONS OF ARITHMETIC TYPE We now begin our discussion of some global properties of differential systems, that is we will study systems

Dy=Gy where G E Mn(K(X)), D = d/dX, and K is an algebraic number field (a finite extension of Q).

1.

The Height.

We fix some notation which will be maintained in the sequel. Let K be an algebraic number field and let d = [K : Q]. We denote by OK the ring of integers of K. By a prime of K we mean a valuation of K. We say that the prime is finite if the corresponding valuation is nonarchimedean, and infinite otherwise. We denote by P (resp. P 1 , Poo) the set of primes of K (resp. the set of finite and infinite primes). If v E P1 then it is the extension of a p-adic valuation of Q for a welldetermined rational prime p, and in this case we write v Ip. We use Kv to denote the completion of K with respect to v E P. If v is finite and v Ip then Kv is a finite extension of Qp and for each fixed rational prime p we have

where dv = [Kv : Qv]· If v E P1 and v Ip we assume v to be normalized so that

222

DIFFERENTIAL EQUATIONS OF ARITHMETIC TYPE

223

for all a E K, or equivalently so that

Similarly, if v E P 00 then we normalize by setting

thus

jadv

= jaji{d

if v is real, that is if Kv = R and

ifv is complex, that is if Kv =C. Here jajR and jajc denote respectively

the classical real and complex magnitudes. With these normalizations we have the Product Formula: for 0 ::j:. a E K

rr

ialv

= 1.

vEP

Let a E R, a> 0. We define log+a=logsup(1,a). If a, b E R, a, b > 0 then

sup(1, ab):::; sup(1, a) sup(1, b) so that log+(ab):::; log+ a

+ log+ b.

If a E K the (logarithmic absolute) height of a is defined by h(a) = l:)og+ jajv .

(1.1)

vEP

In view of our normalizations it is not difficult to see that h( a) depends only on a and not on the extension field containing a. Moreover from the product formula it follows that, for each a f. 0, h(a)

= h(1/a).

CHAPTER VII

224

Similarly we can also define

hJ(a)

=L

log+ lalv ,

vE1'J

For a,

(1.2)

f3 E K we clearly have h( af3)

h( a) + h(f3)

~

and similarly for h 1 and h 00 • If (a1, ... , an) E Kn we define

L

sup log+ la;lv (1.3) 19:5n and similar definitions hold for h1(a1, ... , an) and h00 (a1, ... , an)· If P(X) E K[X1, ... ,Xn] then h(P) is defined to be the height of the coefficient vector defining P. From the Gauss lemma it follows that for P, Q E K[X1, ... ,Xn] we have

h(a1, ... ,an)=

vE1'

hJ(PQ)

~

hJ(P) + hJ(Q) .

Let u be an isomorphism of K into Qalg. Then for a 1, ... , an E K and for any P E K[X 1, ... , X 3 ] we have h(af, ... , a~)= h(a1, ... , an)

h(P 17 ) = h(P) where P 17 is obtained by applying u to the coefficients. Similar equalities hold for h 1 and h 00 • If a1, ... , an E K we write

(a;)

= a;b;

,

i

= 1, ... , n

where a; and b; are integral ideals of K and (a;, b;) = 1. Let b = l.c.m.(b1, b2, ... , bn); the positive integer den (a1, ... , an) defined by

(1.4) (den(a1,··· ,an))= NK/Qb is called the denominator of a 1, ... , an. Consequently if we now set M den( a1, ... , an), we have b I M so that ( M) b b' for a suitable integral ideal b', and

=

=

(Ma;) = (M)(a;) = bb'ai/b; = b'a;(b/b;) and therefore, the last ideal being integral, M a; E () K.

DIFFERENTIAL EQUATIONS OF ARITHMETIC TYPE

225

Lemma 1.1. Let a 1 , ... , an E K. Then

Proof. Let v be a prime of K dividing the rational prime p. Since v is discrete there is an order function ordv associated to v and because of our normalizations, for a E K we have

where fv denotes the relative degree of v. Now let (a;) = a;j&; with a;, &; relatively prime integral ideals. Then

II

=

&;

v-ord.a,

v ordvlki log r

if if

>_ r , j(j < r.

