Lectures on P-Adic L-Functions. (AM-74), Volume 74 9781400881703

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Annals of Mathematics Studies Number 74

LECTURES ON /?-ADIC L-FUNCTIONS BY

KENKICHI IWASAWA

PRINCETON UN IVERSITY PRESS AND UNIVERSITY O F TOKYO PRESS PRINCETON, NEW JE R S E Y 1972

Copyright © 1972 by Princeton University Press All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review. LC Card: 78-39058 ISBN: 0-691-08112-3 AMS 1971: 10.14, 10.65, 12.50

Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

PREFACE These are notes of lectures given at Princeton University during the fall semester of 1969. The notes present an introduction to p-adic Lfunctions originated in Kubota-Leopoldt [10] as p-adic analogues of classi­ cal L-functions of Dirichlet. An outline of the contents is as follows. In §1, classical results on Dirichlet’s L-functions are briefly reviewed. For some of these, a sketch of a proof is provided in the Appendix. In §2, we define generalized Ber­ noulli numbers following Leopoldt [12] and discuss some of the fundamen­ tal properties of these numbers. In §3, we introduce p-adic L-functions and prove the existence and the uniqueness of such functions; our method is slightly different from that in [10].

§4 consists of preliminary remarks

on p-adic logarithms and p-adic regulators. In §5, we prove a formula of Leopoldt for the values of p-adic L-functions at s = 1. The formula was announced in [10], but the proof has not yet been published. With his per­ mission, we describe here Leopoldt’s original proof of the formula (see [1], [7] for alternate approach). In §6, we explain another method to de­ fine p-adic L-functions. Here we follow an idea in [9] motivated by the study of cyclotomic fields. In §7, we discuss some applications of the results obtained in the preceding sections, indicating deep relations which exist between p-adic L-functions and cyclotomic fields. Conclud­ ing remarks on problems and future investigations in this area are also mentioned briefly at the end of §7. Throughout the notes, it is assumed that the reader has basic knowl­ edge of algebraic number theory as presented, for example, in BorevichShafarevich [2] or Lang [11]. However, except in few places where cer­ tain facts on L-functions and class numbers are referred to, no deeper v

vi

PREFACE

understanding of that theory may be required to follow the elementary arguments in most of these notes. As for the notations, some of the symbols used throughout the notes are as follows: Z, Q, R, and C denote the ring of (rational) integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively.

Z

r

and Q

r

will denote the ring of p-

adic integers and the field of p-adic numbers, respectively, p being, of course, a prime number. In general, if R is a commutative ring with a unit, Rx denotes the multiplicative group of all invertible elements in R, and R[[x]] the ring of all formal power series in an indeterminate x with coefficients in R. I should like to express my thanks here to H. W. Leopoldt for kindly permitting us to include his important unpublished results in §5, and also to R. Greenberg, J. M. Masley, and F. E. Gerth for carefully reading the manuscript and making valuable suggestions for its improvement. Kenkichi Iwasawa PRINCETON, OCTOBER 1971

CONTENTS PREFACE ...............................................................................................................

v

§1.

Dirichlet’s L-functions ............................................................................

3

§2.

Generalized Bernoulli Numbers...............................................................

7

§3.

p-Adic L-functions .....................................................................................

17

§4.

p-Adic Logarithms and p-Adic Regulators..........................................

36

§5.

Calculation of L p (l;Y ) ............................................................................

43

§6.

An Alternate Method ...................................................................................

66

§7.

Some Applications ......................................................................................

88

APPENDIX ............................................................................................................. 100 BIBLIOGRAPHY .................................................................................................... 105

vii

LECTURES ON p-ADIC L-FUNCTIONS

§1. DIRICHLET’S L-FUNCTIONS In this section, we shall review some of the well-known classical re­ sults on Dirichlet’s L-functions. For proofs and more details, we refer the reader to [2], [6], [13].

1.1. L et n be a positive integer, n > 1. A map X: Z -> C from the ring of integers Z to the complex field C is called a Dirichlet character to the modulus n if it has the properties that i) X(a) depends only upon the residue class of a mod n, ii) X(ab) = X(a)X(b) for any a, b in Z, and iii) X(a) ^ 0 if and only if a is prime to n: (a, n) = 1. Obviously, there is a natural one-to-one correspondence between such Dirichlet characters to the modulus n and the characters (in the usual sense) of the multiplicative group (Z /nZ )x of the residue class ring Z/nZ. Hence a Dirichlet character to the modulus n is usually identi­ fied with the corresponding character of (Z /n Z )x . L et X

be a Dirichlet character to a modulus m and let m be a

factor of n. Define X(a), a e Z, by X(a) = X (a) , = 0

if (a, n) = 1, ,

if (a, n) > 1.

Then X is a Dirichlet character to the modulus n. We say that the character X is induced from X . A Dirichlet character X to a modulus n is called primitive if X is not induced from any character to a modu­ lus m with m < n. n is then called the conductor of X and is denoted

3

4

p-ADIC L-FUNCTIONS

by fy. In the following, all Dirichlet characters we consider will be assumed as primitive. Let X 1 and X2 be such (primitive) Dirichlet characters and let and f2 be the respective conductors. Then there is a unique (primitive) Dirichlet character X with conductor f dividing fjf2 such that X(a) = X 1(a )X 2(a) for integers a prime to f 1 f2: (a, fjf2) = 1- X is called the product of X± and X2: X = X1X2' Note that X(a) = X 1(a) X2(a) is not necessarily true if (a, f 1f2) > 1. The set of all primitive Dirichlet characters form an abelian group with respect to the above multiplication. The identity of the group is the principal character X ° defined by X°(a) = 1 for every a in Z; this is the unique character with conductor 1. The inverse of X is the character X which is the complex-conjugate map of X: X(a) = X(a), a e Z.

1.2. Let X be a Dirichlet character and let oo L(s; X) = ^ X(n>n -S • n=l The series on the right converges absolutely for all complex numbers s with Re(s) > 1 so that L (s; X) defines a holomorphic function of s in the half-plane where Re(s) > 1. L(s; X) is called Dirichlet’s L-function for the character X. For the principal character X °, L(s; X°) is nothing but the zetafunction of Riemann, £(s). We shall next describe some fundamental properties of L(s; X). First, L(s; X) can be expressed as an infinite product: L(s; X) = l i d - X ( P ) P - S ) - 1 . P

Re(s) > !>

§1. DIRICHLET’S L-FUNCTIONS

5

where the product is taken over all prime numbers p. Hence L (s; X) 4 0 for Re(s) > 1. By analytic continuation, L (s; X) can be extended to a meromorphic function on the entire s-plane; if X 4 x ° , L (s; X) is holomorphic everywhere, but if X = X °, L (s; X) has a unique pole of order 1, with residue 1, at s = 1. Furthermore, on the s-plane, it satisfies a functional equation as follows. Let r(X) denote the Gaussian sum: f

2771a

K x) = 2 x (a )e a=l

f

’

1 = vc i - f = fx -

and let 8 = 8

X

= 0 ,

if X ( - l ) = 1,

= 1 ,

if X ( - l ) = - 1 .

Then T \=* (| ) r ( § ^ ) L ( s ; X ) = Wx ( i ) 2 r ( i ^ ) L ( l - s ; X)

where T denotes the T-function, X is the inverse of X, and

iw* ' - i The same equality can be written also as L (s; X) = M r ( ^ ) S - , L ( l -g i X) . 2 i^ V f ' T(s) cos 7r(s~ g)

2

Actually, all such analytic properties of L (s; X) can be proved for a much wider class of functions similar to L (s; X) (see [11]). However, in the case of our L (s; X), the proof is simpler; a sketch of such a proof is given in the appendix.

