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.