ASPECTS OF THE. ZETA FUNCTION

J. de BRUIJN

, .. r.~"'''''···''''·- ...... ASPECTS OF THE ZETA .FUNCTION..

JOHANNES de BR UIJN

Submitted to the Faculty of Graduate Studies and Re search in partial fulfilxnent of the requirexnents for the degree of Master of Science

McGILL UNIVERSITY

1968 .

'·0 :~ ® Johannes de Bruijn 1969 .1 ABSTRACT

J. de Bruijn

Aspects of the Zeta Function

" Departm.ent of Mathem.atics M.Sc."

After defining a zeta function for an irreducible schem.e

of finite type over Z, we prove som.e elem.entary properties

and study the zeta function of schem.es over finite fields in

greater detail. In particular we prove a rationality statem.ent

for such zeta functions and we investigate the case of a curve

defined over a finite field. Finally we specialize to the case of

an algebraic num.ber field. We show that the zeta function of an

algebraic num.ber field has a sim.ple pole at the point 1. Som.e

conc1uding rem.arks are made about Tate and Weil' s appreach

to the study of zeta functions . ACKNOWLEDGMENTS

l wouldÎike ta thank Professor W. Kuyk for the advice

and encouragement he has given throughout the preparation of

. this t:hesis. He criticized the original manuscriptand Many

improveme.ntsare due to·him.

. . TABLE OF CONTENTS

0.1. Historical note 0.2. Introduction

chapter 1. The definition of the Zeta Function. The classical case. 1.1 Introduction 1. 2 Some lenlmas on schemes I. 3 The definition of the zeta function 1.4 Some special cases; the classical cas,e Appendix l, Some facts about 'schemes " .> Appendix II, The zeta function of spec (:[p [Tl •••• , Ta,J)

Chapter II. Rationality of the 'Zeta Function.

~.l Introduction; statement of the main theorem II.2 A Ierruna about log Z(X, t) II.3 Reduction .to the case of a hypersurface

11:04 Bor~l's criterion of rationality II.5 Additive characters of finite fields, II.6 Traces and determinants of infinite matrices

II.7 The meromorphic char~cter of Z(X., t) II.S Corollaries and remarks

Chapter III. The Zeta Function of a Ctuve defined over a Finite Field III. 1 Introduction lII.2 The Riemann Roch theorem III. 3. The divisor class group IlI.4 The functiona1 equation of l;: (X, e) III. 5 The Riemann hypotheeis

Chapter IV. The Zeta Function of an Algebraic Number Field

lV.1 Intr~duction; statement of the main theorem IV.2 A genera11emxna on series. IV • 3 The logai'ithmic space IV.4 A theorem on fundamenta1 domains IV • 5 Proof of theorem IV.1:2. IV • 6 A volume computation IV.7. Primes of first degree

Chapter V. Conc1uding remarks ·V.1 Local - global ..,

Bibliogr aphy

·e, "

Index of Not~tions , . , ,

N Natural num.bers. , . • Z Natural integers.

Q Rational num.ber s •

'R Real num.ber s .

C Com.plex num.ber s .

R IL Real n- spa ce .

Qp p-adic numbers.

Zp , p-adic integers.

F q field of q elem.ents. :, ?

i .. /

HISTORICAL NOTE AND INTRODUCTION

·0.1 Historica1 Note

Leonard Euler gave the following proof of the fact that

thereare infinite1y many prime. integers. Assume that

. pl, •••• , p. are aU the prime' integer s. Then. an easy computation l )-1 ( . 1 )-1 (' 1 )-1 1 1 shows thât ( 1 - . P ·1 - pa· . •• 1 - P. \ = 1 + 2 + '3 +

The product on the 1eft is finite, while the series on the right

dive.rges. Hence there is a contradiction and there must. be in-

finite1y many prime integers. The interest of this proof lies in the . . ( 1 )-1 fact that it calls attention to products of the type C (s) = n l - -8 .,. . p p where sis a complex·number, and pruns over aU prime numbers.'

. For certain values of s, Euler found a functiona1 equation for

C (s) similar to 1. 4:1.

.-' A century later, the mathematician Riemann defined the

function C(s) =V(1.- p-S)-l, and showed that it was an analytic

function of the complex variable s, for certain values of s. He al80

8 noted that C (s).= n~l n- , and posed questi~ns about the poles and

zeros of C (8). In particu1ar he conjectured that the comp1ex zeros

qf C (s) are of the form 1/2 + iy. This conjecture became known as

the Riemann hypothesis and is still unanswered;' cf chapter 1. o-z

Later Dedekind and others had developed the theory of algebraicnuznbers, and the questi~n arose if a similar function could be defined for algebraic nuznbers. It was Dedekind who defined such a function for an algèbraic nuznber field. Hecke coznputed. a functional equation for a ~odification of the function defined by Dedekind, and quite recently J. Tate gave a totàl1y new proof of Hecke' s results using the theory of ideles, devel-

l ~ oped by Chevalley and A. Weil; cf. chapter L~.

:rn: the field of alge braic geoznetry, E. Artin considered a siznilar type' of function for function fields, which cazne frozn curves defined over finite fields. He, and other znatheznaticians, notably F. Schm.idt, H. Hasse, A. Weil, worked on the Rieznann hypothesis. It was A. Weil who proved the Rieznann hypothesis

.. for aU function fields in one variable over a finite field. His work also l'roduced a functional equationj cf chapter lU.

Afterthe second-'world war, A. Grothendieck developed his theory of scheznes and cohoznology. . of schemes, after siznilar work by J. P. Serre ..Again a sim.ilar type of function was

defined in this general setting, and znany of,the old problezns could

be rephrased in the language of scheznes, CI. chapter 1. D.'w'ork

, , proved that the function in this setting, defined for finite type

. scheznes, is a rational function, cf. chapter n. 0-3

A11 these functions have become known as zeta functions.

o .2. . Introduction

We intend to introduce the zeta function in the cüse of

schemes and then show how this case is related to those of

cur.ve sand number fields.

More precisely, in chapter l, we define the zeta function

of any scheme of finitetype over Z. As an example we make

sorne remarks about Riemann' s zeta function. It wi11 turn out

that the·whole theo~ywi11 depend in an essential way upon the

. classical theory·of Riemann, cf. 1.3:2, 1.4:1. The material of

. chapter 1 is based upon Serre's article in [23] and E. G.

Titchmarsh's· book [24] 0. The proofs in sections 1.2 and 1.3

. were independently supplied by the author.

ln chapter II, we specialize to a scheme over a finite field

and prove ~ork' s theorem of· rationality. This proof depends

on an old criterion of E. Borel, and uses techniques in p-adic '"\.' .

analysis. The presentation is an expanded version of [22] and

ln chapter III, we turn to the case, of a curve defined over a

finite field. This isnot a special case of the first two chapters.

Nevertheless, all the main theorems of these chapters will be

found to be valid in this case. The results of chapter III are quite 0 .. 4

deep b~cause they depend on the Riem.ann Roeh theorem. for

curves. The m.aterial of chapter III follows [6] and [11 ] •

Finally~ in chapter IV, we study the case of a1gebraic

. nUID:ber fields. This is a special case of chapters 1 and II.

Wefollow the presentation of [3], after m.aking som.e changes.

We make som.eïDform.al.rem.arks in chapter V.

\. o 1-1

0.· ~ CHAPTER 1. The definition of the Zeta Function. The clas sical case.

1. 1 Introduction

ln this chapter, we propose to define the zeta-function

in the general setting of schem.es. We will draw special

attention to the special case s to be studied in the later chapter s •

In particular, we will point out the case of a variety over a

finite field and the case of a nUIll.berfield. Finally, we study

some of the properties of the Riemann zeta-function, which

inspired all later investigations into the zeta-function.

1. 2 Som.e lemIll.as on schem.es

Before we can define the zeta-function proper, we need

Some preliminary results on schem.es. The basic facts about

schemes which we will take for granted are colleded in

Appendix l, at the end of this chapter.

Let X be an irreducible schem.e, of finite type over Z,

with generic point'x (App.l, ch.!). The stalk (App. l ch. 1) at x

will be denoted by 0 lt and its unique m.axim.al ideal by nil"

Then the residue field at x is Ox 1 tnx and is denoted by k (x).

Definition 1. 2·:!- Let X be an irreducible schem.e of finite type

'over Z, and let x be its generic point. If k(x) has zero charac-

.. teristic then we define the dim.ension of X to be the transcendence 1-2

degree of k(x) over the prime field augmented by 1. Otherwise, wedefine the dimension of X to be the transcendence degree of

~(x) over the prime field. The dimension of X will be denoted by

dim(X) or dirnX.

LemIna 1. 2 :2. Let X be an irreducible scheme of finite type over

. Z. A point of X is closed if and only if its residue field is a finite

field.

Proof: A point y of X will have an open affine neighbourhood

U = spec (Z CUl, ••• , Un J), since X is of finite type over Z. The

transcendence degree of the ring A = Z [Ul, ••• , Un J is S n.

Now assume that y i~ a closed point. Then y is closedîn

U and thus correspond~ to a maximal ideal tE of A. Since

mn Z =pZ, we conclude that k(y) = AIr:!} = B· CUl, ••• , Un J, which

means that k(y) is finite and algebraic over F~p:; Hence k(y) is a

finite field.

Conversely assume that k(y) is finite, and that y corresponds

to a prime ideal!!l of the 'ring A. Let Vi be the image of Ui under

the map A ~ AIr:!}. Then k(y) = F:p (Vit ••• , Vu ), for sorne prime

integer p. Since k(y) is a finite field, each Vi must be algebraic

over ~, and hence k(y) = :[::ll [Vi' ••• " Vu J,. which means that

k(y) =AIr:!}. Hence m is a maximal ideal of A and y is a closed point. Q.E.D. 1-3·

Notation The set of closed points of X will be denoted by X.

Definition 1. 2 :3. ·Let X be an irreducible scheme of finite type over Z. Let y be a closed point of X. Then N(y) = the nurnber of elements of k(y) and is called the norm of y.

Lemma 1. 2 :4. Let X be an irreducible scheme of finite type over Z. Thel..'e are only finitely many elements of X which have , the same norm.

Proof: Since X is of finite type over Z, it is quasi-compact

(App. l, ch. 1). Thus it i s 'sufficient to show that each open affine .subset of X contains finitely many points having a given norm N. Thus let U = spec (Z [Ul,"" un]) and let y be a closed point of U. Then k(y) = Z [Ul, •• " Un ]/~, where m is the maxi.In.al ideal corresponding to y. Now N will be of the form p. and hence Z [Ul, .•. , un] Il!!.. = F p~ .. Let Vi be the image of

Ui under the map: Z [Ul, "". Un ] ~ Z [Ul, •.. , un] Il!!. .•

Then ~ .•' = F P' CUl, ..• , un]' But there are only finitell many possible choices for the Vi 's, and consequently,. only finitely many possibilities for ~. This proves the lemma.

1. 3 The definition of the zeta function.

Definition 1.3 :1. Let X be an irr,~ducible scheme of finite type over

Z. By the zeta function of X we mean the formaI product 1-4

l:: (X. s) = n (1 - N(x)-& ) -1. xE'X

. where sis a comp1exnumber. c.!. [23J."

If l:: (X. s) is to be a function of a comp1ex variable.

the product l:: (X. s) must converge for aU s in some subset of

c. As usua1. if s is a comp1ex number Re( s) denotes the rea1

part of s. Im( s) its imaginary part.

Theorem 1.3 :2 •

Let X be an irreducib1e scheme of finite type over Zo.

Then the product for l:: (X. s) converges abso1ute1y for aU s.

with Re(s) > dim X.

Fir st we prove a reduction 1emma:

Lemma 1.3:3. (a) Let X satisfy the hypotheses of 1.3:2 and

as sume X is a finite union of schemes Xl. Then if theorem 1.3 :2

is valid for each Xl. it is a1so valid for X.

(b) Let X and Y satisfy the hypothesis of theorem

1.3:2 and let f: X---:;.Y be a finite morphism over k. Thenoif

theorem 1. 3:2 is valid for y! it is valid for X. m Proof: (a) X is certainly a subset of U Xi. and dim X ~ dim Xl 1=1 for each i. This gives the desired result immediately.

o - 1- (b) First we show that X = r (Y). Let yE'Y • and let

.u be an open affine neighbourhood of y. If xE'r1 (y). xE'C1 (U). which 1-5

is an open affine subset of X (App. l, ch 1). Since f: X ~ Y

is a finite morphism over z, r (Cl (U), ° x ) is integral over

~ (U, 0yL (App. l, ch. 1). Since.y is c10sed it corresponds

to a maximal idea1 of r (U, Q..y) and hence x is a maximal idea1

l - - of the ring r (Cl (U), O~), [5 ]. Thus C (Y) ex. Con- . - verse1y if x(X, then f(x)(Y • by the same argument, and we con-

c1ude that X = Cl (Y) .

