Absolute zeta functions

Transcription

1 138 Poc. Japan Acad., 84, Se. A (2008) [Vol. 84(A), Absolute zeta functions By Anton DEITMAR, Shin-ya KOYAMA and Nobushige KUROKAWA (Counicated by Shigefui MORI, M.J.A., Sept. 12, 2008) Abstact Two new concepts of zeta functions fo schees ove the field of one eleent ae poposed. A localization foula and an explicit foula in the affine case ae given. This allows fo a coputation fo evey schee. Key wods Zeta function; field of one eleent. Intoduction. In [2], the fist naed autho has intoduced a zeta function fo F 1 -schees genealizing an idea of Soulé s in [10]. Basically, this zeta function only detects fee ans of the involved goups, so it is insensitive to tosion. In the pesent pape we will intoduce two new inds of zeta functions fo F 1 -schees which ae sensitive to tosion, yet still peseve the infoation on the ans. The fist new zeta function is of Weil type, whee the finite fields ae eplaced by the basic finite onoids. The second is of Igusa type, as inspied by [7]. 1. Soulé zeta function. Soulé [10], inspied by Manin [9], gave a definition of zeta functions ove the field of one eleent F 1. See also [8]. We descibe the definition as follows Let be a schee of finite type ove Z. Fo a pie nube p one sets afte Weil,! Z ðp; T def ¼exp 1 T n n #ðf p n ; n whee F p n denotes the field of p n eleents. This is the local zeta function ove p, and the global zeta function of is given as jz ðs def ¼ Y Z ðp; p s 1 p Soulé consideed in [10] the following condition Suppose thee exists a polynoial Nðx with intege coefficients such that #ðf p n¼nðp n fo evey pie p and evey n 2 N. ThenZ ðp; p s 1 is a ational function in p and p s. The vanishing ode 2000 Matheatics Subject Classification. Piay 11G25. Univesitaet Tuebingen, Matheatisches Institut, Auf de Mogenstelle 10, Tuebingen, Geany. Depatent of Matheatics, Ewha Woans Univesity, Daehyun-dong 11-1, Sedaeoon-u, , Seoul, South Koea. Depatent of Matheatics, Toyo Institute of Technology, Oh-oayaa, Meguo-u, Toyo , Japan. at p ¼ 1 is Nð1. One ay thus define Z ðp; p s 1 jf1 ðs ¼li p!1 ðp 1 Nð1 One coputes that if Nðx ¼a 0 þ a 1 x þþa n x n, then jf1 ðs ¼s a 0 ðs 1 a 1 ðs n a n Based on ideas of [6], in the pape [1] thee is given a definition of a schee ove F 1 as well as an ascent functo Z fo F 1 -schees to Z-schees. Fo the convenience of the eade, we will biefly ecall the definition. Recall a onoid is a set A with an associative coposition and a unit eleent 1 2 A, i.e., one has 1a ¼ a1 ¼ a fo evey a 2 A. In this pape, all onoids will be coutative. An ideal in the onoid A is a subset a A with aa A. Anidealpis a pie ideal if S p ¼ A p is a subonoid. Let Spec A denote the set of all pie ideals with the Kull topology [1]. Fo p 2 Spec A, let A p ¼ Sp 1A be the localization at p and let A p be its unit goup. Then A p is the quotient goup of S p ¼ A p. On the topological space Spec A one has a canonical sheaf O A of onoids with stals being the