Coniveau over $p$-adic fields and points over finite fields
CONIVEAU OVER p-ADIC FIELDS AND POINTS OVER
FINITE FIELDS
HÉLÈNE ESNAULT
Abstract. If the ℓ-adic cohomology of a projective smooth variety, defined
over a p-adic field K with finite residue field k, is supported in codimension
≥ 1, then any model over the ring of integers of K has a k-rational point.
Version française abrégée. Soit X une variété projective et absolument
irréductible sur un corps local K. Rappelons qu’un modèle de X/K sur l’anneau
de valuation R de K est un morphisme X → SpecR projectif et plat, tel que
(X → SpecR)⊗K = (X → SpecK). Nous considérons la cohomologie ℓ-adique
H i(X̄) à coefficients dans Qℓ. Dire qu’elle est supportée en codimension 1 signifie
que toute classe dans H i(X̄) a une restriction nulle dans H i(Ū), où U ⊂ X est
un ouvert non vide. Le but de cette note est de prouver le théorème suivant.
Théorème: Soit X une variété projective lisse et absolument irréductible sur un
corps local K de caractéristique 0 et à corps résiduel fini k. On suppose que la
cohomologie ℓ-adique H i(X̄) est supportée en codimension ≥ 1 pour tout i ≥ 1.
Soit X /R un modèle. Alors il existe un morphisme projectif surjectif σ : Y → X
de R-schémas tel que |Y(k)| ≡ 1 modulo |k|.
On en déduit immédiatement le corollaire suivant.
Corollaire: Sous les hypothèses du théorème, tout modèle X /R possède un point
k-rationnel.
Pour ce qui concerne l’existence du point k-rationnel, ceci affranchit [6, Theo-
rem 1.1], (qui est vrai aussi si K est de caractéristique p > 0), de l’hypothèse de
régularité sur le choix du modèle X , qui était utilisée pour pouvoir appliquer le
théorème de pureté de Gabber [7]. Pour ce faire, nous montrons que d’avoir des
singularités quotient est suffisant, de même que pour l’étude de l’application de
spécialisation. Nous appliquons alors la version plus précise du théorème de Jong
ainsi qu’elle est exposée dans [2].
1. Introduction
Let X be a projective, absolutely irreducible variety defined over a local field K
with finite residue field k. Recall that a model of X/K on the valuation ring R
Date: April 6, 2007.
Partially supported by the DFG Leibniz Preis.
http://arxiv.org/abs/0704.1273v1
POINT 2
of K is a flat projective morphism X → SpecR such that (X → SpecR)⊗K =
(X → SpecK). We consider ℓ-adic cohomology H i(X̄) with Qℓ-coefficents. One
defines the first coniveau level
N1H i(X̄) = {α ∈ H i(X̄), ∃ divisor D ⊂ X s.t. 0 = α|X\D ∈ H
i(X \D)}.(1.1)
As H i(X̄) is a finite dimensional Qℓ-vector space, one has by localization
∃D ⊂ X s.t. N1H i(X̄) = Im
(X̄) → H iX̄)
,(1.2)
where D ⊂ X is a divisor. One says that H iX̄) is supported in codimension 1 if
N1H i(X̄) = H i(X̄). This definition is general, but has good properties only if X
is irreducible and smooth or has only very mild singularities.
In [6, Theorem 1.1] it is shown that if X/K is smooth, projective, absolutely
irreducible over a local field K with finite residue field k, and if ℓ-adic cohomology
H i(X̄) is supported in codimension ≥ 1 for all i ≥ 1, then any regular model
X /R of X/K has the property
X (k) ≡ 1 mod |k|.(1.3)
The purpose of this note is to drop the regularity assumption if K has character-
istic 0.
