Lectures on p-ADIC L-Functions (Annals of Mathematics Studies)

LECTURES ON p-ADIC L-FUNCTIONS BY

KENKICHI IWASA WA

PRINCETON UNIVERSITY PRESS AND UNNERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY

1972

Annals of Mathematics Studies Number 74

Copyright © 1972

�y

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 L­

functions 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 [121 and discuss some of the fundamen­ tal p roperties of these numbers. In

§3,

we introduce p-adic L-functions

and prove the existence and the uniqueness of such functions; our method

[10]. §4 consists of preliminary remarks p-adic regulators. In §S, we prove a formula of

is slightly different from that in on p-adic logarithms and

Leopo1dt for the values of p-adic L-functions at s announced in

[10],

=

1. The formula was

but the proof has not yet been published.

With his per­

mission, we describe here Leopoldt's original p roof of the formula (see

[1], [7] for a1 temate 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 in g 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 al gebraic number theory as presented, for examp le, in Borevich­ Shafarevich [2] or Lang [11]. However, except in few places where cer­ tain facts on L-functions and class numbers ar� referred to, no deeper v

understanding of that theory may be required to follow the el ementary 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 p and Qp 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, i f 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, ] . 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

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

§1. Dirichlet's L-functions

.

. ....

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

§2. Generalized Bernoulli Numbers

§3. p-Adic L-functions

. .

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

...

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

§5. Calculation of L pC1; X)

§6. An Alternate Method

APPENDIX

. ... ...

BIBLIOGRAPHY

......

. ....

.

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

.

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

.

§4. p-Adic Logarithm s and p-Adic Regulators

§7. Some Applications

.

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

. . . ..

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

.

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

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

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

.

. ......

.. .

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

.. .

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

.

...

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

.. .

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

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

.

...

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

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

.

.

.

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

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

vii

.

v

3 7

17

36

43

66

88

............... 100

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

lOS

LECTURES ON p-ADIC L-FUNCTIONS

§1. DIRICHLET'S L-FUNCTIONS

In this section, we shall review some of the well-known classical re­

suIts on Dirichlet's L-functions. For proofs and more details, we refer the reader to [2], [6], [13].

1.1. Let 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) a, b in Z, and iii) X(a)';' 0 if and only if

a

=

X(a) X(b) for any

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/nZ)x. ' Let X be a Dirichlet character to a modulus m and let m be a factor of n. Define X(a), a X(a)

f

=

=

Z, by ' X (a)

,

if (a, n)

=

1,

i f (a, n) > 1.

0

Then X is a Dirichlet character to the modulus n. We say that the ' character X is i nduced 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

--_0'

� ••

'-"ULUler

cnaracters we consider will be

assumed as primitive. Let X 1

and

f2

and X2 be such (primitive) Dirichlet characters and let f1

be the respective conductors.

Then there is a

unique

Dirichlet character X with conductor f dividing £1 £2

for integers Xl

a prime to

and X2: X

true if (a , f1 f2)

=0

f1 f2: (a,

Xl X2·

> 1.

fl f2)

='

Note that X(a)

1. 0=

Xl (a) X2(a) is not necessarily

O

O

€

Z.

52

form an

The identity of

defined by X (a)

Z; this is the unique char acter with conductor

X(a), a

the prodU�t of

The set of all primitive Dirichlet characters

the group is the principal character X

X is the character

such that

X is called

abelian group with respect to the above multiplication.

a in

(primitive)

1.

==

1 for every

The inverse of

which is the complex-conjugate map of X: X(a)

=0

1.2. Let X be a Dirichlet charact er and let L(s; X)

=

I

s X(n)n- .

n=l The series on the right converges absolutely for all complex numbers s with Re(s) >

1

so that

L(s; X)

the half-plane where Re(s) >

1.

defines a holomorphic L(s; X)

is called

function

of s in

Dirichlet's L-function

for the character X.

For the principal character function of Riemann,

O

XO, L(s; X ) is nothing

but the zeta­

�"(s).

We shall next describe some fundamental properties of L(s; X). First, Les; X)

can be expressed as an infinite

L(s; X)

=

II (1 P

-

product:

X(p)p-S)-

l

,

Re(s) >

1,

where the product is taken over all prime numbers p. Hence L(s;

X) '" 0

X) can be extended to a meromorphic function on the entire s-plane; if X;, Xo, L(s; X) is holo­ morphic everywhere, but if X = XC, L(s; X) has a unique pole of order

for Re(s) > 1. By analytic continuation, L(s;

1, with residue 1, at s

=

1. Furthermore, on the s-plane, it satisfies

a functional equation as follows. Let f

reX)

denote the Gaussian sum:

27Tia

-reX) 2. X(a) e f , , a=1

and let if XC-1) = 1, if

X( -1)

=

- 1.

Then �

(tyr(s�3)L(S;X)

=

W X

(�)

I-s -2 r

e-�+O) L(l-s;X)

where r denotes the r�function, 52 is the inverse of

X,

and

The same equality can be written also as L(s; X)

=

§ (217) s

reX 2i

f

X)

Lel-s; res) cos rrCs-8) 2

Actually, all such analytic properties of L(s; much wider class of functions similar to L(s; the case of our

L(s; X),

given in the liippendix.

X)

X)

can be proved for a

(see [11]). However, in

the proof is Simpler; a sketch of such a proof is

Let F

X(z)

be a metomorphic function of. z defined by

F iz)

=

f � �

a X(a)ze z ' e fz 1 a=l _

The argument in the appendix also implies that L(1-ni X) r{n) for

the residue of

FXCz) z-n-l I,

at z

=

0

every integer n > 1. This formula willse 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 p rincipal character

XO,

we already mentioned that L(s; Xo)( = (s»

pole with residue 1 at s

=

1 so that

has a simple

lim (8-1) L(s; Xo) = 1 . s...l On the other hand, if X is not principal, then L(I; X) f. 0, "" and its value is explicitly given as follows Csee [8]): L(I; X)

=

-

rei) � X(a) log (1

ftwhere f

=

fX'

,

a

r(fX)

;-r,

that l:oS a:::; f, (a, £)

==

(-a)

r(fX).� X(a) log 11- ,ai,

2"i

=

_

1.

IT 2. 5(a) a , a

if X C-I) = 1, if X C-I)::

-1,

and the sum is taken over all integers a such

§2. GENERALIZED BERNOULLI NUMBERS

In this section, we define generalized Bernoulli numbers and discuss some of the fund�ental properties of these numbers which are needed in the later sections. 2 . 1 . The definition of (ordinary) Bernoulli numbers is well known; let

be an indeterminate and let

F(t) Exp and F(t) (formally) into a power series of t: F(t) The coefficients Bn, n?: 0, are called Bernoulli numbers. (Sometimes C1F(t) is used as the generating function instead of F(t) . ) It is clear that Bn are rational numbers:

Since Fet) - t ,

F(-t) we also see that

for odd n >

7

1

Let

x

be another indeterminate and let (l+_ x)t F(t, x) '" F(t) e t x = t;:.::e:..,t e -1

As for F(t), let

Since F(t, x) we have n

Bn (x) =

,l (�) Bi xn-i

i=O

n::::: O.

,

Hence B n (x), n? 0, are polynomials of x with rational coefficien ts, and they are called Bernoulli polynomials. As B o 1, Bn(x) is a monic polynomial of degree n: =

It is clear that

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

f � X(a)te(a+x)t xt F (t, x) = F (t)e = � X X a = 1 eft 1 _

Expanding these into power series of t, let

Then n

Bn, X(x) =

� (D Bi, Xxn-i , i�o

Let Q be the rational field and let Q(X) denote the field generated over Q by all values X(a), a Z. It is clear that Bn, X are in QCX) and Bn, XCx) are polynomials in Q(X) [xl. Bn, X and Bn, X(x), n 2: 0, are called generalized Bernoulli numbers and generalized Bernoulli polynomials, respectively, belonging to the Dirichlet character X. If X XO, the princip al character (f = I), then €

=

F XCt)

= F(t),

Fit, x) = F(t, x)

so that

2.2.

