Lecture notes in mathematics

Lecture Notes in Mathematics Subseries: Mathematisches Institut der Universit~.t und Max-Planck-lnstitut fSr Mathematik, Bonn - vol... Analytic arithmetic in algebraic number fields..

Lecture Notes in

Mathematics

Subseries: Mathematisches Institut der Universit~.t und Max-Planck-lnstitut

fSr Mathematik, Bonn - vol 7

Author

B Z M o r o z

Max-Planck-lnstitut fLir Mathematik, Universit~.t Bonn

Gottfried-Claren-Str 26, 5 3 0 0 Bonn 3, Federal Republic of G e r m a n y

M a t h e m a t i c s S u b j e c t Classification (1980): 11 D 5 7 , 11 R 3 9 , 11 R 4 2 , 11 R 4 4 ,

11 R 4 5 , 2 2 C 0 5

I S B N 3 - 5 4 0 - 1 6 7 8 4 - 6 Springer-Verlag Berlin H e i d e l b e r g N e w York

I S B N 0 - 3 8 ? - 1 6 7 8 4 - 6 Springer-Verlag N e w York Berlin H e i d e l b e r g

Library of Congress Cataloging-in-Publication Data Moroz, B.Z Analytic arithmetic in algebraic number fields (Lecture notes in mathematics; 1205) "Subseries: Mathematisches lnstitut der Universit&t und Max-Planck-lnstitut fur Mathematik, Bonn -vol ? " Bibliography: p Includes index

1 Algebraic number theory I Title I1 Series: Lecture notes in mathematics (Springer-Verlag; 1205 QA3.L28 no 1205 [QA247] 510 [512'.74] 86-20335

ISBN 0-38?-16784-6 (U.S.)

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© Springer-Vertag Berlin Heidelberg 1986

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Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr

2146/3140-543210

the h i s t o r y of this p r o b l e m ; o n e m a y r e g a r d this n o t e as a r ~ s u m ~ of

C h a p t e r II, if y o u like) C h a p t e r III d e s c r i b e s a p p l i c a t i o n s of t h o s e

M a x - P l a n c k - I n s t i t u t fur M a t h e m a t i k (Bonn) W e are g r a t e f u l to the

D i r e c t o r of the I n s t i t u t e P r o f e s s o r F H i r z e b r u c h for his h o s p i t a l i t y

the M a t h e m a t i s c h e s Institut U n i v e r s i t ~ t Z~rich, w h e r e parts of the

m a n u s c r i p t have b e e n prepared

Bonn-am-Rhein, im M~rz 1986

{xIP(x)} is the s e t of o b j e c t s x s a t i s f y i n g the p r o p e r t y P(x)

c a r d S, or s i m p l y IsI, s t a n d s for the c a r d i n a l i t y of a f i n i t e s e t S;

§I O n the m u l t i d i m e n s i o n a l a r i t h m e t i c i n the s e n s e of E H e c k e

{x;A,T) = c a r d { P i P 6 A N So, f{p) E Y, IpI < x}

Let x 6 Jk and let Xp be the p-component of x, we set then

IIxll = H IIXpll By the p r o d u c t formula,

is known to be compact The group X* can be identified with the sub-

group {xlx 6 Jk' Xp = 1 for p E S o } of Jk' so that ~ + embedded

diagonally in X* m a y be regarded as a subgroup of C k It follows

then that

There is a natural h o m o m o r p h i s m id: Jk ÷ I(k) of Jk on I(k) given

by the equation

PES o

L e t ; 6 Ck' let ~ ( ~ ) b e the c o n d u c t o r of ~ (defined as, e.g.,

in [93], p 133) and let ~ = ( ~ ) be set of those primes in S 1 at

w h i c h ~ is ramified; w r i t e ~ ( ; ) = { ~ ( ; ) , £ ~ ( ~ ) } One c a n d e f i n e

a c h a r a c t e r Xp on I ( ~ ( ~ ) ) by the e q u a t i o n

X ~ ( ~ ) = ~(x) for ~ e I ( ~ ( ~ ) ) , x 6 id-1(0£),

if one r e g a r d s p as a c h a r a c t e r of Jk (trivial on k~) It follows

from d e f i n i t i o n s that Xp is w e l l d e f i n e d since p is c o n s t a n t on

i d - 1 ( ~ ) for ~ 6 I ( ~ ( ~ ) )

