2 JAN DENEF AND FRANC OS LOESER [S] = [S n S 0 ] + [S 0 ] if S 0 is closed in S and [S S 0 ] = [S] [S 0 ]. We set L := [A 1 k ]. n this setting, the a

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1 MOTVC GUSA ETA FUNCTONS JAN DENEF AND FRANC OS LOESER To appear in Journal of Algebraic Geometry 0. ntroduction Let p be a prime number and let K be a nite extension of Q p. Let R be the valuation ring of K, P the maximal ideal of R, and K = R=P the residue eld of K. Let q denote the cardinality of K, so K ' F q. For z in K, let ord z denote the valuation of z, and set jzj = q?ord z. Let f be a non constant element of K[x 1 ; : : : ; x m ]. The p-adic gusa local zeta function (s) associated to f (relative to the trivial multiplicative character) is dened as the p-adic integral (0.1) (s) = R m jf(x)j s jdxj; for s 2 C, Re(s) > 0, where jdxj denotes the Haar measure on K m normalized in such of way that R m is of volume 1. For n in N, set n = fx 2 R m ord f(x) = ng: We may express (s) as a series (0.2) (s) = n0 vol ( n ) q?ns : Now, if we denote by n the image of n in (R=P n+1 ) m, we may rewrite the series as (0.3) (s) = n0 since vol ( n ) = card ( n ) q?(n+1)m. card ( n ) q?ns?(n+1)m Let k be a eld of characteristic zero. M.Kontsevich recently introduced the concept of motivic integration [13] (see also [6] and 2.5), which is a k[[t]]-analogue of usual p-adic integration. This motivic integration takes values into a certain completion of a localisation of the Grothendieck ring K 0 (Sch k ) of algebraic varieties over k, i.e. reduced separated schemes of nite type over k, (see 2.5 for more details). The ring K 0 (Sch k ) is generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is isomorphic to S 0, Date: revised september

2 2 JAN DENEF AND FRANC OS LOESER [S] = [S n S 0 ] + [S 0 ] if S 0 is closed in S and [S S 0 ] = [S] [S 0 ]. We set L := [A 1 k ]. n this setting, the analogue of Rm, resp. of (R=P n+1 ) m, is the k-scheme L(A m k ), resp. L n(a m k ), which represents the functor R 7! R[[t]] m, resp. the functor R 7! (R[[t]]=t n+1 R[[t]]) m, on the category of k-algebras, i.e. the k-scheme parametrizing m-tuples of series, resp. of series modulo t n+1. Let f be a non constant element of K[x 1 ; : : : ; x m ]. Dene n as the subscheme of L(A m k ) of series ' such that ord t f(') = n and n as the image of n in L n (A m k ), viewed as a reduced subscheme. A natural analogue of the right-hand side of (0.3), which is a series in [p?1 ][[p?s ]], is the following series in K 0 (Sch k )[L?1 ][[L?s ]] (0.4) geom (s) = n0 [ n ] L?ns?(n+1)m : When s takes a xed value in N, this series can be interpreted as a Kontsevich integral (see 2.5). More generally, p-adic gusa local zeta functions involve multiplicative characters. Let be a xed uniformizing parameter of R and set ac(z) = z?ord z for z in K. For any character : R! C, one denes the p-adic gusa local zeta function (s; ) as the integral (0.5) (s; ) = (ac(f(x)))jf(x)j s jdxj; R m for s 2 C, Re(s) > 0 (see [12], [3]). To extend the denition of (0.4) to the more general situation involving characters, it is necessary to replace varieties by motives. More generally, let be a smooth connected separated scheme of nite type over k (a eld of characteristic zero), let W be a reduced subscheme of, and let f :! A 1 k be a morphism. n the present paper, we dene, for a multiplicative character of any nite subgroup of k, motivic gusa functions W (f s ; ). These functions live in a power series ring K 0 (M)[[L?s ]], where K 0 (M) is a Grothendieck ring of Chow motives and L is the standard Lefschetz motive (precise denitions are given in 1.1 and 2.1), and are dened as series quite similar to the right-hand side of (0.4). These motivic gusa functions specialize, in the p-adic case with good reduction, to the usual p-adic gusa local zeta functions (see 2.4). They also specialize to the topological zeta functions top (s) introduced by the authors in [5] (see 2.3). The functions top (s) are, heuristically, obtained as a limit as q goes to 1 of p-adic gusa local zeta functions (in a more provocative way, one can say they are dened by integrals over W (F 1 ), the ring of Witt vectors with coecients in the eld with one element).

3 W MOTVC GUSA ETA FUNCTONS 3 The content of the paper is the following. The motivic integrals (f s ; ) are dened in section 2. Their denition uses variants for schemes with nite group action of recent results of Gillet-Soule [8] and Guillen-Navarro [9] on motivic Euler characteristics of schemes which are given in section 1. n Theorem we give a formula for W (f s ; ) in terms of an embedded resolution of f, a result which implies in particular the rationality of W (f s ; ). We then explain the relationship with topological zeta functions, p-adic gusa local zeta functions and motivic integration. Section 3 is devoted to functional equations. Here = W = A m k and f is a homogenous polynomial. n this situation we are able to prove a functional equation for motivic gusa functions when is the trivial character. For general the result depends upon a conjectural statement on motivic Euler characteristics of quotients, but we are able to prove it holds true if one replaces the Grothendieck group of Chow motives by the Grothendieck group of Voevodsky's \triangulated category of geometrical motives" [28]. These functional equations are analogues in the present setting of the functional equations for p-adic gusa local zeta functions proved in [7]. n section 4 we study the limit for s!?1 of motivic gusa functions and investigate its relation with nearby cycles of f at the origin. Here we are guided by analogy with [4], where the limit when s!?1 was studied for p-adic gusa local zeta functions, and showed to be related to the trace of some liftings of the Frobenius automorphism acting on the cohomology of Milnor bers. More precisely, for x a closed point of the ber f?1 (0), we give a meaning to L m 1? L lim s!?1 (f s ; ): fxg Heuristically this limit is the \motivic incarnation" of c (i xr f; ), where R f; denotes the eigenspace of nearby cycles for the eigenvalue corresponding to the character of the semi-simple part of the monodromy. We prove in Theorem that this holds in particular for the C-Hodge realization. As a corollary it follows that the whole Hodge spectrum of f at x, which is an important invariant of singularities (see [22],[23]), may be deduced from the knowledge of motivic gusa functions. Acknowledgements. We would like to thank O.Villamayor for answering to our questions on equivariant resolution and F.Morel for interesting discussions.

