Derived category invariants and L-series

0.24 MB·English

by  Philippe Cassou-Noguès

DERIVED CATEGORY INVARIANTS AND L-SERIES
PH. CASSOU-NOGUE`S, T. CHINBURG*, B. EREZ, AND M. J. TAYLOR
3
1
0 Abstract. We relate invariants in derived categories associated to tame actions of
2
finite groups on projective varieties over a finite field to zeros of L-functions

n
a
J

1
1. Introduction
3

] A recurring theme in the study of values of L-functions of arithmetic schemes is that
T
these should be related to Euler characteristics of various kinds. Behind the cohomol-
N
ogy groups needed to define such Euler characteristics are hypercohomology complexes
.
h
t in derived categories. In this paper we consider how to determine the additional infor-
a
m mation contained in such complexes beyond what is seen by Euler characteristics. In
[ the geometric situations we consider, this additional information takes the form of ex-

1 tension classes. Our main result is that two natural extension classes constructed from
v
´etale and coherent cohomology differ by a numerical invariant which is the reciprocal of
3
6 the zero of an L-function. This suggests that it may be fruitful to study relationships
6
between derived category invariants and L-functions in more general contexts, e.g. for
7
. projective schemes over Z.
1
0 We will now describe the contents of this paper. In Section 2 we consider the fol-
3
1 lowing geometric situation. Let X be a smooth projective variety over the algebraic
:
v closure k of a finite field k having a tame, generically free action over k of a finite
i
X group G. We suppose that the cohomology groups of OX vanish except in dimen-
r sions 0 and n for some integer n > 0 and that the characteristic p of k divides #G.
a
In this case, the isomorphism class of the hypercohomology complex H•(X,O ) in
X
the derived category of the homotopy category of k[G]-modules is determined by its
Euler characteristic and an extension class β(X,G) in the one-dimensional k-vector
space Extn+1(Hn(X,O ),H0(X,O )). Similarly, the isomorphism class of H•(X,k)
k[G] X X et
in the derived category of the homotopy category of complexes of k[G]-modules is de-
termined by its Euler characteristic together with an extension class γ(X,G) in the
one-dimensional k-vector space Extn+1(Hn(X,k),H0(X,k)).
k[G] et et
In Theorem 2.9 we show that
β(X,G) = 1⊗γ(X,G)
Date: February 1, 2013.
*Supported by NSF Grant #DMS0801030.
1
2 PH. CASSOU-NOGUE`S,T. CHINBURG,B. EREZ, AND M. TAYLOR
relative to a natural isomorphism
k ⊗ Extn+1(Hn(X,k),H0(X,k)) = Extn+1(Hn(X,O ),H0(X,O ))
k k[G] et et k[G] X X
induced by the map k → G of ´etale sheaves on X. One can thus think of
a
k ·β(X,G) = k ·γ(X,G)
as the “´etale k-line” inside Extn+1(Hn(X,O ),H0(X,O )).
X X
k[G]
In Section 3 we make some additional hypotheses on G and X described in Hypoth-
esis 3.1. We assume in particular that the p-Sylow subgroups of G are cyclic and non
trivial. We let C be a p-Sylow subgroup of G. Under these hypotheses we define a
k-line
k ·α(X,G)
inside Extn+1(Hn(X,O ),H0(X,O )) which is associated to a model Y over k of the
X X 0
k[C]
quotient Y = X/G of X by the action of G. One can think of k·α(X,G) as a k-line de-
terminedbycoordinatesfortheone-dimensionalk-vectorspaceExtn+1(Hn(X,O ),H0(X,O ))
X X
k[C]
which arises from the model Y .
0
The restriction map induces an isomorphism of k-vector spaces
Extn+1(Hn(X,O ),H0(X,O )) → Extn+1(Hn(X,O ),H0(X,O )).