1~"1 '>

if r -s 'no~.;.I log sup(R1, I( I) +constant . Oc:-i I,~ IPI, I~ I, = IPip ~flog I~ +mlpL

if s

< (lcp- j)/M

if s

2: (lcp- j)/M.

Therefore

1

v>•

logp

s 1. Let I be an idele of K with V(I) ~ 1. Consider the map

xt---+ Ax.

CHAPTER VIII

286

If I E N, I

f

0, let / 00 be the idele defined by if v is real, if v is complex, if v is finite.

Let us assume that J is such that all elements A;j of A belong to S(J); if A= 0 we take J = 1. Then in any case V(J) 2: 1. Moreover if v is real, if v is complex, if v is finite, and therefore

L j A;j x j

E S(I J N 00 ) so that the image of our map lies

in (S(IJN 00 ))M. From (3.4) it follows that

and #(S(IJNco))M:::; sup

(1, (c2V(IJN

00

))M)

We know that each of the four numbers V(J), V(N 00 ), V(I) and 2: 1, and hence

c2

is

Hence if then by the Pigeon Hole Principle there exist two distinct elements y, z E S(I)N such that Ay = Az. Thus if x = y- z then clearly Ax = 0 and x E S(J2 00 ). Let L = !200 . Then the conditions on L which assure the existence of an x E S(L)N\{0} such that Ax= 0 are the following and

G-SERIES. THE THEOREM OF CHUDNOVSKY

287

where () = c2fc1 > 1 and d = [K : Q]. This last condition is equivalent to

which also includes the first condition. Let J be such that IIJvllv = SUP;,j(1, IIA;jllv)· Then V(J)

= (htA)d.

Thus if

V(L(200)-1) > ..!._()Mf(N-M)(NhtA)dM/(N-M) cl

there exists a non-trivial solution i E S(L)N such that Ai = 0. Let us now fix avo E P 00 , and a c > 2c~l/d. If Lis such that

Lv = 1 for all v # vo Lvo = cd()Mf(N-M)(NhtA)dM/(N-M)' then there exists a non-trivial solution i E S(L)N, so that llillv ~ 1 for all v # vo and

v

= sup(1, llxlllv, · · · , llxNIIv) ~

Therefore

ht i

Lv 0

•

~ C/M/(N-M)(Nht A)Mf(N-M)

where c = 2c~l/d and 'Y = () 1 /d. If

Q.E.D.

00

y = LY•X" •=0 with

y.

E

K[[X]]n

E Kn we define

- v) =sup log + IYdv - . h(s, ~~·

CHAPTER VIII

288

For N E N we define the truncation operator TN as the K-linear endomorphism of K[[X]] such that ifi ~ N ifi > N

0

If I denotes the identity map of K[[X]] we set

In the application we will take yto be the column vector in the statement of Theorem 1.50 Lemma 3.2. Let r E (0, 1) and let

=L 00

Y.X" E K[[X]]n ,

s=O

with Ys E Kno For each N EN there exists a q E K[X]\{0}, depending on N, such that degq ~ N,

ordx'RN(qil)

h(q)

~ const + 1 ~

7

1-T

> 1 + N + N-n

(log N

,

+ L h ( N + [N 1 : vE1'

7

] ,

v))

Proof. Let q = L:f=o qiXi with q0 , 000 , qN indeterminates in Ko Then the second condition in the statement of the lemma is equivalent to requiring that the coefficients of XN+l, XN+ 2 , 000 , XN+(N(l-r)/n] in qy be zero, that is N

LYs-jqj

(305)

=0

i=O

for s = N

+ 1,0 00

,N + (N 1 ~r]o Consider the matrix of blocks

A= (

AN+l

AN+[N~l-r)/n]

)

G-SERIES. THE THEOREM OF CHUDNOVSKY

289

where the block A, is given by

As= ( Ys

Ys-N) E Mn,N+l(K) ·

Then A is an M x (N + 1) matrix where M system (3.5) can be written as

= n [N 1 ~T)

and the linear

Since N + 1 > M for all positive integers N and n

M

[N1: T]

---=----;:-~-.,... < N ( 1 - T) < 1 - T 1 N+1-n [ N

1+N-M

:T

1 +NT -

] -

T

we may apply Siegel's Lemma with N replaced by N + 1 to obtain h(q)

1-T

:S const +--(log 1 +log N + loght A) T

Since htA

:S

II sup (1, !Yo\v, ... , !YN+[N'-;;r]lv) vE'P

we have

thus obtaining the desired inequality.

