Introduction to p-adic Analytic Number Theory 082183262X, 9780821832622

Table of contents :
Contents
Preface
Chapter 1. Historical Introduction
Chapter 2. Bernoulli Numbers
Chapter 3. p-adic Numbers
Chapter 4. Hensel's Lemma
Chapter 5. p-adic Interpolation
Chapter 6. p-adic L-functions
Chapter 7. p-adic Integration
Chapter 8. Leopoldt's Formula for L_p(1, χ)
Chapter 9. Newton Polygons
Appendix: Newton Polygons and Galois Groups
Chapter 10. An Introduction to lwasawa Theory
Bibliography
Index

Citation preview

AMS/IP

Studies in Advanced Mathematics Volume 27

Introduction to p-adic Analytic Number Theory M. Ram Murty

American Mathematical Society

•

International Press

i?

Introduction to p-adic Analytic Number Theory

Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 11-01, 11-02, 11E95, 11Sxx.

The quote on page v from Swami Vivekananda is reprinted with permission, from the Complete Works of Swami Vivekananda, Vol. 2, p. 227, Advaita Ashrama 5, Dehi Entally Road, Kolkatta 700 014, India. @ Advaita Ashrama.

Library of Congress Cataloging-in-Publication Data Murty, Maruti Ram. Introduction to p-adic number theory / M. Ram Murty. p. cm. - (AMS/IP studies in advanced mathematics, ISSN 1089-3288; v. 27) Includes bibliographical references and index. · ISBN 0-8218-3262-X (alk. paper) 1. Number theory. 2. p-adic analysis. I. Title. II. Series. QA241 M85 2002 512'.74-dc21

2002025584

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission©a.ms. org.

© 2002 by the American Mathematical Society and International Press. All rights reserved. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. Printed in the United States of America.

i§

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.a.ms.org/ "visit the I~ternational Press home page at URL: http://www. intlpress. com/ 10987654321

07 06 05 04 03 02

One life runs through all like a continuous chain, of which all these various forms represent the links, link after link, extending almost infinitely, but of the same one chain. Vivekananda

Contents Preface

ix

1 Historical Introduction

1

2 Bernoulli Numbers

9

3 p-adic Numbers

25

4 Hensel's Lemma

43

5 p-adic Interpolation

57

6 p-adic £-functions

71

7 p-adic Integration

87

8

Leopoldt's Formula for Lµ(l, x)

9

Newton Polygons

99

113

10 An Introduction to Iwasawa Theory

135

Bibliography

145

Index

148

vii

Preface Following Hensel's discovery in 1897 of the p-adic numbers, Ostrowski's theorem of 1918 classifying all the possible norms that one can define on the rational numbers has been the starting point of what is now called the adelic perspective. Loosely speaking, this means that the rational numbers should not be thought of as merely a subset of the real numbers but rather as a subset of a spectrum of topological fields obtained by completing the rational number field with respect to each of the possible norms. This adelic perspective has been a unifying theme in number theory ever since its conception in the 20th century. From this vantage point, all completions of the rationals must be treated "equally." It would seem then that what is traditionally called analytic number theory looks at only one completion, and ignores the non-archimedean (or p-adic) completions. In the latter half of the 20th century, this restricted viewpoint was enlarged through the foundational work of Kubota and Leopoldt and later by Iwasawa who established much of the groundwork of a p-adic analytic number theory. Thus, the search for p-adic incarnations of the classical zeta and £-functions is of relatively recent origin and has been a useful motif in the study of special values of various £-functions and their arithmetic significance. This perspective has also been a fertile program of research, largely inspired by the method of analogy with the archimedean context. As such, it is an exciting program to discover to what extent p-adic analogues of classical theorems and in some cases, the classical methods, exist. This monograph is a modest introduction to the p-adic universe. Its aim is to acquaint the non-expert to the basic ideas of the theory and to invite the novice to engage in a fecund field of research. It grew out of a course given to senior undergraduates and graduate students at Queen's University during the fall semester of 2000-2001. That course was based on notes of a shorter course given at the Harish-Chandra Research Institute on p-adic analytic number theory from December 22, 1999 till January 12, 2000. The prerequisites have been kept to a minimum. My goal is to introduce the novice student to a beautiful chapter in number theory. A background in basic algebra, analysis, elementary number theory and at least one course in complex analysis should suffice to understand the first nine chapters. The last chapter requires a little bit more background and is intended to give some inspiration for studying the subject at a deeper level. There are more than a hundred exercises in the book and their level varies. Most of them are routine and the students in the course were able to keep pace by doing them. ix

X

PREFACE

I would like to thank Alina Cojocaru, Fernando Gouvea. Hershy Kisilevsky, Yu-Ru Liu, Kumar Murty, D.S. Nagaraj, and Lawrence Washington for their careful and critical reading of preliminary versions of these lecture notes. I also thank the students and post-doctoral fellows at Queen's who participated in the seminar and course out of which the monograph was born. M. Ram Murty Kingston, Ontario, Canada January 2002

Chapter 1

Historical Introduction There is a remarkable historical parallel between the proof of the prime number theorem and Fermat's Last Theorem. The prime number theorem is the assertion that the number of primes p s; x, denoted 7r(x), is asymptotic to x/logx as x--+oo. Fermat's Last Theorem (FLT) predicts that the equation xP

+ yP = zP

for odd primes p has no non-trivial integer solutions. This was recently resolved by the work of Wiles and Ribet using a galaxy of ideas and techniques that can be traced to diverse areas of mathematics, most notably, number theory, algebraic geometry and representation theory. It was in 1860 that Riemann wrote his fundamental ( and only) paper in number theory outlining how one should prove the prime number theorem. The main tool, he wrote, should be the use of the (-function, defined as 00

((s) =

1

L ns'

n=l

as a function of a complex variable s. Riemann observes that the above series which converges absolutely for Re( s) > 1 is connected to prime numbers via the Euler product: (1.1) where the product is over prime numbers p. (1.1) can be viewed as an analytic formulation of the unique factorization theorem in the ring of integers Z. In the same paper, Riemann shows that ((s) admits an analytic continuation for all complex values of s E (C apart from s = 1 where it has a simple pole. Moreover, he derives the functional equation 7r-s/ 2r(s/2)((s)

= 7r-(l-s)/ 2r((l - s)/2)((1 - s).

He then writes down, without proof, a remarkable explicit formula for 7r(x) in terms of the zeros of the ( function and explains that the main term x / log x

1

CHAPTER 1. HISTORICAL INTRODUCTION

2

really comes from the simple pole at s = 1. As is well-known, Riemann suggested that all the zeros p of ((s) in the region O Re(s) 1 satisfy Re(p) = 1/2. This Riemann hypothesis is still an unsolved problem.

s

s

Most of Riemann's statements were simply assertions and his paper came to be viewed as a program to prove the prime number theorem. In fact, in the three decades after Riemann's paper had appeared, Hadamard had developed his theory of entire functions largely with the goal of proving the prime number theorem. Finally, in 1895, Hadamard and de la Vallee Poussin had independently proved the prime number theorem. The key ingredients in their work were the analytic continuation of ( (s) and the non-vanishing of ((s) on Re(s) = 1.

It is rather striking that around the same time, Kummer was developing his fundamental ideas on cyclotomic fields. His celebrated theorem that FLT holds for all primes p which are coprime to the class number of the cyclotomic field Q((p) was proved in a series of papers beginning in 1847. But how does one determine if p divides the class number hp of Q((p). Kummer derived a very beautiful criterion for this. Define the Bernoulli numbers Bn by the power series

Then, B 2 k+l

= 0 for

Bo= l,

B1

Bs = -1/30,

k

~

l and

= -1/2,

B2

B10 = 5/66,

= 1/6, B12

B4

= -1/30,

= -691/2730,

B6 B14

= 1/42, = 7/6, ...

Earlier, Euler showed that B2k =J O and related them to the value of ((2k). Kummer proved that p divides hp if and only if p divides the numerator of some Bk, k = 2, 4, 6, ... ,p - 3. For example, xl3

+ yl3 = zl3

has no non-trivial integral solutions because 13 does not divide the numerators of B2k for k S 5. A prime pis called a regular prime if pis coprime to hp. It is still unknown if there are infinitely many such primes. Kummer derived his criterion for regularity by proving his famous congruences for the Bernoulli numbers: if

CHAPTER 1. HISTORICAL INTRODUCTION n

3

"¥- O(modp - 1) and m = n(modpa(p - 1)) with m, n even, then

(1.2)

(l _ pm-l) Bm m

= (l _

pn-l) Bn (modpa+l ). n

For a= 0, the Kummer congruences take the simpler form

Bm=Bn (modp ) m n

(1.3) whenever m

= n "¥- O(modp -

1), with m, n even.

Building on Kummer's work, Hensel developed systematically the theory of p-adic number fields in the early 1900's. He discovered the p-adic norm on the field of rational numbers and completed the field with respect to this norm. The full significance of this construction did not become clear until Ostrowski proved in 1918 his celebrated theorem, that essentially the only norms that can be defined on (Ql are the p-adic norms and the usual norm given by the absolute value. The familiar construction of the real numbers IR and the complex numbers

kk /kl,

we deduce that

so that (2.2) as k--+oo. In view of the functional equation for the Riemann zeta function, Euler's formula can be rewritten in a simpler way as

(2.3) and it is this formulation that gains significance in the 1964 work of Kubota and Leopoldt .

CHAPTER 2. BERNOULINUMBERS

12

There is an amusing way of deriving (2.3). We have

1 -1--et - 1

from which (2.3) easily follows. Of course, none of the sums above are convergent and the argument is not rigorous. However, the argument can be made rigorous in the following way. Recall that the f-function is defined for Re(s) > 0 by the integral r(s)

=

1=

e-xxs- 1 dx.

By integration by parts, we see that r( s) satisfies the functional equation r( s + 1) = sf( s) which can be used to derive the meromorphic continuation of it for all s E C. By an application of Stirling's formula and the calculus of residues, we have 1 12+i= e-x = -. x-sr(s)ds,

2-i(X)

21ri

since r(s) has simple poles at s = -k fork= 0, 1, 2, ... and residue (-l)k /k! at s = -k. If now F(s) = I::~=l an/n 8 is a Dirichlet series absolutely convergent for Re(s) > 1, then we get

L ane-nx = -1. 12+i= 2-i= 0()

n=l

In particular,

X- 8 F(s)r(s)ds.

21ri

L e-nx = -.1 12+i= x-sc(s)r(s)ds. 2-i= 0()

n=l

27TZ

The left hand side is

1 eX - l By moving the line of integration to the left to -oo, the right hand side becomes

(2.4)

1

~

-+ L X

k=O

xk(-ll((-k) k'. .

Moreover, by the functional equation and Stirling's formula,

1((-k)I :S

Ceklogk_

CHAPTER 2. BERNOULINUMBERS

13

As kl"' eklogk-k+O(logk), we deduce that the power series in (2.4) converges in a small enough disc so that we can compare coefficients and deduce (2.3). This argument can be extended to obtain similar results for the Dirichlet £-functions, and Hurwitz zeta functions. The precise details are contained in a paper by the author and Marilyn Reece [MR]. Though the 'naive' derivation above leads to the correct formula but requires complex analysis, analytic continuation and functional equation of the (-function for its justification, it is possible to derive the analytic continuation and derive (2.3) in a much more elementary fashion as described in [MR]. Recall that the Riemann (-function, originally defined by the Dirichlet series

((s)

=

f ~'

n=l

n

for Re(s) > 1. Let us denote by [x] the number of positive integers less than or equal to x. By partial summation,

((s)

= s J1

oo xs+l [x] s Joo xs+1 {x} dx = s - l - s dx, 1

where {x} denotes the fractional part of x. This gives the analytic continuation of ((s) for Re(s) > 0. We can now proceed inductively. Writing

{x} oo 1n+l -x -- dn x - oo J oo --dxxs+l - ; n xs+l - ;

11

1

udu

o (u+n)s+l

and integrating the last integral by parts, we get

~

[1

~lo

udu _ !~ 1 ( s + 1) (u+n) 8 +l-2~(n+l) 8 +1 + 2

Joo {x } dx 2

xs+

1

2

and the latter integral converges for Re( s) > -1. That is,

((s) = _s_ - ~(((s + 1) -1) - s(s + 1) 2 2 s -1 from which we infer that ((0)

((s)

= 1+ s ~ 1 -

t

= -1/2.

loo {x}2 dx x + 1

8

2

Thus, inductively, we deduce

s(s + l)(~·}:)7 r - l) ( ((s + r) - 1)

_ s(s + 1) · · · (s + m) ~ (m + 1)!

~

[1

lo

um+ 1 du (u + n)s+m+l

CHAPTER 2. BERNOULINUMBERS

14

and the infinite sum on the right hand side converges for Re( s) > -m. This derivation, though not well-known, is not new. For example, it is found in [Ap]. However, the idea of deriving the special values of ((1 - k) from it, seems to have not been noticed before. Indeed, if in the above, we put s = l - m, and note that for r = m, ((s + m) has a simple pole at s = l - m, we obtain the recurrence

1 (-l)m m-l r m - l 1 ((1-m) = 1- - + - - - "'(-1) ( )-(((l-m+r)-1). m m(m+l) ~ r r+l r=l By integrating the expression

m-1

I: (m;1)(-1rxr

(l-x)m-1 =

r=O

we deduce

l-(1-xr =m-l(m-1) -1

L

m

r

(

r=O

Putting x

rxr+l

)

r+1·

= l gives 1 m

~(m-1)(-lt ~ r r+l r=O

so that we obtain: Theorem 2.1 For positive integers m,

( l)m m-1 1 1 m( (1 - m) = ( )- m -1 r ( m (1 - m + r). m+l r r+l r=l

I: (

Since

we easily deduce the recurrence

for n

~

l. Our goal now is to prove:

Theorem 2.2 For k

~

2,

)-c

15

CHAPTER 2. BERNOULINUMBERS

To this end, we will apply induction using Theorem 2.1. Since

is an even function, the Bernoulli numbers for odd subscripts 2: 3 vanish, and we can rewrite the above equation as

((1- k) = (-l)k-l~k. Then, by induction, we see that the right hand side of the formula in Theorem 2.1 becomes

It is immediate that this reduces to

by virtue of the fact that ((0) = -1/2. The result is now immediate. Our goal now is to derive the Kummer congruences and lay the foundations for the p-adic interpolation of the Riemann zeta function. We will say that a rational number a/b with (a, b) = 1 is p-integral if p is coprime to b. Given any rational number r, we can write it as puai/b 1 with (p, a 1 b1 ) = 1 which determines the integer u uniquely. We define ordvr to be u. Sometimes we also use the notation vp ( r) to designate u. We now begin with: Lemma 2.3

Proof.

Let p be a prime number. Then pBk is p-integral.

We induct on k. We set n =pin (2.1) and write it as

Since pi 2: 2i 2: i + 1, we see that ordv(Pi/i + 1) 2: 0 and by induction, each of pBk-i is p-integral in the sum. Since Sk (p) is an integer, it follows that pBk is p- integral. •

Corollary 2.4

Let Bk

= Uk/Vk with (Uk, Vk) =

l. Then

Vi is squarefree.