6

p-ADIC L-FUNCTIONS

Let F (z) be a meromorphic function of z defined by X

a=l

e

~ 1

The argument in the appendix also implies that - j^Q~rni ■ X) = the residue of F (z )z “"n_1 at z = 0 1 In)

X

for every integer n > 1. This formula will be used in the next section.

The value of L (s; X) at s = 1 (which is not included in the above formula) is particularly important in number theory. For the principal character X °, we already mentioned that L(s; X °)( = £(s)) has a simple pole with residue 1 at s = 1 so that lim ( s - l ) L ( s ;X ° ) = 1 . s->l On the other hand, if X is not principal, then L (l; X) ^ 0, oo and its value is explicitly given as follows (see [8]):

L(l; X) =

^

X(a) log(l - C a)

a

X ( a ) l o g | l - C a |,

if X (—1) = 1,

a

if X ( - l ) = - l , a 27ri where f = f , £ = e, f , and the sum is taken over all integers a such X that 1 < a < f, (a, f ) = 1.

§2. GENERALIZED BERNOULLI NUMBERS In this section, we define generalized Bernoulli numbers and discuss some of the fundamental properties of these numbers which are needed in the later sections.

2.1.

The definition of (ordinary) Bernoulli numbers is well known; let t

be an indeterminate and let

F(t) =

e - 1 Expand F(t) (formally) into a power series of t:

n=0 The coefficients B n, n > 0, are called Bernoulli numbers. (Sometimes t_ 1 F(t) is used as the generating function instead of F (t).) It is clear that Bn are rational numbers: B0 - 1, Bj - ^ , B2 - g , B3 - 0, ... .

Since F (—t) = F(t) - t , we also see that Bn = 0 ,

for odd n > 1 .

7

8

p-ADIC L-FUNCTIONS

Let x be another indeterminate and let

F(t, x) = F(t) e

tx

te^1+x^t el - 1

As for F(t), let

n=0 Since

n=0

n=0

we have

B„(x) = X

( i ) Bi ^

’

n- ° ‘

1=0

Hence Bn(x), n > 0, are polynomials of x with rational coefficients, and they are called Bernoulli polynomials. As B Q = 1, Bn(x) is a monic polynomial of degree n: B 0(x) = 1, B j(x) = x + j ,

B 2( x ) = x2 + x + g-, ... .

It is clear that Bn(0) = Bn ,

n > 0.

We generalize the above definition of Bn and Bn(x) as follows: let X be a Dirichlet character with conductor f = f^, and let f X(a)teat a=l

™A

f

F x ( t ,.) = F x ( .) .« . 2 a=l

e

1

9

§2. GENERALIZED BERNOULLI NUMBERS

Expanding these into power series of t, let O O n=0

n=0 Then n n > 0. i=0 Let Q be the rational field and let Q(X) denote the field generated over Q by all values X(a), a e Z. It is clear that Bn y are in Q(X) and Bn yOO are polynomials in Q(X)[x]. B n y and B n y(x)> n > 0, are called generalized Bernoulli numbers and generalized Bernoulli polynomials, respectively, belonging to the Dirichlet character X. If X = X °, the principal character (f = 1), then F ^ (t) = F(t), F ^ (t, x) = F(t, x) so that B

n > 0.

We shall next describe some simple properties of Bn y and Bn

2.2.

n>0

1) f

if X ^ x°.

2) a=l

10

p-ADIC L-FUNCTIONS

Hence

deg(BjL X ^ K n 3)

Since f

Xtegdz*

V — ) ■2 a=l

e

““ 1

f )te ^ _a+x^ S X ( - l ) X (f-a e * -l

a=l

= X ( - l ) F x (t,x ) ,

if

X^X°,

we have

(-D nBnfX(-x) = X(-l)BnfX(x).

n > 0.

Putting x = 0, we obtain Bn X = 0 , where 8 = 8 4)

X. is

i f X ^ X 0 , n ^ S mod 2,

defined as in §1.

Similarly, f

Fx(t,x) = r 2

a=l

*(a>F(ft'

implies f

Bn ,X « = f 2

X(a)fnBn( * ^ ) ,

n > 0.

(¥)■

n > 0.

a=l

In particular (x = 0),

^

\ x - f i

a=l

x(,>," Bn

§2.

5)

11

GENERALIZED BERNOULLI NUMBERS

For any integer k > 0, let k Sn, X(k) =

S

X(a)a° '

11 ^

a=l For X = X°,

^(k) will be simply denoted by Sn(k): k v k> = 2 a" ’ a=l

n ^ °-

Now, f F ^ (t, x) - F ^ (t, x - f ) = ^

X(a)t e(a+ x_f)t

a=l implies f Bn,

“ Bn,

= n 2 ^ (a) ( a+x“ f )n_1 , a=l

n > 0.

Replacing n by n+1 in the above and adding the equalities for x = f, 2f, ...,k f, we obtain

Sn.X * ^ r ( B n+l ,X < k f) - B n *l,X < 0 » '

In particular (X = X ° ) ,

V k) = HTT(Bn+l ( k ) - Bn+lW )>

n- k > °-

We now prove a formula which motivated the definition of Bn y.

THEOREM 1. For a Dirichlet character X and for any integer n > 1,

L ( l —n; X) = -

n

.

12

p-ADIC L-FUNCTIONS

Proof. As stated in 1.2, — LC1

r(n)

= the residue of F (z)z n 1 at z = 0 .

X

Since f

■a=l 2 ^e

■n=0 2

1

we obtain the formula immediately. The functional equation for L (s; X) shows that if n = 8 mod 2 ,

n > 1, 8 = 8 , A

then U „; X) -

4® fe )n 9 :0 \ t /

n—i

r(n)(-l) 2

2 i' Since L(n; X) ^ 0, n > 1, it follows that Bn X ^ 0 »

if n > 1, n = 8 mod 2.

We already know the cases where Bn y = 0- Since 8^, = 8 — and 8^ q = 0, we may summarize our results as follows:

T h e o r e m 2.

i)

X = X° (the principal character): B q = 1, B1 =

\

Bn

£0,

Bn =

/or even n > 0 ,

*O TocW

n > * '

13

§2. GENERALIZED BERNOULLI NUMBERS

ii) B o ,X = o , B n, X t 0 ,

for n > 1, n = 8

B n, X = 0 ,

for n > 1, n ^ 8

X X

mod 2, mod 2.

We would like to add some remarks on the preceding theorems. For X = X °, the above formula for L(n; X) states n +i g C(n) = ( - 1 ) 2 \ (2n)n , z n;

n > 1, n = 0 mod 2,

or, equivalently,

B

n +i = (—l ) 2

£(n) ,

for even n > 1.

(2 n f Since £(n) > 0, we know more precisely than above that B 2n > 0 ,

for odd n > 1,

®2n < ^ »

f°r even n > 1.

Next, let X be a Dirichlet character with 8

X

Theorem 2 states that

f Using the formula in 4) and

X(a) = 0, we see a=l

= 1 , i.e., X(—1) = —1.

14

p-ADIC L-FUNCTIONS

f B 1 ,X - 2 a=l

- 2 M a=l

t

+i )

f ■ r 2 » * a=l

( - r si ,x « ) ) -

Hence f ^ X(a)a ^ 0 , a=l

if X ( - l) = - 1 .

It seems that no elementary proof of this simple fact is known.

Let X be again an arbitrary Dirichlet character with f = f . We X know that B n y , n > 0, are algebraic numbers in the field Q(X). Various 2.3.

results are known on the arithmetic properties of the numbers Bn y. We shall prove here two elementary lemmas which we need later. Let p be a fixed prime number, and Qp the field of p-adic numbers. Let Qp(X) denote the field generated over Qp by X(a), a e Z (in an algebraic closure of Qp). Qp(X) is a locally compact topological field containing Q(X) as a dense subfield.

L e m m a 1.