X X X X
k[G] k[C]
We consider k.β(X,G) as a k-line of Extn+1(Hn(X,O ),H0(X,O )) via this isomor-
X X
k[C]
phism. Our goal is to compare in Extn+1(Hn(X,O ),H0(X,O )) the´etale k-line with
X X
k[C]
the k-line provided by the model Y . To be more precise our main result, Theorem 3.6
0
and its corollaries, is that
k ·α(X,G) = ζ ·k ·β(X,G)
∗
for a constant ζ ∈ k such that
µ(X,G) = ζ1−#k ∈ k∗
is independent of all choices and is the reciprocal of a zero of an L-function associated
to X. If n is odd, this L-function is the numerator of the mod p zeta function of Y
0
over k. If n is even, the L-function is the denominator of the mod p zeta function of
the variety X which is the quotient of X by the group generated by a lift to X of the
0
arithmetic Frobenius in Gal(Y/Y ). If X is an elliptic curve and k = Z/p then µ(X,G)
0
is simply the Hasse invariant associated to Y (c.f. Example 3.11).
0
In the last section of this paper we provide examples of projective varieties of arbi-
trary large dimension, endowed with an action of a cyclic group of order p for which
the hypotheses of Theorem 3.6 are fulfilled.
Acknowledgments: The first author would like to thank the University of Pennsyl-
vania for its hospitality during work on this paper.
DERIVED INVARIANTS 3
2. Varieties with two non-vanishing cohomology groups
Let k be a finite field of order q = pf, where p is a prime, and let k be an algebraic
closure of k. We will suppose that G is a finite group of order divisible by p acting
tamely and generically freely over k on a smooth projective variety X over k of dimen-
sion d. Let π : X → Y = X/G be the quotient morphism. If F is a coherent G-sheaf
on X, we denote Hi(X,F) by Hi(F).
Hypothesis 2.1. There is an integer n ≥ 1 such that Hi(O ) 6= {0} if and only if
X
i ∈ {0,n}.
Lemma 2.2. The coherent hypercohomology complex H•(O ) is isomorphic in the
X
derived category D(kG) of the homotopy category of complexes of k[G]-modules to
a perfect complex P• of k[G]-modules which has trivial terms outside degrees in the
interval [0,n]. This complex defines an exact sequence
(2.1) 0 → H0(O ) → P → ··· → P → Hn(O ) → 0
X 0 n X
and thereby an extension class β(X,G) in Extn+1(Hn(O ),H0(O )).
X X
k[G]
Proof. By a result of Nakajima [7], H•(O ) is isomorphic to a perfect complex in
X
D(kG) because the action of G on X is tame. Because of hypothesis 2.1, we can
truncate this complex to arrive at P•.
Definition 2.3. Let F : k → k be the arithmetic Frobenius automorphism over k, so
that F(α) = αq for α ∈ k. A k-linear map T : M → M between vector spaces over
1 2
k will be called semilinear (resp. anti-semilinear) if T(α · m ) = F(α)T(m ) (resp.
1 1
T(α·m ) = F−1(α)T(m )) for α ∈ k and m ∈ M .
1 1 1 1
Lemma 2.4. Suppose that ℓ is a field of characteristic p, and that there is an exact
sequence of ℓG-modules
(2.2) 0 → ℓ → P → ··· → P → M → 0
0 n
in which P is projective and finitely generated for all i. Then Extn+1(M,ℓ) is a one-
i ℓ[G]
dimensional ℓ vector space with respect to the multiplication action of ℓ on ℓ. The
degeneration of the spectral sequence Hp(G,Extq(M,ℓ)) → Extp+q(M,ℓ) gives an iso-
ℓ ℓ[G]
morphism
(2.3) Extn+1(M,ℓ) = Hn+1(G,Hom (M,ℓ))
ℓ[G] ℓ
Proof. By dimension shifting via the sequence (2.2), we get an exact sequence
Hom (P ,ℓ) → Hom (ℓ,ℓ) → Extn+1(M,ℓ) → 0.
ℓ[G] 0 ℓ[G] ℓ[G]
4 PH. CASSOU-NOGUE`S,T. CHINBURG,B. EREZ, AND M. TAYLOR
Here Hom (ℓ,ℓ) = ℓ. So either Extn+1(M,ℓ) is a one-dimensional ℓ-vector space, or
ℓ[G] ℓ[G]
the injection ℓ → P splits. However, we have assumed p divides the order of G, so ℓ
0
is not a projective ℓ[G]-module and the latter alternative is impossible.
