Unit L-functions and a conjecture of Katz.

By Matthew Emerton and Mark Kisin

Let p be a prime number, 𝔽_{p} the finite field of order p, and ℤ_{p} the ring of p-adic integers. Suppose that X is a separated finite type 𝔽_{p}-scheme and that L is a lisse ℤ_{p}-sheaf on the étale site of X. One defines in the usual way an L-function L(X, L) attached 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-dimension, and so we may form the L-function L(Spec 𝔽_{p}, f_{!}L), which is also an element of 1 + T ℤ_{p}[[T]] (and in fact a rational function). The ratio L(X, L)/L(Spec 𝔽_{p}, f_{!}L) thus lies in 1 + T ℤ_{p}[[T]], and so may be regarded as a nowhere-zero analytic function on the p-adic open unit disk |T | < 1. If L were a lisse ℤ_{l} sheaf with l ≠ p then in fact this ratio would be identically 1 (this is Grothendieck’s approach to the rationality of the zeta function). One does not have this in the setting of ℤ_{p}-sheaves. However, in this paper we prove the following result, which was conjectured by Katz ([11], 6.1):

Theorem 0.1. The ratio L(X, L)/L(Spec 𝔽_{p}, f_{!}L) extends to a nowherezero function on the closed unit disc |T | ≤ 1.

In particular this implies that the L-function for L(X, L) is p-adic meromorphic in the closed unit disc. According to Katz ([11], 6.1), when X = 𝔸^{n}, this is an old result of Dwork’s.

Katz also conjectured that L(X, L) extends to a meromorphic function on the rigid-analytic affine T -line. The second part of Katz’s conjecture was a generalization of a conjecture of Dwork predicting that L(X, L) was p-adic meromorphic when the unit F -crystal M associated to L was the unit part of an ordinary, overconvergent F -crystal. Wan [13] has found counter-examples to the part of Katz’s conjecture predicting that the L-function is meromorphic for general lisse sheaves. On the other hand, in a recent preprint [14] he has proven that Dwork’s more cautious prediction is actually true. Thus, while the question of when the L-function is meromorphic is by now quite well understood, much less was known about the location of the zeroes and poles. In fact the only prior results in this direction that we are aware of are due to Crew [3], who proved Katz’s conjecture in the special case when X is an affine curve and the sheaf L has abelian monodromy, and Etesse and Le Stum [8], who proved Katz’s conjecture under the strong assumption that L extends 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 residue field. We refer to Corollary 1.8 for the precise formulation.

Our methods also have some applications to results on lifting representations of arithmetic fundamental 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 residue field, and

ρ : π_{1}(X) → GL_{d}(Λ)

a representation of the arithmetic étale fundamental group of X. Consider a finite flat local ℤ_{p}-algebra Λ̃ and a surjection Λ̃ → Λ. There exists a continuous lifting of ρ

ρ̃: π_{1}(X) → GL_{d}(Λ̃).

If X is an open affine subset of ℙ^{1}, then ρ̃ may be chosen so that the L-function of the corresponding lisse sheaf of Λ̃-modules is rational.

Let us now describe the contents of the paper in more detail.

In section 1 we define the necessary L-functions, and state our main results.

In section 2 we give the proof of some standard results on behaviour of L-functions in exact triangles, and under stratification of the underlying space. These are used later to reduce our calculations to a special case.

In section 3 we introduce a key ingredient in our work, which is the relationship between locally constant or lisse étale sheaves and unit F -crystals. We show that if Λ is an artinian (respectively a complete noetherian) finite local ℤ_{p}-algebra with finite residue field and if X is a formally smooth p-adic formal scheme equipped with a lifting F of the absolute Frobenius of its special fibre, then there is a one-to-one correspondence between locally constant étale sheaves of finite free Λ-modules (respectively lisse étale sheaves of Λ-modules) on X and finite free Λ ⊗_{ℤ_{p}} O_{X}-modules E on X equipped 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.

UNIT L-FUNCTIONS 3

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-adic constructible sheaves of Λ-modules. By passing to the inverse limit, it is enough to consider the case where Λ is artinian. Using standard techniques we reduce ourselves to having to verify Katz’s conjecture for a locally constant flat Λsheaf on an open subset of 𝔾_{m}^{d} with the additional property that the associated unit (Λ, F)-crystal E (as explained above) has an underlying locally free sheaf of Λ ⊗_{ℤ_{p}} O_{X}-modules which is actually free. Under this assumption we may choose a surjection Λ̃ → Λ, with Λ̃ a finite flat local ℤ_{p}-algebra, and lift this Λ-sheaf to a lisse Λ̃-sheaf by lifting the associated unit (Λ, F)-crystal.

In this situation we prove Katz’s conjecture directly, by using an explicit trace calculation in the style of Dwork [5]. To reduce to this trace calculation we utilize a technique of Deligne [4] in order to find an Artin-Schreier sequence for the extension by zero of a lisse Λ̃-sheaf under an open immersion, which allows us to relate p-adic 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 methods can be used to lift representations of arithmetic fundamental groups.

Finally, let us mention that the methods used in this paper are a special instance of a more general theory developed by the authors in the papers [6], [7]. This theory describes the étale cohomology of p-power torsion or padic constructible sheaves on finite type 𝔽_{p}-schemes in terms of quasi-coherent cohomology; more precisely, a certain Riemann-Hilbert correspondence is established between the appropriate derived categories of such étale sheaves, and a derived category of certain quasi-coherent analogues of unit F -crystals. However, we have written this paper so as to be independent of these more general techniques.

Acknowledgments. The authors would like to thank Pierre Berthelot for suggesting that they apply their techniques to the study of L-functions. The first author would also like to thank Mike Roth for useful discussions.

1. Notation and statement of results

(1.1) Let p be a prime number, fixed for the remainder of the paper, 𝔽_{p} denote a finite field of order p, and \overline{𝔽}_{p} be a choice of an algebraic closure of 𝔽_{p} (fixed once and for all). For any positive integer n we denote by 𝔽_{p^{n}} the unique degree n extension of 𝔽_{p} contained in \overline{𝔽}_{p}. We denote by σ : \overline{𝔽}_{p} → \overline{𝔽}_{p} the (arithmetic) Frobenius automorphism a ↦ a^{p} of \overline{𝔽}_{p}, and denote by ϕ the geometric Frobenius automorphism of \overline{𝔽}_{p}, that is, the inverse of σ. Both σ and ϕ are topological generators of the Galois group of \overline{𝔽}_{p} over 𝔽_{p}.

We denote by \overline{W} the ring of Witt vectors of \overline{𝔽}_{p}, and by σ and ϕ respectively the canonical Frobenius automorphism of \overline{W} and its inverse. If n is any positive integer, we let W (𝔽_{p^{n}}) denote the Witt ring of 𝔽_{p^{n}}, regarded as a subring of \overline{W} . As usual we will denote by ℤ_{p} the ring W (𝔽_{p}) of p-adic numbers.

(1.2) Let X be a finite type scheme over 𝔽_{p}, and Λ a noetherian ring, with mΛ = 0 for some positive integer m. We denote by D^{−}(X, Λ) the derived category of bounded above complexes of étale sheaves of Λ-modules on X, and we denote by D_{ctf}^{b}(X, Λ) the full subcategory of D^{−}(X, Λ) consisting of complexes of Λ-sheaves which have finite tor-dimension, and which are constructible (in the sense that their cohomology sheaves are constructible Λmodules [1]). Given any object L^{•} of D_{ctf}^{b}(X, Λ), we may define its L-function in the usual way, which we briefly recall.

If L^{•} is a constructible complex of Λ-sheaves of finite tor-dimension on X then by [4], proposition-définition 4.6 (ii), p. 93, we may assume that L^{•} is a finite length complex of flat constructible Λ-sheaves. If x is a closed point of X we let L_{x}^{i} denote the flat Λ-sheaf obtained by restricting L^{i} to the point x. If we choose an identification of the residue field of x with the subfield 𝔽_{p^{d(x)}} of \overline{𝔽}_{p}, and denote by x̄ the \overline{𝔽}_{p}-valued point of X lying over x, then Li ; x̄ is a free Λ-module equipped with an action of the d^{th} power ϕ^{d} of the geometric Frobenius automorphism. We let |X| denotes the set of closed points of X, and define

L(X, L^{•}) = ∏_{i} ∏_{x∈|X|} det_{Λ}(1 − ϕ^{d(x)}T^{d(x)}, Li ; x̄)^{(−1)^{i+1}}.

This is an element of 1 + T Λ[[T]].

