Zeta Functions of Regular Arithmetic Schemes at S = 0 Baptiste Morin

Zeta functions of regular arithmetic schemes at s = 0 Baptiste Morin

To cite this version:

Baptiste Morin. Zeta functions of regular arithmetic schemes at s = 0. Duke Mathematical Journal, Duke University Press, 2014, 163 (7), pp.1263-1336. 10.1215/00127094-2681387. hal-02484903

HAL Id: hal-02484903 https://hal.archives-ouvertes.fr/hal-02484903 Submitted on 19 Feb 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ZETA FUNCTIONS OF REGULAR ARITHMETIC SCHEMES AT s =0

BAPTISTE MORIN

Abstract. In [35] Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme at s =0in terms of Euler-Poincaré characteristics. Assuming the (conjectured) X ﬁnite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over Spec(Z).Inparticular,weobtain(unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the Zeta function ⇣( ,s) at s =0in terms of a perfect complex of abelian groups RW,c( , Z).Then X X we relate this conjecture to Soulé’s conjecture and to the Tamagawa number conjecture of Bloch-Kato, and deduce its validity in simple cases.

Mathematics Subject Classiﬁcation (2010): 14F20 14G10 11S40 11G40 19F27 · · · · Keywords: Weil-étale cohomology, special values of Zeta functions, motivic cohomology, regulators

1. Introduction In [35] Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme at s =0in terms of Euler-Poincaré characteristics. More precisely, we have the following Conjecture 1.1. (Lichtenbaum) On the category of separated schemes of finite type X! Spec(Z),thereexistsacohomologytheorygivenbyabeliangroupsHi ( , Z) and real vector W,c X spaces Hi ( , R˜) and Hi ( , R˜) such that the following holds. W X W,c X (1) The groups Hi ( , Z) are finitely generated and zero for i large. W,c X (2) The natural map from Z to R˜-coefficients induces isomorphisms i i H ( , Z) R H ( , R˜). W,c X ⌦ ' W,c X (3) There exists a canonical class ✓ H1 ( , R˜) such that cup-product with ✓ turns the 2 W X sequence ✓ i ✓ i+1 ✓ ... [ H ( , R˜) [ H ( , R˜) [ ... ! W,c X ! W,c X ! into a bounded acyclic complex of finite dimensional vector spaces.

B. Morin CNRS Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse cedex 9, France e-mail: [email protected]

1 2BAPTISTEMORIN

(4) The vanishing order of the zeta function ⇣( ,s) at s =0is given by the formula X i i ords=0⇣( ,s)= ( 1) i rank H ( , Z) X · · Z W,c X i 0 X (5) The leading coefficient ⇣ ( , 0) in the Taylor expansion of ⇣( ,s) at s =0is given ⇤ X X up to sign by 1 i ( 1)i Z (⇣⇤( , 0) )= det H ( , Z) · X Z W,c X i Z O2 i ⇠ i ( 1) where : R ( i Z detZHW,c( , Z) ) R is induced by (2) and (3). ! 2 X ⌦ Such a cohomology theoryN for smooth varieties over finite fields was defined in [34], but similar attempts to construct such cohomology groups for flat arithmetic schemes failed. In [35] Lichtenbaum gave the first construction for number rings. He defined a Weil-étale topology which bears the same relation to the usual étale topology as the Weil group does to the Galois group. Under a vanishing statement, he was able to show that his cohomology miraculously yields the value of Dedekind zeta functions at s =0. But this cohomology with coefficients in Z was then shown by Flach to be infinitely generated hence non-vanishing in even degrees i 4 [13]. Consequently, Lichtenbaum’s complex computing the cohomology with Z-coefficients needs to be artificially truncated in the case of number rings, and is not helpful for flat schemes of dimension greater than 1. However, the Weil-étale topology yields the expected cohomology with R˜-coefficients, and this fact extends to higher dimensional arithmetic schemes [14]. The first goal of this paper is to define the right Weil-étale cohomology with Z-coefficients for arithmetic schemes satisfying the following conjecture 1.2 (see Theorem 1.3 below), in order to state a precise version of Conjecture 1.1. We consider a regular scheme proper X over Spec(Z) of pure Krull dimension d, and we denote by Z(d) Bloch’s cycle complex. i Conjecture 1.2. (Lichtenbaum) The étale motivic cohomology groups H ( et, Z(d)) are finitely X generated for 0 i 2d.   Using some purity for the cycle complex Z(d) on the étale site (which is the key result of [20]) we construct in the appendix a class (Z) (see Definition 5.9) of separated schemes of L finite type over Spec(Z) satisfying Conjecture 1.2. This class (Z) contains any geometrically L cellular scheme over a number ring (see Definition 5.13), and includes the class A(Fq) of smooth projective varieties over Fq which can be constructed out of products of smooth projective curves by union, base extension, blow-ups and quasi-direct summands in the category of Chow motives (A(Fq) was introduced by Soulé in [43]). In order to state our first main result, we need to fix some notation. For a scheme X separated and of finite type over Spec(Z), we consider the quotient topological space := X1 (C)/GR where (C) is endowed with the complex topology and we set := ( , ). X X X X X1 We denote by the Artin-Verdier étale topos of which comes with a closed embedding X et X u : Sh( ) et in the sense of topos theory, where Sh( ) is the category of sheaves 1 X1 ! X X1 on the topological space (see [14] Section 4). The Weil-étale topos over the archimedean X1 place ,W is associated to the trivial action of the topological group R on the topological X1 space (see [14] Section 6). Our first main result is the following X1 Theorem 1.3. Let be a regular scheme proper over Spec(Z).Assumethat (Z),or X X2L more generally that the connected components of satisfy Conjecture 1.2. Then there exist X complexes RW ( , Z) and RW,c( , Z) such that the following properties hold. X X ZETA FUNCTIONS AT s =0 3

If has pure dimension d,thereisanexacttriangle • X (1) RHom(⌧ 0R( , Q(d)), Q[ 2d 2]) R( et, Z) RW ( , Z). X ! X ! X The complex RW ( , Z) is contravariantly functorial. • X There exists a unique morphism i⇤ : RW ( , Z) R( ,W , Z) which renders the • 1 X ! X1 following square commutative.

R( et, Z) / RW ( , Z) X X

u⇤ i⇤ 1 1 ✏ ✏ R( , Z) / R( ,W , Z) X1 X1 We define RW,c( , Z) so that there is an exact triangle X i⇤ RW,c( , Z) RW ( , Z) 1 R( ,W , Z). X ! X ! X1 The cohomology groups Hi ( , Z) and Hi ( , Z) are finitely generated for all i and • W X W,c X zero for i large. The cohomology groups Hi ( , Z) form an integral model for l-adic cohomology: for • W X any prime number l and any i Z there is a canonical isomorphism 2 i i H ( , Z) Zl H ( et, Zl). W X ⌦ ' X If has characteristic p then there is a canonical isomorphism in the derived category • X R( W , Z) ⇠ RW ( , Z) X ! X where R( W , Z) is the cohomology of the Weil-étale topos [34] and RW ( , Z) is X X the complex defined in this paper. Moreover the exact triangle (1) is isomorphic to Geisser’s triangle ([18] Corollary 5.2 for · = Z). G If = Spec( ) is the spectrum of a totally imaginary number ring then there is a • X OF canonical isomorphism in the derived category

RW ( , Z) ⇠ ⌧ 3R( W , Z) X !  X where ⌧ 3R( W , Z) is the truncation of Lichtenbaum’s complex [35] and RW ( , Z)  X X is the complex defined in this paper. Theorem 1.3 is proven in 2.9, 2.10, 2.11, 2.12, 2.13, 2.15 and 3.1. Notice that the same formalism is used to treat flat arithmetic schemes and schemes over finite fields (see [35] Question 1 in the Introduction). The basic idea behind the proof of Theorem 1.3 can be explained as follows. The Weil group is defined as an extension of the Galois group by the idèle class group corresponding to the fundamental class of class field theory (more precisely, as the limit of these group extensions). In this paper we use étale duality for arithmetic schemes rather than class field theory, in order to obtain a canonical "extension" of the étale Z-cohomology by the dual of motivic Q(d)-cohomology. More precisely, our Weil-étale complex is defined as the cone of a map

↵ : RHom(⌧ 0R( , Q(d)), Q[ 2d 2]) R( et, Z), X X ! X where ↵ is constructed out of étale duality. Here the scheme is of pure dimension d. This X X idea was suggested by works of Burns [8], Geisser [18] and the author [38]. The techniques involved in this paper rely on results due to Geisser and Levine on Bloch’s cycle complex (see [17], [20], [21], [22] and [32]). 4BAPTISTEMORIN

From now on, Conjecture 1.1 is understood as a list of expected properties for the complex RW,c( , Z) of Theorem 1.3. The second goal of this paper is to relate Conjecture 1.1 to X Soulé’s conjecture [42] and to the Tamagawa number conjecture of Bloch-Kato (in the formulation of Fontaine and Perrin-Riou, see [15] and [16]). To this aim, we need to assume the following conjecture, which is a special case of a natural refinement for arithmetic schemes of the classical conjecture of Beilinson relating motivic cohomology to Deligne cohomology. Let be a regular, proper and flat arithmetic scheme of pure dimension d. X Conjecture 1.4. (Beilinson) The Beilinson regulator 2d 1 i 2d 1 i H ( , Q(d))R H ( /R, R(d)) X ! D X is an isomorphism for i 1 and there is an exact sequence 2d 1 2d 1 0 0 H ( , Q(d))R H ( /R, R(d)) CH ( Q)R⇤ 0 ! X ! D X ! X ! Now we can state our second main result. If is defined over a number ring we set X OF F = F and = F . X X⌦OF XF X⌦OF Theorem 1.5. Assume that satisfies Conjectures 1.2 and 1.4. X Statements (1), (2) and (3) of Conjecture 1.1 hold for . • X Assume that is projective over Z.ThenStatement(4) of Conjecture 1.1, i.e. the • X identity i i ords=0⇣( ,s)= ( 1) i rank H ( , Z) X · · Z W,c X i 0 X is equivalent to Soulé’s Conjecture [42] for the vanishing order of ⇣( ,s) at s =0. X Assume that is smooth projective over a number ring F ,andthattherepresenta- • i X 1 i O tions H ( , Ql) of GF satisfies H (F, H ( , Ql)) = 0.ThenStatement(5) of XF,et f XF,et Conjecture 1.1, i.e. the identity 1 i ( 1)i Z (⇣⇤( , 0) )= det H ( , Z) · X Z W,c X i Z O2 2d 2 i is equivalent to the Tamagawa number conjecture [15] for the motive h ( )[ i]. i=0 XF The proof of the last statement of Theorem 1.5 was already given in [14],L assuming expected i ( 1)i properties of Weil-étale cohomology. This proof is based on the fact that i Z detZHW,c( , Z) 2 X provides the fundamental line (in the sense of [15]) with a canonical Z-structure. This result N shows that the Weil-étale point of view is compatible with the Tamagawa number conjecture of Bloch-Kato, answering a question of Lichtenbaum (see [35] Question 2 in the Introduction). We obtain examples of flat arithmetic schemes satisfying Conjecture 1.1. Theorem 1.6. For any number field F ,Conjecture1.1holdsfor =Spec( ). X OF Let be a smooth projective scheme over the number ring ( )= F ,whereF is an X OX X O abelian number field. Assume that admits a smooth cellular decomposition (see Definition XF 5.13) and that (Z) (see Definition 5.9). Then Conjecture 4.2 holds for . X2L X We refer to [11] for a proof of a dynamical system analogue of Conjecture 1.1.

Notations. We denote by Z the ring of integers, and by Q, Qp, R and C the fields of rational, p-adic, real and complex numbers respectively. An arithmetic scheme is a scheme ZETA FUNCTIONS AT s =0 5 which is separated and of finite type over Spec(Z). An arithmetic scheme is said to be X proper (respectively flat) if the map Spec(Z) is proper (respectively flat). X! For a field k, we choose a separable closure k/k¯ and we denote by Gk = Gal(k/k¯ ) the absolute Galois group of k. For a proper scheme over Spec(Z) we denote by := (C)/GR X X1 X the quotient topological space of (C) by G where (C) is given with the complex topology. X R X We consider the Artin-Verdier étale topos given with the open-closed decomposition of X et topoi

' : et et Sh( ):u X ! X X1 1 where is the usual étale topos of the scheme (i.e. the category of sheaves of sets on Xet X the small étale site of )andSh( ) is the category of sheaves of sets on the topological X X1 space (see [14] Section 4). For any abelian sheaf A on et and any n>0 the sheaf X1 X Rn' A is a 2-torsion sheaf concentrated on (R). In particular, if (R)= then ' R' . ⇤ X X ; ⇤ ' ⇤ For (C)= one has et = et.ForT = et, et or Sh( ), or more generally for any X ; X X X X X1 Grothendieck topos T , we denote by (T, ) the global section functor, by R(T, ) its total right derived functor and by Hi(T, ):=Hi(R(T, )) its cohomology. Let Z(n):=zn( , 2n ) be Bloch’s cycle complex (see [3], [17], [19], [31] and [32]), which ⇤ we consider as a complex of abelian sheaves on the small étale (or Zariski) site of .For X an abelian group A we deﬁne A(n) to be Z(n) A.NotethatZ(n) is a complex of ﬂat ⌦ sheaves, hence tensor product and derived tensor product with Z(n) agree. We denote by Hi( ,A(n)) the étale hypercohomology of A(n),andbyHi( ,A(n)) := Hi( ,A(n)) its Xet X XZar Zariski hypercohomology. For (R)= , we still denote by A(n)=' A(n) the push-forward X ; ⇤ of the cycle complex A(n) on ,andbyHi( ,A(n)) the étale hypercohomology of A(n) X et X et (in this paper A = Z, Z/mZ, Q or Q/Z). Notice that the complex of étale sheaves Z/mZ(n) n is not in general quasi-isomorphic to µm⌦ [0] (however, see [17] Theorem 1.2(4)). We denote by the derived category of the category of abelian groups. More generally, we D write (R) for the derived category of the category of R-modules, where R is a commutative D ring. An exact (i.e. distinguished) triangle in (R) is (somewhat abusively) depicted as D follows

C0 C C00 ! ! where C, C0 and C00 are objects of (R). For an object C of (R) we write C n (respectively D i i D  i C n) for the truncated complex so that H (C n)=H (C) for i n and H (C n)=0 i i  i   otherwise (respectively so that H (C n)=H (C) for i n and H (C n)=0otherwise). Let R be a principal ideal domain. For a finitely generated free R-module L of rank r,we r 1 set detR(L)= R L and detR(L) = HomR(detR(L),R). For a finitely generated R-module M, one may choose a resolution given by an exact sequence 0 L 1 L0 M 0, where V ! ! ! ! L 1 and L0 are finitely generated free R-modules. Then 1 detR(M):=detR(L0) R detR(L 1) ⌦ does not depend on the resolution. If C (R) is a complex such that Hi(C) is finitely gener- i 2D i ( 1)i ated for all i Z and H (C)=0for almost all i, we denote detR(C):= i Z detRH (C) . 2 2 For an abelian group A we write Ator and Adiv for the maximal torsion and the maximal N divisible subgroups of A respectively, and we set Acodiv = A/Adiv and Acotor = A/Ator.We denote by A and A the kernel and the cokernel of the map n : A A (multiplication by n n ! n). We write AD = Hom(A, Q/Z) for the Pontryagin dual of a finite abelian group A. 6BAPTISTEMORIN

Contents 1. Introduction 1 2. Weil-étale cohomology 6 2.1. The conjectures L( et,d) and L( et,d) 0 6 X X 2.2. The morphism ↵ 6 X 2.3. The complex RW ( , Z) 13 X 2.4. Relationship with Lichtenbaum’s definition over finite fields 18 2.5. Relationship with Lichtenbaum’s definition for number rings 22 3. Weil-étale Cohomology with compact support 22 3.1. Cohomology with R˜-coefficients 23 3.2. Cohomology with Z-coefficients 24 3.3. The conjecture B( ,d) and the regulator map 28 X 4. Zeta functions at s =0 30 4.1. The main Conjecture 31 4.2. Relation to Soulé’s conjecture 31 4.3. Relation to the Tamagawa number conjecture 32 4.4. Proven cases 40 4.5. Examples 40 5. Appendix 42 5.1. Motiviccohomologyofnumberrings 43 5.2. Smooth projective varieties over finite fields 45 5.3. The class (Z) 46 L 5.4. Geometrically cellular schemes 48 References 51

2. Weil-étale cohomology

2.1. The conjectures L( et,d) and L( et,d) 0. Let be a proper, regular and connected X X X arithmetic scheme of dimension d. The following conjecture is an étale version of (a special case of) the motivic Bass Conjecture (see [26] Conjecture 37). i Conjecture 2.1. L( et,d) For i 2d,theabeliangroupH ( et, Z(d)) is finitely generated. X  X In order to define the Weil-étale cohomology we shall consider schemes satisfying a weak version of the previous conjecture. i Conjecture 2.2. L( et,d) 0 For 0 i 2d,theabeliangroupH ( et, Z(d)) is finitely X   X generated. 2.2. The morphism ↵ . Let be a proper regular connected arithmetic scheme. The goal X X of this section is to construct (conditionally) a certain morphism ↵ in the derived category X of abelian groups , in order to define in Section 2.3 the Weil-étale complex RW ( , Z) as D X the cone of ↵ . X 2.2.1. The case (R)= . In this subsection, denotes a proper, regular and connected X ; X arithmetic scheme of dimension d such that (R) is empty. In particular ' is exact (since i i X i ⇤ (R)= ), hence H ( et, Z(d)) := H ( et,' Z(d)) H ( et, Z(d)). X ; X X ⇤ ' X ZETA FUNCTIONS AT s =0 7

Lemma 2.3. We have i H ( et, Z(d)) = 0 for i>2d +2 X Q/Z for i =2d +2 ' =0for i =2d +1 i H ( , Q(d)) for i<0. ' X Proof. For any positive integer n and any i Z,onehas 2 i 1 i 1 (2) H ( et, Z/nZ(d)) = Ext (Z/nZ, Z/nZ(d)) ,Z/nZ X X i (3) Ext (Z/nZ, Z(d)) ' X 2d+2 i D (4) H ( et, Z/nZ) ' X where (3) and (4) are given by ([20] Lemma 2.4) and ([20] Theorem 7.8) respectively, and 2d+2 i D 2d+2 i H ( et, Z/nZ) denotes the Pontryagin dual of the finite group H ( et, Z/nZ). X X The complex Z(d) consists of flat sheaves, hence tensor product and derived tensor product with Z(d) agree. Therefore, applying Z(d) L ( ) to the exact sequence 0 Z Q ⌦ ! ! ! Q/Z 0 yields an exact triangle: ! (5) Z(d) Q(d) Q/Z(d). ! ! i i Moreover, we have H ( et, Q(d)) H ( , Q(d)) for any i (see [17] Proposition 3.6) and X ' X Hi( , Q(d)) = 0 for any i>2d (see [31] Lemma 11.1). Hence (5) gives X i i 1 i 1 H ( et, Z(d)) H ( et, Q/Z(d)) lim H ( et, Z/nZ(d)) X ' X ' ! X for i 2d +2. The result for i 2d +2then follows from (4). By ([24] X.6.2), the scheme has l-cohomological dimension 2d +1for any prime number X i l (note that (R)= ), hence (4) shows that H ( et, Q/Z(d)) = 0 for i<0. The result for X ; i i X i<0 now follows from H ( et, Q(d)) H ( , Q(d)) and from the exact triangle (5). X ' X It remains to treat the case i =2d +1.Assumethat is flat over Z. Then we have (see X [28] Theorem 6.1(2))

