Unit L-functions and a conjecture of Katz.

Transcription

1 UNIT L-FUNCTIONS 1 Unit L-functions an a conjecture of Katz. By Matthew Emerton an Mark Kisin Let p be a prime number, F p the finite fiel of orer p, an Z p the ring of p-aic integers. Suppose that X is a separate finite type F p -scheme an that L is a lisse Z p -sheaf on the étale site of X. One efines in the usual way an L-function L(X, L) attache to L. This is a power series in a formal variable T, which by construction is an element of 1 + T Z p [[T ]]. If f : X Spec F p is the structural morphism of X then f! L is a constructible complex of Z p - sheaves on the étale site of Spec F p of finite tor-imension, an so we may form the L-function L(Spec F p, f! L), which is also an element of 1 + T Z p [[T ]] (an in fact a rational function). The ratio L(X, L)/L(Spec F p, f! L) thus lies in 1 + T Z p [[T ]], an so may be regare as a nowhere-zero analytic function on the p-aic open unit isk T < 1. If L were a lisse Z l sheaf with l p then in fact this ratio woul be ientically 1 (this is Grothenieck s approach to the rationality of the zeta function). One oes not have this in the setting of Z p -sheaves. However, in this paper we prove the following result, which was conjecture by Katz ([11], 6.1): Theorem 0.1. The ratio L(X, L)/L(Spec F p, f! L) extens to a nowherezero function on the close unit isc T 1. In particular this implies that the L-function for L(X, L) is p-aic meromorphic in the close unit isc. Accoring to Katz ([11], 6.1), when X = A n, this is an ol result of Dwork s. Katz also conjecture that L(X, L) extens to a meromorphic function on the rigi-analytic affine T -line. The secon part of Katz s conjecture was a generalization of a conjecture of Dwork preicting that L(X, L) was p-aic meromorphic when the unit F -crystal M associate to L was the unit part of an orinary, overconvergent F -crystal. Wan [13] has foun counter-examples to the part of Katz s conjecture preicting that the L-function is meromorphic for general lisse sheaves. On the other han, in a recent preprint [14] he has proven that Dwork s more cautious preiction is actually true. Thus, while the question of when the L-function is meromorphic is by now quite well unerstoo, much less was known about the location of the zeroes an poles. In fact the only prior results in this irection that we are aware of are ue to

2 2 MATTHEW EMERTON AND MARK KISIN Crew [3], who prove Katz s conjecture in the special case when X is an affine curve an the sheaf L has abelian monoromy, an Etesse an Le Stum [8], who prove Katz s conjecture uner the strong assumption that L extens to some compactification of X. In fact we prove a more general result showing that an analogue of Katz s conjecture is true even if one replaces Z p by a complete noetherian local Z p - algebra Λ with finite resiue fiel. We refer to Corollary 1.8 for the precise formulation. Our methos also have some applications to results on lifting representations of arithmetic funamental groups. In particular, we show the following Theorem 0.2. Let X be a smooth affine F p -scheme, Λ an artinian local Z p -algebra, having finite resiue fiel, an ρ : π 1 (X) GL (Λ) a representation of the arithmetic étale funamental group of X. Consier a finite flat local Z p -algebra Λ an a surjection Λ Λ. There exists a continuous lifting of ρ ρ : π 1 (X) GL ( Λ). If X is an open affine subset of P 1, then ρ may be chosen so that the L-function of the corresponing lisse sheaf of Λ-moules is rational. Let us now escribe the contents of the paper in more etail. In section 1 we efine the necessary L-functions, an state our main results. In section 2 we give the proof of some stanar results on behaviour of L-functions in exact triangles, an uner stratification of the unerlying space. These are use later to reuce our calculations to a special case. In section 3 we introuce a key ingreient in our work, which is the relationship between locally constant or lisse étale sheaves an unit F -crystals. We show that if Λ is an artinian (respectively a complete noetherian) finite local Z p -algebra with finite resiue fiel an if X is a formally smooth p-aic formal scheme equippe with a lifting F of the absolute Frobenius of its special fibre, then there is a one-to-one corresponence between locally constant étale sheaves of finite free Λ-moules (respectively lisse étale sheaves of Λ-moules) on X an finite free Λ Zp O X -moules E on X equippe with a Λ-linear isomorphism F E E. This generalizes a result of Katz ([12], 4.1), which treats the cases when Λ = Z p or Z/p n for some positive integer n. In section 4 we begin our proof of Katz s conjecture. For the usual reasons of technical convenience, we prove a more general result on complexes of p-aic constructible sheaves of Λ-moules. By passing to the inverse limit, it is enough to consier the case where Λ is artinian. Using stanar techniques we reuce

3 UNIT L-FUNCTIONS 3 ourselves to having to verify Katz s conjecture for a locally constant flat Λ- sheaf on an open subset of G m with the aitional property that the associate unit (Λ, F )-crystal E (as explaine above) has an unerlying locally free sheaf of Λ Zp O X -moules which is actually free. Uner this assumption we may choose a surjection Λ Λ, with Λ a finite flat local Z p -algebra, an lift this Λ-sheaf to a lisse Λ-sheaf by lifting the associate unit (Λ, F )-crystal. In this situation we prove Katz s conjecture irectly, by using an explicit trace calculation in the style of Dwork [5]. To reuce to this trace calculation we utilize a technique of Deligne [4] in orer to fin an Artin-Schreier sequence for the extension by zero of a lisse Λ-sheaf uner an open immersion, which allows us to relate p-aic étale cohomology with compact support to coherent cohomology of formal Z p -schemes. A key part of this calculation is the subject of section 5. Finally, in section 6 we show how our methos can be use to lift representations of arithmetic funamental groups. Finally, let us mention that the methos use in this paper are a special instance of a more general theory evelope by the authors in the papers [6], [7]. This theory escribes the étale cohomology of p-power torsion or p- aic constructible sheaves on finite type F p -schemes in terms of quasi-coherent cohomology; more precisely, a certain Riemann-Hilbert corresponence is establishe between the appropriate erive categories of such étale sheaves, an a erive category of certain quasi-coherent analogues of unit F -crystals. However, we have written this paper so as to be inepenent of these more general techniques. Acknowlegments. The authors woul like to thank Pierre Berthelot for suggesting that they apply their techniques to the stuy of L-functions. The first author woul also like to thank Mike Roth for useful iscussions. 1. Notation an statement of results (1.1) Let p be a prime number, fixe for the remainer of the paper, F p enote a finite fiel of orer p, an F p be a choice of an algebraic closure of F p (fixe once an for all). For any positive integer n we enote by F p n the unique egree n extension of F p containe in F p. We enote by σ : F p F p the (arithmetic) Frobenius automorphism a a p of F p, an enote by φ the geometric Frobenius automorphism of F p, that is, the inverse of σ. Both σ an φ are topological generators of the Galois group of F p over F p. We enote by W the ring of Witt vectors of F p, an by σ an φ respectively the canonical Frobenius automorphism of W an its inverse. If n is any positive integer, we let W (F p n) enote the Witt ring of F p n, regare as a subring of W. As usual we will enote by Z p the ring W (F p ) of p-aic numbers.