4 4 JAN DENEF AND FRANC OS LOESER 1. Grothendieck groups of Chow motives 1.1. Chow motives. n this section we recall material from [14], [15], [21]. We x a base eld k, and we denote by V k the category of smooth and projective k-schemes. For an object in V k and an integer d, d () denotes the free abelian group generated by irreducible subvarieties of of codimension d. We dene the rational Chow group A d () as the quotient of d () Q modulo rational equivalence. For and Y in V k, we denote by Corr r (; Y ) the group of correspondences of degree r from to Y. f is purely d-dimensional, Corr r (; Y ) = A d+r ( Y ), and if = ` i, Corr r (; Y ) = Corr r ( i ; Y ). The category M k of k-motives may be dened as follows (cf. [21]). Objects of M k are triples (; p; n) where is in V k, p is an idempotent (i.e. p 2 = p) in Corr 0 (; ), and n is an integer. f (; p; n) and (Y; q; m) are motives, then Hom Mk ((; p; n); (Y; q; m)) = q Corr m?n (; Y ) p: Composition of morphisms is given by composition of correspondences. The category M k is additive, Q-linear, and pseudo-abelian. There is a natural tensor product on M k, dened on objects by (; p; n) (Y; q; m) = ( Y; p q; n + m): We denote by h the functor h : V k! M k which sends an object to h() = (; id; 0) and a morphism f : Y! to its graph in Corr 0 (; Y ). This functor is compatible with the tensor product and the unit motive 1 = h(spec k) is the identity for the product. We denote by L the Lefschetz motive L = (Spec k; id;?1). There is a canonical isomorphism h(p 1 k ) ' 1 L. We denote by _ the involution _ : M k! M k, dened on objects by (; p; n) _ = (; t p; d? n) if is purely d-dimensional, and as the transpose of correspondences on morphisms. For in V k purely of dimension d, h() _ = h() L?d. Let E be a eld of characteristic zero. Replacing the Chow groups A by A Q E, one denes similarly the category M k;e of k-motives with coecients in E Grothendieck groups of Chow motives. Let K 0 (M k ) be the Grothendieck group of the pseudo-abelian category M k. t is also the abelian group associated to the monoid of isomorphism classes of motives with respect to. The tensor product on M k induces a natural ring structure on K 0 (M k ). Let Sch k be the category of schemes which are separated and of nite type over k. We suppose from now that the characteristic of k is zero. The following result has been proven by Gillet and Soule [8] and also by Guillen and Navarro Aznar [9].

5 MOTVC GUSA ETA FUNCTONS 5 Theorem Let k be a eld of characteristic 0. There exists a unique map such that c : ObSch k?! K 0 (M k ) (1) f is smooth and projective, c () is equal to [h()], the image of h() in K 0 (M k ). (2) f Y is a closed subscheme in a scheme, c ( n Y ) = c ()? c (Y ): Let us remark that c (A 1 k ) = L. Also, for and Y in ObSch k, we have c ( Y ) = c () c (Y ). The following result, due to Guillen and Navarro Aznar [9], gives, dually, the existence of motivic Euler characteristics without supports. A proper relative isomorphism ( ~; ~Y )! (; Y ) consists of the following data : a proper morphism f : ~! between objects of Sch k, reduced closed subschemes ~Y and Y of ~ and respectively, such that ~Y is the preimage of Y in ~, and such that the restriction of f to ~ n ~Y is an isomorphism onto n Y. Theorem Let k be a eld of characteristic 0. There exists a map : Ob Sch k?! K 0 (M k ) such that (1) f is smooth and projective, () = [h()]. (2) f ( ~; ~Y )! (; Y ) is a proper relative isomorphism, () = ( ~) + (Y )? ( ~Y ): (3) f Y is a smooth divisor in a smooth scheme, ( n Y ) = ()? (Y ) L: (4) f is a smooth scheme purely of dimension d, () _ = c () L?d Furthermore, is determined by conditions (1)-(3). The Euler characteristics c and are compatible with realization functors, in particular with Euler characteristics of mixed Hodge structures on cohomology with compact support and cohomology, respectively. By additivity c may be naturally extended to constructible sets.

6 6 JAN DENEF AND FRANC OS LOESER Remark We expect, but do not know how to prove, that K 0 (M k;e ) has no (L? 1)-torsion. This assertion is implied by the conjectural existence (cf. [21] p.185) of additive functors h j : M k;e! M k;e, j 2, such that for any in M k;e, the h j () form a ltration of with h?k () = 0, h k () = for some k, and h j (L) = L h j?2 () for all j. ndeed, for A in K 0 (M k;e ) the h j (A) are well dened in K 0 (M k;e ) and the relation (L?1)A = 0 implies h j (A) = L h j?2 (A), whence A = 0. A similar argument also shows, without using any conjecture, that the etale realization and the Hodge realization kill all (L? 1) i -torsion in K 0 (M k;e ), for each i in N Finite group action. Let G be a nite abelian group and let ^G be its complex character group. We denote by V k;g the category of smooth and projective k-schemes with G-action. Let E be a subeld of C containing all the roots of unity of order dividing jgj. For in V k;g and g in G, we denote by [g] the correspondence given by the graph of multiplication by g. For in ^G we consider the idempotent f := jgj?1 g2g?1 (g)[g] in Corr 0 (; ) E, and we denote by h(; ) the motive (; f ; 0) in M k;e. Clearly, for in V k;g purely of dimension d and in ^G, we have h(; ) _ = h(;?1 ) L?d. We will denote by Sch k;g the category of separated schemes of nite type over k with G-action satisfying the following condition: the G-orbit of any closed point of is contained in an ane open subscheme. This condition is clearly satised for quasiprojective and insures the existence of =G as a scheme. Objects of Sch k;g will be called G-schemes. We will need the following variants of Theorems and They are proved in the appendix as a consequence of [9] and [27]. Theorem will only be used in section 3. Theorem Let k be a eld of characteristic 0. There exists a unique map c : ObSch k;g ^G?! K 0 (M k;e ) such that (1) f is smooth and projective with G-action, for any character, c (; ) = [h(; )]. (2) f Y is a closed G-stable subscheme in a G-scheme, for any character, c ( n Y; ) = c (; )? c (Y; ):

7 MOTVC GUSA ETA FUNCTONS 7 (3) f is a G-scheme, U and V are G-invariant open subschemes of, for any character, c (U [ V; ) = c (U; ) + c (V; )? c (U \ V; ): Furthermore, c is determined by conditions (1)-(2). By a proper relative isomorphism of G-schemes ( ~; ~Y )! (; Y ) we mean the following data : a proper morphism f : ~! of G- schemes, reduced closed G-stable subschemes ~ Y and Y of ~ and respectively, such that ~ Y is the preimage of Y in ~, and such that the restriction of f to ~ n ~ Y is an G-isomorphism onto n Y. Theorem Let k be a eld of characteristic 0. There exists a map : Ob Sch k;g ^G?! K 0 (M k;e ) such that (1) f is smooth and projective with G-action, (; ) = [h(; )]. (2) f ( ~; ~Y )! (; Y ) is a proper relative isomorphism of G-schemes, (; ) = ( ~ ; ) + (Y; )? ( ~ Y ; ): (3) f Y is a smooth G-invariant divisor in a smooth G-scheme, ( n Y; ) = (; )? (Y; ) L: (4) f is a smooth G-scheme purely of dimension d, (5) f is a proper G-scheme, (; ) _ = c (;?1 ) L?d : c (; ) = (; ): (6) f is a G-scheme and U and V are G-invariant open subschemes of, then, for any character, (U [ V; ) = (U; ) + (V; )? (U \ V; ): Furthermore, is determined by conditions (1)-(3). Proposition Let k be a eld of characteristic 0. (1) For any in ObSch k;g, c () = 2 ^G c (; ): (2) Let be in ObSch k;g. Assume the G-action factors through a quotient G! H. f is not in the image of ^H! ^G, then c (; ) = 0.