From the definition we see that L-functions are compatible with change of ring: if Λ, Λ′ are noetherian rings killed by some positive integer, g : Λ → Λ′ a map of rings, and 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 and let i : X_{red} → X be the closed immersion of the underlying reduced subscheme of X into X. Then the functors i_{!} and i^{−1} are quasi-inverse, and induce an equivalence of triangulated categories between D_{ctf}^{b}(X_{red}, Λ) and D_{ctf}^{b}(X, Λ) (preserving the canonical t-structure of both categories). If L^{•} is any object of D_{ctf}^{b}(X, ℤ/p^{n}) then L(X, L^{•}) = L(X_{red}, i^{−1}L^{•}).

(1.4) If f : X → Y is a separated morphism of finite type 𝔽_{p}-schemes then push-forward with proper supports induces a functor f_{!} : D_{ctf}^{b}(X, Λ) → D_{ctf}^{b}(Y, Λ) ([4], théorème 4.9, p. 95).

Theorem 1.5. Let f : X → Y be a separated morphism of finite type 𝔽_{p}-schemes, and Λ a finite, local, artinian ℤ_{p}-algebra, with maximal ideal 𝔪. If L^{•} is any object of D_{ctf}^{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].

(1.6) Suppose that Λ is finite and reduced of characteristic p. Then from theorem 1.5 we conclude that the L-functions of objects of D_{ctf}^{b}(Xét, Λ) are invariant under proper push-forward. This was originally proved 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}-modules, to show that his formula does not hold without the hypothesis that Λ is reduced ([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 modulo the principal ideal (X). This also follows from Deligne’s theorem by “specialization of L-functions” (1.2), and 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 define 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 and an ideal I ⊂ Λ, such that Λ is I-adically complete and p is nilpotent in Λ/I. We define the category D_{I−sm}^{b}(X, Λ) to be the 2-limit of the categories D_{ctf}^{b}(X, Λ/I^{n}), n = 1, 2, . . .. One has a formalism of f_{!}, f^{−1}

and ⊗^{𝕃} in D_{I−sm}^{b}(X, Λ) (see [9] for details).

If L^{•} is an object of D_{I−sm}^{b}(X, Λ) then by construction L^{•} ⊗^{𝕃}_{Λ} Λ/I^{n} belongs to D_{ctf}^{b}(X, Λ/I^{n}) 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}) (mod I^{m}).

Thus we may define 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_{!} : D_{I−sm}^{b}(X, Λ) → D_{I−sm}^{b}(Y, Λ). For each positive integer n there is a canonical isomorphism

(f_{!}L^{•}) ⊗^{𝕃}_{Λ} Λ/I^{n} ⟶^{∼} f_{!}(L^{•} ⊗^{𝕃}_{Λ} Λ/I^{n}),

so that

L(Y, f_{!}L^{•}) ≡ L(Y, f_{!}(L^{•} ⊗^{𝕃}_{Λ} Λ/I^{n})) (mod I^{n}).

If we now suppose that Λ is a complete local ℤ_{p}-algebra with finite residue field (so that in the above discussion I = 𝔪 is the maximal ideal of Λ), by letting n tend to infinity in the above discussion we derive the following corollary of theorem 1.5, which includes Katz’s conjecture as a special case (take Λ to be ℤ_{p} and L^{•} to be a single lisse p-adic sheaf in degree zero):

Corollary 1.8. Let Λ be a complete local ℤ_{p}-algebra, with finite residue field, and maximal ideal 𝔪. Let f : X → Y be a separated morphism of finite type 𝔽_{p}-schemes. If L^{•} is any object of D_{𝔪−sm}^{b}(X, Λ) then the ratio of Lfunctions

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 ⟩ denotes the 𝔪-adic 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 separated morphism f : X → Y of finite type 𝔽_{p}-schemes and any

object L^{•} of D_{ctf}^{b}(X, Λ), let us write

Q(f, L^{•}) = L(X, L^{•})/L(Y, f_{!}L^{•}).

We begin by recalling some standard tools for analyzing such ratios of Lfunctions. Although the proofs are well-known and quite straightforward, for the sake of completeness we recall them.

Lemma 2.2. If f : X → Y is a separated morphism of finite type 𝔽_{p}- schemes, then for any distinguished triangle

⋯ → L_{1}^{•} → L_{2}^{•} → L_{3}^{•} → L_{1}^{•}[1] → ⋯

of objects of D_{ctf}^{b}(X, Λ) we have the equality

Q(f, L_{2}^{•}) = Q(f, L_{1}^{•})Q(f, L_{3}^{•}).

Proof. This follows immediately from the fact that f_{!} takes distinguished triangles to distinguished triangles, together with the multiplicativity of Lfunctions of objects in a distinguished triangle.

If f : X → Y is a morphism of schemes and y is a point of Y we let f_{y} : X_{y} → y denote the fibre of f over the point y.

Lemma 2.3. For any separated morphism f : X → Y of finite type 𝔽_{p}- schemes and any object L^{•} of D_{ctf}^{b}(X, Λ) we have the equality

Q(f, L^{•}) = ∏_{y∈|Y |} Q(f_{y}, L_{|X_{y}}^{•})

Proof. By partitioning the elements of |X| according to their image in |Y |, one sees that

L(X, L^{•}) = ∏_{y∈|Y |} L(X_{y}, L_{|X_{y}}^{•}).

By the proper base-change theorem, for any point y of |Y |,

(f_{!}L^{•})_{y} = f_{y!}L_{|X_{y}}^{•},

and so

L(Y, f_{!}L^{•}) = ∏_{y∈|Y |} L(y, f_{y!}L_{|X_{y}}^{•}).

Thus

Q(f, L^{•}) = ∏_{y∈|Y |} L(X_{y}, L_{|X_{y}}^{•})/L(y, f_{y!}L_{|X_{y}}^{•}) = ∏_{y∈|Y |} Q(f_{y}, L_{|X_{y}}^{•}),

proving the lemma.

Lemma 2.4. If f : X → Y and g : Y → Z are two separated morphisms of finite type 𝔽_{p}-schemes and L^{•} is any object of D_{ctf}^{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-hand side)

Q(g, f_{!}L^{•})Q(f, L^{•}) = \frac{L(Y, f_{!}L^{•})}{L(Z, g_{!}f_{!}L^{•})} \frac{L(X, L^{•})}{L(Y, f_{!}L^{•})} = \frac{L(X, L^{•})}{L(Z, (gf)_{!}L^{•})} = Q(gf, L^{•}).

This proves the lemma.

Lemma 2.5. For any separated quasi-finite morphism f : X → Y of finite type 𝔽_{p}-schemes and any object L^{•} of D_{ctf}^{b}(X, Λ), Q(f, L^{•}) = 1.

Proof. Lemma 2.3 reduces us to the situation in which Y = Spec 𝔽_{p^{d}} is a point, so that X is either empty, in which case there is nothing to prove, or else zero-dimensional. In this second case we may replace X by its underlying reduced subscheme (via (1.4)), and writing this as a disjoint union of points we reduce to the case that X is also a point, say X = Spec 𝔽_{p^{d′}}, with d dividing d′. Using lemma 2.2 one reduces to the case that L^{•} = L is a single flat Λ-module equipped with a Λ-linear ϕ^{d′} action. Then f_{!}L is simply the induced module

f_{!}L = ℤ/p^{n}[ϕ^{d}] ⊗_{ℤ/p^{n}[ϕ^{d′}]} L,

and the equality of L-functions L(X, L) = L(Y, f_{!}L) is an elementary calculation.

Lemma 2.6. Suppose that f : X → Y is separated morphism of finite type 𝔽_{p}-schemes, and that X = S_{0} ∐ S_{1} ∐ ⋯ ∐ S_{n} is a stratification of X by locally closed subsets S_{i} (more precisely, we take this to mean that each S_{i} is closed in the union ∐_{j=0}^{i} S_{j}.) We give each of the S_{i} their reduced induced scheme structure. If L^{•} is an object of D_{ctf}^{b}(X, Λ) then let L_{|S_{i}}^{•} denote the restriction of L^{•} to each of the locally closed subsets S_{i}, and let f_{i} : S_{i} → Y denote the restriction of the morphism f to each of the locally closed subsets S_{i}. Then

Q(f, L^{•}) = ∏_{i=0}^{n} Q(f_{i}, L_{|S_{i}}^{•}).

In particular, this applies if the stratification on X is obtained by pulling back a stratification on Y.

Proof. This follows from lemmas 2.2., 2.4 and 2.5, since immersions are quasi-finite.

3. Etale sheaves and unit F -crystals

(3.1) We want to explain a generalization of a result of Katz relating étale sheaves and unit F -crystals. Although what we do can be done somewhat more generally, we restrict ourselves to smooth ℤ/p^{n}-schemes.

Assume we have a smooth scheme X over ℤ/p^{n}, and let Λ be a ℤ/p^{n}- algebra. We assume that X is equipped with an endomorphism F lifting the absolute Frobenius on its reduced subscheme.