CHAPTER 2. BERNOULINUMBERS

16

Proof. If p2 jVk for some prime p, then pBk will still have a denominator divisible by p. •

We can sharpen the argument of Lemma 2.3 to deduce Lemma 2.5

If k is even and 2 2, then

pBk Proof.

= sk(P)(modp).

If k is even and 2 4, then k - l is odd and 2 3 so that

By an easy argument (see Exercise 3) as before, pi-l /(i + 1) is p-integral and by Lemma 2.3, we obtain sk(P) = pBk(modp), for k 2 4. For k = 2, B 2 = 1/6 and the result is verified directly. • We also have the elementary: Lemma 2.6 -l(modp). Proof.

If (p - l)fk, then sk(P)

= O(modp).

If (p - l)lk, then sk(P)

=

Let g be a primitive root mod p. Then, if p - l does not divide k, p-2

sk(P)

= "'(gjl = L..., j=O

gk(p-1) -

k

g -l

1

(modp)

and the numerator is O mod p. In the case (p- l)lk, we find by Fermat's little theorem that the sum sk(P) = p - l = -l(modp) as desired. • We can now deduce the famous Theorem 2. 7

(von Staudt - Clausen) Fork even,

Proof. If p - l does not divide k, then Lemma 2.6 implies that sk(P) O(modp) so that by Lemma 2.5, p does not divide Vk, the denominator of Bk, If p - l divides k, then again by Lemma 2.6, sk(P) = -l(modp) so that by

CHAPTER 2. BERNOULINUMBERS

17

Lemma 2.5, we get pBk = -l(modp). Therefore, p necessarily divides Vk. Thus, the primes dividing Vk are precisely those primes p such that p - llk. Hence, 1 Bk+P

is p-integral for every p such that (p - 1) Ik which completes the proof.

•

Remark. It is suggestive to think of Bk as having 'simple poles' at primes p such that (p - l)lk and the sum in the theorem can be viewed as the 'polar' part of Bk. Also observe that 6JVk for all even k 2: 2.

We now deduce the Voronoi congruences from which the Kummer congruences will follow. It seems that Voronoi proved them while a student in 1889. Theorem 2.8

Proof.

Let k be even and 2: 2. Then,

As before, we write from (2.1),

(2.5) The sum

is p-integral for p 2: 3 since pi- 2 2: 3i- 2 2: 1 + 2( i - 2) = 2i - 3 2: i + 1. To prove the theorem, we study p-integrality of the expression for primes pin. Also for p 2: 5, the term corresponding to i = 2 is

is p-integral. Multiplying the equation (2.5) by Vk and recalling that 6JVk, we see that Vk(~)Bk-2i

is 3-integral when 3ln. Finally, note that 2i- 2 2: (i

+ 1)/2 for i 2: 4 so that

CHAPTER2.BERNOULINUMBERS

18

is 2-integral when 2ln since 21Vk· In addition,

Vk

(:)Bk-2i

is clearly 2-integral. Therefore, we see that the sum in brackets in (2.5) when multiplied by Vk is p-integral for all pin. This completes the proof. • Theorem 2.9 Then,

(Voronoi, 1889) Let n be a natural number and (a,n)

=

l.

n-1

(ak - l)sk(n)

= knak-l I)Ja/n]/- 1

(mod n 2 ).

j=l

Proof. For each j satisfying 1:::; j Then, qj = [ja/n] and we have

(Jal

< n, write ja = qjn+rj where O:::; rj < n.

+ kqjnrJ- 1 (mod n 2 ) rj + kqjn(Jat- 1 (mod n 2 ). rj

As j ranges over the integers from 1 to n - l so does rj because (a, n) = l. Therefore, summing the above congruence from j = 1 to n - l gives the required result. • Corollary 2.10 n-1

(ak - l)Uk

(2.6)

= kVkak-l L)Ja/n]jk-I (mod n). j=l

Proof.

Obvious from Theorem 2.8 and Theorem 2.9.

Theorem 2.11 p-integral.

•

(Adams congruences) Ifp- l does not divide k, then Bk/k is

Proof. If ptlk, then by Corollary 2.10 with n = pt, we deduce, (ak - l)Uk = O(modpt). Choosing a to be a primitive root mod p, we can ensure that pis coprime to ak - l so that Uk is divisible by pt. • Theorem 2.12 (Kummer congruences) Let k 2 2 be even, p prime such that p - l does not divide k. If k = k'(mod c;/>(pe)), then

CHAPTER 2. BERNOULINUMBERS

Let t

Proof. n

=

=

ordpk and write Bk

19

=

Uk/Vk as before. In (2.6), we set

pe+t to get pe+t_ 1

(ak - l)Uk

= kVkak-l

L

[ja/pe+tJ/- 1(modpe+t).

j=l

Since (p - l)Yk, Vk and pare coprime. By Theorem 2.11, ptlUk so that (2.7) We begin with the case e = 1. On the right hand side of the above congruence, we can omit those j divisible by p since e = 1. Since k = k' (mod p - l), we find jk-l = jk'- 1(modp) for j coprime top. Choosing a to be a primitive root (mod p), we deduce

When e > 1, we write (2.8) pe+t_ 1

:z=

p'+'-l

[ja/pe+tuk-1

j=l

=

:z=

pe+t-1_ 1

[ja/pe+ti/-1

+ pk-1

:z=

[Ja/pe+t-1 Jjk-1 _

j=l

j=l

(j,p)=l

The congruence (2.7) withe replaced bye - 1 gives fork 2". 2,

so that (2.8) becomes

For k = k' (mod ¢(pe) ), we deduce the desired congruences. This completes the proof of the theorem. • Remark.

The value

(l - pk-1) Bk k

can be interpreted as ((1 - k) with the p-Euler factor removed. A prime p will be called regular if it does not divide any of the numerators of the Bk with k even and:=::: p - 3. (Later, we will prove Kummer's theorem that

CHAPTER 2. BERNOULINUMBERS

20

this is equivalent to p being coprime to the class number of the p-th cyclotomic field.) If p is not regular, we will call it irregular. It is still unknown if there are infinitely many regular primes. We have, however, the following.

Theorem 2.13

There are infinitely many irregular primes.

Proof. Suppose not. Let P1, ... , Ps be the set of irregular primes. Let k be even and set n = k(p1 - 1) ... (ps - 1) ifs 2: 1. Otherwise, set n = k. By (2.2), IEn/nl--+oo as k--+oo so we can choose k sufficiently large so that IEn/nl > 1. Let p be a prime so that ordp(Bn/n) > 0. By von Staudt-Clausen theorem (Theorem 2.7), we have (p - l)Yn. Thus, p # Pi for 1 :S: i :S: s and p # 2. We now show that pis irregular. Indeed, let n = m(modp - 1) with m < (p - 1) since p - 1 does not divide n. Moreover, mis even because n is even. By the Kummer congruences,

En=Em( - modp) n m so that ordp(Em/m) proof.

> 0.

Hence, ordp(Em)

> 0 as claimed.

This completes the •

The prime 691 is irregular since it divides the numerator of E 12 . The first few irregular primes are 37, 59, 67, 101, 103, 131, 149, and 157 which in fact divides the numerators of two distinct Bernoulli numbers E 2 k with k ::; 77.

Exercises 1. Let p be prime and define for each non-zero rational number Vp ( x) by the equation x = pvp(xlx 1 where x 1 is a rational number coprime top. Define the p-adic metric by llxll = p-vp(x) and set IIOII = 0. Show that llx + YII :S: max(llxll, IIYII).

2. With II· II as in the previous exercise, does the limit limn---+oo lln!II exist? What is the limit?

3. Prove that V

--

P

for i 2: 2.

(pi-1) + i

1

>O -

CHAPTER2.BERNOULINUMBERS

21

4. Prove that

text _ et -1 . where

Bk(x) denotes the

~

Bk(x)tk

~

kl

k=O

'

k-th Bernoulli polynomial. Deduce that

5. Let A be an abelian group. The exponent of A is defined to be the smallest positive integer e such that xe = l for all x E A. If e is the exponent of A, show that A has an element of order e.

6. Let p be prime and let 1 that

:s; a :s; p -

F(ab)

l. Set F(a)

= (aP-l - l)/p. Prove

= F(a) + F(b) (modp).

7. Prove the following generalization of Wilson's theorem:

(p - k)!(k - 1)! = (-ll (modp) for 1

:s; k :s;p-1.

8. If p is an odd prime, show that 2p-l numerator of 1 1 1

=

l(modp2 ) if and only if the 1

1--+---+ .. ·--2 3 4 p-l

is divisible by p.

9. With F as in Exercise 6, show that

F(a +pt)= F(a) - a't(modp) where aa'

= l(modp).

10. Let [x] denote the greatest integer less than or equal to x. For 1 p - l, show that

aP - a

p-l

l

p

j=l

J

- - = L-:

[aj] p

(modp).

:s; a :s;

CHAPTER 2. BERNOULI NUMBERS

22 11. Let n

= 2k + 1 be

an odd positive integer. Prove by induction that

sin nx

cosnx cosx

= Pn(sinx),

- - = Qn-1(sinx),

for polynomials Pn and Qn of degree at most n, with integer coefficients. [Hint: apply induction.]

12. Show that for n

= 2k + 1 an

sin nx ns1nx

odd positive integer, we have

.

.

. 2k

--.- = 1 + a1 sm x + a2 sm2 x + · · ·a2k sm x for certain rational numbers ai.

13. Using the previous exercise, show that for n

= 2k + 1,

2 x) sinnx sin -- k ( 1-nsinx sin 2 1rr

IT

r=l

•

n

Deduce that sin 1rx

"n(1rx/n)

14. Let n

= 2k + 1.

k

(

=}]

sin 2 l - sin 2

1rx )

:

·

Show that

rr (1 -:::: :: ) (1 -;: )-l - 1 n

r=l

ask-+ oo.

15. From the previous exercise, deduce that sin 7rX = rroo 1fX

(l _

r=l

x2 ) r2

.

16. For O < x :S 1, define the Hurwitz (-function by the series 00

1

((s,x) = ~ ( ) . ~ n+x 8 n=O

CHAPTER 2. BERNOULINUMBERS

23

First, prove that the series converges for Re( s) > 1. Then, by noting that 1 +((s,x)-((s) -8 X

1- - -1 ) , =" (n+x) n 00

~

(

8

8

n=l

and writing the summand as

show that

- :s + ((s, x) - ((s) =

~ ( ~s) ((s + r)xr.

Deduce that (s - l)((s, x) extends to an entire function of s and that lim(s - l)((s, x)

s->1

=

1.

17. Deduce from the previous exercise that

for every integer k 2'. 1. 18. With the notation of Exercise 6, show that

L F(a) = (p-1)'p · + 1 (modp),

p-l

a=l

for p prime.

19. Show that Bp-1(p)

for pa prime, and sk(p)

= lk +

= pBp-1 (modp

2)

2k + · · · (p- l)k, as usual.

20. Deduce from the previous exercises that (p - 1)! + 1 only if the numerator of 1 Bp-1 + - -1 p is divisible by p.

= 0 (modp2 )

if and

Chapter 3

p-adic Numbers A norm on a field F is a map 11 · 11 : F----+lR+ such that (i) llxll = 0 if and only if x = O;

(ii) (iii)

llxyll = llxllllYII; llx + YII :S llxll + IIYII

(triangle inequality).

We can always define a 'trivial' norm by setting IIOII = 0 and llxll = 1 for 0. We usually discard this norm and study non-trivial norms. Let us also make one more 'trivial' remark. Note that in a field F where O =J 1 (that is, the additive identity does not coincide with the multiplicative identity), we have by (ii) that 11111 = 1. Thus, 11-111 2 = 1so that 11-111 = 1. Therefore II-xii= llxllx

=J

Given a norm, we can define a metric on F by setting the distance between x and y as d(x,y) = llx - YII- The usual absolute value is a norm on IQ and the corresponding metric is the usual distance function. But we can construct other norms on IQ in the following way. Let p be a prime number and for each rational number x =J 0, write x = pvp(xlx 1 where x 1 is a rational number coprime top (that is, when x 1 is written in lowest terms, both the numerator and denominator are coprime top). Define the p-adic absolute value of x, denoted lxlP as lxlp = p-vp(x). For x = 0, we set IOIP = 0. It is easily checked that (iii) is satisfied in the sharper form

I· Ip is a norm on IQ.

In fact,

(3.1) as is easily verified. Equality holds in (3.1) whenever lxlp =J IYlp· This property is often phrased as the theorem that "all p-adic triangles are isosceles" or somewhat metaphorically as "the stronger one wins" since lxlp > IYlp implies lx+ylP = lxlp· Observe that in this metric, the sequence pn----+O as n----+oo. Thus, the norm measures divisibility by powers of p and is therefore ideal in the study of congruences. It allows us to bring the notions of analysis into the study of congruences. Two metrics d 1 and d 2 are said to be equivalent if they induce the same

25

26

CHAPTER 3. P-ADIC NUMBERS

topology on F. Two norms are said to be equivalent if their induced metrics are equivalent. If O < c

and

< 1 and pis prime, we can define for llxll = Cvp(x),

x

=I- 0,

I[0 II = 0. It is then easy to see that this norm is equivalent to I · Ip·

We have the following celebrated theorem of Ostrowski which says that up to equivalence the only norms on (Ql are the p-adic norms or the usual norm given by absolute value.

Theorem 3.1 (Ostrowski, 1918) Every non-trivial norm to I · IP for some prime p or the usual absolute value.

11 · I [

on (Ql is equivalent

Proof. Case (1): suppose that llnll ~ 1 for all integers n. Since the norm is non-trivial, there is an n such that Ilnl I < 1. Moreover, since 11 - nl I = Ilnl I, we may assume n is positive. Let n 0 be the least such positive n. Then n 0 must be prime by property (ii). Say n 0 = p. If q is a prime-=/- p, then we claim that llql[ = 1. If not, then llqll < 1 and for sufficiently large N, llqNI[ < 1/2. Similarly, for sufficiently large M, [[PMI[ < 1/2. Since pM and qN are coprime, we can find integers u and v such that upM

+ vqN =

1.