We shall next describe some simple properties of Bn, X and Bn, X(x) 1) n 2: 0, Bn, X(O) = Bn, X •

f

2)

Bo, X

=

} � XCa) a=l

'"

0,

IV

p-ADIC L-FUNCTIONS

H ence deg(Bn ,

X(x»

Since

3)

< n ,

f

-(a-x)t ) (-t)e FX (-t, -x) = I X(a - -e-ft _ 1 a=1 f (f-a+x)t � XC-1) X(f_a)te ft e 1 a=1 _

= XC-I) FX(t, x) , we have n > O. Putting

x = 0,

we

obtain B

where a

4)

= Ox

X=0, is defined as in §l. n,

Similarly, F X(t) ,x =

implies

X(a) F(ft, a-ff+x) a=1

f

X(x) = f � X(a )fn Bn (a-:+x) , a=1 (x 0) , f Bn, X f l X(�)fn Bn( aft) , a= l B , n

In particular

f

1 � f k

n > o.

=

=

n:::: o.

5) For any integer k?: 0, let

Sn , X (k) =

k

�

XCa)a n ,

a=l

For X = XO, Sn X(k) will be simply denoted by Sn (k): , S nCk) =

k

�

an ,

a=l

Now, F (t, x) X

-

Fit, x-f ) =

implies Bn X(x) - B , (x-f ) = n n X ,

�

a=l f

�

a=l

X(a)t e(a+x-f )t

X(a) (a+x_f )n -l ,

Replac i ng n by n+l in the above and adding the equalities for x

=

2f , . . ., kf, we obtain

f,

n , k ?: o. In particular eX

=

XD ), n, k ?: O.

We now prove a formula which motivated the definition o f Bn, X . THEO REM

1. For a Dirichlet character X and for any integer n?: 1, L(1-n; X)

=

Proof.

A s stated in 1.2, X ) - the residue of F (z)z-n-l at z - L(l-n; X r{n) -

'"

0

.

Since

we obtain the formula immediately. The functional equation for Les; X) shows that if n;: 8 mod 2 , then 211")n L(1-n; -X) L(n; X) = r(X) ( n-o 2·8 f 1

=

(-1)

r{n}(-I)

-

2

1+n-o n B ...,.. 2 I.OO. 211" ( ) n,n! X . 2 iO f

Since L(n; X) ;, 0, n ? 1, it follows that if n 2: I, We already k now the cases where Bn, X O. Since 0, we may summarize our results as follows: ""

THEOREM 2. i) X XO =

Ox

(the principal character): 1,

1

2'

Bn;' 0, Bn = 0,

for even for odd

n

�

0,

n>1,

=

n 0"

0

mod 2.

8X and 0XO

=

S:J.

13

GENERALIZED BERNOULLI NUMBERS

BO, X

�

0, tor tor

n;: 1, n :; 8X mod 2, n;: 1, n ,;, Ox mod 2.

We would like to add some remarks on the preceding theorems. For X = XO , the above fonnula for L(n; X) states �+l

t;(n) (-1)2 00

B

1 (2IT)n nr' 2' n

n ;: 1,

n :2

0

mod 2,

or, equivalently, for even n > 1. Since

( n) > 0,

we know more precisely than above that for

Bzn < 0, Next, let X be a Dirichlet character with Theorem 2 states that f

odd n;: 1, for even n 2 1. Ox

B l, X i O.

Using the formula in 4) and � X(a) = 0, we see a=l

= 1, i.e., X(-I) -1. =

f

l

a""l f

l

a=l

= Hence

f

�

a=l

(

X (a)B1 aff ( f X (a) af

fl

a=l

XCa)a

)

+ �) (

=

fSl,

X(a)a f. 0,

)

XC£) .

if XC-I) = -1.

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

2.3. Let X be again an arbitrary Dirichlet character with f = f ' We X know that Bn X , n 2: 0, are algebraic numbers in the field QCX). Various , results are known on the arithmetic properties of the numbers Bn X . 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 Q (X) denote the field generated over Qp by X(a), a � Z (in an p algebraic closure of Qp)' QpCX) is a locally compact topological field

containing QCX) as a dense subfield. LEMMA 1. In QpCX), Bn X

,

Proof.

=

lim -1 Sn (phf ) , X h-><XI P hf '

This follows immediately from the fact that

n

> O.

§2.

15

GENERALIZED BERNOULLI NUMBERS

and that Bn+1,X(x) - Bn+1 X(O) = (n+1)Bn Xx , ,

+

(terms of degree 2: 2 in x).

For X = XO, f = 1, the lemma states n> O.

2. The denominator of the rational number Bn (n � 0) is divisible at most by p, but not by p2.

LEMM A

Proof. By the above lemma, it is sufficient to show that

for all h?: 1. If h = 1, this is trivial. Let h > 1. Since each integer a, 1 s: a s: p h, can be uniquely written in the form

1 s: b .s ph-1, 0 .s c < p,

a = b + cp h -1 , we have

ph-1 an

==

p

�

b=l

bn

==

p Sn(Ph-l) mod ph-I.

Hence the lemma is proved by induction on h. The lemma follows also from the well-known theorem of v. Staudt­ Clausen which states that B2n

==

-

� �

mod Z ,

for n?: 1,

where the su m 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 � Xo, then f B , X is an algebraic x n

16

p-ADlC L-FUNCTIONS

integer for every n?: 0 and that if fX is not a power of a prime, then even n1.Bn, X is an algebraic integer.

§3. p-ADIC L-FUNCTIONS Let p be a prime number and let Up be an algebraic closure of Qp' In the following, we shall fix both p and Up and consider p-adic func­ tions which are defined on sufficiently large domains in np and take values in the same field Up' 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 Up ' In [10], such a function f(s) although the condition fen) -2 ,

. . .

=

L(n; X) for n

, had to be modified slightly, and they named it the p-adic

=

0, -1,

L­

function 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 eJ:C;istence and the uniqueness of the function f(s) as mentioned in the above.

3.1. Let Up be as above and let I �I, for � i Up , denote the absolute value on Up ' normalized so that

P

-1

np 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

q = p,

if P > 2

,

if P = 2 ,

4,

Let

u=

Z�,

D

=

1 + qZ ; P

U is the multiplicative group of all p-adic units and 0 is the subgroup

of U consisting of all elements of the form 1+qa, a

Zp ' For p > 2, let V be the cyclic group of order p-1 consisting of (p-1)-st roots of

unity in Q ' and for p p

=

£

2, let V= !±11. Then

U =VxD topologically. Each a in U can be uniquely written in the form a " w{a) where w{a) and < a > denote the projections of a on V and 0 re­

spectively, under the above direct decomposition of U. We see easily that if p > 2, then

lim ap n�oc>

w{a)

n

Now, let 0 be the field of all algebraic numbers, i.e., the algebraic closure of Q in e. In the following, we imbed 0 in O p once and for all, and consider 0 as a sub field of Op ' The group V in Up is then

identified with the group of roots of unity in n with order p-1 or 2, namely, with a subgroup of ex, Hence, definin g w(a)

=

0

for a in Z with (a, p) > 1, we obtain a map Z

a

....

I-->

C,

w(a) .

Clearly, this is a Dirichlet character in the sense of 1.1, and we denote it again by

0.

Note that the conductor of

0

is

q

and that it induces

an isomorphism v

c

ex.