Corollary 2.5. Suppose ℓ = k and that T : M → M is a semilinear map commuting
with the action of G on M. There is a G-equivariant anti-semilinear endomorphism
T−1 of Hom (M,k) defined by
k
T−1(f)(m) = F−1(f(T(m))) for f ∈ Hom (M,k) and m ∈ M.
k
Via (2.3) this gives an anti-semilinear action of T−1 on Extn+1(M,k) ∼= k.
k[G]
The following result is Lemma III.4.13 of [5]; see also [1, §XXII.1] and [6, p. 143].
Lemma 2.6. Let V be a finite dimensional vector space over k, and let φ : V → V be a
semilinear map. Then V decomposes as a direct sum V = V ⊕V , where V and V are
s η s η
subspaces stable under φ, φ is bijective on V and φ is nilpotent on V . Moreover, V
s η s
has a basis {e ,...,e } such that φ(e ) = e for all i. It follows that Vφ is the k-vector
1 t i i
space having basis {e ,...,e } and φ−1 : V → V is surjective.
1 t
We have an exact Artin-Schreier sequence of ´etale sheaves on X given by
F−1
(2.4) 0−→k−→G −→G → 0
a a
where F : G → G is the arithmetic Frobenius morphism defined by α 7→ αq =
a a
F(α) for α a local section of G . By [5, Remark III.3.8], there is an isomorphism
a
H•(X,O ) → H•(X,G ) in the derived category, giving isomorphisms Hj(O ) →
X et a X
Hj (X,G ) for all j.
et a
Lemma 2.7. The long exact cohomology sequence associated to (2.4) splits into short
exact sequences
(2.5) 0−→Hi (X,k)−→Hi(O )−F→−1Hi(O )−→0
et X X
for all i. The terms of this sequence are trivial if i 6∈ {0,n}. When i = 0, one has
H0(X,k) = k and H0(O ) = k. When i = n, there is a kG-module isomorphism
et X
(2.6) Hn(O ) = Hn(O ) ⊕Hn(O )
X X F,s X F,η
arising from Lemma 2.6 in which F is an isomorphism on Hi(O ) and nilpotent
X F,s
on Hi(O ) . The sequence (2.5) with i = n shows Hn(O )F = Hn(X,k) and
X F,η X et
Hn(O ) = k ⊗ Hn(X,k).
X F,s k et
Proof. The action of F on Hi (X,G ) = Hi(O ) is semilinear for all i. The split exact
et a X
sequences (2.5) arise from the fact that by Lemma 2.6,
F −1 : Hi(O ) → Hi(O )
X X
DERIVED INVARIANTS 5
is surjective for all i. When i = n, the decomposition in (2.6) is a kG-module decom-
position because F commutes with the action of G,
Hn(O ) = ∩ Fm(Hn(O ))
X F,s m≥1 X
and
Hn(O ) = Kernel(Fm : Hn(O ) → Hn(O )) if m >> 0.
X F,η X X
The sequence (2.5) shows Hn(O )F = Hn(X,k) so Hn(O ) = k ⊗ Hn(X,k) by
X et X F,s k et
Lemma 2.6
Lemma 2.8. The complex H•(X,k) is perfect, and Hj (X,k) 6= 0 if and only if j ∈
et et
{0,n}. The sequence (2.4) gives rise to a morphism
(2.7) H•(X,k) → H•(X,G ) = H•(O )
et et a X
in the derived category of complexes of k[G]-modules. Let H•(O )′ be the complex
X
resulting from H•(O ) by truncating H•(O ) in dimensions greater than n and by
X X
replacing Hn(O ) by the submodule Hn(O ) appearing in (2.6). The morphism
X X F,s
(2.7) gives a quasi-isomorphism
(2.8) k ⊗ H•(X,k) = H•(O )′
L,k et X
of perfect complexes of k[G]-modules, where L on the left is the left derived tensor
product. The k[G]-module Hn(O ) in (2.6) is projective.