Hence,

+ vqNII ~ [[u[ll[PMII + [[vlll[qNI[ < 1/2 + 1/2 = 1, a contradiction. Therefore, [lq[I = 1. Write then c = [[p[[. Then, l[n[[ = cvp(n) and this metric is equivalent to I· Ip· Case (2): suppose there is a natural number n such that [In[ I > 1. Let no be the least such. Then no > 1. Write [lno 11 = n 0 l

=

l[upM

for some a. Given a natural number n, let us write it in base n 0 :

n = ao + a1no + · · · + asng, with a 8 =/- 0. Notice that inequality,

llnll

[laill

~

0 ~ ai

< no,

1, by the definition of no. By the triangle

< [lao[[ + [[a1no[[ + · · · + ilasng[[ < [laol[ + [[a1[[ng + · · · + i[as[[ngs < 1 + no + · · · + ngs 1 1 ) < n as ( 1+-+-+··· O

na 0

n2a 0

1 + - 1 + · · · ) = C n 0as < n 0as ( 1 + -2a 22a '

CHAPTER 3. P-ADIC NUMBERS

27

by the choice of n 0 and with Ca constant independent of n 0 . Since n 2 n 0, we deduce for all natural numbers n, llnll

:s;

Cn°',

c

for some absolute constant C 2 l. Thus, llnNII :s; CnN°' and so llnll :s; 1 /Nn°'. Letting N--+oo gives Ilnl I :s; n°' for all natural numbers n. We can also deduce the reverse inequality. Since ng+ 1 > n 2 n 0, we have lln + ng+i - nil

llng+ill


n 0(s+l)a - (n 0s+l - n )°' > no(s+l)a - (nos+l - nos)°' > n6s+l)a ( 1 - ( 1 - n\) °') > C1n°'

for some constant C 1. Arguing as before, we deduce that Ilnl I > n°'. Thus, llnll = n°' which is equivalent to the usual absolute value. • Given a field F with norm 11 · 11, let R be the set of all sequences {an} ~=l, an E F, which are Cauchy with respect to the norm 11 · 11 · One can define addition and multiplication of sequences pointwise, by setting

{an}~=l

X

{bn}~=l

= {anbn}~=l·

Then, (R,+, x) is a commutative ring. Moreover, the subset m of R consisting of null Cauchy sequences, that is sequences converging to zero, is a maximal ideal. Consequently, R/m is a field. We can embed Fin R via the map a - (a, a, ... ) which is clearly a Cauchy sequence. We can therefore view F as a subfield of R/m. R/m is called the completion of F with respect to 11 · 11 · In case, F = Q with the usual absolute value, this construction gives the field of real numbers. If F = Q, with norm I · Ip, we get the field, designated (Qlp, of p-adic numbers. We can extend the concept of norm to (Qlp in the following way. Given any sequence a = {an}~=l in (Qlp (or more precisely an equivalence class of

28

CHAPTER 3. P-ADIC NUMBERS

sequences), we observe that {lanlp}~=l is a Cauchy sequence of real numbers and as IR is complete, the sequence has a limit. So we define

It is easily checked that this is well-defined. Moreover, Qlp is complete with respect to I· Ip. Indeed, let { a (j)} ~ 1 be a Cauchy sequence of equivalence classes in Qlp. We must show that there is a Cauchy sequence to which it converges. To do this, observe that QI is dense in Qlp (exercise). Thus, we can find rational numbers ,,\ (j) so that

lim la(j) - ,,\(j)IP

= 0,

n-H)O

where we interpret ,,\(j) as the constant sequence in Qlp. It is now not hard to show that the sequence of rational numbers ,,\ (j) is a Cauchy sequence in QI and letting ,,\ denote the element of Qlp corresponding to this sequence we deduce that lim a(j) = ,,\. j----4CXJ

We leave the details as an exercise. The set

is a subring of Qlp called the ring of p-adic integers. Sometimes, we write I · I for I · IP when it is clear from the context that we are speaking of the p-adic norm. The subset

is a maximal ideal of Zp· Thus every element x of m satisfies vp(x) ::::: 1 and so we deduce m = pZP. Moreover, every non-trivial ideal of Zp is a power of m. (Commutative rings with only one maximal ideal are called local rings. Zp is thus a local ring.) Given x E QI satisfying lxlp ~ 1 and any natural number i, we can find a positive integer ai satisfying O ~ ai < pi such that Ix - ail ~ p-i. Indeed, write x = a/b with (a, b) = 1. As lxlP ~ 1, p and bare coprime, so we can find integers u and v such that ub + vpi = 1. Let ai = ua. Then

We can translate ai by a multiple of pi to ensure O ~ ai inequalities are not altered.


nPn,

0 ~ bn ~ P - 1.

n=O

Given any x E Qp, we can find pN such that IPN xlp < 1 so that every element of QP can be thought of as a formal series 00

L

bnpn,

0 ~ bn ~ p - 1.

n=-N

This p-adic expansion of x is easily checked to be unique. These remarks allow us to infer a few of the basic topological aspects of Zp and Qp. For example, recall that a topological space Xis called locally compact if every point of X has a compact neighborhood. It is not difficult to see that QP is locally compact and Zp is compact. Indeed, Zp is complete since it is a closed subset of a complete field. It is also totally bounded since for any E = p-n, the pn balls a=O,l,2, ... ,pn-l cover Zp- Now given any x E Qp, we can find some N so that pN x E Zp which has a compact neighborhood and thus x has one. It is useful to observe the analogy with Laurent series and the field of meromorphic functions of a complex variable. At each point z E C, a meromorphic function has a Laurent expansion which is unique. This suggests that if a rational number has denominator divisible by p, we can think of it as a p-adic number having a 'pole' at p. This analogy has been a guiding force for much of the development in p-adic number theory.

Now we consider the convergence of p-adic infinite series. The p-adic metric is much simpler to deal with than the usual absolute value for this study as the following theorem shows. Theorem 3.2

A p-adic series

converges if and only if lcnlp-O as n-----,oo.

CHAPTER 3. P-ADIC NUMBERS

30

Proof. It is clear that if the series converges, then /cnlv--+O since the usual proof carries over. Now suppose lcnlv--+0. Let N

Len.

SN=

n=l

Since (Qlp is complete, it suffices to show that { SN }N=l is Cauchy. We have for

M>N,

/sM - sNlp

/cN+l + ... +CM Ip

< max(/cN+ilv, ... , /cMlv)

which goes to O as N--+oo. This completes the proof.

•

For example, the sequence 00

converges (to (1 - p)- 1 ). Also, the series 00

converges. It is still unknown if this number is a p-adic irrational. That is, can we show that it is an element of (Qlp \(Ql ? By contrast, we have 00

Ln·n!

(3.2)

=

-1

n=O

because N

L n · n! = (N + 1)! -1, n=O

as an easy induction argument shows. Taking p-adic limits gives us (3.2). We can also consider the convergence of double series of the form 00

00

m=On=O

Suppose that for every m, lim

n-->oo

amn

= 0.

CHAPTER 3. P-ADIC NUMBERS

31

Suppose further that limm-+oo amn = 0 uniformly in n. That is, for each n, and given E > 0, there is an N = N(c) (independent of n) such that m 2:: N=}lamnlp
ianlp for n 2: 1. If f(r) = 0 for some r E Zp, we have

so that

a contradiction. This proves the theorem for N = 0. Now assume N 2: 1. If r E Zp is a zero of f(x), we can write f(x)

= f(x) - f(r) n=l oo n-1

(x - r)

LL anxjrn-l-J. n=l j=O

CHAPTER 3. P-ADIC NUMBERS

36

By Theorem 3.3, we can re-arrange our series: 00

f(x)

L bnxn

= (x - r)

n=O

where

00

bn

=

L an+l+krk. k=O

Observe that for every n,

and that lbN-1lv = iaN + aN+1r +···Iv= iaNlv so that if n

~

N, we have

Thus,

g(x)

= f(x)

x-r

satisfies the hypotheses of the theorem with N replaced by N -1. By induction, g(x) has at most N - 1 zeros in Zp so that f(x) has at most N zeros in Zp. • One can use Strassman's theorem to determine zeros of oo

- logp(l - x) =

n

L ~n

n=l

Since the power series converges only in lxlv < 1, we cannot use Theorem 3.4 directly. We therefore consider

If p =/- 2, we see that max lpn n>l n -

I

= IPlv

p

so that - logp(l - px) has at most one zero in lxlv ~ 1. But x = 0 is a zero in this disc and so we deduce that the only Zv-zeros of logp(l - px) is x = 0. If p = 2, we see that - log 2 (1 - 2x) has at most two zeros in lxl 2 ~ 1. x = 0 and x = 1 are the two zeros. ( Compare this with the example above as well as Exercise 5.)

CHAPTER 3. P-ADIC NUMBERS

37

Let us make a few general remarks. It is not difficult to show that a p-adic power series represents a continuous function in its region of convergence. The notion of a p-adic derivative of a function is clear. We define it as

f(x+pn)-f(x) . 1Im

n-->oo

pn

when this limit exists and denote it by f' (x). An elementary verification shows, for instance, that the derivative off (x) = xn is nxn-l for non-negative integers n. It is also clear that if f(x) is a convergent power series, we can differentiate term by term thus f'(x) is given by a convergent power series in this region. (There is something to check and we leave the verification to the reader as an elementary exercise.) Now for a fixed natural number a, consider the function f(x) = ax. Naturally, we will interpret this to be (1 + a - l? = Bx(a - 1) which converges and defines a continuous function for la - 11 < 1. In the complex analytic case, the derivative of f(x) is easily computed to be (loga)ax. A similar result holds in the p-adic case, although the proof of this is not trivial. It requires us to discuss the derivatives of functions of the form

G) which we will not do here. Observe, however that if an analogous result holds, then, we derive the interesting formula:

aPn - l n-->oo pn

logP a= lim

which can be interpreted as the p-adic derivative of ax at x of the form (aPn - l)/pn behave like the logarithm.

= 0.

Thus, quotients

We already witnessed this phenomenon in the Fermat quotient. In the next chapter, we will see a theoretical basis for this behaviour. For the moment, this maybe a good place to discuss the Fermat quotient in the light of what we have done: F(a) = aP-1 - 1 p defined for 1 :::; a:::; p - l. F(a) has similar properties to the p-adic logarithm defined above, and there is a good reason for this, as will be shown later. For instance,

F(ab)

= F(a) + F(b)(modp)

because

(1 + pF(a))(l + pF(b)) 1 + p(F(a) + F(b))(modp 2 ).

38

CHAPTER 3. P-ADIC NUMBERS

One can relate the Fermat quotient to sums of 'Voronoi' type studied in the previous chapter. Observe that

F(a +pt)= F(a) - at(modp)

(3.4) where

aa = l(modp).

Indeed,

+ p(p - l)taP- 2 (modp2 ) 1 + pF(a) + p(p - l)taP- 1 a(modp 2 ) 1 + pF(a) + p(p - l)ta(l + pF(a))(modp 2 ) 1 + p(F(a) + (p - l)ta)(modp2 )

(a+ pt)P- 1

ap-l

from which (3.4) is deduced. [One should think of this as 'Taylor series' of F(a) in a small disc.] Now

p-1

p-1

j=l

j=l

since F(aj)

p-1

= LF(a) + LF(j)(modp)

LF(aj)

= F(a) + F(j)(modp).

j=l

Thus,

p-1

F(a)

p-1

= LF(j)- LF(aj)(modp). j=l

Now write aj

= pqj + rj with

j=l

1 ~ rj ~ p - 1. Then

F(aj) = F{rj

+ pqj) = F(rj) - rjqj(modp)

by (3.4). As j runs through 1 top - 1, so does rj, Therefore, p-1

p-1

F(a) = "qj ~~ j=l

.

1 = "[Ja] ~ - ~(modp) j=l

J

p J

from which we deduce: Theorem 3.5

For 1 ~a~ p-1, aP -a p

p-1

=L

j=l

-p

[ja] -:-(modp). 1 J

In particular, observe that for odd primes p p

L p/2v(a)l!1[K:Qlvl, where NK/Q>v(a) denotes the norm ofa (which, as always, is the product of the conjugate roots of the minimal polynomial of a over Qp). Theorem 4.3

I· Ip is a non-archimedean norm on K.

CHAPTER 4. HENSEL'S LEMMA

48

It is clear that lxlp = 0 if and only if x = 0. It is also clear that To prove the stronger triangle inequality, namely, Ix+ Yip :S:: max(lxlp, IYlp) Proof.

lxylP = lxlplYlp since the norm is multiplicative.

(say, without loss of generality, lxlp :S:: IYlp) we see that upon dividing by y, it suffices to prove that for z EK, and lzlp :s; 1, we have

This will follow if we can show

because lzlp :s; 1 =} I - zip :s; 1 =} I - z - lip

= lz + lip

:S:: 1.

Therefore, it suffices to show that

This reduces to showing

Now,

we

must

use

a

bit

of algebraic

number

theory.

Clearly,

QMz) = (Qp(z - 1). Let

be the minimal polynomial of z in (Qp[x]. The minimal polynomial for z - l is then J(x + 1)

= xn +

Notice that NK/Qv(z)

(an-l + n)xn-l + · · · + (1 + an-l + · · · a1 + ao).

=

(-l)nao, and

NK/Qv (z - 1)

= (-lr(l +

an-l + · · · a1 + ao).

We now use the polynomial form of Hensel's lemma. If all the coefficients of f(x) are in Zp, we are done. So assume that ao E Zp but some ai .j. ZpChoose m to be the smallest exponent such that pmai E Zp for all i and "clear denominators":

with bi= pmai. Since f(x) is monic, bn = pm and we also know that bo By assumption, at least one of the bi is not divisible by p. Thus,

= pmao.

CHAPTER 4. HENSEL'S LEMMA

49

where k 2'. 1 is the smallest index such that bk is not divisible by p. By Theorem 4.2, this factorization extends to Zp[x], which means that g(x) = pm f(x) is reducible, a contradiction, since f(x) is the minimal polynomial of z. This completes the proof of the stronger triangle inequality. • Let us make a few more remarks that are easily verified. First, as K '.:::' Q;, it is evident that K is complete with respect to this extended norm. Second, if a: K---+ K is an automorphism of K, we see that lxlP = la(x)lp as the norms of x and a(x) are the same. If we define for x E K, x -/- 0, the function vp(x) by Ix IP = p-vp(x), this extends to K the definition of Vp defined earlier for x E Q;. The map Vp : Kx ---+ Q is a homomorphism into the additive group of rational numbers, whose image is easily seen to be of the form (1/e)Z for some divisor e of n = [K: Qp] (see Exercise 7). e is called the ramification index of Kover QP. We say that K/Qp is unramified if e = 1, ramified if e > 1, and totally ramified if e = n. An element 1r EK is called a uniformizer if vp(1r) = 1/e. The ring

now plays the analogous role of Zp, and the maximal ideal

p = {x EK:

lxlP < 1}

plays the role of pZp. Sometimes, the role of p is played by a uniformizer. All of our previous theorems extend to K such as Hensel's lemma and Strassman's theorem (see Exercises 5 and 6). The notion of a Teichmi.iller representative (which depended on the residue field) is also easily extended. Let us look at an important special case. Let (p denote a primitive p-th root of unity and let K = Qp((p)· By Exercise 2, [K : Qp] = p - l. By Exercise 3, N K /!Qp ( 1 - (p) = p and therefore I' - 11 I',p p-

p-1/(p-1) --

rp,

where rp was defined in the previous chapter in connection with the radius of convergence of the exponential power series. ( Now we see elements that actually have this absolute value.) Thus, K is totally ramified and 1r = 1 - (p is a uniformizer. As remarked above, our earlier theorems on p-adic power series extend to finite extensions of QP (see Exercise 6). Thus, we may consider the Exp and logP functions in this setting. Consider the problem of finding the roots of logP x in K = Qp((p). Let 1r = 1-(p be a uniformizer. Applying Strassman's theorem

CHAPTER 4. HENSEL 'S LEMMA

50

shows that the power series oo

7fnXn

-logP (1-nx)= " L..., -nn=l

has at most p zeroes (see Exercise 8). logP x = 0, we have found them all.