localizations of A, soo A;p ¼ A p fo evey p 2 Spec A. Aschee ove F 1 is a topological space togethe with a sheaf O of onoids such that ð; O is locally isoophic to ðspec A; O A fo onoids A. Let be an F 1 -schee of finite type. Note that this iplies that has finitely any points and that O ;p is a finitely geneated goup fo evey p 2. An affine F 1 -schee is given by a coutative onoid and its lift to Z is given by the coesponding onoidal ing. This pocedue extends to geneal schees as it espects gluing. We say that a Z-schee is defined ove F 1, if it coes by ascent #2008 The Japan Acadey

2 No. 8] Absolute zeta 139 fo a schee ove F 1. The natual question aising is whethe schees defined ove F 1 satisfy Soulé s condition. Siple exaples show that this is not the case. Howeve, schees defined ove F 1 satisfy a slightly weae condition which seves the pupose of defining F 1 -zeta functions as well, and which we give in the following theoe, poven in [2]. Theoe 1.1 (See [2]). Let be a Z- schee of finite type defined ove F 1. Then thee exists a natual nube e and a polynoial Nðx with intege coefficients such that fo evey pie powe q one has ðq 1;e ) #ðf q ¼Nðq This condition deteines the polynoial N uniquely (independent of the choice of e). We call it the zeta-polynoial of. Using this polynoial Nðx, one defines the Soulé zeta function as above. Poposition 1.2. The Soulé zeta function satisfies the localization foula S ðs; 2 S ðs; O ;p Poof. Let Z be the Z-lift of. Foapie powe q let F q 1 be the onoid ðf q ;. Then j Z ðf q j ¼ jðf q 1 j by [1], Theoe 1.1. Genealizing the pevious notation, fo a natual nube let F be the onoid [f0g, whee denotes the cyclic goup of eleents. The onoid opeation on F is given by 0x ¼ 0 fo evey x 2 F. The poposition follows fo the next Lea. Lea 1.3. Let be an F 1 -schee. Thee is a canonical disjoint decoposition HoðSpec F ;¼ a p2 HoðO ;p ; Hee on the ight HoðO ;p ; coesponds to the set of all hooo-phiss Spec F! with ðc ¼p, wheec is the closed point of Spec F. Poof. The set Spec F consists of two points, the geneic point and the closed point c. Let SpecF! be a hooophis and let U be an open affine subset containing ðc. Then is contained in the open set 1 ðu, sothat indeed, is a hooophis fo Spec F to U ¼ Spec A. Inothewods, is affine. It theefoe suffices to pove the lea in case of affine. In this case is given by a onoid ophis A! F. The set p ¼ 1 ð0 is a pie ideal and induces a hooophis A p!. Thus one gets a disjoint decoposition HoðA; F ¼ a HoðA p; p2spec A The lea and the poposition follow. The factos of Soulé s zeta function can be calculated as S ðt;o ;p ¼sa 0 ðs 1 a 1 ðs a ; whee ¼ anðo ;p ¼di QðO ;p Q is the an of the finitely geneated goup O ;p,and a j ¼ð 1 j j So S ðt;a only depends on the ans of the local goups O ;p and ignoes the finite pats. We will now intoduce a zeta function of Weil type, which caies oe infoation. 