X F,η
Proof. Since X → Y is a tame G-cover, the sheaf π k in the ´etale topology on Y is a
∗
sheaf of projective k[G]-modules. The argument of [5, Theorem VI.13.11] now shows
that H•(X,k) is a perfect complex of k[G]-modules. The isomorphism (2.8) in the
et
derived category results form the calculation of the cohomology groups Hi (X,k) in
et
Lemma 2.7, where the left derived tensor product k⊗ is just the tensor product
L,k
because all k-modules are free. This implies H•(O )′ is perfect because H•(X,k)
X et
is. Because H•(O ) is also perfect, we conclude that Hn(O ) must be projective
X X F,η
because this was the module truncated from H•(O ) in degree n in the construction
X
of H•(O )′.
X
It follows from Lemma 2.8 that H•(X,k) is a perfect complex such that Hi (X,k) 6=
et et
{0}ifandonlyifi ∈ {0,n}. FollowingthelinesofLemmas2.2and2.4weconcludethat
we can attach to H•(X,k) an extension class γ(X,G) in Extn+1(Hn(X,k),H0(X,k))
et k[G] et et
and prove that this k-vector space is of dimension 1.
Theorem 2.9. The morphism (2.7) leads to an isomorphism of one-dimensional k
vector spaces
(2.9) k ⊗ Extn+1(Hn(X,k),H0(X,k)) = Extn+1(Hn(O ),H0(O ))
k k[G] et et k[G] X X
6 PH. CASSOU-NOGUE`S,T. CHINBURG,B. EREZ, AND M. TAYLOR
such that
(2.10) β(X,G) = 1⊗γ(X,G).
The action of F on Hn(O ) and on H0(O ) = k leads to an anti-semilinear action
X X
of F−1 on Extn+1(Hn(O ),H0(O )). Via (2.9), the one-dimensional k-vector space
X X
k[G]
L = 1 ⊗ Extn+1(Hn(X,k),H0(X,k)) is the subspace of Extn+1(Hn(O ),H0(O ))
1 k k[G] et et kG X X
which is fixed by F−1. In particular, β(X,G) is fixed by F−1.
Proof. SinceHn(O ) isaprojectivek[G]-modulebyLemma2.8,inclusionHn(O ) →
X F,η X F,s
Hn(O ) induces an isomorphism of one-dimensional k-vector spaces
X
(2.11) Extn+1(Hn(O ),H0(O )) → Extn+1(Hn(O ) ,H0(O ))
X X X F,s X
k[G] k[G]
which sends the extension class β(X,G) associated to H•(O ) to the extension class
X
β(X,G)′ associated to H•(O )′. In view of Lemma 2.8, the isomorphism
X
k ⊗ H•(X,k) ∼= H•(O )′
L,k et X
in the derived category gives an isomorphism
(2.12) k ⊗ Extn+1(Hn(X,k),H0(X,k)) = Extn+1(Hn(O ) ,H0(O ))
k k[G] et et k[G] X F,s X
ofone-dimensional vector spacesover k whichidentities 1⊗γ(X,G)withβ(X,G)′ when
γ(X,G) is the extension class in Extn+1(Hn(X,k),H0(X,k)) associated to H•(X,k).
k[G] et et et
Combining(2.11)and(2.12)thusleadstoanisomorphism(2.9)whichidentifiesβ(X,G)
with 1⊗γ(X,G).
The action of F on Hn(O ) is via the action of F on O and commutes with the
X X
action of G. (This F is different from the k-linear relative Frobenius automorphism
F of Hn(O ) = Hn(X,G ) described by Milne in [5, §VI.13].) Since F acts semi-
X/k X et a
linearly and fixes both Hn(X,k) ⊂ Hn(O ) and H0(X,k) = k ⊂ k = H0(O ), the
et X et X
remaining assertions in Theorem 2.9 follow from (2.9).
3. Extension class invariants arising from models
In this section we will assume the following strengthening of Hypothesis 2.1. .