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

cally compact field in the topology defined by the absolute value, and an infinite series .i an ' an € K , converges in K if and only if lanl ... 0 n=O Let K[[x]] be the algebra of all formal power series in x. A as n ... 00.

power series A in K[[x]]: A

A(x) =

l

n=O

converges at x = � in Up if and only if I an � n l 0 as n 00. Hence if A converges at �, it also converges at every Tf in Up with ITfI :; I � I ->

->

LEMMA 1. Let both A(x) and B(x) be power series in K[[x]], con· vergent in a neighborhood of 0 in Up' Suppose that

for a sequence of elements �n

f. 0,

n:::

0, in

lim � = O. n ...O n Then A(x) = B(x) . Proof. Let A(x) - B(x)

Up such that

20

p-ADIC L-FUNCTlONS

Assume that A(x) -fo B(x) and let no be the minimum of n such that cn -! O. Then we have

0

for every ';i ' Since .;i a s i -> 00 and since the sum on the right side is then bounded, we get the contradiction ....

cn = O. O For a p ower series A = A(x) =

I: a xn in K [[x]] , define n=O n

IIAII = sup lanl . n Let PK denote the set of all A in K[[x]] with IIAII < algebra of K[[x]] containing the polynomial ring K[x]:

00.

PK is a sub­

K[x] t:; PK f K[[xll , and IIAII defines a norm on PK: IIAII � IIA

+

0;

IIAII =

0

only when A

=

0,

BII :::: max( IIAII, IIBI\) ,

IIcAl! = I c I I\AII. IIABII :::: IIAIIIIBII •

C €

K.

LEMMA 2. PK is complete in the norm IIAII so that it is a Banach alge­ bra over the local field K. Proof.

Let Ak• k ? 0, be a fundamental sequence in PK and let Ak(x)

=

I a�k) xn •

n=O

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

33.

i)

For each n?

ii) A = A(x) iii)

lim

k

....=

p-ADIC L-FUNCTIONS

lim a�k) = an exists k-4oo an xn belongs to PK ,

0,

�

in

21 K,

n=O Ak = A in the n orm topology of P K" =

For each n?

0,

we define a polynomial

(�)

in K[x] by

x(x-1) .... (x-n+1) n! It is obvious that

I\(�)II The following lemma estimates LEMMA

3.

For

:s

I

�I .

Inti:

n ? 1, n n l Iplp - :s I nti :s nplplp-l .

Proof.

Let

It is known in elementary number theory that the highest p ower of p which divides nl is

Hen ce

n-s p p -l

p-ftLH\""'- L".r VI" ...... J.IV1'\t.;:,

n-I (Since s?: 1, we have even I nll 2' IpIP-I.) A s aN?: 1, we see that pN :S n so that N Slog n/ log p. Hence s 'S (p -l) (N+l) 'S (p-l) (log n/ log p

+

1)

and In!1

n-s p I Ipl -

=

q.e.d.

n Note that Iplp -1:s In! I holds also for n

=

O. Hence

for all n > O. Now, let bn, n ?:

0,

be a sequence of elements in K and let n

cn =

�

i=o

(_l)n -i

(i) bi '

n ?: 0,

so that

Then cn

f.

K,

n

2:

0,

and n

bn =

THEOREM 1. Let

r

� (i) Ci ' i=o

1

i be a real number such that 0 < r < IpIP- . Suppose

that for all n 2: 0,

with some C > O. Then there exists

a

which has the following properties;

i) A(x)

converges at every

�

in

unique power series A(x) In PK

Up

such that

1

!�! < !p !P-l [-1 , ii) For every

n:: 0, A(n) = bn

.

1

We first note that since Ip!P-l r-1 > 1, A(x) converges for all � Up with !g!:s: 1 so that A(n) in ii) is well-defined. Now, let "" k Ak(x) = � ci cn I a�k ) xn , i=O n",O Proof. f

=

Since Ak (x) is a polynomial of degree a(k n)= 0,

S

k,

we have if k < n.

By the assumption on cn ' !! cn (�)II where

Hence

s

n

Icnllpl-p-l

:s: C

r� ,

n.2: 0,

p-ADIC L-FUNCTIONS

24

It then follows from Lemma 2 that exists in

PK,

and we have k::: O.

Let A

= A(x) � n=O The proof of Lemma 2 shows that an = klim ... a( =

""

But

la(nk) - a(n-l)1 -< IIAk - An-l II n


n.

Hence converges at t Qp with It! < fl1 Now, fix such t, It! < fl1, and consider

and A(x)

€

A(�) - A CO

k

If k < n, then and if n:S k,

=

1

=

!p!P-l

� (an - a�k) �n .

n=o

n 2: 0, r-1 .

S3.

p-ADIC L-FUNCTIUN:::i

25 ,

where l n C r�+ \�\ � C

�,

k

� C (rl\�\) ,

if

\� \ � 1,

if

\�\ > 1.

Therefore

and we see that

For each integer n � k, we have n

k

A k(n)

;�

i;o

ci

(i); � (i )Ci i=o

bn·

Hence it follows from the above that A(n) = bn,

n> O.

This proves the existence of A(x) having the properties i), ii). The uniqueness of A{x) is an immediate consequence of Lemma 1. We state the formula proved above as the following; C OROLLARY.

Let A( x) 1 be the power series in Theorem 1. For each

� in Up with \�\


o. Since f and fn differ only by a factor which is a power of p, it follows from Lemma 1, §2 that Proof.

33.

p-ADIC L-FUNCTIONS

Hence

because

for (a, p)

=0

1.

Therefore

where

n

II

c , h =0 n

i=O a= l p ya ghf

=

gh f

"

l

a=l

pya

(_l)n-i

CO X(a)i

X(a) «a> - l)n .

27

28

p-ADIC L-FUNCTIONS

We shall next prove by induction on h that in the local field K, for h? 1. Since "" 1 mod q, this is trivial for h "" 1. 1 Sa S qh + l f, (a, p) = 1. Write a in the form a

=

u +

Let

h

>

1 and let

qh fv,

Since a "" u mod q, we have (Ll(a) = (Ll(u) so that

Hence

«a>

-

l)n

n

=

1. (�) «u> - 1)i (qhf(Ll(ur1v)n- i , i""o

where the i-th term in the sum on the right is divisible by

If n-i? 1 , then

i

+

hen- i)

=

n +

(h-l) (n- i) ? n

+

h

-

1 .

Therefore X(a)«a>

-

l)n

""

X(u){

_

l)n mod qn +h-l .

Taking the sum over a, we see that

so that h>l

§3. p-ADIC L-FUNCTIONS

29

This p roves the congruence mentioned above and hence also the lemma, because cn is the limit of q -h f-1 cn , h"

3.4. We may now a pp ly Theorem 1 for the above sequences bn and cn ' n :::: 0, in

K

=

QCX) and for

r The theorem

=

\q\

shows that thete exists




O.

By Theorem 2, §2, we have, in particular,

A iD)

=

C1 - X(p)p-l)Bo, X = 1 =

-

�,

0

where X O denotes the principal character. THEOREM 2. There exists a p-adic meromorphic function L pCs; X) with the following properties: i)

L pCs; X) is given by

00

where a_I

=

=

1 0

-

1, p

if X

=

XO ,

p-ADIC L·FUNCTIONS

30

and where the power series converges in the domain

m

ii)

For

n

=

=

I s I Sf Up' Is-I I < r}, r

1, 2, 3, ...

Lp (l-n; X)

Furthermore, as a

Lp(s; X) Proof.

p

1

1 l Ip IP- Iql- >

=

1.

,

=

- (1

_

B n ' Xn n-l Xn (p) p ) __ n

adic meromorphic function on the domain

-

is uniquely characterized by the above two properties

m,

i) and ii).

Let

with the A <x) mentioned above. Since X Bn,Xn L (I- n; Xn) = - --' n

n?: 1,

by Theorem 1, §2, both i) and ii) follow immediately from the correspond­ ing properties of the power series A <x), The uniqueness of Lp(s; X) X is a consequence of Lemma 1.