In Qp(X), Bn , X = ,h-*» lim ip bi Sn ,X (Ph f ) ’

Proof. This follows immediately from the fact that

n ^ °-

15

§2. GENERALIZED BERNOULLI NUMBERS

and that Bn+i y(x) - Bn+1 ^(0) = (n+l)Bn ^x + (terms of degree > 2 in x).

For X = X °, f = 1, the lemma states Bn = ulim \

h->oo p

LEMMA 2.

Sn(Ph> -

n > °-

The denominator of the rational number B n (n > 0) is divisible

at most by p,

r\

but not by p .

Proof. By the above lemma, it is sufficient to show that Sn(ph) = o for all h > 1.

mod ph_1

If h = 1, this is trivial. Let h> 1. Since each integer a,

1 < a < p*1, can be uniquely written in the form a = b + cph_1 ,

1 < b < p^1 -1 , 0 < c < p,

we have «h P sn(Ph) = 2 a=l

h—1 P an

- p 2 =b”Ps n(ph b=l

mod Ph 1 •

Hence the lemma is proved by induction on h.

The lemma follows also from the well-known theorem of v. StaudtClausen which states that B 2n = -

”

mod Z ,

for n > 1,

where the sum is taken over all prime numbers p such that p—1 divides 2n. For further arithmetic properties of Bn, see the papers [5], [12]. It is proved, for example, that if X 4- x ° , then fx Bn> Y is an algebraic

16

p-ADIC L-FUNCTIONS

integer for every n > 0 and that if f

A

even ^-Bn ^

an algebraic integer.

is not a power of a prime, then

§3.

p-ADIC L-FUNCTIONS

Let p be a prime number and let ft_ be an algebraic closure of Qn. r r In the following, we shall fix both p and Op and consider p-adic func­ tions which are defined on sufficiently large domains in Op and take values in the same field Op. Let L (s; X) be a classical L-function of Dirichlet. The main problem of this section is to find a suitable p-adic function which may be regarded as a p-adic analogue of the classical function L (s; X). To solve this problem, Kubota-Leopoldt looked for a p-adic meromorphic function which takes the same values as L (s; X) at s = 0, —1, —2, ..., observing that by Theorem 1, §2, these values of L(s; X) are algebraic numbers and, hence, may be considered as elements of the algebraically closed field Op. In [10], they succeeded in finding such a function f(s) although the condition f(n) = L(n; X) for n = 0, —1, —2 , . . . , had to be modified slightly, and they named it the p-adic Lfunction for the Dirichlet character X. In the following, we shall first study p-adic (holomorphic) functions which are defined by convergent power series and which take pre-assigned values at s = 0, 1, 2 ,... . Using the results thus obtained, we shall then prove the existence and the uniqueness of the function f(s) as mentioned in the above.

3.1. Let 0

be as above and let |£|, for r value on Op, normalized so that

e 0 , denote the absolute r

Op is a topological field in the metric defined by the absolute value. The topology induced on the subfield Qp is of course the p-adic topology of Qp. Define 17

p-ADIC L-FUNCTIONS

18

q=

P ,

if

= 4 ,

P> 2 ,

if p= 2 .

Let U = Zp ,

D = 1 + qZp ;

U is the multiplicative group of all p-adic units and D is the subgroup of U consisting of all elements of the form 1+qa, a e Zp. For p > 2, let V be the cyclic group of order p—1 consisting of (p—l)-st roots of unity in Qp, and for p = 2, let V = i± li.

Then

U = V x D topologically. Each a in U can be uniquely written in the form a = o)(a) < a > where co(a) and < a> denote the projections of a on V and D re­ spectively, under the above direct decomposition of U. We see easily that if p > 2, then n co(a) = lim ap . n->oo

Now, let Q be the field of all algebraic numbers, i.e ., the algebraic closure of Q in C. In the following, we imbed Q in Qp once and for all, and consider Q as a subfield of Qp. The group V in Qp is then identified with the group of roots of unity in Q with order p -1 or 2, namely, with a subgroup of Cx . Hence, defining o)(a) = 0 for a in Z with (a, p) > 1 , we obtain a map Z - C, a h> co(a)

§3.

19

p-ADIC L-FUNCTIONS

Clearly, this is a Dirichlet character in the sense of 1.1, and we denote it again by co. Note that the conductor of co is q and that it induces an isomorphism (Z/qZ)x ^ V C

3.2.

Cx .

Let K be a finite extension of Qp contained in Qp. K is a lo­

cally compact field in the topology defined by the absolute value, and an oo

infinite series as n

oo.

2 a„, n=0 n Let K[[x]]

an € K, converges in K if and only if |an |-> 0 n 1 be the algebra of all formal power series in x. A

power series A in K[[x]]:

converges at x = f in Qp if and only if |an £ n| 0 as n -> oo. Hence if A converges at 0 , in tip such that lim f

n->0

= 0 .

Then A(x) = B(x) .

Proof. Let oo

A(x) - B(x) = 2 cn x" ’ n=0

cn € K -

p-ADIC L-FUNCTIONS

20

Assume that A(x) ^ B(x) and let nQ be the minimum of n such that cn ^ 0. Then we have

- c nQ - e

S

j L*

c n £n“ no_1

n>n0 for every

Since

• -> 0 as i -> oo and since the sum on the right side

is then bounded, we get the contradiction

For a power series A = A(x) = 2 an xn in K[[x]], define n=0 n

llAli = sup iani ■ n Let Pj^ denote the set of all A in K[[x]] with ||A||
0; ||A || = 0 only when

A= 0 ,

IIA + B|| < max(||A||, ||B||) , IIcA|| = |c| ||A||, ||AB|| < ||A|| ||B|| ,

c * K.

LEMMA 2. Pj^ is complete in the norm ||A|| so that it is a Banach alge­

bra over the local field K. Proof. Let A^, k > 0, be a fundamental sequence in Pj^ and let oo

AkW =

2

ank )x" ’

ank ) < K-

n=0 The lemma can be proved by routine argument in the following steps:

§3.

21

p-ADIC L-FUNCTIONS

i)

For each n > 0 , lim aj^) = an exists in K, k->oo O O ii) A = A(x) = S a x belongs to Pt^, n=0 iii) lim A^ = A in the norm topology of P Kk->oo

in K[x] by

For each n > 0, we define a polynomial ( x \ _ x ( x - l ) .... (x-n+1) W ~ n!

( © = >)•

It is obvious that

T h e follow ing lemma e s tim a te s

L e m m a 3.

|n!| :

F o r n > 1, n

n

|p|p_1 < in!i < np|p|p

Proof. Let N n

=

2

aiP '

0 < aj < p, aN

-

i=0

N s =

2

ai •

i=0 It is known in elementary number theory that the highest power of p which divides n! is n—s pp - 1 • Hence n—s

n

ln ! l = I p I p _ 1 > Ip IP 3 1 •

£

0,

22

p-ADIC L-FUNCTIONS

n -1 (Since s > 1, we have even |n!| > |p|p *.) As a^ > 1, we see that p^ < n so that N < log n/ log p. Hence s < (p -l)(N + l) < (p—1) (log n/ log p + 1) and n—s n s |n!| = |p|p_1 = Ip I ^ 1 p P "1 < np|p|p_1 ,

q.e.d.

n Note that |p|p_1 < |n!| holds also for n = 0. Hence

II(n)II ^ lpl ^

’

for a11 n ^

Now, let bn, n > 0, be a sequence of elements in K and let

2

cn '

( ? ) bi •

"> »'

i=0 so that

e -‘ V

b„ % = Y n nt

n=0

Cn n ni

n=0

Then cn e K, n > 0, and

V-SC?)'!’ i=0

"2°1

THEOREM 1.