8 8 JAN DENEF AND FRANC OS LOESER (3) Let and Y be in ObSch k;g and let G act diagonally on Y. Then c ( Y; ) = 2 ^G c (; ) c (Y;?1 ): Proof. f L L is smooth and projective with G-action, d = 2 ^G f, so h() = 2 ^G h(; ). t is a direct verication that, if the G- action factors through a quotient G! H and is not in the image of ^H! ^G, then f = 0. f Y is another smooth and projective scheme with G-action, then f ( Y ) = 2 ^G f () f?1(y ): Assertions (1), (2) and (3) follow by additivity of c ( ; ) Motivic Kummer sheaves. We x an integer d 1. We denote by d (k) the group of d-roots of 1 in k and by d a xed primitive d-th root of unity in C. We assume from now on that d (k) is of order d. Let f :! G m;k be a morphism in Sch k. For any character of order d of d (k), one may dene an element [; f L ] of K 0 (M k;q[d ]) as follows. The morphism [d] : G m;k! G m;k given by x 7! x d is a Galois covering with Galois group d (k). We consider the ber product e f;d G m;k [d] f G m;k : The scheme e f;d is endowed with an action of d (k), so we can dene [; f L ] := c ( e f;d ; ): Lemma Let f :! G m;k and g :! G m;k be morphisms in Sch k. For any character of order d of d (k), the following holds n particular, [; f d L ] = c (): [; (f d g) L ] = [; g L ]: Proof. The morphism (x; t) 7! (x; tf?1 (x)) induces an isomorphism of d (k)-schemes e f d g;d ' e g;d. For the last assertion remark that the ber product e 1;d is isomorphic as a d (k)-scheme to the product d (k).

9 MOTVC GUSA ETA FUNCTONS 9 Lemma Let f :! G m;k and g : Y! be morphisms in Sch k. Assume that g is a locally trivial bration for the ariski topology with ber. For any character of order d of d (k), the following holds [Y; (f g) L ] = c ()[; f L ]: Proof. mmediate. Lemma Let a be an integer and let d (k) act on G m;k by multiplication by a, 2 d (k). For any non trivial character of d (k), c (G m;k ; ) = 0. Proof. The action of d (k) on G m;k extends to an action on P 1 k leaving xed 0 and 1. So it is enough to verify that if is a non trivial character of d (k), [h(p 1 k ; )] = 0. Now remark that, for any in k, the class of the graph of the multiplication by in A 1 (P 1 k P 1 k ) is equal to the class of the diagonal, hence f = 1 if is trivial, and f = 0 otherwise. f f :! G m;k and g : Y! G m;k are morphisms in Sch k, we denote by f g the morphism Y! G m;k given by multiplication of f and g. We have the following generalization of Lemma Lemma Let g :! G m;k be a morphism in Sch k. For any character of order d of d (k) and any integer n not divisible by d, [G m;k ; ([n] g) L ] = 0: (Gm;k ) [n]g;d. We may identify W with f(x; z; t) 2 Proof. Set W = G m;k G m;k x n g(z) = t d g, the action of d (k) being multiplication on the last factor. Set = gcd(n; d), n = n 0, d = d 0, and choose integers a and b such that an 0 = 1+bd 0. f we set w = t d0 x?n0, x 0 = w a x, t 0 = w b t, we may identify W with f(x 0 ; z; t 0 ; w) 2 G m;k G m;k G m;k g(z) = w and x 0n0 = t 0d0 g: We may rewrite this as an isomorphism W ' g; e G m;k, the action of d (k) being the product of the action on g; e given by composition with the surjection 7! d0, d (k)! (k), with the action on G m;k given by multiplication by a, for 2 d (k). By Proposition (3), c (W; ) = 2 d (k) c (e g; ; ) c (G m;k ;?1 ): Hence, by Lemma and Proposition (1), c (W; ) = c (e g; ; ) c (G m;k ):

10 10 JAN DENEF AND FRANC OS LOESER But c (e g; ; ) = 0, by Proposition (2), because is of order d and < d Quotients. We discuss here motivic Euler characteristics of quotients. This part will only be used in section 3. Lemma Let be a smooth projective scheme with G-action, H a subgroup of G, and a character of G=H. Assume the quotient =H is smooth. Then h(=h; ) ' h(; %), where % is the projection G! G=H. Proof. The projection! =H induces by functoriality a morphism h(=h)?! h() in M k, which induces an isomorphism between h(=h; ) and h(; %). n view of Lemma and Corollary 1.5.4, it seems quite natural to expect the following statement holds. Assertion f is a G-scheme, H a subgroup of G, and a character of G=H, then c (=H; ) = c (; %) and (=H; ) = (; %), where % is the projection G! G=H. n the paper [28], Voevodsky constructs for a perfect eld k a tensor triangulated category DM gm (k) which he calls the triangulated category of geometrical motives. When k is of characteristic zero, he associates to any object in Sch k complexes Mgm() c and M gm () in DM gm (k). Let E be a eld of characteristic zero and denote by DM gm (k) E the category DM gm (k) E. By [28] 2.2, when is proper and smooth M gm () = M c gm() and the restriction of M gm to V k factorizes through an additive functor M k;e! DM gm (k) E. Hence there is a canonical morphism of groups ' : K 0 (M k;e )! K 0 (DM gm (k) E ). The tensor structure on DM gm (k) induces a ring structure on the Grothendieck group K 0 (DM gm (k) E ) and ' is a morphism a rings. By [28] Corollary 3.5.5, the morphism ' is surjective, but it does not seem to be known whether ' is injective or not. t follows directly from the properties of c and Mgm c that, for any in Sch k, '( c ()) = [Mgm()] c (cf. citeg-s 3.2.4). Similarly, it follows by an easy induction on dimension and the properties of and M gm that, for any in Sch k, '(()) = [M gm ()]. Let G be a nite abelian group and assume E contains all the roots of unity of order dividing jgj. Let be an object of Sch k;g. Then it follows from the denition of the complexes Mgm() c and M gm () that G acts on them and