The theorem solves the problem stated at the beginning of this section. A s already mentioned there, we shall call Lp(s; X) the p-adic for the Dirichlet character X.

L-function

Now, let e denote the order of the group V in 3.1; e = p-l or e = 2 according as p > 2 or P = 2. Since e is also the order of the

character

cu,

we have Xn

=X

whenever n

==

0 mod e. Hence

The function L p (s; X) is uniquely characterized also by i) and by the above equalities.

§3. p-ADIC L-FUNCTIONS

31

Let X be a Dirichlet character such that

XC..c.l)

-1,

=0

If n '" 0 mod e, then n is even so that Bn, X

�

0

by Theorem 2, §2. Hence, by the uniqueness mentioned above, if XC-I)

=

-1.

On the other hand, if XC-I)

1,

then

for n

==

0 mod e so that for n;:: 1, n

==

0 mod e.

Hence Lp Cs; X) is certainly not identically O. 3.5. We shall next prove a theorem which states that the .converse of

Theorem 1 is also partially true. Although this will never be used in the

following (exce pt for an elementary lemma below), the result seems inter� esting enough to be mentioned here. For each n ;::·0, let

Ce X

_

Sin ce (eX

-

1) n

i d�:) xk .

1)n

bo n

=

l

1",,0

(_I)n-i .

cn eix ,

p-ADIC L-FUNCTIONS

32

we have

n n ) d� = � i=O

(_l)n-i

( i) ik .

It is also clear from the definition that for 0:; k < n.·· LEMMA 5.

The integer

dkn)

is divisible by

n!

(for all

n,

k 2: 0)

so

that

fo r n � 1. Proof.

This is obvious if

k

=

0. Let

k 2: 1.

Since

we have Hence we see, by induction on k, that d�n) is divisible by n! so that I dkn)1 :; I nti. The second inequality was proved in Lemma 3. Now, let bn, n � 0, be a sequence of elements in Up and let n

cn = I (_l)n-i (i) hi ' i=O

Let

[1

be a real number greater than

1:

n> O.

93.

p.ADIC L·FUl'lCTlUl'lS

33

THEOREM 3. Suppose that there exis ts a power series A(x)

in

Up[[x]]

which converges for all I; in U p with

and which satisfies A(n)

bn,

=

for all

n>

O.

Then lim lcn! r-n

n.... oQ

for any real number

r

=

0

such that

Proof. Since

we can find � in U such that p

1 < \1;\
1 by the choice of g, lim Icnl r-n n-+oo

=

0

f

q.e.d.

Note that since rl > 1, there exists a real number r such that

Now, let K be a finite extension of Qp in Up and suppose that all bn and, hence, all cn(n:: 0) are contained in K. Then Theorems 1, 3 imply the following result; there exists a power seri es A(x) in KUx]] which co nverges in a circle of radius >1 around 0 and which satisfies for all n? 0,

if and only if lim \en \ r-n

n...oo

for a real number .r such that

=

0

1

r < \p \P-l This may be compared with the classical theorem of Mahler which states that there exists a continuous map o


Zp/pZp'

k induces a homomorphism of multiplicative groups

then induces

Let V denote the group of all toots of unity in

prime to p. Then V

plits the extensi?n

s

np

iJ / D :

whose orders are

p-ADIC L-FUNCTIONS

38

Now, it follows from the above that U; p V 0 and that this is a topological direct decomposition of U� . In p articular, the projection x

=

.

7T.

x

nX

��p ...

D�

is continuous. 4.2.

As usual, we define a power series log(1+x) in Qp[[x)] by (_1)n-l n -"-- --'-'n -- x .

It is well known that the power series converges for every It I < 1 and that

t

in Up with

Up log a log(l+(a -1)) defines a continuous homomorphism of 0 into the add itive group of Up ' It is also clear from the definition that a 0, log a(a) a(log a) , for every automorphism of the Galois group Gal(U/ Qp)' ¢: 0 a

...

f-->

=

€

=

a

LEMMA.

The above ¢ can be uniquely extended to a homomorphism

such that

rjJ(p)

=

0.

r/J is continuous and satisfies r/J(o(a)}

Proof. Let

rr:

n�

....

=

o(r/J(a) ,

D be the projection defined in 4. 1 and let r/J

=

� .... 0

¢ 0 17: U

....

Up .

Clearly, r/J i s a continuous extension of ¢, satisfying r/J(p)

be an arbitrary extension of ¢ with A(p) tion of P and V that for each (3 in P

=

x

m, n "0 such that

=

O. Let

O. It follows from the defini­

V, there exist integer s

This implies

so that A({3) =

O.

Hence A(P

x

V)

=

0

and

Thus the uniqueness is proved. Finally, fix A

=

0-1 or/Joo: O

�

0 ->

This is an extension of ¢ and satisfies A(p)

uniqueness, 0/

=

A, i.e., r/J(a)

=

in Gal(Ol Q p) and let

U p' =

O. Hen ce, by the

o-l(r/J(o(a») for every a in

O�;

40

p-ADIC L·FUNCTIONS

In the following, we shall denote ljJ(a) again by log ai when it is necessary to distinguish it from the ordinary (real or comp lex) log, we shall denote it also by log a : p log

a

=

logp a

=

ljJ(a) ,

a t

n�.

We can See easily that the continuous homomorphism

is surjective and that its kernel consists of n-th roots, for all n ? all powers of p in 11' :

n�

...

n�.

We also note that although the proj ection

1, of

fi depends upon the choice of P and hence is , not unique, IjJ = .

¢ 011' is canoni cally defi ned as characterized in the lemma. 4.3. As an immediate application of the above, we

p-adic regulator o f a number field. Let

F

shall next define the .

b e a finite al gebraic number

field, i. e. , a finite extension of degree, say, n, over the rational field Q, contained in

n.

L et

' be the infinite (archimedean) absolute values of

1 ::; i ::; r, l et

F,

and fo r each i,

denote a morphism such that a E

F.

Let ei

=

1 or 2

according as Vi is real or complex, namely, according as ¢/F) is con­ tained or not contai ned in the real field R. Let E be the group of all units in F, and · W the subgroup of all roots of unity in

F.

A theorem

41

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

of Dirichlet states that E/W i s a free abelian group o f rank r-1. Let 2 , " " 2 r _ 1 be a system of fundamental units in F, L e. , a set of ele­ 1 ments in E which represent a basis of E/W, Let 2 r be any rational integer, E r > 1. Define, with ordinary log,

where det denotes the . determinant of the . r

Then R is independent of the choice of

E1,

x

r matrix

. . , 2 r _ 1 ' Er as mentioned .

above and is called the regulator of F. It i s known that R f O.

ei = 1 fo r 1 :s i :S r, and ¢ t ' . . . , ¢r constitute all isomorphisms of F into U. Replacing the real log in the above by the p-adic log, we define We assume now that F is a totally teal field so that r

Co

n,

it

the p-adic regulator of F. Here we choose E r to be prime to p so that lo� E r f O. Note that ¢i (Ej ) are in U and hence in Up so and call

that lo� ¢i(Ej ) are defined. It is easy to see that up to a factor ±1 , � is again independent of the choice of E 1 , . · . , E r_ 1 ' E r and defines an invariant Df the field F, (If F is not totally real , we can still define Rp similarly. However , it may depend upon the choice of rPi which correspond to complex Vi 's.) Now, it was conj ectured by Leopoldt that

for every p and totally real F. The conjecture was first verified by Ax � n some special cases, and later, following the idea

of

Ax and

using a powerful method of Baker, Brumer [4] pro�ed it in the case where . F is an abelian extension of the rational field. However , the conjecture is not yet proved i n the general case.