A unit (Λ, F)-crystal on X is a sheaf E of finite locally free Λ ⊗_{ℤ/p^{n}} O_{X} modules, equipped 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 depends only on the reduction of X modulo p^{n′}.

Proposition 3.2. Suppose that Λ is noetherian, local, and finite over ℤ/p^{n}. There is an equivalence of categories (explicitly described below) between the category of locally constant étale sheaves of finite free Λ-modules on X and the category of (Λ, F)-crystals on X.

Proof. Let L be a locally constant étale sheaf of finite free Λ-modules on X (which is, of course equivalent to the data of such a sheaf on the reduction of X modulo p). We associate to L a (Λ, F)-crystal on X as follows. Consider the étale sheaf Eét =: L ⊗_{ℤ/p^{n}} O_{Xét}. This is a coherent sheaf of Λ ⊗_{ℤ/p^{n}} O_{Xét}- modules, which is in fact locally free of finite rank (since L is locally free of finite rank over Λ). Regarding Eét as a coherent sheaf on O_{Xét}, we see by [12], 4.1, that Eét comes by pull-back from a Zariski coherent O_{X}-module E. Moreover, E is equipped with an isomorphism F^{∗}E ⟶^{∼} E. If we denote by Φ the composite E → F^{∗}E ⟶^{∼} E, then the induced map Φét on Eét has fixed subsheaf equal to L. As the formation of E is functorial in Eét, we see that E is equipped with the structure of a Λ-module, and 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}-module. If x ∈ X denote by x̃ an étale point lying over x. Now E_{ét,x̃} = E_{x} ⊗_{(Λ⊗_{ℤ/p^{n}}O_{X,x})} (Λ ⊗_{ℤ/p^{n}} O_{Xét,x̃}) is certainly a free Λ ⊗_{ℤ/p^{n}} O_{Xét,x̃} module. Since Λ ⊗_{ℤ/p^{n}} O_{Xét,x̃} is faithfully flat over Λ ⊗_{ℤ/p^{n}} O_{X,x} (O_{Xét,x̃} being faithfully flat over O_{X,x}), we conclude by descent 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}, and 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 module M over a semi-local ring A which is locally free of constant rank is in fact free. (To see this, observe that M/rad(A) is locally free of constant rank over the direct sum of fields A/rad(A), and 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 and its Frobenius endomorphism Φ back to the étale site to get a coherent locally free Λ ⊗_{ℤ/p^{n}} O_{Xét}-module Eét, equipped with a Λ-linear endomorphism Φét. We have to show that if L = ker(1 − Φét), then the natural map L ⊗_{ℤ/p^{n}} O_{Xét} → Eét is an isomorphism. Indeed, once we have this, then L is necessarily locally free over Λ, by flat descent, as O_{Xét} is flat over ℤ/p^{n}, and Eét is locally free over Λ ⊗_{ℤ/p^{n}} O_{Xét}.

For this suppose first that Λ is flat (and necessarily finite) over ℤ/p^{n}. In this case we may regard 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 Λ̃ : as Λ is finite over ℤ_{p}, there is a surjection h : ℤ_{p}[x_{1}, …, x_{r}] → Λ, and 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 Λ̃ to be a localization of ℤ_{p}[x_{1}, …, x_{r}]/(p_{1}(x_{1}), …, p_{r}(x_{r})) at a suitable maximal ideal. Then Λ̃ is finite flat over ℤ_{p}, as ℤ_{p} is a complete local ring (so localization does not destroy finiteness).

To show that we have L ⊗_{ℤ/p^{n}} O_{Xét} ⟶^{∼} Eét we may work locally. As E is a locally free Λ⊗_{ℤ/p^{n}} O_{X}-module, after localizing on X, we may assume that E is free over Λ⊗_{ℤ/p^{n}} O_{X}, say E = (Λ⊗_{ℤ/p^{n}} O_{X})^{m}. Write Ẽ = (Λ̃/p^{n} ⊗_{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}-modules to a morphism F^{∗}Ẽ → Ẽ of free Λ̃/p^{n} ⊗_{ℤ/p^{n}} O_{X}-modules, and any such lift is an isomorphism, by Nakayama’s lemma. This gives Ẽ the structure of a unit (Λ̃/p^{n}, F)-crystal. We denote by Φ̃ : Ẽ → Ẽ the induced Λ̃-linear, F^{∗}-semi-linear endomorphism of Ẽ, and by Eét, Ẽét, Φét, Φ̃ét the pull-backs to the étale site of X of E, Ẽ, Φ, Φ̃. If x is any étale point of Xét, we obtain a commutative diagram

\xymatrix{0 \ar[r] & ker(1 − Φ̃ét)_{x} \ar[r] \ar[d] & Ẽ_{x} \ar[r]^{1−Φ̃ét} \ar[d] & Ẽ_{x} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ker(1 − Φét)_{x} \ar[r] & E_{x} \ar[r]^{1−Φét} & E_{x} \ar[r] & 0.}

Here the bottom row is obtained by applying ⊗_{Λ̃/p^{n}}Λ to the top row. By the previous discussion, applied to the finite flat ℤ/p^{n} algebra Λ̃/p^{n}, we know that the top row is exact, hence the bottom one is also, as Ẽ_{x} is flat over Λ̃/p^{n}, O_{X} being flat over ℤ/p^{n}. Thus, letting L̃ denote ker(1 − Φ̃ét), we see that the map L ⊗_{ℤ/p^{n}} O_{Xét} → Eét is obtained by applying ⊗_{Λ̃/p^{n}}Λ to the isomorphism L̃ ⊗_{ℤ/p^{n}} O_{Xét} ⟶^{∼} Eét, and 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 define a unit (Λ, F)-crystal E to be a locally free sheaf of finite rank Λ ⊗_{ℤ_{p}} O_{X_{0},crys}-modules on the crystalline site of X_{0}/ℤ/p^{n} equipped with an isomorphism F^{∗}E ⟶^{∼} E (here F denotes the Frobenius on the crystalline site, and O_{X_{0},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} and a lift F of the absolute Frobenius on X, then the two notions are equivalent (exercise (!), but see also [7]), so our new definition 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-modules and unit (Λ, F)-crystals. Indeed to construct this equivalence, we may work locally, and then we may assume ([10], III) that X_{0} lifts to a smooth ℤ/p^{n} scheme, equipped with a lift of Frobenius. Now we can appeal to our previous results.

The referee has also remarked that there is a connection between the preceding theorem and the results of Berthelot and Messing which relate finite étale group schemes and Dieudonné 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-adic formal scheme over ℤ_{p}, equipped with a lift F of the absolute Frobenius on its reduced subscheme, and Λ a finite flat ℤ_{p}-algebra. A unit (Λ, F)-crystal E is defined as in (3.1).

Namely, it is a locally free, coherent Λ ⊗_{ℤ_{p}} O_{X}-module E equipped with an isomorphism F^{∗}E ⟶^{∼} E. Then we have

Corollary 3.5. Suppose that Λ is local, and finite flat over ℤ_{p}. There is an equivalence of categories between p-adic lisse sheaves of Λ-modules on X and unit (Λ, F)-crystals on X.

Proof. This follows immediately from proposition (3.3) once we note that a unit (Λ, F)-crystal E satisfies E ⟶^{∼} lim_{←}E/p^{n}E, because Λ is finite over ℤ_{p}.

Explicitly, if L is a p-adic lisse sheaf of Λ-modules, write L_{n} = L/p^{n}L. Then L_{n}⊗_{ℤ_{p}} O_{Xét} descends to a coherent, locally free Zariski sheaf of Λ/p^{n}⊗_{ℤ_{p}} O_{X}-modules E_{n} on O_{X}, equipped with an isomorphism F^{∗}E_{n} ⟶^{∼} E_{n}. Then we may attach E = lim_{←}E_{n} to L. It is locally free over Λ ⊗_{ℤ_{p}} O_{X} as each of the E_{n} is locally free over Λ/p^{n} ⊗_{ℤ_{p}} 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}-modules L_{n} to E/p^{n}E, and set L = lim_{←}L_{n}.

If Eét is the pull-back of E to Xét, we sometimes abuse notation, and write Eét = L ⊗_{ℤ_{p}} O_{Xét}.

4. Proof of theorem 1.5

(4.1) In this section we present the proof of theorem 1.5. We denote by Λ a finite, local, artinian ℤ_{p}-algebra. We begin with some preliminaries on formal schemes, and liftings of Frobenius.