Since 1, (p, ... , (i- 1 are roots of

This result has now the curious consequence that the power of p dividing the numerator of the rational number

tends to infinity as n tends to infinity. (Notice that this is the norm of an algebraic number over the field of rational numbers.) One can also derive a lower bound for this power of p of the form

n - clogn, for some constant c > 0 depending upon p. This can be viewed as a generalization of Exercise 5 in the previous chapter which corresponds to the case p= 2. The above discussion extends the metric to any finite extension of (Q)p. It is thus clear that the metric can be extended to (Q)P. However, we have the important theorem: Theorem 4.4

(Q)P is not complete.

Proof. For each n coprime top, let f(n) be a primitive n-th root of unity. If p divides n, let f(n) = l. Now consider the convergent series 00

If (Q)P were complete, the series would converge to some ex E (Q)P. Thus, ex would lie in some finite extension K of(Q)p (say). Observe that f(l) = 1 EK. Suppose we have shown that f(n) E K for all n < m. We will show by induction that f(m) E K also. If m is divisible by p, there is nothing to prove. So we may suppose p is coprime to m. By the induction hypothesis, m-1

(3 :=

P-m( ex - L f(n)pn) EK. n=l

CHAPTER 4. HENSEL'S LEMMA

51

Observe that

= f (m) + pf (m + 1) + p 2 f (m + 2) + · · · . denote the maximal ideal of OK = {x E K : lxlP :S: 1}. Then, /3 = /3

Let j) f(m)(modp) so that the equation xm - 1 = 0 has a solution mod p because f(m) is a primitive m-th root of unity. By Hensel's lemma we can lift this solution to K. Since pis coprime to m, all the m-th roots of unity are distinct mod p (see Exercise 3) and it follows that f(m) EK. This proves that f(n) EK for all n. That is, all the roots of unity of order prime top lie in K. Since the roots of unity of order prime top are distinct mod p, we thus get infinitely many residue classes mod p which is a contradiction since K/Qp is a finite extension.• Now let CP be the completion of QP with respect to the p-adic absolute value. Thus, for a E i())P we define

and it is easily checked that this is well-defined. The definition is motivated by the fact that we expect norms to be preserved under isomorphism. Thus, !alp= ier(a)lp for any automorphism er E Gal(Qp/Qp). We can extend the norm to CP as before and it turns out that Cp is algebraically closed and QP is dense in Cp. To establish this, we will need an important lemma, known as Krasner's lemma, that basically says that if a E Cp is sufficiently close to /3 E Cp and algebraic over Qp(/3), then a E Qlp(/3). Lemma 4.5 (Krasner's lemma) Suppose K is a complete field with respect to a non-archimedean norm I· Ip· Let a,/3 EK, with a separable over K(/3). Suppose for all K(/3)-conjugates ai # a of a we have

Then, K(a) ~ K(/3).

Proof. Consider K(a,/3)/K(/3) and let L/K(/3) be the Galois closure. Let er E Gal( L / K (/3)). Then, er(/3-a) =/3-er(a).

Thus, l/3 - er(a)lp = 1/3 - alp< iai - alp, for all ai

#

a. Therefore,

If er(a) # a, this is a contradiction. Thus, er(a) = a for all er and a E K(/3) as desired. •

52

CHAPTER 4. HENSEL'S LEMMA

We can now prove: Theorem 4.6

Cp is algebraically closed.

Proof. Let K = Cp and I· Ip its p-adic norm. Suppose a is algebraic over K. Let f(x) be its minimal polynomial in Cp[x]. Since QP is dense in Cp, we may choose a monic g(x) E Qp[x] whose coefficients are sufficiently close to those of J(x). Thus, g(a) = g(a) - J(a) is very small in terms of the p-adic norm. Writing /3 = /31 with /31 sufficiently close to a, and

g(x) = Il(x - /3j) j

we see that

1/3- alp


n -1 _ vp(n) p-1

= (n- l) (-1- _ p-1

vp(n)). n-1

If n = pan1 with n1 coprime to p, we have vp(n) _ a a ---- < - n - 1 - pan1 -1 - pa -1

--

= - - - - -a- - - - - < -1-

+ · · · + p + 1) Ilogp(l + x)lp =

(p - l)(pa-l

so that lxn /nip < lxlp for n > 1. This means the domain of definition for Exp .

- p- 1' lxlp < rp is in

Following lwasawa, we now show that the logarithm can be extended to all ofre; =rep\{O}.

Theorem 4. 7 (Iwasawa) There is a unique extension of logP such that logp(xy) = logp(x) + logp(Y) for all x, y Ere; and logPp = 0. Proof. We begin by proving that every element x of re; can be written as pr wu where w is a root of unity of order prime to p and u is in rep satisfies lu - lip< 1. There are a few things to check. Let us observe that the extended norm on QP can take on only rational powers of p. Since QP is dense in rep, and the norm is non-archimedean, we see the same is true for rep- Thus, if r = ordp(x) = a/b E Q, let pr denote any root of xb - pa= 0. Then, x = prx 1 with x1 of norm 1. We can find Y1 E QP so that lx1 - Y1 IP < 1. Then, as lx1 IP = 1, again by the non-archimedean property of the metric, IY1 IP = 1. Since Y1 is algebraic, by the Hensel lemma construction, let w(y1) denote the unique Teichmilller representative of Y1 and set u = yif w(y1). Thus, any element of re; can be written as prwu. This is unique up top-power roots of unity ( for which logP ( = 0. We then define the logarithm by

This is well-defined since we have set logarithms of roots of unity to be 0.

•

With this extended logarithm, let us observe an interesting connection to the Fermat quotients. (Henceforth, we may use the notation 'Log' to indicate the padic logarithm and omit the p if it is clear which prime is meant. Sometimes, we

CHAPTER 4. HENSEL'S LEMMA

54

may write 'log' also when there is no cause for confusion.) Notice that p 2 ILog a (with Log a =f. 0) if and only if the Fermat quotient satisfies F(a) = O(modp). This is because 1

Log a= --Logap-l p-1

=

1

--Log (1 p-1

2 + pF(a)) = pF(a) --(modp ).

p-1

As stated in Chapter 3, the notion of a derivative is easily extended. As before, we define the derivative of a continuous function f to be

f 1( X )

_

l°

-

h~

f (X + h) - f (X) h

'

provided this limit exists. In the p-adic metric, this means

J'(x) = lim f(x

+ pn) - f(x)' pn

n--too

provided this limit exists. Thus, for example, the derivative of xm is mxm-l. The p-adic logarithm has nicer properties than the usual logarithm. It is locally analytic on (in the sense that given x 0 E it has a convergent power series expansion valid in a sufficiently small neighbourhood of x 0 ). Since this series can be differentiated term by term, we find that

re;

re;

d -Logx dx for XE

1

=-

X

re;.

For a E Zp, let us compute the derivative of f(x) to compute the limit aPn -

lim

n--too

=

ax at x

= 0. We need

1

pn

As we noted above, Log a = (Log av- 1 ) / (p- 1) so that it suffices to compute the limit for a= l(modp). Writing a= 1 + tp we have by the binomial theorem,

from which it is clear that

CHAPTER 4. HENSEL'S LEMMA Thus, the derivative of ax at x

55

= 0 is Log a.

Let us make one more observation concerning the p-adic logarithm. If /j E Gal(Qp/Qp) then as 1/j(x)lp = Jxlp for all x E Qp, we may extend (j to a continuous automorphism of Cp. By continuity,

Thus, Log/j(a) = /j(Loga) for a E Logo: E QP and in fact in Qp(a). ·

c;.

In particular, for a E Qp, we have

Exercises 1. In Newton's method, consider the problem of finding a real root of f(x) = x 3 - x. If an= 1/v'5, show that an+l = -1/v'5 and an+ 2 = 1/v'5. Thus,

the sequence generated by the initial value a0

= 1/v'5 does not converge.

2. Let p be prime. Show that the p-th cyclotomic polynomial p(x) = xp-l

+ xP- 2 + · · · + x + l

is irreducible in Qp[x]. 3. Let (m be a primitive m-th root of unity. Show that m-1

m

=

II (1 - ,~).

j=l

4. Given that Eisenstein polynomials factor rriod p (see Chapter 3, Exercise 10), why can't we use Hensel's lemma in the form of Theorem 4.2 to factor them in Zp? 5. (Hensel's lemma for finite extensions) Let K/Qp be a finite extension and 7r be a uniformizer. Let f(x) E OK[x]. Suppose there is an a0 E OK such that f(ao) = O(modrr) and f'(a 0 ) ¢. O(modrr), where f'(x) is the (formal) derivative of f(x). Then, there is a unique a E OK such that a= ao(modrr) and f(a) = 0. 6. Prove that Strassman's theorem extends to finite extensions of Qp.

CHAPTER 4. HENSEL'S LEMMA

56

7. Let K//JJ.p be a finite extension. Prove that the image of the map Vp: Kx --+ /JJ.p is of the form (1/e)Z for some e dividing n = [K: /JJ.p]. 8. Let p be prime and (pa primitive p-th root of unity. Set K 1r = 1 - (p. Prove that i1rn /nip is largest when n = p. 9. Show that x 3

= /JJ.p((p) and

= 2 has a solution in Q. 5 •

10. Prove the following generalization of Hensel's lemma: let f(x) E Zp[x]. Write f'(x) for its formal derivative. Suppose for some non-negative integer N and some integer ao, we have f (ao) = 0( mod p2 N +1) but f' (a0 ) = O(modpN), f'(a 0 ) ¢ O(modpN+ 1 ). Show that there is a unique a E Zp such that f(a) = 0 and a= a0 (modpN+l ).

Chapter 5

p-adic Interpolation We will consider first continuous functions f : Zp----+Qlp and begin with the p-adic analogue of the classical problem of interpolating the factorials. Recall that

1

00

e-xxndx

= n!

and so the problem of finding a continuous f(s) taking the value of n! for s is solved by setting

f(s)

=

1

00

=n

e-xx 8 dx

which is the familiar gamma function r(s + 1). The p-adic analogue of this problem has no solution since n!----+O (in the padic metric) as n----+oo in the "usual" metric. This suggests that it may be more natural to consider j

II

j':5:n

(j,p)=l

as the p-adic analogue of the factorials.

If we are given a sequence of integers {ak}k=l we can ask: when does there exist a continuous function f : Zp----+Qlp such that f (k) = ak? Since Zp is compact, such a function must be uniformly continuous and bounded. Thus, a necessary condition is that for each m, there must exist an integer N = N(m) such that (5.1) That is, whenever k and k' are close p-adically, then ak and ak, should be close p-adically. Conversely, if (5.1) is satisfied, and sup Ian Ip n

is bounded, then we can define a continuous function f : Zp ----+ (Qlp as follows. For x E Zp, let ni be a sequence of integers tending to x and define

f(x)

= Hm f(ni). i-+oo

57

CHAPTER 5. p-ADIC INTERPOLATION

58

The limit exists by virtue of (5.1). Moreover, if n~ is any other sequence tending to x, it is easy to see that

Hm f(ni) = Hm f(n~)

i--+oo

i--+oo

so that the function is well-defined. Thus, the interpolation problem reduces to checking (5.1). We state these observations as: Theorem 5.0 Suppose that a sequence of numbers J(k), k = 0, 1, 2, ... is padically bounded and satisfies the following condition. For each natural number m, there is a natural number N = N (m) such that

k ~ k'(modpN)

=}

= J(k')(modpm).

J(k)

Then, there is a continuous function f: Zp--+Zp such that f(k) .

= J(k). .

.

As an application of the above theorem, notice that if n = l(modp), the function f(k) = nk can be p-adically interpolated since the condition (5.1) is easily verified. We will show below that, in fact, f(s) = n 8 is a p-adically analytic function for s E Cp satisfying Isip < p;=~. This region is larger than the unit disc for odd primes p. · Let us, for now, consider the sequence ak

=

IT

j

Jk :, = e-x L f(n) :, . k=O

n=O

5. Let

00

I'p(x + 1)

be the Mahler series of r P ( x

I:(- l) 00

k=O

k+l

+ 1).

=LG) ak

k=O Show that

akx k _ 1 - x p -----exp k! 1-x

( x+x p) . p

6. Show that r P ( x + 1) cannot be extended to a Cp-analytic function in the domain lxlp:::; 1. (Hint: use the previous exercise and Theorem 5.7) 7. Let x

= ao + a 1p + a2p 2 + · · · E Zp. Prove that I'p(x)

8. Let p

= I'p(ao + a1p + · · · + anpn)(modpn+ 1).

= kf + 1 be prime and a:::;

k. Show that

-~ = af(l + P + P2 + ... + pn)(modpn+l). Deduce that I'p ( 1- ~)

= I'p (1 + af + afp + · · · + afpn) (modpn+ 1 ).

9. With notation as in the previous exercise, prove that rp ( 1 - ~)

= (-1rf+l(af)!(modp)

and I'p (~)

= (a~)!(modp).

10. Deduce Theorem 5.6 from Theorem 5.7.

Chapter 6

p-adic £-functions Let f be a natural number and x be a primitive Dirichlet character mod f. (Sometimes, we may write fx for f to make its dependence on the character explicit.) We extend x to all natural numbers by

x(n) = { x(n~odf)

if (n,f) = 1 otherwise.

The Dirichlet series, given by

L(s, x)

=

I: n=l

x(~) n

is well-known to converge for Re(s) > 0 if x-/=- Xo, the trivial character. Moreover, it extends to an entire function and satisfies a functional equation of the following form. Define the Gauss sum

L

T(X) =

x(b )e21rib/ f

b(modf)

and set

~(s, x) =

7r-s

s +a I 2J8 I 2r(-2-)L(s, x)

where a = 0 or 1 according as x( -1) and odd if x(-1) = -1.) Then,

= 1 or -1. (We say x is even if x( -1) =

1

~(s, X) = Wx~(l - S, X) where Wx = T(X)/ia J1 l 2 . Here, xis the complex conjugate character of X· One can also write the functional equation in an asymmetrical form: L(l - s, x)

ifx(-1)

=

j! :;~~ (2;) ½-s (

cos ~s)r(s)L(s, x)

= 1, and

(27r) ½-s ( 7rS) f sin 2 r(s + l)L(s, x)

ifl/2 L(l - s, x) = -(21r)- 1 l 2 T(X) .

71

CHAPTER 6. p-ADIC L-FUNCTIONS

72

for x( -1)

= -1, as is easily derived by an application of 7f

r(s)f(l - s) = - . Slll7fS

and Legendre's duplication formula: 1

f(2s)vn = 22 s- 1 r(s)f(s + 2). This allows us to derive the estimate for k E N,

IL(-k,x)I::; Ck!(f /21r) 2 k if X ( -1)

= 1 and IL(-k, x)I ::; C(k + l)!(f /21r)k

if x(-1)

= -1, for a certain positive constant C which is 0(../l).

As hinted in Chapter 2, we can deduce the formula

~ ( ) -nx = ~ L(-k, x)(-x)k L., X n e L., k!