4 4 MATTHEW EMERTON AND MARK KISIN (1.2) Let X be a finite type scheme over F p, an Λ a noetherian ring, with mλ = 0 for some positive integer m. We enote by D (X, Λ) the erive category of boune above complexes of étale sheaves of Λ-moules on X, an we enote by Dctf b (X, Λ) the full subcategory of D (X, Λ) consisting of complexes of Λ-sheaves which have finite tor-imension, an which are constructible (in the sense that their cohomology sheaves are constructible Λ- moules [1]). Given any object L of Dctf b (X, Λ), we may efine its L-function in the usual way, which we briefly recall. If L is a constructible complex of Λ-sheaves of finite tor-imension on X then by [4], proposition-éfinition 4.6 (ii), p. 93, we may assume that L is a finite length complex of flat constructible Λ-sheaves. If x is a close point of X we let L i x enote the flat Λ-sheaf obtaine by restricting L i to the point x. If we choose an ientification of the resiue fiel of x with the subfiel F p (x) of F p, an enote by x the F p -value point of X lying over x, then L ī x is a free Λ-moule equippe with an action of the th power φ of the geometric Frobenius automorphism. We let X enotes the set of close points of X, an efine L(X, L ) = et Λ (1 φ (x) T (x), L ī x) ( 1)i+1. i x X This is an element of 1 + T Λ[[T ]]. From the efinition we see that L-functions are compatible with change of ring: if Λ, Λ are noetherian rings kille by some positive integer, g : Λ Λ a map of rings, an L is as above, then we have L(X, L L Λ ) = g(l(x, L )). (1.3) Let X be a finite type F p -scheme an let i : X re X be the close immersion of the unerlying reuce subscheme of X into X. Then the functors i! an i 1 are quasi-inverse, an inuce an equivalence of triangulate categories between Dctf b (X re, Λ) an Dctf b (X, Λ) (preserving the canonical t-structure of both categories). If L is any object of Dctf b (X, Z/pn ) then L(X, L ) = L(X re, i 1 L ). (1.4) If f : X Y is a separate morphism of finite type F p -schemes then push-forwar with proper supports inuces a functor f! : Dctf b (X, Λ) Dctf b (Y, Λ) ([4], théorème 4.9, p. 95). Theorem 1.5. Let f : X Y be a separate morphism of finite type F p -schemes, an Λ a finite, local, artinian Z p -algebra, with maximal ieal m. If L is any object of Dctf b (X, Λ) then the ratio of L-functions L(X, L )/L(Y, f! L ), a priori an element of 1 + T Λ[[T ]], in fact lies in 1 + mt Λ[T ].

5 UNIT L-FUNCTIONS 5 (1.6) Suppose that Λ is finite an reuce of characteristic p. Then from theorem 1.5 we conclue that the L-functions of objects of Dctf b (X ét, Λ) are invariant uner proper push-forwar. This was originally prove by Deligne ([4], théorème 2.2, p. 116). An interesting point is that Deligne gives a counterexample involving a locally constant sheaf of free rank one F p [X]/X 2 -moules, to show that his formula oes not hol without the hypothesis that Λ is reuce ([4], 4.5, p. 127). Thus if we take Λ = F p [X]/X 2, then Deligne s counterexample shows that, in general, the quotient of the L-functions in theorem 1.5 is not equal to 1. Nevertheless, our theorem asserts that it is equal to 1 moulo the principal ieal (X). This also follows from Deligne s theorem by specialization of L-functions (1.2), an in fact this same argument shows that the ratio of L-functions L(X, L )/L(Y, f! L ) occurring in the statement of theorem 1.5 always lies in 1 + mt Λ[[T ]]. Thus the key result of theorem 1.5 is that this ratio is in fact a polynomial. (1.7) We want to efine L-functions for lisse sheaves, or more precisely, the lisse analogue of constructible sheaves. For this, suppose that we are given a noetherian ring Λ an an ieal I Λ, such that Λ is I-aically complete an p is nilpotent in Λ/I. We efine the category DI sm b (X, Λ) to be the 2-limit of the categories Dctf b (X, Λ/In ), n = 1, 2,.... One has a formalism of f!, f 1 an L in DI sm b (X, Λ) (see [9] for etails). If L is an object of DI sm b (X, Λ) then by construction L L Λ Λ/I n belongs to Dctf b (X, Λ/In ) for each positive integer n. If n m then the compatibility of formation of L-functions with change of rings shows that L(X, L L Λ Λ/I n ) L(X, L L Λ Λ/I m ) (mo I m ). Thus we may efine L(X, L ) 1 + T Λ[[T ]] to be the limit of the L-functions L(X, L L Λ Λ/I n ). Now suppose that f : X Y is a separable map of finite type F p -schemes. As before, we have a functor f! : DI sm b (X, Λ) Db I sm (Y, Λ). For each positive integer n there is a canonical isomorphism so that (f! L ) L Λ Λ/I n f! (L L Λ Λ/I n ), L(Y, f! L ) L(Y, f! (L L Λ Λ/I n )) (mo I n ). If we now suppose that Λ is a complete local Z p -algebra with finite resiue fiel (so that in the above iscussion I = m is the maximal ieal of Λ), by letting n ten to infinity in the above iscussion we erive the following corollary of theorem 1.5, which inclues Katz s conjecture as a special case (take Λ to be Z p an L to be a single lisse p-aic sheaf in egree zero):

6 6 MATTHEW EMERTON AND MARK KISIN Corollary 1.8. Let Λ be a complete local Z p -algebra, with finite resiue fiel, an maximal ieal m. Let f : X Y be a separate morphism of finite type F p -schemes. If L is any object of Dm sm(x, b Λ) then the ratio of L- functions L(X, L )/L(Y, f! L ), a priori an element of 1 + T Λ[[T ]], in fact lies in 1 + mt Λ T, where Λ T enotes the m-aic completion of the polynomial ring Λ[T ] 2. Preliminaries on L-functions (2.1) Below Λ will be a noetherian Z/mZ-algebra for some integer m. For any separate morphism f : X Y of finite type F p -schemes an any object L of Dctf b (X, Λ), let us write Q(f, L ) = L(X, L )/L(Y, f! L ). We begin by recalling some stanar tools for analyzing such ratios of L- functions. Although the proofs are well-known an quite straightforwar, for the sake of completeness we recall them. Lemma 2.2. If f : X Y is a separate morphism of finite type F p - schemes, then for any istinguishe triangle L 1 L 2 L 3 L 1[1] of objects of Dctf b (X, Λ) we have the equality Q(f, L 2) = Q(f, L 1)Q(f, L 3). Proof. This follows immeiately from the fact that f! takes istinguishe triangles to istinguishe triangles, together with the multiplicativity of L- functions of objects in a istinguishe triangle. If f : X Y is a morphism of schemes an y is a point of Y we let f y : X y y enote the fibre of f over the point y. Lemma 2.3. For any separate morphism f : X Y of finite type F p - schemes an any object L of Dctf b (X, Λ) we have the equality Q(f, L ) = Q(f y, L X y ) y Y