(4.2) Let d be a positive integer. We let ℙ^{d} denote d-dimensional projective space over 𝔽_{p}, equipped with homogeneous coordinates Z_{0}, …, Z_{d}. Write z_{i} := Z_{i}/Z_{0} for the affine coordinates corresponding to our choice of homogeneous coordinates; then we have

𝔾_{m}^{d} = Spec 𝔽_{p}[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}] ⊂ ℙ^{n}.

We denote by ℙ̂^{d} the formal, d-dimensional projective space over ℤ_{p}. It is obtained by completing the d-dimensional projective space ℙ_{ℤ_{p}} over ℤ_{p} along the special fibre p = 0. The underlying topological spaces of ℙ̂^{d} and ℙ^{d} are equal. We denote by O_{ℙ̂^{d}} the structure sheaf of ℙ̂^{d}, thought of as a sheaf on the underlying topological space of ℙ^{d}, and if U is any open subset of ℙ^{d}, we write OÛ = O_{ℙ̂^{d}}|_{U}. We will sometimes write Û to denote U considered 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}^{d} ⊂ ℙ^{d}.

(4.3) Let X be of finite type over 𝔽_{p}. If x is a closed point of such a scheme we let κ(x) denote the residue field of x. This is a finite field of some degree d(x) over 𝔽_{p}. We denote the ring of Witt vectors of κ(x) by W_{x}. We again use σ (respectively ϕ) to denote the Frobenius automorphism of W_{x} (respectively its inverse automorphism). If we choose an isomorphism of κ(x) with 𝔽_{p^{d(x)}}, this induces an isomorphism of W_{x} with W (𝔽_{p^{d(x)}}), which is equivariant with respect to the automorphism σ of each of these rings.

Consider the endomorphism F of ℙ̂^{d} given by Z_{i} ↦ Z_{i}^{p}. This is a lift of the absolute Frobenius to ℙ̂^{d}, which induces an endomorphism of each formal open Û ⊂ ℙ̂^{d}, and in particular of the formal d-dimensional multiplicative group

𝔾̂_{m}^{d} = Spf(ℤ_{p}⟨z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}⟩) ⊂ ℙ̂^{d},

where it is given by F^{∗}(z_{i}) = z_{i}^{p}. (Here ℤ_{p}⟨z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}⟩ denotes the p-adic completion of the ℤ_{p}-algebra ℤ_{p}[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}]).

If k is a finite field, and x is a k-valued point of 𝔾̂_{m}^{d} corresponding to the morphism

𝔽_{p}[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}] → k

of 𝔽_{p}-algebras, then x lifts to a morphism

ℤ_{p}⟨z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}⟩ → W (k)

(the Teichmüller lifting), characterized by the property that the diagram

\xymatrix{ℤ_{p}⟨z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}⟩ \ar[r] \ar[d]^{F^{∗}} & W (k) \ar[d]^{σ} \\ ℤ_{p}⟨z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}⟩ \ar[r] & W (k)}

commutes. We denote this W (k)-valued point of 𝔾̂_{m}^{d} by x̃.

In particular this applies if x is a closed point of X and we take k to be κ(x), the residue field at x, so that x is naturally a κ(x)-valued point of 𝔾_{m}^{d}. If we choose an embedding of κ(x) into a subfield 𝔽_{p^{n}} of \overline{𝔽}_{p} (for some positive integer n) then we may regard x as an 𝔽_{p^{n}}-valued point of 𝔾_{m}^{d}, and the corresponding embedding of W_{x} as a subring of W (𝔽_{p^{n}}) realizes x̃ as a W (𝔽_{p^{n}})-valued point of 𝔾̂_{m}^{d}.

(4.4) We now begin the proof of theorem 1.5. Consider an arbitrary separated morphism f : X → Y of finite type 𝔽_{p}-schemes. We may find a stratification of Y by locally closed 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} obtained by restricting f to the subschemes S_{i}.

Since each T_{i} is affine, the structural morphisms g_{i} : T_{i} → Spec 𝔽_{p} and h_{i} = f_{i}g_{i} : S_{i} → 𝔽_{p} are separated, and lemma 2.4 shows that for any object L^{•} of D_{ctf}^{b}(S_{i}, Λ),

Q(f_{i}, L^{•}) = Q(h_{i}, L^{•})/Q(g_{i}, f_{i!}L^{•}).

Thus it suffices to prove theorem 1.5 for the morphisms g_{i} and h_{i}.

(4.5) The preceding section shows that we are reduced to considering theorem 1.5 in the case that f : X → Spec 𝔽_{p} is the structural morphism of a finite-type separated 𝔽_{p}-scheme. We will prove theorem 1.5 by induction on the dimension of X. Thus we assume that the result holds for all separated 𝔽_{p}-schemes of finite type and of dimension less than that of X.

Nothing is changed if we replace X by its reduced subscheme. We find a dense open subscheme U of X such that each connected component of U′ admits a quasi-finite and dominant morphism to 𝔾_{m}^{d} for some natural number d (which is necessarily the dimension of this connected component, and so is less than or equal to the dimension of X). To find such a U we choose a dense smooth open subscheme of X. Each connected component of this smooth open subscheme is irreducible, and contains a non-empty (and so dense) open subset which admits an étale map to some 𝔾_{m}^{d}. We take U be the union of these open subsets. Since X \ U has lower dimension than X, lemma 2.6 together with our inductive hypothesis shows that it suffices to prove theorem 1.5 for each connected component of U. Fix one such connected component, V say, equipped with a quasi-finite dominant morphism g : V → 𝔾_{m}^{d}. Let h : 𝔾_{m}^{d} → Spec 𝔽_{p} be the structural morphism of 𝔾_{m}^{d}, and k = gh : V → Spec 𝔽_{p} the structural morphism of V. For any object L^{•} of D_{ctf}^{b}(V, Λ), lemmas 2.4 and 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}^{d}, and Y = Spec(𝔽_{p}).

(4.6) Let L^{•} be any object of D_{ctf}^{b}(𝔾_{m}^{d}, Λ). As observed in 1.2, we may assume that each of the finitely many non-zero L^{i} is a constructible étale sheaf of flat Λ-modules. 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-modules.

We may find a non-empty (and hence dense) open affine subscheme U of 𝔾_{m}^{d} such that L is locally constant when restricted to U. We form the tensor product E = L_{|U} ⊗_{ℤ_{p}} OÛ. Note that E is a coherent OÛ/p^{n}-sheaf for sufficiently large integers n. Hence it descends to a coherent Zariski sheaf of locally free Λ ⊗_{ℤ_{p}} OÛ-modules. In particular, U contains a non-empty (and hence dense) open subset V over which E becomes free as a Λ⊗_{ℤ_{p}} OÛ-module. We may find a global section a of O_{𝔾_{m}^{d}} such that the zero-set Z(a) of a is a proper subset of 𝔾_{m}^{d} containing the complement of V. Writing D(a) = 𝔾_{m}^{d} \ Z(a) ⊂ V, we see that D(a) is non-empty and that E is free over Λ ⊗_{ℤ_{p}} OÛ, when restricted to D(a). Since Z(a) has dimension less than d, our induction hypothesis, together with lemma 2.6, shows that it suffices to prove that Q(f, L_{|D(a)}) lies in 1 + 𝔪T Λ[T].

(4.7) Since we will only be dealing with the locally constant sheaf L_{|D(a)} from now on, we write simply L and E rather than L_{|D(a)} and E_{|D(a)}, so that E = L ⊗_{ℤ_{p}} OD̂(a).

Recall the endomorphism F of D̂(a) defined in section (4.3) by restricting the endomorphism F of ℙ̂^{d}. The tensor product of the identity morphism on L and the endomorphism 1 ⊗ F^{∗} of Λ ⊗_{ℤ_{p}} OD̂(a) induces an 1 ⊗ F^{∗}-linear endomorphism of E, which we denote by Φ, and endows E with the structure of a unit (Λ, F)-crystal, from which L may be recovered 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}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}]. Let ã denote a lift of a to a global section of O_{𝔾̂_{m}^{d}}. Then a priori ã is an element of ℤ_{p}⟨z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}⟩, but we may and do choose ã to be an element of ℤ_{p}[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}] (since our only concern is that it reduce modulo p to a).

Choosing a basis of E induces an isomorphism Λ ⊗_{ℤ_{p}} O_{D̂(a)}^{m} ⟶^{∼} E. With respect to this basis the F^{∗}-linear endomorphism Φ may be written in the form Φ = (r_{ij}) ∘ F^{∗}, where (r_{ij}) is an invertible m × m matrix of elements of Λ ⊗_{ℤ_{p}} ℤ_{p}[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}, ã^{−1}].