Next we remark that if y(Y, Cl(y) is a finite subset of X.

This again follows from the fact that r (Cl (U), 0:x) is integra1

overr (U, O~, [7]. Thus,let Cl(y) =' (Yl' .•• ' Yn).

Obvious1y n depends upon y.

Then we may write.

n s l:: (X, s) = n n (1- N(ytl- ) -1

Since r is a finite morphism o x,. X is integra1 over 0 f(x), y

. (App. l, ch. 1), and the transcendence degree of~(x) is equal

to that of k( f(x». Since this holds for any point of X, it is in

particu1ar true for a generic point of X. From this we conc1ude

that dim X = dim Y.

Hence if Re( s) > dim Y,. the product n_ (1 _ N(y)-S )-1 y(Y o converges by hypothesis, which now implies the convergence of 1-6

l;:(X, a).

Q.E.D.

Proof of Theorem 1.3 :2: By part (a) of 1emma I. 3:3, we mayas sume that X = spec (F·p [Ul, "" U III J) or spec r (Z CUl, •.. , u J). - . III

In::he first case, by Noethe~'s normalization 1emma

(App. l, ch. 1), there is a finite surjective morphism from X onto Y = spec (F··p [Tl, "" T n J). In Appendix II, ch. l, we compute l;: (Y, s) to be (1 - pn-s) -l, which converges if Re( s) >n.

Now Y has as generic point the zero ideal and hence its residue field is F); (Tl, "', T n ) (App. l, Ch.I) and thus dim Y = n.

Thus 1.3:3 is now valid for Y, hence for spec (~. CUl •.. , u lII ]) by the (b) part of I. 3:3.

In the second case, we again app1y Noether' s normalization lemma, to get a finite morphism over Z from X = spec

(Z CUl, ... , u lII J) onto Y = spec (Z [Tl, .• " T" J). In appendix

II ch. l, l;: (Y, s) is computed to be l;: (n - s), the Riemann zeta function, which converges if Re(s) > n + 1 (1.4). But in this case, the residue field of the generic point of Y is Q(Tl, .•. , T n )

(App. l, ch.I), which means that dim Y = n + 1 (def'n I. 2:1)" Hence, the theorem has been proved for both cases. 1-7

Remark. It has been conjectured that C(X, s) is a mero- morphic function on the entire complex plane, or rather that

.C (X, s) can be continued analyti.cally as' a meromorphic function on it. This has been shown for the case of spec (Z)j for the full proof Bee (1. 4:1). 1t also has been shown for schemes over finite fields, c.f. (2.8). We have the following results:

Theorem 1. 3:4.

Let X be an irreducible scheme of finite type over Z.

Then C(X, s) can be continued analytically as a meromorphi~ function in the hal! plane Re( s) > dim X - i.

Theorem 1.3:5.

Let X satisfy the hypothesis of the,orem 1. 3:4. Let K be the residue field of its generic point. Then

(a) If K has characteristic zero, the only pole

in Re( s) > dim X- i is at the point s = dirn X, and it

is a simple pole.

(b) If K has characteristic p, let F n be the -p largest finite field contained in K. Then the only poles

of C(X, s) in the hal! plane Re( s) > dim X - i are at the

• ~1T im points s = dlm X ... 1 n(pn) mE' Z,' and they are simple

poles. '. 1-8

Proof~ The proofs of these theorexns are to be found in [18 ]

and [27 J. The proof:of 1.3:4 is a1so in [18 J.

In chapter III, we will prove both theorexns in detail for

the case of a c'urve defined over a finite field. In chapter IV,

we show that the zeta function of a nuxnber field has a sixnp1e

pole at s =1.

::L4 Soxne special cases; the c1assica1 case.

In the latter chapters we will be interested in three ca·ses.

First we take X to be a schexne of finite type over:[.q. Since

there is a xnap Z ~ :[.q, which xnakes:[.q a finite Z-xnodu1e,

. we see that X is still of finite type over Z.

Second1y we study the case of a curve, whose defining

equations have their coefficients in a finite field. Unfortunate1y

) su ch a schexne is not of finite type ove~ ~, but we do have

theorexns ana1ogous to 1.3:2, 1.3:4 and 1.3:5; c.f. section II1.l.

The third :case will be that of a nuxnber field. Let K be

a finite extension of 9 and let A be its ring of a,lgebraic integers ..

Since A is a Dedekind ring, all its prixne idea1s are xnaxixna1 and

thus all the points of spec (A) are c1osed. Thus C(spec A, s) is

identica1 with the zeta function defined by. Dedekind for the nuxnber

field K. In fact Dedekind defined the zeta function of K to be 1-9

1 n ( 1 - 'ocar,d (AI p )-8 r • p which is easily seen to be the SaIne as our definition.

At this point we consider the case X = spec (Z). In this case C(X, s) reduces to the product C (s) = n . (1 _ p-8)-l, P prlme which is the 'c1assica1 Riemann zeta function.

Theorem 1. 4:1.

, The function C(s) is ana1ytic for all s:in the haU plane

Re(s) > 1. Itcan be continued a~a1ytically as a meromorphic function in·the entire comp1ex plane,' with one simple pole at . s = l, havingresidue 1. Furthermore C(s) satisfies the func- tiona1 equation:

8 :1t. -81:a r (s 12) C(os) =~..: -k + 1:a r (~ - ~) C(1 - s).

Proof: after [24], [14].

(A) First we prove that the p~oduct C (s) converges for

co Re(s) > 1. Consider the series f(s) = E n-s. 0.=1

Let s = u + iv .. Then 1 n - B 1 = n ~u. 50 if u ~ 1 + 6, we have 6 -8 1 -1- h· ) . if 1n ~ n . T 1.S means that f( s converges uniform1y u ~ 1 + 6 , for any fixed 6 > 0, by the Weierstrass M-test,

Hence f(s) represents an ana1ytic func~ion in the haU plane Re(s»1.

Next we show that for Re(s) >1, f(s) = C(s). Again set 1-10 '

, < el'...• . s = u + iv and asswne that Pl, ••. ,pli are an the primes less

than an integer N. For each of the Pi' s we have

(1 -8r1 -8.1, since u 1 - Pi =j=Oof Pl >

Each of these series is absolutely convergent if u > 1, and it

is permissible to multiply a finite ~wnber of them. This gives:

")0) -sJI) Jl0 J.O)'-S ( ,I;-O Pl = (Pl •.• P. J1- Jl-of_o ].-:~-O

Il _ P1-S )-1 -S Then f()s - i=1n (1 = ~ n where F is the set of nEF

aH positive integers whichhave at least one prime factor

u u larger than N. Thus if nE"F, n > N and n- ,; < N- and we get

Il 1 f ( s ) iI!I (1 - P i -s r lis: of n -u n=N+1

since Re(s) = u.

0) Il But n~1 n - converges., Hence as N tends to infinity,

, -u n tends to zero and we conc1ude that n=N+1

lim = f(s) li Re(s) > 1. N

(B) The functional equation. If Re(s) > 0, then

11 rx S/'irl e-n 2 :Ji. dx :: n- s ft -s/2 r' (s/2).

Hence if Re(s) > 1 we have . 1-11

'ft -s/a r (s/2) C(s) = f 1 XS /a-l exp(-n27t.x)dx n=l J o By abso1ute convergence of C (s), we interchange the integra- t~on and summation sign to get

a a r(s/2)C(s) n.--/ = Jx / a-l g(x) dx, o . ~ afTx where g(x) = n~ e""'ll •

It is known [10], that if x > 0,

1 f i.e:. = x n=-CIO

2 gx( ) + 1 =JX 1 (2 g (lx) + 1)

Hence we. obtain .

. c· l" CIO S 2 1 = 1 + if xa/2-~/ag(1/x) + X / - g(x)dx {s-l)s ~ Il

CIO 1 =~~~ ( s-l)s + L {x-î -s /a + xs/a -l} g(x) dx

But this has the SaIne value for sand l-s, ~nd the integral on the right is convergent for aU values of s. Hence

V' -a/ a r / )" . . l .I..a/ al' .fT· (s 2).,.(s) ='it~-al r(-a ~ s/2) C(l- s). 1-12

(C) . M eromorphi c nat ure 0 f 'a,.. ()S . Th e f unct' lon n l-s -:'( n+1)1-11. lS entire and has a zero at s = 1. Hence the function

(s - '1)-1 (n1 -11 - (n + l)l-s) is entire, and equa1 to r.+~-s dt., Il

CI) S This holds for aU s(C. Also Il t- dt. = (s - 1)-1. Now if .

CI) , Re(s) > 1 C(8) - 1 = rP:1 (n + l)-s Hence

CI) CI) s C(s) - 1 .; (s - 1)-1 = I; (n + lr - t-S dt. n=l J1

We break the integra1 up into a sum to yie1d:,

C(8) - l - (s - 1)-1 = jl { (n + l)-S -l~ (n + t)-II dt }

CI:) 1 S 1 - - s n~l Jo (n+tr - dt

The series of integra18 converges abso1utely for Re(s) > 0, and

~hus we have continued C(s) onto the half plane Re( s) > o. (In 1 fact the integra1s have ab solute. value less thann- - Ô if

Re( s) ~ Ô > '. 0 • ) 1t i s also c1ear that C(s) has a simple pole with residue 1 at s =1.

Now we make use of the functiona1 equation

C(~) = 7f.s-i r(t - ts) C(l;- s). r (s/2)

If Re(s) <0, r (i - is) is analytic, since it is analytic for aU com- 1-,13

plex nuinbers with positive real part. Also r (s/Z)-l is an entire function and we already know that C(1 - a) is an analytic function if Re(s) < 0 by 1. 3:Z. Hence for Re(s) < 0, the functional equation determ.ine â C( s) uni que ly , and this

definea C(s) as a m.erom.orphic functiol'l on C .

. Corollary 1. 4:Z . The function ç (s) has no zeros in the half plane Re(s) > 1, no real zeros in th~ 'strip 0 < Re(a) < l, and no zeros in the hal! plane Re(s) < 0, except for the pointa

s = -Z, -4, ..• , -Zn, ne:N.

I:>roof: We consider each case separately.

Case' 1. Re(a) >1. Let Pl, ... , Pli be the prim.ea

leas than N. Then

~8) -8 )1' 1 -Re(s) 1(1 - Pl. •• (1 - Pm . ) ... ( s ) ~ 1 - (N + 1) ,

_ (N + Z)-Re(s) ... , and this is greater than zero if Re( s) > 1

CaseZ. Re( s) < o. We m.ake use of the functional

, ' 8-1. l / Z ) equation C (a) = ,TT Z r ('2 - S C<1- 06) r (s/Z)

Now if Re( a) < 0, then Re(l - s) > 1 and hence 1C (1 - s) I~ 0

by case 1. The function,T (!- - s/Z) is neve~ zero, and r (s/Z)-~

has zeros at the points -Z, -4,., ,., since r (z) has sim.ple

poles at -l, -Z, •.•. This prove s case Z. 1-14

Case 3., 0 < Re(.s) < 1. If s is rea1, and lies between o and 1, (1 -2 l-S) 1:: (s) = 1 - ~s + ;s ... > 0,

[24]. Hence 1:: (s) has no rea1 zeros between 0 and 1.

Q.E.D.

Re:rnark. Hence any non-rea1 zeros ~ust occur in the region o < Re(s) < 1. Now if u + iv is a zero, so is I - u - iv by the functionai equation. Si:rnilarly the co:rnplex conjugates are zeros as weIl. Hence, the zeros occur in qua,drupIes: i + u + iv,

1" +. 1 • dl. a - u lV, a';: u - lV an a - u - lV, where 1u 1< i.

Conjecture (Rie:rnann Hypothesis). The zeros of 1:: (s), apart fro:rn -2, -4, .•• , lie on the line Re(s) = i.

The question is still undecided. 1-15

Appendix 1. Sorne facts about schemes.

Let A be a comm.utative ring with unity . Let spec (A)

denote the set of prime ideals of A. If E is a subset of A,

set- V(E) ~ (x( spec A: E ,~,_ x ). Then spec (A) can be made

into a topo1ogica1 space, by 1etting sets of the form V(E) be the

c10sed sets of spec (A). The basic open sets are of the form:

(x( spec AI f f x ), where f(A. There is a contravariant

functo,r from the' category of open sets of spec A, with inclusions as

morphisms, into the category of comm.utative rings with 1. If

U is an open set, the ring as sociated with U is denoted by

r (U, 0 x.> and is called the ring of sections above U. Then if Xf

is the basic open set ('x( spec A : f ~ x ), it turns out that

r (X , 0 x) = (l, f, .•. f' ) -1 A. The sta1k at x is the ring f ~ r (U, 0 x) and is Ax = (A"x )-1 A. The above structure xf: U fs ~alled an affine scheme. Also r (spec A, 0 x) = A, and A can

be viewed as the ring of rational functions on spec A.