2d d (6) H ( , Q(d)) CH ( ) Q =0 X ' X ⌦ i d hence H ( et, Q(d)) = 0 for i 2d. Here CH ( ) denotes the Chow group of cycles of X X codimension d.Weobtain 2d+1 2d H ( et, Z(d)) H ( et, Q/Z(d)) X ' X1 D lim (H ( et, Z/nZ) ) ' X ! D (lim Hom(⇡1( et), Z/nZ)) ' X ab D Hom(⇡1( et) , Z) ' X =0 b where the ﬁrst isomorphism (respectively the second) is given by (5) (respectively by (4)), and the last isomorphism follows from the fact that the abelian fundamental group ⇡ ( )ab is 1 X et ﬁnite (see [28] Theorem 9.10 and [48] Corollary 3). 8BAPTISTEMORIN

Assume now that is a smooth proper scheme over a finite field (hence = ). One has X X X 2d 1 D (7) H ( et, Q/Z(d)) lim (H ( et, Z/nZ) ) X ' X ! D (8) lim (Hom(⇡1( et), Z/nZ) ) ' X ! d (9) lim (CH ( ) Z/nZ) ' X ⌦ ! d (10) CH ( ) Q/Z ' X ⌦ where (7) is given by (4) while (9) follows from class field theory (see [48] Corollary 3). The exact triangle (5) yields an exact sequence 2d p 2d 2d+1 (11) H ( , Q(d)) H ( et, Q/Z(d)) H ( et, Z(d)) 0. X ! X ! X ! Moreover, the isomorphism (10) is the direct limit of the maps d 2d 2d 2d CH ( )n = H ( , Z(d))n H ( , Z/nZ(d)) H ( et, Z/nZ(d)). X X ! X ! X It follows that the map p in the sequence (11) can be identified with the morphism d d CH ( ) Q CH ( ) Q/Z X ⌦ ! X ⌦ induced by the surjection Q Q/Z. Hence p is surjective, since CHd( ) is finitely generated ! 2d+1 X of rank one ([28] Theorem 6.1), so that H ( et, Z(d)) = 0. X ⇤ j j Notation 2.4. We set H ( et, Z(d)) 0 := H (R( et, Z(d)) 0). X X Lemma 2.5. Assume that satisfies L( et,d) 0.Thenthenaturalmap X X i 2d+2 i H ( et, Z) Hom(H ( et, Z(d)) 0, Q/Z) X ! X is an isomorphism of abelian groups for i 1. Proof. The map of the lemma is given by the pairing i 2d+2 i 2d+2 H ( et, Z) Ext (Z, Z(d)) H ( et, Z(d)) Q/Z. X ⇥ X ! X ' 1 For i =1the result follows from Lemma 2.3, since one has H ( et, Z)=Homcont(⇡1( et,p), Z)= X X 0 because the fundamental group ⇡1( et,p) is profinite (hence compact). X i i The scheme is connected and normal, hence H ( et, Q)=H ( et, Q)=Q, 0 for i =0 X X X and i 1 respectively. We obtain i i 1 i 1 H ( et, Z)=H ( et, Q/Z)=lim H ( et, Z/nZ) X X ! X for i 2, since is quasi-compact and quasi-separated. For any positive integer n the X canonical map 2d+2 i i Ext (Z/nZ, Z(d)) H ( et, Z/nZ) Q/Z X ⇥ X ! is a perfect pairing of finite groups (see [20] Theorem 7.8). The short exact sequence 0 Z Z Z/nZ 0 ! ! ! ! yields an exact sequence j 1 j 1 j H ( et, Z(d)) H ( et, Z(d)) Ext (Z/nZ, Z(d)) X ! X ! X j j H ( et, Z(d)) H ( et, Z(d)). ! X ! X ZETA FUNCTIONS AT s =0 9

We obtain a short exact sequence j 1 j j 0 H ( et, Z(d))n Ext (Z/nZ, Z(d)) nH ( et, Z(d)) 0 ! X ! X ! X ! for any j. By left exactness of projective limits the sequence j 1 j j 0 lim H ( et, Z(d))n lim Ext (Z/nZ, Z(d)) lim nH ( et, Z(d)) ! X ! X ! X j j is exact. The module lim nH ( et, Z(d)) vanishes for 0 j 2d +1since H ( et, Z(d)) is X   X j assumed to be finitely generated for such j.WehavelimnH ( et, Z(d)) = 0 for j<0 since X j H ( et, Z(d)) is uniquely divisible for j<0 (see Lemma 2.3). This yields an isomorphism of X profinite groups j 1 j 2d+2 j D lim H ( et, Z(d))n ⇠ lim Ext (Z/nZ, Z(d)) ⇠ (lim H ( et, Z/nZ)) , X ! X ! X ! j where the last isomorphism follows from the duality above (and from the fact that Ext (Z/nZ, Z(d)) is finite for any n). This gives isomorphisms of torsion groups X 2d+2 (j 1) 2d+2 j DD H ( et, Z) ⇠ (lim H ( et, Z/nZ)) X ! ! X j 1 D j 1 ⇠ (lim H ( et, Z(d))n) ⇠ Hom(H ( et, Z(d)) 0, Q/Z) ! X ! X for any j 2d +1 (note that 2d +2 (j 1) 2 j 2d +1). The last isomorphism above  j ,  follows from the fact that H ( et, Z(d)) is finitely generated for 0 j 2d and uniquely X   divisible for j<0. Hence for any i 2 the natural map i 2d+2 i H ( et, Z) Hom(H ( et, Z(d)) 0, Q/Z) X ! X is an isomorphism. ⇤ Recall that an abelian group A is of cofinite type if A Hom (B,Q/Z) where B is a ' Z finitely generated abelian group. If A is of cofinite type then there exists an isomorphism r A (Q/Z) T , where r N and T is a finite abelian group. If satisfies L( et,d) 0 then i' 2 X X H ( et, Z) is of cofinite type for i 1 by Lemma 2.5. X Theorem 2.6. Assume that satisfies L( et,d) 0.Thereexistsauniquemorphismin X X D ↵ : RHom(R( , Q(d)) 0, Q[ 2d 2]) R( et, Z) X X ! X such that Hi(↵ ) is the following composite map X 2d+2 i 2d+2 i Hom(H ( , Q(d)) 0, Q) ⇠ Hom(H ( et, Z(d)) 0, Q) X ! X 2d+2 i i Hom(H ( et, Z(d)) 0, Q/Z) ⇠ H ( et, Z) ! X X for any i 2. Proof. In order to ease the notation, we set D := RHom(R( , Q(d)) 0, Q[ 2d 2]).We X X consider the spectral sequence (see [45] III, Section 4.6.10) p,q p i q i p+q (12) E2 = Ext (H (D ),H ( et, Z)) H (RHom(D , R( et, Z))). X X ) X X i Z Y2 An edge morphism from (12) gives a morphism 0 0 i i (13) H (RHom(D , R( et, Z))) Ext (H (D ),H ( et, Z)) X X ! X X i Z Y2 10 BAPTISTE MORIN such that the composite map 0 0 i i Hom (D , R( et, Z))) ⇠ H (RHom(D , R( et, Z))) Ext (H (D ),H ( et, Z)) D X X ! X X ! X X i Z Y2 is the obvious one. i By Lemma 2.5, H ( et, Z) is of cofinite type for i =0. It follows that, if both i =0 pX i q i 6 i q 6 and p =0, then Ext (H (D ),H ( et, Z)) = 0, since H (D ) is uniquely divisible. 6 p i q rX X r X Indeed, Ext (H (D ), (Q/Z) )=0for p 1, since (Q/Z) is divisible. Moreover, if T is X p i q p i q a finite abelian group of order n then Ext (H (D ),T)=0since Ext (H (D ),T) is i q X X both uniquely divisible (since H (D ) is) and killed by n (since T is). For i =0,onehas p 0 q 0 X 0 Ext (H (D ),H ( et, Z)) = 0 as long as p 2, because H ( et, Z)=Z has an injective X X p,q X resolution of length one. In particular, E =0for p =0, 1, hence the spectral sequence (12) 2 6 degenerates at E2. Moreover, one has 1, 1 1 i+1 i 1 1 0 E2 = Ext (H (D ),H ( et, Z)) = Ext (H (D ),H ( et, Z)) = 0 X X X X i Z Y2 since H1(D )=0. It follows that the edge morphism (13) is an isomorphism. X i i i i For i 1,anymapH (↵ ):H (D ) H ( et, Z) must be trivial since H (D )=0.For  X X ! X X any i 2, we consider the morphism i i 2d+2 i 2d+2 i ↵ : H (D )=Hom(H ( , Q(d)) 0, Q) ⇠ Hom(H ( et, Z(d)) 0, Q) X X X ! X 2d+2 i i Hom(H ( et, Z(d)) 0, Q/Z) ⇠ H ( et, Z) ! X X where the last isomorphism is given by Lemma 2.5. But (13) is an isomorphism, hence there exists a unique map in D ↵ : RHom(R( , Q(d)) 0, Q[ 2d 2]) R( et, Z) X X ! X i i such that H (↵ )=↵ . ⇤ X X 2.2.2. The general case. In this subsection, we allow (R) = .So denotes a proper, X 6 ; X regular and connected arithmetic scheme of dimension d. It is possible to define a dualizing complex Z(d)X on et, to prove Artin-Verdier duality in this setting and to define ↵ in the X X very same way as in Section 2.2.1. For the sake of conciseness, we shall use a somewhat trickier construction, which is based on Milne’s cohomology with compact support [37]. We denote by ˆ ˆ (14) Rc( et, ):=Rc(Spec(Z)et, Rf ) X F ⇤F ˆ Milne’s cohomology with compact support, where Rc(Spec(Z)et, ) is defined as in ([37] Section II.2) and f : Spec(Z) is the structure map. Milne’s definition yields a canonical ˆ X! map Rc(Spec(Z)et, ) R(Spec(Z)et, ) inducing ! ˆ (15) Rc( et, Z) R( et, Z). X ! X We need the following lemma. ˆ c Lemma 2.7. There is a morphism Rc( et, Z) R( et, Z) satisfying the following prop- X ! X erties. The composite map ˆ c '⇤ Rc( et, Z) R( et, Z) R( et, Z) X ! X ! X coincides with (15), Hi(c) is an isomorphism for i large and Hi(c) has finite 2-torsion kernel and cokernel for any i Z. 2 ZETA FUNCTIONS AT s =0 11

Proof. Recall from ([14] Section 4) that there is an isomorphism of functors u⇤ ' ⇡ ↵⇤ 1 ⇤ ' ⇤ where ↵ : Sh(GR, (C)) et and ⇡ : Sh(GR, (C)) Sh( ) are the canonical maps X !X X ! X1 while u : Sh( ) et and ' : et et are complementary closed and open embeddings 1 X1 ! X X ! X (see [14] Section 4). Here Sh(GR, (C)) (respectively Sh( )) denotes the topos of GR- X X1 equivariant sheaves of sets on (C) (respectively the topos of sheaves on ). It is easy to X X1 see that ↵⇤ sends injective abelian sheaves to ⇡ -acyclics ones. We obtain ⇤ u⇤ R' R(u⇤ ' ) R(⇡ ↵⇤) (R⇡ )↵⇤. 1 ⇤ ' 1 ⇤ ' ⇤ ' ⇤ Consider the map >0 R' Z u , u⇤ R' Z u , R⇡ Z u , ⌧ R⇡ Z. ⇤ ! 1 ⇤ 1 ⇤ ' 1 ⇤ ⇤ ! 1 ⇤ ⇤ 0 Since ⌧  R⇡ Z is the constant sheaf Z = u⇤ Z put in degree 0, we obtain an exact triangle ⇤ 1 >0 (16) ZX R' Z u , ⌧ R⇡ Z. ! ⇤ ! 1 ⇤ ⇤ >0 where ZX denotes the constant sheaf Z on et. Moreover, we have R( et,u , ⌧ R⇡ Z) >0 >0 X >0X 1 ⇤ ⇤ ' R( ,⌧ R⇡ Z) R( (R),⌧ R⇡ Z) because u , is exact and ⌧ R⇡ Z is concentrated X1 ⇤ ' X ⇤ 1 ⇤ ⇤ on (R) . Therefore, applying R( et, ) to (16), we get an exact triangle X ⇢X1 X >0 (17) R( et, Z) R( et, Z) R( (R),⌧ R⇡ Z). X ! X ! X ⇤ On the other hand there is an exact triangle ˆ ˆ (18) Rc( et, Z) R( et, Z) R(G , (R), Z) X ! X ! R X where Rˆ(G , (R), ):=Rˆ(G , (R( (R), )) denotes equivariant Tate cohomology (which R X R X was introduced in [44]). Notice that the canonical map Rˆ(G , (C), Z) Rˆ(G , (R), Z) R X ! R X is an isomorphism [44]. ˆ >0 We shall define below a canonical map c : R(GR, (R), Z) R( (R),⌧ R⇡ Z) such 1 X ! X ⇤ that Hi(c ) is an isomorphism for i large and Hi(c ) has finite 2-torsion kernel and cokernel 1 1 for any i Z. The map c is compatible (in the obvious sense) with the identity map of 2 1 ˆ R( et, Z), hence there exists a morphism c : Rc( et, Z) R( et, Z) such that (c, Id, c ) X X ! X 1 is a morphism of exact triangles from (18) to (17). Hence the result will follow. It remains to define c . We denote by R(GR, (R), ):=R(GR, (R( (R), )) usual 1 X X equivariant cohomology. We have isomorphisms ˆ (19) R(G , (R), Z)>d R(G , (R), Z)>d R X ' R X (20) R( (R),R⇡ Z)>d ' X ⇤ (21) R( (R), (R⇡ Z)>0)>d ' X ⇤ (22) R( (R), R(G , Z)>0)>d ' X R ˆ (23) R( (R), R(G , Z)>0)>d ' X R ˆ (24) R( (R), R(G , Z))>d ' X R where we consider Rˆ(G , Z) and R(G , Z) as complexes of constant sheaves on (R). R R X The canonical map R(G , (R), Z) Rˆ(G , (R), Z) and the corresponding morphism of R X ! R X hypercohomology spectral sequences give the isomorphism (19), since (R) is of topological X dimension d. Similarly (21) and (24) can be deduced from the corresponding morphisms of 12 BAPTISTE MORIN spectral sequences. Finally, (22) is given by (R⇡ Z) (R) R(GR, Z).SinceGR is cyclic, we ⇤ |X ' have an isomorphism Rˆ(G , Z) ⇠ Rˆ(G , Z)[2] in .ApplyingR( (R), ) we obtain R ! R D X R( (R), Rˆ(G , Z)) ⇠ R( (R), Rˆ(G , Z))[2]. X R ! X R The same is true for equivariant Tate cohomology: we have an isomorphism Rˆ(G , (R), Z) ⇠ Rˆ(G , (R), Z)[2]. R X ! R X Hence for any integer k 1, (24) induces an isomorphism ˆ ˆ R(GR, (R), Z)>d 2k R( (R), R(GR, Z))>d 2k. X ' X Taking k large enough, we obtain a canonical isomorphism ˆ ˆ (25) R(G , (R), Z)>0 R( (R), R(G , Z))>0. R X ' X R ˆ ˆ The map R(G , Z) R(G , Z)>0 induces R ! R ˆ ˆ (26) R( (R), R(G , Z)) R( (R), R(G , Z)>0). X R ! X R ˆ Since R( (R), R(G , Z)>0) is acyclic in degrees 0, (26) is induced by a unique map X R  ˆ ˆ (27) R( (R), R(G , Z))>0 R( (R), R(G , Z)>0) X R ! X R and c is defined as the composition 1 ˆ ˆ (25) ˆ R(G , (R), Z) R(G , (R), Z)>0 R( (R), R(G , Z))>0 R X ! R X ! X R (27) ˆ >0 R( (R), R(GR, Z)>0) R( (R),⌧ R⇡ Z). ! X ' X ⇤ ⇤

If (R)= , then the maps c and '⇤ of Lemma 2.7 are isomorphisms in (this follows X ; D from (18) and from the fact that ' is exact in this case). Therefore, the following result ⇤ generalizes Theorem 2.6.

Proposition 2.8. Assume that satisﬁes L( et,d) 0.Thenthereexistsauniquemorphism X X in D ↵ : RHom(R( , Q(d)) 0, Q[ 2d 2]) R( et, Z) X X ! X such that Hi(↵ ) is the following composite map X 2d+2 i 2d+2 i Hom(H ( , Q(d)) 0, Q) ⇠ Hom(H ( et, Z(d)) 0, Q) X ! X i 2d+2 i ˆ i H (c) i Hom(H ( et, Z(d)) 0, Q/Z) ⇠ Hc( et, Z) H ( et, Z) ! X X ! X for any i 2. Proof. We consider the composite map p RHom (Z, Z(d)) RHomSpec( )(Rf Z, Rf Z(d)) RHomSpec( )(Rf Z, Z(1)[ 2d + 2]) X ! Z ⇤ ⇤ ! Z ⇤ q ˆ ˆ RHom (Rc(Spec(Z)et, Rf Z), Q/Z[ 2d 2]) = RHom (Rc( et, Z), Q/Z[ 2d 2]) ! D ⇤ D X where p is induced by the push-forward map of ([20] Corollary 7.2 (b)) and q is given by 1-dimensional Artin-Verdier duality (see [37] II.2.6) since Z(1) Gm[ 1] (see [20] Lemma ' 7.4). This composite map induces ˆ R( et, Z(d)) 0 RHom(Rc( et, Z), Q/Z[ 2d 2]) X ! X ZETA FUNCTIONS AT s =0 13

ˆ ˆ i since Rc( et, Z) is acyclic in degrees > 2d +2. Indeed, H (Spec(Z)et,F)=0for any i>3 X c and any torsion abelian sheaf F (see [37] Theorem II.3.1(b)); hence by proper base change the ˆ i hypercohomology spectral sequence associated to (14) gives H ( et, Q/Z)=0for i>2d +1. c X By adjunction, we obtain the product map L ˆ (28) R( et, Z(d)) 0 Rc( et, Z) Q/Z[ 2d 2]. X ⌦Z X ! For any positive integer n and any i Z, the canonical map 2 2d+2 i ˆ i (29) Ext (Z/nZ, Z(d)) Hc( et, Z/nZ) Q/Z X ⇥ X ! is a perfect pairing of finite groups (see [20] Theorem 7.8). Note that (29) is compatible with (28). The arguments of the proof of Lemma 2.5 show that the morphism ˆ i 2d+2 i (30) Hc( et, Z) ⇠ Hom(H ( et, Z(d)) 0, Q/Z), X ! X induced by (28), is an isomorphism of abelian groups for i 2. The argument of the proof of Theorem 2.6 provides us with a unique map ˆ ↵˜ : RHom(R( , Q(d)) 0, Q[ 2d 2]) Rc( et, Z) X X ! X such that Hi(˜↵ ) is the following composite map X 2d+2 i 2d+2 i Hom(H ( , Q(d)) 0, Q) ⇠ Hom(H ( et, Z(d)) 0, Q) X ! X 2d+2 i ˆ i Hom(H ( et, Z(d)) 0, Q/Z) ⇠ Hc( et, Z) ! X X for any i 2. Then we define ↵ as the composite map X ↵˜ ˆ c RHom(R( , Q(d)) 0, Q[ ]) X Rc( et, Z) R( et, Z) X ! X ! X where c is the map of Lemma 2.7. It remains to show that ↵ does not depend on the X choice of c (note that c is not uniquely defined). The map ' c is well defined (since it ⇤ coincides with (15) by Lemma 2.7), hence so is '⇤ ↵ : D R( et, Z), where D := X X ! X X RHom(R( , Q(d)) 0, Q[ 2d 2]). By (17) and using the fact that D is a complex of X X Q-vector spaces while R( (R),⌧>0R⇡ Z) is a complex of Z/2Z-vector spaces, we see that X ⇤ composition with '⇤ induces an isomorphism

'⇤ : Hom (D , R( et, Z)) ⇠ Hom (D , R( et, Z)) D X X ! D X X so that ↵ does not depend on the choice of c. The result follows. ⇤ X

2.3. The complex RW ( , Z). Throughout this section denotes a proper regular con- X X nected arithmetic scheme of dimension d satisfying L( et,d) 0. X Definition 2.9. There exists an exact triangle ↵ RHom(R( , Q(d)) 0, Q[ 2d 2]) X R( et, Z) RW ( , Z) X ! X ! X well defined up to a unique isomorphism in . We define D i i H ( , Z):=H (RW ( , Z)). W X X The existence of such an exact triangle follows from axiom TR1 of triangulated categories. Its uniqueness is given by Theorem 2.11 and can be stated as follows. If we have two objects 14 BAPTISTE MORIN