Q.E.D.

4.

Conclusion of the Proof of Chudnovsky's Theorem.

Let

y satisfy the hypotheses of Theorem 1.5.

then from Lemma 3.2 it follows that

If we take

CHAPTER VIII

290

and so or dx

I P;pi

I-

y;

W

deg p- > [N(1n

r)]

-+

oo

as N -+ oo. By Proposition 2.3 this implies that if N > N 0 then det R( 0 ) f. 0 provided that P f. 0. This is indeed the case since otherwise we would have ordx (qii) = ordx (RN(qY)) 2:: 1 + N

+ [ N(1n

r)]

so that since deg q :::; N we would obtain

1- T] ordxy> [N-nand this is impossible for N sufficiently large ( ordx dependent of N).

y is a constant in-

Let Q E OK[X] be such that QG E Mn(K[X]) and let

t

= 1 + sup(degQG,degQ -1)

Our goal is now to calculate

fl.

.

in terms of y and q.

Lemma 4.1. Let r, n, N and q be as in Lemma 3.2, and let TN(qii). Then for any integers such that

P=

N1-r s r; recall that if p(r) > r, then D- G has only trivial singularities in D(O, 1-)). Thus in all cases we have

Thus, for sEN, j E Z,

Hence, for s E Nx, z = jfs, we have log M { 1

i.e.

forM~

IA•,sz I

1 } :::; slog-() 1 s, n- P pr

M1,

1

!M(z):::; log-() p r

1

Hence

log-() ~sup pr zEQ which gives u(r)

~

p(r).

1

+ sz log-r 1

+ zlog-. r

(fM(z)- z r1) log-

1

=log-() u r

Q.E.D.

Corollary. Let Ro:::; r1 < r2:::; 1. If p(r;) = r; (i = 1, 2), then p(r) = r for all r E [r1, r2].

Proof. Let l; be the line of contact of slope log (1/r;) ( i = 1, 2). Let P be the point of intersection of £1 and £2, and let lr be the line through P of slope log ( 1/ r). Then lr lies above the polygon and hence lies above the line of contact of the same slope. Thus if q is the Y -intercept of lr then 1 q ~log u(r) .

APPENDIX I

306

In the present situation both £1 sand £2 pass through ( -1, 0). Hence q =log (1/r) and so o-(r) ~ r. But o-(r) > r implies o-(rt) = o-(r) > r > r 1 a contradiction. Hence r = o-(r) as asserted. Q.E.D. Remark. By the same argument one can deduce that log (1/ p(r)) is a convex function of log (1/r) for r E [R 0 , 1] (see Christol [2]) and indeed the Corollary is a consequence of this convexity (to avoid any ambiguity regarding what we mean by a convex function, we cite f( x) = x 2 as an example of such a function). It seems reasonable to believe that Christol's convexity theorem implies the Theorem of this Appendix. It would be enough to define

g(z) =

inf

mE[m,,m 2 ]

(bm

+ mz)

for all z E R. Then show that if m --+ bm is a convex function on [m 1 , m 2 ] then g(z) is a convex function of z and then finally show that, formE [mt, m2], we have bm = supz(g(z)- mz). The variation of p(r) with r has been studied by P. Robba (see Robba [2]) with a view towards calculation of index. A posthumous work of Philippe Robba, Cohomologies p -adiques et sommes exponentielles, should appear shortly in the series "Travaux en cours" of Hermann and Co. We refer the reader to that work for further discussion of index calculations.

APPENDIX II

ARCHIMEDEAN ESTIMATES In this Appendix 1-1 will denote the classical complex absolute value. If A= (Aij) E Mn(C) we shall denote by lA I the sup of the IAij I· Likewise if v E en' we put lv I = SUP; lv; I· But when viewing A as a transformation of en' we must recognize that

IAvl = IIAIIIvl with

IIAII :S niAI.