As in the proof of proposition 3.2, there exists a finite flat local ℤ_{p}- algebra Λ̃, and a surjection Λ̃ → Λ. We write 𝔪̃ for the maximal ideal of Λ̃. Now we may lift each of the rational functions r_{ij} to an element r̃_{ij} of Λ̃[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}, ã^{−1}]. Using these lifts we may form the following unit (Λ̃, F)-crystal on the open subset D̂(a) of 𝔾̂_{m}^{d}:

\xymatrix{Φ̃ : Λ̃ ⊗_{ℤ_{p}} Om ; D̂(a) \ar[r]^{(r̃_{ij})∘F^{∗}} & Λ̃ ⊗_{ℤ_{p}} Om ; D̂(a),}

which reduces to E after ⊗Λ̃Λ. Denote this unit (Λ̃, F)-crystal by Ẽ, and let L̃ denote the corresponding lisse étale ℤ_{p}-sheaf on D̂(a) obtained as the Φ̃- fixed étale subsheaf of the étale sheaf induced by Ẽ. We saw in section 3 that L̃ is a lisse sheaf of Λ-modules of rank m. We will prove that the ratio L(D(a), L̃)/L(Spec 𝔽_{p}, g_{!}L̃), belongs to 1 + 𝔪̃T Λ̃⟨T ⟩. Specializing via the map Λ̃ → Λ, we conclude 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 and its lift Ẽ in order to ensure that the matrices (r_{ij}) and (r̃_{ij}) have certain properties. Slightly abusing notation, we will again denote by ã the image of ã in Λ[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}]. We begin by observing that

F^{∗}ã ≡ ã^{p} (mod p)
≡ 0 (mod 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

(F^{∗}ã)^{N} ≡ 0 (mod ã^{N+1}, p^{n}),

and so

(F^{∗}ã/ã)^{N} ∈ ãΛ[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}].

If we let M be the maximal power of ã occurring in the denominators of the elements r_{ij} of Λ[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}, ã^{−1}] then we see that

(F^{∗}ã/ã)^{(M+1)N}r_{ij} ∈ ãΛ[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}]

for each pair i, j. Consider the commutative diagram

\xymatrix{Λ ⊗_{ℤ_{p}} Om ; D̂(a) \ar[d]^{ã^{(M+1)N}} \ar[r]^{(F^{∗}ã/ã)^{(M+1)N} (r_{ij})∘F^{∗}} & Λ ⊗_{ℤ_{p}} Om ; D̂(a) \ar[d]^{ã^{(M+1)N}} \\ Λ ⊗_{ℤ_{p}} Om ; D̂(a) \ar[r]^{(r_{ij})∘F^{∗}} & Λ ⊗_{ℤ_{p}} Om ; D̂(a)}

in which the horizontal arrows are F^{∗}-linear maps and the vertical maps are Λ ⊗_{ℤ_{p}} OD̂(a)-linear isomorphisms (since they are simply multiplication by

ã^{(M+1)N}, a unit on D̂(a)). 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}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}], and from now on we assume that we have made such a choice, so that L is described by the short exact sequence

\xymatrix{0 \ar[r] & L \ar[r] & Λ ⊗_{ℤ_{p}} O_{D̂(a)}^{m} \ar[r]^{1−(r_{ij}(z))∘F^{∗}} & Λ ⊗_{ℤ_{p}} O_{D̂(a)}^{m} \ar[r] & 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 r̃_{ij} of ãΛ̃[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}], to obtain a lift Ẽ of E and a lift L̃ of L. Let s be an integer chosen so that each (z_{1} ⋯ z_{d})^{s}r̃_{ij} belongs to Λ̃[z_{1}, …, z_{d}], and choose a second integer t so that (p − 1)t > s. Then the diagram

\xymatrix{Λ̃ ⊗_{ℤ_{p}} Om ; _{𝔾̂_{m}^{d}} \ar[d]^{(z_{1}···z_{d})^{t}} \ar[r]^{(z_{1}···z_{d})^{(p−1)t}(r̃_{ij})∘F^{∗}} & Λ̃ ⊗_{ℤ_{p}} Om ; _{𝔾̂_{m}^{d}} \ar[d]^{(z_{1}···z_{d})^{t}} \\ Λ̃ ⊗_{ℤ_{p}} Om ; _{𝔾̂_{m}^{d}} \ar[r]^{(r̃_{ij})∘F^{∗}} & Λ̃ ⊗_{ℤ_{p}} O_{𝔾̂_{m}^{d}},}

in which the horizontal arrows are F^{∗}-linear morphisms and the vertical arrows are Λ̃⊗_{ℤ_{p}} O_{𝔾̂_{m}^{d}}-linear isomorphisms (since (z_{1} ⋯ z_{d})^{t} is invertible on 𝔾̂_{m}^{d}), commutes. The matrix (z_{1} ⋯ z_{d})^{(p−1)t}(r̃_{ij}) consists of elements of

(4.9)  (z_{1} ⋯ z_{d})Λ̃[z_{1}, …, z_{d}] ∩ ãΛ̃[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}],

by construction. Thus we see that we may choose a basis of Ẽ so that the matrix r̃_{ij} describing the F^{∗}-linear endomorphism Φ̃ of Ẽ consists of elements of the intersection (4.9), and from now on we assume that we have done this, and that the matrix r̃_{ij} is chosen with respect to this basis.

Let g denote the structural morphism g : ℙ^{d} → Spec 𝔽_{p}, let h denote the open immersion of 𝔽_{p}-schemes h : D(a) → ℙ^{d}, underlying the open immersion of p-adic formal schemes D̂(a) → ℙ̂^{d}, so that f = gh, and let O(−1)_{ℙ̂^{d}} denote the ideal sheaf of the formal hyper-plane at infinity of ℙ̂^{d} (that is, the hyperplane described by the equation Z_{0} = 0). Let u be an integer chosen so that (p − 1)u is greater than the degree of each of the polynomials r̃_{ij}. By virtue of this choice of u, the matrix (r̃_{ij}) induces an F^{∗}-linear endomorphism (r̃_{ij}) ∘ F^{∗} of Λ̃ ⊗_{ℤ_{p}} O(−u)_{ℙ̂^{d}}. Furthermore, since the r̃_{ij} are divisible by each z_{i} as well as by ã, and since (p − 1)u is in fact greater than (rather than just equal to) the degree of any of the r̃_{ij}, we see that the étale sheaf of (r̃_{ij}) ∘ F^{∗}-invariants of Λ̃ ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}} is exactly equal to h_{!}L̃. More precisely, write L̃_{n} = L̃/p^{n}L̃, and Λ̃_{n} = Λ̃/p^{n}Λ̃. Then for each n ≥ 1, we have have a short exact sequence of étale sheaves

\xymatrix{0 \ar[r] & h_{!}L̃_{n} \ar[r] & Λ̃_{n} ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}} \ar[r]^{1−(r̃_{ij})∘F^{∗}} & Λ̃_{n} ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}} \ar[r] & 0.}

This is the exact sequence which we will use to compute f_{!}L̃ = g_{!}h_{!}L̃, and hence to compute its L-function, in order to compare it with the L-function of L̃. (The construction of this exact sequence is an application of the technique of [4], lemma 4.5, p. 120. It is greatly generalized in [6], [7].)

(4.10) If we apply g_{∗} to the short exact sequence of étale sheaves constructed above then keeping in mind that for coherent sheaves étale pushforward agrees with the coherent push-forward, and that for i < d,

R^{i}f_{∗}(Λ̃_{n} ⊗_{ℤ_{p}} O(−u)_{ℙ̂^{d}}) = H^{i}(ℙ̂^{d}, Λ̃_{n} ⊗_{ℤ_{p}} O(−u)_{ℙ̂^{d}}) = 0,

we obtain the exact sequence

\xymatrix{0 \ar[r] & R^{d}f_{!}L̃_{n} \ar[r] & H^{d}(ℙ^{d}, Λ̃_{n} ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}})}
[figure: 1−H^{d}((r̃_{ij})∘F^{∗}); H^{d}(ℙ^{d}, Λ̃_{n} ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}}); R^{d+1}f_{!}L̃_{n}; 0]

of étale sheaves on Spec 𝔽_{p}. If we take the stalks of this exact sequence over the geometric point Spec \overline{𝔽}_{p} of Spec 𝔽_{p}, and pass to the inverse limit over n, then we obtain the exact sequence

\xymatrix{0 \ar[r] & (R^{d}f_{!}L̃)_{\overline{𝔽}_{p}} \ar[r] & H^{d}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} \overline{W}}
[figure: 1−H^{d}((r̃_{ij})∘F^{∗})⊗σ; H^{d}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} \overline{W}]
[figure: (R^{d+1}f_{!}L̃)_{\overline{𝔽}_{p}}; 0.]

The morphism 1−H^{d}((r̃_{ij})∘F^{∗})⊗σ is surjective, and so we see that R^{d+1}f_{!}L̃ = 0.