7 UNIT L-FUNCTIONS 7 Proof. By partitioning the elements of X accoring to their image in Y, one sees that L(X, L ) = L(X y, L X y ). y Y By the proper base-change theorem, for any point y of Y, an so Thus Q(f, L ) = proving the lemma. y Y (f! L ) y = f y! L X y, L(Y, f! L ) = y Y L(y, f y! L X y ). L(X y, L X y )/L(y, f y! L X y ) = y Y Q(f y, L X y ), Lemma 2.4. If f : X Y an g : Y Z are two separate morphisms of finite type F p -schemes an L is any object of Dctf b (X, Λ) then we have the equality Q(gf, L ) = Q(g, f! L )Q(f, L ). Proof. We compute (beginning with the right-han sie) Q(g, f! L )Q(f, L ) = L(Y, f!l ) L(X, L ) L(Z, g! f! L ) L(Y, f! L ) = L(X, L ) L(Z, (gf)! L ) = Q(gf, L ). This proves the lemma. Lemma 2.5. For any separate quasi-finite morphism f : X Y of finite type F p -schemes an any object L of Dctf b (X, Λ), Q(f, L ) = 1. Proof. Lemma 2.3 reuces us to the situation in which Y = Spec F p is a point, so that X is either empty, in which case there is nothing to prove, or else zero-imensional. In this secon case we may replace X by its unerlying reuce subscheme (via (1.4)), an writing this as a isjoint union of points we reuce to the case that X is also a point, say X = Spec F p, with iviing. Using lemma 2.2 one reuces to the case that L = L is a single flat Λ-moule equippe with a Λ-linear φ action. Then f! L is simply the inuce moule f! L = Z/p n [φ ] Z/p n [φ ] L, an the equality of L-functions L(X, L) = L(Y, f! L) is an elementary calculation.

8 8 MATTHEW EMERTON AND MARK KISIN Lemma 2.6. Suppose that f : X Y is separate morphism of finite type F p -schemes, an that X = S 0 S1 Sn is a stratification of X by locally close subsets S i (more precisely, we take this to mean that each S i is close in the union i j=0 S j.) We give each of the S i their reuce inuce scheme structure. If L is an object of Dctf b (X, Λ) then let L S i enote the restriction of L to each of the locally close subsets S i, an let f i : S i Y enote the restriction of the morphism f to each of the locally close subsets S i. Then n Q(f, L ) = Q(f i, L S i ). i=0 In particular, this applies if the stratification on X is obtaine by pulling back a stratification on Y. Proof. This follows from lemmas 2.2., 2.4 an 2.5, since immersions are quasi-finite. 3. Etale sheaves an unit F -crystals (3.1) We want to explain a generalization of a result of Katz relating étale sheaves an unit F -crystals. Although what we o can be one somewhat more generally, we restrict ourselves to smooth Z/p n -schemes. Assume we have a smooth scheme X over Z/p n, an let Λ be a Z/p n - algebra. We assume that X is equippe with an enomorphism F lifting the absolute Frobenius on its reuce subscheme. A unit (Λ, F )-crystal on X is a sheaf E of finite locally free Λ Z/p n O X moules, equippe with a Λ Z/p n O X -linear isomorphism F E E. Note that if the map Z/p n Λ factors through Z/p n for some positive integer n < n, then the notion of a unit (Λ, F )-crystal epens only on the reuction of X moulo p n. Proposition 3.2. Suppose that Λ is noetherian, local, an finite over Z/p n. There is an equivalence of categories (explicitly escribe below) between the category of locally constant étale sheaves of finite free Λ-moules on X an the category of (Λ, F )-crystals on X. Proof. Let L be a locally constant étale sheaf of finite free Λ-moules on X (which is, of course equivalent to the ata of such a sheaf on the reuction of X moulo p). We associate to L a (Λ, F )-crystal on X as follows. Consier the étale sheaf Eét =: L Z/p n OXét. This is a coherent sheaf of Λ Z/p n OXét - moules, which is in fact locally free of finite rank (since L is locally free of

9 UNIT L-FUNCTIONS 9 finite rank over Λ). Regaring Eét as a coherent sheaf on OXét, we see by [12], 4.1, that Eét comes by pull-back from a Zariski coherent O X -moule E. Moreover, E is equippe with an isomorphism F E E. If we enote by Φ the composite E F E E, then the inuce map Φét on Eét has fixe subsheaf equal to L. As the formation of E is functorial in Eét, we see that E is equippe with the structure of a Λ-moule, an that the isomorphism F E E is Λ-linear. To see that this gives E the structure of a unit (Λ, F )-crystal we have to check that E is locally free as a Λ Z/p n O X -moule. If x X enote by x an étale point lying over x. Now Eét, x = E x (Λ Z/p no X,x ) (Λ Z/p n OXét, x ) is certainly a free Λ Z/p n OXét, x moule. Since Λ Z/p n OXét, x is faithfully flat over Λ Z/p n O X,x (OXét, x being faithfully flat over O X,x ), we conclue by escent that E x is locally free of constant rank over Λ Z/p n O X,x. Since Λ is finite over Z/p n we see that Λ Z/p n O X is finite over the local ring O X,x, an so is semi-local. Thus the freeness of E x over Λ Z/p n O X follows from the following (simple) commutative algebra fact: a finite moule M over a semi-local ring A which is locally free of constant rank is in fact free. (To see this, observe that M/ra(A) is locally free of constant rank over the irect sum of fiels A/ra(A), an so is certainly free. Lifting generators, Nakayama implies that M is itself free.) Next we construct the quasi-inverse functor. Given a unit (Λ, F )-crystal E of rank m, we pull it an its Frobenius enomorphism Φ back to the étale site to get a coherent locally free Λ Z/p n OXét -moule E ét, equippe with a Λ-linear enomorphism Φét. We have to show that if L = ker(1 Φét ), then the natural map L Z/p n OXét E ét is an isomorphism. Inee, once we have this, then L is necessarily locally free over Λ, by flat escent, as OXét is flat over Z/p n, an Eét is locally free over Λ Z/p n OXét. For this suppose first that Λ is flat (an necessarily finite) over Z/p n. In this case we may regar E as a unit (Z/p n, F )-crystal, so the map above is an isomorphism by Katz s theorem [12], 4.1. In general, we can write Λ as a quotient of a finite flat local Z p -algebra Λ : as Λ is finite over Z p, there is a surjection h : Z p [x 1,..., x r ] Λ, an for i = 1,..., r there exists a monic polynomial p i with coefficients in Z p such that h(p i (x i )) = 0. Thus Λ is a quotient of Z p [x 1,..., x r ]/(p 1 (x 1 ),..., p r (x r )), which is a finite flat Z p -algebra, hence in particular semi-local. Thus we may take Λ to be a localization of Z p [x 1,..., x r ]/(p 1 (x 1 ),..., p r (x r )) at a suitable maximal ieal. Then Λ is finite flat over Z p, as Z p is a complete local ring (so localization oes not estroy finiteness). To show that we have L Z/p n OXét Eét we may work locally. As E is a locally free Λ Z/p n O X -moule, after localizing on X, we may assume that E