Theorem 1.1. Let X be a smooth, projective, absolutely irreducible variety de-
fined over a local field K of characteristic 0 with finite residue field k. Assume
that ℓ-adic cohomology H i(X̄) is supported in codimension ≥ 1 for all i ≥ 1. Let
X be a model of X over the ring of integers R of K. Then there is a projective
surjective morphism σ : Y → X of R-schemes such that
|Y(k)| ≡ 1 mod |k|.
As an immediate corollary, one obtains
Corollary 1.2. Under the assumptions of the theorem, every model X /R has a
k-rational point.
The regularity of the model X in the proof of [6, Theorem 1.1] (which is shown also
when K has characteristic p > 0) was used to apply Gabber’s purity theorem
[7]. We show that for the piece of regularity one needs, it is enough to have
quotient singularities. Likewise, for the properties needed on the specialization
map, quotient singularities are good enough. The more careful use of de Jong’s
theorem as exposed in [2] allows then to conclude.
Acknowledegment: This note relies on de Jong’s fundamental alteration theorems.
T. Saito suggested to us the use of them in the shape formulated in [2]. We thank
him for this, and for many subsequent discussions on the subject. We exposed a
weaker version of Theorem 1.1 at the conference in honor of S. Bloch in Toronto
in March 2007. Discussions with him, A. Beilinson and L. Illusie contributed to
simplify our original exposition.
POINT 3
2. Proof of Theorem 1.1
Let K be a local field of characteristc 0 with finite residue field k. Let R ⊂ K be
its valuation ring. Let X → SpecR be an integral model of a projective variety
X → SpecK. We do not assume here that X is absolutely irreducible, nor do
we assume that X/K is smooth. Then by [2, Corollary 5.15], there is a diagram
SpecR
(2.1)
and a finite group G acting on Z over Y with the properties
(i) Z → SpecR and Y → SpecR are flat,
(ii) σ is projective, surjective, and birational,
(iii) Y is the quotient of Z by G,
(iv) Z is regular.
So Y → SpecR is not quite a model of X → SpecK, but is close to it. We show
in the sequel that σ in (2.1) does it in Theorem 1.1. Set
Y = Y ⊗K, Z = Z ⊗K.
For an open U ⊂ X let us set YU = U ×X Y, ZU = U ×X Z.
Let us assume now that X/K is smooth. This implies that
H i(Ū)
σ∗ inj
−−−→ H i(YU).(2.2)
Moreover, one has a trace map from Y to X
H i(YU)
(trace)Y/X
// H i(Ū)(2.3)
which splits σ∗ in (2.2). Let i ≥ 1 and let D ⊂ X be a divisor such that
(X̄) ։ H i(X̄) and such that σ|X\D : Y \σ
−1(D) → X \D is an isomorphism.
Then (2.3) yields the commutative diagram
H i(Ȳ )
(trace)Y/X
// H i(Y \ σ−1(D))
H i(X̄)
// H i(X \D)
(2.4)
and we conclude
X/K smooth =⇒ N1H i(X̄) = H i(X̄) ⊂ N1H i(Y ) = H i(Y ).(2.5)
We endow all schemes considered (which are R-schemes) with the upper subscript
u to indicate the base change ⊗RR
u or ⊗KK
u, where Ku ⊃ K is the maximal
POINT 4
unramified extension, and Ru ⊃ R is the normalization of R in Ku. Likewise, we
write ? to indicate the base change ⊗RR̄, ⊗KK̄, ⊗kk̄, where K̄ ⊃ K, k̄ ⊃ k are
the algebraic closures and R̄ ⊃ R is the normalization of R in K̄. We consider as
in [6, (2.1)] the F -equivariant exact sequence ([5, 3.6(6)])
. . . → H i
−→ H i(B̄) = H i(Yu)
−−→ H i(Y u) → . . . ,(2.6)
where F ∈ Gal(k̄/k) is the geometric Frobenius, and B = Y ⊗ k.
One has
Claim 2.1. The eigenvalues of the geometric Frobenius F ∈ Gal(k̄/k) acting on
H i(Xu) and on H i(Y u) lie in q · Z̄ for all i ≥ 1.