Let r be a real number such that 0 < r < |p|p _ 1 - Suppose

that lc nl ^ C r11 ,

for all n > 0 ,

§3.

with some C > 0.

p-ADIC L-FUNCTIONS

23

Then there exists a unique power series A(x) in Pj^

which has the following properties: i)

A(x) converges at every f in Qp such that 1

Ifl < Ip IP _ 1 r - 1 , ii) For every n > 0, A(n) = bn . 1

Proof. We first note that since |p|^ 1 r_1 > 1, A(x) converges for all with

|< 1

so that A(n) in ii) is well-defined.

Now, let k

oo

Ak(x) = 2 ci ( i ) = 2 4 k) x" * 1=0 n=0

Since A^(x) is a polynomial of degree < k, we have a^) = 0 ,

if k < n.

By the assumption on cn,

II cn (n )ll < lcnl Ip I H

< Cri ’

n > °
0) are contained in K. Then Theorems 1, 3 imply the following result: there exists a power series A(x) in K[[x]] which converges in a circle of radius >1 around 0 and which satisfies A(n) = bfl ,

for all n > 0,

§3.

35

p-ADIC L-FUNCTIONS

if and only if lim |c |r

= 0

n-^ 0,

if and only if lim |c | = 0 .

n->oo

(If exists, f is unique and is given by OO

fW = S Cn ( n ) ’ n=0

x f Zp >

§4. p-ADIC LOGARITHMS AND p-ADIC REGULATORS In the preceding section, we defined the p-adic L-function Lp(s; Y) and found the values of Lp(s; X) for s = 0, —1, —2 ,... . Hence it is natural to ask the values of the same function at s = 1, 2, 3 , ..., and to see how these values are related to L(s; X) for the same s. For s = 1, Leopoldt obtained a remarkable formula for Lp(l; X) and we shall prove it later in §5. In the present section, we discuss a generalization of the p-adic log function which we need in the computation of Lp(l; X), and define p-adic regulators.

4.1.

Let Qp denote as before a fixed algebraic closure of the p-adic

field Qp and let Lf = \a\a € Qp, \a\ = 1} , D = \a\ a € Qp, |a—1| < 1} . These are subgroups of the multiplicative group Q*: D C U C Q^ . We consider the structure of the factor groups. Let a e Q^ and let |a| = p ~ r .

Then r is a rational number: r e Q. The map np

Q

a h> r 36

37

§4. p-ADIC LOGARITHMS AND p-ADIC REGULATORS

defines a homomorphism of Qx onto the additive group of Q and it in­ duces QX / O ^

Q.

For each r 6 Q, we can choose pr in Qx in such a way that PrPs = Pr+s '

r, s f Q,

|prl = P- r

.

t f Q,

Pn = P"

,

« e Z.

Let P denote the set of all pr, r f Q. Then P is a subgroup of Q*, which splits the extension Qx / U:

= p x u,

p^

q

.

Note that the choice of !pf! such as above is not unique. Next, let 0 = \a\ a e Qp, \a\ < 1} , P = {a\ a e Qp, \a\ < l ! , k = o/ip . Since p is a maximal ideal of the subring 0 in Qp, k is a field; in fact, it is an algebraic closure of the prime field Zp/pZp. The canonical map 0 -> k induces a homomorphism of multiplicative groups U - kx and it then induces U /D ^ = > kx . Let V denote the group of all roots of unity in Qp whose orders are prime to p. Then V splits the extension U / D :

38

p-ADIC L-FUNCTIONS

U = V

X

D, V ^

kx .

Now, it follows from the above that

B*

P

=

V

X

X

D

and that this is a topological direct decomposition of Bp. In particular, the projection

is continuous.

4.2.

As usual, we define a power series log(l+x) in Qp[[x]] by oo

log(l+x) =

— xn . n=l

It is well known that the power series converges for every f in Bp with |< 1 and that 0 : D - Bp a i-» log a = log(l+(a —1)) defines a continuous homomorphism of D into the additive group of Bp. It is also clear from the definition that log a (a) = a(log a) ,

a e D,

for every automorphism a of the Galois group Gal(Bp/ Qp).

LEMMA.

The above can be uniquely extended to a homomorphism

- Op such that W ) = o •

39

§4. p-ADIC LOGARITHMS AND p-ADIC REGULATORS

if/ is continuous and satisfies x

a €Q

a,

if/(o(a)) = a (iff(a)) , for every o in Gal(Op/Qp).

Proof. Let n: 0 * -> D be the projection defined in 4.1 and let if/ = (f>om Q* -> D -> Qp . Clearly, if/ is a continuous extension of cf>, satisfying if/(p) = 0. Let A: Op -> flp be an arbitrary extension of cf> with A(p) = 0. It follows from the defini­ tion of P and V that for each /3 in P x V, there exist integers m, n 4 0 such that /3m

= pn .

This implies mAQ8) = A(/6m) = A(pn) = nA(p) = 0 so that A(j8) = 0. Hence A(P x V) = 0 and A = Ao 77 = 0 o77 = if/ . Thus the uniqueness is proved. Finally, fix o

inGal(Op/Qp) and let

A = o ~ l °if/oa\ Op -> Op . This is an extension of cf> and satisfies A(p) = 0. uniqueness, if/ - A, i.e ., if/(a) = o ~ 1(ifz(o(a)))

Hence,by the

forevery a in 0 * .

40

p-ADIC L-FUNCTIONS

In the following, we shall denote if/(a) again by log a; when it is necessary to distinguish it from the ordinary (real or complex) log, we shall denote it also by logp a : log a = logp a = ijj(a) ,

a e Q*.

We can see easily that the continuous homomorphism log: Q*

Qp

is surjective and that its kernel consists of n-th roots, for all n > 1, of all powers of p in ft*. We also note that although the projection 77: Qp

D depends upon the choice of P and hence is not unique, if/ =

f>°77 is

canonically defined as characterized in the lemma.

c

4.3. As an immediate application of the above, we shall next define the p-adic regulator of a number field. Let F be a finite algebraic number field, i.e ., a finite extension of degree, say, n, over the rational field Q, contained in ft. Let vr . . . , v f be the infinite (archimedean) absolute values of F, and for each i, 1 < i < r, let { : F -> ft C C denote a morphism such that Vj(a) = |0j(a)| ,

a e F.

Let e- = 1 or 2 according as v^ is real or complex, namely, according as •••> er - l

ke a system of fundamental units in F , i.e ., a set of ele­

ments in E which represent a basis of E/W. Let £f be any rational integer, £ f > 1. Define, with ordinary log, R = |(n log er)- 1 detCej lo g ^ e ^ D I where det denotes the determinant of the r x r matrix (ei log |^i(£j> l)i,j==i , r

•

Then R is independent of the choice of £ j , ..., £r_ p

as mentioned

above and is called the regulator of F. It is known that R / 0 . We assume now that F is a totally real field so that r = n, e* = 1 for 1 < i < r, and

..., 0 r constitute all isomorphisms of F into fl.

Replacing the real log in the above by the p-adic log, we define Rp = (n logp e r)_1 det(logp ^ (E j )) and call it the p-adic regulator of F. Here we choose £ f to be prime to p so that logp £f / 0. Note that -(ej) are in D and hence in Qp so that logp 0j(£j) are defined. It is easy to see that up to a factor ±1, Rp is again independent of the choice of

...,

and defines an

invariant of the field F. (If F is not totally real, we can still define Rp similarly. However, it may depend upon the choice of

which

correspond to complex v-’s.) Now, it was conjectured by Leopoldt that Rp

4

0

for every p and totally real F. The conjecture was first verified by Ax in some special cases, and later, following the idea of Ax and using a powerful method of Baker, Brumer [4] proved it in the case where F is an abelian extension of the rational field. However, the conjecture is not yet proved in the general case.