RW ( , Z) and RW ( , Z)0 given with exact triangles as in Definition 2.9, then there exists X X a unique map f : RW ( , Z) RW ( , Z)0 in which yields a morphism of exact triangles X ! X D ↵ RHom(R( , Q(d)) 0, Q[ 2d 2]) X / R( et, Z) / RW ( , Z) X X X Id Id f ✏ ↵ ✏ ✏ RHom(R( , Q(d)) 0, Q[ 2d 2]) X / R( et, Z) / RW ( , Z)0 X X X Proposition 2.10. The following assertions are true. (1) The group Hi ( , Z) is finitely generated for any i Z.OnehasHi ( , Z)=0for W X 2 W X i<0, H0 ( , Z)=Z and Hi ( , Z)=0for i large. W X W X (2) If (R)= then there is an exact sequence X ; i i 2d+2 (i+1) 0 H ( et, Z)codiv HW ( , Z) HomZ(H ( et, Z(d)) 0, Z) 0 ! X ! X ! X ! for any i Z. 2 (3) For any and any i Z,thereisanexactsequence X 2 i i i+1 0 H ( et, Z)codiv HW ( , Z) Ker(H (↵ )) 0 ! X ! X ! X ! i+1 2d+2 (i+1) where Ker(H (↵ )) HomZ(H ( et, Q(d)) 0, Q) is a Z-lattice. X ✓ X Proof. In order to ease the notation we set := 2d +2. The exact triangle of Defintion 2.9 yields a long exact sequence i i i ... Hom(H ( , Q(d)) 0, Q) H ( et, Z) HW ( , Z) ! X ! X ! X (i+1) i+1 Hom(H ( , Q(d)) 0, Q) H ( et, Z) ... ! X ! X ! and a short exact sequence i i i+1 (31) 0 CokerH (↵ ) HW ( , Z) KerH (↵ ) 0. ! X ! X ! X ! Assume for the moment that (R)= .Fori 1,themorphism X ; i i i H (↵ ):Hom(H ( , Q(d)) 0, Q) H ( et, Z) X X ! X is the following composite map: i i Hom(H ( , Q(d)) 0, Q) ⇠ Hom(H ( et, Z(d)) 0, Q) X ! X i i Hom(H ( et, Z(d)) 0, Q/Z) H ( et, Z) ! X ' X i Since H ( et, Z(d)) 0 is assumed to be finitely generated, the image of the morphism i X i i H (↵ ) is the maximal divisible subgroup of H ( et, Z), which we denote by H ( et, Z)div. X X X For i =0one has 0 0 H (↵ ):0=Hom(H ( , Q(d)), Q) H ( et, Z)=Z X X ! X i i and H ( , Q(d)) = H ( et, Z)=0for i<0.Weobtain X X i i i i CokerH (↵ )=H ( et, Z)/H ( et, Z)div =: H ( et, Z)codiv X X X X for any i Z. By definition of ↵ , the kernel of Hi(↵ ) can be identified with the kernel of 2 i X i X the map Hom(H ( et, Z(d)) 0, Q) Hom(H ( et, Z(d)) 0, Q/Z). In other words, one X ! X has i i KerH (↵ ) Hom(H ( et, Z(d)) 0, Z). X ' X ZETA FUNCTIONS AT s =0 15

The exact sequence (31) therefore takes the form i i 2d+2 (i+1) 0 H ( et, Z)codiv HW ( , Z) HomZ(H ( et, Z(d)) 0, Z) 0. ! X ! X ! X ! We obtain the second claim of the proposition. Assume now that (R) = . Then for i 2,themorphismHi(˜↵ ) is the map X 6 ; X i i Hom(H ( , Q(d)) 0, Q) ⇠ Hom(H ( et, Z(d)) 0, Q) X ! X i ˆ i Hom(H ( et, Z(d)) 0, Q/Z) Hc( et, Z) ! X ' X and Hi(↵ ) is defined as the composite map X Hi(˜↵ ) i i X ˆ i i H (↵ ):Hom(H ( , Q(d)) 0, Q) Hc( et, Z) H ( et, Z). X X ! X ! X i ˆ i The same argument as in the case (R)= shows Im(H (˜↵ )) = Hc( et, Z)div. Let us ˆ i i X ; X X show that h : Hc( et, Z)div H ( et, Z)div is surjective for any i Z. This is obvious for i X ! X ˆ i 2 i i 1, since H ( et, Z)div =0for i 1.Fori 2,themapsHc( et, Z) H ( et, Z) and ˆi Xˆ i  ˆ iX ! X Hc( et, Z)div Hc( et, Z) have both finite cokernels (note that Hc( et, Z) is of cofinite type X ! X ˆ i X i for i 2 by (30)). It follows immediately that the cokernel of h : H ( et, Z)div H ( et, Z)div c X ! X is of finite exponent and divisible, hence vanishes. So h is indeed surjective. We obtain i i Im(H (↵ )) = H ( et, Z)div hence X X i i CokerH (↵ )=H ( et, Z)codiv X X for any i Z. We now compute KerHi(↵ ). By definition of ↵˜ , the kernel of Hi(˜↵ ) can be 2 X i X i X identified with the kernel of the map Hom(H ( et, Z(d)) 0, Q) Hom(H ( et, Z(d)) 0, Q/Z). X ! X Hence one has i i (32) KerH (˜↵ ) Hom(H ( et, Z(d)) 0, Z). X ' X ˆ i i We denote by Ti the kernel of the surjective map h : H ( et, Z)div H ( et, Z)div so that c X ! X there is an exact sequence ˆ i h i (33) 0 Ti H ( et, Z)div H ( et, Z)div 0. ! ! c X ! X ! Consider the obvious injective map f : KerHi(˜↵ ) KerHi(↵ ), and the following morphism X ! X of exact sequences:

Hi(˜↵ ) i i X ˆ i 0 / KerH (˜↵ ) / Hom(H ( , Q(d)) 0, Q) / Hc( et, Z)div / 0 X X X f = h ✏ ✏ Hi(↵ ) ✏ i i i 0 / KerH (↵ ) / Hom(H ( , Q(d)) 0, Q) X / H ( et, Z)div / 0 X X X The snake lemma then yields an isomorphism T = Ker(h) Coker(f).Using(32),weobtain i ' an exact sequence i i 0 Hom(H ( et, Z(d)) 0, Z) KerH (↵ ) Ti 0. ! X ! X ! ! Moreover, Ti is finite and 2-torsion. We obtain the third claim of the proposition. It follows from the third claim that Hi ( , Z) is finitely generated for any i Z. Indeed, W X 2 KerHi+1(↵ ) is finitely generated since it is a Z-lattice in a finite dimensional Q-vector space. Xˆ i Moreover, Hc( et, Z) is of cofinite type for i 2 and finitely generated for i 1. Hence ˆ i X i  H ( et, Z)codiv is finite for any i Z and so is H ( et, Z)codiv by (33). c X 2 X 16 BAPTISTE MORIN

i 0 Moreover, one has H ( et, Z)=0for i<0 and H ( et, Z)=Z. Moreover one has i ˆ i X ˆ i X H ( et, Z) Hc( et, Z) for i large (by Lemma 2.7) and Hc( et, Z)=0for i>2d +2 by (30), X i ' X i+1 X 2d+2 (i+1) hence H ( et, Z)=0for i large. Finally, KerH (↵ ) Hom(H ( , Q(d)) 0, Q)= X X ⇢ X 0 for i 0 as well as for i 2d +2. The ﬁrst claim of the proposition follows.  ⇤ Theorem 2.11. Let f : be a morphism of relative dimension c between proper X!Y regular connected arithmetic schemes, such that L( et,d ) 0 and L( et,d ) 0 hold, where X X Y Y d (respectively d )denotesthedimensionof (respectively of ). We choose complexes X Y X Y RW ( , Z) and RW ( , Z) as in Deﬁnition 2.9. Assume either that c 0 or that L( et,d ) X Y X X holds. Then there exists a unique map in D f ⇤ : RW ( , Z) RW ( , Z) W Y ! X which sits in a morphism of exact triangles

RHom(R( , Q(d )) 0, Q[ 2d 2]) / R( et, Z) / RW ( , Z) X XO X XO O X

fet⇤ fW⇤

RHom(R( , Q(d )) 0, Q[ 2d 2]) / R( et, Z) / RW ( , Z) Y Y Y Y Y In particular, if satisfies L( et,d ) 0 then RW ( , Z) is well defined up to a unique X X X X isomorphism in . D Proof. Let and be connected, proper and regular arithmetic schemes of dimension d and X Y X d respectively. We set := 2d +2and := 2d +2. We choose complexes RW ( , Z) Y X X Y Y X and RW ( , Z) as in Definition 2.9. Let f : be a morphism of relative dimension Y X!Y c = d d . The morphism f is proper and the map X Y n n c z ( , ) z ( , ) X ⇤ ! Y ⇤ induces a morphism f Q(d ) Q(d ))[ 2c] ⇤ X ! Y of complexes of abelian Zariski sheaves on . We need to see that Y (34) f Q(d ) Rf Q(d ). ⇤ X ' ⇤ X Localizing over the base, it is enough to show this fact for f a proper map over a discrete valuation ring. Over a discrete valuation ring, Zariski hypercohomology of the cycle complex coincides with its cohomology as a complex of abelian group (see [32] and [17] Theorem 3.2). This yields (34) and a morphism of complexes R( , Q(d )) R( ,f Q(d )) R( , Q(d ))[ 2c]. X X ' Y ⇤ X ! Y Y If c 0 this induces a morphism R( , Q(d )) 0 R( , Q(d )) 0[ 2c]. X X ! Y Y i If c<0 then L( et,d ) holds by assumption. It follows that H ( , Q(d )) = 0 for i<0. Xi X i X X Indeed, we have H ( et, Z(d)) H ( , Q(d )) for i<0 (by Lemma 2.3 for (R)= and by X ' X X i i X ; (29) for the general case). Hence for i<0, H ( , Q(d )) H ( et, Z(d)) is both uniquely X X ' X ZETA FUNCTIONS AT s =0 17 divisible and finitely generated by L( et,d ), hence must be trivial. So we may consider the X X map

R( , Q(d )) 0 ⇠ R( , Q(d )) R( , Q(d ))[ 2c] R( , Q(d )) 0[ 2c]. X X X X ! Y Y ! Y Y In both cases we get a morphism

RHom(R( , Q(d )) 0, Q[ ]) RHom(R( , Q(d )) 0, Q[ ]) Y Y Y ! X X X such that the following square is commutative: ↵ RHom(R( , Q(d )) 0, Q[ ]) Y / R( et, Z) Y Y Y Y

✏ ↵ ✏ RHom(R( , Q(d )) 0, Q[ ]) X / R( et, Z) X X X X Showing that the diagram above is indeed commutative is tedious but straightforward, using the push-forward map deﬁned in [20] Corollary 7.2 (b). Hence there exists a morphism

f ⇤ : RW ( , Z) RW ( , Z) W Y ! X inducing a morphism of exact triangles. We claim that such a morphism fW⇤ is unique. In order to ease the notations, we set

D := RHom(R( , Q(d )) 0, Q[ ]) and D := RHom(R( , Q(d )) 0, Q[ ]). X X X X Y Y Y Y The complexes RW ( , Z) and RW ( , Z) are both perfect complexes of abelian groups, X Y since they are bounded complexes with ﬁnitely generated cohomology groups. Applying the functor Hom ( , RW ( , Z)) to the exact triangle D X D R( et, Z) RW ( , Z) D [1] Y ! Y ! Y ! Y we obtain an exact sequence of abelian groups:

Hom (D [1], RW ( , Z)) Hom (RW ( , Z), RW ( , Z)) D Y X ! D Y X Hom (R( et, Z), RW ( , Z)). ! D Y X On the one hand, Hom (D [1], RW ( , Z)) is uniquely divisible since D [1] is a complex D Y X Y of Q-vector spaces. On the other hand, the abelian group Hom (RW ( , Z), RW ( , Z)) is D Y X finitely generated as it follows from the spectral sequence p i q+i p+q Ext (H ( , Z),H ( , Z)) H (RHom(RW ( , Z), RW ( , Z))) W Y W X ) Y X i Z Y2 since RW ( , Z) and RW ( , Z) are both perfect. Hence the morphism X Y Hom (RW ( , Z), RW ( , Z)) Hom (R( et, Z), RW ( , Z)) D Y X ! D Y X is injective, which implies the uniqueness of the morphism fW⇤ sitting in the morphism of exact triangles of the theorem. Assume now that satisfies L( et,d ) 0, and let RW ( , Z) and RW ( , Z)0 be two X X X X X complexes given with exact triangles as in Definition 2.9. Then the identity map Id : X!X induces a unique isomorphism RW ( , Z) RW ( , Z)0 in . X ' X D ⇤ i i ⌫ We denote by Hcont( et, Zl):=H (RlimR( et, Z/l Z)) continuous l-adic cohomology. X X 18 BAPTISTE MORIN

Corollary 2.12. Let be a proper regular connected arithmetic scheme of dimension d sat- X isfying L( et,d) 0.Foranyprimenumberl and any i Z,thereisanisomorphism X 2 i i H ( , Z) Zl H ( et, Zl). W X ⌦ ' cont X Proof. Consider the exact triangle

RHom(R( , Q(d)) 0, Q[ 2d 2]) R( et, Z) RW ( , Z) X ! X ! X Let n be any positive integer. Applying the functor L Z/nZ we obtain an isomorphism ⌦Z L L R( et, Z/nZ) R( et, Z) Z/nZ ⇠ RW ( , Z) Z/nZ X ' X ⌦Z ! X ⌦Z L since RHom(R( , Q(d )) 0, Q[ 2d 2]) Z/nZ 0. Let l be a prime number and let ⌫ X X ⌦Z ' be a positive integer. We obtain a short exact sequence i i ⌫ i+1 (35) 0 H ( , Z)l⌫ H ( et, Z/l Z) l⌫ H ( , Z) 0 ! W X ! X ! W X ! for any i Z. By left exactness of projective limits we get 2 i i ⌫ i+1 0 lim HW ( , Z)l⌫ lim H ( et, Z/l Z) lim l⌫ HW ( , Z). ! X ! X ! X i+1 i+1 But lim l⌫ HW ( , Z)=0since HW ( , Z) is finitely generated. Moreover, we have X X i i ⌫ Hcont( et, Zl) lim H ( et, Z/l Z) X ' X i ⌫ since H ( et, Z/l Z) is finite by (35). X ⇤ 2.4. Relationship with Lichtenbaum’s definition over finite fields. Let Y be a scheme of finite type over a finite field k. We denote by Gk and Wk the Galois group and the Weil sm group of k respectively. The (small) Weil-étale topos YW is the category of Wk-equivariant sheaves of sets on the étale site of Y k. The big Weil-étale topos is defined [14] as the fiber ⌦k product Y := Y ( ) Y sm B W et Spec Z W et BG Wk ⇥Spec(Z)et ' ⇥ k Bsm B G where Gk (resp. Wk ) denotes the small classifying topos of k (resp. the big classifying sm topos of Wk). The topoi YW and YW are cohomologically equivalent (see [14] Corollary 2). Therefore, by [18] one has an exact triangle in the derived category of abelian sheaves on Yet Z R Z Q[ 1] Z[1] ! ⇤ ! ! where : Y Y is the first projection. Applying R(Y , ) and rotating, we get W ! et et aY R(Yet, Q[ 2]) R(Yet, Z) R(YW , Z) R(Yet, Q[ 2])[1]. ! ! ! Theorem 2.13. Let Y be a d-dimensional connected projective smooth scheme over k satisfying L(Yet,d) 0.Thenthereisanisomorphismin D (36) R(YW , Z) ⇠ RW (Y,Z) ! where R(YW , Z) is the cohomology of the Weil-étale topos and RW (Y,Z) is the complex defined in this paper. Moreover, the exact triangle of Definition 2.9 is isomorphic to Geisser’s triangle ([18] Corollary 5.2). ZETA FUNCTIONS AT s =0 19

Proof. We shall define a commutative square in D aY R(Yet, Q[ 2]) / R(Yet, Z) Id ' ✏ ↵Y ✏ RHom(R(Y,Q(d)) 0, Q[ 2d 2]) / R(Yet, Z) where the vertical maps are isomorphisms. The existence of the isomorphism (36) and the compatibility with Geisser’s triangle ([18] Corollary 5.2 for · = Z) will immediately follow. G Moreover, the uniqueness of (36) will follow from the argument of the proof of Theorem 2.11. One is therefore reduced to define the commutative square above. Replacing k with a finite extension if necessary, one may suppose that Y is geometrically connected over k.Onehas H2d(Y,Q(d)) = CHd(Y ) Q ([28] Theorem 6.1) and Hi(Y,Q(d)) = 0 for i>2d. This Q ' yields a map R(Y,Q(d)) Q[ 2d]. The morphism ! RHomY (Q, Q(d)) RHom(R(Y,Q), R(Y,Q(d))) RHom(R(Y,Q), Q[ 2d]) ! ! induces a morphism

(37) R(Yet, Q) R(Y,Q) RHom(R(Y,Q(d)) 0, Q[ 2d]). ' ! i It follows from Conjecture L(Yet,d) 0, Lemma 2.5 and from the fact that H (Yet, Z) is ﬁnite i for i =0, 2,thatthegroupH (Yet, Z(d)) 0 is ﬁnite for i =2d, 2d+2 and torsion for i =2d+2. 6 6 6 It follows easily that (37) is a quasi-isomorphism. It remains to check the commutativity of the square above. The complex

DY := RHom(R(Y,Q(d)) 0, Q[ 2d 2]) R(Yet, Q[ 2]) ' is concentrated in degree 2. It follows that both aY and ↵Y uniquely factor through the truncated complex R(Yet, Z) 2. It is therefore enough to show that the square 

R(Yet, Q[ 2]) / R(Yet, Z) 2  Id ✏ ✏ RHom((Y,Q(d)) 0, Q[ 2d 2]) / R(Yet, Z) 2  commutes. The exact triangle D Z[0] R(Yet, Z) 2 ⇡1(Yet) [ 2] Z[1] !  ! ! induces an exact sequence of abelian groups D Hom (DY , Z[0]) Hom (DY , R(Yet, Z) 2) Hom (DY ,⇡1(Yet) [ 2]) D ! D  ! D which shows that the map D Hom (DY , R(Yet, Z) 2) Hom (DY ,⇡1(Yet) [ 2]) D  ! D is injective. Indeed, Z[0] [Q Q/Z] has an injective resolution of length one and DY is ' ! concentrated in degree 2, hence Hom (DY , Z[0]) = 0. In view of the quasi-isomorphism D 2d 0 DY Hom(H (Y,Q(d)), Q)[ 2] H (Y,Q)[ 2], ' ' 20 BAPTISTE MORIN one is reduced to show the commutativity of the following square (of abelian groups):

d0,1 0 2 2 H (Yet, Q) / H (Yet, Z)

Id ✏ H2(↵ ) ✏ 2d Y 2 Hom(H (Y,Q(d)), Q) / H (Yet, Z)