2. The absolute Weil zeta function. In the theoy of schees ove F 1, the onoids F, 2 N play the ole of finite fields. Analogous to the above definition of the local Weil zeta function we define the Weil zeta function of a schee of finite type ove F 1 asfoalpoweseiesint by W ðt;¼ def exp 1 jhoðspec F ;j T! In the case when is equal to q 1 fo a pie powe q, the onoid F can be identified with the ultiplicative onoid of the finite field F q.inthat case thee is a natual bijection HoðSpec F1 F q 1 ;! ¼ HoðSpec F q ; Z ; whee Z denotes the Z-ascent of. Inthecasewhen is affine, i.e., ¼ Spec A fo soe onoid A, we will also wite W ðt;a instead of W ðt;spec A. By Lea 1.3, the zeta function W satisfies the sae decoposition foula as the Soulé zeta function, W ðt;2 W ðt;o ;p We will now copute W ðt;a fo a finitely geneated abelian goup A. Any such goup is isoophic to Z C n1 C n fo soe 0 and n 1 ;...;n 2 N. Hee C n denotes the cyclic

3 140 A. DEITMAR, S. KOYAMA, andn. KUROKAWA [Vol. 84(A), goup of n eleents. Since fo copie nubes n; n 0 one has C nn 0 ¼ C n C n 0, one can aange the cyclic goups in a way that n j divides n jþ1 fo evey j ¼ 1;...; 1. Assuing this, we can pove Poposition 2.1. Fo A ¼ Z C n1 C n one has 8 Y ð1 T jdj ðdd 2 1 d 1 fo ¼ 0, >< W djn ðt;a¼ Y exp g ðt jdj ðdd þ 1 d 1 1 > fo 1, ð1 T jdj djn whee we have used the following notation. The poducts un ove all tuples d ¼ð ;...;d 2N such that jn 1, d 2 j n 2 n, and so on until d j d 1. These conditions ae suaized in the notation djn. Futhe, jdj ¼ d, and ðd ¼ ð ðd,whee is the Eule -function. We denote by g ðt 2Z½TŠ the Eule polynoials which ae defined ecusively by g 1 ðt ¼T and ð1 g þ1 ðt ¼ ðt 1 g ðt 1 Poof. Let A ¼ Z C n1 C n as above, then jhoða; j ¼ ð; n 1 ð; n ; whee ð; n denotes the geatest coon diviso of and n. Using the popety of the -function, P djn ðd ¼n, we infe that the logaith of W ðt;a equals þ 1 ð; n 1 ¼ 1 þ 1 jð;n 1 ð; n 2 ð; n s T ð ð; n T ¼d ¼ 1 ð 1 ð þ 2 d1jn1 ð; n 2 ð; n T d1 The su ove is now of the sae fo as thefistsu.sowecaniteatetheaguentto each ð d1 þ 2 jn 1 d j ðd 2 d þ 3 2 d 2 j n 2 ðd d þ 1 n d 1 1 T d The aguent is finished with the following lea which yields! exp 1 1 T g ðt ¼ exp ð1 T Lea 2.2. Fo ¼ 1; 2; 3;...,wehave 1 T ¼ g ðt ð1 T ; whee g ðt 2Z½TŠ is defined by (1). Poof. Put S ¼ 1 1 T Since we copute ð2 If we put TS þ1 ¼ 1 ð1 T S þ1 ¼ 1 ¼ 1 T þ1 ¼ 1 ð 1 T ; ð ð 1 T ¼ ¼ 1 ¼0 ¼ 1 S ¼ ð 1 þ1 T ð 1 þ1 S þ1 ð 1 1 S þ1 ð 1 S f ðt ð1 T and substitute it to (2), we find that f ðt is a polynoial in T with f ðt ¼g ðt, becausef ðt also satisfies (1).

4 No. 8] Absolute zeta 141 Exaple 2.3 (Eule polynoials). g 1 ðt ¼T; g 2 ðt ¼T; g 3 ðt ¼T 2 þ T; g 4 ðt ¼T 3 þ 4T 2 þ T 3. The absolute Igusa zeta function. We also intoduce anothe type of zeta function which has an additive localization foula, I ðs; ¼ I ðs; O ;p p2 It is defined fo s 2 C with Reðs 0 as I 1 def jhoðspec F ;j ðs; ¼ s This is analogous to the global Igusa zeta function of a Z-schee, which is defined by Z I ðs; ¼ 1 jhoðspecðz=z;j s It has an Eule poduct expession