42

p-ADIC L-FUNCTIONS

4.4,

Let E'l ' " " E'r_ l be a system of independent units in F, i.e., a set of r-l units in F such that the subg�oup E' generated by E'1' " " E'r_l ' and W, has a finite index in E. Choosing an integer E 'r > 1, we define R' is called the regulator of E'l' ,.,' E'r_l' and is denoted by R( E '1 ' " . , E'r_l); in fact, R' is independent of the choice of E 'r' and a simple group-theoretical argument shows R' [E : E'] R , It is now clear how to define the p-adic regulator of E'l ' . , . , E 'r_l' Rp(E'l ' " " E'r_l)' in the case of totally real F . Furthermore, for SUGh Rp(E'l ' , . . , E'r_ l )' we have again =

with a suitable choice of factors ±1 for Rp and Rp(E'l' . '" E 'r_ l ) ' Hence Rp /:. 0 if and only if Rp(E'l' ' ' ' ' E �_ l ) /:. O.

§S. CALCUL ATION OF Lp Cl; 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 Up and let CK

�enote the set of all continuous maps

CK is obviously a commutative algebra over K. For f in CK , let Il fll = max I f(s) I . SEZp Then Ilfll defin es a norm on CK such that

Ilf + g il s max( II £ II , Il g ll ) , l I fgll s lI f ll I I gil ,

l I afll

=

I al 1\ fll

a E K.

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

in the norm topology in

CK , then res) = lim fn es) n ...""

in

K for every s in Zp '

43

44

p-ADIC L-FUNCTIONS

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

f->

S

res)

,

and f(x) is uniquely determ,ined by this map . Hence we may identify f(x) with the map it defines and consider K [x] as a subalgebra of CK . Thus, for example, the polynomial n 2: 0, defined in §3, belongs to

(�)

CK, and since

for the norm

f

Zp for

(�).

S f

Zp and

I (�)I\

in

=

(�1)

=

±1, we have

1

CK .

We now define a ma p ¢ : Zp

x

Zp

(x, s )

->

Zp

f->

¢(x , s)

by ¢(x , s)

=

if x I U (Le. x

0

E

U,

P

>

2,

x €

U,

P

=

2.

if x

if

f

p Zp)'

U , <x> (resp. x) is an element of 1 + p Z (see the p definition of <x> in 3.1) and <x>s (resp. xS) is defined by In the case x

(

s <x>

(1 + <x> - 1)s

(resp .

l (�) «

n=O

x> - 1 )n

00

xS

(1

+

x - 1 )s

=

l (�) (x n=O

_

1)n

).

§s.

CALCULATION OF

L

p

(I; X)

45

Since

(resp.

)

,

the power series for <x>s (resp . xS) converges unif ormly for (x, s) in U

x

Zp' Si nc

e

is continuous.

U

x

Zp

is open

in

Z p

x

Zp'

it follows that ¢: Z x Zp p

4

For each integer n � 0, let

n

Yn(s)

=

l (_l)n-i (�) ¢(i , s) , i=O

It is clear that Yn belongs to CK . LEMMA

1.

I I Yn l1 .::: I n !1 '

n > 0.

Proof. We have to sh ow that ! Yn(s)! .::: l n f ! for all s integer, m � 0, such that

p-1 I m , Then n

yn(m)

""

2: i=o n

=

(_1)n -i

(i) ¢(i, m)

I (-1 l-i (n im i=o pri




O.

It satisfies

1! r( A)\! ::; \\ All ,

and

1((1

+

n x) ) (s)

=

¢(n, s) ,

fot all

A E QK

for all

n ::: O_�

p-ADIC L-FUNCTIONS

48

Let r: K [x] CK be the linear map which maps a polynomial m m A(x) = n=O � an xn to ['(A) = � an Yn · Using Lemma 1 , we see n=O ->

Proof.

Since K [x] is everywhere de,nse in QK' [' can be extended to a linear map QK '" CK with I ['(A) I ::; IIAII for every A in QK' By Lemma 2, n i ['( 1 + x)n ) (s) l ( i) n x ) (s ) =

i=o n

=

=

l (i) Yi(S) = I (�)Yi (S ) i=o

hoD

¢(n, s)

n > O.

,

Finally, the uniqueness of [': QK CK is obvious. For A A(x) in QK' ['(A) (s) is a continuous function of s with values in K. We write ['A(s) for [,(A) (s): ->

=

E

Zp

and call ['A(s) the ['-transform of A in QK' We shall next prove several lernmas which describe the properties of ['-transforms. As in 3.5, let n > O. Given any A(x) = "l n=O

§5. in

CALCULATION OF

Lp( l ; X)

49

K[[x]], consider the fonnal power series in K[[t]]: ""

00

00

=� where

n

0n(A) = I. ai d�i ) . i=o If

A(x) beloogs to Q K ' it follows from Lemma 5, §3 that xaxCl ai d�) I) S �a 1> -

0

-

0

Therefore, for each n ? 0,

defines a linear map with LEMMA

3.

For

A

E

QK '

s f

Zp ,

where the limit is taken over any sequence of integers ni' i that

n i � 0, p-1 1 ni'

and such that

as

i

�

""

.

::: 0, such

50

p-ADlC L-FUNCTIONS

(Since the integers n

�0

with p-1 1 n are everywhere dense in Zp ' such a sequence always exists for any given s in Zp ' )

Proof. Let m > 0 be fixed and let A(x)

=

(1 + x)m .

By Theorem 1 ,

O n the other hand,

n=O so that n ::: O. Hence,

if

p I m, then lim 8n . cA) 1

Let p

Y

==

n· lim m 1

0

=

=

¢(m, s) .

n· m . If p > 2, then p-l I ni implies m 1

lim

8 n . (A) 1

n·

=

lim <m>

1

=

<m> s

=

=

<m>

n·

1

so that

¢(m, s) .

If p :; 2, we have similarly lim 8 . (A) n1

=

n· lim m 1

:;

m S :; ¢(m, s )

Hence the formula of the lemma is proved for A(x) hence for all polynomials in K [x].

0,

Now, given any A in QK and any E > such that II A - 0 as n is defined; A('; ) € K(O f Up . It also follows from the above that �

00 .

=

Hence, with .; fixed, A(';) dep ends continuously upon A. LEMMA 5.

For

A


1. L (s; X) is then a holomorphic function of s p < r! and hence is continuous in that do­ in the domain Is I s ( Dp ' X be a Dirichlet character with conductor f

=

Is-I I

main (Theorem 2, §3).

We fix an integer N > 1 such that (N, fp)

=

X(N) P. 1

1,

and let

{"\ l r; r eX) 52

= the set of all N-th roots of unity in Dp ' 2 1Ti -= e f ( U S; Op ,

f

=

l

(i

X(a) r; a

(Gaussian sum, see 1 . 2)

a=l = the conjugate character of X(a) = X(a)=

We also fix

r;,

a

finite extension

and X(a), a

(

'" y=I),

Z.

0

l

,

X:

f = f and X if (a, f ) = 1 , if (a, £ ) � l .

K of Qp in Up which contain s all

A,

54

p-ADIC L-FUNCTIONS \

N ow, let z be an indeterminate and let

This is a rational

function in t

for the

K(z) and it has the property

et G( et) = F (t) = X

00

l

Bn, X

n=O

�

F (t) introduced in 2. 1 . Let X

g(z)

f

l

=

a=I

X(a)z a - l .

By a well-known formula in al geb ra, f

G(z)

=

l

� = zf 1

a=l

_

where

a ( g«( ) = a

f

l

�� , f z _ "",

it

for every s in Zp ' If s f. 0 , then this implies rA(s) = (1 - XCN) s) Lp (l-s; X) . However, the same holds also at s = 0 because the both sides are con­ tinuous functions of s Zp . If P = 2 , we obtain a similar formula with N S instead of s . Therefore the following theorem is proved: €

TH EOREM

2.

be a non-principal Dirichlet character with

(N , pf) = 1, X(N) f. 1 N-th roots of unity in Dp ' Let

Fix an integer the set of all

X N ::: 1

Let

such that

f fX. =

and let {AI denote

with

f

reX) = � X(a) , a , a=l

2171 -, =; e f .