P r o p o s i t i o n 2 The f u n c t i o n ~ ~ X ~ ( ~ ) is a p r o p e r g r o s s e n c h a r a c t e r

and ~ ( X ~ ) = ~(~); it s a t i s f i e s (5) w i t h ~ = ~(~) and l equal

to the r e s t r i c t i o n of ~ to X ~ (regarded as a s u b g r o u p of Jk ), in

p a r t i c u l a r , X~ is n o r m a l i s e d if and only if ~ + ~ Ker g If X is a

p r o p e r g r o s s e n c h a r a c t e r , there is one and only one ~ in Ck such that

P r o p o s i t i o n I d e f i n e s a f i b r a t i o n of gr(k) over the set of (generalised)

conductors L e t X 6 gr(k) and s u p p o s e that X s a t i s f i e s (5) w i t h

of the s h a p e (6); we call a p , t p a p p e a r i n g in (6) e x p o n e n t s of X and

L e t G be a c o m p a c t g r o u p a n d let p b ~ the H a a r m e a s u r e o n G nor-

But y 6 H if and only if uyu -I 6 H for u 6 H, t h e r e f o r e ~ ( u y u -I)

= ~(y) w h e n e v e r u 6 H Thus (4) gives

the r e p r e s e n t a t i o n s O,8 and

it is e n o u g h to show that

Ci' respectively In view of lemma 1,

n

G G~G = Z ~i "

one obtains from (9) an e q u a t i o n

(Ci)~G(x) = f d~ (u)$ (uxu -1) ~((wtiv)uxu -I (wtiv)-l)

be the c h a r a c t e r i s t i c functions of A and B, and c o n s i d e r an i n t e g r a l

Ji(x) := f fA (Y) fB (z) • ((YtiZ) x (YtiZ)-I) d~ (Y) dP (z) •

for I ~ j ~ r For j = I e q u a t i o n (14) is obvious

holds for some j in the i n t e r v a l I < j < r-1 Since

Suppose (I 4)

IndG(j) (X (j)) @ IndGj+ I (Xj+ I) = IndG(j+1) (X (j+1))

by P r o p o s i t i o n 2 (in v i e w of the c o n d i t i o n H(J)Jj+ I = G), e q u a t i o n (14) implies that

Pl @ "'" ~ Pj+I = IndG('+1 (x(J+I))

H 3 )

This proves (14) for any j, in particular, we o b t a i n (13)

If G is a finite group, the f o l l o w i n g r e l a t i o n holds:

E^ X ( g l ) x ( g 2 ) = I O w h e n gl ~ {g2 }

l{gl}l

w h e r e {g} = {hgh-11h E G} denotes the conjugacy class of g in G

T h e o r e m I Every c h a r a c t e r of a finite g r o u p is a linear c o m b i n a t i o n

of m o n o m i a l characters

Proof See, e.g., [83], §10

d e t ( l + A t ) i I A A i n = (I+At) ZI ^ "'" ^ (1+At) i n

a n d

( 1 + A t ) £ j = ( 1 + a j t ) £ j , I ! J ! n

L e t K b e a f i n i t e G a l o i s e x t e n s i o n of k The r e l a t i v e W e i l g r o u p

W(KIk) is d e f i n e d as the e x t e n s i o n of G(KIk) b y C d e t e r m i n e d b y

the f u n d a m e n t a l class of class f i e l d theory If K ) K' a n d K' Ik

dim p < [K:k] (5)

1 ] = [K:k ]

Proof Since W I (KIk) is compact and [W I (KIk) :C K

(5) follows from lemma 2.3

of W(KIk) This subgroup is denoted by W(KpIkp) ; thus W(K~Ikp)

G(K;Ikp) One defines two subsets of W(K~Ikp):

l(p) = {~Io6w(K~Ikp) , so = ~(~a) for e 6 v;}

and

~p = {(~I ~ E W(K~Ikp), C~ (~ - eIPI(~a) for c~ E v;}

Let us recall that

=

V p {xlx 6 V, ~ ( T ) X = X for T 6 l(p)}, p 6 So(k),

of V We say that p is u n r a m i f i e d at p if V = V

P

p r o p o s i t i o n 2 L e t p E R(KIk) and let p 6 So(k) If p is un-

r a m i f i e d in Klk and if U~ ~ Ken p w h e n e v e r ~IP, ~ 6 So(K), then

p is u n r a m i f i e d at p Here U ~ d e n o t e s the s u b g r o u p of units in