Hypothesis 3.1. The p-Sylow subgroups of the group G are cyclic and non trivial and
the k-vector spaces Hn(G,k¯) and Hn+1(G,k¯) are of dimension one. The variety X is
of dimension n and Hi(O ) = {0} if and only if i 6∈ {0,n}. There exists a smooth
X
projective variety Y over k for which the following is true.
0
a. Y = X/G = k ⊗ Y .
k 0
b. The morphism π˜ : X → Y which is the composition of π : X → Y with the
0
projection Y → Y is Galois.
0
DERIVED INVARIANTS 7
We fix once for all a p-Sylow subgroup C of G. Since k is of characteristic p, for any
integer m, the restriction map induces an injection
(3.1) ResG : Hm(G,k¯) 7→ Hm(C,k¯).
C
Since we have assumed C to be cyclic and non-trivial, the k¯-vector spaces Hm(C,k¯)
are of dimension 1 for all m. Therefore, Hypothesis 3.1 requires that (3.1) is an
isomorphism for m ∈ {n,n+1}.
Example 3.2. Suppose that G is the semi-direct product of a normal subgroup H of
order prime to p with a non-trivial cyclic p-group C. Then the groups Hi(H,k¯) are
trivial for i ≥ 0. Therefore the inflation homomorphisms
Inf : Hi(G/H,k¯) → Hi(G,k¯)
are isomorphisms and
Hi(C,k¯) ≃ Hi(G,k¯), for i ≥ 0.
We conclude that, in this case, the dimensions of Hn(G,k¯) and Hn+1(G,k¯) are both
equal to one as required in Hypothesis 3.1.
The aim of this section is to show that the model Y for Y over k leads to a class
0
α(X,G) ∈ Extn+1(Hn(O ),H0(O ))
X X
k[C]
which is well defined up to multiplication by an element of k∗, and which is different in
general from the class obtained by restriction from the class β(X,G) constructed in the
previous section. This new class should be understood as an obstruction to a descent
problem, and to be more precise, the descent of the action X ×G −→ X defined over
k to an action X ×G → X defined over k. The key to constructing α(X,G) is given
0 0
by the following three results.
Proposition 3.3. Let Γ = Gal(X/Y ). The morphism of sheaves on Y given by
0 0
O → (π˜) O leads to a homomorphism
Y0 ∗ X
Hn(Y ,O ) → Hn(Y ,(π˜) O ) = Hn(O )
0 Y0 0 ∗ X X
and an exact sequence
(3.2) 0−→W−→Hn(Y ,O )−→Hn(O )Γ−→W′−→0
0 Y0 X
in which W and W′ are k vector spaces of dimension 1 with a trivial action of G.
Tensoring O → (π˜) O with k over k gives the natural homomorphism O → π O
Y0 ∗ X Y ∗ X
of sheaves on Y. Tensoring (3.2) with k over k gives the exact sequence
(3.3) 0−→k ⊗ W−→Hn(Y,O )−→Hn(O )G−→k ⊗ W′−→0.
k Y X k
associated to the homomorphism Hn(Y,O ) → Hn(Y,π O ) = Hn(O ) which results
Y ∗ X X
from O → π O .
Y ∗ X
8 PH. CASSOU-NOGUE`S,T. CHINBURG,B. EREZ, AND M. TAYLOR
Proposition 3.4. Suppose that n is odd. Then the trace map Tr : Hn(O ) →
∗ X
Hn(O ) and the inclusion k ⊗ W → Hn(O ) induce k¯-linear maps
Y k Y
Extn+1(Hn(O ),k¯) → Extn+1(Hn(O ),k¯)
k¯[C] Y k¯[C] X
and
Extn+1(Hn(O ),k¯) → Extn+1(k ⊗ W,k¯)
k¯[C] Y k¯[C] k
respectively. These maps are both surjective with the same kernel, giving an isomor-
phism
(3.4) Extn+1(Hn(O ),k¯) → Extn+1(k ⊗ W,k¯) = k ⊗ Extn+1(W,k).