Proof. For H i(Xu), this is [6, Theorem 1.5(ii)]. One has H i(Ȳ ) = H i(Z̄)G, thus
in particular, π∗ : H i(Ȳ ) → H i(Z̄) is injective. By (2.5) one has
H i(Ȳ )
π∗ inj
−−−→ N1H i(Z̄).(2.7)
Since K has characteristic 0, and Z is regular by (iv), Z is smooth. Thus we can
apply again [6, Theorem 1.5(ii)]. This finishes the proof.
Claim 2.2. The eigenvalues of the geometric Frobenius F ∈ Gal(k̄/k) acting on
ι(H i
(Yu)) ⊂ H i(B̄) lie in q · Z̄ for all i ≥ 1.
Proof. By (iii), one has H i
(Yu) = H i
(Zu)G ⊂ H i
(Zu), where C = π−1(B).
Since by (iv), Z is regular, we can apply [6, Theorem 1,4], which is a consequence
of Gabber’s purity theorem [7], to conlude.
Proof of Theorem 1.1. Claims 2.1 and 2.2 together with (2.6) show that the eigen-
values of F acting on H i(B̄) lie in q · Z̄ for all i ≥ 1.
We apply the Lefschetz trace formula |B(k)| = TrF |H∗(B̄). As B is absolutely
connected and defined over k, F |H0(B̄) = Identity. By the discussion, one has
|B(k)| ∈ N ∩ (1 + q · Z̄) ⊂ 1 + q · Z. �
3. Remarks
Starting from Theorem 1.1, and Corollary 1.2, we may ask what happens if
K has equal characteristic p > 0 and whether or not the congruence of the
theorem is true on all models. We have no counter-examples for either question.
What K is concerned, characteristic 0 is used in the proof of Claim 2.1: if K
has characteristic p > 0, we only know that Z is regular, thus we can’t apply
immediately [6, Theorem 1.5(ii)]. Going up to a strict semi-stable model does not
help as for this, one has to ramify R and one loses regularity of Z and Z. What the
POINT 5
congruence is concerned, instead of going to one birational model Y (or birational
up to some inseprable extension in characteristic p > 0), one should go up to a
hypercover built out of such Y . In doing Deligne’s construction of hypercovers
with resolutions of singularities being replaced by de Jong’s morphisms of the type
σ in (2.1), one creates components which do not dominate X , the cohomology of
which is very hard to control. So one perhaps loses the coniveau property.
References
[1] de Jong, A. J.: Smoothness, semi-stability and alterations, Publ. Math. IHES 83 (1996),
51-93.
[2] de Jong, A. J.: Families of curves and alterations, Ann. Inst. Fourier 47 no2 (1997),
599-621.
[3] Deligne, P.: Théorème d’intégralité, Appendix to Katz, N.: Le niveau de la cohomologie
des intersections complètes, Exposé XXI in SGA 7, Lect. Notes Math. vol. 340, 363-400,
Berlin Heidelberg New York Springer 1973.
[4] Deligne, P.: Théorie de Hodge III, Publ. Math. IHES 44 (1974), 5-77.
[5] Deligne, P.: La conjecture de Weil, II. Publ. Math. IHES 52 (1981), 137-252.
[6] Esnault, H.: Deligne’s integrality theorem in unequal characteristic and rational points
over finite fields, with an appendix with P. Deligne, Annals of Mathematics 164 (2006),
715-730.
[7] Fujiwara, K.: A Proof of the Absolute Purity Conjecture (after Gabber), in Algebraic Ge-
ometry 2000, Azumino, Advanced Studies in Pure Mathematics 36 (2002), Mathematical
Society of Japan, 153-183.
Universität Duisburg-Essen, Mathematik, 45117 Essen, Germany
E-mail address : esnault@uni-due.de
	1. Introduction
	2. Proof of Theorem ??
	3. Remarks
	References