11 MOTVC GUSA ETA FUNCTONS 11 that they decompose in DM gm (k) E into direct sums of isotypic components M c gm() ' L 2 ^G M c gm() and M gm () ' L 2 ^G M gm(). One derives similarly as before that '( c (; )) = [M c gm() ] and '((; )) = [M gm () ]. Lemma f is a G-scheme, H a subgroup of G, and a character of G=H, then there are canonical isomorphisms M c gm(=h) ' M c gm() % and M gm (=H) ' M gm () % : Proof. This follows directly from the denition (such a statement is already true at the level of the Nisnevich sheaves L c and L of [28] 4.1). Corollary f is a G-scheme, H a subgroup of G, and a character of G=H, then and '( c (=H; )) = '( c (; %)) '((=H; )) = '((; %)): 2. Motivic gusa zeta functions 2.1. We will consider the ring of formal series K 0 (M k;q[d ])[[L?s ]]. n this ring we will write L j (L?s ) i = L j?si, when j 2 and i 2 N. We will also consider the subring K 0 (M k;q[d ]) [L?s ] loc of the ring K 0 (M k;q[d ]) [[L?s ]] generated by K 0 (M k;q[d ]) [L?s ] and the series (1? L?Ns?n )?1 = i2n L?Nsi?ni ; for N and n in N n f0g. Let be a smooth and connected separated k-scheme of nite type of dimension m, f :! A 1 k be a morphism, and W be a reduced subscheme of. We assume that d (k) is of order d. For any character of d (k) of order d, we dene the motivic gusa zeta function W (f s ; ) in K 0 (M k;q[d ]) [[L?s ]] as follows. We denote by L n () the k-scheme which represents the functor, dened on the category of k-algebras, R 7! Mor k?schemes (Spec R[t]=t n+1 R[t]; ); for n 0 (cf. p.276 of S. Bosch, W. Lutkebohmert and M. Raynaud, Neron models, Ergeb. Math. Grenzgeb. (3) 21, Springer-Verlag, Berlin, 1990). We denote by L() the projective limit in the category of schemes of the schemes L n (), which exists since the transition maps are ane. Note that for any eld K containing k the

12 12 JAN DENEF AND FRANC OS LOESER K-rational points of L() are the morphisms Spec K[[t]]!. For f :! A 1 k = Spec k[x] a morphism, and n in N, we dene n;f;w as the reduced subscheme of L() whose K-rational points, for any eld K containing k, are the morphisms ' : Spec K[[t]]! sending the closed point of Spec K[[t]] to a point in W, and such that f ' is exactly of order n at the origin. We denote by n;f;w the image of n;f;w in L n (), viewed as a reduced subscheme of L n (), and by f the morphism f : n;f;w! G m;k which associates to ' in n;f;w the constant term of the series t?n x(f '). We dene, for any character of d (k) of order d, W (f s ; ) := n2n [ n;f;w ; f L ] L?ns?(n+1)m in K 0 (M k;q[d ]) [[L?s ]]. When is the trivial character, we write instead of Remarks. W the series (f s ; ). 1. When is the trivial character, f s := [ n;f;w ] L?ns?(n+1)m W n2n in K 0 (Sch k )[L?1 ][[L?s ]] by the natural morphism K 0 (Sch k )[L?1 ][[L?s ]]?! K 0 (M k ) [[L?s ]] W f s W f s is the image of induced by c. 2. t would be interesting to investigate whether motivic gusa functions already exist at a ner level than a Grothendieck group of Chow motives, for instance at the level of complexes of Chow motives, or objects of DM gm (k), or \mixed motives" Let D be the divisor dened by f = 0 in. Let (Y; h) be a resolution of f. By this, we mean that Y is a smooth and connected k-scheme of nite type, h : Y! is proper, that the restriction h : Y n h?1 (D)! n D is an isomorphism, and that (h?1 (D)) red has only normal crossings as a subscheme of Y. Let E i, i 2 J, be the irreducible (smooth) components of (h?1 (D)) red. For each i 2 J, denote by N i the multiplicity of E i in the divisor of f h on Y, and by i? 1 the multiplicity of E i in the divisor of h dx, where dx is a local non vanishing volume form, i.e. a local generator of the sheaf of

13 MOTVC GUSA ETA FUNCTONS 13 dierential forms of maximal degree. For i 2 J and J, we consider the schemes E i := E i n[ j6=i E j, E := \ i2 E i, and E := E n[ j2jn E j. When = ;, we have E ; = Y. Now denote by J d the set of J such that d j N i for all i in and by U d the union of the E, with in J d. Let be locally closed in U d. For any character of d (k) of order d, we will construct an element [ f; ] in K 0 (M k;q[d ]) as follows. f on we may write f h = uv d with u non vanishing on, we set [ f; ] = [; u L ]. t is well dened by Lemma n general we cover by a nite set of r 's for which the previous condition holds, and we set [ f; ] = [( r ) f; ]? [( r1 \ r2 ) f; ] + ; r r 1 6=r 2 which is well dened by additivity of c ( We can now state the following result. ; ). Theorem For any character of d (k) of order d, Y (f s ; ) = L?m [(E \ h?1 (W )) f; ] W 2J d i2 in K 0 (M k;q[d ]) [[L?s ]]. K 0 (M k;q[d ]) [L?s ] loc. n particular W (L? 1) L?N is? i 1? L?N is? i (f s ; ) belongs to the ring Proof. We set L(; D) := L() n L(D), and we dene similarly the scheme L(Y; h?1 (D)). We denote by n the projections L(; D)! L n () and L(Y; h?1 (D))! L n (Y ), and by the projections onto and Y respectively. f U is a reduced subscheme of Y we set L U (Y; h?1 (D)) =?1 (U) and L U (; D) = h (L U (Y; h?1 (D))), and we dene similarly L n;u (Y ). Moreover, for n 0 n, we denote by n 0 ;n the projections L n 0()! L n () and L n 0(Y )! L n (Y ). The morphism h being proper, composition with h induces a bijective morphism h : L(Y; h?1 (D)) ' L(; D), and we have a commutative diagram L(Y; h?1 (D)) n L n (Y ) h L(; D) For U a reduced subscheme of Y, we set n h n L n (): n0 n;f;w;u := n 0( n;f;w \ L U (; D)):

14 14 JAN DENEF AND FRANC OS LOESER We dene M n;f;w;u := L?(n?n0 )m [ n0 n;f;w;u; ( f n 0 ;n) L ]; where n 0 is big enough with respect to n. By Lemma 1.4.2, the definition of M n;f;w;u does not depend on n 0 because n 0 ;n is a locally trivial bration for the ariski topology with ber A m(n0?n) k and because n;f;w \L U (; D) is a union of bers of n 0 : L(; D)! L n 0() when n 0 n, since h?1 has \only powers of f in the denominator". For any character of d (k) of order d, we dene U\h?1 (W ) h (f s ; ) = n2n M n;f;w;u L?ns?(n+1)m in K 0 (M k;q[d ]) [[L?s ]]. The result is a direct consequence of the following proposition, by additivity of c ( ; ). Proposition Assume U E and f h = uq i2 yn i i on a neighborhood of U, where u is a unit on U, and y i = 0 is an equation for E i on a neighbourhood of U. (1) f d divides N i, for all i 2, then U\h?1 (W ) h (f s ; ) = L?m [(U \ h?1 (W )) f; ] Y i2 in K 0 (M k;q[d ]) [[L?s ]]. (2) f d does not divide N i, for some i 2, then U\h?1 (W ) h (f s ; ) = 0: (L? 1) L?N is? i 1? L?N is? i Proof. We may from the beginning assume U \ h?1 (W ) = U and we will write n;f;u n0 instead of n;f;w;u. n0 We will use the following lemma. Let, Y and F be algebraic varieties over k, and let A, resp. B, be a constructible subset of, resp. Y. We say that a map : A! B is piecewise trivial bration with ber F, if there exists a nite partition of B in subsets S which are locally closed in Y such that?1 (S) is locally closed in and isomorphic, as a variety over k, to S F, with corresponding under the isomorphism to the projection S F! S. We say that the map is a piecewise trivial bration over some constructible subset C of B, if the restriction of to?1 (C) is a piecewise trivial bration.