Let f: A -..:;. B be a ring homomorphisme This induces a

map f* : spec B ~ spec A, by mapping xf: spec B onto f-1(x)

'of spec A. Such a map is continuous and i!'l called a rhorphism.

from spec B into spec A.

The above can be generalized. A topo1ogical space is

called a ringed space if there is a contravariant functor from the 1-16 '

categoryof open sets into the category of commutative rings, satisfying several compatibility conditions. Then by a scheme we will mean a ringed space, such that each point has an affine scheme as a neighbourhood.

The class o~ schemes can be made into a category. If

X and Y are two schemes, and f: X ~ ,y is a morphism in this category, then on affine subsets of X, f reduces to the mor- phisn;t of affine schemes indicated above.

Schemes are Kolmogorov spaces, i. e. given two points of a sb.heme, there exists an open set containing the one but not the other. I~ general, schemes are not Hausdorff spaces.

Affine schemes are compact, but not Hausdorff. We' caU this property quasi-compact. In general a scheme is not quasi- compact.

The functor, which makes a scheme into a ringed space, is denoted by r ( ,Qx). Thus if U is an open set, r (U, Ox) is the corresponding ring of sections.

Also if f: X ~ Y is a morphism between schemes, and if

V is an open subset of Y, then there is a ho;rnomorphism f* r (V, Q)') into r (r1 (V), Qx). If x is a point of a scheme X, con- tained in an affine neighbourhood spec A, then the stalk at x is

9c, x and is Ax = (.A:'\x )-1 A., Then if y = f(x) in the above morphism, 1-1. '

we have an induced hOInoInorphisIn 0 7 , Y ~ Ox, x of local ring s, . i. e. the, InaxiInal ideal ~ of 0 7 , Y is Inapped into the maxiInal ideall!!x of 0 x, x. This induces an injection

0,., y 1 lE'1 ---;;>~ 0 x, x Il!!x of fields. The field 0 x, xl ~x is denoted by k(x} and is called 'the residue field of x.

Now let A be a fixed rkg. A scheIne X is said to be a

scheIne over A, if there exists a InorphisIn f: X ~ spec (A).

The class of scheInes over A, forIns a sub-category of the

category of scheInes. A Inor phi SIn in this category will be called

a InorphisIn over A.

A scheIne X fs said to be of finite type over A if X is a

~cheIne over A and if Cl (spec A) is a finite union of open affine

subsets U 1 = spec Al 1 ~ i ~ n of finite type. An open affine

subset spec B of X is said to be of finite type over A if B is a

, finitely generated A-algebra. In,this case X is quasi-coInpact.

Let X and Y be scheInes of finite type over A. A morphisIn

f: X ~ Y is said to be a finite InorphisIn over A, if for all open

1 affine subsets spec B of Y, Cl (spec B) is an affine subset spec C

of X and r (spec C, Ox ) is a finitely gener ate'd r (spec B, 0 y ) -Inodule.

In particular, r (spec C, Ox) is integral over r (spec B, 0 y ).

A theorem about finite InorphisIns is the following: 1-18

Lem.rna' (Noether' s Normalization Iem.rna). Let Z CUl, ••• , up J be a finite1y generated Z-a1gebra, with transcendence degree n ~ m over Z. Then there exists a finite surjective morphism over Z from. spec ( Z CUl, ... , u J) onto spe c ( Z [T l, "" T!1 J) lb Proof: See [5 J.

A point x of a scheme X is said to be g,eneric point if X is the closure of the set (x). Since schemes are Kol~orogov spaces, a generic point must be necessarily unique. Schemes that have a generic point ~re said to be irreducib1e.

, A ring A isan integrai domain if and only if the zero ideai is a generic point ç.f spec A.

The 'reader interested in more details about affine schemes and schemes shouid consu1t: [12 J, [7 J, [8 J, and [19}:.

The affine case is aiso treated in [5 J. 1-19

Appendix II The zeta function of spec (E p [Tl, ... ,

. Let F p bethe field of p elem.ents and let Tl, ••. , T n b~ transcendental elem.ents over F p. In this case we can com.pute c (X, s) if X = spec (F p [Tl, ... , T n J).

Let ad denote the num.ber of points of X of degree d. Let

N m be the num.ber of points of X in the field F p. • Then Nil equals the num.ber of hom.om.orphism.s from. ,Ep. [ Tl, •.. , T D ] into F ., and N. is thus pD Il • P

However each point of X in the field F Il also induces a p hom.om.orphism. from. k(x) into Fm' If x is a point of degree d, p • then k(x) = F d and there are precisely d em.beddings of k(x) into . p

F Il Clearly d Inust divide m.. p

Thus the nwnber of points of X into F Il is just the sum.. . p

Hence IJ d. ad = p m D ; à\/m.

Let g(p-s ) be the function

co co 1 k(n-s) IJ = IJ k P k=l k=l

This converges absolutely if 1 pD-S <·1 Le. Re(n - s) < 0, or

Re{ s) > n. For the se· values of s 1-20

co ( . I;' (ad.. d) ) p -le CI = l; . k=l d\k k

.....

p -ka) .

Let k.= d i and keep d fixed. We obtain' .

= i a (f ~p-(d 1) a ) d . 1 • d=l . 1=1

- l; ~ p-1 S deg(x) ) ( ~ 1 . xf:X i=l

= _ I; , log (1 _ P -8 deg (JI:) ) xf:X

The above steps are permis sible because we started from a convergent series. We have thus established that

log (1 _ p-s deg (x» if Re(s) > n. But

co g(p-S)= I; 1 plll(n-s) =_log(1_pn-s) m=l p:1. 1-21

since Ipn -a 1 < 1. -a deg(x» Rence - log (1 _ pn - a) = - I; 1og (1 - p • But since xE'X

the right side converges, we can put it equa1 to . 1 )" log n (1 _ p-a deg (x) )-1 Rence 1 n-a = ... (X, s) if xEX -p

Re( s) > n. Now it is an easy matter to compute the zeta function

of spec (Z [Tl, ••. , T n J) = Y. The product for C (Y, s) can

be broken up into subproducts as follows. Consider aU points

yE'Y such that N(y) is' a multiple of p. Let us first compute the

product coming from these points ~

n (1 _ N(x)-& )-1. p\Nlx) xE'X

Since p!N(x), the idea1 corresponding to x, contains pZ. But the

idea1s of Z- [Tl, ... , T n ] containing pZ- correspond in a one to

one manner with the idea1s of F~' [Tl, ... , T n ] which have zero

intersection with F p • But this means that the above product is the

sarne as the zeta function of spec F p [Tl, ... , T n J. Rence

C(Y, s) = n (1 - pli - a )-1 = C(n - s) • p prime .. II-l

CHAPT ER II. Rationality of the Zeta Function.

il.l Introduction.

We. keep the terrninology and definitions of Chapter l,

in particular that of sections 1. 2 and 1. 3. In section 1. 4 we

m.entioned that a schem.e of finite type over a finite field is

. also of finite type over Z. Hence su ch schem.es are special

cases of the schem.es which were studied in Chapter l, and

aIl the results of sections 1.2 and 1.3 hoId for these scheInes.

In this chapter we restrict ourselves to the case of a scheIne

of finite type over F q, the field with q eleInents.

In Chapter l, section 3, we reInarked that it has been

conjectured that C(X, s) can be continued as a meroInorphic

function on C. The theoreIn, which we stated for schemes of

finite type over Z, was Inuch weaker. In this chapter we will

show that if X is an irreducible: scheIne of finite type over

~q, C(X, s) can be continued as a IneroInorphic function on C.

It will be Inore· convenient to work with the function Z(X, t),

obtained froIn C(X, s) by the çhange of variables t = qS. This

change of variable is convenient, because the norm. N(x) of any

point x of X is a power of q, and we write N(x) = q deg(x), where

deg(x) denotes the degree of x. Since C (X, s) converges if II-Z

Re{s ) > 'dÎln. ()X ,Z(X ,t) converges if. 1t 1< q -dimX

, Then the precise resu1t which we will prove is:

Theorem. 11.1:1.

Let X be a schem.e of finite type over F q' Then Z(X, t) is a rational function of t.

The rationality of Z(X, t) was fir st conjectured by Weil

[25 J, after he had co~puted Z(X, t) for the hypersu~face given by the equation

= b, with ah b€F q' For this case he showed Z(X, t) to be rational.

The proof which we give follows Serre' s exposé, [2~ J,

of lliv10rks original paper [9 J .

It falls into severa1 sections. After a prelim.inary 1em.m.a, y.rhich will enab1e us to com.pute Z(X, t), we show that it is

sufficient to consider the case of a hypersurface. Second1y, we

recall a ge~era1 criterion of rationality due to E. Borel, and

finally after som.e technica1 devices, we show how this criterion

can be applied to Z(X, t). These argum.ents will be presented

in severa1 se'ctions.

Il.2 . A 1ernm.a about log Z(X, t) .

In order to m.ake com.putations of Z{X, t) possible, we m.ust II-3

put Z(X, t) in a. different form.

Let N be the nurnber of morphisms (App. l, ch I) from ID spec (:E' ID) into X. Then we have: q .

Lernrna II.2~1. If X is an irreducib1escheme of finite type over F q' the function log Z(X, t) is equal to

I; N,t'. m=l m

Proof: Let ai be the number of points of X of degree i, i. e.

. . i whose norms are q. Now if xE'X,

Hence we can write

log Z(X, t) = - I; log (1 _ tdeg{x) ) xE'X

= - I;_. C~ xE'Xl=l al .It then follows that the coefficient of t lll is just

d + ..• + ad + ... + a' where" the sum runs over aU m I!I

integers d that divide m. Now the nurnber Nm of morphisms from spec (F m) into X is equal to the surn: I; (nurnber of q xE'X

embeddings of k(x) into Fm) which in turn equals q

al + •.. + dad + ... + mam , since there are cl. embeddings of

F" d, i.ttto F • , dlffi. Hence the coefficient of el is just N m lm. q q .Q.E.n. II-4'

~ ~ II.3 Reduction to the case of a hyper surface.

Let Xl and Xa be subschemes of X such that X = X1U Xa

Z(Xl, t) Z(X:a, t) . ?-nd y = Xl n X:a. Then Z(X, t) = Z(Y, t) . Smce X

is a finite union of open affine subset, there is no loss of

generality in as suming X to a an affine scheme. Hence

X = spec F Cl ([Tl, ••• , Tu] /1). ), where the ideal 'a is generated - ,

by the polynomials f l , "" f r of F Cl [Tl, "" Tu ]. Let Y 1

Then X = Y ln •.•. n Y:r, where Y 1 is a hyper surface defined by the

polynomial fi' If r = 2, then Z(X, t) = Z( y 1, t) Z(Y:a, t) . This Z(Yl U Y~lt t)

formula, and an easy induction, permits us to assume without loss

of generality. that X is the hyper surface ... , Tu J/(f)),

where fis a polynomial in Fil [Tl, "" Tu ].

Note that in this case, the number N m is equal to the number

of solutions of f(Xl, .•. , X u ) = 0 in the field of qlll elements.

Also, since F D is of finite type over F p, we can assume p that q = p.

Hence, from now on in this chapter, we assume that X is the

hypersurface spec (F p [Tl, ... , Tu J/(f) )', II-S

11.4 Borel' s criterion of rationality.

In this section we discuss the criterion of rationality, which will be applied to Z(X, t) later on in section II. 7 .

Let k be a field, and let F be the formaI power series

CIO F(t) = ~ Aee, A'SE'k. For a fixed s, let m be any integer s=O

:

Proposition II. 4~ 1. Let F(t) be a formaI power series in t, with coefficients in k.. Then F(t) is a quotient of two polynomials of k [t J,iff there exists an integer m :

This is Borel' s criterion of rationality, whose proof is to be found in [2 J and [4 J .

If the As are rational integers we can consider their usual absolute value (since Z is a subring of C ) . Similarly we can consider the p-adic absolute value of the As since Z is also embedded in the p field Q p of p-adic number 5 • We denote the p-adic absolute value by 1 1 p, normalized in the usual way:

From now on Kp will denotean algebraically closed, com- 1I-6

pIete field eontaining ,Q p, whose absolute value e~ends that of

co Definition II. 4:2. A series f(t) = I; i=O holomorphie in the disk (tE'Kp : ~ Ip < l' J, if it converges absolutely in the disk. A quotient of two series, holomo;rphie in a disk D, will be ealled a meromorphic function in the disk

D.

We need proposition II.4:l in a different form. We first

state a lernrna. co Lemma II. 4:3. Let f = I; alti, alE'Kp , be holomorphie in the i=O

disk (tE'Kp : It 1p < r). .If r 1 < r, then there exists a poly-

nomial PE'Kp [t J and a series fi, holoInorphie in

(tE'Kp : It Ip < ri), su ch that f = P. fi. In addition the inver'se

of fi is holomorphie in (tEKp : \t 1p' < ri).