(6.1)

k=O

n=l

by considering the vertical line integral 1 12+ioo -2 . L(s, x)f(s)x-sds 7fZ 2-ioo

and moving the path of integration to Re(s) can be simplified as follows: 00

I:x(n)e-nx n=l

I: b(mod f)

L

= -oo. The left hand side of (6.1)

x(b) n=b(mod f),n>O 00

x(b) (

b(mod f)

r=O

-bx

~ e-fx

I:

x(b\

I:

x(b)efx_1

b(mod f)

b(mod f)

Recall that

L e-(fr+b)x)

eU-b)x

73 .

CHAPTER 6. p-ADIC L-FUNCTIONS where Bk(x) denotes the k-th Bernoulli polynomial. Observe that e(l-b/ f)fx

eU-b)x

1

efx -

1

efx -

can be expanded as oo

L

~ )xk-1 Jk-1

Bk(l _

k!

k=O

When we substitute this series in the calculation above, we get

L x(n)e-nx = L ( L oo

b

oo

k=O

n=l

x(b)Bk ( 1 -

y))

(fx)k-1 k! .

b(mod f)

Notice that fork= 0, Bo(x) = 1 and as

L

x(b)

= o,

b(modf)

the polar term (that is, the term corresponding to k = 0) disappears. We deduce

(-1r- 1

L(l-n,x)=

L

n

b x(b)Bn(1-y)·

b(modf)

Also note that

(-t)ex(-t) e-t - 1

te(l-"'.)t

et -1 so that and we deduce

(6.2)

1n-l L(l-n,x)=-~

L

b x(b)Bn(y)·

b(mod f)

Defining the generalized Bernoulli numbers Bn,x by the formula

Bn,x =

r-l

L

x(b)Bn(y),

b(modf)

we can rewrite (6.-2) as (6.3)

Bnx L (1-n,x) = --'n

which can be thought of as a generalisation of the corresponding result (Theorem 2.2) derived for the Riemann zeta function in Chapter 2.

74

CHAPTER 6. p-ADIC L-FUNCTIONS

This formula can also be deduced without contour integration by making use of exercises 16 and 17 in Chapter 2. In those exercises, we defined the Hurwitz zeta function as 1 CX)

((s,x) = ~ ( ) ~ n+x s n=O

for O < x s; l. By noting that 1 +((s,x)-((s) = ~. = ( . 1 - -1 ) . -xs ~ (n + x) s n s and writing the summand as

we use the binomial theorem to deduce

-:s +

((s, x) - ((s) =

~ ( ~s) ((s + r)xr.

From this expression, we easily deduce that (s - l)((s, x) extends to an entire function and that lim(s - l)((s, x) = l. s---->l

Indeed, for all r sufficiently large, ( (s + r) is absolutely bounded so that the series involving the binomial coefficients is easily seen to converge. Putting s = l - k in the penultimate expression gives ((1- k,x)

= - Bk?)

for every integer k :::: 1. A Dirichlet £-function can be written as a linear combination of Hurwitz zeta functions. (In fact, historically, Hurwitz introduced his zeta function for the explicit purpose of deriving an analytic continuation of the L-series L( s, x).) Indeed, 1 L(s,x) = x(b) ns'

L

b(mod f)

n=b(mod f)

and the inner sum is really CX)

rsI: (· 1=0

Therefore,

L(s, x)

=

J

1

b)s =f-s((s,b/f).

+f

rs L b(mod f)

x(b)((s, b/ f)

CHAPTER 6. p-ADIC £-FUNCTIONS

75

from which we deduce the analytic continuation of L(s,x). As a consequence of the above formula, we also deduce (6.3). Also note that

L x(b)En(7)

L x(b)Bn(1-7)

(-lt

b(modf)

b(modf)

(-ltx(-1)

L x(f-b)Bn( 1 7b) b(modf)

(-ltx(-1)

~

x(b)Bn(7)

b(modf)

so that L(l - n, x)

= (-ltx(-l)L(l - n, x).

Thus, if x and n have opposite parity, then 'L(l - n, x) = 0. Ihhey have the same parity, the functional equation shows that for n ~ 1,

IL(n, x)I = lr(x)I Since L(n, x) -/=- 0 for n

~

IB;(-1 . (27r)n f

1, it follows that Bn,x -/=- 0 in this case.

Notice that if xis odd, B 1 ,x -/=- 0 comes from the non-vanishing of L(l, x). This means that for x(-1) = -1, 1

B1,x

=

f

.

J LX(a)a-/=- 0 ·a=l

and no elementary proof of this simple fact is known. Fo.r n ~ 1 therefore, L(l - n, x) -/=- 0 if and only if x(-1) L(l - n, x) -/=- 0 if and only if x and n have the same parity.

= (-lt. That is,

The significance of the equation (6.2) lies in the fact that the right hand side is essentially a linear combination of terms of the form

which makes sense when n is replaced by a p-adic variable and

pJf.

76

CHAPTER 6. p-ADIC L-FUNCTIONS More precisely, we define the p-adic Hurwitz zeta function for plF:

This will be the p-adic analogue of the classical Hurwitz zeta function: b

00

H(s,b,F)=

L(n+ F)

-s

O 1, and ( a fixed primitive f-th root of unity. Set f

T(X) =

L x(a)(a. a=l

Choose a positive integer N such that (N, fp) = 1 and x(N) -=/:- 1. Let {.X} be the set of all N-th roots of unity (in some algebraic closure of the p-adic field Qp)- We will denote by Bn,x the generalized Bernoulli number defined in Chapter 6. We will also denote by x(a) the quantity x(a)- 1 if (a,!)= 1 and 0 otherwise. Lemma 8.4

Proof.

Let f

g(X) =

Z:x(b)xb-1_ b=l

We have the partial fraction expansion:

g(X) t-1

so that

f

Ca

=Z:xa=l

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x)

103

Thus,

f x(b)tebt Left_ 1

tet g(X)

Xi-I

b=l

t

~ (a

t

e L....,

I X=et

g ((a) I

f X -

(a X=et

a=l

by (8.1) used with

x instead of x.

Recall that (see Exercise 1, Chapter 6)

Therefore, CXJ tn L(x(N)Nn - l)Bn,xn!

x(N)

n=O

X

f x(b)NtebNt efNt _ 1 -

f x(b)tebt eft -1

L

L

b=l

b=l

(N)eNt Nt T(X)

f

~

L....,

x(a) eNt _ (a

a=l

- tet T(X)

f

~

L....,

x(a) . et _ (a

a=l

In the first sum on the right hand side, we may replace a by aN since (N, f) = l. Thus,

Now,

xN - (aN =

IT (X -

>.(a)

.\

so that logarithmic differentiation gives

and the above simplifies to

t tT(X) e

f

'°'~ et x(a) - >,ra . L...., L....,

.\cfl a=l

',

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x)

104 Writing

we obtain that the sum can be rewritten as

T(X) " ~

= f

L...,

tet

L...,X(a\ _ ).(a

>.,fl a=l

=-T(X)"~-(a)~

1

~ (>.(a 1-

~

L...,

l)i L...,

i=O

J=O

.

(

i) · ·

j

·t

(-l)"-JeJ

~(i)(-lt-j~(j+lrtm+l j ~i m!

{ti;:iX 6' (>.(a -1)'+1 f='o

f

since (et -1 )i has a power series expansion starting with ti, the summation over m above can begin at m = i as all the earlier terms must vanish. This simplifies to

We may rewrite the inner two sums as

0L..., (>.(a 1i=l

-~(i-1)(-l)i-l-j(j+lr L..., j

1)'

J=O

and observe that ( i-1) j

=

(

i-1 ) i-(j+l)

(

i

)j+l

= j+l - i -

so that our expression becomes (

- TX

f

)

m+l LL t m+l m! I:x(a) Li). oo

>.,flm=O

f

a=l

l a -

i=l ( (

i ( .) l i i_ (-l)i-jjm+l. ) j=l J

L

The lemma now follows by comparing coefficients of tn. Now we can prove Theorem 8.3. We will make use of Euler's theorem

in the weaker form of a p-adic limit: lim n--->oo

aq,(pn)

= 1,

•

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lv(l, x)

or as

105

llv :::; p-n which is needed in Exercise 5. Thus,

la(pn) -

(x(N) - l)Lv(l, x)

=

lim (x(N)N(vn) - l)L(l - oo

which is by Theorem 6.2 equal to

- lim (x(N)N(vn) - 1)(1 - x(p)p(vn)-l) B¢,(pn),x. n->oo

.,ila=l wP=lj=O f ¢,(pn) . _ lim """"""""x(a)T(X)"""""""";: ( w-1 )' n---+oo L...t L...t fp L...t L...t i ).(a - 1 >.c;il a=l w i=l

Lt

>.,ila=l

since (>.(a)f P

-:f.

x(a)T(X) Llog fp

w

1 and thus, (>.(a)p

(1- Awa-_\) (

-:f.

1. We therefore have

l(w - 1)/(>.(a - l)lv < 1

wj

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x)

106

so that the series for the logarithm converges. Taking into account the sum over the p-th roots of unity, we obtain that the limit in question is

=Lt .\/1

x(a)T(X) (log(),P(ap - 1) - plog(,\(a - 1)) .

a=l

fp

If Pl f, we may by the definition of the conductor f, choose b = 1 ( mod f / p) such that x(b) =J 1 and (b, f) = 1 (see Exercise 2). Thus, · f

f

LX(a) log(,\P(ap - 1)

L x( a) log(

a=l

a=l

),P (apb -

1)

f

x(b) LX(a) log(V(ap - 1) a=l and as x(b)

=J

1, the above sum vanishes and we deduce that if Plf,

Lt

(x(N) - l)Lp(l, x) =

x(a)T(X) log(,\(a - 1).

f

.\# a=l

If (p, f)

= 1, then f

f

LX(a) log(,\P(ap - 1) = X(P) LX(a) log(,\P(a - 1). a=l

a=l

As ,\ runs through non-trivial N-th roots of unity, so does ,\P, and we can write f

x(p)

f

L I:x(a) log(,\P(a .\/1

1) = x(p)

a=l

L I:x(a) log(,\(a .\/1

1) .

a=l

In either case, we obtain

(x(N) - l)Lp(l, x)

=

Lt

f

p

~ X(a)T(X)

(X(P) _

~

1) log(,\(a - 1)

x(a)T(X) (x(p) -

.\# a=l

f

P

l)

l

(aN - 1 og (a -1 f

(x(N) - 1) ( X~) - 1) TjX) ;x(a) log((a - 1).

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x) Since x(N)-/- 1 and log((a - 1)

= log(l -

107

(-a), we obtain f

Lp(l, X) = - ( 1 - xr;)) TjX) ;x(a) log(l - Ca),

•

which is Leopoldt's theorem. The above derivation also gives us a new analytic expression for Lp(s, x): Theorem 8.5 Suppose x Then, for s E Zp, we have

Lp(s,x) =

-/-

1, f -/- p, and ( is a primitivef-th root of unity.

T(X) f _ -T ;x(a)

8 00

1 i((a - l)i

~

( .) ; (-l)i-j(j)l-s_

(p,j)=l

Proof. We have Lp(s, x) = limn_, 00 Lp(l -n, x) where n m---+ ( 1 - s) / (p - 1) p-adically. So

Lp(s,x)

= -

= (p- l)m---+oo and

lim (1- x(p)pn-i)Bn,x = lim Bn,x. n n---+00 n

n--t(X)

By Exercise 6, we have

Since

for (p, j) = 1, the result follows. The justification of the termwise evaluation of the limit is the same as indicated in Exercise 5. • We can use Theorem 8.3 to prove: Theorem 8.6 (Ankeny, Artin and Chowla, 1952) Let p be a prime= l(mod4). Let h be the class number of Q( ytp) and E = (t + uytp) /2 > 1 be its fundamental unit. Then, uh = B(p-i); 2 (modp)

t

where Bk denotes the k-th Bernoulli number.

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lv(l, x)

108

Proof.

By our remark following Theorem 8.1, we have p-1 2h log E =

L x(a) log(l - (;)

-

a=l so that

p--1 E-2h

=

IT (l _ (;)x(a).

a=l In QP we may choose (p, E and v1P so that T(X) = v1P and the above relation holds. Fix an embedding of Q into Cp. Since the above inequality holds in Q, it holds in Cp. Taking p-adic logarithms, we get 2hLogE

= -

p-1 LX(a)Log(l - (;) a=l

I:

- T(X)vP x(a)Log (1 - (;) p a=l (1-

x;) )- 1 v'PLv(l, x)

y'pLv(l, x)

by Theorem 8.3. Now,

x = wCv- 1)/ 2 since xis a quadratic character of conductor

p. By Corollary 6.5 and Theorem 6.2, we obtain

p-l

Lv(l- - 2 -,x) ( l _ p(v-1)/2) Bcv-1)/2 (p - 1)/2 2B(p-l)j 2(modp).

By Exercise 7, Logr

= O(modp) for any rational number r so that

LogE

so that

Log(t/2)+Log (1+~v'P) u ty'p(modp)

hu

-t = Bcv-1);2(modp), which completes the proof.

•

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x)

109

As mentioned earlier, the Ankeny-Artin-Chowla (AAC) conjecture predicts that u ¢. O(modp). One can show (see for example, Exercise 6.5.8 on p. 77 of [EM] ) that h < ,jp. Thus, the significance of the conjecture is that if u ¢. O(modp), the congruence determines h. Moreover, we see that u ¢. O(modp) is equivalent to the numerator of B(p-l)/ 2 being coprime top. In particular, if p is regular, then the AAC conjecture is true. However, we still do not know if their conjecture is true for infinitely many primes. If p is a prime of the form n 2 + 1, and n > 2, then one can show that 2 + 1 is a fundamental unit of Q(,jp) (see Exercise 9). Hence for E = n + such primes, the AAC conjecture is true.

vn

Some heuristic reasoning can be given against the the numerator of B(p-l)/ 2 as a random number, then is divisible by p is 1/p. Thus, the expected number p = l(mod4) and p dividing the numerator of B(p-l)/ 2

1

p'5cx P=l(mod4)

conjecture. If we view the probability that it of primes p < x with is

1

- "' - log log x. p 2

This heuristic reasoning predicts that the AAC conjecture is false. For p < 6,270, 713, the conjecture has been verified. However, this range is too small a range for any tangible counterexample to show up. This conjecture is quite analogous to the existence of primes p such that (8.3) As we saw in Chapter 1, the ABC conjecture predicts that there are infinitely many p for which (8.3) holds. Recall that this would follow if had the following. Writing, 2n -1 = UnVn with (un,Vn) = 1, Un squarefree and Vn squarefull, it suffices to show that Un -,oo as n-,oo. Now we consider the question of whether tionally. If n = 2k 2:: 5, we get

As v 2 k is squarefull, we can write it as

and one can show that

Un -,oo

can be proved uncondi-

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x)

110

is the fundamental unit of the quadratic field Q( .ju2kb 2k). Thus, this leads to the question of whether, given a real quadratic field Q( .JD), the fundamental unit satisfies

UD

= O(modD).