Likewise, if A, B lie in Mn(C) then

IABI :S niAIIBI. Let G E Mn(C[[X]]) and suppose that G converges in the closed disk D(O, R+) in C. We assert that the matrix W E Mn(C[[X]]) constructed in the proof of Proposition 111.8.5 converges in the open disk D(O, R-). We use the notation of that proof. By hypothesis there exists M > 0 such that IGj I :S M / Ri for all j 2: 0. For m 2: 0 let

the choice of O:m to be made later. (Recall that ml - 1/Ja 0 is a transformation of Mn(C), that is of en,.) From the relation

we deduce

IWml :S O:m (IG1 Wm-11 + · · · + IGm-1 W1l + IGml) :S O:m (niG1IIWm-1l + · · · + niGm-1IIW1I + IGm I)

(nM

M)

nM - 0. Thus form> m 0 we have

ll(ml- 'lfaa)- 1 11 ~ n2 l(ml- 'lfaa)- 1 1

=


mo, where k'

and as desired.

= 2kn 2 • Thus

2kn 2

k'/m

APPENDIX III CAUCHY'S THEOREM

Let K be a field of characteristic zero and consider the system of ordinary differential equations dY;

dX =/;(X, Yt, ... , Yn)

with /;(X, Yt, ... , Yn) E K[[X, Yt, ... , Yn]J, fori= 1, ... , n. We seek solutions Yt(X), ... , Yn(X) E K[[X]] of the system with initial conditions y;(O) = 0 for i = 1, ... , n. We will write the system in vector form

dY

dX

~

~

= f(X,Y)

(1)

where

f(x, Y) = ( Then we seek

Y= such that

dfi dX

en ~

= /(X, ii)

ft(X,

:

Y)) ~

.

fn(X, Y)

E (K[[X]])"

'

fi(O) = 0.

We assert that there exists one and only one such formal solution. Clearly, taking ii = lim, iis in the X -adic topology, this is equivalent to the existence and uniqueness of a sequence {ii. }s>o, where iio = 0 and iis E (K[XJt is a polynomial of degrees, such that

ii.

=iis-t

310

mod

x•

APPENDIX III

311

and that

dfi. - ) dX -= f-(x ,y,

mo d x•

(2)

for all s > 1. To prove the existence and uniqueness of such a sequence we proce;d by induction. Let fis = fi,_ 1 +B. X• with B. E Kn. Then

dfi. - dfis-1 jj x•-1 dX- dX +s • so that, since clearly

j{x, ii.)

s B- s x•-

=j(x, fi,_

- )= f-(X ,Ys-1

1 -

mod

1)

1 dfisdX

x•,

we have

mod x• .

From (2) it follows that

We now write

_ -

A'2 2 A'3 3

y=A1X+-X +-X+··· 2! 3! and determine the

A'1 A2

=

Aj 's using Taylor's formula.

j(o, o, ... ,o)

d2fi I d 2 X=O = dX = dX

() f(X, iJ)

Clearly we have

I

X=O

ol ~ of dy; ) = ( oX (X, iJ) + ~ oY; (X, iJ) dX

I

X=O

- +~ - .!'lv. of (X, Y) - ) = ( of X (X, Y) L..J /;(X, Y) .Q

0

V~i

i= 1

V

o)-1

= ( ox + ~ f; oY; n

(3)

I

.-. _

(X,r )-0

l(x,Y)=o '

and, in general, by induction, we find

(4)

312

APPENDIX III

From ( 4) it easily follows that if we write

J(x, Y) then, for each s

~

=

"" CI' Xl'oyl'l L..J 1

... yl'n n

(5)

'

1,

- - ( - )h, ... )

A, = P,

... , ( C ~"

whereP, ( ... ,(CI')h, ... ) isauniversalpolynomialinZ[ ... ,(C:)h, ... ] with positive coefficients.

The Classical Case. We take K = C, the field of complex numbers. For i = 1, ... , n we consider functions /;(X, Y1, ... , Yn), which are assumed to be analytic in the polydisk D(O, a+) x (D(O, b+))n C C"+ 1 where a, b E R+, the positive real numbers. Cauchy's problem consists in proving the convergence in a non-trivial disk of the formal solution y obtained above. After scaling the variables we may assume that a= b = 1. Let J(x, Y) be as in (5) and let

g(X, Y) =

L

BI'XI'DY{' · · · Y,i'n E R+[[X, Y1, ... , Yn]]

1

I'ENn+l to be explicit, (BI')j is a positive real number for all 11 and j. We say that g dominates land write l-< g if for each 11 we have

for j = 1, 2, ... , n, that is each component of every coefficient in g is greater than or equal to the absolute value of the corresponding coefficient component in {. Let us suppose that

l-< g and let

x•

z= L:A.-, s. 00

•=1

-

313

APPENDIX III

be the formal solution of the system

dY

-

(6)

dX = g(X,Y)

with initial conditions z(O) = 0. Therefore

A, = ( a8x + and clearly

t.