Proof: This lernrna ean be dedueed from the theory of Newton

polygons of power series.. See [15 J.

Theorem II.4:4. co Let F(t) = I; AstS be a series with AsE' Z and let p be s=O

a prime nuznber. Suppose there exist two real nurnbers Rand

r with R. r > 1 sueh that F is meromorphic in the disk II-7

( z~ C : 0\ z \ < R) of C and in the disk { z~KR: \ z \ R < r) of

KR' Then F is a rational funetion of t.

Proof: E. Borel proved that if R > l, then F is eertain1y rational, cf [2 J., Renee we mayas sume that R ~ land r > 1.

By hypothesis there exist two series A(t) and

00 1 B(t) = ~ B 1t , where al. Bl~Kpo, holomorphie in ~Ip < r, i=O suehthat B(t) = F(t) A(t). By lemmaII.4:3, we eanmake r smaller, such that Rr is still> 1,. and we then may assume

A to be a polynomial. . We ean as sume Ao = 1. If we make r smaller still, We obtain,

(1) lBs Ip ~ r- B if s is sufficiently large.

(2) and lAs 1 ~ R- B if s is sufficiantly large.

Now let d be the degree of A, then sinee Rr > l, there exists an integer m sueh that (R~)III+l (d = k > 1. Now we show that

II.4:1 ean be applied. Sinee B = AF, we have

+ ••. + adA . Thus in the determinant s 0 N o· = det (aS+l +.1) 0 s: i s: m,OS: j s: m, we ean replace s, III

Now we apply the inq,qualities (1) and (2) and note that

This gives IN s il Ip < r - (III h-d) s if s is large , ':1 II-8

CIII 1 enough.' Sitnilar1y we can obtain, IN8, III 1 ~ R- + )(8+:am) i8 8 is, large enough.

W h ere k ' = R- ( III +1) :.?III • Now k > 1, and thu8

INs • IR liNs, IR Ip < 1 for s. large enough. Since N s , IR is an integer, we conc1ude that N s • IR = 0 if sis large enough.

Q.E.D.

Rernark. The previou8 theorern is applied to Z(X, t) as follows.

Since X has dimension n,' say, it is contained in the n-ditnensiona1 affine space over F p and we have thus that N 1 ~ pn 1 and hence

co This shows that the series I; N,t1 is majorized by i=O i co I; 1 (pn t) 1 ;;: _ log (1 _ pn t). But log (1 - pn t) converges if i=O i

It 1 < P -n and hence Z(X, t) is ho1omorphic in It 1 < p-n

Thu8 in the notation of 11.4:4 we have shown that R = p-n In order to prove that Z(X, t) is a rational function, we must show that

Z(X, t) is meromorphic in a disk (t~Kp: It 1p < r), where

r > pn. In fact, we will show that Z(X, t) is a quotient of two series

of infinite radius of convergence in Kp. First we need sorne tech-

nica1 preliminaries, which make up the next two sections.

II.5 Additive char acter 8 of finite fields. II-9

'" f~(.':.~ .. ' ' ~ Let K be the residue class fielc;1 of Kp. It will contain

F D for all n > o. Each element of K has a m.ultiplicative p representative in Kp, cf [21]. From now on, we will fix. a

set of multiplicative representatives of K in Kp and will call

this set the set of multiplicative representatives. The multi-

plicative representatives of F D are thep -l'th roots of unity . -p

in Kp and the element o. Let Ip be the ring of i~teger s of Kp.

Then Ip n Op, = Zp.

Let T and Y be two indeterminates and consider the

formaI expression ~(T, Y) = (1 + y)T =i=o (~) yll ,

. 'T ), d ' T(T - 1) •• ~. (T - m + 1) w h ere ( m. enotes as usual: . m!

Thus (~).~O [T ,J. H(T, Y)~O [[T, Y J J. Now if T~~,

then ( ~) ~b and H(T, Y) converges if ord (Y) > 0, y~Op.

Here ·ord denote s the valuation on 0 p. Next we set

TP - T F(T, Y) = H(T, Y) H ( P ,

T 2 i.e. (l+.YP)

Lernrna II. 5:1.· The coefficien:ts of the series F(T, Y) are in Zp •

Proof: The proof is to be found in [9 J . II-lO

Let us consider F(T, Y) as a power series in

CIO Y. F(T, Y) = !; B (T) ylII • Tp.en one sees easily that the m m=O

B' (T) arepolynomials in T of degree ~ m. This enables us to m CIO write F(T, Y) ::!; am (Y) Till where a (Y) is a polynomial in m m=O Y,· starting with a term of degree :

We fix a primitive p'th root of unity, denoted by (, and set À = ( - 1. Then we have ord (À ) = (p'- 1)-1, cf· [3 ] .

Then we substitute À for Y in F(T; Y) and set a (T) = F(t, À ), and denote am (À ) by Since am (Y) begins with a power of . p'.m

Y with degree : - (p - 1)-1 }.

Take ~ non-zero element t' (F s Ç;K and let t be the multi­ p. . 8 plicative representatiye of t'in K. Then t P = t. Set

8 -1 P Tr(t) = t + tp + ... + t ,and then Tr(t)( Z p. An easy calcu-

lation shows

(1 + Y) T r ( t) = F (t, Y) F (tp, Y ) .•. F (tp 8-1, . Y), where we have substituted t for T in F(T, Y). The two sides of

this expression are series with coefficients in Ip, and without

endangering the convergence, we are allowed to substitute À for

Y. This finally yields: II-lI

Tr(t) 8-1 € . = a (t) •••• a (e ). Tr(t) Note that € . Illakes sense .because Tr(t)€ b. Now since 8-1 ~ P = l, we can reduce Illod p to obtain 'n(.t) = t' + . .. + t 'P , , . and the Illap t ~ € Tr(t ) lS" an a dd"ltlve . c h aracter on t h e field F s. Putting aH this together we have: p Proposition II. 5:2. Let s ;:: 0 be an integer. The additive T (t ') . . P char acter € r can be factored as a product a (t). a (t ) ••• 8-1 P ... a (t ), where t is the Illultiplicative representative of co lll t'in K p , and a is a series I; {Jill t , with {Jill € Op and Ill=O or d ({Jill) ;:: III (p - 1)-1 •

ReIllark. This proposition is the only part of this section which

will be needed ~ater on. The functions H(T, Y) and F(T, Y) will

not be used anyIllore explicitly.

II.6 Traces and deterIllinants of infinite Illatrices.

This section is independent of the previous ones. We will

only Illake use of the field K p , defined earlier.

Let L be a field and n an integer. If u = (Ul, .•• , un)e- Zn ,

U U U we set X = Xl l ••• Xn n to be the Illono~ial in the indeterminOates

X 1 • We say that u <;:: 0 if U1 ;:: 0 for 1 s: i s: n.. We denote

Ul + ... + Un by c(u).

Let V = L[ [Xl, ••. , X n ] ] = L[ [X ] ] and view V as a II-12

vector space over L. Each power series G in 'y defines an additive endomorphism of y, by multiplication. More precisely, for a fixed G€Y, define

for all F€Y. The dot denotes Cauchy multiplication of power series. Thenthe distributive law yields

eG(F + H) = G(F + H) = G F + G H

= eG{F) + eG(H).

Hence e is an endomorphism. G We define one more endomorphism of V. Let q be an integer ~ 2, and define f q : E ~E by the formula

= :E u~O

Next we compute the matrix of the endomorphism fqO e • G Suppose G =:E qv Xv. Then the infinite matrix of the endomorphism v~O f 0 e is denoted by M(q, G) and has(u, vJth entry gqv_u. q G It is also easy to computé that

(1) fof, =f, cr. , q qq

Let Gq denote the series obtained from G, by raising each X to the q'th power. Then

';L'he above will now be applied to the case where L = Kg and' a II-13

fixed se'ries G = ~ gvXV satisfying the conditicm v~O (3) There exists a constant M> 0, su ch that ord (gv )~M c(v).

Note that if G1 and Ga satisfy (3), so does their product G1Ga, h a~d if G(X) satisfies (3) so does G(X ) for any integer h > O.

Let x = (Xl, ••• , xn)~ KQ.n, where each Xi is a q8 -1 'th root of unity •

Proposition II. 6:1. Assume G is a power series in Ka [ [xl, ... X n ] ] satisfying condition (3). For an integers s ~ 0, the series which 8 gives the trace of the matrix M(g, G)8 converges to a sum Tr(M ) and we have: 8-1 8 Q. (q8 _1)n Tr(M ) = ~ G(x) G(x ) G(xQ. ). 8 xQ. =x

ProDf: First consider the case s = 1. Then Tr(M) = ~ g( . 1) , u~O q- u which is a convergent series because of condition (3). It is easy to check that

~ XV = (q _ l)n if q -1 1 v xQ.=x = 0 othel.Wise

Thus

~ G(x) = (q - l)n ~ g(q-1)u' xQ.=x u~O

and this gives the formula if s = 1. Next if we replace q by q8 8-1 and G(X) by G(X) ••••• G(XQ. ), we may conc1ude the genera1 II-14

/ . result since G(X) .... again s ati sfie s (3) and

Q.E.D.

Now when we compute Z(X, t) in the next section, we will

need some facts about the determinant of the matrix

I - TM(q, G), whereT is an indeterminate. For convenience

we denote M(q, G) by M. Then det (1 - TM) is a formaI series

in T:'

We can compute the Cil 's . Let Ul, •• " UII be di stinct m tuple s

of integer sand Vl •• " VII be a permutation of U1' sand ((u, v)

'denote the signature of that permutation. Then

(4) = (- i)1I ~ ((u, v) Mu v ••• Mu v 1 1 Il Il

where Mu iV 1 denotes the ut. V1 'th en~ry of M, and the sum runs

. \ over aH m'tuples of integers > o.

Let us look at â~typicai term in the expression of Cil"

Denote a typicai term by M(u). Th~n because G satisfies condition

Il (3) we have ord (M(u» ~ M (q - 1) ~ C(utl! This shows that i=1 ôrd (M(u» tends to infinity ensuring the convergence of Cil .

Proposition II. 6:2. Assume G satisfies condition (3). Then

.,-~ . . ~,il ' II-15

CIO (a) det (1 - TM) = exp { !; s=l

(b) The series for det (1 - TM) converges everywhere on

Proof: Formula (a) can be computed easily for finite matrices.

If we then truncaœ M" just keeping the entIiies Mu, v with u, v s: r, then one can show that det Ü - TMr ) tends to det (1 - TM), in the topo1ogy of simple convergence of the entries. Here we have denoted the truncated matrix by M'r'. ,This will give (a).

To prove (b) we .must show that o'rd (C.) lm tends to infinity as m ~,CIO. Now from (4)

ord (~.) è! M(q - 1) inf {' i C(utl l . , i=l where the infimUrn is taken over an sequences Ul, ... , u lll , of distinct n tup1es U1 > 0 (see the beginning of this section).

III '" Thus let d lll ;", inf {!; C(u1)}, and (b) will be proved if i = 1 \ ' lim dlll'/m = + CIO. We can arrange the n'tup1es U1 in a m -?CD

sequence Ul, •.• , U1, ••. , such that C(U1) s: C(U1 + 1), and

III heIice d. = !; C(U1)' . Now the .C(ud' stend to CIO and even their i=l arithrnetic lneans C(Ul) + . .. + C(ua ) tend to + CIO. Hence dll lm m

tends to + CD as m --:;, CIO which proves the proposition.

Q.E.D. n-l6

n.7. The m.erom.orphic char acter of 'Z(X, t)

Now we com.plete the proof of n.l:l by showing that

·Z(X, t) is m.erom.orphic everywhere. on Kp. See the rem.ark at the end of section n.4.

We ret':!.rn to the hypersurface X defined by the polynom.ial

f(Xp [Xl,"', XDJ.

Y Let X be the hyperplane defined by the polynom.ial g = Xl X:a, ••• XD

= O. We com.pute Z(X\x: t). The num.bers M. are then the num.ber of solutions to the system.

, B ' Tr(t') If s ~ 0 is an integer and t (F , then let s,(t) = ( -ps 0-1 P P . cf. sectionn.S. By n:S:2 Be(t') = B(t)B(t ) ••• 6 (t ). Since

6 s is a non-trivial char acter on F .. we have . -P s

!; \. 6 s (Xo u) = pB if u = 0 Xo(F . -pS =Oifu f O •

.!et ks denote F for convenience -ps

. k s * = k s '", {o}.