42 4.4.

p-ADIC L-FUNCTIONS

Let

be a system of independent units in F , i.e ., a

set of r—1 units in F such that the subgroup E ' generated by ...,

and W, has a finite index in E. Choosing an integer

e ' > 1, we define R' = |(n log 8'r) _1 det(ej log |^ ( e j) |) | . R' is called the regulator of s ^ , ..., £'r_j» and is denoted by ...,

in fact, R' is independent of the choice of e'f, and a

simple group-theoretical argument shows R' = [E : E l R . It is now clear how to define the p-adic regulator of RpCe'l* •••> 6 r_i)> in Ibe case of totally real F. Furthermore, for such R p C e ^ ..., e r - l ) ' we have a Sain Rp( 6 i »•••> 6 r - 1^ = [E : E ] Rp , with a suitable choice of factors ±1 for Rp and RpCs^, ..., £/r_ 1)Hence Rp ^ 0 if and only if RpC6 ^ •••> e r_ i ) ^ 0*

§5. CALCULATION OF L (1; X) In this section, we shall prove a formula of Leopoldt for the value of the p-adic L-function Lp(s; X) at s = 1.

5.1.

Let K be a finite extension of Qp contained in Op and let Cj^

denote the set of all continuous maps

Cj£ is obviously a commutative algebra over K. For f in Cj^, let ||f|| = max |f(s)| .

SfZp Then ||f|| defines a norm on Cj^ such that ||f + g|| < max(||f||, ||g||) , llfgll < llfll llsll.

||af|| = |a| ||f||

a e K.

Furthermore, Cj^ is complete in this norm so that it is a Banach algebra over K. Note that if

f = lim f n-oo

n

in the norm topology in Cj^, then f(s) = lim f (s) n-^oo in K for every s in Zp.

43

44

p-ADIC L-FUNCTIONS

Every polynomial f(x) in K[x] defines a continuous map

Zp

K

S H> f(s) , and f(x) is uniquely determined by this map. Hence we may identify f(x) with the map it defines and consider K[x] as a subalgebra of C^. Thus, for example, the polynomial ^ ^p ^or s €

Cj^, and since

n > 0, defined in §3, belongs to anc* (n*) =

we ^ave

i ( ;)» - * for the norm in Cj^. We now define a map 2,

= xs

if

xe U, p = 2.

(x, s) = 0

,

In the case x e U, (resp. x) is an element of 1 + pZp (see the definition of in 3.1) and s (resp. xs) is defined by oo

s = (1 + - l ) s = ^ ( S) « x > - l ) n n=0 (resp.

oo

xs = (1 + x - I f = ^ (n ) n n=0

>•

§5. CALCULATION OF L (1;)C)

45

Since

t(n) « x> - Dn| < IPl" (resp.

[I) ( *

- l ) n l < |2|

the power series for s (resp. xs) converges uniformly for (x, s) in U x Zp. Since U x Z p is open in Zp x Zp, it follows that 0 : ZpX Zp is continuous. For each integer n > 0, let n

yn(s) = 2

( - 1)n_i( i ) # i.s>.

s e Zp.

i=0

It is clear that yn belongs to Cj^.

L e m m a 1.

Ilynll < l«!l >

n>

Proof. We have to show that |yn(s)| < |n!| for all s e Zp. Let m integer, m > 0, such that p - l |m,

|pm| < |n!| .

Then

Kn(m) = ^ 1=0

1=0

P^i

(-1 )” 1( i )

m>

be an

Zp

46

p-ADIC L-FUNCTIONS

because 1,

Since m jN , m2 |f, and (N, fp) = 1, we have m = m1m2 and pfm j. Thus A£a is a root of unity, of which the order is not a power of p. It is known that for such a root of unity in flp, we have i i - A 1,

I(n -l)! | ,

and we see that A(x) belongs to the algebra Qj^. By the definition, f DA(x) = (1+x) log(l+x)

^

V

X(a) V

a=l U l

^ x) a n

n=l (1 -

}

By Lemma 6, we then have

DACe1 - 1) = e1

V

V

X(a) — i -----

a=l A^l

e ~ *

= e* • t • (N X(N) e(N -l)t G(eNt) - GCe1)) = X(N) F x (Nt) - F x (t)

2 W n=0

^ - ^ n . X S

so that Sn(DA) = (X(N) Nn - 1) Bn> x

n > 0.

It follows from Lemmas 3, 4 that for s e Zp, s r A(s) = r DA(s) = lim (X(N) Nn - 1) Bn>x where the limit is taken over a sequence of integers as described in Lemma 3. Suppose first that p > 2. If p—1 |n, then Nn = n so that lim Nn = lim n = S . n n For such n, we also have (Theorem 2, §3)

CALCULATION OF L p( l ; X )

57

n - ! ) B n, X L p( l —n; X) = - ( 1 - X(p)pn_1)

n > 1.

§5.

n

Since Lp(s; X) is continuous on Zp and since lim pn = 0 as n

oo, it

follows that

Hence

s r A(s) = (1 - X(N) S) s L p( l —s; X) for every s in Zp. If s ^ 0, then this implies

r A(s) = (1 - X(N) S) L p( l —s; X) . However, the same holds also at s = 0 because the both sides are continuous functions of s e Zp. If p = 2 ,

we obtain a sim ila r formula with Ns in stea d o f S.

T h e re fo re the fo llow ing theorem i s proved:

TH EO REM 2.

L et X be a non-principal Dirichlet character with f = f .

Fix an integer N > 1 such that (N, pf ) = 1, X(N) ^ 1 and let {A! denote the set of all N-th roots of unity in Qp. L et f

oo

a=l

n=l

with

r(X ) = ^

X (a )C a ,

Then the power series A(x) belongs to

C = e f

and

r A(s) = (1 - X(N) S) L p( l —s; X) , = (1 - X(N)NS) L p( l —s: X) ,

p > 1, p = 2,

58

p-ADIC L-FUNCTIONS

for all s in Zp. In particular,

r A(0) 5.4.

= (1 - x m

L p(l; X)

.

In order to evaluate L (1; X), it is now sufficient to compute TA(0).

For this, we shall use the formula in Lemma 5. Clearly A(0) = 0. Let be a p-th root of unity in Op. Since

l£-i|

< i.

we have

a=l U\

f

n=l

\1 - A4 /

2 2 *•>log(1+r ^ )

a=l A^l

\

’

1 “ A = n A/i

& -

A^ap> •

(p^ n)>

CALCULATION OF Lp(l;>0

§5.

59

we obtain

S = S2 —pS1 where f

Ca)j ,

Sx = ^ a=l

X(a) l o g ( j J (1 - A A^l

f

s2 = s

iog(n ^ - ,v£ap)) •

Suppose first that p f f .

Then X(p) = X(p)_1 implies

a=l

V l

7

f S2 = X(p) ^

X(ap)

a=l

(1 - A£ap)^ = X(p)S1 . V l

Suppose next that p | f so that X(p) = 0. Since f is the conductor of X, there is an integer b such that (b, f ) = 1,

b = 1 mod

X(b) ^ 1.

P

With such b, abp = ap mod f implies f s2 = 2 X(ab) a=l

log(n

(!

-

A£abpA 7

V i

f = x(b) ^ *(*> i o g ( n a=l V l

(i - A^ap) ) 7

= X(b)S2 so that

s2 = 0

= X(p)Sx

60

p-ADIC L-FUNCTIONS

Hence

s2 - X(p)Sj in both cases, and we obtain f

s = (x(P) - p) 2 iog ( n (i ■ ^ a>) • a=l

V l

Let (a, f) = 1 so that £ a is also a primitive f-th root of unity. Then

JJ

(i _ A-Na

l_^a

Hence f

S = 0 C (p )-p )

2

X ( a ) ( l o g ( l - C aN) - l o g ( l —C3)) •

a=l

(a,f)=l Using (f, N) = 1, X(a) = X(N)X(aN), we finally obtain f

s = (X(p) - P) (X(N) - 1)

^

X (a) log (1 - O

.

a=l (a ,f)= l

Since log (1 - O

= log (1 - Ca) + log ( - £ - * ) = log (1 - £ ~ a) ,

we may write S also in the form f

s

=

(X(p)

-

p) (X(N)

- 1)

2

a=l (a ,f)= l

* (a) lQg (1 - C ~ a) -

§5.