2 By construction the map H (↵Y ) is the following composition 2d d d D 2 Hom(H (Y,Q(d)), Q) Hom(CH (Y ), Q) Hom(CH (Y ), Q/Z) ⇠ ⇡1(Yet) H (Yet, Z) ' ! ' where the isomorphism CHd(Y )D ⇠ ⇡ (Y )D is the dual of the map CHd(Y ) ⇡ (Y )ab 1 et ! 1 et given by class field theory, which is injective with dense image (see [48] Corollary 3). The top 0,1 horizontal map in the last commutative square is the differential d2 of the spectral sequence i j i+j H (Yet,R ( )Z) H (YW , Z). ⇤ ) There is a canonical isomorphism 0 1 H (Yet,R ( )Z)=lim Hom(Wk , Z)=Hom(Wk, Q), k0/k 0 ⇤ ! k0/k runs over the finite extensions of k, as it follows from the isomorphism of pro-discrete groups ⇡ (Y ,p) ⇡ (Y ,p) W , which is valid for any Y connected étale over Y . Then 1 W0 ' 1 et0 ⇥Gk k 0 the left vertical map in the last square above is the map d deg⇤ : Hom(Wk, Q) Hom(CH (Y ), Q) ! induced by the degree map d deg : CH (Y ) Z Wk ! ' 0,1 and the differential d2 is the following map: D Hom(Wk, Q)=Hom(⇡1(YW ,p), Q) Homc(⇡1(YW ,p), Q/Z) ⇡1(Yet,p) ! ' where Hom ( , ) denotes the group of continuous morphisms. One is therefore reduced to c observe that the square d0,1 2 D Hom(Wk, Q) / ⇡1(Yet)

deg⇤ Id ✏ H2(↵ ) ✏ d Y D Hom(CH (Y ), Q) / ⇡1(Yet) commutes. ⇤ We shall need the following result in Section 4.3. Proposition 2.14. Let f : Y be a morphism of proper regular arithmetic schemes, such !X that is ﬂat over Spec(Z) and Y has characteristic p.Assumethat has pure dimension d X X and that L( et,d) 0 holds. Then there exists a unique map in X D ˜ f ⇤ : RW ( , Z) R(YW , Z) W X ! ZETA FUNCTIONS AT s =0 21 which renders the following square commutative

R(Yet, Z) / R(YW , Z) O O ˜ fet⇤ fW⇤

R( et, Z) / RW ( , Z) X X where f ⇤ is induced by the map Yet et and R(YW , Z) is the cohomology of the Weil-étale et ! X topos. Proof. We shall deﬁne a morphism of exact triangles

aY R(Y,Q)[ 2] / R(Yet, Z) / R(YW , Z) O O O

0 fet⇤ ↵ RHom(R( , Q(d)) 0, Q[ ]) X / R( et, Z) / RW ( , Z) X X X ˜ where the left vertical map is the zero map and := 2d +2. The existence of fW⇤ will follow ˜ from the commutativity of the left square and the uniqueness of fW⇤ will follow from the facts that R(YW , Z) and RW ( , Z) are both perfect (by [34] Theorem 7.4 and Proposition 2.10 X respectively) and that RHom(R( , Q(d)) 0, Q[ ]) is a complex of Q-vector spaces, as in X the proof of Theorem 2.11. In order to show that such a morphism of exact triangles does exist, we only need to check that the composite map

↵ fet⇤ (38) RHom(R( , Q(d)) 0, Q[ ]) X R( et, Z) R(Yet, Z) X ! X ! is the zero map. The complex of Q-vector spaces D := RHom(R( , Q(d)) 0, Q[ ]) is X X 0 concentrated in degrees 3, because is flat (see (6)). Moreover, we have H (Yet, Z)= ⇡0(Y ) 1 2 X ⇡0(Y ) Z , H (Yet, Z)=0, H (Yet, Z) (Q/Z) A is the direct sum of a divisible group i ' and a finite group A,andH (Yet, Z) is finite for i>2. The spectral sequence p i q+i p+q Ext (H (D ),H (Yet, Z)) H (RHom(D , R(Yet, Z))) X ) X i Z Y2 then shows that 0 Hom (D , R(Yet, Z)) = H (RHom(D , R(Yet, Z))) = 0. D X X Hence (38) must be the zero map, and the result follows. ⇤ In cases where Theorem 2.11 and Proposition 2.14 both apply, we have a commutative diagram: ˜ fW⇤ RW ( , Z) / R(YW , Z) X

f ' W⇤ ' ✏ RW (Y,Z) ˜ where fW⇤ is the map defined in Theorem 2.11, fW⇤ is the map defined in Proposition 2.14 and the vertical isomorphism is defined in Theorem 2.13. Indeed, up to the identification given by Theorem 2.13, the map fW⇤ sits in the commutative square of Proposition 2.14, hence must ˜ coincide with fW⇤ . 22 BAPTISTE MORIN

2.5. Relationship with Lichtenbaum’s definition for number rings. In this section we consider a totally imaginary number field F and we set = Spec( ). The complex X OF RW ( , Z) is well defined since L( et, Z(1)) holds (see Theorem 5.1). X X Theorem 2.15. There is a canonical isomorphism in D RW ( , Z) ⇠ R( W , Z) 3 X ! X  where R( W , Z) 3 is the truncation of Lichtenbaum’s complex [35] and RW ( , Z) is the X  X complex defined in this paper. Proof. By [38] Theorem 9.5, we have a quasi-isomorphism

R( et,RZ) ⇠ R( W , Z) X ! X inducing

(39) R( et,RW Z) ⇠ R( W , Z) 3 X ! X  where RZ is the complex defined in [38] Theorem 8.5 and RW Z := RZ 2. The complex  R( W , Z), defined in [35], is the cohomology of the Weil-étale topos W which is defined in X X [39]. We have an exact triangle 2 Z[0] RW Z R Z[ 2] ! ! W in the derived category of étale sheaves on .RotatingandapplyingR( , ) we get an X X et exact triangle 2 R( et,R Z)[ 3] R( et, Z) R( et,RW Z). X W ! X ! X We have canonical quasi-isomorphisms (see [38] Theorem 9.4 and [38] Isomorphism (35)) 2 R( et,R Z)[ 3] Hom ( ⇥, Q)[ 3] RHom(R( , Q(1)), Q[ 4]). X W ' Z OF ' X 2 It follows that the morphism R( et,R Z)[ 3] R( et, Z) in the triangle above is deter- X W ! X mined by the induced map 0 2 3 Hom ( ⇥, Q)=H ( et,R Z) H ( et, Z)=Hom ( ⇥, Q/Z) Z OF X W ! X Z OF and so is the morphism ↵ . In both cases this map is the obvious one. Hence the square X 2 R( et,RW Z)[ 3] / R( et, Z) X O XO Id '

RHom(R( , Q(1)), Q[ 4]) / R( et, Z) X X is commutative. Hence there exists an isomorphism RW ( , Z) R( et,RW Z). The X ' X uniqueness of this isomorphism can be shown as in the proof of Theorem 2.11. Composing this isomorphism with (39), we obtain the result. ⇤

3. Weil-étale Cohomology with compact support We recall below the definition given in [14] of the Weil-étale topos and some results concerning its cohomology with R˜-coefficients. Then we define Weil-étale cohomology with compact support and Z-coefficients and we study the expected map from Z to R-coefficients. ZETA FUNCTIONS AT s =0 23

3.1. Cohomology with R˜-coefficients. Let be any proper regular connected arithmetic X scheme. The Weil-étale topos is defined as a 2-fiber product of topoi

W := et Spec(Z)W . X X ⇥Spec(Z)et There is a canonical morphism

f : W Spec(Z) B X ! W ! R where BR is Grothendieck’s classifying topos of the topological group R (see [24] or [14]). Consider the sheaf yR on BR represented by R with the standard topology and trivial R- action. Then one deﬁnes the sheaf R˜ := f⇤(yR) on . By [14], the following diagram consists of two pull-back squares of topoi, and the X W rows give open-closed decompositions:

i W / W o 1 ,W X X X1

✏ ' ✏ u ✏ et / et o 1 Sh( ) X X X1 Here the map

W := et Spec(Z)W et X X ⇥Spec(Z)et ! X is the first projection, Sh( ) is the category of sheaves on the space and X1 X1 ,W = BR Sh( ) X1 ⇥ X1 where the product is taken over the final topos. As shown in [14], the topos has the right X W R˜-cohomology with and without compact supports. We have ˜ ˜ ˜ RW ( , R):=R( W , R) R(B , R) R[ 1] R. X X ' R ' Concerning the cohomology with compact support, one has ˜ ˜ (40) RW,c( , R):=R( W ,!R) R( et,'!R)[ 1] R( et,'!R) X X ' X X where the complex Rc( et, R):=R( et,'!R) is quasi-isomorphic to X X Cone(R[0] R( , R))[ 1]. ! X1 1 ˜ Cup-product with the fundamental class ✓ H ( W , R) yields a morphism 2 X ˜ ˜ (41) ✓ : RW,c( , R) RW,c( , R)[1] [ X ! X such that the induced sequence i 1 i i+1 ... H ( , R˜) H ( , R˜) H ( , R˜) ! W,c X ! W,c X ! W,c X ! ˜ is a bounded acyclic complex of finite dimensional R-vector spaces. In view of RW,c( , R)[1] = X R( et,'!R) R( et,'!R)[1], the map (41) is simply given by projection and inclusion X X ˜ ˜ RW,c( , R) R( et,'!R) , RW,c( , R)[1]. X ⇣ X ! X 24 BAPTISTE MORIN

3.2. Cohomology with Z-coefficients. In the remaining part of this section, denotes a X proper regular connected arithmetic scheme of dimension d satisfying L( et,d) 0.If has X X characteristic p then we set RW,c( , Z):=RW ( , Z) and the cohomology with compact X X support is defined as Hi ( , Z):=Hi ( , Z). The case when is flat over Z is the case of W,c X W X X interest. The closed embedding u : Sh( ) et induces a morphism u⇤ : R( et, Z) 1 X1 !X 1 X ! R( , Z). X1 Proposition 3.1. There exists a unique morphism i⇤ : RW ( , Z) R( ,W , Z) which 1 X ! X1 makes the following diagram commutative:

RHom(R( , Q(d)) 0, Q[ 2d 2]) / R( et, Z) / RW ( , Z) X X X

u⇤ ! i⇤ 1 9 1 ✏ ✏ ✏ 0 / R( , Z) / R( ,W , Z) X1 X1 Proof. Recall from [14] that the second projection

,W := BR Sh( ) Sh( )= ,et X1 ⇥ X1 ! X1 X1 induces a quasi-isomorphism R( , Z) ⇠ R( ,W , Z). Hence the existence of the map X1 ! X1 i⇤ will follow (Axiom TR3 of triangulated categories) from the fact that the map 1 ↵ u⇤ (42) : RHom(R( , Q(d)) 0, Q[ 2d 2]) X R( et, Z) 1 R( , Z) X ! X ! X1 is the zero map. Again, we set D = RHom(R( , Q(d)) 0, Q[ 2d 2]) for brevity. Then X X the uniqueness of i⇤ follows from the exact sequence 1 0 Hom (D [1], R( ,W , Z)) Hom (RW ( , Z), R( ,W , Z)) D X X1 ! D X X1 Hom (R( et, Z), R( ,W , Z)), ! D X X1 whose exactness follows from the fact that Hom (D [1], R( ,W , Z)) is divisible while D X X1 Hom (RW ( , Z), R( ,W , Z)) is finitely generated. D X X1 It remains to show that the morphism (42) is indeed trivial. Since D is a bounded complex X of Q-vector spaces acyclic in degrees < 2, one can choose a (non-canonical) isomorphism k D k 2 H (D )[ k]. Then is identified with the collection of maps (k)k 2 in , with X ' X D k k L k : H (D )[ k] H (D )[ k] D R( , Z). X ! X ' X ! X1 k 2 M It is enough to show that =0for k 2.Wefixsuchak, and we consider the spectral k sequence p,q p k q+k p+q k E2 = Ext (H D ,H ( , Z)) H (RHom(H (D )[ k], R( , Z))). X X1 ) X X1 The group HkD is uniquely divisible and Hq+k( , Z) is finitely generated (hence has an X p X1 injective resolution of length 1), so that Ext (Hk(D ),Hq+k( , Z)) = 0 for p =1. Hence X X1 6 the spectral sequence above degenerates and gives a canonical isomorphism k 1 k k 1 Hom (H D [ k], R( , Z)) Ext (H D ,H ( , Z)). D X X1 ' X X1 k Moreover, the long exact sequence for Ext⇤(H D , ) yields X 1 k k 1 1 k k 1 Ext (H D ,H ( , Z)) Ext (H D ,H ( , Z)cotor) X X1 ' X X1 ZETA FUNCTIONS AT s =0 25

k 1 k since the maximal torsion subgroup of H ( , Z) is ﬁnite and H D is uniquely divisible. X1 X The short exact sequence k 1 k 1 k 1 0 H ( , Z)cotor H ( , Q) H ( , Q/Z)div 0 ! X1 ! X1 ! X1 ! k 1 is an injective resolution of the Z-module H ( , Z)cotor. This yields an exact sequence X1 k k 1 k k 1 0 Hom(H D ,H ( , Q)) Hom(H D ,H ( , Q/Z)div) ! X X1 ! X X1 1 k k 1 Ext (H D ,H ( , Z)cotor) 0. ! X X1 ! Let us deﬁne a natural lifting ˜ k k 1 k Hom(H D ,H ( , Q/Z)div) 2 X X1 1 k k 1 ˜ of k Ext (H D ,H ( , Z)cotor) and show that this lifting k is already zero. One 2 X X1 k k can assume k 2. Recall that H D = Hom(H ( , Q(d)) 0, Q). We have the following X X commutative diagram:

k Hom(H ( , Q(d)) 0, Q) X Hk(↵ ) X ✏ * k 1 k H ( et, Q/Z) ' / H ( et, Z) X X

✏ ✏ k 1 k H ( ,et, Q/Z) ' / H ( ,et, Z) XQ XQ

✏ ✏ k 1 k H ( , Q/Z) ' / H ( , Z) XQ,et XQ,et

' ✏ ✏ k 1 k 1 k H ( (C), Q) / H ( (C), Q/Z) / H ( (C), Z) X X X The morphism given by the central column of the diagram above factors through k 1 k 1 k 1 H ( , Q/Z)div H ( , Q/Z) H ( (C), Q/Z) X1 ✓ X1 ! X and yields the desired lifting ˜ k k 1 k : Hom(H ( , Q(d)) 0, Q) H ( , Q/Z)div. X ! X1 Here the morphism k 1 k 1 (43) H ( , Q/Z) H ( (C), Q/Z) X1 ! X is induced by the projection (C) . Using standard spectral sequences for equivariant X !X1 cohomology, it is easy to see that the kernel of the morphism (43) is of ﬁnite exponent (more precisely, this kernel is ﬁnite and killed by a power of 2). It follows that ˜k =0if and only if the map k k 1 (44) Hom(H ( , Q(d)) 0, Q) H ( (C), Q/Z), X ! X given by the central column of the previous diagram, is the zero map. Moreover, the map k 1 k 1 H ( ,et, Q/Z) H ( , Q/Z) XQ ! XQ,et 26 BAPTISTE MORIN

k 1 G ˜ factors through H ( , Q/Z) Q hence so does the map (44). In order to show that k =0 XQ,et it is therefore enough to show that k 1 G k 1 G (H ( , Q/Z) Q )div = (H ( , Ql/Zl) Q )div =0. XQ,et XQ,et Ml Let l be a fixed prime number. Let U Spec(Z) on which l is invertible and such that ✓ U is smooth. Let p U. By smooth and proper base change we have: XU ! 2 k 1 Ip k 1 H ( , Ql/Zl) H ( , Ql/Zl). XQ,et ' XFp,et k 1 Recall that H ( , Zl) is a finitely generated Zl-module. We have an exact sequence XFp,et k 1 k 1 k 1 0 H ( , Zl)cotor H ( , Ql) H ( , Ql/Zl)div 0. ! XFp,et ! XFp,et ! XFp,et ! We get k 1 G k 1 G 0 (H ( , Zl)cotor) Fp H ( , Ql) Fp ! XFp,et ! XFp,et k 1 GFp 1 k 1 (H ( , Ql/Zl)div) H (G p ,H ( , Zl)cotor). ! XFp,et ! F XFp,et 1 k 1 Again, H (G p ,H ( , Zl)cotor) is a finitely generated Zl-module, hence we get a sur- F XFp,et jective map k 1 G k 1 G H ( , Ql) Fp ((H ( , Ql/Zl)div) Fp )div 0. XFp,et ! XFp,et ! Note that k 1 G k 1 G ((H ( , Ql/Zl)div) Fp )div =(H ( , Ql/Zl) Fp )div. XFp,et XFp,et Hk 1( , ) k 1 > 0 But Fp,et Ql is pure of weight by [10], hence there is no non-trivial element k 1 X in H ( , Ql) fixed by the Frobenius. This shows that XFp,et k 1 G k 1 G k 1 G (H ( , Ql/Zl) Qp )div =(H ( , Ql/Zl) Fp )div = H ( , Ql) Fp =0. XQ,et XFp,et XFp,et k 1 G A fortiori, one has (H ( , Ql/Zl) Q )div =0and the result follows. XQ,et ⇤

Deﬁnition 3.2. There exists an object RW,c( , Z), well deﬁned up to isomorphism in , X D endowed with an exact triangle

i⇤ (45) RW,c( , Z) RW ( , Z) 1 R( ,W , Z). X ! X ! X1 i ( 1)i The determinant det RW,c( , Z):= det H ( , Z) is well defined up to a canon- Z X i Z Z W,c X ical isomorphism. 2 N The cohomology with compact support is defined (up to isomorphism only) as follows: i i H ( , Z):=H (RW,c( , Z)). W,c X X To see that det RW,c( , Z) is indeed well defined, consider another object RW,c( , Z)0 of Z X X D endowed with an exact triangle (45). There exists a (non-unique) morphism u : RW,c( , Z) X ! RW,c( , Z)0 lying in a morphism of exact triangles X RW,c( , Z) / RW ( , Z) / R( ,W , Z) X X X1 u Id Id 9 ' ✏ ✏ ✏ RW,c( , Z)0 / RW ( , Z) / R( ,W , Z) X X X1 ZETA FUNCTIONS AT s =0 27

The map u induces

det (u):det RW,c( , Z) ⇠ det RW,c( , Z)0. Z Z X ! Z X By [29] p. 43 Corollary 2, detZ(u) does not depend on the choice of u, since it coincides with the following canonical isomorphism 1 detZRW,c( , Z) detZRW ( , Z) det R( ,W , Z) X ' X ⌦ Z X1 det RW,c( , Z)0. ' Z X Given a complex of abelian groups C, we write C for C R. R ⌦ Proposition 3.3. We set := 2d +2.Thereisacanonical and functorial direct sum decomposition in : D RW ( , Z)R R( et, R) RHom(R( , Q(d)) 0, R[ ])[1] X ' X X such that the following square commutes:

i⇤ R 1⌦ RW ( , Z)R / R( ,W , Z)R X X1

' ' ✏ (u⇤ R,0) ✏ R( et, R) RHom(R( , Q(d)) 0, R[ ])[1] 1⌦ / R( , R) X X X1 Proof. Applying ( ) R to the exact triangle of Deﬁnition 2.9, we obtain an exact triangle ⌦ RHom(R( , Q(d)) 0, R[ ]) R( et, R) RW ( , Z)R. X ! X ! X But the map RHom(R( , Q(d)) 0, R[ ]) R( et, R) X ! X is trivial since R( et, R) R[0] is injective and RHom(R( , Q(d)) 0, R[ ]) is acyclic in X ' X degrees 1. This shows the existence of the direct sum decomposition. We write D :=  R RHom(R( , Q(d)) 0, R[ ]) and R( et, R) R[0] for brevity. The exact sequence X X ' Hom (DR[1], R[0]) Hom (RW ( , Z), R[0]) D ! D X

Hom (R[0], R[0]) Hom (DR, R[0]) ! D ! D yields an isomorphism Hom (RW ( , Z), R[0]) ⇠ Hom (R[0], R[0]). Hence there exists a D X ! D unique map s : RW ( , Z) R[0] such that s ⇤ = Id [0] where ⇤ : R[0] RW ( , Z) X R ! R ! X is the given map.X The functorial behavior of s X followsX from the factX that it is the unique X map such that s ⇤ = IdR[0]. The direct sum decomposition is therefore canonical and X functorial. The commutativityX of the square follows from Proposition 3.1. ⇤

Recall that we denote Rc( et, R):=R( et,'!R). X X Proposition 3.4. We set := 2d +2.Thereisanon-canonical direct sum decomposition

