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Math.tifr.res.in
Mathematische Annalen
Congruences for rational points
on varieties over finite fields
N. Fakhruddin and C. S. Rajan
Received: March 4, 2004 / Revised: April 7, 2005Published online: October 28, 2005 – SpringerVerlag 2005
Abstract. We prove the existence of rational points on singular varieties over finite fields aris
ing as degenerations of smooth proper varieties with trivial Chow group of 0cycles. We alsoobtain congruences for the number of rational points of singular varieties appearing as fibres of aproper family with smooth total and base space and such that the Chow group of 0cycles of thegeneric fibre is trivial. In particular this leads to a vast generalization of the classical ChevalleyWarning theorem. The above results are obtained as special cases of our main theorem whichcan be viewed as a relative version of a theorem of H. Esnault on the number of rational pointsof smooth proper varieties over finite fields with trivial Chow group of 0cycles.
1. Introduction
In this paper we prove the existence of rational points on singular varieties overfinite fields arising as degenerations of smooth proper varieties with trivial Chowgroup of 0cycles. We also obtain congruences for the number of rational pointsof singular varieties appearing as fibres of a proper family with smooth total andbase space and such that the Chow group of 0cycles of the generic fibre is trivial.
In particular this leads to a vast generalization of the classical ChevalleyWarningtheorem. The above results are obtained as special cases of our main theoremwhich can be viewed as a relative version of the following theorem of H. Esnault[10]: if
X is a smooth proper variety over a finite field
k with
CH0
(X
then 
X(k) ≡ 1 mod 
k.
One of the main points of the paper is that our results are valid for proper
morphisms of smooth varieties rather than for morphisms of smooth proper varieties. It is possible to extend Esnault’s theorem to a relative version for propervarieties using the classical theory of correspondences. To deal with nonpropervarieties, we introduce the notion of proper correspondences which plays a keyrole in our proofs. This requires the use of the refined intersection theory developedby Fulton and Macpherson [11]. The use of Bloch’s decomposition of the diagonalin Esnault’s proof is replaced here by a relative version. The other ingredients of
N. Fakhruddin and C. S. RajanTata Institue of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai400005, India(email: naf@math.tifr.res.in,
our proofs are the GrothendieckLefschetz trace formula and Deligne’s integralitytheorem [6] for the eigenvalues of Frobenius on
ladic cohomology.
The methods of this paper can be applied in other contexts where the yoga of
weights is applicable. For a Hodge theoretic illustration see Theorem 4.3.
The main result of this paper is the following:
Theorem 1.1. Let fi :
Xi →
Y , i = 1
, 2
be proper dominant morphisms of
smooth irreducible varieties over a finite field k and let g :
X1 →
X2
be a dominant morphism over Y . Let Zi be the generic fibre of fi, let Zi
k(X1
), and assume that g∗ :
CH0
(Z1
phism. Then for all y ∈
Y (k),

f −1
(y)(k) ≡ 
f −1
(y)(k) mod 
k
.
We remark that there are no flatness assumptions in the above theorem. Spe
cialising to the case
X2 =
Y and
f2 =
I dY , we obtain:
Corollary 1.2. Let f :
X →
Y be a proper dominant morphism of smooth irre
ducible varieties over a finite field k. Let Z be the generic fibre of f and assume
that CH0
(Z
k(X) Q = Q
. Then for all y ∈
Y (k), 
f −1
(y)(k) ≡ 1
When
Y = Spec
(k), the corollary reduces to the theorem of H. Esnault cited
above. Since we do not assume that
f is smooth, we are able to obtain congruenceseven for singular varieties.
An immediate consequence of Corollary 1.2 is that for
f :
X →
Y as above,

X(k) ≡ 
Y (k) mod 
k. By taking
Y to be a point in Theorem 1.1 we obtain:
Corollary 1.3. Let g :
X1 →
X2
be a dominant morphism of smooth proper
varieties over a finite field k. If g∗ :
CH0
(X1
isomorphism, then 
X1
(k) ≡ 
X2
(k) mod 
k
.1
The above corollary as well as the Hodge theoretic analogue (see Section 4)
applies to arbitrary proper birational morphisms of smooth varieties (see alsoCorollary 3.1 for a slightly stronger version in positive characteristics). This givesnew restrictions on the cohomology of varieties which can occur as fibres of suchmorphisms.