10 10 MATTHEW EMERTON AND MARK KISIN is free over Λ Z/p n O X, say E = (Λ Z/p n O X ) m. Write Ẽ = ( Λ/pn Z/p n O X ) m. In this case we can lift the isomorphism F E E of free Λ Z/p n O X -moules to a morphism F Ẽ Ẽ of free Λ/p n Z/p n O X -moules, an any such lift is an isomorphism, by Nakayama s lemma. This gives Ẽ the structure of a unit ( Λ/p n, F )-crystal. We enote by Φ : Ẽ Ẽ the inuce Λ-linear, F -semilinear enomorphism of Ẽ, an by E ét, Ẽét, Φ ét, Φét the pull-backs to the étale site of X of E, Ẽ, Φ, Φ. If x is any étale point of Xét, we obtain a commutative iagram 0 ker(1 Φét 1 Φét ) x Ẽ x Ẽ x 0 0 ker(1 Φét ) x E x 1 Φét E x 0. Here the bottom row is obtaine by applying Λ/p nλ to the top row. By the previous iscussion, applie to the finite flat Z/p n algebra Λ/p n, we know that the top row is exact, hence the bottom one is also, as Ẽx is flat over Λ/p n, O X being flat over Z/p n. Thus, letting L enote ker(1 Φét ), we see that the map L Z/p n OXét E ét is obtaine by applying Λ/p nλ to the isomorphism L Z/p n OXét Eét, an so is an isomorphism. (3.3) We may formulate a version of the above theorem for étale sheaves on a smooth F p -scheme X 0. We may efine a unit (Λ, F )-crystal E to be a locally free sheaf of finite rank Λ Zp O X0,crys-moules on the crystalline site of X 0 /Z/p n equippe with an isomorphism F E E (here F enotes the Frobenius on the crystalline site, an O X0,crys is the structure sheaf on the crystalline site). If there exists a smooth Z/p n scheme X whose special fibre is X 0 an a lift F of the absolute Frobenius on X, then the two notions are equivalent (exercise (!), but see also [7]), so our new efinition is consistent with the previous one. Thus even if no global lift exists we get an equivalence of categories between locally constant étale sheaves of free Λ-moules an unit (Λ, F )-crystals. Inee to construct this equivalence, we may work locally, an then we may assume ([10], III) that X 0 lifts to a smooth Z/p n scheme, equippe with a lift of Frobenius. Now we can appeal to our previous results. The referee has also remarke that there is a connection between the preceing theorem an the results of Berthelot an Messing which relate finite étale group schemes an Dieuonné crystals (see 2 of [2]). (3.4) It will be convenient to have an analogue of proposition 3.2 for lisse sheaves. For this let X be a formally smooth p-aic formal scheme over Z p, equippe with a lift F of the absolute Frobenius on its reuce subscheme, an Λ a finite flat Z p -algebra. A unit (Λ, F )-crystal E is efine as in (3.1).

11 UNIT L-FUNCTIONS 11 Namely, it is a locally free, coherent Λ Zp O X -moule E equippe with an isomorphism F E E. Then we have Corollary 3.5. Suppose that Λ is local, an finite flat over Z p. There is an equivalence of categories between p-aic lisse sheaves of Λ-moules on X an unit (Λ, F )-crystals on X. Proof. This follows immeiately from proposition (3.3) once we note that a unit (Λ, F )-crystal E satisfies E lime/p n E, because Λ is finite over Z p. Explicitly, if L is a p-aic lisse sheaf of Λ-moules, write L n = L/p n L. Then L n Zp OXét escens to a coherent, locally free Zariski sheaf of Λ/pn Zp O X -moules E n on O X, equippe with an isomorphism F E n En. Then we may attach E = lime n to L. It is locally free over Λ Zp O X as each of the E n is locally free over Λ/p n Zp O X. (In fact if U X is an open formal subscheme then E becomes free over U as soon as E 1 is free over U). Conversely, if E is a unit (Λ, F )-crystal, then we may attach a locally constant étale sheaf of free Λ/p n -moules L n to E/p n E, an set L = lim L n. If Eét is the pull-back of E to Xét, we sometimes abuse notation, an write Eét = L Zp OXét. 4. Proof of theorem 1.5 (4.1) In this section we present the proof of theorem 1.5. We enote by Λ a finite, local, artinian Z p -algebra. We begin with some preliminaries on formal schemes, an liftings of Frobenius. (4.2) Let be a positive integer. We let P enote -imensional projective space over F p, equippe with homogeneous coorinates Z 0,..., Z. Write z i := Z i /Z 0 for the affine coorinates corresponing to our choice of homogeneous coorinates; then we have G m = Spec F p [z 1, z 1 1,..., z, z 1 ] Pn. We enote by ˆP the formal, -imensional projective space over Z p. It is obtaine by completing the -imensional projective space P Zp over Z p along the special fibre p = 0. The unerlying topological spaces of ˆP an P are equal. We enote by the structure sheaf of OˆP ˆP, thought of as a sheaf on the unerlying topological space of P, an if U is any open subset of P, we write OÛ = OˆP U. We will sometimes write Û to enote U consiere with the formal scheme structure given by OÛ. We will employ these notations especially in the case that U is an open subset of G m P. (4.3) Let X be of finite type over F p. If x is a close point of such a scheme we let κ(x) enote the resiue fiel of x. This is a finite fiel of some egree

12 12 MATTHEW EMERTON AND MARK KISIN (x) over F p. We enote the ring of Witt vectors of κ(x) by W x. We again use σ (respectively φ) to enote the Frobenius automorphism of W x (respectively its inverse automorphism). If we choose an isomorphism of κ(x) with F p (x), this inuces an isomorphism of W x with W (F p (x)), which is equivariant with respect to the automorphism σ of each of these rings. Consier the enomorphism F of ˆP given by Z i Z p i. This is a lift of the absolute Frobenius to ˆP, which inuces an enomorphism of each formal open Û ˆP, an in particular of the formal -imensional multiplicative group Ĝ m = Spf(Z p z 1, z 1 1,..., z, z 1 ) ˆP, where it is given by F (z i ) = z p i. (Here Z p z 1, z1 1,..., z, z 1 enotes the p-aic completion of the Z p -algebra Z p [z 1, z1 1,..., z, z 1 ]). If k is a finite fiel, an x is a k-value point of Ĝ m corresponing to the morphism F p [z 1, z1 1,..., z, z 1 ] k of F p -algebras, then x lifts to a morphism Z p z 1, z1 1,..., z, z 1 W (k) (the Teichmüller lifting), characterize by the property that the iagram Z p z 1, z 1 1,..., z, z 1 F W (k) Z p z 1, z1 1,..., z, z 1 W (k) commutes. We enote this W (k)-value point of Ĝ m by x. In particular this applies if x is a close point of X an we take k to be κ(x), the resiue fiel at x, so that x is naturally a κ(x)-value point of G m. If we choose an embeing of κ(x) into a subfiel F p n of F p (for some positive integer n) then we may regar x as an F p n-value point of G m, an the corresponing embeing of W x as a subring of W (F p n) realizes x as a W (F p n)-value point of Ĝ m. (4.4) We now begin the proof of theorem 1.5. Consier an arbitrary separate morphism f : X Y of finite type F p -schemes. We may fin a stratification of Y by locally close affine schemes T i. If S i = f 1 (T i ) then lemma 2.6 shows that it suffices to prove theorem 1.5 for each of the morphisms f i : S i T i obtaine by restricting f to the subschemes S i. Since each T i is affine, the structural morphisms g i : T i Spec F p an h i = f i g i : S i F p are separate, an lemma 2.4 shows that for any object L of Dctf b (S i, Λ), Q(f i, L ) = Q(h i, L )/Q(g i, f i! L ). σ