(46) RW,c( , Z)R Rc( et, R) RHom(R( , Q(d)) 0, R[ ])[1] X ' X X inducing a canonical isomorphism 1 detRRW,c( , Z)R detRRc( et, R) det RHom(R( , Q(d)) 0, R[ ]). X ' X ⌦ R X 28 BAPTISTE MORIN

Proof. Again we write DR := RHom(R( , Q(d)) 0, R[ ]) and R( et, R) R[0] for brevity X X ' (recall that is connected). Consider the following morphism of exact triangles: X i⇤ R 1⌦ RW,c( , Z)R / RW ( , Z)R / R( ,W , Z)R X X X1 u 9 ' ' ✏ ✏ ✏ Rc( et, R) DR[1] / R( et, R) DR[1] / R( , R) X X X1 Here all the maps but the isomorphism u are canonical. The non-canonical direct sum decomposition (46) follows. A choice of such an isomorphism u induces

det (u):det RW,c( , Z) ⇠ det (Rc( et, R) D [1]). R R X R ! R X R But detR(u) coincides (see [29] p. 43 Corollary 2) with the following (canonical) isomorphism 1 detRRW,c( , Z)R detRRW ( , Z)R det R( ,W , Z)R X ' X ⌦ R X1 1 det (R( et, R) D [1]) det R( et, R) ' R X R ⌦ R X det (Rc( et, R) D [1]) ' R X R hence does not depend on the choice of u. ⇤

3.3. The conjecture B( ,d) and the regulator map. Let be a proper flat regular X X connected arithmetic scheme of dimension d with generic fibre X = Q. The "integral part 2d 1 i X in the motivic cohomology" HM (X/Z, Q(d)) is defined as the image of the morphism 2d 1 i 2d 1 i H ( , Q(d)) H (X, Q(d)). X ! p Let H (X/R, R(q)) denote the real Deligne cohomology and let D i 2d 1 i 2d 1 i ⇢ : HM (X/Z, Q(d))R H (X/R, R(d)) 1 ! D be the Beilinson regulator. According to a classical conjecture of Beilinson, the map ⇢i should be an isomorphism for i 1 and there should be an exact sequence 1 0 2d 1 ⇢ 2d 1 1 0 0 HM (X/Z, Q(d))R H (X/R, R(d)) CH (X)R⇤ 0 ! ! D ! ! for i =0. Moreover, the natural map 2d 1 i 2d 1 i H ( , Q(d)) H (X, Q(d)) X R ! R is expected to be injective for i 0. This suggests the following conjecture, where we consider the composite map 2d 1 i 2d 1 i 2d 1 i H ( , Q(d))R H (X, Q(d))R H (X/R, R(d)). X ! ! D Conjecture 3.5. B( ,d) The map X 2d 1 i 2d 1 i H ( , Q(d))R H (X/R, R(d)) X ! D is an isomorphism for 1 i 2d 1 and there is an exact sequence   2d 1 2d 1 0 0 H ( , Q(d))R H (X/R, R(d)) CH (X)R⇤ 0 ! X ! D ! ! for i =0. ZETA FUNCTIONS AT s =0 29

By [23], there is a canonical morphism of complexes

(X, Q(d)) C (X/ , R(d)) ! D R inducing Beilinson’s regulator on cohomology [7], where the complex C (X/ , R(d)) computes D R real Deligne cohomology. Let j : X be the natural embedding. Consider the map Zar !XZar R( , Q(d)) R( ,j Q(d)) R( , Rj Q(d)) R(X, Q(d)) (X, Q(d)) X ! X ⇤ ! X ⇤ ' ' where the last isomorphism follows from the fact that, over a field, the Zariski hypercohomology of the cycle complex coincides with its cohomology. Then we consider the composite map ⇢ : R( , Q(d)) (X, Q(d)) C (X/ , R(d)). 1 X ! ! D R and we denote by (R) the derived category of R-vector spaces. D Theorem 3.6. Let be a proper flat regular connected scheme of dimension d satisfying X L( et,d) 0 and B( ,d).Achoiceofadirectsumdecomposition(46)induces,inacanonical X X way, an isomorphism in (R): D ˜ RW,c( , Z) R ⇠ RW,c( , R). X ⌦ ! X Proof. Duality for Deligne cohomology (see [6] Corollary 2.28) i 2d 1 i (47) H (X/R, R(p)) H (X/R, R(d p)) R D ⇥ D ! yields an isomorphism in (R) D C (X/ , R) RHom(C (X/ , R(d)), R[ 2d + 1]). D R ! D R Composing with R( , R) ⇠ C (X/ , R) we obtain X1 ! D R R( , R) ⇠ RHom(C (X/ , R(d)), R[ 2d + 1]). X1 ! D R 0 One has a morphism of complexes C (X/R, R(d)) CH (X)R⇤ [ 2d + 1] and we define ˜ D ! C (X/ , R(d)) to be its mapping fiber. Applying the functor RHom( , R[ 2d + 1]),we D R obtain an exact triangle (note that X is irreducible) ˜ R[0] RHom(C (X/ , R(d)), R[ 2d + 1]) RHom(C (X/ , R(d)), R[ 2d + 1]). ! D R ! D R We have a morphism of exact triangles

R[0] / R( , R) / Rc( et, R)[1] X1 X ! 9 ✏ ✏ ✏ ˜ R[0] / RHom(C (X/ , R(d)), R[ 2d + 1]) / RHom(C (X/ , R(d)), R[ 2d + 1]) D R D R where the vertical map on the right hand side is uniquely determined. This yields a canonical quasi-isomorphism ˜ (48) Rc( et, R)[ 1] ⇠ RHom(C (X/ , R(d)), R[ 2d 1]). X ! D R It follows from Conjecture B( ,d) that the morphism of complexes ⇢ : R( , Q(d)) 0 X 1 X ! C (X/ , R(d)) induces a quasi-isomorphism D R ˜ ⇢˜ ,R : R( , Q(d)) 0,R ⇠ C (X/ , R(d)). 1 X ! D R 30 BAPTISTE MORIN

Applying the functor RHom( , R[ 2d 1]) and composing with the map (48), we obtain an isomorphism: ˜ Rc( et, R)[ 1] ⇠ RHom(C (X/ , R(d)), R[ 2d 1]) ⇠ RHom(R( , Q(d)) 0, R[ 2d 1]). X ! D R ! X

The inverse isomorphism RHom(R( , Q(d)) 0, R[ 2d 1]) ⇠ Rc( et, R)[ 1] together with X ! X a direct sum decomposition (46) provide us with the desired map:

RW,c( , Z)R ⇠ Rc( et, Z)R RHom(R( , Q(d)) 0, R[ 2d 1]) X ! X X ˜ ⇠ Rc( et, R) Rc( et, R)[ 1] ⇠ RW,c( , R). ! X X ! X ⇤

Corollary 3.7. Let be a proper regular connected scheme of dimension d satisfying L( et,d) 0 X X and B( ,d).Thenthereisacanonical isomorphism X ˜ det RW,c( , Z) ⇠ det RW,c( , R). R X R ! R X Proof. If is flat over Z, the result follows from Proposition 3.4 and Theorem 3.6. So we X may assume that is smooth and proper over a finite field. Then we have RW,c( , Z)= X X RW ( , Z) and a canonical isomorphism X

RW ( , Z)R R( et, R) RHom(R( , Q(d)) 0, R[ 2d 1]). X ' X X

It follows from L( et,d) 0 that the natural map X R( et, R)[ 1] RHom(R( , Q(d)) 0, R[ 2d 1]) X ! X is a quasi-isomorphism (see the proof of Theorem 2.13). This yields a canonical map

(49) RW ( , Z)R ⇠ R( et, R) RHom(R( , Q(d)) 0, R[ 2d 1]) X ! X X ˜ (50) ⇠ R( et, R) R( et, R)[ 1] ⇠ RW ( , R). ! X X ! X ⇤

4. Zeta functions at s =0 Recall that the zeta-function of a scheme of finite type over Spec(Z) is defined as an X infinite product 1 (51) ⇣( ,s):= s X 1 N(x) x 0 Y2X where denotes the set of closed points of and N(x) is the cardinality of the residue X0 X field of x . The product (51) converges for (s) > dim( ). It is expected to have a 2X0 < X meromorphic continuation to the whole complex plane. We refer to [29] for generalities on the determinant functor. ZETA FUNCTIONS AT s =0 31

4.1. The main Conjecture. The following theorem summarizes some results obtained pre- viously (see [14] for a) and b), Proposition 2.10 and Definition 3.2 for c) and Theorem 3.6 and (50) for d)). Theorem 4.1. Let be a proper regular arithmetic scheme of pure dimension d. X a) The compact support cohomology groups Hi ( , R˜) are finite dimensional vector W,c X spaces over R,vanishforalmostalli and satisfy i i ( 1) dim H ( , R˜)=0. R W,c X i Z X2 b) Cup product with the fundamental class ✓ H1 ( , R˜) yields an acyclic complex 2 W X ✓ i ✓ i+1 ✓ [ H ( , R˜) [ H ( , R˜) [ ···! W,c X ! W,c X ! ··· i c) If L( et,d) 0 holds, then the compact support cohomology groups HW,c( , Z) are X X finitely generated over Z and they vanish for almost all i. d) If B( ,d) also holds then there are isomorphisms X i i H ( , Z) R ⇠ H ( , R˜). W,c X ⌦Z ! W,c X Consider now a proper regular connected arithmetic scheme of dimension d satisfying X L( et,d) 0 and B( ,d). By Corollary 3.7 we have canonical isomorphisms X X (det RW,c( , Z)) det (RW,c( , Z) ) Z X R ' R X R ˜ det RW,c( , R). ' R X Moreover, the exact sequence b) above provides us with i ˜ ( 1)i ˜ : R ⇠ det H ( , R) ⇠ det RW,c( , R) ⇠ (det RW,c( , Z)) . ! R W,c X ! R X ! Z X R i Z O2 The following conjecture presupposes that ⇣( ,s) has a meromorphic continuation to a neigh- X borhood of s =0. Conjecture 4.2. e) The vanishing order of ⇣( ,s) at s =0is given by X i i ords=0⇣( ,s)= ( 1) i rank H ( , Z). X · · Z W,c X i Z X2 f) The leading coefficient ⇣ ( , 0) in the Taylor expansion of ⇣( ,s) at s =0is given ⇤ X X up to sign by 1 Z (⇣⇤( , 0) )=det RW,c( , Z). · X Z X 4.2. Relation to Soulé’s conjecture. Let be a regular connected arithmetic scheme of X dimension d. We assume in this subsection that is moreover flat and projective over Spec(Z). X The following conjecture is Bloch’s reformulation (see [3] Section 7) of Soulé’s conjecture [42] in terms of motivic cohomology (thanks to [31] Theorem 14.7 (5)). It presupposes that ⇣( ,s) X has a meromorphic continuation near s =0and that the Q-vector space Hi( , Q(d)) is finite X dimensional for all i and zero for almost all i. Conjecture 4.3. (Soulé) One has i+1 2d i ords=0⇣( ,s)= ( 1) dim H ( , Q(d)). X Q X Xi 32 BAPTISTE MORIN

i If satisfies L( et,d), then H ( , Q(d)) = 0 for i<0 (see the proof of Theorem 2.11). X X X Hence the conjunction of L( ,d) and B( ,d) is equivalent to the conjunction of Conjecture Xet X 1.2 and Conjecture 1.4 for . X Proposition 4.4. Assume that satisfies L( ,d) and B( ,d).ThenConjecture4.2e)is X Xet X equivalent to Conjecture 4.3. Proof. Assuming L( ,d) and B( ,d) one has Xet X i i i i ( 1) i rank H ( , Z)= ( 1) i dim H ( , Z) · · Z W,c X · · Q W,c X Q i Z i Z X2 X2 i i 2d+2 i 1 = ( 1) i (dim H ( et, Q)+dim H ( , Q(d))⇤) · · Q c X Q X i Z X2 i 2d i 2d+1 i = ( 1) i (dim H ( , Q(d))⇤ + dim H ( , Q(d))⇤) · · Q X Q X i Z X2 i 2d+1 i = ( 1) dim H ( , Q(d)). Q X i Z X2 i The second equality follows from Proposition 3.4. Since H ( , Q(d)) = 0 for i<0 by L( et,d), X X the third equality follows from Conjecture B( ,d) and from duality for Deligne cohomology, X see the proof of Theorem 3.6. The fourth equality follows from L( et,d) since it implies that 2d i X H ( , Q(d)) is finite dimensional and zero for almost all i. The result follows. X ⇤ 4.3. Relation to the Tamagawa number conjecture. In this section we consider the Tamagawa number conjecture of Bloch and Kato in the formulation of Fontaine and Perrin- Riou (see [15] and [16]). Throughout this section we let be a smooth projective scheme over a number ring ,and X OF we assume that is connected of dimension d. We set F := F F and p := F Fp X X X⌦O X X⌦O where Fp := F /p for any maximal ideal p F . We assume further that satisfies O ⇢O X L( et,d) 0 and B( ,d). In order to ease the notation, we also assume F = ( ).In X X O OX X particular, and (for any finite prime p) are geometrically irreducible (see [36] Corollary XF Xp 5.3.17). Note that there is no loss of generality caused by the assumption F = ( ). Indeed, O OX X if is connected, smooth and projective over F , then ( ) is a number ring (finite over X O OX X F )and is also smooth and projective over ( ). O X OX X 4.3.1. Motivic L-functions. Recall that for any connected, smooth and projective scheme X/F and 0 i 2 dim(X) one defines the L-function   · i i i s 1 (52) L(h (X),s)= Lp(h (X),s)= Pp(h (X),N(p) ) p p Y Y as an Euler product over all finite primes p of F where N(p) is the cardinal of Fp and

i 1 i Ip Pp(h (X),T)=det 1 Fr T H (X , Ql) Ql p · | F,et is a polynomial (conjecturally) with rational coeﬃcients and independent of the chosen prime l as long as p does not divide l. Here Frp denotes a Frobenius element. The product (52) is known to converge for (s) > i +1by [10]. < 2 ZETA FUNCTIONS AT s =0 33

Assume now that X/F is the generic fiber F of the smooth proper scheme over X⌦OF X . By smooth and proper base change one has OF i i i Ip H ( , Ql) H ( , Ql) H ( , Ql) XFp,et ' XF,et ' XF,et and by Grothendieck’s formula, one has 2 dim( ) XF 1 i ( 1)i ⇣( ,s):= s = ⇣( p,s)= L(h ( F ),s) X 1 N(x) X X x 0 p i=0 Y2X Y Y where is the set of closed points in . X0 X 4.3.2. Statement of the Tamagawa number conjecture. Let / be a scheme satisfying the X OF assumptions of the introduction of this section 4.3. We set X := F . Recall that the "in- j X tegral part in the motivic cohomology" HM (X/Z, Q(d)) is defined as the image of the map Hj( , Q(d)) Hj(X, Q(d)). But this map is injective for j 0 by B( ,d), hence we may X j ! j X identity H (X/ , Q(d)) = H ( , Q(d)) for j 0. Define the fundamental line M Z X i 1 i + 2d 1 i f (h (X)) = det (H (X(C), Q) ) det H (X/ , Q(d))⇤ Q ⌦Q Q M Z for 0

1 1 Fr Ip Ip p Ip Rf (Fp,V)=R(Fp,V )=V V ! where Ip is the inertia subgroup at p.Forp dividing l define (1 ,◆) R (F ,V)=D (V ) D (V ) D (V )/F 0D (V ) f p cris,p ! cris,p dR,p dR,p where D (V )=(B V )GFp and D (V )=(B V )GFp (see [15] and [16]). In cris,p cris,p ⌦Qp dR,p dR,p ⌦Qp both cases there is a map of complexes R (F ,V) R(F ,V) and one defines R (F ,V) f p ! p /f p as the mapping cone. Then one defines a global complex Rf (F, V ) as the mapping fibre of the composite map R(U ,V) R(F ,V) R (F ,V). et ! p ! /f p p/U p/U M2 M2 34 BAPTISTE MORIN

There is an exact triangle in the derived category of Ql-vector spaces

(53) R (U ,V) R (F, V ) R (F ,V) c et ! f ! f p p/U M2 where the primes p / U include archimedean p with the convention Rf (R,V)=R(R,V) 2 and Rf (C,V)=R(C,V).

Notation 4.6. For an archimedean prime p of F ,wechooseacomplexembeddingp : F C ! representing p,andweset p := F,p (C)/GFp ,where F,p (C) := HomSpec(F )(Spec(C), F ) X X X X is the space of complex points of F lying over p.Onehas = p p.Finally,weset X X1 |1 X := Sh( ) and := Sh( ) B . Xp,et Xp Xp,W Xp ⇥ R ` The following result is proven in [14] Proposition 9.1. Note that ([14] Conjecture 7) is known in the smooth case.

Proposition 4.7. Let ⇡ : Spec( ) be as above and let be its Artin-Verdier étale X! OF X et topos. Let U Spec( ) be an open subscheme in which the prime number l is invertible. ✓ OF Then there is an isomorphism of exact triangles in the derived category of Ql-vector spaces:

Rc( U,et, Ql) R( et, Ql) R( p,et, Ql) X ! X ! p/U X L2

2d 2 ? 2d 2 ? 2d 2 ? ? i ? i ? i Rc(Uyet,Vl )[ i] Rf (yF, Vl )[ i] Ryf (Fp,Vl )[ i] i=0 ! i=0 ! p/U i=0 L L L2 L i i Here Vl := H (XF,et, Ql) is viewed as a Ql-representation of GF ,thebottomexacttriangleis asumovertriangles(53)andRc( U,et, Ql):=R( et,j!Ql) where j : U,et et is the X X X ! X canonical open embedding.

The statement of the Tamagawa number conjecture requires the following

1 i Conjecture 4.8. (Bloch-Kato) We have Hf (F, Vl )=0for any i.

One can show that H0(F, V 0) = CH0( ) , H0(F, V i)=0for i>0 and H3(F, V i)=0. f l ⇠ XF Ql f l f l Therefore, by Corollary 2.12 and Proposition 4.7, Conjecture 4.8 induces an isomorphism:

i 2 i i+2 i+2 2d 1 i (54) ⇢ : H (F, V ) ⇠ H ( et, Ql) ⇠ H ( , Z) ⇠ H ( , Q(d))⇤ . l f l ! X ! W X Ql ! X Ql Moreover Artin’s comparison isomorphism yields

i + i + (55) H ( (C), Q) ( H ( F,(C), Q)) X Ql ' X Ql :F, C M! i Gp i Gp (56) H ( , Ql) = (V ) ' XF,et l p p M|1 M|1 ZETA FUNCTIONS AT s =0 35 where runs over the complex embeddings of F and F,(C)=Hom (F )(Spec(C), F ) with X Spec X respect to the map : Spec(C) Spec(F ). We obtain an isomorphism (for 0 i 2d 2): !   i i i 1 i (57) # :f (h (X)) ⇠ det Rf (F, V ) det R(Fp,V ) l Ql ! Ql l ⌦ Ql l p O|1 (58) ⇠ det R (U ,Vi) det R (F ,Vi) ! Ql c et l ⌦ Ql f p l p Z O2 (59) ⇠ det R (U ,Vi) ! Ql c et l where Z =Spec( ) U is the closed complement of U. Indeed, (57) is induced by (54) and OF (56), (58) is induced by (53), and (59) is induced by the isomorphism (for p Z) 2 ◆ :det R (F ,Vi) p Ql f p l ⇠= Ql

i Ip i which is in turn induced by the identity map on (Vl ) and Dcris,p(Vl ). Below is the Tamagawa number conjecture in the formulation of Fontaine and Perrin-Riou (see [15] and [16]). It presupposes Conjecture 4.5. Conjecture 4.9. (l-part of the Tamagawa number conjecture) There is an identity of free i rank one Zl-submodules of detQl Rc(Uet,Vl ) i i i 1 i Zl #l # (L⇤(h (X), 0) )=detZl Rc(Uet,Tl ) · 1 i i i for any constructible Zl-sheaf T such that T Ql = V . l l ⌦Zl l i This conjecture is independent of the choice of Tl , as well as of the choice of U in which l is invertible [15].