k¯[C] X k¯[C] k k k[C]
Proposition 3.5. Suppose n is even. Then the inclusion Hn(O )G → Hn(O ) and
X X
the surjection Hn(O )G → k ⊗ W′ coming from (3.3) lead to k¯-vector space maps
X k
Extn+1(Hn(O ),H0(O )) → Extn+1(Hn(O )G,H0(O ))
k¯[C] X X k¯[C] X X
and
Extn+1(k ⊗ W′,H0(O )) → Extn+1(Hn(O )G,H0(O ))
k¯[C] k X k¯[C] X X
respectively. These maps are both injective with the same image, leading to an isomor-
phism
(3.5) Extn+1(Hn(O ),H0(O )) = Extn+1(k ⊗ W′,k) = k ⊗ Extn+1(W′,k).
k¯[C] X X k¯[C] k k k[C]
We will give the proof of these Propositions in the next section.
Our aim is now to use these propositions to construct α(X,G) and a numerical
invariant µ(X,G). The restriction map induces an injective homomorphism of k-vector
spaces
Extn+1(Hn(O ),H0(O )) → Extn+1(Hn(O ),H0(O )).
X X X X
k[G] k[C]
Since these vector spaces are of dimension 1 this is an isomorphism. We identify in
what follows the class β(X,G) and the k-line L defined in Section 2 with their images
1
in Extn+1(Hn(O ),H0(O )).
X X
k[C]
Theorem 3.6. Let L be the k-line in Extn+1(Hn(O ),H0(O )) which is the image of
0 X X
k[C]
1⊗Extn+1(W,k) (resp. Extn+1(W′,k)) under the isomorphism in (3.4) (resp. (3.5))
k[C] k[C]
if n is odd (resp. if n is even). Let α(X,G) be any generator of L over k, so that
0
α(X,G) is defined only up to multiplication by an element of k∗. Then
(3.6) α(X,G) = ζ ·β(X,G)
∗
for an element ζ ∈ k which is well-defined up to multiplication by an element of k∗.
The constant
(3.7) µ(X,G) = ζ1−q ∈ k∗
∗
lies in k and is an invariant of the action of G on X.
DERIVED INVARIANTS 9
Proof. ThisfollowsfromPropositions3.3,3.4and3.5andthefactthatExtn+1(Hn(O ),H0(O ))
X X
k[C]
has dimension 1 over k.
Corollary 3.7. LetF−1 bethe anti-semilinearendomorphismof Ext (Hn(O ),H0(O ))
k[C] X X
induced by the action of F on Hn(O ) and the action of F−1 on H0(O ) = k, de-
X X
scribed as in Theorem 2.9. Then F−1 acts on L by multiplication by µ(X,G). The
0
constant µ(X,G) lies in k∗.
Proof. Let L be the k-line of Extn+1(Hn(O ),H0(O )) introduced in Theorem 2.9.
1 X X
k[C]
It follows from the definitions of Theorem 3.6 that
L = ζL
0 1
foranyζ satisfying(3.6). Supposethatc isak-basisofL sothatζ.c isak-basisofL .
1 1 1 0
The endomorphism F−1 fixes L by Theorem 2.9. Hence since F−1 is anti-semilinear,
1
we have
F−1(τ ·ζ ·c ) = τ ·F−1(ζ)·c = ν ·(τ ·ζ ·c )
1 1 1
for τ ∈ k, where
F−1(ζ)
ν = .
ζ
This proves that F−1 acts as multiplication by ν on the k-line L , so ν ∈ k∗. Hence
0
ζ
ν = F(ν) = = ζ1−q = µ(X,G)
F(ζ)
which completes the proof in view of (3.7).
Corollary 3.8. The action of F on O and on O induces a k-linear action on W
Y0 X
and W′. If n is odd (resp. even) then F acts on W (resp. W′) by multiplication by
µ(X,G) ∈ k∗.