Then the power series A(x) belongs to Q K and rA

(s)

=;

=

X(N) s) Lp(1-s; X) (1 - XCN) NS) Lp(1-s: X) ,

(1

-

,

p > 1,

S8 for all

p-ADIC

s

in Z ' In

p

particular,

rA(0)

=

(1

L-FUNCTIONS

-

Lp (l; X) .

X(N»

5. 4.

In order to evaluate Lp(I; X), it is now sufficient to compute rA(0). For this, we shall use the formula in Lemma 5. Clearly A(O) O. Let t be a p-th root of unity in 0p ' Since =

I 1 t- -A(IaI

we have A(�

-

1)

If

=

-

11 < 1 ,

""

f

- ( -t -s )n

-l l (_I)n X) =� X(a) l � �(f a= l Ah n=l n

with log = logp ' Hence, by Lemma S, rA(o) =

1

I _ V- a

- pf S reX)

where

Since

II

A�l

(1

- 1I'p( ap)

=

II Afo l

(1 - ACap) ,

(p y N),

we obtain

where f

I

S1

a=1

X(a)

10g(II

M1

f

I

S2

a=1

a (1 - A ( )

X(a) 10 g

(rr

Ah

)

,

)

(1 - A( ap) .

Suppose first that p r f. Then X(p) = X(pr 1 implies f

2

S 2 = X(p)

Suppose next that p

a�l

(rr (1

X(ap) 10g

A�l

\ f so that X(P)

�

X, there is an integer b such that (b, f )

�

)

AC aP»

= X(P ) S l . .

O. Since f is the conductor of

b ;: 1 mod

1,

-

t,

X(b) � 1.

p

With such b, abp ;: ap mod f implies f

S2 =

I

a=1

(n

X(ab) IOg

A�1

f

= X(b)

so that

I

8=1

X(a) log

(1

-

(n

Ml

)

A( abp)

(1 - A C aP)

)

bU

p-ADIC L-FUNCTIONS

Hence

in both cases, and we obtain f

S = (X(p) - p)

Let (a, f )

""

I.

a=1

)( a) 10 g

(II

(1

A;fI

_

,\,a») .

1 so that , a is also a prim itive f-th root of unity. Then

Hence

f

I.

S = (X(p ) - p )

a=1 (a, f )=1

Using (f, N) = 1, )( a)

=

X(a) ( l og (1 - , aN)

- log

(1

_ , a»

X(N) X(aN), we finally obtain f

S = (X(p)

- p) (X(N) - 1)

.I

a=1

X(a) log (1

_

, a) .

(a,f )=1 Since

we m ay write S also

in

the form f

S

=

(X(p) - p ) (X(N)

- 1)

�

a=1

(a,£)=1

X(a) log (1

_

, -a) .

.

'3:>.

Now, by Theorem

CALCU LATION OF Lp(l;

X)

61

2,

( 1 - X(N» Lp (1; X) = ['A(0)

reX)

=

-Pi S .

Since 1 - X(N) -I 0, the following theorem is proved: TH EOREM 3. For a non-principal Dirichlet character

X,

f

� X�») r(�O l X(a) logp (1

Lp (l; X) = - with the

a=l

_

,- a)

p-adic log function logp defined in §4.

we .compare this formula of Leopoldt with the classical formula f(}t L(I; X) (see 1 . 2): If

L(I; X) =

-

r(}) l

X(a) log(1

_

,- a)

,

a=1

we find a remarkable similarity between the two. In Theorem proved

2, §3,

we

for n = 1, 2, . The above theorem shows that the saine formula holds also for n = 0 if only log in L(I; X) is replaced by the p-adic lo �. . . .

It is an interesting open problem to find similar expression!> for the values Lp (n; X), n 2 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].) Let

p

>

2,

2m

' = ePCi H), and let =

62

p-ADIC l...-FUNCTIONS

F is the cyclotomic field of p,-th 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 ( Sj F) IlLes; X) , X + ( s; F+) = II L(s; X) x

where the product II (resp. II+) is taken over all Dirichlet characters X such that fX lp Crespo such that fX lp and XC- I) = 1). Motivated by such a fact, Leopoldt Clefined the p-adic z eta-fun ctions of F and F+ by (p Cs; F) (p Cs; F+)

=

=

II Lp (sj X) ,

X

+

II Lp (s; X)

X where the products are taken over the same X's as mentioned above. It follows from Theorem 2, §3 that both (p(s; F) and 'pCs; F+) are p-adic meromorphic functions of s in the domain described in that theorem. Fur­ thermore, since Lp Cs; X) 0 for X(- I) - 1 , (p Cs; F) is also identically O. Hence 'p Cs; F+) is the one which is really important. More generally, let K be a finite algebraic number field which is an abelian extension of the rational field Q. The zeta-function of K, ( s, K), is again a product of Dirichlet L-functions L(sj X) with X ranging over a certain family of Dirichlet characters. Hence the p-adic zeta-function of K, (pCs; K), can be defined in the same way as above, replacing each factor L(s; X) for ( Sj K) by its p-adic analogue Lp (sj X). t.::/s; K) is again a p-adic meromorphic function, and (p Cs; K) 0 unless K is a real field. ==

=

==

§5.

63

X)

CALCULATION OF LpCl ;

Let p denote the residue of the zeta-function �(s; F+) at s classical formula of Dedekind states that 2m- l p Vd h R

=

1. A

=

where m = D"21 = [F+: Q], d is the absolute value of the discriminant of F+: d pm- I , h the class number of F+, and R the regulator of F+. On the other ha nd, si nce �(s; F+) is a p roduct of L(s; X), it follows from the result stated in §l that =

p =

+

II

Xl-X

II

L(I; X) O

+

xlox

O

p- l

(- � }2 X(a) log \ 1 _ (al) . a=l

Using the fact +

r(X) =

II

(yp )m-l ,

xl-x 0

we obtain

( Let

E

m

_

� � X(a) log« l

_

a=l

denote the group of units in the real field

(a) (1 F+

_

)

( -a)) .

and let

E

where a i s primiti ve root mod p; a

E

is a unit in the group

E. A

64

p-ADIC L-FUNCTIONS

theorem on circular determinants then shows that the product above formula for h is equal to the regulator (see 4 . 4)

II+ in the O xf.X

of any m-l elements E 1 , . . . , Em _ l in the set of all conjugates of E ; E O, 0

�

in L emma

1 . Then

I L.

p-ADIC L-FUNCTIONS

Proof. Let f(x) '" l+x. Then � = y(l+q o) so that

¢t(y( I+qo » = ( I+qo)-t = f«((I+qort

-

1) .

This implies the same formula for arbitrary f( x) in o [x] and, hence, for

fex) in A.

6. 4. Let e be a Dirichlet character of the first kind with eC-l)

let

==

1, and

Fix a finite extension K of Qp in Up containing all values of e(a), a f Z, and define qn ' Gn , R n , etc. , in 6. 2 by means of this m o and K. For each n ::: 0, let

where the sum ranges over all integers a such that 0 S. a < q n' (a, qo)

=

1. Let

Both �n and 1"fn are elements of K[rn]' L et � 'n+ l denote the image of �n+ l under the K[rn l] -> K[rn] : +

e-'n+l Let

==

-

morphism

� I b e1(b) yn(b)- l , qn +l

b

. 0 S. a < qn ' (a, qo )

=

1 , 0 S.

i < p.

sO.

AN ALTERNATE METHOD

13

Then b '" a mod qn implies Yn(b) = yn(a), Since fe and fw (= q) divide q o ' the conductor fe also divides q o ' Hence b '" a mod qn again im­ l plies e 1(b) 8 1 (a). It follows that =

=

fn

-

P41 l e1 (a) yn(a)- l a

.