15 MOTVC GUSA ETA FUNCTONS 15 Lemma Let and Y be connected smooth schemes over a eld k and let h : Y! be a birational morphism. For e in N, let e be the reduced subscheme of L(Y ) dened by e (K) := f' 2 Y (K[[t]]) ordt detj ' = eg; for any eld K containing k, where J ' is the jacobian of h at '. For n in N, let h n : L n (Y )! L n () be the morphism induced by h, and let e;n be the image of e in L n (Y ). f n 2e, the following holds. a) The set e;n is a union of bers of h n. b) The restriction of h n to e;n is a piecewise trivial bration with ber A e k onto its image. Proof. This is a special case of Lemma 3.4 of [6]. Let m i, i 2, be strictly positive integers with P i2 m in i = n. We denote by e (mi );U the reduced subscheme of L U (Y; h?1 (D)) whose K-rational points ' : Spec K[[t]]! Y, for any eld K containing k, satisfy the condition that y i ' is exactly of order m i at the origin, for i 2. We denote by Y (mi );U the image of e (mi );U in L n;u (Y ). By Lemma 2.2.3, for n 0 big enough with respect to n, the set n;f;u n0 is the disjoint nite union of the sets h n 0([ Ee?1 n 0 ;n (Y (m i );U)) for e = 0; 1; 2; : : :, where P E P e is the set of all (m i ) i2 with m i > 0, i2 m in i = n and i2 ( i? 1)m i = e. Hence we deduce from Lemma and Lemma U\h?1 (W ) h (f s ; ) = m i >0 L? P i2 ( i?1)m i [Y (mi );U; ( f h n ) L ] L?ns?(n+1)m ; with n = P i2 m in i. (Actually we need here the slightly stronger version of Lemma obtained by replacing e by e \ L U (Y; h?1 (D)). But the proof of this version is the same.) Now remark that : Y (mi );U! U is a locally trivial bration for the ariski topology with bre G m A nm?p i2 m i. On Y (mi );U the function f h n coincides with the product (u ju ), with (') is the constant term of [ Q i2 y i('(t)) N i ]t? P i m in i. So, if d divides N i for all i 2, we deduce from Lemma and Lemma that [Y (mi );U; ( f h n ) L ] = [U f; ](L? 1) jj L nm?p i2 m i ; and the result follows from the previous relation.

16 16 JAN DENEF AND FRANC OS LOESER Assume now that some N i with i 2 is not divisible by d. We may assume, shrinking U if necessary, that is a product. We may then identify Y (mi );U with a product G m;k in such a way that f h n = [m] g, with m not divisible by d (notations of 1.4) and now the result follows from Lemma Relation with the topological zeta functions of [5]. Let us denote by K 0 (M k;q[d ]) [L?s ] 0 loc the subring of K 0(M k;q[d ]) [L?s ] loc generated by the ring of polynomials K 0 (M k;q[d ]) [L?s ] and by the quotients (L? 1)(1? L?Ns?n )?1, for N and n in N n f0g. By expanding L?s and (L?1)(1?L?Ns?n )?1 into series in L?1, one gets a canonical morphism of algebras ' : K 0 (M k;q[d ]) [L?s ] 0 loc?! K 0 (M k;q[d ]) [s][(ns + n)?1 ] n;n2nnf0g[[l? 1]]; where [[L?1]] denotes completion with respect to the ideal generated by L? 1 and where K 0 (M k;q[d ]) is the largest quotient of K 0 (M k;q[d ]) with no (L? 1)-torsion, cf. remark Taking the quotient of K 0 (M k;q[d ]) [s][(ns+n)?1 ] n;n2nnf0g[[l?1]] by the ideal generated by L? 1, one obtains the evaluation morphism ev L=1 : K 0 (M k;q[d ]) [s][(ns + n)?1 ] n;n2nnf0g[[l? 1]]?! (K 0 (M k;q[d ])=L? 1) [s][(ns + n)?1 ] n;n2nnf0g: For in V k, we denote by top () the usual Euler characteristic of (say in etale Q`-cohomology). This induces by a morphism top : K 0 (M k;q[d ])?! ; which induces, since top (L) = 1, a morphism top : (K 0 (M k;q[d ])=L? 1) [s][(ns + n)?1 ] n;n2nnf0g?! [s][(ns + n)?1 ] n;n2nnf0g: By Theorem 2.2.1, for any character of d (k) of order d, the motivic gusa function W (f s ; ) belongs to K 0 (M k;q[d ]) [L?s ] 0 loc consider the rational function ( top ev L=1 ')( W, hence we can (f s ; )) in C(s). Proposition For any character of d (k) of order d, ( top ev L=1 ') W (f s ; ) Y = top (E \ h?1 (W )) 2J d i2 1 N i s + i :

17 MOTVC GUSA ETA FUNCTONS 17 Proof. By Theorem 2.2.1, it is enough to check that top ([(E \ h?1 (W )) f; ]) = top (E \ h?1 (W )): This is clear, since the etale `-adic realization of [(E \ h?1 (W )) f; ] is given by cohomology with compact support of a rank one lisse sheaf on E \ h?1 (W ). Remarks. 1. t follows from Proposition that the topological zeta functions of [5] are obtained by specialization of motivic gusa zeta functions. This gives another proof, not using p-adic analysis, of the main results of [5] on the invariance of topological zeta functions in the algebraic case. n fact it is easily checked that similar arguments work also in the complex analytic case. 2. t might be interesting to study the functions (ev L=1 ')( which belong to (K 0 (M k;q[d ])=L? 1) (s). W (f s ; )) 2.4. Relation with p-adic gusa local zeta functions. Let p be a prime number and let K be a nite extension of Q p. Let R be the valuation ring of K, P the maximal ideal of R, and K = R=P the residue eld of K. Let q denote the cardinality of K, so K ' F q. For z in K, let ord z denote the valuation of z, and set jzj = q?ord z and ac(z) = z?ord z, where is a xed uniformizing parameter of R. Let f be an element of R[x 1 ; : : : ; x m ] which is not zero modulo P. For any character : R! C, one denes the p-adic gusa local zeta function (s; ) as the integral (s; ) = R m (ac(f(x)))jf(x)j s jdxj; for s 2 C, Re(s) > 0, where jdxj denotes the Haar measure on K m normalized in such of way that R m is of volume 1. Let (Y; h) be a resolution of f as in 2.2. We say the resolution (Y; h) has good reduction mod P, if Y has a smooth model Y R over Spec R such that h extends to a morphism Y R! A m R and such that the closure of h?1 (D) red in Y R is a relative divisor with normal crossings over Y R. For closed in Y, we denote by the ber over the closed point of the closure of in Y R. Hence Y and all the E i 's are smooth, [ i2j E i is a divisor with normal crossings, and the schemes E i and E j have no component in common for i 6= j. Let (Y; h) be a resolution with good reduction mod P. For J we have E = \ i2 E i and we set E := E n [ j2jn E j. Assume now the character is of nite order d and is trivial on 1+P. Choose a prime number ` 6= p and denote by L the Kummer Q`-sheaf