61

CALCULATION OF L ( 1 ; ) 0

Now, by Theorem 2, (1 - X(N)) L p(l; X) = T a (0) =

S.

Since 1 — X(N) ^ 0, the following theorem is proved:

TH EO REM 3.

For anon-principal Dirichlet character

X,

f

2

L p(l; X) = - ( l -

* (a) lo g P(1 " ^ _ 3 )

a= l

with the p-adic log function logp defined in §4. If we compare this formula of Leopoldt with the classical formula for L (l; X) (see 1.2): f L (l; X) = -

log d “ ^

’

a=l we find a remarkable similarity between the two. In Theorem 2, §3, we proved L p( l —n; X) = (1 - Xn(p)pn _1) L (l- n ; Xn) for n = 1, 2, ...

. The above theorem shows that the same formula holds

also for n = 0 if only log in L (l; X) is replaced by the p-adic logp. It is an interesting open problem to find similar expressions for the values Lp(n; X), n > 2.

5.5.

We shall next briefly discuss a consequence of the above formula in

Theorem 3. (For the results below stated without proof, see [2], [8].) 2ni Let p > 2, £ = e ^ (i = \f—1), and let

62

p-ADIC L-FUNCTIONS

F = Q (0 ,

F+ = Q (C + r1) ;

F is the cyclotomic field of prth roots of unity and F + is the maximal real subfield of F. Let £(s; F) and £(s; F +) denote the zeta-functions of F and F + respectively. It is known that these are meromorphic func­ tions of s on the entire s-plane, satisfying certain functional equations, and that t f s ;F )

= ] J l ( s; X ) , X +

£(s; F +) =

U

L (s; X)

X where the product II (resp. II+) is taken over all Dirichlet characters X such that f |p (resp. such that f |p and X(—1) = 1). Motivated by X

X

such a fact, Leopoldt defined the p-adic zeta-functions of F and F + by

= [E: E(£>]R '

and we obtain the classical class number formula h = [E: E(e)] . Now, it follows from 4.4 that = [E: E(e)] Rp = hRp for the p-adic regulators Rp for F + and Rp(s1, ..., e m_ 1) for £ 1 , ..., em_ 1- However, the same theorem on circular determinants shows that +

«Vel

m

em-l> - II f I 2 * 2 or p = 2. Let

X be a Dirichlet character with conductor f . Since (Z /2 Z )X = 1, either

s\.

(f , p) = 1 or q I f . Hence f^, — itIq

or

f^, — mQqp

with (mQ, p) = 1 and e > 0. Now, a Dirichlet character if/ will be called a character of the first kind (with respect to p) if either f^ = mQ or f^ = mQq with (mQ, p) = 1; and if/ will be called a character of the second kind (for p) if f^ is a power of p (i.e., f^. = 1 or f^. = qpe, e > 0) and if i/r(a)f for a € Z, (a, p) = 1, depends only upon . Using the natural isomorphisms (Z/m 0qpnZ)x (Z /q p nZ)x -

(Z/m 0Z)x x (Z /q p nZ)x ,

(Z/qZ)x x (1 + qZp) / (1 + qpnZp) ,

we see that each Dirichlet character X is, uniquely decomposed into a product of a character of the first kind, say, 6, and a character of the second kind, say, n: X = 6 tt . 0 will be called the first factor of X, and n the second factor of X. 66

67

§6. AN ALTERNATE METHOD

6.2. Let mQ be an integer such that mQ > 1, (mQ, p) = 1. For each n > 0, define

qn = " W " • Since qQ= mQq and qn = q0pn, n > 0, we see that (a, qn) = 1 if and only if (a, qQ) = 1. For such an integer a, let an(a) = the residue class of a mod qn . Then these ^n(a), (a, qQ) = 1, form the multiplicative group Gn = (Z /q nZ)>< . For m > n > 0, we have a natural surjective homomorphism Gm

Gn h

’,

an(a) ,

(a, qQ) = 1.

Let T n denote the kernel of Gn -> GQ, namely,

rn =

|a = 1 mod qQ! .

This is a cyclic group of order pn. Define another subgroup An of Gn by An = !^n(a)| ap_1 = ±1 mod qpn! . Note that if p > 2, then there exists no integer a satisfying ap_1 = — 1 mod qpn so that the above condition is the same as ap_1 = 1 mod qpn. The natural isomorphisms mentioned in 6.1 imply that ^n = ^n x A n *

Furthermore, Gm -> G , m > n > 0, induces

rm m

r„, Am n' m

an„.

68

p-ADIC L-FUNCTIONS

Let G = lim Gn , r = lim Tn , A = lim Afl with respect to those homomorphisms. These are profinite (compact) abelian groups and G = T x A, A ~ A0 = G0 . For a € Z, (a, qQ) = 1, let CTn(a > = >n(a ) S n(a > -

f r n- S n(a ) f An ’

under the directdecomposition Gn = Tn x An, n > 0. Let

o{a), y(a),

and S(a) denote the limits of an(a), yn(a), and 0,

in G,

T,

and A respectively. Then o(a) = y(a) S(a) under G = T x A. Let a, b £ Z, (a, qQ) = (b, qQ) = 1. Then yn(a) = yn(b) if and only if = mod qpnZp. Hence we have an isomorphism rn

^

(l + qZp) / ( l + qpnZp) ,

yn(a) I

>

(l + qpnZp) ,

It follows that y(a) = y(b) if and only if = , and r

^

y(a) i

l

+ qzp ,

>

Let K be a finite extension of Qp, contained in Qp, and let 0 = \a |a e K, \a\ < 1} , P = {a |a e K, \a\ < 1\ .

n>0.

§6.

AN A LTER N A TE METHOD

69

0 is the ring of local integers in the local field K and p is the maximal ideal of 0. Let A denote the ring of all formal power series in x with coefficients in 0: A = o[[x]] . A is a local domain and its maximal ideal m is generated by x and p. The powers of m, mn, n > 0, define the Ttt-adic topology on A which makes A into a compact topological ring. For each n > 0, let Rn denote the group algebra of Tn over 0 :

Rn = o[rn] . This is a subring of K [r n], the group algebra of Tn over K. For m > n > 0, the group homomorphism r m -> Tn induces morphisms of algebras Rm

RnJ .

K L[ rmJ ] -> K L[r n] .

Let R = lim Rn . The p-adic topology on 0 defines a natural compact topology on each free o-module Rn, n > 0. Hence the inverse limit R is also a compact topological algebra over 0. Clearly, we may consider the inverse limit r = lim T n as a subgroup of the multiplicative group of R: T C Rx .

LEMMA 1.

There exists a unique isomorphism of compact 0-algebras A

r

such that 1+x i-> /(I + cJq) . Proof. Since o[x] is everywhere dense in A = o[[x]], the uniqueness is obvious. It is easy to see that for each n > 0, there exists an isomor­ phism of o-algebras

70

p-ADIC L-FUNCTIONS

A /(l - (i+x)pn)

Rn = o [ r n]

such that 1+x mod (l-(l+ x )P ) h* yn(l+qQ) . The limit of these isomorphisms, defined for all n > 0, gives us the iso­ morphism stated in the lemma.

6.3.

Let 77 be a Dirichlet character of the second kind. By the definition, in = 1

or

tn = qpe ,

e > 0.