Instead of u we substitute

f(X1, ••. , x D) = f(x), where x( (kS*)D

For each solution (Xl, ••. , xD) of f(X1 , .•. , XD) = 0 in (ks *)D we II-17

have

~ f) s(Xc f(x)) = pa. Xo( ks

Since there are N s such solutions, we have

paNs =.(pa -lt + ~ f) s(Xo f(x)), Xc (kli~ the sum. now being over ks*" and x overall n!tup1es of (ks*t' •.

.t W Write X~ f(X1 , ••• , X n ) = ~ 'alX 1 where X now denotes i=l the vector (Xo , ••• , X I1 ), and W l~ Zn +1 • • • J and the al( KP .

This glÎ.ves us

Let Al be them.u1tiplicative (cf. II. 5) representatives of the al in b, and that of x be y. Then_ .t s-l .. J 1 paNs = (pa _1)11 + l;{ , n n 8 (Al y r- ) x P ·=x i=l j=O lIy II. 5:2. .t ' W1 (5) Let G(x) = n f) (A1X ). Then i=l 8-1 paN s = (pa _1)11 + ~s G(y) ,..• G(yP )r by II. 6. x P =x II-lB

and henèe we can app1y the resu1ts of section II. 6, with q = p

to obtain

( 6)

where M denotes thematrix of the endomorphism e of th~ G . series G given by (5). We can expand the above binomia1s to

get

(7)

Mu1tip1y (7) by t 8 1 sand form the sU:m co log Z(X, pt) = ~ (pt)8 Nsl s s=l

= ~ (_1)1 (~) ~ (pn -1 t)8 i=O 1 s=l s

Now from II. 6 co 8 .'"' ~ Tr(M ) t8/s = log ~ (t) = log det (1 -. tM). 8=1

Rence

log Z(X, pt)

and hence II-19

Il Z(X" pt) = n i=O

Since the series for A converges everywhere in Kp by II. 6, we

conc1ude that Z(X, pt) is m.erom.orp~c everywhere in Kp.

This shows that the r of section II. 4 ia. infinite and that Rr > l, which com.p1ete s the proof of II.1 :1.

Q.E.D.

n.8 - Corollaries and rem.arks.

Since the fun~tion ~ = q-Sis one to one, we 'have imm.ëdiate1y

that C(X, s) is a m.erom.orphic function. If we com.bine the

rationa1ity of Z(X" t) and the inform.ation we have about the po1es

of C(X, s), (cf. 1. 3:5), we can m.ake the following observation:

Z(X t) = ao (1 - al tH1 - aat ) ... (1 - art) , (1 - bl t)(l - bat) .•. (1 - ap t)

Since C (~ s) has po1es on the line Re(s) = dim. X, Z(X, t) m.ust / . - dim. X ia have po1es at the pOlnts t = q e, (1. 3 :5), and a can be

computed exp1icit1y from 1. 3:5~

If to is a pole of Z(X, t), then bito = 1 for one of the il S. -dim X Henc~ we conc1ude that 1b i 1= q •

The Riem.ann hypothesis (1. 4) ,can now be generalized to this

case, andits present form.u1ation is simp1y that 1 ail =~

A trivial exam.p1e here is the affine n-space over F p (cf. App. II II-20

. 1 . ch. 1). For this schem.e the Z function is 1 _ pD. t' and the

above rem.arks are certain1y true, even though trivially, in this case.

Finally we rem.aik that the im.portance of theorem.

II.1:1 lie s in the fact that for each schem.e X of finite type over

:['«1' there is one rational function, which gives all the num.bers

N., which are essentially num.bers of solutions of the equations

defining X. III-1

CHAP'rER III. The Zeta Function of a Curve defined over a Finite Field.

III.1 Intr oduction.

In chapters I and II we studied sorne of the properties of

the zeta functions of schemes of finite type over Z. The con-

dition of being of finite type, however, is too strong to inc1ude

all interesting cases. Thus the question is raised wh ether a

. zeta. function exists underweaker conditions. An examinatic;m

of sections 1. 2 and 1. 3 shows that"the.product t (X, s) of a scheme

X, as we defined it in 1. 3, is well defined if three conditions are

satisfied:

1) Residue fields of c10sed points are finite.

2) There are finite1y many points with the same norme

~) A condition "ensuring the convergence of the product t (X, s),

for sorne sE'C. /

Conditions 1) and 2) ensure that if l; (X, s) converges, then it can co be expanded into a Dirichlet series of the form !; In 1. 3, n=l

condition 3 depended heavi1y on X being of finite type over Z.

The qu,estion is, whether these tJ'lree conditions will ensure reaeon-

able properties, such as the rationality of Z(X, t).

The point of this chapter is to study in detail a case where the \ 1II-2

answer is affirmative •.It is the case of a curve defined over a finite field. To fix our ideas le~ t be transcendental over F ~ and let K = F ~(t) [Ul, •• " Un _1 J be a field. Thus the U1' sare algebraic over ~q(t). Let A = F ~ [t, Ul, ..• , Un _1], and aS,sume

A is an integral domain. Then there exists a prime ideal ~ ofFq[Tl , .•. , TnJ, suchthatA=Fq[Tl,"" TnJ/IE-' We are interested in the curve defined by the polynomials generating the ideal m.

We will show in this chapter that for such a curve, there are theorems analogous to 1. 3 :2, 1. 3:5 and II.l:l, even though the corresponding scheme is not of finite type over Z.

From now on" a function field will be a finite algebraic

extension of F~ (t), t being transcendental over F~.

Recall that a discrete valuation on a field K, is a homo­ morphism v from the multiplic~tive group K* into Z, such that

v(x +. y) ~ min {v(x), v(y)) and v(l) = 0

v(O) is usually set equal to + co. If the field K contains a field

k, then v is said to be trivial on k if v(x) =,0 for all xE'k*.

Finally recall that the set

Av' = {xE'K: v(x) ~ 0) is a local ring of K,

with unique maximal ideal ~v == {XE' K: v(x) > 0). The ring Av III-3

is called the valuation ring of the valuation v, cf.. [5].

Let K be a function field, and let the valuation rings of

K be denoted by Av. By a curve defined over Fq, we will mean the scheme X = U spec (Av), where v runs over all discrete v' valuations ofK, trivial on F q. The c10sed point!!lv coming from spec (Av) will be denoted by v as welle These are the on1y c10sed points. Then in the notation of 1. Z, 1. 3 and appendix l, ch. l, OV,:II: = Av and k(v) = Av /mv. The fields k(v) are always finite, and hence the norrn N(v) of v is defined as in 1. Z; indeed

N(\r.)~ = card (k(v».

Then the zeta function of X is defined in exact1y the same way as in 1.3:

t (X, s) = n (1 - N(v)-S)-l, where s~C . v -

We will show (II1.4:1) that ~ (X, s) converges abso1ute1y if Re(s) > 1.

The proof of this will depend on the Riemann Roch theorem for the function field K. In fa ct the Riemann Roch theorern'~ill ensure that the zeta function converges.

III. 2 The Riemann Roch theorern.

Let K be a function field, i. e. a finite' extension of F q(t).

Let k be the a1gebraic c10sure of F q in K. Then k will be finite, and will be called the field of constants of K. The nurnber of e1ements III~4

of k will be denoted by, q';

Definition III. 2 :1. A divisor of K is said to be an elem.ent of the free abelian group, generated by the discrete valuations of

K, which are trivial on k. This grO'l.ip is denoted by D. If

a (D, we write a = n v a (V), where alm.o st aU the integers . v a (v) are zero. The discrete valuation v itself is called a prim.e divisor. The divisor a is caUed an integral divisor if

a (v) ~ 0 for all v.

If a and a l ~re two diV1sors, . 'we say tha t a divides a 1

1 and we write a 1a " if a 1 (v) ~ a (v) for aU v.

Next let x( K. Then we can com.pute v{x) for each v and con­

sider the divisor n VV (x). Clearly this is a divisor, since v{x) = 0 v for almost aU v.

Definition III~ 2:2. A divisor a (D is called a principal divisor

if a = fi VV (x) for som.e x( K. 'l'he divisor induced by x is denoted v by (x).

·The, principal divisors form. a subgroup P of D. The factor

group D/P is called the group of divisor classes and is denoted by

C. Two divisors a and a 1 arecalled equivalent if a.IX 1-1 ( P.

To each divisor a (D, we attach a vector space over k as

foUows. III-5

Definition III.2:3. Let a be a divisor of K. Then L(a) denote s the set·

( x( K: v(x) ~ a (v) for all v }

L( Cl ) is a vector space over k, and the dirnension of L( a ) over k is denoted by :& ( a ).

In section II .1, we defined the degree of a c10sed point of a schem.e over Fq. We now rnodify this definition to suit the case of a function field. Any discl"ete valuation v of K which is trivial on F q, will also be trivial on k, and thus there is an injection frorn k into..k(v) . This rneans that N(v) (cf. III. 1) will . - - deg(v) be a power of q, and we write N(v) = q ; deg (v) being the degree of v. Next we.extend these ideas to the divisors of K in a natural way.

Definition III.2:4. Let a be a divisor of K. The norrn of a denoted by N( a ) will be the nurnber n N(v) a (v). The degree . v of a, denoted by d( a), will be the nurnber:E a(v) deg (v). v . We need one more construction, before we can state the rnain theo'reIns.

For each v, let Kv be a copy of K and let l be the subring

of n Kv,' consisting of all sequences (av)v of n Kv such)that v v v(av) < 0 for at rnost"finitely rnany v. An eleInent of l is denoted

by ex = (av). III-6

.We define v: l ~ Z for each discrete valuation v of K in a natural way:

v( a) = v( av ).

Definition II!. 2:5. Let a; f3n,and (l a divisor of K. We say that a is eguivalent to {3 :modulo a , and write a == {3 rnod· (l if! v( a - (3) è!: (l (v) for aU v i.e. v( av - bv ) è!: a (v) for a1lv.

We set l( (l) = ( a€ l: a == 0 rnod (l ). Now if a€K, we can identify the sequence (a, a, ...• ) € n Kv with a and consider K v a subring of 1. Both land l( (l ) are vector spaces over k, and l( (l ) + K is a subspace of I.

Definition II!. 2:6. Let (l be a divisor of K. We set i( a -1)

= dim.( l/l( (l ) + K) as a vector space over k.

Lernrna III. 2 :7. Let a be a divisor of K. Then t «(l ) is finite.

The nurnber s d( b ) + 1,( b ) as bruns over all divisor s of K are

bounded below.

Proof: The proof is to be found in [6].

Wesetl-g=inf (i.{b )+d(b»). b€D

Definition III. 2~ 8. Let ~ be the set of elernents w( Ho~(I, k),

such that there exists a divisor (l of K with the property that

I( (l ) + K.> ker(w) This set ~ , is called the space of differentials III-7

'of K.

We say that w~~ is congruent to zero modulo Il -1 if

w~ Ho~o(I/I( Il ) + K, k).

' . Theorem nI. 2:9.

Let K be a function field, and Il a divisor of K. Then

1,( Il ) + d( Il, ) + g = 1 + i( Il -1).

Proof: cf[6]and[17].

Remark. If Q = (1), then ~(l) = 0, 1,(1) = 1 and g = i(l).

We state thrOee more resu1ts which we will need in thi' following

sections.

Proposition Ill. 2 :10. Let w~~. If w -:j. 0 and w == 0 mod Il then

dl Il ) :;; 2g - 2.

Proposition III. 2 :11. There exists a unique divisor b which has

maximal degrée, such that w == 0 mod b . This is independent

ofw. Furthermore d(b ) = 2g - 2, i(b) = l, and i( b -1) = g.

Proposition III. 2:12. Let Il be a divisor of K. If d(b ) > 2g-2

then i( Il ) = O. The c1ass of divisors with maximal degree is

called the canonica1 clas s .

Theorem Ill.o2 :13.

Let K be a function field, Il a divisorof K and Il 0 a

o -l) 01 -l -1 ) divisor in the canonica1 class. Then i( Il = "" Il Il 0 and III-8

-1 .t( Q ) + d( Q ) + 9 = 1 + i{ Q - 1 Q 0 ).

For proofs of III.2:l0, IIL2:ll, III.2:l2, and IIL2ü3 see [17 J.

Theore:m.s III. 2 ~ 9 and III. 2:13 are for:m.s of the Rie:m.ann Roch theore:m., ,·.;/hich is a deep relation b,etween the nu:m.bers i{ Q ) and d( Q , ). The nu:m.ber g is called the genus of K. m.3 Thé divisor class group.

In this section, ·we discuss condition Z, of section IlL 1.

More precisely, -,,",:e will prove that the nu:m.ber of divisor classes of a given degree is finite. We ':'l'ill also co:m.pute the nu:m.ber of integral divisors in a divisor class.