In contrast to Corollary 1.2, the assumption that
X1 and
X2 be proper can
not be omitted here. For example, consider a pencil
h :
X → P1 of high genus
1 This can also be proved using results B. Kahn; see Remark 1 of [13].
Congruences for rational points on varieties over finite fields
curves in P2 with smooth generic fibre. Let
X2 be any affine open subset of
P1,
X1 =
h−1
(X2
), and let
g :
X1 →
X2 be the induced map. In this case
CH0
(X1
Q = 0, but the fibres need not have rational
The triviality of the Chow group of zero cycles of degree 0, or even rational
chain connectedness is not sufficient to guarantee the existence of a rational pointfor proper varieties over finite fields which are not smooth (see Remark 3.4).
However, using alterations, we give a criterion for the existence of rational pointswhich can be applied to all degenerations of smooth rationally chain connectedvarieties.
Corollary 1.4. Let f :
X →
Y be a proper dominant morphism of irreducible
varieties over a finite field k with Y smooth. Let Z be the generic fibre of f
and assume that Z is smooth and CH0
(Z
k(X) Q = Q
. Then for any y ∈
Y (k),
The corollary below generalises the Chevalley–Warning theorem [12], which
is the special case
P = P
n,
r = 1 and
L1 =
O(d) with
d ≤
n.
Corollary 1.5. Let P be a smooth projective geometrically irreducible variety
over a finite field k. Let L1
, · · ·
, Lr be very ample line bundles on P such that
(KP ⊗
L1 ⊗ · · · ⊗
Lr)−1
is ample, where KP is the canonical bundle of P . For
i = 1
, 2
, . . . , r, let si ∈
H 0
(P , Li). Then
x ∈
P (k) si(x) = 0
, i = 1
, 2
, . . . , r
Note that the congruence formula of Katz [6], when it applies, only gives con
gruences modulo
p = char
(k). Examples of varieties to which Corollary 1.5 canbe applied include toric Fano varieties and homogenous spaces
G/P (with
G asemisimple algebraic group and
P a reduced parabolic subgroup) of index
> 1,since in these cases ample line bundles are always very ample.
In [1], Bloch, Esnault and Levine formulate and prove a motivic version of the
ChevalleyWarning theorem. In their work, they use the embedding of the hypersurface in the smooth ambient variety P
n and their theory of motivic cohomologywith modulus. In contrast, we work with the family of all hypersurfaces, the totalspace of which is smooth, and use ordinary Chow groups.
It would be interesting to know whether an analogue of the Ax–Katz theorem
[14] holds in the above situation or whether all low degree intersections as aboveover
C1 fields always have rational points.
We now give a brief indication of the methods of our paper. For simplicity we
begin by describing the proof specialised to Corollary 1.3: let
g :
X1 →
X2 be as
in Corollary 1.3, and let
W be a multisection of
g, i.e. an irreducible subvarietyof
X1 mapping dominantly and generically finitely onto
X2 with degree
d. Let
g be the graph of
g, and let
W be the transpose of the graph of
g
W embedded
g be the correspondence in
X1 ×
k X1 given by
the composition of the correspondences
W and
g. Consider the two classes
η1and
η2 in
CH0
(k(X1
) ×
k X1
) given respectively by the pullback of the diagonal
1
/d . By construction both these classes project to the same class
in
CH0
(k(X1
) ×
k X2
) given by the pullback of the graph of
g. The injectivityhypothesis on the Chow groups implies that
η1 =
η2. Hence there exists a correspondence 2 in
X1 ×
k X1 whose support maps to a proper subvariety of
X1 viathe first projection such that
Deligne’s integrality theorem then implies that the eigenvalues of the (geometric) Frobenius acting on the cokernel of the map
g∗ :
H ∗
((X
l ) are algebraic integers divsible by 
k. The proof of Corollary 1.3 is
completed by appealing to the GrothendieckLefschetz trace formula.
For the proof of Theorem 1.1, we apply the diagonal decomposition described
above to the induced morphism on the generic fibres. The induced decomposition is then spread out to obtain a relative decomposition of the diagonal
in
X1 ×
Y X1. Since the varieties need not be proper, we introduce the notion ofproper correspondences. This can be applied in our context since the morphismsinvolved are all proper. This gives rise to a relative diagonal decomposition consisting of proper correspondences (see the following section for the definition andproperties of proper correspondences). The rest of the proof proceeds as indicatedabove.