Let mQ, qn, Gn, etc., be the same as in 6.2. We choose a field K which contains all values of 77(a), a e Z, and define Rn, n > 0, and R as stated above. We denote by nQ the least integer > 0 such that

f^|qn for every n > noNow, fix an integer t and consider the residue class of ;r(a)—1~* mod qR0 for a f Z, (a, qQ) = 1. Since 77 is of the second kind, if n > nQ, this residue class depends only upon yn(a). Hence there exists a mor­ phism of o-algebras, depending upon 77, t, and n > nQ, ^t,n = ^ t,n: Rn = 0 [ r n] - o /V such that a e Z, (a, q0) = 1.

yn(a) i-> ff(a)- 1 - t mod qn0 , Since the diagrams Rm ^ 0/clm0 I 4 ,

m ^ n ^ ^0*

Rn ^ 0 / % ° are obviously commutative, we then obtain a continuous morphism of o-algebras 4>t = 4% = lim

R -* 0

§6. AN ALTERNATE METHOD

71

su ch th at n( - a ) = >n(_1) Xn(a) = ^ n ^ ■ Hence

2

0t (a) yn(a)—1 = 0

a so that ^ n+1 = ^n'

^n+l ^ ^n *

Thus, in general, ^ ^n'

H ^n

under K|Tm] -> K[rn], m > n > 0 . We shall next show that rjn is contained in Rn. Since =

74

p-ADIC L-FUNCTIONS

0«l+qo)a) yn( d + % ) * r ' • n

a

For each integer a, 0 < a < qn, (a, q0) = 1, define integers a' and a" by

0 < a'< qn-

(l+q0)a = a'+ a"qn , Since (l+qQ)a = a' mod qn, we have

(a'), 0((l+qo)a) = 0(a'), yn((l+q0)a) = yn(a") ’ = w((l+q 0)a )- 1 (l+q0)a = + « ( a' r la "q„) 0(a') yn(a ')—1

Since

(i+q0) (qn-a ) = ^n ~ a'+

*

^i(qn— a) =- Va'>, yn(qn-a'>=y^3')» we obtain from the above that

*n = I 2

a

= 2

(a'^ i(a° yn(a0_1 + (% -a">V q n- a'> yn(qn-a')_1)

yn(a')-1 - T ^ (a0 yn(a') _ 1 ) •

75

§6. AN ALTERNATE METHOD

As qQ is even when p = 2, this shows that rjn belongs to Rn = ° [ r n]. Now, since rjm h rjn under Rm -> Rn, m > n > 0, there exists an ele­ ment 77 in R such that I oo

Let g (x ; 6>) h j £ under the isomorphism A = o[[x]]

R in Lemma 1. Let

denote

the field generated over Qp by all values of (9(a), a e Z, and let 0 q be the ring of local integers in K^. By the choice of K,

is a subfield

of K and Oq = K q fl o. It is easy to see from the definition that g(x; 6) is in fact a power series in o^[[x]] and that it depends only upon 9 and is independent of the choice of the auxiliary field K which is used to de­ fine A and R. We now assume that 9 is non-principal: 6 ^ X°. We shall prove that in this case,’ ^n is also contained in R,.. n Since

= "

1V

0(qn~a) =

1

we obtain as in the above that

Fix an integer a Q, (a Q, qQ) = 1, and denote by S " the partial sum taken over all integers a such that

0 < a < - j- ,

(a , q 0) = 1,

yn(a ) = yn( a Q) .

p-ADIC L-FUNCTIONS

76

Since = mod qn for such integers a, we have

X ^(a)yn® _1 = ( X _1

mod

However, we see from Gn = Tn x An that when a takes the values as mentioned above, the elements &n(a) and ^n(qn—a) precisely fill out the group A . Since 0 is essentially nothing but a non-principal character of A , it then follows that

2

6® =

\

X ” (0(a) + ^ n - a» = 0 •

Therefore 0(a ) yn(a )- 1

= 0

mod qnR n

0 (a ) y n(a )_ 1 = 0

mod qnR n .

and hence ^

a For p > 2, this already proves that

is an element of Rn- If p = 2,

then 0j(a) = co(a)- 1 0(a) = ±(9(a) = (9(a)

mod 2o

so that 0 1(a) =

0(a) = 0

mod 2o.

This implies X

X n ®

"

1

s

0

m o d

2 R n>

and Rn, m > n > 0, let

£=> CO = £=* OO ° = lim £n , -’ n and let

£ o o( R

77

§6. AN ALTERNATE METHOD

f(x; 0) k under the isomorphism

R.

Just as g(x;0), f(x; 0) is then a power

series in 0 /q[[x]] and it depends only upon 0. Clearly lim (l - (i+q0) yn(i+q0) 1) = 1 - (i+q0) Ki+qo) 1 • Hence ^

= (1 - (l+q0) y(l+qor

)^ (

Let

h(x; 6) = 1 - ^

= l _ (l+q0) £ (-* )" • n=0

Then h(x; 0) h> 1 - (l+q0) y(l+q0) under A

-1

R. Hence g(x; 0) = h(x; 6) f(x; 6) .

If 0 is the principal character: 0 = X °, then mQ = 1, qQ= q,

=

Qp, and g(x; X°) is a power series in Zp[[x]]. As in the above, let

h(x; X°) = 1 -

,

q0 = q.

We define f(x; X °) by f(x; X°) = g(x; X°) h(x; X 0) - 1 . Thus f(x; X °) is not a power series but is an element of the quotient field of Zp[[x]] satisfying g(x; X°) = h(x; X °) f(x; X°) . We shall see later that f(x; X°) is, in fact, not contained in Z [[x]].

p-ADIC L-FUNCTIONS

78

6.5. Let X be a Dirichlet character with X(—1) = 1 and let X = Qn be the decomposition of X into the product of the first factor 6 and the second factor n. Let ft< * J = g(C(i+q0) _ t - i; 0) • On the other hand,

h(C(l+q0)_t - 1;0) = 1 - C- 1 d+q0)t+1 t 0 for t > 0. Hence it follows from the above that

2f(i(l«,or ' - 1;» . - j l , Hm L X xt+. 1. Since f^, with some e'> 0, this implies (a, L

Xt+i

^n

2

*w w u‘ -

0 0,

It is easy to see that

because |b |< 1, n > 0, and the coefficients of u(s—1) satisfy the same condition as above. Therefore the power series

n=0

84

p-ADIC L-FUNCTIONS

converges in the domain ® defined above and O O A(c + u ( s - l ) ) = B(u(s—1)) =

c n( s - l ) n n=0

for every s in ® . Note that |u(s 1)| < 1 ,

|c + u(s—1)| < 1 ,

s 1.

It follows that h(£(l+q0)s — 1; X °) ^ 0 for every s in ® and that the function

h(^(l+q0)S - 1; X°)

86

p-ADIC L-FUNCTIONS

is defined in 2) and is given by a power series

F (s; X) = 2 an ,x ( s - D n n=0

an,X f K- lan,xl < Cr_n

with C = 1 1 - t T r 1 > 0. Finally, let d = X ° and also X = n = X °. In this case, £ = 1 and

h((l+q0)s - 1 ; X° ) = log(l+q0) ( s —1) ^ ^ ^ (n+1)! n=0

(l-s )n

where, by the same remark in §3,

(n+1)! Therefore h((l+qQ)s — 1; X °) vanishes in ® only at s = 1, and the function F (s; X °) = 2f((l+q0)s - 1; X °) is now defined for all s ^ 1 in ® and is given by

F (s; X °) = ^

^ an(s —1)n n=0

where a _ l > an f K >

lanl - Cr n >

n > 0 (C > 0 ).

By the remark at the end of 6.5, we also see that

a j = lim ( s - l ) F ( s ; X °) = (1 _1 s-»l

p } |°g(1+qo) = 1 _ I . log(l+q0) P

87

§6. AN ALTERNATE METHOD

6.8.