Here

l a

(8 1 (a) yn(a) -

1

+

8 1 (qn - a) Yn(qn-a) - I )

where the sum on the ri ght is taken over all integers a such that qn O s a < 2' Ca, q o) = 1. However,

Hence

I. 8 1 (a) yn(a)- l

=

0

a

so that

Thus, in general,

We shall next show that In is contained in Rn , Since < l+qo> l+q o ' e (l+q o) 1, =

=

74

p-ADIC L-FUNCTIONS

lln

=

�

fn + 2

n

l « I+ 8((1+qo)a) yn«(1+QO)a)- 1 a

For each integer a, 0 :s a < qn ' (a, q o)

w« I+qo)a) "" wCa'), eC(I+qo)a) « I+qo)a>

=

=

=

=

.

1 , define integers a' an d a" by

&(a ') , YnC(I+qo)a)

w« I +qo)a)- 1 (l+qo)a

=

=

yn(a')

,

w(a ,)- l (a '+ a"qn)

l1n under Rm -> Rn ' m 2: n 2: 0, there exists an ele­ ment 11 in R such that ""

= 11 o 00

11 ""

l 1' m l1n .

=

Let g(x; g)

I->

� 11

under the isomorphism A = o Ux]] � R in Lemma

1.

Kg

Let

denote

the field generated over Qp by all values of 8(a), a < Z. and let 0 be 8 the ring of local integers in Kg. By the choice of K, Kg is a subfield of

K

and 00 =

Kg

n 0.

It is easy to see from the definition that g(x; 8)

is in fact a power series in 0 0[[xl] and that it depends only upon

g

and

is independent of the choice of the auxili ary field K which i s used to de­

fine A and R.

o

We now assume that 8 is non-principal: 8 /0 X . We shall prove that in this case,

e-n

is also contained in Rn ' Since

we obtain as in the above that

1 e- = n __ 2 qn =

_l

l a

« a> 8(a) yn(a)-

l

+

l 8(qn -a) Y n(qn-a) - )

l a

;

81 (a) yn(ar

1

.

Fix an integer a D , (a O ' qo) = 1. and denote by �,". the partial sum taken over all integers a such that

ib

p-ADIC L-FUNCTIONS

Since '" mod qn for such integers a, we have

However, we see from G n = rn l'l. n that when a takes the values as mentioned above, the elements 0nCa) and 0n(qn-a) precisely fill out the group l'l. n. Since () is essentially nothing but a non-principal character of l'l. n ' it then follows that x

Therefore

and hence

� a

For p > 2,

this

()(a) Yn(a)- l

already proves

that

_

0

fn is an element of Rn . If p

=

2,

then

mod 20

so that mod 20. This implies

and fn is again contained in Rn· Now, since fm en under Rm f->

and let

.... Rn ,

m

�

n

� 0,

let

S6.

AN ALTERNATE METHOD

f(x; 8)

�

77

t!

under the isomorphism A � R. Just as g(x; 8), f(x; 8) is then series i n

0e[[x]]

and it depends only upon

e.

Clearly

a

power

Hence

Let

Then

under A � R. Hence g(x; 8)

=

hex; 8) f(x; 8) .

o If e is the principal character: e X , then m o 1 , qo q, K e o Qp ' and g(x; X ) is a power series in Zp [[x]]. As in the above, let =

=

=

o We define f(x; X ) by

o Thus f(x; X ) is not a power series but is an element of the quotient

field of Zp [[x]] satisfying

o g(x; X )

=

o o hex; X ) f(x; X ) .

O We shall see later that f(x; X ) is, in fact, not contained in Zp [[x]l

=

p-ADIC L-FUNCTIONS

'/'6

6 . 5 . Let X be a Dirichlet character with XC-I) X

=

=

1 and l et

e 'lT

be the decomposition of X into the product o f the first factor 8 an d the second factor fe qo Since

'IT.

=

=

=

Let fe

or

rna

(

m o q,

1, 'lTC- I)

=

moq ,

=

X(I+q O ) - l

1, w e have

=

eC-l)

=

XC-I)

=

1 .

Hence f(x; e) can be defined as explained in 6 . 4 . e or f mo q p , e :::: o. X

Note also that f X

=

ll!o

=

LEMMA

3. 1

2f (S (I+qo) -

n

- 1;

e)

=

-

Bn ,X n 1 ( 1 - Xn(p)p - ) --n , Xn n

=

n

Xw-

,

for every integer n 2: 1. Proof.

We fix a finite extension K of Qp in U p ' containing all values

of X(a), 8(a), 'IT(a), a

Ii

Z, and apply the results of the preceding parts.

Let t be a fixed integer, t 2: O. We app ly the morphism ¢t , n

n 2: 0, in 6 . 3 to the both sides of 2 TJ�

=

I a" 81 (a') yn(a,) - l . a

By the definition o f ¢ t, n ' 2 ¢t, n (TJn)

=

I a " 81(aj 1T(a') t a

=

2. a " X 1 (a') t a

� a" X a ') a , t = � t+l ( a

=

¢f, n'

§6.

with X 1 = X - I Xt+1 = XUJ -t -1 UJ

79

AN ALTERNATE METHOD

,

•

Now, (1 +qo) a

=

a , + a ,,qn implies

If n is large and fX I qn' then Hence it follows from the above congruence that Xt+ l ( l+qo) (1+qo)t+ 1 2. Xt+l (a) at+1 a a

a so that Therefore

2(t+1) ¢lfJoo) 2 (t+1 ) lim ¢,t n(1n) - (1 - ,- 1 (1+qo)t+ l ) J�� q1 l Xt+ 1 (a) at+1 . n a As g(x; () 1� under A � R, it follows from Lemma 2 that '00

=

!->

On the other hand, for t > O. Hence it follows from the above that

Now, suppose that (a, p) = 1 but Ca, qo) > 1. Since f Xt+l

=

m Op

e'

with some e' 2: 0, this implies (a, f t ) > 1 so that Xt l (a) = O. Hen ce + X +1

�

O�a (1 a" a,-1 qn) so that

and, consequently,

rex; X

=

-

3

-1

=

=

=

+

Hence, taking the sum over the integers a such that 1, (qo q), we obtain =:;

(1 - �)qn log(l+Qo) � log(l+aHa,-lqn) a =:;

0 :;

a < qn ' (a, qo)

=

· p -ADIC L-FUNCTIONS

82 so that

(1

-

�) 10g(1+QO) � a"'a,- 1 '"

a

m od p n + 1 Z . p

Therefore

namely,

These results will be used later.

6.6.

Let X, qo ' C;, K, etc . , be defined as in 6 . 5. Consider a power

series uCs-l) in K[[s- I]] defined by

Since qo is divisible by q and I log(1+qo) i S i q i , we have, by Lemma

3 , §3 ,

with

r

�

1 I q i -1 i p l Fl > I .

Hence u Cs-I) converges in the domain

so and

Let

�

Is i s

f

U p ' I s-1 1 < d ,

luCs-I)1

< 1,

for s

f

SO .

20.

AN ALTl!:RN ATJ:£ Ml!:THUD

c is contained in the maximal ideal f' of the ring

0

of local integers in

K. Let A(x)

be any power series in A ideal

m

=

o[[x)]. Since c+x is contained in the maximal

of A,

�� exists in A. For each a

=

in

m

�

n=O

an (c+x)n

Bex)

0p ' lal < 1, we then have A(c+a)

Let

=

Bea) .

00

Since uCs-l) is a power series in s - 1 without the constant term, we obtain by a formal computation that

It is easy to

see

that

because \bn l S. 1, n 2:: 0, and the coefficients of uCs-l) satisfy the same condition as above. Therefore the power series

84

p·ADIC L-FUNCTIONS

converges in the domain � defined above and A(c + u(s-I»

�

B(uCs-l»

'"

�

cn(s-l)n

for every s in 'il . Note tha t lu(s-I)1 < 1, so that A(c

+

u(s-I»

Ic + u(s-I)\ < 1 ,

and B(uCs-l»

s (�

are well defined in . 'il .