K~ r e g a r d e d as a s u b g r o u p of C K

Proof L e t T 6 1(p) T h e n T H ~(~) for ~ e v; D K~; since K~Ik p

is unramified, w e h a v e s T = ~ for e 6 K~ Thus

By v i r t u e of local class field theory, it follows from (6) that l(p) =

U~; since U ~ ~ Ker p, w e c o n c l u d e that p is u n r a m i f i e d at p

N o t a t i o n I L e t p 6 R(k) We denote by So(P) the set of those

primes in So(k) at w h i c h p is ramified

P r o p o s i t i o n 3 The set So(P) is finite for any p in R(k)

Proof S u p p o s e that p C R(KIk) The r e s t r i c t i o n of p to C K is a

c o n t i n u o u s h o m o m o r p h i s m of C K in GL(n,~) for some n in ~ ; there-

fore there is a finite s u b s e t S3(P) of primes in So(K) such that

U ~ ~ Ker p for ~ 6 S o ( K ) \ S 3 ( P ) (7)

By P r o p o s i t i o n 2 and (7),

So(P) _~ { P I ~ E S o ( k ) , ~ I P for some ~ in S3(@)}

i (p,t) = K i (p',tf(~)),

where ~ range over primes in So(k') lying above p, and Nk,/k ~ =

p ; here Ik is a finite field extension

Proof Assertion 2) is a reformulation of the Artin's reciprocity law: equation (11) follows from the definitions when one recalls that the inertia subgroup and the Frobenius class in G(k~Ikp)_ may be identified with Up and ZUp in Ck, where ~ C k ~ and w (~) = I Let us

where pp(~p) is defined as the restriction of pp(T) for T 6 ~p to

Vp, the subspace of l(p)-invariant vectors in the representation space

of pp If, in particular,

pp = p l W ( K ~ ikp) for p 6 R(KIk),

then £p(pp, t) = £p(p,t) independently of the choice of ~ above p

(I) and _(2) of W ( K ~ Ikp) we observe

that

V (1)p • V (2)p = Vp,

w h e r e Vp, V (I) and V (2) denote the subspaces of ] (p)-invariant

vectors in the r e p r e s e n t a t i o n spaces of OP'P (1)p and PP(2) , respec-

(I) ~ (2) T h e r e f o r e tiveiy; pp:= pp pp

Zp(p(1)p • pp(2),t) = £p(p;t),t)lp(p;2),t) (13)

(i) = Pi I (K~ ?I i = Identity (10) is a special case of (13) w i t h Pp W kp)'

I, 2 To prove (I 2) w e need the following lemma

Proof L e t t (~) = I (~) n W ( K ~ Ik~) be the inertia s u b g r o u p of

W ( K ~ Ik~) and let

e 6 e n=1

L e t

for any y in W ( K ~ Ikp) and any (finite d i m e n s i o n a l continuous) repre-

A n easy c a l c u l a t i o n shows that

Proof See [87], p 10, and [91]

We r e m a r k t h a t Ro(Klk) ~ Ro(k) N R(KIk) L e t p 6 R(k) and let

29

= Z 8 Ind (K k) _ 7 ~ indG(K )~j

for some finite extensions kj and g r o s s e n c h a r a c t e r s ~j in gr(kj),

I < j < m It follows from (11) that Pl (Op) = XI (p) w i t h X I = tr Pl'

If If(b+it) l < 1

> O, @ + a > O,

a n d

t h e n If(a+it) i < A I Q + a + i t l ~ w h e n e v e r t E ] R w i t h

b - R e u if(u) I < (AI~+ul~) b - a for u • S ( a , b ) (8)

[~(s,e) l > IQ+s[ Yb -J~ f o r s 6 S(a,b)

M o r e o v e r , ~(s,Q) = O IIm sl c ) for s e S ( a , b ) (with a r e a l c o n s t a n t

M a k i n g use of L e m m a 4 we c o n s t r u c t two functions

w i t h the following properties:

R e l a t i o n s (26), (27), (14), (6) and (18) give:

IF(a+it)[ < (1+q-1) nd(X) , t CIR (28)

S u p p o s e now that p d o e s n ' t c o n t a i n the i d e n t i c a l r e p r e s e n t a t i o n , then