Lemma 4.11. The ratio

L(Spec 𝔽_{p}, f_{!}L̃)/detΛ̃(1 − H^{d}((r̃_{ij}) ∘ F^{∗})T, H^{d}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}}))^{(−1)^{d+1}}

is an element of 1 + 𝔪̃T Λ̃⟨T ⟩, and is a rational function.

Proof. Let us denote the finite rank free Λ̃-module H^{d}(ℙ^{d}, Λ̃⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}}) by M, for simplicity of notation, and let us denote the Λ̃-linear endomorphism H^{d}((r̃_{ij}) ∘ F^{∗}) of M by Ψ. A little linear algebra shows that M has a canonical direct sum decomposition M = M_{unit} ⊕ M_{nil}, determined by the property that M_{unit} is the maximal Ψ-invariant Λ̃-submodule of M on which Ψ acts surjectively (or equivalently, bijectively), while M_{nil} is the maximal Ψ-invariant Λ̃-submodule of M on which Ψ acts topologically nilpotently. Indeed, it is clear that such decomposition exists for M as a ℤ_{p}-module. However, as it is canonical M_{unit} and M_{nil} are Λ̃ stable. Thus they are projective, and hence free Λ̃-modules.

We extend the endomorphism Ψ to the σ-linear endomorphism Ψ ⊗ σ of M ⊗_{ℤ_{p}} \overline{W} . Since Ψ acts topologically nilpotently on M_{nil}, 1 − Ψ ⊗ σ acts bijectively on M_{nil} ⊗_{ℤ_{p}} \overline{W} . Thus we may replace the exact sequence

0 → (R^{d}f_{!}L̃)_{\overline{𝔽}_{p}} → M ⊗_{ℤ_{p}} W̄ ⟶^{1−Ψ⊗σ} M ⊗_{ℤ_{p}} W̄ → 0

of section (4.10) by the exact sequence

0 → (R^{d}f_{!}L̃)_{\overline{𝔽}_{p}} → M_{unit} ⊗_{ℤ_{p}} W̄ ⟶^{1−Ψ⊗σ} M_{unit} ⊗_{ℤ_{p}} W̄ → 0.

Since Ψ acts bijectively on M_{unit}, we see that Ψ makes M_{unit} into a unit (Λ̃, F)- crystal, and that R^{d}f_{!}L̃ is the corresponding lisse p-adic étale sheaf on Spec 𝔽_{p}. Thus

L(Spec 𝔽_{p}, f_{!}L̃) = detΛ̃(1 − ϕT, (R^{d}f_{!}L̃)_{\overline{𝔽}_{p}})^{(−1)^{d+1}}
= detΛ̃(1 − ΨT, M_{unit})^{(−1)^{d+1}}
= detΛ̃(1 − ΨT, M)^{(−1)^{d+1}}/detΛ̃(1 − ΨT, M_{nil})^{(−1)^{d+1}}.

Thus the lemma will be proved once we show that detΛ̃(1 − ΨT, M_{nil})^{(−1)^{d+1}} belongs to 1 + 𝔪̃Λ̃⟨T ⟩. But this follows from the fact that Ψ acts topologically nilpotently on M_{nil}, and so nilpotently on M/𝔪̃, so that the characteristic polynomial of Ψ on M_{nil} satisfies the congruence

detΛ̃(1 − ΨT, M_{nil}) ≡ 1 (mod 𝔪̃).

(4.12) We now turn to describing the L-function of L in terms of the matrix (r̃_{ij}).

Lemma 4.13. The L-function of L̃ on D(a) is determined by the formula

L(D(a), L̃) = ∏_{x∈|D(a)|} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r̃_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}.

(Recall from section 4.3 that x̃ denotes the Teichmüller lift of the closed point x.)

Proof. This follows from the fact that Om ; D̂(a) equipped with the F^{∗}-linear

endomorphism (r̃_{ij}) ∘ F^{∗} is the unit F -crystal Ẽ corresponding to L̃. More precisely, the Artin-Schreier short exact sequence

\xymatrix{0 \ar[r] & L \ar[r] & Om ; D̂(a) \ar[r]^{1−(r̃_{ij})∘F^{∗}} & Om ; D̂(a) \ar[r] & 0}

shows that for any point x of D(a), if x̄ denotes an étale point lying over x, then

det(1 − ϕ^{d(x)}T^{d(x)}, Lx̄) = det(1 − ((r̃_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m}).

Corollary 4.14. The L-function of L̃ on D(a) agrees with the product

∏_{x∈|𝔾_{m}^{d}|} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r̃_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}

up to multiplication by an element of 1 + pT Λ̃⟨T ⟩.

Proof. Lemma 4.13 shows that the ratio of L(D(a), L̃) and the product in the statement of the corollary is equal to

∏_{x∈|Z(a)|} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r̃_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}.

Thus it suffices to show that this product belongs to 1 + pΛ̃⟨T ⟩.

For any positive integer δ, let |Z(a)|_{δ} denote the (finite!) subset of |Z(a)| consisting of those closed points x for which d(x) = δ. By construction, each r̃_{ij} is an element of ãΛ̃[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}]. Thus for each x in |Z(a)|_{δ}, r̃_{ij}(x̃) lies in pΛ̃ ⊗_{ℤ_{p}} W_{x}, so that the entries of

((r̃_{ij}(x̃)) ∘ σ)^{d(x)} = ((r̃_{ij}(x̃)) ∘ σ)^{δ}

lie in p^{δ}Λ̃, and the finite product

∏_{x∈|Z(a)|_{d}} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}

lies in 1 + p^{δ}T Λ[[p^{δ}T]] ⊂ 1 + p^{δ}T^{δ}Λ̃⟨T^{δ}⟩. From this we conclude that

∏_{x∈|Z(a)|} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}

= ∏_{δ>0} ∏_{x∈|Z(a)|_{δ}} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}

lies in 1 + pT Λ̃⟨T ⟩, thus proving the corollary.

(4.15) Theorem 1.5 now follows from lemma 4.11 and corollary 4.14, together with the following result, whose proof is the subject of section 5.

Proposition 4.16. Let P (T) denote

detΛ̃(1 − H^{d}((r̃_{ij}) ∘ F^{∗})T, H^{d}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}})) ∈ 1 + T Λ̃[T].

Then

∏_{x∈|𝔾_{m}^{d}|} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r̃_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}

= ∏_{i=0}^{d} P (p^{i}T)^{(−1)^{d+1−i}\binom{d}{i}},

and so in particular, the ratio

( ∏_{x∈|𝔾_{m}^{d}|} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r̃_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}) /P (T)^{(−1)^{d+1}}

lies in 1 + pT Λ̃⟨T ⟩, and is a rational function.

5. Proof of proposition 4.16

(5.1) In order to prove proposition 4.16, we begin by describing the morphism which is Grothendieck-Serre dual to the map 1 − H^{d}((r̃_{ij}) ∘ F^{∗}).

The free Λ̃-module H^{d}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} O(−u)m ; _{ℙ̂^{d}}) ⟶^{∼} H^{d}(ℙ^{d}, O(−u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} Λ̃ has

Grothendieck-Serre dual equal to H^{0}(ℙ^{d}, Ω^{d}(u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} Λ̃. Let C denote the Cartier operator on Ωd ; _{ℙ̂^{d}} corresponding to the lift of Frobenius F, defined by the formula

C(z_{1}^{i_{1}} ⋯ z_{d}^{i_{d}} \frac{dz_{1} ∧ ⋯ ∧ dz_{d}}{z_{1} ⋯ z_{d}})

= \{ \begin{matrix}z1^{\frac{i_{1}}{p}} ⋯ zd^{\frac{i_{d}}{p}} \frac{dz_{1} ∧ ⋯ ∧ dz_{d}}{z_{1} ⋯ z_{d}} & if i_{j} ≡ 0 (mod p) for all j \\ 0 & if i_{j} ≢ 0 (mod p) for some j. \end{matrix}

By virtue of our choice of u, the composite C ∘ (r̃_{ij}) defines an endomorphism of Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u) :

\xymatrix{Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u) \ar[r]^{C∘(r̃_{ij})} & Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u),}

and the dual of the endomorphism H^{d}((r̃_{ij})) ∘ F^{∗} of H^{d}(ℙ^{d}, O(−u)m ; _{ℙ̂^{d}}) is the endomorphism H^{0}(C ∘ (r̃_{ij})) of H^{0}(ℙ^{d}, Ω^{d}(u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} Λ̃. Since a matrix and its adjoint have the same characteristic polynomial we see that

P (T) = detΛ̃(1 − H^{0}(C ∘ (r̃_{ij}))T, H^{0}(ℙ^{d}, Ω^{d}(u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} Λ̃).