18 18 JAN DENEF AND FRANC OS LOESER on G m; K associated to viewed as a character of K (here we choose an isomorphism between the group of roots of unity in C and in Q`). Set U = Y n[ i2j E i and denote by : U,! Y the open immersion and by : U! G m; K the map induced by f h. Set F = L. Denote by K a the algebraic closure of K and by F the geometric Frobenius automorphism. n the good reduction case the following result gives a cohomological expression for p-adic gusa local zeta functions. Theorem ([1][2]). Let (Y; h) be a resolution of f with good reduction mod P. Assume the character is of nite order d and is trivial on 1 + P. Then with (s; ) = q?m 2J d c ; Y c ; = i i2 (q? 1) q?n is? i 1? q?n is? i (?1) i Tr (F; H i c(e K a ; F )): n conclusion, in view of Theorem and Theorem 2.4.1, one can state that \in the good reduction case, the p-adic gusa local zeta functions are given by the trace of the Frobenius action on the `-adic etale realization of the corresponding motivic ones". As in the p-adic case (see e.g. [3]), there is the intriguing question whether R (f s ; ) always belong to K 0 (M k;q[d ])[(1?L?Ns?n )?1 ] (N;n)2M; where M is the set of all pairs (N; n) in (N n f0g) 2 with exp(2in=n) an eigenvalue of the monodromy action on the complex R f of nearby cycles on f?1 (0). For some recent work in the p-adic case, see [24], [25] Relation with motivic integration. M.Kontsevich introduced in [13] the completion b K0 (Sch k ) of K 0 (Sch k )[L?1 ] with respect to the ltration F m K 0 (Sch k )[L?1 ], where F m K 0 (Sch k )[L?1 ] is the subgroup of K 0 (Sch k )[L?1 ] generated by f[s] L?i i? dim S mg, and dened, for smooth over k, a motivic integration on L() with values into bk 0 (Sch k ). n the paper [6], we extended Kontsevich's construction to semi-algebraic subsets of L() and also to the non smooth case. The following statement is proved in [6] (Denition-Proposition 3.2). Denition-Proposition Let be an algebraic variety over k of pure dimension m. Denote by n the natural morphism L()! L n (). Let B be the boolean algebra of all semi-algebraic subsets of L(). There exists a unique map : B! b K0 (Sch k ) satisfying the following three properties.

19 MOTVC GUSA ETA FUNCTONS 19 (2.5.2) f A 2 B is stable at level n, then (A) = [ n (A)]L?(n+1)m : (2.5.3) f A 2 B is contained in L(S) with S a closed subvariety of with dim S < dim, then (A) = 0. (2.5.4) Let A i be in B for each i in N. Assume that the A i 's are mutually disjoint and that A := S i2n A i is semi-algebraic. Then P i2n (A i) converges in b K0 (Sch k ) to (A). We call this unique map the motivic volume on L() and denote it by L() or. Moreover we have (2.5.5) f A and B are in B, A B, and if (B) belongs to the closure F m ( b K0 (Sch k )) of F m K 0 (Sch k )[L?1 ] in b K0 (Sch k ), then (A) 2 F m ( b K0 (Sch k )). Hence, for A in B and : A! [ f+1g a simple function, we can dene A L? d := n2 (A \?1 (n)) L?n in b K0 (Sch k ), whenever the right hand side converges in b K0 (Sch k ), in which case we say that L? is integrable on A. f the function is bounded from below, then L? is integrable on A, because of (2.5.5). Semi-algebraic subsets of L() and simple functions on semi-algebraic subsets are dened in [6], as well as the notion of stable semialgebraic subsets of L() of level n. n particular, L() is a semialgebraic subset of L() and, for any morphism g : Y! of algebraic varieties over k, the image of L(Y ) in L() under the morphism induced by g is a semi-algebraic subset of L(). When is smooth, a semi-algebraic subset of L() is stable of level n if and only if it is a union of bers of n : L()! L n (). Consider a coherent sheaf of ideals on and denote by ord t the function ord t : L()! N [ f+1g given by ' 7! min g ord t g('), where the minimum is taken over all g in the stalk 0 (') of at 0 ('). The function ord t is a simple function. When is smooth and is Rthe ideal sheaf of an eective divisor D on, the motivic integral L() L?ordt R d was rst introduced by Kontsevich [13] and denoted by him [ ed ]. n particular, for a morphism f :! A 1 k with divisor D R and a natural number d in N, we can consider the motivic integral?1 0 (W ) L?ordtO(?dD) d, for any reduced subscheme W of, because?1 0 (W ) is a semi-algebraic subset of L(). t follows from Theorem 5.1 of [6] that R?1 0 (W ) L?ordtO(?dD) d belongs to the image

20 20 JAN DENEF AND FRANC OS LOESER of K 0 (Sch k )[L?1 ; ((L i? 1)?1 ) i1 ] in b K0 (Sch k ). On the other hand, the motivic gusa function R W f s belongs to K 0 (M k ) [L?s ] loc and is the nat- R ural image of a well dened element f s in K W 0 (Sch k )[L?1 ][[L?s ]], cf. Rremark 1 in 2.1. Moreover the proof of Theorem also shows that f s belongs to K W 0 (Sch k )[L?1 ][L?s ] loc. Hence for any natural number d in N, we can formally replace s by d and obtain by evaluation a well dened element ( R W f s ) js=d in b K0 (Sch k ). The following statement is a direct consequence of the denitions. Proposition Let be a smooth and connected separated k- scheme of nite type of pure dimension m, f :! A 1 k be a morphism, and W be a reduced subscheme of. For any natural number d in N, the equality L?ordtO(?dD) d = f s?1 0 (W ) W js=d holds in b K0 (Sch k ). 3. Functional equation 3.1. We denote by K 0 (M k;q[d ]) [L s ; L?s ] loc the localization of the algebra of Laurent polynomials K 0 (M k;q[d ]) [L s ; L?s ] with respect to the multiplicative set generated by the polynomials 1? L?Ns?n, for N and n in N n f0g. One may consider K 0 (M k;q[d ]) [L?s ] loc as embedded in K 0 (M k;q[d ]) [L s ; L?s ] loc. The involution M 7! M _ extends to K 0 (M k;q[d ]). One can extend it to a K 0 (M k;q[d ])-algebra involution on K 0 (M k;q[d ]) [L s ; L?s ] loc by setting (L s ) _ = L?s, (L?s ) _ = L s, and ((1? L?Ns?n )?1 ) _ =?L?Ns?n (1? L?Ns?n )?1. n this section we assume = A m k of degree r. Theorem (1) The equality f s _ = L?rs and f is a homogenous polynomial f s holds in K 0 (M k ) [L s ; L?s ] loc. (2) Assume holds. Then, for any character of d (k) of order d, (f s ; ) _ = L?rs in K 0 (M k;q[d ]) [L s ; L?s ] loc. We begin with the following lemma. (f s ;?1 )