F(s) s a t i s f i e s c o n d i t i o n s of L e m m a 2 w i t h ~ = 0 in v i e w of (25), (28)

and (19) T h e r e f o r e it follows that

IF(s) I < (1+q-1) nd(X) for -q < Re s < l+q (AW 29)

I n e q u a l i t y (17) (with g(x) = O) follows from (29) in view of d e f i n i -

tions (22), (23) and i n e q u a l i t i e s (20) To c o m p l e t e the proof of p r o p o -

s i t i o n 2 we make use of the f o l l o w i n g i n e q u a l i t y (see, [80], p 200,

T h e o r e m 4):

l+sl ( / ~ ~ ll+sl ) n l 2 ) 1 + n - R e S ~ ( l + ~ ) n

[Ck(s)l i 3[T~ 2~ (3o)

in the s t r i p -q < Res < 1+q We w r i t e

Since

ILCs,x)-II ! ~k (Re s) for Re s > I, X 6 gr(k),

it follows from (I) that one can find a p o s i t i v e c o n s t a n t c 2 s a t i s f y i n g three inequalities:

I S - S ( t l ) I < r ( t I)

L e t us r e c a l l that a(x) = IDI'I F ( x ) I a n d d e f i n e

b(X): = K (3+Itp(X) I) K (3+ IaP(X) I+ItP(X) I 2 2 ) (29)

w i t h c 8 > O, w h e r e p ranges over p r i m e ideals of k Here ~ d e n o t e s

the p o s s i b l e e x c e p t i o n a l zero of L(s,x) in the r e g i o n d e f i n e d by (2)

Proof Let, for T > I,

~ T ~ - ds - ~ (s,x) = O ( x I-~(T) (log T ) ~ ( T ) - 2 + X I+~(T) (T~(T))

we remark also that

-I ) (36)

co

Z m=2 ipmi<x

and c h o o s e T to s a t i s f y the e q u a t i o n

lo 9 T

E s t i m a t e s (32), (34), (36) - (38) c o m b i n e d w i t h (30) give:

R I (x,T) = O ( x exp (-c 9 log x )), c 9 > O (39)

log (a (x)b ( X ) ) + ~

M a k i n g use of the p a r t i a l s u m m a t i o n (cf., e.g., [78], p 371, Satz 1.4

w i t h A(x) = Z X(P) loglpl and g(() = (log ()-I) we deduce the

E s t i m a t e (41) follows now from t h e o r e m I a p p l i e d to each of the charac-

ters Xj in (44) w h e n one takes into a c c o u n t r e l a t i o n s (45) and (46)

If A is c h o s e n to b e l a r g e e n o u g h , say

f o l l o w s f r o m (10) t h a t

c(o I ) I/o I log A > ( -~ I) , i t

!log f(s)! < ~ I log A,

a n d (9) f o l l o w s

C o r o l l a r y 2 S u p p o s e t h a t L(s,X) s a t i s f i e s (3.32) T h e n

L 1-s g (Xir] ~

9~4 9 ( U ) d u = I 2~ 9 ei@

Since the c i r c l e Is-~ I = 9/4 is c o n t a i n e d in the s t r i p a ~ Re s ~ b,

w h e r e ~ ranges o v e r the p r i m e divisors in kj E s t i m a t e (30) follows

f r o m P r o p o s i t i o n 3 a p p l i e d to each of the c h a r a c t e r s Xj and r e l a t i o n s

(31), (5.46)

c-(£):=

j=1 +

w i t h

n

b ( x ) : = C(T)n+ -~ iH[a(x) 1_(n+1)-1 (kn+1)-1 b k ( X ) (n+1 )-1 (kn+1) -I

for any n - s m o o t h s u b s e t T; k > I

Proof E s t i m a t e (19) f o l l o w s from (14), (17) and p r o p o s i t i o n I

L e t now J = I (k), H = H ( ~ ) , and let T be d e f i n e d by (I 3)

for any smooth subset T of 9 Here Co(k) and y(n) are e x a c t l y

c o m p u t a b l e c o n s t a n t s d e p e n d i n g on the field k and its degree n, re-

spectively

Secondly, let J = So(k), let H = G(KIk) for a finite Galois

e x t e n s i o n KIk , and let ~ be the torus d e f i n e d by (1.3) We d e n o t e

by ( K ~ ) the A r t i n symbol c o r r e s p o n d i n g to the e x t e n s i o n K[k and

recall that, for p 6 S (k),