Z I ðs; ¼ Y Zp I ðs; p pie with the local Igusa zeta function Zp I jhoðspecðz=p Z;j ðs; ¼1 p ¼0 s Weefeto[3 5]fodetailsonZp I ðs; containing the ationality in p s. The analytic natue of the global Igusa zeta function Z I ðs; is not so wellnown, and it has a natual bounday even fo a athe siple as is shown in [7]. Poposition 3.1. Let A ¼ Z C n1 C n be a finitely geneated abelian goup. Then its Igusa zeta function I ðs; A equals ðs Y pjn p v pðnð1þ s þð1 p s 1 v p ðn jþ1 1 p v pðn 1 n j l¼v p ðn j p ðþ j sl!; whee ðs denotes the Rieann zeta function, n ¼ n 1 n,andv p is the p-adic valuation. We also put n 0 ¼ 1. Poof. We copute I ðs; A ¼ 1 ð; n 1 ð; n s þ 1 l¼0 ðp l ;n 1 ðp l ;n p s l¼v p ðn v p ðn jþ1 1 l¼v p ðn j p v pðn 1 n p lð s p v pðn 1 n j p lð j p lð s! p v pðn p v pðn ð s 1 1 p s þ 1 v p ðn jþ1 1 p v pðn 1 n j l¼v p ðn j p ðþ j sl! This iplies the clai. Recall the Huwitz zeta function, whichfo 2 C with Reð > 0 is defined by Hu ðs; ¼ 1 1 ðn þ s n¼0 It extends to a eoophic function with a siple pole at s ¼ 1 of esidue 1. Poposition 3.2. The Igusa zeta function of a finitely geneated goup A as above can also be expessed as I ðs; A ¼n s n l ðl; n 1 ðl; n Hu ðs ; l n As an application one gets the identity 1 n ðl; n 1 ðl; n n l 1 ¼ Y 1 þ 1 1 p pjn Poof. We copute I ðs; A ¼ 1 ¼ n p v pðn 1 n j ð; n 1 ð; n s l ¼0 n ¼ n s l v p ðn jþ1 1 l¼v p ðn j ðl; n 1 ðl; n ðl þ n s ðl; n 1 ðl; n p ð j 1l! Hu ðs ; l n ; as claied. The application coes about by copaing the esidues of the two expessions at s ¼ þ 1.

5 142 A. DEITMAR, S. KOYAMA, andn. KUROKAWA [Vol. 84(A), Poposition 3.3. The Igusa zeta functions oftheaffineline,thepojectiveline,andgl n ae. I ðs; A 1 ¼ðsþðs 1,. I ðs; P 1 ¼2ðsþðs 1,. I ðs; GL n ¼n!ðs n. Poof. This follows fo the localization foula and the nown stuctue of A 1 ; P 1 ; GL n. Refeences [ 1 ] A. Deita, Schees ove F 1, in Nube fields and function fields two paallel wolds, , Pog. Math., 239, Bihäuse, Boston, Boston, MA, [ 2 ] A. Deita, Reas on zeta functions and K- theoy ove F 1, Poc. Japan Acad. Se. A Math. Sci. 82 (2006), no. 8, [ 3 ] F. Denef, Repot on Igusa s local zeta function (Séinaie Boubai Exp. No. 741), Astéisque, (1991), [ 4 ] J. Denef, The ationality of the Poincaé seies associated to the p-adic points on a vaiety, Invent. Math. 77 (1984), no. 1, [ 5 ] J. Igusa, An intoduction to the theoy of local zeta functions, Ae. Math. Soc., Povidence, RI, [ 6 ] N. Kuoawa, H. Ochiai and M. Waayaa, Absolute deivations and zeta functions, Doc. Math (2003), Exta Vol., (Electonic). [ 7 ] N. Kuoawa, Analyticity of Diichlet seies ove pie powes, in Analytic nube theoy (Toyo, 1988), , Lectue Notes in Math., 1434, Spinge, Belin. [ 8 ] N. Kuoawa, Zeta functions ove F 1,Poc.Japan Acad.Se.AMath.Sci.81 (2005), no. 10, [ 9 ] Y. Manin, Lectues on zeta functions and otives (accoding to Deninge and Kuoawa), Astéisque 228 (1995), [ 10 ] C. Soulé, Les vaiétés sulecopsàun éléent, Mosc. Math. J. 4 (2004), no. 1, , 312.