We conclude by noting that by an old theorem of Brahmagupta (8th century AD) (see [We, p. 19]) there are infinitely many such quadratic fields. The Ankeny- Artin-Chowla conjecture predicts that for D = p = l(mod4), this should not happen. Heuristic reasoning would suggest however that it is a sparse phenomena much like the case primes not satisfying (8.3). Returning to Lp(l, x) and the problem of its non-vanishing, we introduce the p-adic regulator. Let K be a number field. Fix an embedding of Cp into C. Then any embedding of K into Cp becomes an embedding into C, and hence may be considered real or complex depending on the image of K. Let r 1 be the number of real embeddings and 2r 2 the number of complex embeddings for K in the customary notation. We list these, as usual, as

and set r = r 1 + r 2 - 1. Let 1\ = 1 if O'i is real and 2 if E1 , ... , Er be independent units of K. Then, we define

O'i

is complex. Let

Of course, this may depend upon the embedding of Cp into C but this will not affect its p-adic valuation which is what we will be interested in. Also, the regulator defined is only up to sign.

If {E1 , ... ,Er} is a basis for the units of K modulo roots of unity, then we define Rp (K) to be RK,p(E1, ... , Er) to be the p-adic regulator of K. We state: Theorem 8. 7 (Leopoldt) Let K be a totally real abelian number fleld of degree n corresponding to a group X of Dirichlet characters. Let h(K) be the class number of K and dK its discriminant. Then,

up to signs.

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x)

111

If we define the p-adic (-function of K as (K,p(s)

IT Lp(s, x)

=

xEX

then we obtain . ( - l)r ( ) = 2n- 1 h(K)Rp(K) 1Im S ',,K p S .~ .

s---+1

vdK

'

Thus, if Rp(K) =/. 0, then (K,p(s) has a simple pole at s = 1 with a residue which is the p-adic analogue of the residue in the complex case. As mentioned earlier, Brumer proved that Rp(K) =/. 0. The notion of the p-adic zeta function can still be defined in the case K is an arbitrary totally real number field and it is an unsolved problem to show Rp(K) =/. 0 in this case.

Exercises 1. Let

x be a

non-trivial character mod f. Show that f

x(a)T(X) = I:x(b)e2niab/f_ b=l

2. Let x be a primitive character mod f. Show that for any proper divisor d off, there is ab= l(modd) such that x(b) =/. 1. 3. Prove that 00

I:

e2nian/f

n

n=l

converges to log(l - e2naff) whenever (a, f)

= 1.

4. Justify the derivation of the formula for L(l, x) at the beginning of this chapter. 5. Justify the limit process in formula (8.2). You will have to determine precisely the power of p that may enter into the denominators of the expression in (8.2). 6. Let x =/. 1 be a Dirichlet character of conductor f =/. p. Let ( be a primitive f-th root of unity. Prove the following variation of Lemma 8.4:

B~,x = _ T(X)

f

f=t :~\it(~) i

a=l i=l

((

)

j=l

J

(-l)i-jjn.

112

CHAPTER 8. LEOPOLDT'S FORMULA FOR Lp(l, x)

7. If r E Q, show that Logr = O(modp). 8. Let p be a prime= l(mod4) and 2. Show that x(-1) = 1.

x the unique character

9. Suppose n 2 + 1 is squarefree with n > 2. Show that the fundamental unit of Q( ,/n 2 + 1).

E

(mod p) of order

= n + ,/n 2 + 1 is

10. Let D be squarefree and= l(mod4). Suppose that E =(a+ bv'J))/2 > 1 is the fundamental unit of Q( v'J)). If (u + vv'J)) /2 is any other unit, show that blv.

Chapter 9

Newton Polygons Let f(x) = 1 + I:~=l aixi E i2 xi, i=l

the Newton polygon given in Figure 4 is of this type. Second, at some point, the line we are rotating hits infinitely many points, forcing us to stop. In such a case, the Newton polygon has only finitely many line segments. For example, for f(x) = 1 xi, the polygon consists of the x-axis.

I:;:,

A third possibility is that the line we are rotating has not hit any of the points (i,ordpai) which are farther out, but if rotated any further, the line would "miss" some points. This is seen in the example, f(x) = 1 + 1 pxi.

I:;:,

CHAPTER 9. NEWTON POLYGONS

117

Fig. 4: Newton Polygon of f(x)

= 1 + z:=:, 1 pi

2

xi.

6

5 4 3

2 1 0 0

1

2

3

4

In the case of polynomials, the Newton polygon is useful since we can see at a glance where its roots are located. It turns out that in the case of power series also, similar information is given by the Newton polygon. In many cases, such information is sharper than that given by Strassman's theorem that we met in Chapter 3.

Theorem 9.2 polygon of

Let b equal the supremum of all the slopes of the Newton 00

f(x)

= 1 + Laixi

E Cp[[x]].

i=l

Then, the radius of convergence of f(x) is pb. Ifb

= +oo,

then f(x) converges

for all x E Cp.

Proof. Let lxlp = pc, with c < b. Then, ordpx = -c. Now, ordp(ciixi) = ordpai - ci. Since b is the supremum of all the slopes and c < b, it is clear that, sufficiently far out, (i, ordpai) lies arbitrarily far above (i, ci). That is, ordp(aixi)-oo as x-oo so that the series converges. If lxlp = pc with c > b, then the same reasoning shows ordp (aixi) = ordpai - ci is negative for infinitely many values of i so that the series for f(x) does not converge. Therefore, the radius of convergence is exactly pb. • Let us make a few remarks about convergence on the boundary. It is not hard to see that such convergence can only take place when the Newton polygon is of the third type described above. The reason for this is that if lxlp = pb, then we need, ordpai - bi-oo as i-oo for convergence to take place. In case the Newton polygon is of type 1 or 2, ordpai gets very close to bi. Even if the polygon has type 3, convergence can only take place if ordpai - bi-oo. For

118

CHAPTER 9. NEWTON POLYGONS

example, this is the case if 00

f(x)

= 1 + I>ixpi i=l

whose Newton polygon is the horizontal line from (0, 0) consisting of the positive x-axis. (See Exercise 4.) Another remark worth making concerns how the Newton polygon changes under a variable change. Let c E ... Let f ( x) be a given power series and set g (x) = f (x / c). The Newton polygon of g is obtained from that of f by subtracting the line y = .>..x which passes through (0, 0) from the polygon off. To see this, write 00

f(x)

= 1+

L aixi i=l 00

g(x)

= 1+

L bixi i=l

and observe that ordpbi Lemma 9.3

= ordpai -

.>..i from which the result is clear.

Suppose that .>.. 1 is the flrst slope of the Newton polygon of 00

f(x)

= 1 + Laixi. i=l

Let c E . :S .>.. 1 . Suppose that f(x) converges in lxlp :Sp>-. Let g(x) = (1 - cx)f(x). Then, the Newton polygon of g(x) is obtained by joining (0, 0) to (1, .>..) and then translating the Newton polygon of f(x) by 1 to the right and.>.. upward. If f(x) has last slope AJ and converges on lxlp :S p>-t then g( x) also converges in the same disc.

Remark. In other words, the Newton polygon of g(x) is obtained by 'joining' the Newton polygon of 1- ex to the Newton polygon of f(x). Observe that the convergence is assured if .>. < .>.. 1 or if the Newton polygon off has more than one segment.

Proof. We first reduce the proof to the case c = 1, .>. = 0. Suppose the lemma holds in this case. Then fi(x) = f(x/c) and g 1 (x) = (1 - x)fi(x) satisfy the conditions of the lemma with c, .>.., .>.. 1 replaced by 1, 0, .>.. 1 - .>. respectively. Since the lemma holds in this case, we obtain the Newton polygon of g 1 ( x) and the convergence in the disc Ix IP :S p>-t->- when f converges in the disc lxlp :S p>-t. As g(x) = g 1 (cx), by the remarks made earlier about how the Newton polygon

CHAPTER 9. NEWTON POLYGONS

119

changes under variable change, we get the Newton polygon of g. Therefore it suffices to prove the lemma for c = 1 and A = 0. Let g(x) i 2 0. Now,

= 1+

I::

1

bixi. Since g(x)

=

(1 - x)f(x), bi+ 1

=

ai+l - ai, for

with equality if and only if ordpai+l -=/- ordpai. Since (i, ordpai) and (i,ordpai+ 1 ) lie on or above the Newton polygon of f(x), so does (i,ordvbi+ 1 ). If (i, ordpai) is a vertex, then ordpai+l > ordpai so that ordpbi+l = ordpai, This means that the Newton polygon of g(x) must have the shape described in the lemma as far as the last vertex of the Newton polygon of f (x). It now remains to show that in the case when the Newton polygon has a final infinite slope AJ, so does g(x). Moreover, we need to show that if f(x) converges on lxlv = p>-t, then so does g(x). Since ordpbi+l 2 min(ordpai+ 1 ,ordpai), it immediately follows that g( x) converges when f (x) does. We must rule out the possiblity that the Newton polygon has slope )- 9 > AJ, Indeed, if it did have such a slope, then for some large i, the point (i + 1, ordpai) would lie below the Newton polygon of g(x). Then, for all j 2 i + 1, ordpbj > ordpai which implies ordpai+ 1 = ordpai for all j > i. This contradicts the assumed convergence of f(x) in lxlv S 1. •

I::

Lemma 9.4 Let f(x) = 1 + 1 aixi have Newton polygon with first slope .>- 1 . Suppose that f(x) converges in lxlv S p>- 1 and that the line through (0, 0) with slope .>- 1 passes through a point (i, ordpai). Then, there exists an x 0 for which ordpxo = ->-1 and f (xo) = 0. Remark. Notice that both the conditions imposed on fare automatic if the Newton polygon has more than one slope. Proof. For simplicity, we first consider .>- 1 = 0 and reduce the general case to this one. Thus, ordpai 2 0 with ordpai-+oo as i-+oo. Let N be the greatest i for which ordpai = 0. This N is the length of the first segment (of slope 0) unless the Newton polygon off has only one horizontal line. Set n

fn(x)

= 1 + Laixi. i=l

By Theorem 9.1, fn(x) has precisely N roots Xn,l, ... , Xn,N with ordpxn,i = 0 for n 2 N. Define a sequence {xn}~=N inductively by XN = XN,l and for n 2 N, let Xn+l be any one of Xn+1,1, .. ,,Xn+1,N with lxn+l,i - xnlv minimal. Thus, lxn+l,i - Xnlp 2 lxn+l - Xnlp for 1 S i S n + 1. We claim that the sequence { Xn} is Cauchy and its limit has the desired properties. Indeed, let Sn denote the set of roots offn(x). Then, for n 2 N,

CHAPTER 9. NEWTON POLYGONS

120

IT

aESn+l

Xn - I I1 -Xn- I = ITN I1 - p

0:

i=l

Xn+l,i

P

< 0, we have

because for o: E Bn+1 with ordpo:

1 - : ; Ip= 1.

1

Thus, N

lfn+1(xn) - fn(xn)lp

=

IT lxn+l,i -

Xnlp since lxn+1,ilp = 1,

i=l

>

lxn+l - Xnl:

by choice of Xn+i

so that lxn+l - xnl: S lan+lx~+l IP = lan+ilp· Since ordpai-oo, we deduce that the sequence {xn} is Cauchy. Put xo = limn->oo Xn· Then, f(xo) = limn->oo f(xn). In addition

since lailp S 1 and l(xi - x~)/(x - Xn)lp S 1. Therefore

lfn(xo)lp S

lxo -

Xnlp

and letting n-oo, we see that f(xo) = 0. This completes the proof in the case ..\1 = 0. For the general case, let z E Cp be any number such that ordpz = ..\1. Such a z exists since we may take the i-th root of ai for which (i, ordpai) lies on the line through (0, 0) with slope ..\1. Set g(x) = f(x/z). Then g(x) satisfies the conditions of the lemma with .X 1 = 0. By what has already been proved, there is an x~ with ordpx~ = 0 and g(x~) = 0. If we let Xo = x~/ z then, ordpxo = -.X1 and f(xo) = 0. This completes the proof of the lemma. • Lemma 9.5

Let

00

J(x) = 1 +

L aixi E Cp[[x]] i=l

CHAPTER 9. NEWTON POLYGONS converge and have value O at x

121

= a. Let 00

g(x)

= 1 + Lbixi i=l

be obtained from f(x) by dividing it by 1 - x/a. Then, g(x) converges in

lxlP::; lalP' Proof.

Let

n

fn(x)

= 1+

L aixi. i=l

Then,

1 bi = ----;a"

a1 a;.-1 + -. - + .. · + - + ai, ai- 1 a

so that biai fi(a). Hence, lbiailP = lfi(a)lp---+0 as i---+oo since f(a) Therefore g(x) converges in the region indicated.

0. •

We are now ready to prove the p-adic version of a classical theorem in algebraic geometry known as the Weierstrass preparation theorem. The classical version gives a factorization of a formal power series as a polynomial and an invertible formal power series. This factorization is then used to show that the ring of formal power series over any field is a unique factorization domain. A variant of the theorem (see Chapter 10) can then be used to show that the ring of formal power series over Zp, namely, Zp[[x]], is a unique factorization domain. More generally, the same applies to O[[x]] where O is the ring of integers of a finite extension of (Q)p, One can then develop a structure theory of finitely generated modules over such rings and use it to study the growth of class numbers in cyclotomic fields and analyse the zeros of p-adic £-functions. One consequence of the theory (as we shall see in Chapter 10) is that a p-adic £-function has only finitely many zeros.

Theorem 9.6

Let 00

f(x)

= 1 + Laixi i=l

converge in lxlp ::; pA. Let N be the total horizontal length of all the segments of the Newton polygon off having slope::; A, if this length is finite. Otherwise, if f(x) has last slope A, let N be the largest i such that (i, ordpai) lies on the last segment. (Such an i must exist because the series for f converges on Ix IP = pA.) Then, there is a polynomial h(x) of degree Nanda power series 00

g(x) = 1 + Lbixi

i=l

122

CHAPTER 9. NEWTON POLYGONS

which converges and is non-zero in !xiv :::; p>-. such that

h(x) = f(x)g(x). The polynomial h(x) is uniquely determined by these properties and its Newton polygon coincides with f upto (N, ordvaN ).

Proof.

We induct on N. For N = 0, we must show that the power series for

g ( x) = 1/ f (x) converges and is non-zero in Ix Iv :::; p\ By changing variables

(as before in the proof of Lemma 9.3), we can reduce to the case,\= 0. Write 00

f(x) = 1 + Laixi. i=l

Then N = 0 implies ordvai ordvai-----,oo as i-----,oo, Write

> 0. Since f (x) converges in !xiv :::; 1, we have 00

g(x) = 1 + Lbixi. i=l

The relation f(x)g(x) = 1 implies for i 2: 1,

From this, we deduce, by an easy induction on i, that ordvbi > 0. To complete the proof, we must show ordvbi-----,oo as i-----,oo. Since ordvai-----,oo, we can, for any given M, choose m so that i > m implies ordvai > M. If i > m, we may write

from which we see that the terms bi-jaj, j > m have ordv(bi-jaj) 2: ordvaj NI. If we let v = min(ordva1, ordva2, ... , ordvam) > 0,

>

we see that the terms with j :s; m have ordv(bi-jaj) 2: ordv(bi-j) + v. This suggests another induction argument. Namely, we will prove by induction on n that i > nm implies ordvbi > min(M, nv), from which will follow the desired conclusion that ordvbi____,CXJ, To prove the claim, we induct on n. For n = 0, the result is clear. For i > nm and j :::; m, we see that i - j > (n - l)m and so by induction., ordv(bi-j) > min(M, (n - l)v). Thus, ordv(bi-jaj)

> > >

This completes the proof for N = 0.

ordv(bi-j) + v min(M, (n - l)v) min(M, nv).