13 UNIT L-FUNCTIONS 13 Thus it suffices to prove theorem 1.5 for the morphisms g i an h i. (4.5) The preceing section shows that we are reuce to consiering theorem 1.5 in the case that f : X Spec F p is the structural morphism of a finite-type separate F p -scheme. We will prove theorem 1.5 by inuction on the imension of X. Thus we assume that the result hols for all separate F p -schemes of finite type an of imension less than that of X. Nothing is change if we replace X by its reuce subscheme. We fin a ense open subscheme U of X such that each connecte component of U amits a quasi-finite an ominant morphism to G m for some natural number (which is necessarily the imension of this connecte component, an so is less than or equal to the imension of X). To fin such a U we choose a ense smooth open subscheme of X. Each connecte component of this smooth open subscheme is irreucible, an contains a non-empty (an so ense) open subset which amits an étale map to some G m. We take U be the union of these open subsets. Since X \ U has lower imension than X, lemma 2.6 together with our inuctive hypothesis shows that it suffices to prove theorem 1.5 for each connecte component of U. Fix one such connecte component, V say, equippe with a quasi-finite ominant morphism g : V G m. Let h : G m Spec F p be the structural morphism of G m, an k = gh : V Spec F p the structural morphism of V. For any object L of Dctf b (V, Λ), lemmas 2.4 an 2.5 show that Q(k, L ) = Q(h, g! L )Q(g, L ) = Q(h, g! L ). So it suffices to prove theorem 1.5 when X = G m, an Y = Spec(F p ). (4.6) Let L be any object of D b ctf (G m, Λ). As observe in 1.2, we may assume that each of the finitely many non-zero L i is a constructible étale sheaf of flat Λ-moules. Lemma 2.2 allows us to verify theorem 1.5 for each of the L i separately. Thus we assume that L = L is a single constructible étale sheaf of flat Λ-moules. We may fin a non-empty (an hence ense) open affine subscheme U of G m such that L is locally constant when restricte to U. We form the tensor prouct E = L U Zp OÛ. Note that E is a coherent OÛ/p n -sheaf for sufficiently large integers n. Hence it escens to a coherent Zariski sheaf of locally free Λ Zp OÛ-moules. In particular, U contains a non-empty (an hence ense) open subset V over which E becomes free as a Λ Zp OÛ-moule. We may fin a global section a of O G m such that the zero-set Z(a) of a is a proper subset of G m containing the complement of V. Writing D(a) = G m \ Z(a) V, we see that D(a) is non-empty an that E is free over Λ Zp OÛ, when restricte to D(a). Since Z(a) has imension less than, our inuction hypothesis, together with lemma 2.6, shows that it suffices to prove that Q(f, L D(a) ) lies in 1 + mt Λ[T ].

14 14 MATTHEW EMERTON AND MARK KISIN (4.7) Since we will only be ealing with the locally constant sheaf L D(a) from now on, we write simply L an E rather than L D(a) an E D(a), so that E = L Zp OD(a) ˆ. Recall the enomorphism F of D(a) ˆ efine in section (4.3) by restricting the enomorphism F of ˆP. The tensor prouct of the ientity morphism on L an the enomorphism 1 F of Λ Zp OD(a) ˆ inuces an 1 F -linear enomorphism of E, which we enote by Φ, an enows E with the structure of a unit (Λ, F )-crystal, from which L may be recovere as the étale subsheaf of Φ-invariants (see section 3). For convenience we will often abbreviate 1 F by F. The global section a is an element of F p [z 1, z1 1,..., z, z 1 ]. Let ã enote a lift of a to a global section of OĜ m. Then a priori ã is an element of Z p z 1, z1 1,..., z, z 1, but we may an o choose ã to be an element of Z p [z 1, z1 1,..., z, z 1 ] (since our only concern is that it reuce moulo p to a). Choosing a basis of E inuces an isomorphism Λ Zp O mˆd(a) E. With respect to this basis the F -linear enomorphism Φ may be written in the form Φ = (r ij ) F, where (r ij ) is an invertible m m matrix of elements of Λ Zp Z p [z 1, z1 1,..., z, z 1, ã 1 ]. As in the proof of proposition 3.2, there exists a finite flat local Z p - algebra Λ, an a surjection Λ Λ. We write m for the maximal ieal of Λ. Now we may lift each of the rational functions r ij to an element r ij of Λ[z 1, z1 1,..., z, z 1, ã 1 ]. Using these lifts we may form the following unit ( Λ, F )-crystal on the open subset D(a) ˆ of Ĝ m: Φ : Λ Zp O mˆ D(a) ( r ij ) F Λ Zp O mˆ, D(a) which reuces to E after ΛΛ. Denote this unit ( Λ, F )-crystal by Ẽ, an let L enote the corresponing lisse étale Z p -sheaf on D(a) ˆ obtaine as the Φfixe étale subsheaf of the étale sheaf inuce by Ẽ. We saw in section 3 that L is a lisse sheaf of Λ-moules of rank m. We will prove that the ratio L(D(a), L)/L(Spec F p, g! L), belongs to 1 + mt Λ T. Specializing via the map Λ Λ, we conclue that Q(f, L) 1 + mt Λ/p n [T ], completing the proof of theorem 1.5 (4.8) In this section we make a series of changes of basis of the unit F - crystal an its lift Ẽ in orer to ensure that the matrices (r ij) an ( r ij ) have certain properties. Slightly abusing notation, we will again enote by ã the