For each Dirichlet character X, X(—l) = 1, we have thus obtained

a function F (s; X) defined in the domain 1 r = Iq rM p ^ ” 1 > 1,

® = Is |s e Qp, |s 1 1 < r} ,

excluding s = 1 in the case X = X°, with the analytic property as ex­ plained above. By Lemma 3, it also satisfies

F (l-n ; X) = - (1 - X ^ p " - 1) ^

^ = Xft)-n

for every integer n > 1. Hence F (s; X) possesses the properties i), ii) of Theorem 2, §3, which uniquely characterize the p-adic L-function Lp(s; X). Therefore F (s; X) = Lp(s; X) . Thus we see that Lp(s; X) is constructed by a method different from that in §3. Note that we have always assumed in the above that X(—1) = 1. However, if Y(—1) = —1, then Lp(s; X) is identically zero. Hence nothing essential is lost by the new approach. For X(—1) = 1, we now have the important formula

H(C(l+q0)s - 1; 6) for s f2 ) (s ^ 1 if X = X°). We recall that 6 is the first factor of X, Iq = mQ or Iq = mQq with (mQ, p) = 1, qQ = mQq, £ = X iU q ^

1, that

g(x; d) and h(x; 6) are power series in o^[[x]] associated with 6, and that f(x; 6) is also such a power series if 6 ^ X°.

§7. SOME APPLICATIONS As we have already seen in 5.5, p-adic L-functions are closely re­ lated to the arithmetic of cyclotomic fields. In this section, we shall dis­ cuss some applications of the results obtained in §§5, 6 to the theory of cyclotomic fields.

7.1.

As in 6.2, we fix an integer mQ > 1, (mQ, p) = 1, and define % = m0cipn >

n > °-

Let kn, n > 0, denote the cyclotomic field of qn-th roots of unity, i.e ., the number field generated by all qn-th roots of unity over the rational field Q. kn is obviously a Galois extension of Q. For each automor­ phism a of kn/Q, there exists an integer a, (a, qQ) = 1, such that < 0 = C3 for every qn-th root of unity £ in kn< The residue class of a mod qn, namely, crn(a), is then uniquely determined by a, and the map a h> a n(a )

defines a canonical isomorphism of the Galois group of kn/Q onto the group Gfl defined in 6.2: Gal(kn/Q) ^

Gn = (Z /q nZ)x .

If m > n > 0, kn is a subfield of km, and the restriction of an automor­ phism in Gal(km/Q) on the subfield kn defines a natural homomorphism Gal(km/Q) - Gal(kn/Q)

88

89

§7. SOME APPLICATIONS

such that the diagram

Gal(km/Q) ^

Gm

i

i

Gal(kn/Q)

Gn

is commutative. It follows in particular that Gal(kn/k 0) ^ Now, let

T

n = Ker(Gn -> GQ) ,

n > 0.

denote the union of the increasing sequence of number

fields: k0 c kj C . . . C kn c . . . . k^ is the (infinite) cyclotomic field generated over Q by qn-th roots of unity for all n > 0. Since GaKk^/Q) = lim Gal(kn/Q) , it follows from the above that GaKk^/Q) ^

G = lim Gfl .

Similarly GalCk^/kjj)

T = lim Tn

.

Thus we see that the various groups introduced in 6.2 are nothing but Galois groups of certain cyclotomic fields (finite or infinite over Q).

7.2. For each n > 0, let k^ denote the maximal real subfield of kn, and let hn and h+ be the class numbers of kn and kJ respectively. It is known (see [8]) that hn is divisible by h^ and the quotient hn = hn/h i

90

p-ADIC L-FUNCTIONS

is given by the formula:

X

\

X a=l

where Q = 1 or 2 according as mQ = 1 or mQ > 1, wn is the number of roots of unity in kn, i.e ., wn = qn or wn = 2qn according as qn is even or odd, and the product is taken over all Dirichlet characters X such that fy |qn, X(—1) = —1. We write h_n = h“ n htn and call h“

the first factor of hn, and h+ the second factor of hn-

Now, by a remark in 2.2,

Since wn = wQpn, we may write the above formula in the form

X where X now ranges over all characters such that f^ |qn, i^]( q0, and X(—1) = —1. For such a character X, let Xcl>

= On

be the decomposition of Xco into the product of the first factor 0 and the second factor n, and let = "(l+ q o r1 where q'Q is defined by

q 0 = m0*^ %= m0 or fA .

(

n > A,

p-ADIC L-FUNCTIONS

94

Thus the following theorem is proved:

denote the first factor of the class number of the e cyclotomic field defined above and let p be the highest power of 6 p which divides hn : |hn | — (p |, en > 0 (n > 0). Then there exist inTH EO REM 1.

Let hn

tegers A, p, v, and nQ; A, p, nQ > 0, such that en = An + fipn + v for every n > nQ (i.e., for every sufficiently large n > 0).

Note that with p fixed, A, p, and v depend only upon the integer m0 > 1, (m0, p) = 1, which we also fixed at the beginning of this section. It is conjectured that V = 0 for every choice of mQ, but this is not yet proved or disproved. The formula for en given in Theorem 1 is a special case of a similar formula proved by class field theory for so-called Zp-extensions of alge­ braic number fields, of which the extension k^/kQ in 7.1 is a typical example.

7.4. We continue to use the same notations as in 7.3. It follows from the definition of A and p that A= p = 0 if and only if A(0) is a p-adic unit, i.e ., IA(0)| = 1 . If £ is a root of unity with order a power of p, then |A(0)| = 1 is equivalent with |A (£-1)| = 1

§7.

because = 1,

|£ — 1| < 1.

C^

prime to p.

95

SOME A P P L IC A T IO N S

Hence it follow s from

|h^~| = |h^"| |IIA(£ — 1)|,

that A = /x = 0 in Theorem 1 if and only if h ^ / h ^

is

Of course, a sim ilar result holds with h“ /h~, m > n > 0, in­

stead of h^“/h^. We shall next consider the special case where mQ = 1, another condition equivalent with A =

Q = 1> and if X and

fi =0.

w n = 2pn + 1 , then p |

Hence we obtain, as in 7.2, that

I I f(C- ho)

h- = 2pn+1 n

e where

In this case,

is a Dirichlet character such that f^|qn» ^ ^ X °,

X (p) = 0.

and find

9ranges

C

over all characters such that f^|q0» 0(—l ) =

over a ll pn-throots of unity, including

1.

It w as proved

I IlfCC-i;xr°)l = |pnl_1 , £

an 2 or q = 4. Since h0

for kQ= Q(\/--l),

=

h0

=

h0

= ^

we see that A = p = 0 in the case where

Now, let p > 2. As is

p = 2, mQ= 1.

well-known, the prime number p is called

regular (resp. irregular) if the class number hQ of the cyclotomic field of p-th roots of unity, kQ, is prime to p (resp. divisible by p). A theorem of Kummer also tells us that hQ is prime to p if and only if its first factor h^" is prime to p. Hence the above theorem actually states that A = p = 0 (for mQ = 1) if and only if p is regular. For example, it is known that for three irregular primes less than 100, p = 37, 59, 67, we have A = 1,

p = 0 .

It is an interesting problemto study the values of A and p when p is irregular.

7.5. We shall next briefly sketch a proof of the theorem of Kummer men­ tioned above. Let p > 2. Let F (= kQ) denote the cyclotomic field of p-th roots of unity, and F + (= k^) the maximal real subfield of F. We have shown in 5.5 that when 0 ranges over all Dirichlet characters 4 X° with fglp, 0 ( - 1) = 1,

0

v

97

§7. SOME APPLICATIONS

where m =

= [F +: Q], d is the absolute value of the discriminant of

F +, h (= h p the class number of F +, and Rp the p-adic regulator of F +. On the other hand, putting s = 1 in the formula of 6.8: L p(s; 9) = 2f((l+p)S - 1 -,9),

(qQ= p)

we obtain L p( l ;0 ) = 2f(p;9 ) . Hence

A(p) = II f(p;