8 g; 8 Y; )'-' g

l(x,i')~o

.4. = ft. ( ... , (.8~-' )h, ... )

whence

I(A.)j I ~

z

(A.)j ,

and so if the solution of system (6) is convergent on some disk centered at zero, then the same is true of fj. By hypothesis f is convergent for lx I ~ 1, IY1I ~ 1, · · · , IYn I ~ 1, so that there exists a positive M such that

I(Cp)hl ~ M for all J.L E Nn+l and h = 1, ... , n. If we set

g= then clearly

(~) ~

1 (1 - X)(1- Yl) .. · (1- Yn)

g dominates f

The system (6) then becomes M (1- X)(1- Y1) .. · (1 - Yn)

dYi dX

(7)

fori= 1,2, ... ,n, Since the right hand side of (7) is independent of i, if z1 , ... , Zn are the formal solutions to (7) with initial conditions z;(O) = 0, then we have Zl = · · · = Zn = Z where z E C[[X]] satisfies the equation dZ dX

M

(1 - X)(1 - z)n with initial condition z(O) = 0. Therefore we obtain (1-

zt+ 1 = 1 + M(n + 1) log(1- X)

so that z = 1- {1

+ M(n + 1) log(1- X)} 1 /(n+l)

which clearly converges on a suitably small disk centered at zero.

APPENDIX III

314

The p-adic Case. We now assume f{ to be a complete p-adic field. By a function analytic on the disk D(O, a+), a> 0, we mean a power series

L qjXj E K[[X]] 00

f(X) =

j=O

such that Jqi Jai --+ 0

as j --+ oo .

The key point in what follows is that the set of all functions analytic on D(O, a+) forms a Banach space Ba under the norm

IlL I = s~p 00

qjXi

j=O

Jqi Jai .

J

The operator

is a linear map and

so that

A function analytic on the polydisk D(O, a+) is a power series quX"'oylul ... Yn"'n uENn+l

X

D(O, b+)n, a, b > 0,

L

satisfying

JquJauabul+··+un--+ 0

as uo + u1 + · · · + Un --+ oo. The set Ba,b of functions analytic in the above polydisk is a Banach space under the norm

II

L

uENn+l

q.. xuaytl ...

y;n II = s~p Jq.. Ja"'obul+··+un

APPENDIX III

315

and

Moreover the evaluation map

Ba,b f(X, Y) is a linear map with norm

~

-n I-+

f(O, 0)

1, and the multiplication map f

I-+

fg

where g E Ba,b is a linear map with norm equal to

IIYII·

We now assume the J;(X, Y) in (1) to be elements of Ba,b· Therefore there exists M > 0 such that llfll ~ M, that is llf;ll ~ M for i = 1, 2, ... , n. Then the linear map

f/X + t J; {/¥; satisfies i=l

so that, by (4) we obtain

IAil ~

M))i-l

1 M ( sup ( ~'b

If, as usual, 1r is defined by the condition it follows that

1rp-l

so that fi(X) converges for

lxl < Remark 1.

'

sup(~,~)

In the case of a linear system

aY

-

dX = G(X)Y

= -p, then from (11.4.6)

316

APPENDIX III

where G(X) is ann x n matrix analytic in D(O, r) one may find a solution

with an arbitrary choice of the vector A0 as one sees by setting = y- A0 and applying the result just proved (classical or non-archimedean) to the system

z

di = G(X)Z- + G(X)Ao -

dX

.

Remark 2. The first treatment of the non-archimedean Cauchy Theorem was given by Lutz [1). Later the topic was considered by lgusa [1), Clark [1) and Serre [1).