It i s clear that i f s i s an integer, s = t E Z , then

Hence, by continuity we also have 1

for every s in Z p ' where C1+qo)s is defined as in 5. 1 . Therefore we may write ( I+Qo)S - 1 instead of c + u(s-l) also for any s in 'il.

Thus we have A«( ( l+ QO)S - 1)

=

for every s in the domain � .

I.

n=O

cn(s-l)n ,

6.7. We now assume that 0, the first factor of X , is non-principal : e ,b. Xc . In this case, f(x; 8), defined in 6. 4, is a power series in A =

oUx]l Hence, by the above remark, the function F(s; X) = 2f«( (1+Qo)s

-

1; 0)

is defined in the domain SD and is given by a p ower series: F(sj X)

=

I

n=O

an X(s-l)n , ,

§6.

85

AN ALTERNATE METHOD

Assume next that e is principal: e

=

a power series in A, the function

Xo, X

=

Since g(x; Xo ) is

17 .

is a gain defined in 5D and is given by a power series similar to that for F(s; X) in the above. On the other hand,

=

�

Since

�

1

_

}" 1

':, -

l

n

(l o g( l+q )) n.

o ----:I-=--

(I-s)n .

is a root o f unity with order a power o f p,

Hence, by the remark in the proof of Lemma 3, §3,

I

1

1-(

(log( I+Q o))

nl

n

I .:s

It follows that h«((I +Q )S o function

I p I-P�i I QI n I P I-�=t 1 ; X o) f 0

�

_

r-n ,

n > 1.

for every s in 5D and that the

86

p-ADIC L-FUNCTIONS

is defined in 5D and is given by a power series F(s; X) =

Finally, let e

=

l

n=O

an ,X(s-1)n ,

X O an d also X

= 1l '"

O X . In this case, , = 1 and

where, by the same remark in §3, =

Therefore h«(1+Qo)S function

-

-n

r

n > O.

�

o 1; X ) vanishes in 5D only at s = 1 , and the

is now defined for all s f, 1 in 5D and is gi ven by

where n 2: O (C > O). By the remark at the end of 6.5, we also see that °

. a_I = lIm (s - 1) F(s,. X ) = s ...1

1 log(1+Q ) (1 - p) o I og(1 +qo)

1

-

1

p .

§6.

87

AN ALTERNATE METHOD

6 . 8 . For each Dirichlet character X, XC-I)

=

1, we have thus obtained

a function FCs; X) defined in the domain

'l) excluding s

""

=

{ s ] s ( Up ' ]s-l] < rl ,

1 in the case X

plained above. By Lemma

3,

=

XO, with the analytic property as ex­

it also satisfies n Xn - X(j)-

for every integer n � 1 . Hence FCs; X) possesses the properties i), ii)

of Theorem 2, §3, which uniquely characterize the p-adic L-function L p (s; X). Therefore

FCs; X)

=

Lp Cs; 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 XC-I) 1. =

-1, then L p (s; X) is identically zero. Hence nothing essential is lost by the new approach.

However, if XC-I)

For XC-I) =

1,

=

we now have the important formula

for s ( 'l) (s � 1 if X = XO). We recall that 61 is the first factor of X,

mo or f8 m o q with (m o' p) 1 , qo moq, , X(I+qO)-l, that g(x; 8) and hex; 8) are power series in 0 e[[x]] associa�ed with 8, and fO

=

=

=

=

that f(x; e) i s also s uch a power series if 8

=

�

XO.

§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 cyclotomic fields .

7.1.

As i n 6 . 2, w e fix an integer m o � 1, (mo' p)

§§S,

=

6 to the theory of

1 , and define n > O.

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. k n is obviously a Galois extension of Q. For each automor­ phism

a

of kn /Q, there exists an integer a, (a. qo)

=

1, such that

for every qn -th root of unity (; in kn . The residue class of a mod qn ' namely , Gn (a). is then uniquely determined by 0, and the map

defines a canonical isomorphism of the Galois group of kn/Q onto the group Gn defined in 6 . 2 :

I f m � n � 0, k n i s a subfield o f km ' and the restriction o f a n automor­ phism in Gal(km /Q) on the subfield kn defines a natural homomorphism

88

§7.

SOME APPLICA TrONS

89

such that the diagram

1 is commutative. It follows in particular that

Now, let koo denote the union of the increasing sequence of number

fields:

ko

C

kl

C

. . .

C

kn

C ... .

k,.., is the (infinite) cyclotomic field generated over Q by qn-th roots of unity for all n > O. Since Gal(k /Q) 00

=

lim GaICkn/Q) , 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

where Q

=

1 or 2 according as m a

=

=

such that fX

I qn' X(-l)

=

-1 . We write

and call h� the first factor of hn , and h� the second factor of hn . Now, by a remark in 2 . 2 ,

Since w n

=

w opn , we may write the above formula in the form

where X now ranges over all characters such that fX I qn ' fx t qo' and X( 1 ) -1. For such a character X, let -

=

XU)

=

e77

be the decomposition of XU) into the product of the first factor e and the second factor

77,

and let

where q'a is defined by (m'a, p)

=

1

§7. SOME APPLICATIONS

Since p \ fX and X(p)

=

0, it follows from Lemma

91

3, §6, that

Furthermore, when X ranges over all characters such as mentioned

above, the corresponding first factor e of XU) ranges over all characters such that fO \ q ' 8(-1) 1 , and 'X over all p n_th roots of unity .;,. 1. o Therefore we see that =

h�

=

ho p n

II II f C' - 1; e) e

,

where the product is taken over all e and .; such that fe \ qOT 0(- 1) n 1, .; .;,. 1 . and cP

=

1,

=

Let

A(x)

=

II

f(x ; e) ,

e.;,.x o

fe \ qo' e(-I) = 1.

be any automorphism of U p over Qp . For each e in the above product, let eO denote the Dirichlet character such that

Let

a

a

E

Z.

fe , Oa i s again one of the e characters which appear in the above product. On the other hand, it follows from the definition in 6.4 that for such e .;,. X O , f(x; 8) is a power series

(Recall that 8(a)

E

Q � Up ') Since f o

=

in 0e[[x]] and that the power series f(x; eO) is obtained from f(x; 8) by replacing each coefficient of f(x; e) by its image under a. Hence we see

that A(x) is a power series in Zp[[x]], depending only upon the integer rna :::: I:

It is now dear from the above that h�

=

ho (p n

II f('; - 1; Xo» II A (, - 1) , .;

.;

92

p-ADIC L-FUNCTIONS

We shall next calculate the p-adic absolute values of the factors on the right hand side.

7.3. By the remark at the end of 6 . 5 , g(x; Xo) is a power series in Zp[[x]] of which the constant term g(O; X o) is a p-adic unit. Hence

for every ( such that .;:p

n

=

1

« f. 1). On the other hand

so that

1

II (C (

-

1) 1 - 1

=

n I p l -l ,

Therefore we have l h�1

I ho l I

II A(( C

-

1) I

,

It then follows that for m ::: n 2' 0, I h;1 = I h;1 I

II A(( (

-

1)1

m n where C now ranges over all roots of unity such that CP 1, ( P .,£. 1. Note that this implies in particular that A(x) is not identically zero. =

Now, as A(x) is a non-zero power series in Z p [[x]], there exist integers A and fl' A, fl 2' 0, such that A(x)

=

pfl B(x)

§7,

SOME APPLICATIONS

93

where B{ x)

=

l

bn Xn ,

n=O

for 0




\( )- 1 p 1 - m = ,, I p i < I p l l1. p - l

no' then \

-

111\ .

Hence , for such " n I bn C' - l ) l � I p i I b n