+v

CHAPTER 9. NEWTON POLYGONS

123

Now suppose N 2: 1 and that the theorem holds for N - 1. Let .>11 :S A be the first slope of the Newton polygon of f(x). Using Lemma 9.4, we can find an a such that f(a) = 0 and ordpa = -A1. Let fi(x)

= f(x) (1 + x/a + (x/a) 2 + ... ) I ' = 1 + 's"OO L..,i=l a;x".

By Lemma 9.5, fi(x) converges in lxlp :S pA 1 • Let c = 1/a so that f(x) = (1 - cx)fi(x). If the Newton polygon of fi(x) has first slope Ai < A1, then it would follow by Lemma 9.4 that fi(x) has a root with p-adic ordinal -Ai and then, so would f(x), which is not the case. Hence, Ai 2: A1. By Lemma 9.3, the Newton polygon of Ji (x) is the same as the one for f (x) minus the segment from (0, 0) to (1, A1). Moreover, Lemma 9.3 tells us that if f (x) converges in Ix IP :S pA then so does fi(x). Thus, fi(x) satisfies the conditions of the theorem with N replaced by N - 1. By induction, there is a polynomial h1(x) E 1 + xCCp[x] of degree N - l and a series g(x) E 1 + xCCp[[x]] which converges in lxlP :S pA such that h1(x) = fi(x)g(x). Multiplying both sides of this equation by (1 - ex) and setting h(x) = (1 - cx)h 1(x) gives the final result. Finally, since f has precisely N zeros in lxlp :S p\ it follows that h(x) is unique, since its zeros must be the same as the N zeros of f(x). • Corollary 9. 7

If a

segment of the Newton polygon of

f(x) E 1 + xCCp[[x]] has finite length N and slope A, then there are precisely N values of a for which f(a) = 0 and ordpa = -A.

Observe that in Theorem 9.6, we can also write

f(x) = h(x)G(x) where G(x) E 1 + xCCp[[x]], and h(x) is a polynomial of degree N. (Sometimes, the theorem is also stated in this form in the literature.) In addition, the proof also shows that if we write G(x)

= 1 + d1x + d2x 2 + · · ·

then ldnlPmn < 1, for all n so that IG(x) - lip< 1. An important consequence of Theorem 9.6 is the p-adic analogue of the Weierstrass and Hadamard factorization theorem for entire functions. Indeed let

CHAPTER 9. NEWTON POLYGONS

124

be a power series which converges for all x E Cp. This implies (see Exercise 10) that ordp( an) /n-+oo. We may suppose a0 =f. 0 (otherwise, we can factor out a power of x and consider the quotient which has a non-zero constant term). By dividing by ao, we can assume that J(x) = 1 + a1x + a 2 x 2 + · · · . We can consider this series for lxlP:::; 1 and apply Theorem 9.6 to deduce

where h1(x) is a polynomial and G 1(x) does not vanish in lxlp:::; 1. Since Cp is algebraically closed, we can factor h 1 (x) into linear factors N

h1(x) = 11(1- .Xix), i=l

with IG1(x) - lip < 1. We can now consider larger discs of the form lxlp :::;

pm. This is easily done by changing variables. Set fi(x) = f(x/p). Applying Theorem 9.6 to fi(x) gives

with IG2(x) - lip < 1. Thus,

J(x) = h2(px)G2(px) = h2(x)G2(x) where IG2(x) - lip < 1/p. In this way, we obtain successively

f(x)

= hm(x)Gm(x)

with hm(x) a polynomial consisting of zeros of f(x) lying in the disc lxlp:::; pm and IGm(x) - lip:::; p-m. That is lim Gm(x)

m--too

= 1.

This proves: Theorem 9.8 Let f(x) be a power series defining an entire function in Cp. Then, f(x) can be written uniquely as 00

J(x) = axr

Il (1 i=l

AiX)

CHAPTER 9. NEWTON POLYGONS

125

where a E 2.

3. Show that the discriminant of xn +ax+ b is given by

4. If f(x) E Z[x], show that if the congruence f(x) root mod p then plD(f).

= O(modp) has a double

5. Prove that the x-co-ordinates of the p-adic Newton polygon of x2

f n (x)+l+x+-+ 2!

xn .. · + n!

are given by where

n = b1pn 1 + b2pn 2 + · · · , and 1 ::; bi ::; p - 1.

n1

> n2 > · · ·

Chapter 10

An Introduction to lwasawa Theory As mentioned in the previous chapter, the Weierstrass preparation theorem implies that F[[x]], the ring of formal power series over a field is a unique factorization domain. The proof of this fact can be modified to show that O[[x]] is a unique factorization domain whenever O is a complete local ring, such as Zp. It is customary in Iwasawa theory to designate by A, the ring Zp[[x]]. Then, the p-adic Weierstrass preparation theorem can be used to show that for any f (x) E A, f not zero, we can write it uniquely as

f(x) = pµ P(x)U(x) whereµ?: 0, U(x) is a unit in A and P(x) is a polynomial of the form

P(x) = xn

+ an-1Xn-l + · · · + ao,

pJai,

0 :S: i :S: n - l.

Such polynomials are called distinguished. For instance, Eisenstein polynomials are distinguished. Manin noticed that a "semi-Euclidean" algorithm holds for the ring A. This fact can then be used to derive a variant of the p-adic preparation theorem. This can, in turn, be used to study the structure of A and finitely generated modules over A. We follow Manin below. Lemma 10.1 Let f, g E A = Zp [[x]] and assume f for O :S: i :S: n - l, PY an. Then, we can write

g(x) = q(x)f(x)

= ao + a1x + · · ·, with pJai

+ r(x),

with q EA, and r E Zp[x] is a polynomial of degree at most n - l. Proof.

We first prove uniqueness which means we must consider the equation qf + r = 0. If q, r =/=- 0, we may suppose that either pyr or PY q. Reducing mod p gives that pJr since qf (mod p) is a power series whose first term begins with q(O)anxn. Since py f, we deduce that pJq, a contradiction.

135

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

136

The existence is a little more involved. We define the "head" and "tail" operators H and T given by

and

Clearly Hand Tare Zp-linear. Notice that (i) T(xnh(x)) = h(x) \/h(x) E Zp[[x]]; (ii) T(h(x)) = 0 if and only if h(x) is a polynomial of degree ::; n - l. Also observe that since to find a q such that

PY an, T(f) T(g)

Since f

= H(f) + xnT(f),

is a unit in A. In our notation, we want

= T(qf).

we can write this as

T(g) = T(qH(f))

+ T(qxnT(f))

= T(qH(f))

+ qT(f),

by property (i). Since T(f) is invertible, setting Z = qT(f), we can rewrite this as T(g) = T ( H(f) T(f) Z ) + Z = ( 1 + To H(f)) T(f) Z. Since H(f)

= O(modp), we can invert the operator and get Z

H(f))-1 = ( 1 +To T(f) T(g),

.

from which q can be obtained because T(f) is invertible. This completes the ~~

Theorem 10.2

(p-adic Weierstrass preparation theorem) Any power series

f EA, f(x)=ao+a1x+···

with plai for O::; i::; n - 1, PY an can be written uniquely as f(x)

= (xn + bn-1Xn-l + · · · + bo)u

where plbi and u is a unit in A. Proof.

By Lemma 10.1, we can write uniquely

xn

= qf +r,

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

137

where r is a polynomial of degree less than n. If we write q =Co+ C1X + · · ·

we obtain by comparing coefficients of xn on both sides of the equation, eoan

Since PY an, we find

= l(modp).

PY co so that q is invertible in A.

Therefore,

Since f = anxn+. · · (modp), we must haver= O(modp). This proves existence. Uniqueness is immediate. • Theorem 10.3 A is a unique factorization domain whose irreducible elements are p and the irreducible distinguished polynomials. The prime ideals of A are 0, (p), (p,x) and (P(x)), where P(x) is a distinguished irreducible polynomial. A is a local ring. The ideal (p, x) is the unique maximal ideal. Remark. Thus, A is almost a principal ideal domain. It is also immediate that A is Noetherian. Recall that finitely generated modules over principal ideal domains admit a beautiful structure theorem. Namely, any such module is isomorphic to a direct sum of a finite number of cyclic submodules. lwasawa discovered a similar structure theorem for finitely generated A-modules. To describe this, we define two A-modules Mand M' to be pseudo-isomorphic (written M,....., M') if there is a homomorphism M --t M' with finite kernel and cokernel. This relation is not reflexive (see Exercise 3). However, one can show that for finitely generated torsion A-modules that M,....., M' if and only if M',....., M. The structure theorem for finitely generated A-modules is: Theorem 10.4 module. Then,

(Iwasawa) Let A

= Zp[[x]] and M a finitely generated A-

where the r, ni, mj are non-negative integers and /j(x) are distinguished and irreducible.

Remark. Thus, Mis pseudo-isomorphic to a direct sum of cyclic submodules. This is similar to the case that M is a finitely generated module over a PID,

138

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

except that isomorphism has been replaced by pseudo-isomorphism. A proof modelled along classical lines can be found in several books such as [W, p. 270-276]. Iwasawa applied his theorem to study the growth of the p-part of the ideal class groups of the cyclotomic extensions Ql((pn ). More generally, suppose we have a tower of number fields

with Gal(Kn/K0 ) c:::c 'lL/pnz. (For example, Kn= Ql((pn+1) is one such tower.) Then,

The extension K 00 /K0 is called a Zp-extension (or sometimes, a r-extension). Let pen be the exact power of p dividing the class number of Kn. Iwasawa proved:

Theorem 10.5 that

(Iwasawa, 1958) There exist integersµ~ 0, >.

~

0 and v such

for all sufliciently large n. Remark. µ, >. and v are usually called the Iwasawa invariants of the 'lLp extension K 00 / K 0 • They may depend on p but are independent of n. Proof. (Sketch) Let r = Gal(K00 / Ko) c:::c Zp. This is a pro-cyclic group. Let Ln be the maximal unramified abelian p-extension of Kn. By class field theory, Xn = Gal(Ln/ Kn) is isomorphic to An, the p-Sylow subgroup of the ideal class group of Kn. Let L 00 = U~=lLn and X = X 00 = Gal(L 00 /K00 ). Each Ln is Galois over Ko since Ln is maximal, so that L 00 / K 0 is Galois. Let G = Gal(L 00 / K 0 ). The idea is to make X into a r-module, and therefore, a A-module. Iwasawa [Iw] shows that this can be done, and that it is finitely generated and A-torsion. By the structure theorem, it is pseudo-isomorphic to a direct sum of modules of the form and

A/(P(xl).

One can study the structure of Xn from the structure of X. For instance, if J(,0 / Ko is a Zp-extension in which exactly one prime is ramified and totally ramified (like in the case Ko= Ql((p) and K 00 = Un~1Ql((pn+1) where only pis ramified and totally ramified), one can show

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

139

By Exercise 2, A/(!) is infinite for any f E A. Since An is finite, we must therefore have r = 0. Thus, X is pseudo-isomorphic to a direct sum of modules:

where each gj ( x) is distinguished (but not necessarily irreducible). Since ( 1 + x )Pn - 1 is distinguished, we can apply the division algorithm to deduce that (see Exercise 5) for n large,

Thus, these quotients contribute µpn, where µ = I:;;=l ki to en. To deal with the other factors, note that (1 + x)Pn = 1 + p(polynomial) + xPn so that if gj (x) has degree dj, then xdj = p(polynomial)(modgj) since g1 is distinguished. By induction,

for k 2: dj. Therefore, if pn 2: dj, then (1

+ x)Pn = 1 + p(poly.)(modgj)-

Thus,

(g1 (x), (1 + x)Pn - 1) = (gj(x),p(poly.)), from which it is not hard to see (by an application of Lemma 10.1) that

for n sufficiently large. These terms therefore give a contribution of An with A = I:;j d1 to en. Putting it all together, we obtain en = µpn +An+ z; with z; bounded. This completes our sketch of the proof. • Iwasawa conjectured for cyclotomic Zp-extensions (that is, extensions K= c:;; UnKo((pn) ) we must haveµ= 0. He constructed non-cyclotomic ZP-extensions which have µ > 0. His conjecture has now been proved if K 0 /Q is abelian by Ferrero and Washington [FW]. It is still open in the general case. As a consequence of the Ferrero-Washington theorem, we have: Corollary 10.6 Let hn be the class number of Q( (pn+1). Let pen be the power of p dividing hn. Then, there are constants A 2: 0 and z; such that en = An + z; for all n sufficiently large. Remark. In other words, thanks to the fact that µ = 0, the growth of the power of pin hn is linear inn. It is interesting to observe that by using analytic

140

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

class number formulas or the Brauer-Siegel theorem, such a sharp estimate on the growth of the p-part cannot be deduced (see Exercises 7 and 8). If Fis a number field and we consider a Zp-extension of F, then some results are known. For example, Iwasawa proved that if F has class number coprime top and. there is only one prime in F lying above p, then any Zp extension of F hasµ=,\= v = 0. In particular, if pis a regular prime, the class number of Q( (pn+1) is coprime to p for all natural numbers n.

Iwasawa derived a "Wieferich-type" congruence involving the Bernoulli numbers whenever µ =I= 0. More precisely, he showed that if µ > 0, then there is an even integer 2 ::; j ::; p - 3 such that p divides the numerator of Bj and Bj+p-l = Bi ( d 2) ·+ 1 _ . mo p ,

J

p-

J

where the second condition is recognized as a stronger congruence than the one implied by the Kummer congruences. The converse, however, does not hold. It can happen that the stronger Kummer congruence holds. For example, 7 · 13 2 16 4 5 · 17 so that the "stronger" Kummer congruence holds for the pair (p,j) = (13,4). Still, the above criterion allows one to check that µ = 0 for all primes p < 8,000,000 for the classical cyclotomic Zp-tower. Bl6 B4 ---

25 •

The motivation for making the conjecture that µ = 0 comes from algebraic geometry. We will try and describe it below. The inspiration comes from the situation in function fields. Recall that if lF q is a finite field, we may consider the function field lF q ( x) which in many ways is analogous to the field of rational numbers. We may consider a finite algebraic extension k = lF q ( x, y) of lF q ( x) and think of it as analogous to a finite extension of the rational numbers. Such an extension is given by a curve of the form f(x, y) = 0 and we may consider the zeta function of k which can be written as

P(q-s)

(k(s) = (1- q-s)(l - ql-s) where P(x) is a polynomial E Z[x] of degree 2g, with g being equal to the genus of the curve. If we wanted to construct a Zp-extension in the function field case, the natural tower to consider is

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY so that if kn

= klF

qPn ,

141

then we have the tower k

= ko

C k1 C · · · C

kn C · · · C k00

is also a Zp-extension. It turns out that the zeta function of kn can be written as (see [W, p.129])

where 2g

Pn(x)

=

II (1 - of xPn) = II j=l

P((x).

(Pn=l

Several facts are known about P(x) and we review them now.