4.3.3. One can reformulate the Tamagawa number conjecture in terms of the L-function

i i LU (h (X),s)= Lp(h (X),s) p U Y2 i associated to the smooth l-adic sheaf Vl over U, using a second isomorphism i ˜◆p :det Rf (Fp,V ) Ql Ql l ' which satisﬁes

i 1 i ri,p (60) ◆p = Pp⇤(h (X), 1) ˜◆p = Lp⇤(h (X), 0) log(N(p) )˜◆p i i where ri,p = ordT =1 Pp(h (X),T)= ords=0 Lp(h (X),s). The isomorphism ˜◆p is deﬁned i as follows. We shall see below that the complex Rf (Fp,Vl ) is semi-simple at 0;inother i Ip i words, the identity map (on (V ) for p - l and Dcris,p(V ) for p l) induces an isomorphism l l | H0(F ,Vi) ⇠ H1(F ,Vi) hence a trivialization f p l ! f p l i 0 i 1 1 i ˜◆p :det Rf (Fp,V ) det H (Fp,V ) det H (Fp,V ) Ql. Ql l ' Ql f l ⌦Ql Ql f l ' Then we deﬁne

(61) #˜i : (hi(X)) ⇠ det R (U ,Vi) det R (F ,Vi) ⇠ det R (U ,Vi) l f Ql ! Ql c et l ⌦ Ql f p l ! Ql c et l p Z O2 36 BAPTISTE MORIN for 0 i 2d 2, where the first isomorphism is (58) and the second is induced by the ˜◆p’s.   ˜i i i By (60) we have #l = p Z Pp⇤(h (X), 1) #l and we need to define 2 · (62) Q #˜i := log(N(p))ri,p #i . 1 1 p Z Y2 0 In our situation, we have ri,p =0for i>0 (for weight reasons) and r0,p =1(because Vl = Ql with trivial G -action, see below) for any p Z. The Tamagawa number conjecture becomes F 2 i Conjecture 4.10. There is an identity of free rank one Zl-submodules of detQl Rc(Uet,Vl ) ˜i ˜i i 1 i (63) Zl #l # (LU⇤ (h (X), 0) )=detZl Rc(Uet,Tl ). · 1 i Now we observe that Rf (Fp,Vl ) is indeed semi-simple at 0, and explain the compatibility between ˜◆p and the canonical trivialization of (det R( p,W , Z)) . Let p Z. By Proposition Z X Ql 2 4.7 we have i (64) Rf (Fp,V )[ i] R( p,et, Ql) R( p,W , Z) . l ' X ' X Ql i 0 M i i But H ( p,W , Z)Ql = Ql for i =0, 1 (since p is connected) and H ( p,W , Z)Ql =0otherwise X X i X (by [34] Theorem 7.4). It follows that Rf (Fp,Vl ) is acyclic for i>0, hence semi-simple at 0 0.Fori =0,wehaveVl = Ql with trivial GF -action (since F is geometrically connected), 0 Ip 0 X 0 hence (V ) = Ql for p - l and Dcris,p(V )=(Fp)0 for p l. In both cases, Rf (Fp,V ) is l l | l semi-simple at 0. Moreover, (60) is given by ([9] Lemma 1). Indeed, for i>0 and p l,one | has i i i P (V , 1) = P ((V )0, 1) = det (1 D (V )) p l l l Ql | cris,p l i i i where (Vl )0 := IndFp/Ql (Vl ) is the l-adic representation of GQl induced by Vl (seen as a representation of GFp ). The case p - l is similar and the case i =0is obvious. i 0 i Note also that Hf (Fp,Vl ) H ( p,W , Z)Ql for i =0, 1. Under this identification, the 0 0 1' X0 0 isomorphism Hf (Fp,Vl ) ⇠ Hf (Fp,Vl ) given by semi-simplicity of Rf (Fp,Vl ) corresponds ! e 0 [ 1 to the map H ( p,W , Z)Ql H ( p,W , Z)Ql given by cup-product with the fundamental class 1 X ! X 1 e H ( p,W , Z). Recall that e is the pull-back of the homomorphism in H (Spec(Fp)W , Z)= 2 X Hom(WFp , Z) which sends Frp to 1. It follows that we have a commutative square of isomorphisms

( 1)i i ( 1) i ˜◆p det R (F ,Vi) i 0 Ql f p l Ql N ! (65) N k 1 ? µ p, ? X Ql det R(y p,W , Z) Ql Ql X Ql ! where the left vertical isomorphism is induced by (64), and the lower horizontal isomorphism ⇠ is induced by µ p : Q detQR( p,W , Z)Q which is in turn induced by the exact sequence X ! X (see [34] Theorem 7.4)

e i e i+1 e (66) ... [ H ( p,W , Z) [ H ( p,W , Z) [ ... ! X Q ! X Q ! ZETA FUNCTIONS AT s =0 37

4.3.4. Let / be a smooth projective scheme satisfying the assumptions of the introduction X OF of this section 4.3. Let U Spec( ) be an open subscheme on which the prime number l is ✓ OF invertible. Consider the morphism

RW ( , Z) R( p,W , Z) X ! X p/U M2 given by Proposition 2.14 and Proposition 3.1, where is the Weil-étale topos of . Here Xp,W Xp the primes p / U include archimedean primes, see Notation 4.6. We deﬁne RW,c( U , Z) such 2 X that the triangle

(67) RW,c( U , Z) RW ( , Z) R( p,W , Z) X ! X ! X p/U M2 is exact. We do not show nor use that RW,c( U , Z) only depends on U . In fact RW,c( U , Z) X X X is only defined up to a non-canonical isomorphism but det RW,c( U , Z) is canonically defined Z X and we have a canonical isomorphism 1 (68) det RW,c( U , Z) det RW,c( , Z) det R( Z,W , Z) Z X ' Z X ⌦ Z X where Z = Spec( ) U. We define OF ˜ ˜ RW,c( U , R):=R( W ,j!R) X X where j : is the obvious open embedding of topoi (i.e. induced by ). XU,W ! X W XU,et ! X et The triangle ˜ ˜ ˜ RW,c( U , R) RW ( , R) R( p,W , R) X ! X ! X p/U M2 is exact, where the map on the right hand side is induced by the closed embedding p/U p,W 2 X ! (which is the closed complement of j). We obtain a canonical isomorphism X W ` ˜ ˜ 1 ˜ (69) det RW,c( U , R) det RW,c( , R) det RW ( Z , R). R X ' R X ⌦ R X By Corollary 3.7, (68) and (69) we have a canonical isomorphism ˜ (70) det RW,c( U , Z) ⇠ det RW,c( U , R) R X R ! R X and we obtain

⇠ ˜ ⇠ (71) U : R detRRW,c( U , R) detRRW,c( U , Z)R X ! X ! X such that the following square of isomorphisms commutes: 1 X ⌦ XZ 1 R R det RW,c( , Z) det R( Z,W , Z) ⌦ ! R X R ⌦ R X R (72)

? ? XU R? det RW,c?( U , Z) y ! R y X R Here we identify with its dual, the left vertical map is induced by the product map and XZ the right vertical map is induced by (68). Theorem 4.11. Let / be a smooth projective scheme over .Assumethat is con- X OF OF X nected of dimension d and that satisﬁes L( et,d) 0 and B( ,d).Assumemoreoverthat 1 i X X X H (F, H ( , Ql)) = 0 for all i,andthat⇣( ,s) has a meromorphic continuation to s =0. f XF,et X 38 BAPTISTE MORIN

2d 2 i Then the Tamagawa number conjecture (Conjecture 4.9) for the motive h (X)[ i] and i=0 all l is equivalent to statement f) of Conjecture 4.2 for . X L Proof. We consider an open subscheme U Spec( ) on which l is invertible and we let Z ✓ OF be the closed complement of U. By [34] satisﬁes Conjecture 1.1 for any p Z. Hence Xp 2 the factorization ⇣( ,s)=⇣( ,s) ⇣( ,s) together with (68), (69) and (72) show that X XU · XZ Conjecture 4.2 f) for is equivalent to Conjecture 4.2 f) for . X XU By Proposition 3.4 and by deﬁnition of , we have a canonical isomorphism X 2d 2 i ( 1)i (73) det RW,c( , Z) f (h (X)) Q X Q ' Oi=0 i ( 1)i which is compatible (in the obvious sense) with and i(# ) . In particular, Conjecture 2d 2 i X ⌦ 1 4.2 f) implies Conjecture 4.5 for i=0 h (X)[ i]. Moreover, for any prime p Z, cup-product 1 2 with the canonical class e H ( p,W , Z) yields a trivialization (induced by (66)) 2 LX

(74) µ p : Q = detQR( p,W , Z)Q. X ⇠ X Then (68), (73) and (74) yield an isomorphism 1 #W :det RW,c( U , Z) =det RW,c( , Z) det R( p,W , Z) Q X Q ⇠ Q X Q ⌦ Q X Q p Z O2 2d 2 i ( 1)i ⇠= f (h (X)) . Oi=0 We claim that the following diagram of isomorphisms is commutative: XU R det RW,c( U , Z) ! R X R

(75) #W,R k i ˜i ( 1) ? i i(# ) 2d 2 ? i ( 1) R ⌦ 1 f (yh (X)) ! i=0 R where (respectively #˜i ) is deﬁned in (71)N (respectively in (62)). Similarly, for any prime XU l invertible on U we have1 a commutative diagram of isomorphisms

det RW,c( U , Z) det Rc( U,et, Ql) Ql X Ql ! Ql X # (76) W,Ql i ? i (#˜i)( 1) ?i 2d 2 ? i ( 1) ⌦i l 2d 2 ( 1)? i f (yh (X)) det y Rc(Uet,V ) i=0 Ql ! i=0 Ql l where the top horizontalN isomorphism is induced byN the isomorphism

(77) det RW,c( U , Z) Zl = det Rc( U,et, Zl) Z X ⌦Z ⇠ Zl X given by (68) and Corollary 2.12, while the right vertical isomorphism is induced by the isomorphism 2d 2 ( 1)i i (78) detZl Rc( U,et, Zl) = detZl Rc(Uet,R⇡ Zl) = det Rc(Uet,Tl ) X ⇠ ⇤ ⇠ Zl Oi=0 ZETA FUNCTIONS AT s =0 39

i i i 2d 2 i ( 1) where T := R ⇡ . Hence the -lattice of (h (X)) given by the images of l Zl Zl i=0 f Ql ⇤ i 2d 2 ( 1) i det RW,c( U , Z) and det Rc(Uet,T ) coincide. This shows that Conjecture 4.2 Zl X Zl i=0 Zl N l 2d 2 i f) implies Conjecture 4.10N (hence Conjecture 4.9) for i=0 h ( F )[ i] and all l. Conversely, 2d 2 i X Conjecture 4.10 for h ( )[ i] and all l implies the l-primary part of Conjecture 4.2 i=0 XF L f) for [1/l] and all l, hence the l-primary part of Conjecture 4.2 f) for and all l, hence X X Conjecture 4.2 f) forL . Here by the l-primary part of Conjecture 4.2 f) for we mean an X X identity 1 Z(l) (⇣⇤( , 0) )=detZ(l) RW,c( , Z)Z(l) · X X X where Z(l) is the localization of Z at the prime ideal lZ. It remains to check that the squares (75) and (76) are indeed commutative. Consider the following diagram.

1 µ , b det R ( , ) ⌦ XZ Ql det R ( , ) det R( , ) det R ( , ) Ql W,c XU Z Ql ⌦ Ql ! Ql W,c XU Z Ql ⌦ Ql XZ,W Z Ql ! Ql W,c X Z Ql

a 1 a f d ⌦ ⌦

? ( 1)i+1 ? ? i ? 1 ◆˜ ? ? i ( 1) y i ⌦ p,i p i y i c yi ( 1) det Rc(Uet,V ) l det Rc(Uet,V ) det Rf (Fp,V ) f (h (X)) i Ql l ⌦ Q N ! i Ql l ⌦ p,i Ql l ! i Ql N N N N Here we identify Weil-étale cohomology tensor Ql with l-adic étale cohomology. Then the left vertical isomorphism a is induced by (78), the map f in the central vertical isomorphism is given by (64), and b, c and d are induced by (68), (58) and (73) respectively. The commutativity of the left square follows from the commutativity of (65). The commutativity of the right square follows from Proposition 4.7. By deﬁnition of #W we have d b (1 µ Z ,Ql )=#W,Ql . i ( 1)i+1 ⌦ X By deﬁnition of #˜i we have (#˜i)( 1) =(c (1 ˜◆ ) 1. The commutativity of (76) l ⌦i l ⌦ p,i p follows immediately. The square (75) can be decomposed as follows:N

1 ⌦ X XZ 1 b det RW,c ( , ) det R( Z,W , ) det RW,c ( U , ) R ⌦ R ! R X Z R ⌦ R X Z R ! R X Z R

1 1 ⌘ µ , #W, ⌦ ⌦ XZ R R ? i ( 1)i ? ? ? ( (# ) ) ( log(N(p))) i ? ? i y ⌦i ⌦ p i ( 1) y Id yi ( 1) 1 ( f (h (X)) ) f (h (X)) R ⌦ R Q ! i R ⌦ R ! i R Indeed, under the canonical identificationN R R = R, the composition ofN the top horizontal ⌦ ˜i ( 1)i maps (respectively of the lower horizontal maps) is U (respectively i(# ) ). Here Id is X ⌦ 1 the obvious identification, ⌘ is induced by (73) and b is induced by (68). The right hand side square is commutative by definition of #W . The left hand side square is the tensor product of two squares; it is therefore enough to check commutativity of both factors separately. The commutativity of the square corresponding to the first factor follows from the fact that (73) i ( 1)i is compatible with and i(# ) . The commutativity of the square corresponding to X ⌦ 1 the second factor boils down to the identity 1 1 = log(N(p)) µ :det R( , ) . (79) p p,R R p,W Z R R X · X X !

Identity (79) follows from the fact that µ p,R is obtained by cup-product with the class e 1 X 2 H (WFp , R) = Hom(WFp , R) sending Frp WFp to 1, while p is obtained by cup-product 1 2 X with the class ✓ H (WFp , R) sending Frp to log(N(p)). This is an easy computation in view 2 i ˜ i ˜ of the fact that both complexes [... H ( p,W , R) H ( p,W , R) ...] are isomorphic to ! X ! X ! [R Hom(W p , R)] put in degrees 0, 1, because p/Fp is geometrically connected. Identity ! F X 40 BAPTISTE MORIN

(79) is of course compatible with Z ( , 1) = log(N(p)) ⇣ ( , 0). This concludes the proof ⇤ Xp · ⇤ Xp of the theorem. ⇤

4.4. Proven cases. Let Fq be a finite field and let A(Fq) be the class of smooth projective varieties over Fq defined in Section 5.2.

Theorem 4.12. Let be a smooth projective variety over the ﬁnite ﬁeld Fq.If lies in X X A(Fq) then Conjecture 4.2 holds for . X Proof. The variety lies in A(Fq) hence L( et,d) L( W ,d) holds (see Proposition 5.7). X X , X In view of Theorem 2.13 the result follows from [34] Theorem 7.4. ⇤ Theorem 4.13. If = Spec( ) is the spectrum of a number ring, then Conjecture 4.2 X OF holds for . X Proof. The result follows from the explicit computation of the Weil-étale cohomology in this case (see Section 4.5.1) and from the analytic class number formula

ords=0⇣( ,s)=] 1 and ⇣F⇤ (0) = hR/w X X1 where h (respectively w) is the order of Cl(F ) (respectively of µF )andR is the regulator of the number ﬁeld F . ⇤

Theorem 4.14. Let be a smooth projective scheme over the number ring ( )= F , X OX X O where F is an abelian number field. Assume that admits a smooth cellular decomposition XF (see Definition 5.13) and that p (Z) for any finite prime p of F .ThenConjecture4.2 X 2L holds for . X Proof. The scheme is connected and we set d = dim( ). The scheme satisfies Conjec- X X X ture L( et,d) (resp. Conjecture B( ,d)) by Proposition 5.14 (resp. by Proposition 5.15). X X 0 Moreover, F is a cellular variety hence h( F ) is of the form k h (F )(rk) (in the cate- X X i gory of Chow motives, see [5] Theorem 3.1). Hence L(h ( F ),s) is a product of shifts of the X L Dedekind zeta function ⇣ (s); in particular L(hi( ),s) satisfies meromorphic continuation F XF and the functional equation. By Theorem 3.6 and [14] Theorem 1.1, Statement e) of Con- i jecture 4.2 for follows. Moreover, the Galois representations H ( , Ql) is a sum of Ql(r) X XF (with r 0), hence satisfy Conjecture 4.8 (see [1] Lemme 4.3.1). Finally, Conjecture 4.9 is  i 0 known for h ( F ) (which is a sum of h (F )(r)) since F/Q is abelian [12]. Hence Statement X f) of Conjecture 4.2 for follows from Theorem 4.11. X ⇤ The simplest non-trivial example of a scheme satisfying the assumptions of the previous n F/ theorem is P F where Q is abelian. The result then gives a cohomological interpretation of ⇣ (n) for nO 0 (see Section 4.5.2). F⇤  4.5. Examples.