We have a :m.ap d: D ~"Z, given by d( Q) = the degree of

Q • for all Q ~ D. If x~ K then d( (x) ) = 0, cf. [6 J. Hence all divisors in a divisor class have the sa:m.e degree. This :m.eans that d can be viewed as a function on the divisor, classes, and we have d: D/P--,>~ Z . From. now on we can talk about the degree of a divisor class. The :m.ap d :m.ay not be onto, and we have d( C) = e Z , e ~ N •

The following lenuna is exactly condition 2 of section IlL 1 and also le:m.:m.a 1. 2,:4 of chapter 1.

Lenuna III. 3:1. Let K be a function field. The nu:m.ber of dis- crete valuations v of K satisfying the inequality deg (v) ::; d, for a III-9

given nurnber d, is finite.

Proof: Mter [6] and [11] .

Let u be a transcendental element of K over k. Then K is finite algebraic over k(u). 'Thus each discrete valuation v on k(u) extends to at most CK.: k(u) ] valuations on K. We will show that there are a finite nurnber of discrete valuations von kru) of degree less than d. ,Let this nurnber be M. Then there are at most M [K: k(u) ] discrete valuations on K with degree le s s than d, since the degree of a valuation extending a discllete valuation v of k(u) is always ~ deg (v). Thus the lerruna will be proved, if we prove the statement about the valuations, on k(u), Now:iL v is a discrete valuation on k(u), then v cornes from an irreducible polynomial f(u) or from l/u.

In the first case, deg (v) = deg (f) and there are but finitely many polynomials of degree less than d. The one valuation coming from l/u does not affect the finiteness, no matter what its degree is. Hence there are, finitely many discrete valuations of k(u) of degree less than d, and the lerruna is proved .. An irrunediate corolloryis the fact that there are finitely many integral divisors having a given degree d. Ill-IO

Theorem III. 3 :2.

The number of divisor classes, having a given degree d" is finite. This number is either 0 or a fixed positive integer h, independent of d.

Proof: After [6] and [11].

Fir st we show that there is a bijection between the set of divisors of degree d and of degree 0,. Indeed let {a 111=1, ••• n •• be tq.e set of divisors of degree d, and let a 1 be a fixed divisor of degree d. Then the map a 1 ~ a 1-1 a 1 is a bijection onto the set of divisors of 'degree zero. Since aU divisors in a divisor class have the same degree, this induces a bijection between the divisor classes of degreed and the divisor classes of degree O.

Thus the nurnber of divisor classes of given degree is independent of d.

We show that this nurnber fs finite. By the above discus sion it is no 10ss to assume d> g. Assume a divisor a has degree d > g. From the Riemann Roch theorem, III. 2:9, JJ.. a -1}. + d( il -1)

= l - g + i( a ) and hence since - g - d( a -1) = d - g > 0, we have i( a -1} > O. Hence there is an x~ L( a -1). This means that a is

'equivalent to the divisor (x) a , which, is integral. We therefore

can restrict ourselves to integral divisors, and by lernrna III. 3:1

there are but finitely many integral divisors of degree d;

Q.E.D ili.ll

Next we compute explicitly the number of integral divisors in a divisor class.

Proposition III. 3 :3. Let Qbe a divisor of K. and n( Q) the number of integral· divisors in the class of Q. Then we have

_ _.e( Q -1) (q - 1). n( Q) = q - 1.

Proof: After [6 ] and [11] •

The cla~s of Q consists of all divisors (x). Q , where x~K*. Now (x) Q is integral iff x~L( Q-1~. Since L( Q-1) has dimension .e( Q -1) over k, we conclude that there are at most _ .e( Q -1) q - l integral diyisors in,this class. Sorne of thern will coinéide. In fact (x) Q = Y ( Q) iff ( ~) is the class (1), that is , y , if x/y~k*. Thus (x)Q = (y) Q iff x is one of the elernents ya, a~k*.

Since k*has q -1 elernents, we conclude that there are

~ .e ( Q -1) _ l integral divisors. q - l Q.E.D.

III. 4. The functional eguation an d pole s of l;: (X. s) . • We now return to the scherne X of section IlL l and continue

our study of C (X, s). At this point we do n'ot know yet if ~ (X, s)

converges. However, the preceding section allows us to conclude

that l;: (X, s) can be expressed as a Dirichlet series, if either III-I2

c (X. s) converges or the Dirichlet series converges.

Indeed let an be the num.ber of integrai divisors of degree n. Let F(s) De the formaI series:E ann-s.sE'C. Let Vl •..•• VIII n=I be the prime divisors of degree Iess than d, and consequently of norm less than N. Then

m co ln (1 - N(vd-B)-l_ F(s) 1 < ":E ann-s i=l n=N+l

As N ~ co. so does m. Thus if the series F(s} converges, so ' does C (X. s) and

m s lim...-.::b... n (1 - N(vd- ) = F( s}. m~co i=l

By a sirnilar argument. one can show that if C(X. s) converges.

so does F(s} and C (X, s) = F(s), cf. ci. similar proof.in chapter IV r

section 1. and the sarne argument in the proof of theorem 1. 4:1.

Theorem III.4:1.

Let K be a function field, and X the scheme U spec (Av). v Then Ç(X, s) converçes absolutely if Re(s) > l, and satisfies a

functional equati0Il:r

C(X, 1- s) = g(g - 1)(2s -l)C(X, s).

Let 'F(:s:) be the formaI series E N( a )-S, Proof: daselg> O. a where the sum runs over an integral divisors of K, sE" C.

- -~a)s ' Then formally F( s) = :E n( a ) q • where the sum runs over C III-13

one representative. Q in each divisor class of c. Since the

degree depends ·only on the class (III. 3 :2), we have (again form.ally) ,

QO .. ~ -res F(s)' = ~ (q ) r=O C 1.. q - 1 d( Q )=er'"

Hence

il... Il -l)_res ] _ h (q-l) F(s) = =es 1 - q

Now if d( Il)> 2g-2, then.i(1l ) = 0 by (III.2:12). Hence

by m. 2:9, Riem.ann Roch theorem. j ( Il -1) = d( Q ) - g + 1 = er - .g + 1.

Furthermore if Il 0 is a fix,ed divisor in the canonical class (cf. III.2)

then d( tl. o ) = 2g·- 2, which shows that 2g - 2 = e i. for som.e integer

From. these rem.arks, we break up the suin F(s). Let

_ il... Il -1) - d( Il ) S f( s) = ~ q

d( Il )~ 2g-2

and

~ -q 1 - g + er(l-s)]._ h(l:' _ -q -es)-l g( s) = h [ i='=i.t1

f(s) is a polynom.ial in q-S and g(s) converges when Re(s) >1,

since the series in the bracket is a geom.etric series with com.m.on III-14

. - e(l-a) ratl.o q • Thua we have, if Re( a) > 1, F(a) = C (X, a)

1 = (f(a) + g(s) ).

An easy computation now showa that g(l - s)

= -(gq - 1)(2s - 1) g (')ses R ( ) > 1 • """hiJ. S lS. t h e f unctl.ona . 1 equatl.on . f0:t: g( s) • Next we derive the functiona1 equation for :f( s). If

a runs over all divisor classes with d( a ) ~ 2g - 2, so does

a -1 a 0, where a 0 is a fixed divisor in the canonica1 c1as s •

This allows ua to SUIn f( a) over the index

VVZl! obtain:

f(:b- s) = ~ 1 a- a 0

1 _ (1,( a~) + d( 0- ) + g - 1) - (2g - 2 - d( a)(l - s» = ~ q a -1 a 0

= ~ q (~ - 1)(2s - 1) f,(s) a -1 a o

Rence f(s) a1so,satisfies the functional equation, and so does

. C (X, 's) by additivity. Case 2, g = 0, e = 1. In this case K =k(t)

and h = 1, cf [20]. Also 2g - 2 = - 2, and hence f(s) = O. Then (- 1)>" (X ) ~ - 1 - g - r(.l-s) _ (1 _ -q-S)-l q - '.:0 ,s = r=O q

1 =

and aU the statements of the theorem foUow trivially.

Q.E.D. III-15 .

Corollary III. 4:2. With the above notation, 1: (X, s) is m.ero- . m.orphic in C and Z(X, t) is a rational function of t.

Proof: f(s) is a polynom.ial in q-S = t and

ëj (1 - g + 2g - 2 + eHl - s) g(s} = h [ 1 -e(l-s) - q ,

This shows that g(s) is m.erom.orphic in C. If t = q-S is substituted, it is also clear that g is a rational function in t. Hence Z(X, t), being the sum. of a polynom.ial and a rational function, is a rational function in t.

Corollary III. 4:3. Under the above notation 1: (X, s) has sim.ple poles at the points: 2'ITri s = 1 + 2'ITri _ and s e log q = e log Ci r~ Z.

Proof: The poles are found by setting the denom.inators of g(s)

- e~l-s) d _e.s equal to zero .. There are two cases q = 1 an q' = 1. In the first case, let s =x + iy and we obtain q:( l-x)e. q:-le:y = 1. 2'ITir. Thus le.= 1 and exp ( - iey logq ) = 1. Hence s = 1 + -- - r~Z. e log q

· '1 1 if _-es 1 . b' 21Tir Z SIm.l ar y q . we 0 tal.n s 1 _, r ~ . To show that = , = e og q the pole at s = 1 + 2~ir _ is sim.ple, m.ultiply g( s) by the factor e og q

-e(l-S ) 21Tir 1 - q and take the lim.it as s tends to 1 + e log q . The sam.e is true for the other pole.

Q.E.D. III-l6

Remark. Let·Q be a divisor of K. Define a Mobius function as follows:

is a product of r distinct valuations. tJ.( Q.) = . fo-l)r if II otherwise.

ThenC(X, s)-l = ~ J.L(Q)N(Q )-8 if Re(s)> O. . Q

This series converges absolutely and is analytic if Re(s) > 1.

Hence C(X, s) has no zeros if Re(s) > 1 and the functional equation now irnplies that C (X, s) has no zeros if Re(s) < O.

III.5 The Riemann hypothesis.

The' zeta function of a curve defined over a finite field was first studied by E. Artin in his thesis, cf [l J. He proved the

Riernann hypothesis for· certain quadratic function fields. Later

F. Schmidt, redefined the zeta functionïn terrns of valuations, cf. [20]. Hasse proved the Ri~rnann hypothesis for function fields of genus one, cf. [13 J, Finally, Andr é Weil proved it for any function field in his famous book: "Les courbes algebriques, et les variétés qui s'en deduisent", cf. [27].

Lernrna III. 5:1. Let X be as defined in Ill .. 1. Then t'eX, s) satis- fies the Riernann hypothesis iff Z '(X, t) has radius of convergence 1 Z(X, t) El..--=.--z • III-17

Remark. Z '(X, t) denotes the derivative of Z(X, t) with respect to t.·

Proof: If Re( s) > 1 ~ (X, s) = n (1 _ :,' q - s deg(v) )-1. The v iun~tion t'= q-S maps Re( s) > 1 onto the interior of the disk

1u 1 < -q-l. Then Z(X, t) = n (1 .. tdeg(V»-l if 1t 1 < q-l. v 1 The Riemann hypothesis is true if Z(X, t) vanishes at 1u 1= (:ca.

Hence the Riemann hypothesis holds if Z '(X, t) has radius of Z~X, t)

1 convergence q-a (cf. [14J ).

Q.E.D.

Remark. From section III.4 we know that Z(X, t) has simple poles at t = 1 and t= q-l. Hence the function H(t) = (t-l)(t-

00, Z' (X, t) = :E (section II.2) Z(X, t) n=l

, -n. Let bD = 1 + q - ND' Then by an easy computation

H' (t) = 1 H(t) t

Thus the Riemann hypothesis ho1ds if this series has radius of 1 convergence -q-a, that is if limb _-D < 00 n~oo n q /a • 1II-18

Theorem III. 5:2.

With theabove notation 1 b n 1 :;; 2g. ëj "11 /a, 'where gis the genus of X.

Remark. This is the famous result of A. Weil [27J. The prooi is very deep and relies heavily on the theoryof abelian varieties, cf. [16 J and [27 J. It turns out that 2g ..Jl ~ ln N n= 1 +q I.J,V! i=l l where Iw 11 = t:(~. This SUIn. can be interpreted as the trace of the

Frobenius automorphism of the curve, cf., [ 16 ] for details.

We remark that in the c'ase g :::: 0 Z(X, t) = [(1 - qt)(l - t) Tl and the result is trivial.

In the case g = l, Z(X, t) can be shown to be

2 gt + (h - ëj + l)t -+- 1 (1 - qt)(l - t) __ ëj - h - 1 ± i -vi 4q - (h _ ëj + 1)2 and the zeros are t - 2q --- • If t is com- plex then 1t 12 :: :~:a = q-l, wmch yiélds the Riemann hypothesis.

The'se' computations for g = 1 are due to F. Schmidt, cited' above. IV-l

CHAPT ER IV. The Zeta Function of an A1gebraic Number Field.