= IT~! 1(1 - ajx), lajl = q112 (Riemann hypothesis for (b) P(l) = IT~! 1(1- aj) = h(k), the class number of k; (c) h(kn) = I1~!1 (1 - ar) = ho IT.;Pn=l,(#l P(().

(a) P(x)

curves);

It easily follows from (c) that if pen is the exact power of p dividing h (kn), then en = >.n + v, (see Exercises 9 and 10) with ).. :; 2g. In particular, µ = 0. Similarly, if£ #- p is another prime, and £in is the exact power of £ dividing h(kn), then f n is bounded. By analogy, we may conjecture a similar result in the number field case. Washington [W2] proved that if k/!Q is abelian, then fn is bounded for the cyclotomic Zp-extensions.

The function field case also suggests that the Iwasawa invariant ).. is analogous to the genus of a curve. Motivated by this analogy, Kida [K] derived for the >.-invariant the analog of the Riemann-Hurwitz formula for the genus of a curve. This formula has some nice applications to the study of zeros of p-adic £-functions. There is a further analogy that motivated Iwasawa and this has to do with the Riemann hypothesis. The Galois group of Fp/lFP is procyclic generated by the Frobenius automorphism (x f----+ xP). The Frobenius automorphism acts on the lFP points of the curve and thus on its Jacobian. P( x) above is essentially the characteristic polynomial of the Frobenius automorphism acting on the Jacobian of C. This interpretation of the zeta function was an essential factor in the resolution of the Riemann hypothesis for the zeta functions of curves. Returning to the case ofµ= 0, we note that An is approximately (Z/pnz/·. One can construct a direct system by defining a map An----, Am for n < m given by sending the class of an ideal Cl to the class of the ideal generated by Cl in the

142

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

ring of integers of Fm, so that

This reminds one of the p-power torsion of the Jacobian of a curve or more generally for an abelian variety. Perhaps, the p-adic zeta function can be written as a "characteristic polynomial or power series" of some topological generator of ZP acting on X. It was this analogy that motivated Iwasawa to formulate the so-called "main conjecture." More precisely, let K = (Q)( (p + (; 1 ) be the maximal real subfield of (Q)( (p). We can define the p-adic zeta function of K as

I wasawa proved that there is a formal power series

A(x) =

IT

f(x, x)

such that

A((l

(K,p(s)

E

+ p)s -

Zp[[x]]

1)

= h((l + p)s - 1)'

with h(x) = (x-p)/(x+l). This is analogous to the formula for the zeta function of a curve with A(x) playing the role of P(x). It is possible to interpret A(x) as the "characteristic polynomial" of some generator of Gal(K=/ K 0 ) '::::' Zp. A(x) is essentially the characteristic polynomial of X= and this fact came to be known as the "main conjecture of Iwasawa theory." More precisely, with X as above, we note that complex conjugation acts on each An and therefore on X. Let x-={xEX:

x=-x}.

This group plays the analogous role of the Jacobian of a curve in I wasawa theory. Then x- is a A-module which is Noetherian and torsion and so by the structure theorem,

x-

cv

ffi 1 A/(!J(x)aj).

Let F(x) = IT~ !J(x)aj. The main conjecture states that there is a unit u(x) EA such that A(xJ = F(x)u(x). This was first proved in 1980 by Mazur and Wiles using high-powered techniques from algebraic geometry. Later work of Thaine, Rubin and Kolyvagin has led to simpler proofs (see Rubin's appendix to Lang

[L]).

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

143

Mazur extended Iwasawa theory to study the Mordell-Weil groups of elliptic curves (and more generally of abelian varieties) in Zp-extensions. In 1970, he made the following conjecture. Suppose A is an abelian variety over a number field F. Assume that pis a prime such that A has good, ordinary reduction at all primes of F lying above p. Let F 00 / F be the cyclotomic Zp extension. Then A(F00 ) is finitely generated. He succeeded in proving this conjecture under the assumption that A(F) and IIIA(F)p are both finite. Here, IIIA (F)p denotes the p-primary subgroup of the Tate-Shafarevich group of A over F. These investigations have had significant applications towards the proof of the Birch and Swinnerton-Dyer conjectures in the theory of elliptic curves. Since the p-adic £-functions have now been interpreted as some form of characteristic polynomials of some operator, it may be interesting to study their zeros and inquire into their significance. It is fair to say that the study of the zeros of p-adic £-functions is still in its infancy. Washington [W3] has done some work on this question and has related it to the Buniakowski-Schinzel hypothesis about the existence of primes in polynomial sequences. It is possible that sieve theory may have some application to such questions and shed further light on the matter.

Exercises 1. Let K be a finite extension of QP and O its ring of integers. Prove Lemma 10.1 with A replaced by O[[xl].

2. Let f E A. Using Theorem 10.3, show that A/(f) is infinite. (Hint: By the structure theorem, it suffices to consider the two cases f = p and f distinguished and irreducible. In the former case, A/(p) c::: IFp[x] and the latter case, apply the division algorithm.)

3. Show that pseudo-isomorphism is not reflexive. Namely, show that M rv M' does not imply M' rv M. (Hint: notice that (p, x) rv A. If A rv (p, x), consider the image f of 1 under this map. Now apply the previous exercise.)

4. Prove Theorem 10.2 (Weierstrass preparation theorem) directly by writing

u

= co +c1x + · · ·

144

CHAPTER 10. AN INTRODUCTION TO IWASAWA THEORY

and solving the system of equations that result from

f(x)

= (xn + bn-1Xn-l + · · · + bo)u.

5. Let f(x) be distinguished of degree n. Show that

6. If f(x) is distinguished, show that f(x)m is distinguished. 7. Let dm denote the discriminant of Q((m) and d;t_ the discriminant of Q((m+(;:;;,1 ).. Show that log ldml ""¢(m) logm and log ld;t.l "" ½¢(m) logm as m---+oo. 8. Let hn be the class number of Q( (pn+1) and ht, the class number of its totally real subfield. It is known that hn = ht, h:;; for some integer h:;;. The Brauer-Siegel theorem (see [W, p. 44]) implies log h;;-

""

pn(p- 1) (n 4

+ 1) log p

as n---+oo. Write hn = pen h~ with h~ coprime to p. Using the BrauerSiegel theorem and Iwasawa's theorem, show that h~ tends to infinity as n---+oo. 9. Let r = ordp(a - 1). If r ~ 1, show that ordp(aPn - 1) = r

+ n.

10. If ordp(a - 1) = 0, show that ordp(aPn - 1) = 0. Deduce that

ordp

(rr

(ar -

1))

=An+ V

J=l

with)..= #{j : ordp(aj - 1) > O}

~

2g and v bounded.

Bibliography [Ap]

T. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, 1976.

[Bo]

R. Bojanic, A simple proof of Mahler's theorem on approximation of continuous functios of a p-adic variable by polynomials, J. Number Theory, 6 (1974)pp. 412-415.

[CJ

J. Cassels, Local Fields, London Mathematical Society Student Texts, 3, Cambridge University Press., 1986.

[Cl]

W. E. Clark, The Aryabhatiya of Aryabhata, University of Chicago Press, Chicago, Illinois, 1930.

[Co]

R. Coleman, On the Galois groups of the exponential Taylor polynomials, Enseign. Math., 33(2) (1987), no. 3-4, 183-189.

[EM]

J. Esmonde and M. Ram Murty, Problems in Algebraic Number Theory, GTM, Vol. 190, Springer-Verlag, 1999.

[FW] B. Ferrero and L. Washington, The Iwasawa invariant µP vanishes for abelian number fields, Annals of Math., 109 (1979) pp. 377-395. [G]

F. Gouvea, p-adic Numbers, Universitext, Springer, 1997.

[GS]

D. Goldfeld and L. Szpiro, Bounds for the order of the Tate-Shafarevich group, Comp. Math., 97 (1995), no.1-2, pp. 71-87.

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R. Greenberg, Iwasawa theory, Past and Present, in Advanced Studies in Pure Mathematics, 30 (2001), pp. 335-385.

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K. Iwasawa, On some properties off-finite modules, Annals of Math., 70(2) (1959), pp. 530-561.

[J]

G.J. Janusz, Algebraic Number Fields, Second Edition, Graduate Texts in Mathematics, 7, American Math. Society, Providence, Rhode Island, 1996.

[K]

N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions, Vol. 58, Springer-Verlag, 1977.

[K2]

N. Koblitz, p-adic analysis: a short course on recent work, London Math. Society Lecture Notes, 46 (1980), Cambridge University Press, Cambridge, New York.

[Ki]

Y. Kida, £-extensions of CM fields and cyclotomic invariants, Journal of Number Theory, 12 (1980), pp. 519-528. 145

146 [L]

BIBLIOGRAPHY S. Lang, Cyclotomic Fields, second edition, Springer-Verlag, 1990.

[MW] B.Mazur and A. Wiles, Class fields of abelian extensions of Ql, Invent. Math., 76 (1984), no. 2, 179-330. [MO] J. Oesterle, Nouvelles approches du "theoreme" de Fermat, Sem. Bourbaki, 694, Asterisque 161-162, S.M.F., (1988), pp. 165-186. [MaMu] L. Mai and M. Ram Murty, The Phragmen-Lindeli:if theorem and modular elliptic curves, Contemp. Math., 166 (1994), pp. 335-340, American Math. Society, Providence, R.I. [M]

M. Ram Murty, Problems in Analytic Number Theory, GTM/RIM, Vol. 206, Springer-Verlag, 2001.

[MM] M. Ram Murty and V. Kumar Murty, Non-vanishing of L-functions and Applications, Progress in Mathematics, 157, Birkhaiiser-Verlag, 1997. [MR]

M. Ram Murty and Marilyn Reece, A simple derivation of ((1 - k) = -Bk/k, Functiones et Approximatio, 28 (2000), 141-154.

[OJ

H. Osada, The Galois groups of the polynomials xn + ax 1 + b, J. Number Theory, 5 (1987), no. 2, 230-238.

[Ra]

C. S. Rajan, On the size of the Shafarevich-Tate group of elliptic curves over function fields, Comp. Math., 105 (1997), no. 1, pp. 29-41.

[Ro]

A. Robert, A course in p-adic analysis, GTM, Vol. 198, Springer-Verlag, 2000.

[Se]

E. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956), 287-302.

[Si]

C. L. Siegel, Zu zwei Bemerkungen Kummers, Nachr. Akad. Wiss., Gottingen, Math.-Phys. Kl. (1964), no. 6, 51-57. (See also, Gesammelte Abhandlungen, Springer-Verlag, Berlin, 1966, Vol. 3, 436-442.)

[SJ

J. H. Silverman, Wieferich's criterion and the abc-conjecture, J. Number Theory, 30 (1988), no. 2, 226-237.

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L. Washington, Introduction to Cyclotomic Fields, GTM, Vol. 83, Second Edition, Springer-Verlag, 1997.

[W2]

L. Washington, The non-p-part of the class number in a cyclotomic Zpextension, Inv. Math., 49 (1978), 87-97.

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L. Washington, A family of cubic fields and zeros of 3-adic L-functions, Journal of Number Theory, 63 (1997), no. 2, 408-417.

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A. Weil, Number Theory, An Approach Through History, From Hammurapi to Legendre, Birkhaiiser, 1984.

BIBLIOGRAPHY

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147

A. Wiles, Modular elliptic curves and Fermat's last theorem, Annals of Math., 141(2) (1995), no. 3, 443-551.

Index p-adic p-adic p-adic p-adic

r-function, 67 integers, 33 interpolation, 65 Weierstrass preparation theorem, 159, 161 p-adic distribution, 103 p-adic regulator, 130

exponential function, 37, 60 Fermat quotient, 43, 98 Fermat's Last Theorem, 1, 2 Ferrero, 8, 100, 164 Frobenius automorphism, 167 functional equation, 84 functional equation of ((s), 2 fundamental unit, 120

ABC conjecture, 5 Adams congruences, 22 Ankeny, 120, 127 Artin, 120, 127 Artin-Hasse exponential, 40 Aryabhata, 11

Galois group, 149 Gauss sum, 83, 117 generalized Bernoulli number, 121 Greenberg, 100 Gross and Koblitz, 7, 79, 100

Bernoulli distribution, 105 Bernoulli numbers, 2, 11 Bernoulli polynomial, 12, 85, 106 Birch, 168 Brahmagupta, 129 Brauer-Siegel theorem, 170 Brumer, 120, 131

Haar distribution, 105 Hensel, 3 Hensel's lemma, 49, 51, 64 Hensel's lemma for polynomials, 54 Hurwitz (-function, 5, 27, 86 imprimitive character, 117 index of irregularity, 5 inertia group, 150 irregular primes, 23 Iwasawa, 4, 162 Iwasawa's logarithm, 61

Cauchy sequences, 32 Chebotarev density theorem, 152 Chowla, 120, 127 Clausen, 20 Coleman, 154 completion, 32 conductor, 117 cyclotomic polynomial, 63

Jacobian, 167 Kida, 166 Koblitz, 114 Krasner's lemma, 59 Kubota, 14 Kubota and Leopoldt, 4 Kummer congruences, 3, 20, 22, 92, 111 Kummer criterion, 46, 95 Kummer-Vandiver conjecture, 5

decomposition group, 149 Dirac distribution, 105 Dirichlet, 118 Dirichlet character, 83 discriminant, 153 Eisenstein polynomials, 159 Eisenstein's criterion, 48, 135 elliptic curves, 4 equivalent norms, 30 Euler, 13 Euler product, 1

Laurent series, 34 Leopoldt, 14, 117, 120 local ring, 33, 159 148

INDEX locally analytic, 62 locally compact, 34 locally constant, 104 logarithm, 38, 61 Mobius function, 40, 54 Mahler's theorem, 68 Manin, 114, 159 Mazur, 9, 99, 103, 168 Mazur measure, 108, 111 measure, 106 Mellin transform, 99, 103 Minkowski's theorem, 150 Newton polygon, 133, 155 Newton's method, 51, 63 norm, 29 Ostrowski, 3 Ostrowski's theorem, 30 prime number theorem, 2 primitive character, 117 pseudo-isomorphic, 162 quadratic reciprocity, 151 radius of convergence, 37 ramification index, 56 ramified prime, 56, 150 regular prime, 3 regularized measure, 107 regulator, 130 relative discriminant, 150 Riemann, 1 Riemann ( function, 1, 16 Riemann hypothesis, 2 Riemann hypothesis for curves, 166 Riemann-Hurwitz formula, 166 Schur, 154 Shafarevich-Tate groups, 6 Siegel, 4 squarefull number, 6 Stickelberger, 79 Stirling numbers, 75 Strassman's theorem, 41, 42

149 Swinnerton-Dyer, 168 symmetric group, 153 Teichmiiller representative, 53 Tiechmiiller character, 53 totally ramified, 56 trinomials, 153 uniformizer, 56, 57 unramified, 56 Vandiver, 96 von Staudt, 20 von Staudt-Clausen theorem, 46, 112 Voronoi congruences, 20 Voronoi's theorem, 21 Washington, 8, 164 Wieferich, 98 Wieferich criterion, 5 Wiles, 4, 9 Wilson's theorem, 25, 41, 67, 98

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