4.5.1. Number rings. Let F be a number ring and let = Spec( F ).NotethatL( et, Z(1)) O X O X holds (see Theorem 5.1). The cohomology H⇤( et, Z) is computed in ([38] Proposition 6.6). X By Proposition 2.10 one has i H ( , Z)=0for i<0 and i>3. W X By Proposition 2.10 again one has H0 ( , Z)=Z, H1 ( , Z)=0, an exact sequence W X W X 2 2 1 (80) 0 H ( et, Z)codiv H ( , Z) Hom(H ( et, Z(1)), Z) 0 ! X ! W X ! X ! ZETA FUNCTIONS AT s =0 41 and an isomorphism 3 3 D H ( , Z) H ( et, Z)codiv = Hom( ⇥, Q/Z)codiv = µ . W X ' X OF F 0 since H ( et, Z(1)) = 0. The sequence (80) reads as follows ([38] Proposition 6.6.) X D 2 0 Cl(F ) H ( , Z) Hom( ⇥, Z) 0 ! ! W X ! OF ! where Cl(F ) is the class group of F , ⇥ is the unit group and µ := ( ⇥) . The map OF F OF tor i i H ( , Z) R ⇠ H ( , R˜) W,c X ⌦ ! W,c X is trivial for i =1, 2, the obvious isomorphism for i =1and the inverse of the dual of the 6 classical regulator map 2 H ( , Z) R Hom( ⇥, R) ( R)/R W,c X ⌦ ' OF ! YX1 ˜ for i =2. The acyclic complex (H⇤ ( , R), ✓) is reduced to the identity map W,c X [ 1 Id 2 H ( , R˜)=( R)/R ( R)/R = H ( , R˜). W,c X ! W,c X YX1 YX1 We obtain (see [35] Section 7) 1 (detZRW,c( , Z)) = (w/hR) Z. X X · 4.5.2. Projective spaces over number rings. Let be the ring of integers in a totally imagi- OF nary number ﬁeld F and let n 1. We set = Pn and d =dim( )=n+1. It follows easily X F X from Proposition 5.14 (or directly from (81) belowO and Theorem 5.1) that L( ,d) holds. Xet Proposition 4.15. For 2 i 2d +1,wehaveanexactsequence   D i 0 (Ki 2( F )tor) HW ( , Z) HomZ(Ki 1( F ), Z) 0. ! O ! X ! O ! Proof. The localization sequence (see [20] Corollary 7.2(a)) i 2 n 1 i n i n ... H (P ,et, Z(d 1)) H (P F ,et, Z(d)) H (A F ,et, Z(d)) ... ! OF ! O ! O ! i n i i n is split by H (A , Z(d)) ⇠ H (Spec( F )et, Z(d)) H (P , Z(d)) (see Lemma 5.11). F ,et O ! F ,et By induction, thisO yields the projective bundle formula O n i i i 2j (81) H ( et, Z(d)) H ( et, Z(d)) H (Spec( F )et, Z(d j)). X ' X ' O Mj=0 i 2j If 0 j n 1, then H (Spec( F )et, Z(d j)) = 0 for i 2j =1, 2 (by Lemma 5.4) and   O 6 i 2j i 2j K2d i( F )=K2(d j) (i 2j)( F ) H (Spec( F ), Z(d j)) H (Spec( F )et, Z(d j)) O O ' O ' O for i 2j =1, 2 by Theorem 5.1(6) and (83). i 2d We obtain H ( et, Z(d)) K2d i( F ) for 1 i 2d 1 and H ( et, Z(d)) Cl(F )= X ' O   X ' K ( ) . The result then follows from the exact sequence (see Proposition 2.10) 0 OF tor i i 2d+2 (i+1) 0 H ( et, Z)codiv HW ( , Z) HomZ(H ( et, Z(d)) 0, Z) 0 ! X ! X ! X ! i 2d+2 i D and from H ( et, Z)codiv (H ( et, Z(d)) 0,tor) for i 1 (see Lemma 2.5). X ' X ⇤ 42 BAPTISTE MORIN

Recall from [4] that K ( ) is finitely generated for any i 0 and finite for i =0even. By i OF 6 Proposition 4.15 and Proposition 2.10, H⇤ ( , Z) is given by the following identifications and W X exact sequences: i H ( , Z)=Z for i =0, W X =0for i =1, D 2 0 Cl(F ) H ( , Z) Hom( ⇥, Z) 0, ! ! W X ! OF ! D = µF for i =3, D 4 0 K2( F ) H ( , Z) Hom(K3( F ), Z) 0, ! O ! W X ! O ! =(K ( ) )D for i =5, 3 OF tor ... D 2n+2 0 K2n( F ) H ( , Z) Hom(K2n+1( F ), Z) 0, ! O ! W X ! O ! =(K ( ) )D for i =2n +3, 2n+1 OF tor =0for i>2n +3. The long exact sequence i i i ... HW,c( , Z) R HW ( , Z) R H ( , Z) R ... ! X ⌦ ! X ⌦ ! X1 ⌦ ! i and Proposition 3.3 show that, for i 2, HW,c( , Z) R is canonically isomorphic to either i i 1 X ⌦ HW ( , Z) R or H ( , Z) R depending on the parity of i. Hence the acyclic complex X ⌦ X1 ⌦ ✓ i ✓ i+1 ✓ ... [ H ( , Z) R [ H ( , Z) R [ ... ! W,c X ⌦ ! W,c X ⌦ ! is canonically isomorphic to

G R0⇤ 0 1 G R1⇤ 0 0 ( R) R /R Hom( ⇥, R) ( (2i⇡) R) R Hom(K3( F ), R) ... ! ! OF ! ! O ! F, C F, C Y! Y! 0 n G Rn⇤ 0 ... ( (2i⇡) R) R Hom(K2n+1( F ), R) 0 ! ! O ! F, C Y! where the isomorphisms Rm⇤ are dual to the regulator maps. It follows (see [35] Section 7) 1 that maps detZRW,c( , Z) to X X (w/hR0) (]K3( F )tor/]K2( F ) R1) ... (]K2n+1( F )tor/]K2n( F ) Rn) Z · O O · · · O O · · where Rm is the Beilinson regulator (we follow the indexing of [33]). In view of n ⇣⇤(P , 0) = ⇣F⇤ (0) ⇣F⇤ ( 1) ... ⇣F⇤ ( n) OF · · · we see that Conjecture 4.2 for = Pn gives a cohomological reformulation of the classical X F version of Lichtenbaum’s conjecture (seeO [33] 4.2). Note that Weil-étale cohomology gives (conjecturally) the right 2-torsion for any number ﬁeld, i.e. possibly with some real places (see [33] 2.6). For example, if F/Q is abelian, then Conjecture 4.2 holds for = Pn and X F any n 0 by Theorem 4.14. O 5. Appendix The goal of this section is to establish simple cases of Conjectures L( ,d) and B( ,d) Xet X which are used in Section 4.4. ZETA FUNCTIONS AT s =0 43

5.1. Motivic cohomology of number rings. Let F be a number field and let F be its ring i O of integers. In order to ease the notations, in this subsection we denote by H ( F , Z(n)) := p i p O H (Spec( F )Zar, Z(n)) and H ( F , Z(n)) := H (Spec( F )et, Z(n)) Zariski and étale hyper- O et O O cohomology of the cycle complex Z(n) over Spec( F ). We consider the spectral sequence O constructed by Levine (see [32] Spectral Sequence (8.8)) p,q p (82) E2 = H ( F , Z( q/2)) K p q( F ). O ) O The aim of this subsection is to prove Theorem 5.1. It is well-known (see for example [30] Proposition 2.1) that this result follows one way or another from the Bloch-Kato conjecture (relating Milnor K-theory to Galois cohomology), which is proven in [46]. However, we could not find a proof of Theorem 5.1 in the literature. Theorem 5.1. The following assertions are true. i (1) For i 2, Het( F , Z(1)) is finitely generated.  O i (2) For any n 0 and any i Z, H ( F , Z(n)) is finitely generated. 2 i O (3) For any n 2 and any i Z, H ( F , Z(n)) is finitely generated. 2 et O (4) The edge morphisms from the spectral sequence (82) yield maps i ci,n : K2n i( F ) H ( F , Z(n)) O ! O for n 2 and i =1, 2. (5) The kernel and the cokernel of ci,n are both finite and 2-primary torsion. (6) If F is totally imaginary, then the maps ci,n are isomorphisms. Proof. The proof of Theorem 5.1 requires the following lemmas. Lemma 5.2. Let n 2.Ifi =1, 2,then 6 i i 1 H ( F , Z(n)) H ( F , Q/Z(n)). et O ' et O Proof. Applying the exact functor Q to the spectral sequence (82), we obtain a spectral ⌦Z sequence with Q-coefficients. By [31] Theorem 11.7, (82) degenerates with Q-coefficients and yields i (n) H ( F , Q(n)) K2n i( F ) . O ' O Q (n) By Borel’s theorem [4], K2n i( F )Q =0for i even, i =2n. Moreover, one has K2n i( F ) = O 6 O Q 0 i i =1 K ( ) = K ( )(0) Hi( , (n)) = for odd, (see [47] Theorem 47). Since 0 F Q 0 F Q , this gives F Q 6 i O O O 0 for i even, except for (i, n)=(0, 0),andH ( F , Q(n)) = 0 for i odd, except for i =1. Lemma O 5.2 then follows from the long exact sequence i i i ... H ( F , Z(n)) H ( F , Q(n)) H ( F , Q/Z(n)) ... ! et O ! et O ! et O ! i i and from the isomorphism H ( F , Q(n)) H ( F , Q(n)) (see [17] Proposition 3.6). O ' et O ⇤ For a prime number p, we write j :Spec( [1/p]) Spec( ) for the obvious open p OF ! OF embedding, jp, for the direct image functor with respect to the étale topology and Rjp, for ⇤ ⇤ its total derived functor. Lemma 5.3. For n 2 one has an isomorphism in the derived category of étale sheaves on Spec( F ): O n Q/Z(n) lim Rjp, µp⌦r ' ⇤ p ! M 44 BAPTISTE MORIN

r where µpr is the étale sheaf of p -th roots of unity and the direct sum (respectively the colimit) is taken over all prime numbers (respectively over r). r n Proof. It is enough to showing that Z/p Z(n)=Rjp, µp⌦r for any prime number p.Using[20] ⇤ Corollary 7.2, [21] Theorem 8.5 and [17] Theorem 1.2(4), we obtain an exact triangle n 1 r n ip, (⌫r )[ (n 1) 2] Z/p Z(n) Rjp, (µp⌦r ) ⇤ ! ! ⇤ n n n where ⌫r = ⌫pr = W ⌦r,log is the logarithmic de Rham-Witt sheaf, and ip is the closed n 1 immersion of the points lying over p. But ⌫r is trivial on the finite field Fp := F /p (for r O p p) because n 1 1. The result then follows from Q/Z(n) p lim Z/p Z(n). ⇤ | ' ! Lemma 5.4. The following assertions are true. L i If n 1 and i 0,thenwehaveHet( F , Z(n)) = 0. •  i O If n 2 and i 3,thenHet( F , Z(n)) is finite and 2-torsion. • O i Assume that F is totally imaginary. If n 2 and i 3,thenH ( F , Z(n)) = 0. • et O Proof. In view of Z(1) Gm[ 1] where Gm is the multiplicative group (see [20] Lemma 7.4), i ' the fact that H ( F , Z(n)) = 0 for n 1 and i 0 follows immediately from Lemma 5.2 et O  and Lemma 5.3. 2 n We assume now that n 2. By [41] Theorem 5, the group H ( [1/p], lim µ⌦r ) is of finite et OF p 3 n! exponent for any p =2(the colimit is taken over r). But Het( F [1/p],µp⌦r )=0for p =2by 6 2 O n 6 Artin-Verdier duality (see [37] Corollary 3.3), hence Het( F [1/p], lim µp⌦r ) is divisible. This 2 n O yields H ( [1/p], lim µ⌦r )=0for p =2, since this group is both! divisible and of finite et OF p 6 exponent. Moreover,! we have i n Het( F [1/p], lim µp⌦r )=0 O ! for any i 3 and p =2, again by Artin-Verdier duality. By [40] Corollary 4.4 and [40] 6 i n r1 Proposition 4.6, for any i 2,wehaveHet( F [1/2], lim µ2⌦r )=(Z/2Z) for n i odd and i n O H ( [1/2], lim µ⌦r )=0for n i even, where r is the! number of real places of F . et OF 2 1 i ! i n Using Het( F , Q/Z(n)) p lim Het( F [1/p],µp⌦r ) (see Lemma 5.3), it then follows that i O ' O i H ( F , Q/Z(n)) is finite 2-torsion! for i 2,andH ( F , Q/Z(n)) = 0 for F totally imaginary et O L et O and i 2. We obtain the result using Lemma 5.2. ⇤ i Proof of Theorem 5.1. The finite generation of Het( F , Z(1)) for i 2 follows from Z(1) i O  0 ' Gm[ 1] (see [20] Lemma 7.4). Indeed, Het( F , Gm)=0for i<0, Het( F , Gm)= F⇥ is 1 O O O finitely generated and the class group H ( F , Gm)=Cl(F ) is finite. et O By [46] Theorem 6.16 and by [17] Theorem 1.2(2), the map i i (83) H ( F , Z(n)) ⇠ H ( F , Z(n)), O ! et O induced by the morphism from the étale site to the Zariski site, is an isomorphism for i n+1. i  Note that, for dimension reasons, we have H ( F , Z(n)) = 0 for i>n+1. By Lemma 5.4, i O we have H ( F , Z(n)) = 0 for n 1 and i 0. It follows that the edge morphisms from the O  spectral sequence (82) give maps i ci,n : K2n i( F ) H ( F , Z(n)) O ! O for n 2 and i =1, 2. This yields assertion (4) of Theorem 5.1. ZETA FUNCTIONS AT s =0 45

Let us prove assertion (2) of Theorem 5.1. The codomain of any non-trivial differential p,q dr of the spectral sequence (82) at the Er-page for r 2 is a finite 2-torsion abelian group. p,q p,q Moreover, for (p, q) fixed, the differential dr is zero for r>>0 big enough. Since E is a p,q 1 sub-quotient of some K-group, it is finitely generated by [4]. It follows that E2 is finitely i generated for any p and any q. Hence H ( F , Z(n)) is finitely generated for any i and any n. O If F is totally imaginary, the spectral sequence (82) degenerates by Lemma 5.4 (any differential at the E2-page has either trivial domain or trivial codomain), and the maps ci,n are isomorphisms. This proves assertion (6) of Theorem 5.1. For an arbitrary number field F , the spectral sequence (82) degenerates with Z[1/2]-coefficients by Lemma 5.4, and yields isomorphisms i ci,n Z[1/2] : K2n i( F ) Z[1/2] ⇠ H ( F , Z(n)) Z[1/2] ⌦ O ⌦ ! O ⌦ for n 2 and i =1, 2. This proves assertion (5) of Theorem 5.1. i Finally, H ( F , Z(n)) is finitely generated for n 2 and any i Z. Indeed, (83) is an et O 2 isomorphism for i n +1and Hi (X, Z(n)) is finite for i n +2 4 by Lemma 5.4. This  et gives assertion (3) of Theorem 5.1. ⇤ Remark 5.5. Using arguments of Levine [31],onecandeterminemostofthedifferentialsof the spectral sequence (82) (in particular (82) degenerates at E4), and obtain precise informa- tion on the kernel and the cokernel of the map ci,n (see [31] Theorem 14.10). 5.2. Smooth projective varieties over finite fields. For a smooth projective scheme Y i over a finite field Fq, we consider the cohomology H (YW , Z(n)) of the Weil-étale topos YW with coefficients in Bloch’s cycle complex (see [34] and [18]). The following conjecture is due to Lichtenbaum and Geisser. i Conjecture 5.6. L(YW ,n) For any i Z, H (YW , Z(n)) is finitely generated. 2 Following [43] and [18], we consider the full subcategory A(Fq) of the category of smooth projective varieties over Fq generated by products of curves and the following operations: (1) If X and Y are in A(Fq) then X Y is in A(Fq). (2) If Y is in A(Fq) and there are morphisms c : X Y and c0 : Y X in the category of ` ! ! Chow motives, such that c0 c : X X is multiplication by a constant, then X is in A(Fq). ! m m m (3) If Fq /Fq is a finite extension and X q Fq is in A(Fq ), then X is in A(Fq). ⇥F (4) If Y is a closed subscheme of X with X and Y in A(Fq), then the blow-up X0 of X along Y is in A(Fq).

Proposition 5.7. Let Y be a connected smooth projective scheme over a finite field Fq of dimension d.Thefollowingassertionsaretrue. We have L(Y ,d) L(Y ,d). • W , et If Y belongs to A(Fq) then L(Yet,d) holds. • i If Y belongs to A(Fq) and n>d,thenH (Yet, Z(n)) is finitely generated for any i Z. • 2 Proof. By [18] Theorem 7.1, there is an exact sequence (for any n) i i i 1 i+1 (84) ... H (Yet, Z(n)) H (YW , Z(n)) H (Yet, Q(n)) H (Yet, Z(n)) ... ! ! ! ! ! which yields isomorphisms i i i i 1 (85) H (YW , Z(n)) Q H (YW , Q(n)) H (Yet, Q(n)) H (Yet, Q(n)). ⌦ ' ' 46 BAPTISTE MORIN

Assume now that Conjecture L(YW ,d) holds. By [18] Theorem 8.4, it follows from L(YW ,d) that we have an isomorphism i i H (YW , Z(d)) Zl H (Y,Zl(d)) ⌦ ' cont i for any prime number l and any i. But for i =2d, 2d +1, Hcont(Y,Zl(d)) is finite for any l and 6 i zero for almost all l (see [27] Proof of Corollaire 3.8 for references). Hence H (YW , Z(d)) is i i finite for i =2d, 2d +1. Then (85) gives H (Yet, Q(d)) = 0 for i<2d, hence H (Yet, Q(d)) i 6 i ' H (Y,Q(d)) = 0 for i =2d. The exact sequence (84) then shows that H (Yet, Z(d)) i 6 i ! H (YW , Z(d)) is injective for i 2d+1. Hence H (Yet, Z(d)) is finitely generated for i 2d+1.   This yields L(Y ,d) L(Y ,d). Conversely, we have L(Y ,d) L(Y ,d) by Theorem 2.13. W ) et et ) W Consider now a variety Y A(Fq) of pure dimension d. The fact that L(YW ,d) holds is 2 given by [18] Theorem 9.5. Let n>d. It follows from the proofs of [18] Theorems 9.4 and 9.5 that Hi(Y,Q(n)) = 0 for any i<2n. Moreover Hi(Y,Q(n)) = 0 for any i 2n>n+ d. i The last claim of the proposition then follows from (84) since H (YW , Z(n)) is known to be finitely generated for any i Z by [18] Theorem 9.5. 2 ⇤ 5.3. The class (Z). In this section we follow Geisser’s notation (see [20] Section 2) for the L cycle complex Zc(n).If is a d-dimensional connected scheme which is proper over Z then X we have c (86) Z (n)=Z(d n)[2d]. Definition 5.8. Let be a separated scheme of finite type over Spec(Z).Wesaythat c X X satisfies L ( et) if one has: i X c H ( et, Z (0)) is finitely generated for any i 0; • i X c  H ( et, Z (n)) is finitely generated for any i Z and any n<0. • X 2 Note that if is a regular scheme connected of dimension d which is proper over Z, then c X L ( et) L( et,d) (see (86) above). We define below a class of (simple) arithmetic schemes X ) X c satisfying Property L ( et). Let SFT(Z) be the category of separated schemes of finite type X over Spec(Z). Definition 5.9. We denote by (Z) the class of schemes of SFT(Z) generated by the following L objects: the empty scheme ; • ; varieties Y A(Fq) for any finite field Fq; • 2 spectra of number rings Spec( ); • OF and the following operations: ( 0) Let Z, X be a closed immersion with open complement U such that Z is regular • L ! and proper. If two object of (Z, X, U) belong to (Z) then so does the third. L ( 1) Let Z, X be a closed immersion with open complement U (Z).Then • L ! 2L X (Z) if and only if Z (Z). 2L 2L ( 2) We have Xi (Z) for 0 i p if and only if 0 i p Xi (Z). • L 2L     2L ( 3) If V U is an affine bundle and U belongs to (Z),thensodoesV . • L ! L` ( 4) Let Ui X, i I be a finite surjective family of étale morphisms. If Ui0,...,ip • L { ! 2 } p+1 belongs to (Z) for any (i0,...,ip) I and any p 0,thensodoesX. L 2 In the statement of ( 4), we write U := U ... U as usual. In practice, we use L i0,...,ip i0 ⇥X ⇥X ip ( 4) for a finite étale Galois cover U X, in which case it is enough to check that U (Z). L ! 2L ZETA FUNCTIONS AT s =0 47

c Proposition 5.10. Any object in the class (Z) satisfies L ( et). X L X We say that the property Lc is stable under operation ( i), if any scheme X SFT(Z) L 2 constructed out of schemes X satisfying Lc(X ) by operation ( i) also satisfies Lc(X ). ↵ ↵,et L et c Proof. By Theorem 5.1 (see also (86)), any number ring Spec( F ) satisfies L (Spec( F )et). c O O By Proposition 5.7, any variety Y A(Fq) satisfies L (Yet). It remains to check that the 2 property Lc is stable under operations ( i) for i =0,...,4. By purity (see [20] Corollary 7.2), L we have a long exact sequence i c i c i c i+1 c ... H (Zet, Z (n)) H (Xet, Z (n)) H (Uet, Z (n)) H (Zet, Z (n)) ... ! ! ! ! ! for any open–closed decomposition U, X -Zand any n 0. Moreover, if Z is regular 1 c ! 1  c proper, then H (Zet, Z (0)) is finitely generated. Indeed H (Zet, Z (0)) = 0 if Z(R)= and 1 c ; H (Zet, Z (0)) is a finite dimensional Z/2Z-vector space otherwise (see Lemma 2.3). It follows that the property Lc is stable under operations ( 0) and ( 1). L L The fact that Lc is stable under operation ( 2) is obvious since cohomology respects finite L direct sums. It is stable under operation ( 3) by Lemma 5.11. L Let Ui X, i I be a finite étale covering family. We write Xp = p+1 Ui ,...,i { ! 2 } (i0,...,ip) I 0 p for p 0. The Cartan-Leray spectral sequence 2 ` p,q q c p+q c (87) E = H (Xp,et, Z (n)) = H (Xet, Z (n)) 1 ) q c converges by Lemma 5.12. Indeed Lemma 5.12 implies that H (Xp,et, Z (n)) is a Q-vector q c space for q< 2 dim(X), which must be trivial since H (Xp,et, Z (n)) is assumed to be · a finitely generated abelian group. The spectral sequence (87) then shows that Lc is stable under operation ( 4). L ⇤ r Lemma 5.11. Let X be separated of finite type over Spec(Z) and let f : A Xet be the X,et ! natural map. Then one has Rf Zc(n) Zc(n r)[2r] for any n 0. ⇤ '  Proof. Since Zc(n) satisfies étale cohomological descent for n 0 (see [20] Theorem 7.1), one is  reduced to show the analogous statement for the Zariski topology. Using [17] Corollary 3.4, the c c 1 result follows from the homotopy formula Rp Z (n) Z (n 1)[2], where p : AY,Zar YZar ⇤ ' ! is the natural map and Y is defined over a field (see [17] Corollary 3.5). ⇤ One defines Z/mZc(n)=Zc(n) Z/mZ Zc(n) L Z/mZ (since Zc(n) is a complex of flat ⌦ ' ⌦ sheaves) and Q/Zc(n)=lim Z/mZc(n). Note that in the situation of Lemma 5.11 one has ! c c (88) Rf Q/Z (n) Q/Z (n r)[2r]. ⇤ ' Indeed, applying Lemma 5.11, the functor L Z/mZ and passing to the limit, we obtain (88). ⌦ Here we use the fact that Rf commutes with filtered inductive limits (since X is separated ⇤ of finite type over Spec(Z)). Lemma 5.12. Let X be separated of finite type over Spec(Z) and let n 0.Then  i c H (Xet, Q/Z (n)) = 0 for i< 2 dim(X). · n Proof. Replacing X with AX ,wemayassumen =0by (88). There exists a finite étale covering family Vi Spec(Z) such that Vi(R)= , hence a finite étale covering family { ! } ; 48 BAPTISTE MORIN