IV.1 Introduction and statement of main theorem.

In this chapter we consider the case of an a1gebraic number field. We consider a fixed field K which is a finite extension of the rational field Q, of degree n. We recall that an e1ement x of K is integra1 if it satisfies an equation

where the ai' sare ratio;na1 integers. The set of aU integra1 e1ements of K is a ring A called the integra1 c10sure of Z in K, or the ring of integer s of K. It turns out that A is a finite Z module [3 J, and in fact there is a basis Xl, :Ka, •• " Xn of K over Q such that A = ZXl + . .. + ZXn •

A is a1so a Dedekind ring, which means that each ideal

Q of A is a finite product of power s of .prime idea1s.

Q =

This expression is unique. AlI non-zero priIn.e idea1s of A are

maximal, [5]. We will be interested in the zeta function of

spec (A). Since A is a finite A module, dim (spec (A» = 1. In

fact the generic point of spec A is the zero idea1, and the residue .- field is just K, which has transcendence degree zero over Q. IV-2

This is the biggest difference between this chapter and chapter m. Fro:rn chapter l, section 3, we conclude that C (spec A, s) converges if Re(s) > 1. We also conclude (theore:rn 1. 3:5) that

C (spec A, . s) has a si:rnple pole at s = 1. The bulk of this chapter will be devoted to the co:rnputation of the residue at s = 1. In particular we will co:rnpute the followingli:rnit:

li:rn (s - 1) C(spec A, s). s~l+

Recall that a subset a of K is a fractional idealof K if a is an A-sub:rnodule of K such that there exists an ele:rnent d~ A with the pr.operty that d. a ~ A. An ideal of A is then so:rneti:rnes called an integral ideal of K. We can define an equivalence re1a­

.tion on the fractional ideals. Let a, a 1 be two fractional ideals of K. We call a equivalent to a 1 if there exists an x~ K such

1 that a = (xA) a . Then we have:

Theore:rn IV.l:·1.

Let K be an algebraic nu:rnber field and A its ring of integers.

The fractional ideals of K for:rn a group G, under ideal :rnultipli- cation, and the principal fractional ideals for:rn a group P. The group

G/p is finite.

Proof: See [3 ] •

This group is caUed the ideal c1ass group. The order of G/P is IV-3

denoted by h and is called the class nUInber of K. Orie notices

sim.ilarity between this theorem. and t}1eorem. III. 3:2. In this

case the proof is m.uch deeper. The fractional ideals here play

the sam.e role that the divisors played in chapter III.

We are now in CI: position to state the m.ain theorem. of this

chapter.

Theorem. IV .1:2.

Let K be an algebraic num.ber field of degree nover Q.

Let A be its ring of integers, X = spec (A). Then

li::l (8 - 1) I;(X, s) = k. h, s ~l+

where k is a num.ber depending only on K. and h is the c1ass

nUInber.

The num.ber k will be com.puted explic~tly in the course of the

, proof. It will be convenient to express 1; (X,s) in a

different form..

Lem.m.a IV .1:3., Let K be an algebraic num.ber field of degree n

over Q, A its ring of integers. Then if Re( s) > 1

1; (spec A., s) = I; u N( Il )-S, where Il runs over all integral

ideals of A, and N( Il ) = card (AI Il ).

Proof: W,e use the fact 1.2:4 that there,are only finitely m.any

closed points of spec A having given norm.. Indeed since A is a , IV-4

finitely generated Z module, . spec (A) is of finite type over Z and 1. 2:4 applie s. Also the prime ideals of A, which are maximal, are the points of spec (A). Let N be a given integer. Let fN{s) = l; N{1l )-8 when the SUIn runs over all ideals of N{ Il )< N norm 1ess than N. Then if s = u + iv Re{s) = u > l, we get

1 c (X, s) - fN{s) 1 ~ n (l - N(P )-U)-l N{P )~ N

Since C (X, s) converges, we have tha,t as N ~CD )

lim fN{s) = C(X, s) N~CD

The proof is exactly the sarne as 1. 4:1, except there we knew t~at

l; n-B converged, for Re{s) > land deduced the convergence of the product n {l - p -8)-1 • p Q.E.D.

Remark. Note the sirni1arity between this result (iL~4:1), and

section III. 4.

IV. 2. A generallernrna on series.

The proof of theorem IV .1:2 will depend heavily on the lernrna

which we prove in thi's section. Rn stands for Euclidean n-space.

Definition IV .2:1. A subset X of Rn is called a cone if it satisfie s

the following two conditions: IV-S

1) o = (0, •• " 0) ~ X

2)

Definition IV. 2 :2. A 1attice in Rn is a free Z module generated by m linear1 y inde pendent vector s Xl, ••• , Xm. m Sn. The fundamenta1 paralle10piped of the 1attice, with respect to the basis Xl, •• " X m is the set of all ve'ctor s ? in Rn satisfying

X = a ,-Xl + ... + am X m

It will be 1ebesgue measurab1e and will have measure J.l. ( !In ), where!IR denotes the 1attice.

Theorem IV. 2:3.

Let X be a· cone in Rn, and !In be an n dimensiona1

1attice, whose fundamenta1 paralle10piped has volume EL ( !IR ).

Let F: X ~R be a positive function on X satisfying the conditions:

1) If x€X, a€R a > 0, then F( a~) = an F(x)

2) The set T = {'X€X 1 F(x) S 1 } is bounded and

measurab1e, having volume fL (T).

Thenthe series :E F(x)-S converges if Re(s) > 1 and xE'·$,n X

fL (T) lim (s - 1) { :E F(x)""'s } = J.l.(!Dl ) s ~ 1+ xE' !Dl n X

Thus the idea of the proof of IV .1:2 is to reduce that case to the

above theorem. This process will be carried out in various stages in 1ater. sections. IV-6

Proof of IV. 2:3. Let r be a real num.ber > o. Let Wl r be the lattice obtained by contracting 9Jl by a factor of r.

Then the volume of the fundam.ental parallelopiped of 9Jl r

n is IJ. (!ID )/r • Now let N(r) be the num.ber of elem.ents of 9Jl r

contained in T. Since T is bounded N(r) is finite. Then

IJ.(T) = lim., N( r) IJ. (9Jl ) = IJ. ( 9Jl) Hni N(r) n n r~ co r r_;:>CO r

Sim.ilarly we can also consider the sets rT, consisting of aIl vectors r. x where xf:T. Then N(r) is also equal to the number

of points of 9Jl contained in T. Finally N(r) is equal to the

n number of points xf: 9Jl n X such that ~(x) S r • Let us arrange

the points of 9Jl n X in a sequence Xl, X2 •.•• , such that

o < F(Xl) S F(x:a) S •••• S F(:;x~) S •••• and let

Then Xl. X2 •..• Xk 1;>elong to rk T. but Xk does not belong to

(rlc - f: ) T. for any positive real num.ber f:. This m.eans that

N(rk - f:) < k S N(rlc). Thus consider the following inequality.

and take the lim.it as k tends to infinity.

Now as k ~co so does rlc. and hence IV-7,

lim k/F(Xk) = lim k -";'.0) k~O)

i. e. lim k = H..ill k~ , J.1.(!Dl ).

'1'0 get what we want, we need to ùse the fact that

a ~ (s) = 1:; n- converges if s > t and that it has a simple n = l pole with residue l at s ;::: 1.

a (1. 4:1) Set f( s) = 1:; F(xr • f(s) converges or x€!Dl n X diverges with ~ (s). :since lim = [ J.1. Ùr) Ja k~O) J.1.(!Dl) which is non-zero" Now it is easy to compute the residue of the pole of f(s). Let € be a positive real nw:nber. ,Then there is a ko such that

~r· ..1..( J.1. (T) ) l < l ( J.1. (T) + € )\ k J.1.(!Dl) - € .. < F(Xk) k J.1.( !Dl ~ for all k è! ko.

ifs>1.

(Xl Now lim (s - 1) ~ (s) = lim (s - 1) 1:; k-B = 1 s ~1+ s~1+ k=ko .

ko -1 . and lim (s - 1) 1:; F(Xk ra = o. Hence if we multip1y by s - l s ~1+ k=1 IV-8

and let s tend ta one from. the right we obtain . co = Il (T) lim. + (s sinee 1l(!IR ) s~l

the inequality holds for aIl € > O.

Q.E.D.

IV. 3. The logarithm.ie spaee.

Sinee [K: Q] = n, and K is separable over Q, K has n

em.beddings into C.

Definition IV ~ 3 :1. Let Cl : K ~ C be an em.bedding of fields.

If Cl (K) !:; R then Cl is said to be a rea1 em.bedding; otherwise

Cl is ealled a eom.plex em.bedding.

Now if Cl : K ~ C· is a eom.plex em.bedding so is the

em.bedding cr defined by cr (x) = Cl (x), the bar denoting eornp1ex

eonjugation. Henee K will have r rea1 em.beddings and 2t eom.-

plex ones. Obvious1y r + 2t = n. Let Cil, .•. , Cl r be the real

. em.bedding's and Cl r+1 , ... , Cl r+t the non=eonjugate eom.plex ones.

The rem.aining em.beddings there are Ci D +1 1 s: i s: t.

Let V be the linear spaee R r X C:' We ean eonsider V to be RD

as a veetor space over R. We m.ap K into V as follows:

Cl (x) = (Cl 1 (x), ..• , Cl r(x), Cl r+l(x) , ••. , Cl r +t(x))

This is also a ring hom.om.orphism. of K into V if addition and

m.ultiplication in V are done eom.ponentwise. IV-9

Next let V*.= R*r X C*t. ,We define a map t: V* _>Rr+t

Let x = (Xl, ••• , xr+t)f: V*. Define

i,. (Xi) = log 1 Xi 1 if 1 ~ i ~ r.

tdxd = log 1 Xi 12 if r + 1 ~ i ~' r + t

This gives a map t: V* ~.....;>~R s+t by setting t (x) =(k(Xl), ••• ,

Now if xf: X, we get

(t.. O' )(x) = (log 10' l(X) \, ••• , log 10' r+éx)2)

Here we write tdx) '= t i (O'(x»

r+t r+t Tl:ten ~ ~t~~: (:0' (x» = ~ log 10' dx) \ ei where e1 = 1 if 1 ~ i s: r, i=l i=l

Z otherwise, and we ha'(e

r+t 2 ~ tdO' (x» = log 10' i(X) •.• 0' r(x) 11 0' r+l(x) ..• 0' r+t(X) 1 i=l

We now examine the effect of the map t- 0' on the units of A. If u is a unit of A, ,N(u) = l, which means that 0' (u)f: V*, and t .. 0' is r+t well defined. ~ t dO' (u» = log 1 N(u) 1 = o. Thus the units i = 1 r+t of A are mapped into the hyperp1ane ~ Y 1 = 0 of R r+t i=l Dr-lO

Theorern IV. 3 :1 (Dirichlet Unit Theorern)

,Let [K: Q ] = n, A be the integer s of K and A''f. the group of units of A. Then (R, ~ cr )(A*) is an r+t-l dirnensionallattice contained in the hyperplane

Yi + '... + y r+t = O.

Proof: See [3 J.

This rneans that there exist units Ul, •• " u + - such that r t l the vectors (R,a cr )(Ui) 1 s: i s: r+t-l are a basis for the hyper- plane Yl + ... + Yr +t = O. Let R,* be the vector

(1, ... ,1,2, ... ,2). Then R,* doe s not lie on the hyperplane anÇl. ,'---v---.l ~ r t hence R r+t is generated by 1#, R, ocr (Ut) 1 s: i s: r+t-l. For convenience we denote r+t-l by q. IV.4 A theorern on .iundarnental dornains.

The space V still denotes R r X ct • ,

Definition IV.. 4:1. A subset X of V is called a fundarnental dornain of V lif it conslsts of points x of V satisfying the conditions:

1) x(V *

2)

then 0 s: ai < 1 for 1 s: i s: q.

3) If Xl is the fir st cornponent of x, then Xl > 0

2 'fT ~ Xl li r 1 and if r = 0, 0 s: arg < , -,In where rn is the

order of the group of roots of unity of K. IV-ll

Lernrna IV.4:2. The set X of definition IV .4:1 is a cone of V.

Proof: Let xE'X and Cl! E'R Cl! > O. Clearly Cl!xE'V * .

.13. Let 1. (x) = fJ 1. * +'E fJ 1 1.. 0' (Ui) i=l

1. (Cl!x) = (log Cl! + log 1 xli, •• ~, log Cl! + log 1 x r l, 2 log Cl!

+ log 1 xr+ll, •.•• , 2 log Cl! + log 1 xr+t la)

= log Cl! 1. * + 1. (x) Il. 1. ( Cl!x) = (log Cl! + fJ ) 1. * + E fJ 1 1. • 0' (ud i=l and 0 S; fJ 1 < l, s:ince xE' X, and:. the components of the 1. <> 0' (Ui) are not changed.