Proof. The action of F on O and on O is via the map α → αq on local sections,
Y0 X
and is k-linear. Thus F respects the homomorphism O → (π˜) O in Proposition
Y0 ∗ X
3.3, so it follows that F acts k-linearly on the one dimensional k-vector spaces W and
W′. Suppose n is odd. Since C is cyclic there are isomorphisms
Extn+1(W,k) = Hn+1(C,Hom (W,k))
k[C] k
(3.8) = Hˆ0(C,Hom (W,k))
k
= Hom (W,k)/Tr Hom (W,k)
k[C] C k
Because W ∼= k with trivial C-action, this gives an F−1-equivariant isomorphism be-
tweenExtn+1(W,k)andHom (W,k). RecallthatF−1 sendsanelementf ∈ Hom (W,k)
k[C] k k
to the homorphism F−1f defined by (F−1f)(w) = F−1(f(F(w))) for w ∈ W. Since
10 PH. CASSOU-NOGUE`S,T. CHINBURG,B. EREZ, AND M. TAYLOR
dim W = 1 the action of F on W is given by multiplication by some α ∈ k, and F
k
fixed k. Hence
(F−1f)(w) = f(αw) = αf(w)
so F−1 acts on Hom (W,k) by multiplication by α. Thus α is also the eigenvalue of
k
F−1 on L = Extn+1(W,k), so α = µ(X,G) by Corollary 3.8. The proof when n is
0 k[G]
even is similar.
We now relate µ(X,G) to zeta functions. Let ζ(V/k,T) be the zeta function of a
smooth projective variety V over k. Then ζ(V/k,T) ∈ Z [[T]], and the congruence
p
formula in [1, Expos´e XXII, 3.1] is
dim(V)
(3.9) ζ(V/k,T) = Y det(1−FT|Hi(V,OV))(−1)i+1 mod pZp[[T]].
i=0
Write this formula as
ζ (V/k,T)
(3.10) ζ(V/k,T) mod pZ [[T]] = 1
p
ζ (V/k,T)
0
where
(3.11) ζj(V/k,T) = Y det(1−FT|Hi(V,OV))
i≡j mod 2
for j = 0,1. Note that ζ (V/k,T) and ζ (V/k,T) could have a common zero, so the
1 0
formula (3.10) does not imply that a zero of ζ (V/k,T) (resp. of ζ (V/k,T)) is a zero
1 0
(resp. pole) of ζ(V/k,T) mod pZ [[T]].
p
Corollary 3.9. If n is odd then µ(X,G)−1 is a zero of ζ (Y /k,T). Suppose n is even.
1 0
Let X be the quotient of X by the the action of a lift φ to Gal(X/Y ) of the arithmetic
0 X 0
Frobenius of Gal(Y/Y ) ≡ Gal(k/k). Then X is a smooth projective variety over k
0 0
such that X = k ⊗ X , and µ(X,G)−1 is a zero of ζ (X /k,T).
k 0 0 0
Proof. If n is odd, we have shown µ(X,G) is the eigenvalue of F acting on the one-
dimensional k-space W inside Hn(Y ,O ). Hence 1−µ(X,G)−1F is not invertible on
0 Y0
Hn(Y ,O ), so µ(X,G)−1 is a zero of ζ (Y /k,T). Suppose n is even. We have shown
0 Y0 1 0
µ(X,G) is the eigenvalue of F acting on a one-dimensional k-space W′ which is a
quotient of Hn(O )Γ, where Γ = Gal(X/Y ). It is shown in Lemma 4.1 below that Γ is
X 0
the semidirect product of thenormal subgroup G withthe closure hφ i of thesubgroup
X
generated by φ . Since X → Y is a pro-´etale cover, it follows that X is smooth and
X 0 0
projective over k, and that X = k ⊗ X . Hence Hn(O ) = k ⊗ Hn(X ,O ), so
k 0 X k 0 X0
Hn(O )hφXi = Hn(X ,O ). Thus W′ is a subquotient of Hn(X ,O ), so as above,
X 0 X0 0 X0
1 − µ(X,G)−1F is not invertible on Hn(X ,O ). Since we assumed n is even, this
0 X0
shows µ(X,G) is a zero of ζ (X /k,T).
0 0