15 UNIT L-FUNCTIONS 15 image of ã in Λ[z 1, z1 1,..., z, z 1 ]. We begin by observing that F ã ã p (mo p) 0 (mo ã p, p). Thus, if n is an integer such that p n = 0 in Λ, then for some sufficiently large natural number N we have an so (F ã) N 0 (mo ã N+1, p n ), (F ã/ã) N ãλ[z 1, z 1 1,..., z, z 1 ]. If we let M be the maximal power of ã occurring in the enominators of the elements r ij of Λ[z 1, z1 1,..., z, z 1, ã 1 ] then we see that (F ã/ã) (M+1)N r ij ãλ[z 1, z 1 1,..., z, z 1 ] for each pair i, j. Consier the commutative iagram Λ Zp O mˆ D(a) ã (M+1)N Λ Zp O mˆ D(a) (F ã/ã) (M+1)N (r ij ) F Λ Zp O mˆ D(a) ã (M+1)N (r ij ) F Λ Zp O mˆ D(a) in which the horizontal arrows are F -linear maps an the vertical maps are Λ Zp OD(a) ˆ -linear isomorphisms (since they are simply multiplication by ã (M+1)N, a unit on D(a)). ˆ This shows that we may choose a basis for E with respect to which the matrix (r ij ) of Φ has entries in ãλ[z 1, z1 1,..., z, z 1 ], an from now on we assume that we have mae such a choice, so that L is escribe by the short exact sequence 0 L Λ Zp O mˆd(a) 1 (r ij (z)) F Λ Zp O mˆd(a) 0, in which (r ij (z)) is a matrix of of polynomials vanishing precisely at the points of Z(a). Now as in (4.7) we may lift the polynomials r ij, this time to elements r ij of ã Λ[z 1, z1 1,..., z, z 1 ], to obtain a lift Ẽ of E an a lift L of L. Let s be an integer chosen so that each (z 1 z ) s r ij belongs to Λ[z 1,..., z ], an choose a secon integer t so that (p 1)t > s. Then the iagram Λ Zp O m Ĝ m (z 1 z ) (p 1)t ( r ij ) F Λ Zp O m Ĝ m (z 1 z ) t Λ Zp O m Ĝ m ( r ij ) F (z 1 z ) t Λ Zp OĜ m,

16 16 MATTHEW EMERTON AND MARK KISIN in which the horizontal arrows are F -linear morphisms an the vertical arrows are Λ Zp OĜ m -linear isomorphisms (since (z 1 z ) t is invertible on Ĝ m), commutes. The matrix (z 1 z ) (p 1)t ( r ij ) consists of elements of (4.9) (z 1 z ) Λ[z 1,..., z ] ã Λ[z 1, z 1 1,..., z, z 1 ], by construction. Thus we see that we may choose a basis of Ẽ so that the matrix r ij escribing the F -linear enomorphism Φ of Ẽ consists of elements of the intersection (4.9), an from now on we assume that we have one this, an that the matrix r ij is chosen with respect to this basis. Let g enote the structural morphism g : P Spec F p, let h enote the open immersion of F p -schemes h : D(a) P, unerlying the open immersion of p-aic formal schemes D(a) ˆ ˆP, so that f = gh, an let enote O( 1)ˆP the ieal sheaf of the formal hyper-plane at infinity of ˆP (that is, the hyperplane escribe by the equation Z 0 = 0). Let u be an integer chosen so that (p 1)u is greater than the egree of each of the polynomials r ij. By virtue of this choice of u, the matrix ( r ij ) inuces an F -linear enomorphism ( r ij ) F of Λ O( u)ˆp. Zp Furthermore, since the r ij are ivisible by each z i as well as by ã, an since (p 1)u is in fact greater than (rather than just equal to) the egree of any of the r ij, we see that the étale sheaf of ( r ij ) F -invariants of Λ Zp O( u) mˆp is exactly equal to h! L. More precisely, write Ln = L/p n L, an Λ n = Λ/p n Λ. Then for each n 1, we have have a short exact sequence of étale sheaves 0 h! Ln Λn Zp O( u) mˆp 1 ( r ij ) F Λn Zp O( u) mˆp This is the exact sequence which we will use to compute f! L = g! h! L, an hence to compute its L-function, in orer to compare it with the L-function of L. (The construction of this exact sequence is an application of the technique of [4], lemma 4.5, p It is greatly generalize in [6], [7].) (4.10) If we apply g to the short exact sequence of étale sheaves constructe above then keeping in min that for coherent sheaves étale pushforwar agrees with the coherent push-forwar, an that for i <, R i f ( Λ n Zp O( u)ˆp) = H i (ˆP, Λ n Zp O( u)ˆp) = 0, we obtain the exact sequence 0. 0 R f! Ln H (P, Λ n Zp O( u) mˆp) 1 H (( r ij ) F ) H (P, Λ n Zp O( u) mˆp) R +1 f! Ln 0 of étale sheaves on Spec F p. If we take the stalks of this exact sequence over the geometric point Spec F p of Spec F p, an pass to the inverse limit over n,

17 UNIT L-FUNCTIONS 17 then we obtain the exact sequence 0 (R f! L)Fp H (P, Λ Zp O( u) mˆp) Zp W 1 H (( r ij ) F ) σ H (P, Λ Zp O( u) mˆp) Zp W (R +1 f! L)Fp 0. The morphism 1 H (( r ij ) F ) σ is surjective, an so we see that R +1 f! L = 0. Lemma The ratio L(Spec F p, f L)/et Λ(1! H (( r ij ) F )T, H (P, Λ ( 1)+1 Zp O( u) mˆp)) is an element of 1 + mt Λ T, an is a rational function. Proof. Let us enote the finite rank free Λ-moule H (P, Λ Zp O( u) mˆp) by M, for simplicity of notation, an let us enote the Λ-linear enomorphism H (( r ij ) F ) of M by Ψ. A little linear algebra shows that M has a canonical irect sum ecomposition M = M unit M nil, etermine by the property that M unit is the maximal Ψ-invariant Λ-submoule of M on which Ψ acts surjectively (or equivalently, bijectively), while M nil is the maximal Ψ-invariant Λ-submoule of M on which Ψ acts topologically nilpotently. Inee, it is clear that such ecomposition exists for M as a Z p -moule. However, as it is canonical M unit an M nil are Λ stable. Thus they are projective, an hence free Λ-moules. We exten the enomorphism Ψ to the σ-linear enomorphism Ψ σ of M Zp W. Since Ψ acts topologically nilpotently on M nil, 1 Ψ σ acts bijectively on M nil Zp W. Thus we may replace the exact sequence 0 (R f! L)Fp M Zp W 1 Ψ σ M Zp W 0 of section (4.10) by the exact sequence 0 (R f! L)Fp M unit Zp W 1 Ψ σ M unit Zp W 0. Since Ψ acts bijectively on M unit, we see that Ψ makes M unit into a unit ( Λ, F )- crystal, an that R f! L is the corresponing lisse p-aic étale sheaf on Spec Fp. Thus L(Spec F p, f! L) = et Λ(1 φt, (R f! L)Fp ) ( 1)+1 = et Λ(1 ΨT, M unit ) ( 1)+1 = et Λ(1 ΨT, M) ( 1)+1 /et Λ(1 ΨT, M nil ) ( 1)+1.