Finally, multiplication by Cl! does not change arg Xl'

Rence Cl!xE'X and X is a cone of V considered as Rn.

Lemma IV. 4:3. Let yE' V *. Then y has a unigue representation of the form: y = x 0' (u) where xE' X and u is a unit of A.

Proof: Let 1. (y) = 'Y 1. * + 'Y l 1.00' (Ui) + ••• + 'Y Il. 1. e 0' (uq ).

Then there exist integers kJ such that 'YJ = kJ + Cl! J and O' s; Cl!J < 1. k k . Il. l Let u = Ul l ••• • ll:-q . and c~nsider Z = y 0' (u- )

. the k i ' s being cancelled out. Now z does not yet belong to X, for

arg z l may not belong to the required interval. Let arg Zl = I/J.

Then there exists an integer k such that IV -12

o ~ c/J _ 2TTk. < 2 'TT rn rn

Let ~ be the m'th root Qi unity, with the property that , 2'TTi cr l (~ ) = e ~ in C. Then we perforrn a rotation by setting

Since;' is a root oi unity 1. b cr(~-k) = 0, and hence 1. (x) has the sarne coeificients a t belonging to the 1. 0 cr (ud. Also

= c/J _ 2'TTk rn

2 'TT ' k HenceO~ argxl <- andx€X. ,Nowy=z. cr(u) =xcr(~ u). rn Thus we have iound one representation. It rernains to prove

the uniquene s spart.

1 1. el cr (u ) are integral'linear cornbinations oi the vectors1." cr(ud.

l ~ i ~ q. Also the coefficients of the 1. Co cr (Ut) in the expansion

of 1. (xl) and 1. (x) are non-negativeand less than one. This rneans

l 1 that 1.œ cr (u ) = 1.,t:J cr (u) and thus u = a u , where a is an rn'th

1 root ~f unity. Then cr(u) = cr (u ). cr(a) and we have xl = x cr(a).

o ~ arg Xl < 2'TT and sirnilarly for arg x. Hence 0 ~ 1 arg cr l(a ) 1 rn < 2'TT and this is possible only if a = l since a is an rn'th root of rn unity. Hence x = xl, which proves the lernrna. IV-13

RecaU that two elements of K * are caUed associates, li their quotient is a unit of the ring A. This yields an equivaleneerelation.

Theorem IV .4:4 ..

In every class of associate numbers of K *, there is a unigue x~K * su ch that cr (x) lie s in the fundarnental domain X of

Proof: Let y f; 0 in K. Then cr (y) == xcr (u) where X~X, u is a unit of A by the lernrna IV. 4:3, and this representation is

1 unique. Let z = y. u- • Then cr (z) = cr (x) and cr (z)~ X. By the uniqueness of IV. 4:3, z is the only such nurnber, ·which completes the proof.

IV.5 Proof of theorern IV .1:2.

. 1 -1 Let i. be one of th,e ideal classes of A. Let Q ~ i. and let

!lJ! be the lattice of V consisting of aU cr (x), x~ Q 1. This is a lattice, since it is a sublattice of cr (A). Let X be the fundarnental

domain of section IV.4, and let F: X >'.c ~R be the function 2 1 N (x) 1 = 1 xli. .. 1 x r 1. • .• 1 xr+112 ..• 1 Xr+t 1 . Then F satisfies

condition 1 of theorem IV. 2:3, and in the next section we show

that it satisfies condition 2 and we compute J.L (T). Rence X, !IR

and F satisfy the conditions of theorem IV.2:3 and we have IV-14

lim (s- 1) ~ \ N(x) \-s. = ,.,,(T) 1,." (!Dl ). s-? 1+ x€!Dl n X

Next .we prove that

N( (ll)S ~. \ N(x) lB x€!Dl n X where the sum on the right runs over an integral ideals of the class t .

By theorem IV. 4:4, we get

N( (l l)B ~ 1 N(x) 1 -s = N( (l l)B ~ 1N(xA) 1-8 x€!Dl n X xA x€ (ll

N( (ll) 8 ~ = N(xA) .xA x€(l 1

Now the map (l. l-l-~>~ (l (ll

gives a bijection between the integral ideals of t and the principal

j,deals of A, dividing (l 1. Hence as xA runs over an principal

ideals dividing (ll, the factor . N( (l 1 ) runs over an factors N(xA)

1N( (l ) 1 -1 = N(. (l )-1 as (l runs over an integral ideals of t.

Hence we get:

. x€!Dl n X

Hm (s -1) ~ N«l ) -B =,." (T) N( (ll) s-71+ (l€t ""(!Dl ) IV-15

t Now Il ( !ID) = 2- N( Q 1),fiI)I where D is the discrim.inant of

K. We will show in the next section that J.! (T) is independent of h. Finally

C (spec A, s) = ~ N( Q )-6 = ~ (~ N( a ) -8 ) ~ aE'~

Rence lim. + C(spec A, s) (5 - 1) = h J!:..J!1 • N( a 1) 5-3>1 . 1l(!Dl )

IV. 6 A volum.e computation.

In order to complete the proof oi theorem. IV.l:2, we m.ust show tha~ the set T = {XE' X: 1 N(x) 1 S; 1} is bounded, m.easurable and we m.ust com.pute its m.easure to show that it is independent of the class nuznber.

Lenuna IV. 6:1. The set T = {xE' xl: 1N(x) 1s; 1 } is bounded.

Prooe.- Let S = {XE' X: N(x) = ± l }

Then T = {CIlx: 0 < CIl S; -1, xE' S }. Thu~ if S is bounded, 50 is T.

l q. If x is in V *, ~(;k) = - log 1N(x) 1 ~ * + ~. CIli ~ 0 0' (Ui)' n . i=l

q This follows from. the fact that 1:: ~ 1 (x) = (r + 2t) a, where i=l

CIl is the com.ponent of ~ *.

Now there exists an integer M such that ail the com.ponents of the vectors ~ ~ 0' (ud l'S; i s; q have absolute value less than M.

Then if xE' S, IV-16

~.\ q ~ ..e, (x) = ~ Cl 1 ..e, '" 0' (ud, 0 ~ Cl 1- < 1. i=l

Hence the components of ..e, (x) have absolute value les s than qM.

Hence the components of x have absolute value less than

..e, qM or ..e, qM/2 and since this holds for aU x in S, S is bounded.

Q.E.D.

Lemma IV. 6:2. Let u be a unit of the ring A of K. The linear

transformation of V into V given by x ~ x 0' (u) is volume

preserving .

. Proof: Let U be a set·of measur.e,.,. in V and T be a linear

endomorphism of V. Then

,.,. (T (U» = 1 det (T) 1,.,. (U) .

In ou'~ case the determinant is N(u) which is just ± 1, . and hence

the transformation is volume preserving.

Now let T k be the set obtained from T, by rotating through 2k'IT an angle of fu"" This is the same as multiplying each element i'lT of T by O'(Ç;~ where ç; is that m'th root of 1 such that 0'1(Ç;) =.e""1lï. m-l Let U be the subset of. U T k consisting of aU x with positive k=O

components., Next consider the vector e = ( ± 1, ... ±l, l, 1, ... ,1) '\..--y-----' \.. r

r of V and perform multiplication by e to u'. There are 2 such IV-17

possible e' s and the images of U under their respective­ m-l multiplications are disjoint sets with union U 'l;'k • Thus k=O " m-l r '2 J.L (U) = J.L ( U T k ) = m J.L (T), if T is measurable. T.he k=O remainder of the argument is a calculation of J.L (U). We perform two changes of variables.

First we introduce polar co-ordinates. Let

i çb 1 lS:iS:r Xl = Pl e r+l s: i s: r+t.

The jacobian then becomes Pr+ .... P + . Then U is now l r t determ.ined by the conditions 1 N(x} 1 s: l and if .t-~x} = ex..t* + ~ ex. l .t .. o(Ul} i then 0 s: ex. 1 < l for aU xf:U. This now become s: .' r+t e- Pl > 0 ••••• Pr+t > 0 and n P1 1 s: l i=l - . ej !U otherwise 2; and log pj - = log n must have coefficients 0 s: ex. k < 1. The çb l' sare independent.

The second change of variable s is as follows: ,r+t è j Let y = n p j and in the above equations ex. k = Yk be new j=l variables. Now U is determined Dy the conditions 0 < y s: l o s: Yi < lIS: i s: q. Hence U is measurable. Also

~ = E...1.. oy ny IV-lB

Hence the jacobian becornes the deterrninant

P;r+t ny

Now add aH rows to the first row,

Since I; /, 1 (CI (u)) = 0 for any unit u of A we get

(pl, " Pr+t) • ny2t

, . .

=(Pl ".' Pr+t)· n -- - ny2t .. IV-19

R = ~,~t~--~----- 2 Pr+l'" Pr+t

Hence Jo ... ' - dx1 .••... dx + J \J r t U

1 'f -t dy. d d J',,. -- R2 Y1 •••• yq o 0

r Hence J.I.( U) = '/Tt R J.I. (T) 2 '/Tt R = In

t Also /-L (!Dl ) is known to be 2- N( Q 1) 1iDï where D is the dis- crirni.nant of K.

Hence; IL (TO) = N( Q 1) /-L(!Dl ) and this does not involve h.

IV. 7. PriInes of first degree.

TheoreIn IV. 7 :1.

Any algebraic nUInber field K has an infinity of priInes of degree one.

Proof: ~ (X, s) = V (1 - N( P ra) -1 • Thus if Re( s) > 1,

~ (X, s) ~ 0 and we can take logs. IV -2.0

log ~ (X, s) = - ~ log (1 - N( P )""11)

co = ( -~ In=l~

Now let P(s) = ~ N(p ')-8 where p' runs over the primes of degree one.

Let log ~ (X, s) = P(s) -1" Q( s). Let f denote the degree of p.

If f ~ 2 we get:

co ~ 1 = (p2a _ 1)< 2 p2Sm 7 In=l In=l

co l co 1 1 2 and ~ ~ --;s = < if InN(P ) -m 8 p2a In=2 . In=2 P f = 1.

~ 2n Henee Q( s) < p < 2n p 2s ainee there are at Inost n priInes above p. Thus Q( s) is bounded as s~l+. Sinee ~(X, s) has a pole there, P(s) Inl.B t beeoIne infinite at l, and henee Inust eontain an infinite nUInber of terIns.

Corollary IV.1:2. There are infinitely Inany priIne integers.

We ean a1so give Euler 1 s proof whieh depends upon the zeta fun etion. Suppose Pl .•. Pr are aU the priInes. Then

r l 1 .-1 TI (1 - P1- r = ~ J. i=l j=l IV-21

since the 'product on the left has a finite nu:m.ber 'of ter:m.s and

since by hypothesis, each integer is a product of powers of

,Pl•...• , Pr· But the series on the right (the har:m.onic series)

diverges. Hence a contradiction. V-1

CHAPTER V. Conc1uding Remarks.

Fir st1y we remark that this chapter is intended to be

·more informa1. In particu1ar we will not give forma1 defini- tions and theorems. The discussion is more in the style of the raconteur.

V .1' Local - Global.

Consider l;; (spec Z, s) ,;" l;; (s). For each comp1etion of

Q, i. e. R or Qp we can define a local zeta function. Indeed, in the case of Qp the natura1 thing is to take

."~( spec Zp, s ) = (1 - P ~ )~ • T~en we s ee that

l;; (spec Z, s) = ~. l;; (spec Zp, s). What about R? Well,. con­

sider the functiona1 equation for l;; (s) (cf; I.4:1)

,;:.. .. -ts (~) ~ ., -t+ts l l '.i' (s) =.'TT. r. \2 . ." (s) = .TT· r ('2 - '2 s) l;; (1 - s).

s Suppose we set l;;(R, s) = :TT. -t r (~). We have q;(s) = q;(1 - s)

and

4}(s) = [n l;;(spec Zi>, s) Jl;; (B, s) p

. Perhaps q; (s) is a more natura1 function to attach to Z than ~ (s).

In any case John Tate carried out the above remarks for

a nu:mber field K. He defined, by means of fourier series on the V-2

idele group. local zeta functions for eàch completion oi K.

'Then he showed that each of these satisfied a functional

equation and taking the product. he obtained a zeta function

ior the global field K.

A. Weil extended this [Basic Number TheoryJ and

applied the same procedure to w:hat he called A-fields. A­

fields are simply:

1) Algebraic number iields and '

2) Finite algebraic extensions of F q(t).

Thus he gave one set oi proofs for chapters III and IV. also

using fourier analysis on the locally compact idele group.

V.2

The above survey also leads us to believe in the es sential

unity between algebraic geometry and algebraic number theory.

This was also part of the motivation for Grothendieck to intro­

duce schemes. ' However. the deepest theorems are still proved

using other means such, as analysis. B-l

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