BIBLIOGRAPHY Adolphson, A., Dwork, B. and Sperber, S. [1] Growth of Solutions of Linear Differential Equations at a Logarithmic Singularity, Trans. Amer. Math. Soc. 271 (1982), 245-252. Amice, Y. [1] Les nombres p-adiques, Presses Universitaire de France, collection SUP 14, Paris, 1975. Andre, Y. [1] G-functions and Geometry, Aspects of Mathematics E3, Viehweg, Braunschweig/Wiesbaden, 1989. Artin, E. [1] Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York, 1967. Baldassarri, F. and Chiarellotto, B. [1] On Andre's transfer theorem, Contemporary Math. 133 (1992), 2537. Berthelot, P. and Messing, W. [1] Theorie de Dieudonne Cristalline, I, in Journees de Geometrie Algebrique de Rennes (Juillet 1978), Asterisque 63, Soc. Math. de France, 1979, 17-38. Bombieri, E. [1] On G-functions, in Recent progress in Analytic Number Theory, vol. 2, Academic Press, New York, 1981, 1-67. Christo!, G. [1] Un theoreme de transfert pour les disques singuliers reguliers, in Cohomologie p-adique, Asterisque 119-120, Soc. Math. de France, 1984, 151-168. [2] Modules differentielles sur des couronnes, Ann. Inst. Fourier (Grenoble), to appear. 317

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Christol, G. and Dwork, B. [1] Effective p-adic bounds at regular singular points, Duke Math. J. 62 (1991), 689-720. Chudnovsky D. and Chudnovsky G. [1] Applications of Pade Approximations to diophantine inequalities in values of G-functions, in Number Theory. A Seminar held at the Graduate School and University Center of the City of New York 1982, Lect. Notes in Math. 1052, Springer-Verlag, Berlin, 1984, 1-51. [2] Applications of Pade Approximations to the Grothendieck conjecture on linear differential equations, in Number Theory. A Seminar held at the Graduate School and University Center of the City of New York 1983-84, Lect. Notes in Math. 1135, Springer-Verlag, Berlin, 1985, 52-100. Clark, D.N. [1] A note on the p-adic convergence of solutions of linear differential equations, Proc. Amer. Math. Soc. 17 (1966), 262-269. Davenport, H. [1] Multiplicative number theory, Grad. Texts in Math. 74, SpringerVerlag, Berlin, 1980. Deligne, P. [1] Equations differentielles a points singuliers reguliers, Lect. Notes in Math. 163, Springer-Verlag, Berlin, 1970. Dwork, B. [1] On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631-648. [2] On the zeta function of a hypersurface, Inst. Hautes Etudes Sci. Pub/. Math. 12 (1962), 5-68. [3] Differential operators with nilpotent p-curvature, Amer. J. Math. 112 (1990), 749-786. Dwork, B. and van der Poorten, A. [1] The Eisenstein constant, Duke Math. J. 65 (1992), 23-43. Dwork, B. and Robba, P. [1] Effective p-adic bounds for solutions of homogeneous linear differential equations, Trans. Amer. Math. Soc. 259 (1980), 559-577. Hardy, G.H. and Wright,E.M. [1] An introduction to the theory of numbers, Clarendon Press, Oxford, 1960.

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319

Honda, T. [1] Algebraic Differential Equations, Symposia Mathematica 24 (1981), 169-204. Igusa, J. [1] Analytic groups over complete fields, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 540-541. Katz, N. [1] Nilpotent connections and the Monodromy Theorem: application of a result of Turritin, Inst. Hautes Etudes Sci. Publ. Math. 35 (1970), 175-232. [2] Algebraic solutions of Differential Equations (p-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1-118. Lang, S. [1] Algebraic Number Theory, Addison-Wesley, Reading, 1970. Leopoldt, H.-W. [1] Zur Approximation des p-adischen logarithmus, Abh. Math. Sem. Hamburg 25 (1962), 77-81. Lutz, E. [1] Sur l'equation y 2 = x 3 - Ax- B dans les corps p-adiques, J. Reine Angew. Math. 177 (1937), 238-247. Poole, E.G.C. [1] Introduction to the Theory of Linear Differential Equations, Oxford University Press, Oxford, 1936. Robba, P. [1] Index of p-adic differential operators, III: Application to twisted exponential sums, in Cohomologie p-adique, Asterisque 119-120, Soc. Math. de France, 1984, 191-266. [2] lndice d'un operateur differentiel p-adique, IV: Cas des systemes. Mesure de l'irregularite dans un disque, Ann. Inst. Fourier (Grenoble) 35 (1985), 13-55. Schmitt, H. [1] Operators with Nilpotent p-curvature, Proc. Amer. Math. Soc., to appear. Serre, J.P. [1] Lie Algebras and Lie Groups, Benjamin, Reading, 1965.