18 18 MATTHEW EMERTON AND MARK KISIN Thus the lemma will be prove once we show that et Λ(1 ΨT, M nil ) ( 1)+1 belongs to 1 + m Λ T. But this follows from the fact that Ψ acts topologically nilpotently on M nil, an so nilpotently on M/ m, so that the characteristic polynomial of Ψ on M nil satisfies the congruence et Λ(1 ΨT, M nil ) 1 (mo m). (4.12) We now turn to escribing the L-function of L in terms of the matrix ( r ij ). Lemma The L-function of L on D(a) is etermine by the formula L(D(a), L) = et Λ Zp W x (1 (( r ij ( x)) σ) (x) T (x), Λ Zp Wx m ) 1. x D(a) (Recall from section 4.3 that x enotes the Teichmüller lift of the close point x.) Proof. This follows from the fact that O mˆ equippe with the F -linear D(a) enomorphism ( r ij ) F is the unit F -crystal Ẽ corresponing to L. More precisely, the Artin-Schreier short exact sequence 0 L O mˆ D(a) 1 ( r ij ) F O mˆ D(a) 0 shows that for any point x of D(a), if x enotes an étale point lying over x, then et(1 φ (x) T (x), L x ) = et(1 (( r ij ( x)) σ) (x) T (x), Λ Zp W m x ). Corollary The L-function of L on D(a) agrees with the prouct et Λ Zp W x (1 (( r ij ( x)) σ) (x) T (x), Λ Zp Wx m ) 1 x G m up to multiplication by an element of 1 + pt Λ T. Proof. Lemma 4.13 shows that the ratio of L(D(a), L) an the prouct in the statement of the corollary is equal to et Λ Zp W x (1 (( r ij ( x)) σ) (x) T (x), Λ Zp Wx m ) 1. x Z(a) Thus it suffices to show that this prouct belongs to 1 + p Λ T. For any positive integer δ, let Z(a) δ enote the (finite!) subset of Z(a) consisting of those close points x for which (x) = δ. By construction, each

19 UNIT L-FUNCTIONS 19 r ij is an element of ã Λ[z 1, z1 1,..., z, z 1 ]. Thus for each x in Z(a) δ, r ij ( x) lies in p Λ Zp W x, so that the entries of lie in p δ Λ, an the finite prouct (( r ij ( x)) σ) (x) = (( r ij ( x)) σ) δ x Z(a) et Λ Zp W x (1 ((r ij ( x)) σ) (x) T (x), Λ Zp W m x ) 1 lies in 1 + p δ T Λ[[p δ T ]] 1 + p δ T δ Λ T δ. From this we conclue that et Λ Zp W x (1 ((r ij ( x)) σ) (x) T (x), Λ Zp Wx m ) 1 x Z(a) = et Λ Zp W x (1 ((r ij ( x)) σ) (x) T (x), Λ Zp Wx m ) 1 δ>0 x Z(a) δ lies in 1 + pt Λ T, thus proving the corollary. (4.15) Theorem 1.5 now follows from lemma 4.11 an corollary 4.14, together with the following result, whose proof is the subject of section 5. Proposition Let P (T ) enote Then x G m et Λ(1 H (( r ij ) F )T, H (P, Λ Zp O( u) mˆp)) 1 + T Λ[T ]. et Λ Zp W x (1 (( r ij ( x)) σ) (x) T (x), Λ Zp Wx m ) 1 x G m = P (p i T ) ( 1)+1 i ( i ), i=0 an so in particular, the ratio et Λ Zp W x (1 (( r ij ( x)) σ) (x) T (x), Λ Zp Wx m ) 1 /P (T ) ( 1)+1 lies in 1 + pt Λ T, an is a rational function. 5. Proof of proposition 4.16 (5.1) In orer to prove proposition 4.16, we begin by escribing the morphism which is Grothenieck-Serre ual to the map 1 H (( r ij ) F ). The free Λ-moule H (P, Λ Zp O( u) mˆp) H (P, O( u) mˆp) Zp Λ has

20 20 MATTHEW EMERTON AND MARK KISIN Grothenieck-Serre ual equal to H 0 (P, Ω (u) mˆp) Zp Λ. Let C enote the Cartier operator on Ω ḓ corresponing to the lift of Frobenius F, efine by P the formula C(z i 1 1 z i z 1 z ) z 1 z i 1p i z1 = z p z 1 z if i j 0 (mo p) for all j z 1 z 0 if i j 0 (mo p) for some j. By virtue of our choice of u, the composite C ( r ij ) efines an enomorphism of Λ Zp Ω (u) : Λ Zp Ω (u) C ( r ij ) Λ Zp Ω (u), an the ual of the enomorphism H (( r ij )) F of H (P, O( u) mˆp) is the enomorphism H 0 (C ( r ij )) of H 0 (P, Ω (u) mˆp) Zp Λ. Since a matrix an its ajoint have the same characteristic polynomial we see that P (T ) = et Λ(1 H 0 (C ( r ij ))T, H 0 (P, Ω (u) mˆp) Zp Λ). (5.2) The usual formula for the logarithm of a eterminant shows that (5.3) log P (T ) = trace Λ(H 0 (C ( r ij )) n, H 0 (P, Ω (u) mˆp) T Zp Λ) n n. n=1 For any sections a of Λ Zp O ˆ P an ω of Ω ˆ P acω = CF (a)ω, an so for any positive integer n we have n 1 (C ( r ij )) n = C n ((F ) i r ij ). i=0 we have the ientity This allows us to rewrite (5.2) in the form (5.4) n 1 log P (T ) = trace Λ(H 0 (C n (F ) k ( r ij )), H 0 (P, Ω (u) mˆp) T Zp Λ) n n. n=1 k=0 (5.5) Fix a positive integer n, an suppose that (f ij ) is any m m matrix of elements of (z 1 z ) Λ[z 1,..., z ], such that each of the f ij is of egree less than (p n 1)u. (For example, n 1 k=0 ((F ) k r ij ) is such a matrix.) We have a morphism Λ Zp Ω (u) mˆp C n (f ij ) Λ Zp Ω (u) mˆp,

21 UNIT L-FUNCTIONS 21 which inuces an enomorphism H 0 (C n (f ij )) of H 0 (P, Λ Zp Ω (u) mˆp). We may also form the sum of matrices (f ij ( x)), x G m(f p n) which will be an m m matrix of elements of Z p. In this situation we have the following lemma, which is a slight moification of lemma 2 of [5]: Lemma 5.6. For any matrix (f ij ) as above we have the following formula: trace Λ(H 0 (C n (f ij )), H 0 (P, Λ Zp Ω (u) mˆp)) 1 = (p n 1) trace Λ( x G m(f p n) (f ij ( x)), Λ m ). (Recall from section 4.3 that x enotes the Teichmüller lift of the F p n-value point x of G m.) Proof. We first reuce the proof of the formula to the case m = 1, as follows. It is clear that the trace on the left han sie is given by the formula trace Λ(H 0 (C n (f ij )), H 0 (P, Λ Zp Ω (u) mˆp)) m = trace Λ(H 0 (C n f ii ), H 0 (P, Λ Zp Ω (u)ˆp)), i=1 while the trace on the right han sie is given by the formula 1 m (p n 1) f ii ( x). i=1 x G m(f p n) Thus if we knew that for any f in z 1 z Λ[z1,..., z ] of egree less than (p n 1)u we ha the equation (5.7) trace Λ(H 0 (C n f), H 0 (P, Λ Zp Ω (u)ˆp)) 1 = (p n 1) f( x), x G m(f p n) the lemma woul follow. This is what we now prove. The set of all such f form a Λ-submoule of Λ[z 1,..., z ], an both sies of equation (5.7) are Λ-linear in f. Thus it suffices to prove the formula in the case that f is a single monomial with f = z α 1 1 zα, (5.8) 0 < α 1,..., α, α α < (p n 1)u.