U X such that U is defined over Spec( ) for some totally imaginary number field K . { i ! } i OKi i If the result is known for the Ui0,...,ip ’s, then it follows for X by the spectral sequence p,q q c p+q c E = H (Xp,et, Q/Z (n)) = H (Xet, Q/Z (n)) 1 ) X = p+1 U U = U ... U (X) where p (i0,...,ip) I i0,...,ip and i0,...,ip i0 X X ip is of dimension dim . 2 ⇥ ⇥  So we may assume that X is defined over Spec( ) for some totally imaginary number field ` OK K. We have an isomorphism of finite groups ([20] Theorem 7.8) 1 i c i D (89) H (Xet, Z/mZ (0)) = Hc(Xet, Z/mZ) i i where the right hand side is the Pontryagin dual of H (Xet, Z/mZ) (here H (Xet, ) denotes c c usual étale cohomology with compact support). Take a Nagata compactification of X over Spec( ), i.e. an open immersion j : X, X with dense image such that X is proper over OK ! 0 0 Spec( K ).NotethatX0(R)= hence X0 is of l-cohomological dimension 2 dim(X)+1for O ; · any prime number l (see [24] X Theorem 6.2). We obtain i i H (Xet, Z/mZ)=H (X0 ,j!Z/mZ)=0for i>2 dim(X)+1 c et · i c hence H (Xet, Z/mZ (0)) = 0 for i< 2 dim(X) by (89). Now the result follows from · i c i c H (Xet, Q/Z (0)) = lim H (Xet, Z/mZ (0)) ! which is valid since the étale site of X is notherian (X is separated of finite type). ⇤ The class (Z) contains singular schemes. For example, it easy to see that any proper curve L (possibly singular) over a finite field lies in (Z). L 5.4. Geometrically cellular schemes. Definition 5.13. Let k be a field and let Y be a scheme separated and of finite type over k. We say that the k-scheme Y has a cellular decomposition if there exists a filtration of Y by reduced closed subschemes red (90) Y = YN YN 1 ... Y 1 = ◆ ◆ ◆ ; ai such that Yi Yi 1 A is k-isomorphic to an affine space over k.Wesaythat(90) is a \ ' k smooth cellular decomposition if Y is moreover smooth over k for any i 0. i The k-scheme Y is geometrically cellular if Y k¯ has a cellular decomposition, where k¯ ⌦k is a separable closure of k. An S-scheme S separated and of finite type is geometrically cellular if the fiber is X! Xs geometrically cellular for any s S. 2 One can easily show that a k-scheme Y is geometrically cellular if and only if there exists a finite Galois extension k /k such that Y k is cellular. It follows from the proof of Proposition 0 ⌦k 0 5.14 below that any geometrically cellular scheme over a finite field belongs to (Z).More L generally, any geometrically cellular scheme over a number ring belongs to (Z). L Proposition 5.14. Let Spec( ) be flat, separated and of finite type over a number X! OF ring ,suchthat is geometrically cellular. The following conditions are equivalent. OF XF For any finite prime p of F , p (Z). • X 2L (Z). •X2L Here we set F := F and p := ( F /p). X X⌦OF X X⌦OF O ZETA FUNCTIONS AT s =0 49

Proof. By assumption there exists a ﬁnite Galois extension K/F such that is cellular. We XK write red ( K) = YN YN 1 ... Y 1 = X⌦OF ◆ ◆ ◆ ; ai such that Yi Yi 1 A and we consider the closure Yi of Yi in F K , where Yi is \ ' K X⌦O O endowed with its structure of reduced closed subscheme. We obtain an isomorphism

ai (Yi Yi 1) K K Yi Yi 1 A \ ⌦O ' \ ' K and it follows that there exist an open Spec( ) Spec( ) and an isomorphism over OK,Si ✓ OK Spec( K,Si ) O ri (Yi Yi 1) K K,Si A . \ ⌦O O ' OK,Si Now we take a finite set S S big enough so that Spec( ) Spec( ) is an étale ◆[ i OK,S ! OF,S Galois cover of group G. Here S also denotes its image in Spec( ). Then we set = OF Yi Yi K,S and we obtain a filtration by (reduced) closed subschemes ⌦OK O red ( K,S) = N N 1 ... 1 = X⌦OF O Y ◆Y ◆ ◆Y ; ri ri such that i i 1 A . But Spec( K,S) belongs to (Z) by ( 0), hence so does A Y \Y ' OK,S O L L OK,S by ( 3), hence so does F K,S by induction and ( 1). Notice that a scheme X SFT(Z) L X⌦O O L 2 belongs to (Z) if and only if Xred does: this follows from ( 1) since (Z). L L ;2L Moreover K,S F,S X⌦OF O ! X⌦ OF O is a finite étale Galois cover, hence F F,S lies in (Z) by ( 4). Therefore, we have X⌦O O L L p S, p (Z) S = p (Z) (Z) 8 2 X 2L ()X X 2L ()X2L p S a2 by ( 2) and ( 1). One may enlarge the finite set S so as to include any given p. L L ⇤ In order to obtain a more functorial formulation of Conjecture 1.4 (which implies B( ,d)) X we use results and notations of [25]. We denote by Hn( , R(p)) Arakelov motivic cohomology X with real coefficients in the sense of [25] Remark 4.7. These groups are defined following the construction of [25] using the spectrum HB Rb instead of HB so that there is an exact ⌦ sequence (see [25] Theorem 4.5 (ii)) n n n ... H ( , R(p)) H ( , Q(p))R H (X/R, R(p)) ... ! X ! X ! D ! at least for an l.c.i. scheme over a number ring in the sense of ([25] Definition 2.3), where X b n (p) X = Q. Here we use the identification H ( , Q(p))R K2p n( ) for regular ([32] X X ' X R X Theorem 11.7). Let be an l.c.i. scheme over a number ring. If is moreover proper, X X regular, connected and d-dimensional then Conjecture 1.4 for is equivalent to X n 2d (91) H ( , R(d)) = 0 for n =2d and H ( , R(d)) = R. X 6 X If the (proper, regular, connected and d-dimensional) scheme lies over a finite field, we say b b X that satisfies Conjecture 1.4 if (91) holds, i.e. if one has X n 2d H ( , Q(d)) = 0 for n =2d and H ( , Q(d)) = Q. X 6 X Note that for (proper, regular, connected and d-dimensional) over a finite field one has X (92) L( ,d) Conjecture 1.4 for . Xet ) X 50 BAPTISTE MORIN

Proposition 5.15. Let be a smooth and projective scheme over the number ring ( )= X OX X F .Assumethat (Z) and that F admits a smooth cellular decomposition. Then O X2L X X satisﬁes Conjecture 1.4. In particular, B( ,d) holds, where d =dim( ). X X Proof. By assumption there exist a ﬁltration

F = YN ! YN 1 ! ... ! Y0 ! Y 1 = X ; ai by smooth closed subschemes Yi and isomorphisms Yi Yi 1 A .Asintheproofof \ ' F Proposition 5.14, one can show that there exist an open subscheme U Spec( ),afiltration ⇢ OF U = N ! N 1 ! ... ! 0 ! 1 = X Y Y Y Y ; ai and U-isomorphisms i i 1 A such that the following holds. One has i U F Yi, Y \Y ' U Y ⌦ ' the scheme is a closed subscheme of ,and is smooth over U. Yi Yi+1 Yi By ([36] Corollary 5.3.17) F is smooth and connected, hence irreducible. This gives 0 X CH ( F )=Z. On the other hand, there is a direct sum decomposition h( F )= h(F )( ai) X X 0 i N in the category of Chow motives ([5] Theorem 3.1). It follows that there exists a unique  index a0 L 0 i0 N such that ai0 =0. But F is proper (over F ), hence so is Y0 A , hence i0 =0.   X ' F The (absolute) dimension yields a locally constant map di : i Z. We define ci : i Z Y ! Y ! similarly: Let y and let (resp. ) be the connected component of (resp. of 2Yi Yi,y Yi+1,y Yi )containingy. Then we define c (y)=codim( , ) and Yi+1 i Yi,y Yi+1,y n 2ci n 2ci(y) H ( i, R(di)) := H ( i,y, R(di(y))) Y Y y ⇡0( ) 2MYi b b For any 0 i N we have a long exact sequence (see [25] Theorem 4.16 (iii))   n 2ci 1 n n ... H ( i 1, R(di 1)) H ( i, R(di)) H ( i i 1, R(di)) ... ! Y ! Y ! Y \Y ! ai Notice that the locally constant function di is constant on i i 1 A with constant value b b Yb\Y ' U a +1. But for i 1,onehasa > 0 and i i n n ai n H ( i i 1, R(di)) H (A , R(ai + 1)) = H (U, R(ai + 1)) = 0 for all n. Y \Y ' U Hn(U, (a + Indeed, theb second equality is givenb by ([25] Theorem 4.16b (ii)) and the vanishing of R i 1)) for a > 0 follows from the fact that U is of the form Spec( ) since the Beilinson reg- i OF,S ulator b

n ⇠ n ⇠ n H (Spec( F,S), Q(ai + 1))R H (Spec(F ), Q(ai + 1))R H (Spec(F )/R, R(ai + 1)) O ! ! D is an isomorphism for ai > 0 and all n. We obtain an identity

n 2ci 1 n H ( i 1, R(di 1)) H ( i, R(di)) for all n. Y ' Y i=N 1 Notice that V and that d and c are both constant on with constant value Y0 ' b 0 i=0 i b Y0 1 and d 1 respectively. An induction on i yields P n n 2d+2 n 2d+2 (93) H ( U , R(d)) H ( 0, R(1)) H (U, R(1)) for all n. X ' Y ' Using (93), the fact that U is of the form Spec( F,S) (with S = ) and well known facts b b n O b 6 ; concerning the Dirichlet regulator, we obtain H ( U , R(d)) = 0 for n =2d 1 and an exact X 6 sequence 2d 1 0 H ( U , R(d))b R R 0 ! X ! ! ! p S Y2 b ZETA FUNCTIONS AT s =0 51 where S denotes the closed complement of U in Spec( ).Foranyfiniteprimep of F , OF Xp is (geometrically) connected (see [36] Corollary 5.3.17) of dimension d 1. By Propositions 5.14 and 5.10, p satisfies L( p,et,d 1) hence p satisfies Conjecture 1.4 by (92). We set X Xn 2 X S := p. The fact that H ( S, R(d 1)) = 0 for n =2d and the long exact sequence X S X X 6 n 2 n n ` ... H ( S, R(d 1)) H ( , R(d)) H ( U , R(d)) ... ! X b ! X ! X ! Hn( , (d)) = 0 n =2d, 2d 1 then give R b for b. The result followsb from the fact that there is a morphism of exactX sequences 6

2bd 1 2d 1 2d 2 2d 0 / H ( , R(d)) / H ( U , R(d)) / H ( S, R(d 1)) / H ( , R(d)) / 0 X X X X

= b b b ' b ✏ ✏ ✏ 2d 1 ✏ 0 / 0 / H ( U , R(d)) / R / R / 0 X p S Y2 b ⇤

Acknowledgments. I am very grateful to C. Deninger for comments and advices and to M. Flach for many discussions related to Weil-étale cohomology. I would also like to thank T. Chinburg, T. Geisser and S. Lichtenbaum for comments and discussions. This work is based on ideas of S. Lichtenbaum. Finally, I would like thank the referees for many constructive suggestions and comments. During the preparation of this work, the author was supported by the Marie Curie fellowship 253346.

References [1] Benois, D.; Nguyen Quang Do, T.: Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs Q(m) sur un corps abélien. Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 5, 641–672. [2] Bienenfeld, M.: An étale cohomology duality theorem for number ﬁelds with a real embedding. Trans. Amer. Math. Soc. 303 (1) (1987), 71–96. [3] Bloch, S.: Algebraic cycles and the Beilinson conjectures. The Lefschetz centennial conference, Part I (Mexico City, 1984), 65–79, Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1986. [4] Borel, A.: Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). [5] Brosnan, P.: On motivic decompositions arising from the method of Bialynicki-Birula. Invent. Math. 161 (2005), no. 1, 91–111. [6] Burgos Gil, J. I.: Semipurity of tempered Deligne cohomology. Collect. Math. 59 (1) (2008), 79–102. [7] Burgos Gil, J. I., Feliu, E., Takeda, Y.: On Goncharov’s Regulator and Higher Arithmetic Chow Groups. To appear in Int. Math. Res. Not. IMRN. [8] Burns, D.: Perfecting the nearly perfect. Pure Appl. Math. Q. 4 (4) (2008), 1041–1058. [9] Burns, D., Flach, M.: On Galois structure invariants associated to Tate motives. Amer. J. Math. 120 (1998), no. 6, 1343–1397. [10] Deligne, P.: La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 273–307. [11] Deninger, C.: Analogies between analysis on foliated spaces and arithmetic geometry. Groups and analysis, 174–190, London Math. Soc. Lecture Note Ser., 354, Cambridge Univ. Press, Cambridge, 2008. [12] Flach, M.: The equivariant Tamagawa number conjecture: a survey. With an appendix by C. Greither. Contemp. Math., 358, Stark’s conjectures: recent work and new directions, 79–125, Amer. Math. Soc., Providence, RI, 2004. [13] Flach, M.: Cohomology of topological groups with applications to the Weil group. Compositio Math. 144 (3) (2008), 633–656. 52 BAPTISTE MORIN

[14] Flach, M., Morin, B.: On the Weil-étale topos of regular arithmetic schemes. Doc. Math. 17 (2012) 313– 399. [15] Fontaine, J-M.: Valeurs spéciales des fonctions L des motifs. Séminaire Bourbaki, Vol. 1991/92. Astérisque No. 206 (1992), Exp. No. 751, 4, 205–249. [16] Fontaine, J-M., Perrin-Riou, B.: Autour des conjectures de Bloch et Kato : cohomologie galoisienne et valeurs de fonctions L. Motives (Seattle, WA, 1991), 599–706, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. [17] Geisser, T.: Motivic cohomology over Dedekind rings. Math. Z. 248 (4) (2004), 773–794. [18] Geisser, T.: Weil-étale cohomology over finite fields. Math. Ann. 330 (4) (2004), 665–692. [19] Geisser, T.: Motivic cohomology, K-theory and topological cyclic homology. Handbook of K-theory. Vol. 1, 193–234. [20] Geisser, T.: Duality via cycle complexes. Ann. of Math. (2) 172 (2) (2010), 1095–1126. [21] Geisser, T., Levine, M.: The K-theory of fields in characteristic p. Invent. Math. 139 (2000), no. 3, 459–493. [22] Geisser, T., Levine, M.: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math. 530 (2001), 55–103. [23] Goncharov, A. B.: Polylogarithms, regulators, and Arakelov motivic complexes. J. Amer. Math. Soc. 18 (1) (2005), 1–60. [24] Grothendieck, A., Artin, M., Verdier, J.L.: Théorie des Topos et cohomologie étale des schémas (SGA4). Lectures Notes in Math. 269, 270, 305, Springer, 1972. [25] Holmstrom, A., Scholbach, J.:Arakelov motivic cohomology I.Preprint2011. [26] Kahn, B.: Algebraic K-theory, algebraic cycles and arithmetic geometry. Handbook of K-theory. Vol. 1, 351–428, Springer, Berlin, 2005. [27] Kahn, B.: Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini. Ann. Sci. école Norm. Sup. (4) 36 (6) (2003), 977–1002. [28] Kato, K., Saito, S.: Global class field theory of arithmetic schemes. Contemp. Math., 55, 255–331, Amer. Math. Soc., Providence, RI, 1986. [29] Knudsen, F., Mumford, D.: The projectivity of the moduli space of stable curves I: Preliminaries on ‘det’ and ‘Div’. Math. Scand. 39 (1976), 19–55. [30] Kolster, M., Sands, J.W.: Annihilation of motivic cohomology groups in cyclic 2-extensions. Ann. Sci. Math. Québec 32 (2) (2008), 175–187. [31] Levine, M.: K-theory and motivic cohomology of schemes. Preprint (1999). http://www.math.uiuc.edu/K- theory/336/ [32] Levine, M.: Techniques of localization in the theory of algebraic cycles. J. Algebraic Geom. 10 (2) (2001), 299–363. [33] Lichtenbaum, S.: Values of zeta-functions, étale cohomology, and algebraic K-theory. Lecture Notes in Math., Vol. 342, 489–501, Springer, Berlin, 1973. [34] Lichtenbaum, S.: The Weil-étale topology on schemes over finite fields. Compositio Math. 141 (3) (2005), 689–702. [35] Lichtenbaum, S.: The Weil-étale topology for number rings. Ann. of Math. (2) 170 (2) (2009), 657–683. [36] Liu, Q. Algebraic geometry and arithmetic curves. Oxford Graduate Texts in Mathematics, 6. Oxford Science Publications. Oxford University Press, Oxford, 2002. [37] Milne, J.: Arithmetic duality theorems. Perspectives in Mathematics 1, Academic Press, Inc., Boston, Mass., 1996. [38] Morin, B.: On the Weil-étale cohomology of number fields. Trans. Amer. Math. Soc. 363 (2011), 4877–4927. [39] Morin, B.: The Weil-étale fundamental group of a number field II. Selecta Math. (N.S.) 17 (1) (2011), 67–137. [40] Rognes, J.; Weibel, C.: Two-primary algebraic K-theory of rings of integers in number fields. Appendix A by Manfred Kolster. J. Amer. Math. Soc. 13 (2000), no. 1, 1–54. [41] Soulé, C.: K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math. 55 (1979), no. 3, 251–295. [42] Soulé, C.: K-théorie et zéros aux points entiers de fonctions zéta. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 437–445. [43] Soulé, C.: Groupes de Chow et K-théorie de variétés sur un corps fini. Math. Ann. 268 (1984), no. 3, 317–345. ZETA FUNCTIONS AT s =0 53

[44] Swan, R.G.: A new method in fixed point theory. Comment. Math. Helv. 34 (1960), 1–16. [45] Verdier, J.L.: Des catégories dérivées des catégories abéliennes. Astérisque No. 239 (1996). [46] Voevodsky, V.: On motivic cohomology with Z/l-coefficients. Ann. of Math. (2) 174 (2011), no. 1, 401–438. [47] Weibel, C.: Algebraic K-theory of Rings of Integers in Local and Global Fields. Handbook of K-theory. Vol. 1, 139–190. [48] Wiesend, G.: Class field theory for arithmetic schemes. Math. Z. 256 (2007), no. 4, 717–729.