(5.2) The usual formula for the logarithm of a determinant shows that

(5.3) log P (T) = − ∑_{n=1}^{∞} traceΛ̃(H^{0}(C ∘ (r̃_{ij}))^{n}, H^{0}(ℙ^{d}, Ω^{d}(u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} Λ̃)\frac{T^{n}}{n} .

For any sections a of Λ̃ ⊗_{ℤ_{p}} O_{ℙ̂^{d}} and ω of Ωd ; _{ℙ̂^{d}} we have the identity

aCω = CF^{∗}(a)ω,

and so for any positive integer n we have

(C ∘ (r̃_{ij}))^{n} = C^{n} ∘ ∏_{i=0}^{n−1}((F^{∗})^{i}r̃_{ij}).

This allows us to rewrite (5.2) in the form (5.4)

log P (T) = − ∑_{n=1}^{∞} traceΛ̃(H^{0}(C^{n} ∘ ∏_{k=0}^{n−1}(F^{∗})^{k}(r̃_{ij})), H^{0}(ℙ^{d}, Ω^{d}(u)m ; _{ℙ̂^{d}}) ⊗_{ℤ_{p}} Λ̃)\frac{T^{n}}{n} .

(5.5) Fix a positive integer n, and suppose that (f_{ij}) is any m×m matrix of elements of (z_{1} ⋯ z_{d})Λ̃[z_{1}, …, z_{d}], such that each of the f_{ij} is of degree less than (p^{n} − 1)u. (For example, ∏_{k=0}^{n−1}((F^{∗})^{k}r̃_{ij}) is such a matrix.) We have a morphism

\xymatrix{Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)m ; _{ℙ̂^{d}} \ar[r]^{C^{n}∘(f_{ij})} & Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)m ; _{ℙ̂^{d}},}

which induces an endomorphism H^{0}(C^{n} ∘ (f_{ij})) of H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)m ; _{ℙ̂^{d}}). We may also form the sum of matrices

∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})}(f_{ij}(x̃)),

which will be an m × m matrix of elements of ℤ_{p}. In this situation we have the following lemma, which is a slight modification of lemma 2 of [5]:

Lemma 5.6. For any matrix (f_{ij}) as above we have the following formula:

traceΛ̃(H^{0}(C^{n} ∘ (f_{ij})), H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)m ; _{ℙ̂^{d}}))

= \frac{1}{(p^{n} − 1)^{d}} traceΛ̃( ∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})}(f_{ij}(x̃)), Λ̃^{m}).

(Recall from section 4.3 that x̃ denotes the Teichmüller lift of the 𝔽_{p^{n}}-valued point x of 𝔾_{m}^{d}.)

Proof. We first reduce the proof of the formula to the case m = 1, as follows. It is clear that the trace on the left hand side is given by the formula

traceΛ̃(H^{0}(C^{n} ∘ (f_{ij})), H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)m ; _{ℙ̂^{d}}))

= ∑_{i=1}^{m} traceΛ̃(H^{0}(C^{n} ∘ f_{ii}), H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)_{ℙ̂^{d}})),

while the trace on the right hand side is given by the formula

\frac{1}{(p^{n} − 1)^{d}} ∑_{i=1}^{m} ∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})} f_{ii}(x̃).

Thus if we knew that for any f in z_{1} ⋯ z_{d}Λ̃[z_{1}, …, z_{d}] of degree less than (p^{n} − 1)u we had the equation

(5.7) traceΛ̃(H^{0}(C^{n} ∘ f), H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)_{ℙ̂^{d}})) = \frac{1}{(p^{n} − 1)^{d}} ∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})} f(x̃),

the lemma would follow. This is what we now prove.

The set of all such f form a Λ̃-submodule of Λ̃[z_{1}, …, z_{d}], and both sides of equation (5.7) are Λ̃-linear in f. Thus it suffices to prove the formula in the case that f is a single monomial

f = z_{1}^{α_{1}} ⋯ z_{d}^{α_{d}},

with

(5.8)  0 < α_{1}, …, α_{d}, α_{1} + ⋯ + α_{d} < (p^{n} − 1)u.

The Λ̃-module H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)_{ℙ̂^{d}}) has as a basis the differentials

z_{1}^{β_{1}} ⋯ z_{d}^{β_{d}} \frac{dz_{1} ∧ ⋯ ∧ dz_{d}}{z_{1} ⋯ z_{d}} ,

with

(5.9)  0 < β_{1}, …, β_{d}, β_{1} + ⋯ + β_{d} < u.

We see that

C^{n}(z_{1}^{α_{1}} ⋯ z_{d}^{α_{d}} z_{1}^{β_{1}} ⋯ z_{d}^{β_{d}} \frac{dz_{1} ∧ ⋯ ∧ dz_{d}}{z_{1} ⋯ z_{d}})

= C^{n}(z_{1}^{α_{1}+β_{1}} ⋯ z_{d}^{α_{d}+β_{d}} \frac{dz_{1} ∧ ⋯ ∧ dz_{d}}{z_{1} ⋯ z_{d}})

= \{ \begin{matrix}z1^{\frac{α_{1}+β_{1}}{p^{n}}} ⋯ zd^{\frac{α_{d}+β_{d}}{p^{n}}} \frac{dz_{1} ∧ ⋯ ∧ dz_{d}}{z_{1} ⋯ z_{d}} & all α_{i} + β_{i} ≡ 0 (mod p^{n}) \\ 0 & some α_{i} + β_{i} ≢ 0 (mod p^{n}). \end{matrix}

From this we see that the matrix of H^{0}(C^{n} ∘ f) with respect to the given basis of H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)_{ℙ̂^{d}}) has entries which are either 0 or 1, and so the trace of H^{0}(C^{n} ∘ f) is equal to the number of 1’s on the diagonal, which is given by the number of d-tuples β_{1}, ⋯, β_{d} which satisfy

(5.10)  α_{i} + β_{i} = p^{n}β_{i}, for 1 ≤ i ≤ d,

together with (5.9). We see immediately that the system of inequalities and equations (5.9), (5.10) has no solutions unless

α_{1} ≡ ⋯ ≡ α_{d} ≡ 0 (mod p^{n} − 1),

in which case there is a unique solution (here we are using the fact that α_{1}, …, α_{d} satisfies (5.8)), so that (5.11)

traceΛ̃(C^{n} ∘ f, H^{0}(ℙ^{d}, Λ̃ ⊗_{ℤ_{p}} Ω^{d}(u)_{ℙ̂^{d}})) = \{ \begin{matrix}1 & all α_{i} ≡ 0 (mod p^{n} − 1) \\ 0 & some α_{i} ≢ 0 (mod p^{n} − 1). \end{matrix}

On the other hand, as x ranges over all the 𝔽_{p^{n}}-valued points of 𝔾_{m}^{d}, the set of Teichmüller lifts x̃ ranges over all d-tuples of (p^{n} − 1)^{st} roots of unity in W (𝔽_{p^{n}}). Thus

(5.12)

\frac{1}{(p^{n} − 1)^{d}} ∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})} f(x̃) = \frac{1}{(p^{n} − 1)^{d}} ∑_{ζ_{1},··· ,ζ_{d}∈μ_{p^{n}−1}} ζ_{1}^{α_{1}} ⋯ ζ_{d}^{α_{d}}

= \{ \begin{matrix}1 & all α_{i} ≡ 0 (mod p^{n} − 1) \\ 0 & some α_{i} ≢ 0 (mod p^{n} − 1). \end{matrix}

Comparing (5.11) and (5.12), we see that (5.7) is proved, and with it the lemma.

(5.13) Applying lemma 5.6 to the matrices ∏_{k=0}^{n−1}((F^{∗})^{k}r̃_{ij}) which appear in the expression on the right side of equation (5.4) we obtain the equation

log P (T) = − ∑_{n=1}^{∞} traceΛ̃( ∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})} ∏_{k=0}^{n−1}(r̃_{ij}(σ^{k}(x̃))), Λ̃^{m})\frac{T^{n}}{n(p^{n} − 1)^{d}}

= − ∑_{n=1}^{∞} traceΛ̃( ∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})}((r̃_{ij}(x̃)) ∘ σ)^{n}, Λ̃^{m})\frac{T^{n}}{n(p^{n} − 1)^{d}} .

This in turn implies that

log ∏_{i=0}^{d} P (p^{i}T)^{(−1)^{d+1−i}\binom{d}{i}} = ∑_{n=1}^{∞} traceΛ̃( ∑_{x∈𝔾_{m}^{d}(𝔽_{p^{n}})}((r̃_{ij}(x̃)) ∘ σ)^{n}, Λ̃^{m})\frac{T^{n}}{n} .

Exponentiating both sides yields proposition 4.16.