21 MOTVC GUSA ETA FUNCTONS 21 Lemma Let be a character of d (k) of order d. f d does not divide r, then (f s ; ) = 0: Proof. t is enough to prove that [ n;f;a m k ; f L ] = 0 if d does not divide r. From the second displayed formula in the proof of Proposition below, which actually holds for any d, it follows that it suces to prove that [ n;f;f0g; f L ] = 0. Thus we have to show that (f s ; ) = 0: f0g Let D, resp. D, be the divisor in P m?1 k, resp. A m k, dened by f = 0, and let h : Y! P m?1 k be a resolution of D P m?1 k (in the sense of, and by p the 2.2). Denote by : B! A m k the blowing up of f0g in Am k natural map p : B! P m?1 k which is the identity on?1 f0g = P m?1 k. Note that p is a locally trivial bration for the ariski topology with ber A 1 k. One veries that the natural map h : Y P m?1 k B! B! A m k is a resolution of D A m k. Moreover h?1 (D) ' Y is a component (hence equal to some E i ) of h?1 (D) on which f h has multiplicity r. Thus (f s ; ) = 0 when d does not divide r, by Theorem f0g 3.2. Proof of Theorem By Lemma we may assume d divides r. We consider the canonical projection : A m k n f0g! P m?1 k and denote by D the image of D n f0g in P m?1 k. Let h : Y! P m?1 k be a resolution of D (in the sense of 2.2). As in 2.2 we denote by E i, i 2 J, the irreducible (smooth) components of (h?1 (D)) red. We dene similarly integers N i and i, and E, J d, U d, etc. We denote by U j the open x j 6= 0 in P m?1 k. The restriction of h to h?1 (U j ) is a resolution of f j = f in U x r j. For locally closed in U d, j [( \ h?1 (U j )) fj ;] has been dened in 2.2, and by Lemma [( \ h?1 (U j ) \ h?1 (U j 0)) fj ;] = [( \ h?1 (U j ) \ h?1 (U j 0)) fj 0 ;]: Thus we may dene without ambiguity [ f; ] = j [( \h?1 (U j )) fj ;]? j6=j 0 [( \h?1 (U j )\h?1 (U j 0)) fj ;]+ : Set E (d) = E n [ fjg=2jd E j.

22 22 JAN DENEF AND FRANC OS LOESER Proposition Assume d divides r. For any character of d (k) of order d, (f s ; ) = (L? 1) L?m 1? L?rs?m [(E (d) 2J d ) f; ] Y i2 Proof. Let us rst remark that (f s 1 ; ) = (f 1? L nf0g s ; ):?rs?m (L? 1) L?N i s?n i 1? L?N is?n i? 1 : ndeed, by homogeneity of f, multiplication by t induces an isomorphism between n;f;a m k and n+r;f;f0g (notations of 2.1), from which one deduces the relation and the equality follows. L m [ n+r;f;f0g; f L ] = L rm [ n;f;a m k ; f L ]; Write n f0g as the disjoint union of the W j 's, for 1 j m, with W j = fx 2 A m k xi = 0 for i < j and x j 6= 0g: Now (W j ) U j and the restriction of to W j is a trivial bration onto its image, with bre G m;k. As the valuation of f('(t)) only depends on ('(t)) we deduce W j (f s ; ) = (1? L?1 ) (W j ) (f s j ; ): Since Y is the disjoint union of the h?1 ((W j ))'s, we deduce from Theorem 2.2.1, by adding up, that nf0g(f s ; ) = (1? L?1 ) L?(m?1) 2J d [(E ) f; ] Y i2 The result follows, because with 2J d [(E ) f; ] Y i2 (L? 1) L?N is?n i 1? L?N is?n i = [(E (d) 2J d ) f; ] Y i2 By Proposition we may write A = (L? 1) L?N is?n i 1? L?N is?n i : (L? 1) L?N i s?n i 1? L?N is?n i? 1 : (f s ; ) = A 2J d [(E (d) ) f; ] Y i2 B i ; (L? 1) L?m 1? L?rs?m and B i = (L? 1) L?Nis?ni 1? L?N is?n i? 1:

23 MOTVC GUSA ETA FUNCTONS 23 Remark that A _ = L?rs L m?1 A and B _ i = L?1 B i. When = 1 the result follows because E being proper and smooth c (E ) _ = L jj?m+1 c (E ). For general the result follows from the following lemma. Lemma Assume holds. For any in J d, [(E (d) ) f; ] _ = L jj?m+1 [(E (d) ) f;?1]: Proof. Let be locally closed in U d. f on a neighborhood of we may write f h = uv d with u non vanishing, [; u L ] = c (e u;d ; ), where e u;d is the cyclic cover dened in 1.4. n general the covers e u;d can be glued together (cf. the proof of Lemma 1.4.1) to give a Galois cover e d! with group d (k), such that [ f; ] = c (e d ; ), for any character of order d. Set W = E (d) and W = E. Lemma The cover f W d! W extends to a ramied d (k)-cover : f Wd! W which satises the following conditions. (1) The scheme f Wd is proper and is locally for the ariski topology quotient of a smooth scheme by a nite abelian group G, the d (k)-action on f Wd being induced from a d (k)-action on commuting with the G-action. (2) The morphism ramies on f Wd nf W d and the d (k)-action on the restriction of to f Wd n f W d factors locally for the ariski topology trough a d 0(k)-action, for some d 0 < d dividing d. Proof. Let x be a closed point of W n W. On a ariski neighborhood 0 of x in Y, Y f = u h N i i v d i2j x with u nonvanishing on 0, J x = fi 2 J n x 2 Ei ; d 6 jn i g and h i local equations for E i near x. Put = 0 \ W. Since W f dj\w is given by y d = u Q i2j x h N i i in G m (\W ), we may extend f W dj\w! \W to f Wdj! by taking f Wdj to be the normalization of the subscheme W dj of A 1 given by y d = u Q i2j x h N i i, and the d (k)-action extends naturally. The schemes f Wdj! glue together (cf. the proof of Lemma 1.4.1) to give a scheme f Wd! W with d (k)-action. Let d 0 be the gcd of d and the N i 's, i 2 J x. We have d 0 < d. Locally for the etale topology near Q x, Wd f is the disjoint union of the normalizations of y d=d0 = " j u 0 i2j x h N i=d 0 i, for 1 j d 0, for " a