INDEX

accessory parameter, 158 additive character, 56 trivial, 56 additive radius of convergence, 42 Amice ring, 129 analytic element, 128, 129 analytic in an annulus, 131 analytic function, 117 bounded, 117 annulus of convergence, 47 apparent singularity, 171 Artin-Hasse exponential, 55 associated scalar operator, 87

derivation, 77 non-trivial, 77 Dieudonne's Theorem, 54 differential extension, 79 differential field, 77 differential module, 86 cyclic, 87 nilpotent, 88 differential operator, 78 globally nilpotent, 98 nilpotent, 81 trivial, 79 differential system, 86 nilpotent, 88 reducible, 280 trivial, 87 dilogarithm, 125 discrete valuation, 21 Dwork-Frobenius Theorem, 156

binomial series, 49, 97, 140 Bombieri condition, 226 local, 234 Bombieri estimate, 261 boundary seminorm, 117 bounded polygon, 303

effective growth Theorem of, 120 entire function, 68 equivalent norms, 9 valuations, 4 exponential series, 50 exponents of an operator, 84, 98 of a system, 113 extension of a valuation, 17

Cauchy estimates, 114 Cauchy sequence, 6 characteristic of a field, 4 Chudnovsky's Theorem, 267 complete field, 6 completion, 7 cyclic differential module, 87 cyclic vector Theorem of, 89 denominator, 224 dense, 7

field, 3 321

322

Frobenius automorphism, 32 Fuchs' Theorem, 101 G-function, xiv G-operator, xiv irreducible, xiii G-series, xiii, 264 Galockin condition, 227 Gauss norm, 10 Gauss' Lemma, 10 generic disk, 93 generic global inverse radius, 226 generic point, 93 global inverse radius, 234, 241 generic, 226 globally nilpotent operator, 98 Hadamard product, 202 height (logarithmic absolute), 223 (multiplicative), 284 Hensel's Lemma, 11 Heun's equation, 157 hyper geometric equation, 150 function, 150 indicial polynomial, 84, 98 induced topology, 3 inertial subfield, 30 irregular singularity, 101 Katz-Honda Theorem, 84 Katz's Theorem, 99 Lame equation, 158 Laurent series, 46 line of contact, 304 line of support, 28 Liouville number (p-adic), 200 logarithm, 38, 102

INDEX

matrix solution, 87 maximum modulus principle, 116 meromorphic function, 68, 117 monodromy map, 100 Newton's Lemma, 13 Newton polygon of a polynomial, 24 of a power series, 42 of a Laurent series, 47 nilpotent differential module, 88 differential operator, 81 differential system, 88 see also globally nilpotent operator norm, 8 normed space, 8 order function, 24 of an operator, 79 ordinary point of a system, 170 Ostrowski's Defect Theorem, 21 p-adic analytic function, 117 entire function, 68 integer, 7 Liouville number, 200 meromorphic function, 68 number, 7 valuation, 6 prepared eigenvalues, 106 prime, 222 complex, 223 finite, 222 infinite, 222 real, 223 product formula, 223

INDEX

radius of convergence additive, 42 of a differential system, 94 of a Laurent series, 254 of a power series, 41 ramification index, 19 reducible system, 280 regular singularity of an operator, 84, 98 of a system, 101, 113 residue field, 5 relative degree, 19 Rouche's Theorem, 131 shearing transformation, 105, 169 Shidlovsky's Lemma, 270 Siegel's Lemma, 284 singular point of a system, 171 singularity, 171 apparent, 171 trivial, 171

323

Teichmiiller representative, 56 totally ramified extension, 30 trivial character, 56 differential operator, 79 differential system, 87 singularity, 171 valuation, 3 truncation operator, 288 type of a number, 199 uniform part, 108 unimodular matrix, 162 unramified extension, 30 valuation, 3 p-adic, 6 trivial, 3 valuation group, 5 valuation ideal, 5 valuation ring, 5

SlZe

of a local solution, 243 of a matrix, 227 splitting of an operator, 81 tamely ramified, 34 Taylor's formula, 12

wildly ramified, 34 Wronskian determinant, 77 matrix, 77 zeta function, 61

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