22 22 MATTHEW EMERTON AND MARK KISIN The Λ-moule H 0 (P, Λ Zp Ω (u)ˆp) has as a basis the ifferentials with z β 1 1 zβ z 1 z z 1 z, (5.9) 0 < β 1,..., β, β β < u. We see that C n (z α 1 1 zα zβ 1 1 z zβ 1 z ) z 1 z = C n (z α 1+β 1 1 z α +β z 1 z ) z 1 z α 1 +β 1 α +β p z n p 1 z n z 1 z all α i + β i 0 (mo p n ) = z 1 z 0 some α i + β i 0 (mo p n ). From this we see that the matrix of H 0 (C n f) with respect to the given basis of H 0 (P, Λ Zp Ω (u)ˆp) has entries which are either 0 or 1, an so the trace of H 0 (C n f) is equal to the number of 1 s on the iagonal, which is given by the number of -tuples β 1,, β which satisfy (5.10) α i + β i = p n β i, for 1 i, together with (5.9). We see immeiately that the system of inequalities an equations (5.9), (5.10) has no solutions unless α 1 α 0 (mo p n 1), in which case there is a unique solution (here we are using the fact that α 1,..., α satisfies (5.8)), so that (5.11) { 1 all trace Λ(C n f, H 0 (P, Λ αi Zp Ω (u)ˆp)) 0 (mo p n 1) = 0 some α i 0 (mo p n 1). On the other han, as x ranges over all the F p n-value points of G m, the set of Teichmüller lifts x ranges over all -tuples of (p n 1) st roots of unity in W (F p n). Thus (5.12) 1 (p n 1) x G m (F p n) f( x) = = 1 (p n 1) ζ α1 1 ζ α ζ 1,,ζ µ p n 1 { 1 all αi 0 (mo p n 1) 0 some α i 0 (mo p n 1). Comparing (5.11) an (5.12), we see that (5.7) is prove, an with it the lemma.

23 UNIT L-FUNCTIONS 23 (5.13) Applying lemma 5.6 to the matrices n 1 k=0 ((F ) k r ij ) which appear in the expression on the right sie of equation (5.4) we obtain the equation log P (T ) = = trace Λ( n=1 x G m (F p n) k=0 n 1 ( r ij (σ k ( x))), Λ m T n ) n(p n 1) trace Λ( (( r ij ( x)) σ) n, Λ m T n ) n(p n 1). x G m (F p n) n=1 This in turn implies that log P (p i T ) ( 1)+1 i ( i ) = trace Λ( i=0 n=1 x G m(f p n) Exponentiating both sies yiels proposition (( r ij ( x)) σ) n, Λ m ) T n n. 6. Lifting representations of arithmetic funamental groups (6.1) In this section we iscuss some applications of our metho of lifting locally constant étale sheaves of finite free Λ moules by lifting the associate (Λ, F )-crystals. For the uration of this section Λ will enote an artinian local Z p -algebra with finite resiue fiel an X will enote a smooth affine F p -scheme. We have seen above that there exists a surjection Λ Λ with Λ a finite flat local Z p -algebra. With these notations, we have the following theorem: Theorem 6.2. Let ρ : π 1 (X) GL (Λ) be a continuous representation of the arithmetic étale funamental group of X. There exists a continuous lifting of ρ ρ : π 1 (X) GL ( Λ). Proof. The representation ρ correspons to a locally constant étale sheaf of finite free Λ-moules on X. We enote this sheaf by L. Now we claim that X can be lifte to a formally smooth p-aic formal scheme ˆX over Z p equippe with a lift F of the absolute Frobenius. Inee by [10], III the obstructions to the existence of such a lifting are containe in the cohomology of certain coherent sheaves on X, an so vanish, since X is affine. Fix ˆX an F, an enote by E the unit (Λ, F )-crystal on ˆX corresponing to L, so that E is a finite free Λ Zp O ˆX sheaf equippe with an isomorphism F E E. We have alreay seen in the proof of proposition 3.2, an also in

24 24 MATTHEW EMERTON AND MARK KISIN (4.7), that locally on ˆX, E can be lifte to a finite free Λ Zp O ˆX-moule Ẽ equippe with an isomorphism F Ẽ Ẽ lifting F E E. The obstruction to the existence of such a global lift is containe in the cohomology of certain coherent O ˆX-moules (because Λ Zp O ˆX is O ˆX coherent), an so vanishes, since O ˆX is formal affine. This proves the proposition, because by proposition 3.2, Ẽ correspons to a lisse sheaf of Λ-moules on ˆX, which gives the require representation ρ. Theorem 6.3. With the notation of theorem 6.2, suppose that X A 1 is an open subset of the affine F p -line. Then ρ may be chosen so that the corresponing lisse sheaf of Λ-moules L has rational L-function. Proof. As in section 4, we let F enote the lift of Frobenius on ˆP1 given (in affine coorinates) by z z p, an let ˆX ˆP1 enote the open formal subscheme corresponing to X P 1. We may exten the unit (Λ, F )-crystal E attache to ρ to a sheaf E + of locally free Λ OˆP1-moules Zp on P 1. Inee to exten across a point of P 1 X, we may work in a neighbourhoo of this point, an hence assume that E is free over Λ Zp O ˆX over this neighbourhoo, when the existence of the extension is clear. Moreover, if D enotes the ivisor on P 1 corresponing to P 1 X, an ˆD enotes the ivisor on ˆP1 corresponing to the Teichmüller lift of D, then replacing E + by E + ( n ˆD), for some large integer n, we may assume that isomorphism F E E extens to a map F E + E + (which cannot be an isomorphism unless ρ is trivial). Now set M = Hom Λ Zp OˆP1 (F E +, E + ). Replacing E + by E + ( n ˆD) for some large integer n replaces M by M((p 1)n ˆD), an so we may assume that we have H 1 (ˆP1, M) = 0. (This will be use later). Now we claim that E + can be lifte to a locally free Λ OˆP1-moule Zp Ẽ +. Inee, such a lifting exists locally, an the obstruction to a global lifting is containe in H 2 of certain coherent sheaves on P 1, hence vanishes (cf. [10] III, 7.1). Let n be the kernel of Λ Λ. We claim that F E + E + lifts to a morphism F Ẽ + /n 2 Ẽ+ /n 2 which is an isomorphism when restricte to X. To see this, we begin by observing that such a lift exist locally. More precisely, such a lift exists over any open U P 1 which has the property that Ẽ+ /n 2 becomes a free Λ OˆP1-moule Zp when restricte to U. Now let U 0, U 1,... U r be a collection of open subsets of P 1 such that the U i cover P 1 an such that Ẽ+ /n 2 becomes a free Λ OˆP1-moule Zp when restricte to each U i. Let φ i : F Ẽ + Ẽ+ be a lift of F E + E + over each U i. The elements φ i φ j on the intersections U i U j give rise to an element of H 1 (P 1, N), where N = Hom (F Λ Zp E +, OˆP1 nẽ+ /n 2 Ẽ + ). We claim that H 1 (P 1, N) = 0. Inee,