24 24 JAN DENEF AND FRANC OS LOESER xed primitive d 0 -th root of unity and u 0 such that u 0d0 = u. This implies that on a ariski neighborhood of x in W n W, the d (k)- action on j f Wd nf W d : f Wd n f W d! W n W factors through a d 0(k)- action, since the normalization of a local complete domain is again local. We still have to verify that locally for the ariski topology f Wd is the quotient of a smooth scheme by a nite abelian group G, the d (k)-action being induced from a d (k)-action on commuting with the G-action. t is enough to check this for the scheme Wdj f which is Q the normalization of the subscheme W dj of A 1 given by y d = u i2j x h N i i. We may assume u = 1. ndeed, consider the etale cyclic cover of degree d, p : 0! given by u = u 0d. Since f Wdj is the quotient of the normalization of the subscheme of A 1 0 given by y d = u 0d Q i2j x (h i p) N i, we are done by using the isomorphism A 1 0! A 1 0 given by (y; x) 7! (yu 0?1 ; x). When u = 1 the scheme f Wdj is the disjoint union of the normalizations of y d=d0 = " j Q i2j x h N i=d 0 i, for 1 j d 0, for " a xed primitive d 0 -th root of unity and d 0 the gcd of d and the N i 's, i 2 J x. Hence we may nally assume Q that fw dj is the normalization of the scheme W dj given by y d = i2j x h N i i in A 1, and that the gcd of d and the N i 's, i 2 J x, is 1. Now consider Q the subscheme W 0 of A 1 A jjxj given by h i = t d i and y = i2j x t N i i. t is easily seen that W 0 is smooth. Let us denote by : W 0! W dj the morphism given by (y; x; (t i )) 7! (y; x) and by G the kernel of the morphism d (k) jjxj! d (k) given by ( i ) 7! Q i2j x N i i. The canonical action of G on A jjxj induces an action on W 0 for which the morphism is equivariant, hence factorizes through a morphism G : W 0 =G! W dj. Since G is of degree 1, the result follows. We are now able to nish the proof. By Lemma (2) and Proposition (2), we have [(E (d) ) f; ] = c (f Wd ; ) and [(E (d) ) f;?1] = c (f Wd ;?1 ); hence the result follows from the following proposition. Proposition Assume holds. Let W be a proper G-scheme of pure dimension m, with G a nite abelian group. Assume that, locally for the ariski topology, W is isomorphic as a G-scheme to a quotient =H with H a nite abelian group and a smooth G H- scheme. Then, for any character of G, c (W; ) _ = L?m c (W;?1 ):

25 MOTVC GUSA ETA FUNCTONS 25 Proof. By additivity of Euler characteristics (Theorem (3) and (6)) and by Theorem (5), we are reduced to prove that if W is a G-scheme of pure dimension m which is isomorphic as a G-scheme to a quotient =H with H a nite abelian group and a smooth GHscheme, then, for any character of G, (W; ) _ = L?m c (W;?1 ). This follows directly from Assertion and Theorem (4). Let us denote by L 0 the image of L by the morphism ' : K 0 (M k;q[d ])?! K 0 (DM gm (k) Q[d ]): One then denes a ring K 0 (DM gm (k) Q[d ]) [L 0s ; L 0?s ] loc similarly as we dened the ring K 0 (M k;q[d ]) [L s ; L?s ] loc, and ' extends to a morphism K 0 (M k;q[d ]) [L s ; L?s ] loc! K 0 (DM gm (k) Q[d ]) [L 0s ; L 0?s ] loc, which we still denote by '. Theorem For any character of d (k) of order d, _ ' (f s ; ) = ' L?rs (f s ;?1 ) in K 0 (DM gm (k) Q[d ]) [L 0s ; L 0?s ] loc. Proof. The proof is the same as the one of Theorem 3.2.1, using Corollary instead of Assertion Limit for s!?1 and nearby cycles 4.1. We consider in this section the subring K 0 (M k;q[d ]) [L?s ] 00 loc of K 0 (M k;q[d ]) [[L?s ]] loc generated by the subring K 0 (M k;q[d ]) and the series L?Ns?n (1? L?Ns?n )?1, for N and n in N n f0g. Lemma There is a well dened ring homomorphism CT : K 0 (M k;q[d ]) [L?s ] 00 loc?! K 0 (M k;q[d ]) which induces the identity on K 0 (M k;q[d ]), and which sends the series L?Ns?n (1? L?Ns?n )?1 to?1. Proof. Similar to the one for the constant term of the power series expansion in T?1 of usual rational functions of degree 0 in T, considering L?s as a variable T. (Heuristically this amounts to taking the \limit" for s!?1.) Denition Let be a smooth and connected k-scheme of nite type of dimension m, f :! A 1 k be a morphism and x be a closed point of f?1 (0). Let be a character of d (k) of order d. We set S ;x := Lm 1? L fxg CT (f s ; ):

26 26 JAN DENEF AND FRANC OS LOESER By Theorem and Lemma 4.1.1, S ;x is an element of K 0 (M k;q[d ]) which is well dened modulo (L? 1)-torsion, cf. remark Furthermore, for any resolution of f, modulo (L? 1)-torsion. S ;x = 2J d [(E \ h?1 (x)) f; ](1? L) jj?1 ; Remark that, for almost all d, S ;x = 0. We now assume for simplicity that k contains all roots of unity and that the group of roots of unity in k is embedded in C. Hence all the groups d (k) are canonically embedded in Q=. We denote by the section Q=! [0; 1) and by i x the inclusion of fxg in f?1 (0). We believe that S ;x is the \motivic incarnation" of c (i xr f; ). Here R f; denotes the eigenspace of nearby cycles for the eigenvalue exp(2i()) of the semi-simple part of the monodromy. We will verify in the next subsection that this is true for the C-Hodge realization Hodge realization. We will use freely the theory of mixed Hodge modules developped by M.Saito in [16], [18]. n particular, for a scheme of nite type over C, we denote by MHM() the abelian category of mixed Hodge modules on. n the denition of mixed Hodge modules it is required that the underlying perverse sheaf is dened over Q. To allow some more exibility we will also use the category MHM 0 () of biltred D-modules on which are direct factors of objects of MHM() as biltred D-modules. We denote by D b (MHM()) and D b (MHM 0 ()) the corresponding derived categories. Let f :! A 1 C be a morphism. We denote by f H and H f the nearby and vanishing cycle functors for mixed Hodge modules as de- ned in [18] and T s the semi-simple part of the monodromy operator. One should note that f H and H f on mixed Hodge modules correspond to f [?1] and f [?1] on the underlying perverse sheaves. f M is a mixed Hodge module on we denote by f; H M the object of MHM 0 () which corresponds to the eigenspace of T s for the eigenvalue exp(2i()). These denitions extend to the Grothendieck group of the abelian category MHM 0 (). Let us recall the denition of complex mixed Hodge structures. A C- Hodge structure of weight n is just a nite dimensional bigraded vector space V = L p+q=n V p;q, or, equivalently, a nite dimensional vector space V with decreasing ltrations F and F such that V = F p F q when p + q = n + 1. A mixed C-Hodge structure is a nite dimensional vector space V with an increasing ltration W and decreasing ltrations