6. Lifting representations of arithmetic fundamental groups

(6.1) In this section we discuss some applications of our method of lifting locally constant étale sheaves of finite free Λ modules by lifting the associated (Λ, F)-crystals. For the duration of this section Λ will denote an artinian local ℤ_{p}-algebra with finite residue field and X will denote a smooth affine 𝔽_{p}-scheme. We have seen above that there exists a surjection Λ̃ → Λ with Λ̃ a finite flat local ℤ_{p}-algebra. With these notations, we have the following theorem:

Theorem 6.2. Let

ρ : π_{1}(X) → GL_{d}(Λ)

be a continuous representation of the arithmetic étale fundamental group of X. There exists a continuous lifting of ρ

ρ̃: π_{1}(X) → GL_{d}(Λ̃).

Proof. The representation ρ corresponds to a locally constant étale sheaf of finite free Λ-modules on X. We denote this sheaf by L. Now we claim that X can be lifted to a formally smooth p-adic formal scheme X̂ over ℤ_{p} equipped with a lift F of the absolute Frobenius. Indeed by [10], III the obstructions to the existence of such a lifting are contained in the cohomology of certain coherent sheaves on X, and so vanish, since X is affine.

Fix X̂ and F, and denote by E the unit (Λ, F)-crystal on X̂ corresponding to L, so that E is a finite free Λ ⊗_{ℤ_{p}} OX̂ sheaf equipped with an isomorphism F^{∗}E ⟶^{∼} E. We have already seen in the proof of proposition 3.2, and also in (4.7), that locally on X̂, E can be lifted to a finite free Λ̃ ⊗_{ℤ_{p}} OX̂-module Ẽ equipped with an isomorphism F^{∗}Ẽ ⟶^{∼} Ẽ lifting F^{∗}E ⟶^{∼} E. The obstruction to the existence of such a global lift is contained in the cohomology of certain coherent OX̂-modules (because Λ ⊗_{ℤ_{p}} OX̂ is OX̂ coherent), and so vanishes, since OX̂ is formal affine.

This proves the proposition, because by proposition 3.2, Ẽ corresponds to a lisse sheaf of Λ̃-modules on X̂, which gives the required representation ρ̃.

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 ρ̃ may be chosen so that the corresponding lisse sheaf of Λ̃-modules L has rational L-function.

Proof. As in section 4, we let F denote the lift of Frobenius on ℙ̂^{1} given (in affine coordinates) by z ↦ z^{p}, and let X̂ ⊂ ℙ̂^{1} denote the open formal subscheme corresponding to X ⊂ ℙ^{1}. We may extend the unit (Λ, F)-crystal E attached to ρ to a sheaf E^{+} of locally free Λ ⊗_{ℤ_{p}} O_{ℙ̂^{1}}-modules on ℙ^{1}. Indeed to extend across a point of ℙ^{1} −X, we may work in a neighbourhood of this point, and hence assume that E is free over Λ ⊗_{ℤ_{p}} OX̂ over this neighbourhood, when the existence of the extension is clear. Moreover, if D denotes the divisor on ℙ^{1} corresponding to ℙ^{1} −X, and D̂ denotes the divisor on ℙ̂^{1} corresponding to the Teichmüller lift of D, then replacing E^{+} by E^{+}(−nD̂), for some large integer n, we may assume that isomorphism F^{∗}E ⟶^{∼} E extends to a map F^{∗}E^{+} → E^{+} (which cannot be an isomorphism unless ρ is trivial). Now set

M = \underline{Hom}_{Λ⊗_{ℤp}O_{ℙ̂1}} (F^{∗}E^{+}, E^{+}).

Replacing E^{+} by E^{+}(−nD̂) for some large integer n replaces M by M((p − 1)nD̂), and so we may assume that we have H^{1}(ℙ̂^{1}, M) = 0. (This will be used later).

Now we claim that E^{+} can be lifted to a locally free Λ̃ ⊗_{ℤ_{p}} O_{ℙ̂^{1}}-module Ẽ^{+}. Indeed, such a lifting exists locally, and the obstruction to a global lifting is contained in H^{2} of certain coherent sheaves on ℙ^{1}, hence vanishes (cf. [10] III, 7.1).

Let 𝔫 be the kernel of Λ̃ → Λ. We claim that F^{∗}E^{+} ⟶^{∼} E^{+} lifts to a morphism F^{∗}Ẽ^{+}/𝔫^{2} → Ẽ^{+}/𝔫^{2} which is an isomorphism when restricted 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 Λ̃ ⊗_{ℤ_{p}} O_{ℙ̂^{1}}-module when restricted 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} and such that Ẽ^{+}/𝔫^{2} becomes a free Λ̃ ⊗_{ℤ_{p}} O_{ℙ̂^{1}}-module when restricted 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 = \underline{Hom}_{Λ⊗_{ℤp}O_{ℙ̂1}} (F^{∗}E^{+}, 𝔫Ẽ^{+}/𝔫^{2}Ẽ^{+}). We claim that H^{1}(ℙ^{1}, N) = 0. Indeed,

choosing a set of generators n_{1}, …, n_{s} ∈ Λ̃ for the ideal 𝔫, we can define a surjection of Λ ⊗_{ℤ_{p}} O_{ℙ̂^{1}}-modules

\xymatrix{(E^{+})^{s} \ar[r]^{(e_{1},...,e_{s})↦e_{1}n_{1}+···+e_{s}n_{s}} & 𝔫E^{+}/𝔫^{2}E^{+}.}

Since F^{∗}E^{+} is a locally free Λ⊗_{ℤ_{p}} ℙ̂^{1}-module, the above surjection yields a surjection M^{s} → N. Taking cohomology and recalling that H^{2} of a coherent sheaf of O_{ℙ̂^{1}}-modules vanishes, we see that H^{1}(ℙ^{1}, N) is a quotient of H^{1}(ℙ^{1}, M)^{s}, which vanishes by assumption. The vanishing of H^{1}(ℙ^{1}, N) implies that we can modify the ϕ_{i} so that they agree on overlaps. This gives the required lift, since over each of U_{i} the ϕ_{i} are automatically isomorphisms by Nakayama’s lemma, as this is true modulo 𝔫.

Let M_{2} = \underline{Hom}_{Λ̃⊗_{ℤp}O_{ℙ̂1}} (F^{∗}Ẽ^{+}/𝔫^{2}, Ẽ^{+}/𝔫^{2}). We also have H^{1}(ℙ̂^{1}, M_{2}) = 0,

as we by taking the long exact cohomology sequence attached to the short exact sequence 0 → N → M_{2} → M → 0, and recalling that H^{1}(ℙ^{1}, M) = H^{1}(ℙ^{1}, N) = 0.

Thus we can replace Λ by Λ̃/𝔫^{2} and E^{+} by Ẽ^{+}/𝔫^{2}. Repeating the above arguments and taking an inverse limit we obtain a lifting F^{∗}Ẽ^{+} → Ẽ^{+} of F^{∗}E^{+} → E^{+} which is an isomorphism when restricted to X. Now this restriction Ẽ = Ẽ^{+}|_{X} gives rise to a lisse étale sheaf L̃ of finite free Λ̃-modules on X which induces the required representation ρ̃. To see that the L-function of ρ̃ has the required property, first note we may replace X̂ by a non-empty open subset, and L by its restriction to this open subset, as this changes the L-function only by a rational function. Thus we may assume that X ⊂ 𝔾_{m}, and that E is free over Λ⊗_{ℤ_{p}} OX̂ (then Ẽ is free over Λ̃⊗_{ℤ_{p}} OX̂, as one can lift a basis for E.) Now we are in exactly the situation of (4.8) (note that Grothendieck’s algebraization theorem guarantees that the matrix entries describing the morphism F^{∗}Ẽ → Ẽ are rational functions, since by construction this extends to a morphism of coherent sheaves F^{∗}Ẽ^{+} → Ẽ^{+}) and the argument of section 4 shows that L(X, L̃) is a rational function over Λ̃. Indeed in the calculation of this L-function in section 4 the only possible non-rational contribution comes from corollary 4.14, where (in the notation of the proof of that corollary) one encounters the product ∏_{x∈Z(a)} det_{Λ̃⊗_{ℤp}W_{x}}(1 − ((r̃_{ij}(x̃)) ∘ σ)^{d(x)}T^{d(x)}, Λ̃ ⊗_{ℤ_{p}} W_{x}^{m})^{−1}. In the general context of that corollary, this product could be infinite. However, in the current situation Z(a) is finite (being a proper closed subset of a curve), and so this is a finite product, and hence is indeed a rational function!

University of Michigan, Ann Arbor, MI (M.E.)

University of Sydney, Sydney, NSW

SFB 478, Westfälische Wilhelms Universität, Münster (M.K.)

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