2. Proper Correspondences
In this section we introduce the concept of proper correspondences and provesome of its properties.2 Most of the proofs are simple modifications of those inFulton’s book [11] and we prove only what we need for later use; several otherresults in [11, Chapter 16] for (usual) correspondences have analogues for propercorrespondences.
Definition 2.1. Let X and Y be smooth irreducible varieties over a field k. The
group of proper correspondences
from X to Y , P Corr(X, Y ), is the free abelian
2 A. Nair has informed us that a variant of this definition has been considered, in a different
context, by E. Urban in the preprint: Sur les repr´esentations
padiques associ´ees aux repr´esentations cuspidales de
GSp4
/Q, 2001.
Congruences for rational points on varieties over finite fields
group generated by irreducible subvarieties
⊂
X ×
Y which are proper over
Y modulo the subgroup generated by elements of the form
{div
(f )
f ∈
k(Z)∗
, Z ⊂
X ×
Y irreducible and proper over Y }
.
This is clearly a graded abelian group; we shall grade it by dimension (lower
indices) or codimension (upper indices) as is convenient. For a cycle
essarily irreducible) in the free abelian group as above we shall denote its class in
P Corr(X, Y ) by [ ].
If
f :
X →
Y is a proper morphism, then the graph of
f gives an element
[
f ] of
P Corr(X, Y ). We shall show below that proper correspondences inducemaps on cohomology generalising the maps induced by proper morphisms.
We recall some properties of ´etale (co)homology and cycle class maps that we shall
need. The main reference is [7, Expos´es VIIX]; we note that the quasiprojectivityhypotheses there can be removed using the methods of [11]. The compatibilityof refined intersection products and refined cycle class maps stated below can bededuced using the methods of [11, Chapter 19]; see also [3].
Let
K be an algebraically closed field and fix a prime number
l =
char(K).
For a variety
X over
K let
H i(X) :=
H i (X, Q
l ) for
Z a closed subvariety of
X. We also let
Hi (X) denote
the locally finite (or BorelMoore) homology
H et (X, Q
groups
H , we denote by
H (n), for an integer
n, the corresponding Tate twistedgroup.
i Zi , on a variety
X, we denote by 
α the support ∪
i Zi
of
α. The following properties are proved in the references cited above:
(1) (Projection formula) For any variety
X there are cap product maps
H i(X) ⊗
Hj (X) →
Hj−
i(X) such that if
f :
X →
Y is a proper morphism,
u ∈
H i(Y ),
v ∈
Hj (X) then
f∗
(f ∗
(u) ∩
v) =
u ∩
f∗
(v) in
Hj−
i(Y ).
(2) For
Z an irreducible variety of dimension
n,
H2
n(Z)(−
n) is one dimensional.
(3) For
α a cycle of dimension
k on a variety
X the fundamental class defines
a canonical element
cl(α) ∈
H2
k(
α
)(−
k). This maps to an element, alsodenoted by
cl(α), in
H2
k(X)(−
k) which is zero if
α = 0 ∈
CHk(X).
(4) For
X smooth of pure dimension
n there is a canonical isomorphism
H2
k(
α
) (−
k) ∼
(X)(n −
k), so we also get an element
cl(α) ∈
(5) For
f :
X →
Y a morphism and
α a cycle on
X of dimension
n whose
support is proper over
Y ,
f∗
(cl(α)) =
cl(f∗
(α)) in
H ∗
(6) For
α and
β cycles of pure dimension
k and
l respectively in a smooth irre
ducible variety
X of dimension
n
cl(α ·
β) =
cl(α) ∪
cl(β) ∈
H 4
n−2
(k+
l)
(X)(n −
k −
l) ,
where the product on the left is the refined intersection product.
Let dim
(X) =
n, dim
(Y ) =
m and let
induces a linear map ∗ :
H i(X) →
H 2
m−2
r+
i(Y )(m−
r)
as the composite of the sequence of linear maps:
−→
Hi(X ×
Y ) −→
Hi+2
(n+
m−
r)
H2
r−
i( 
)(−
r) −→
H2
r−
i(Y )(−
r) ∼
=
H2
m−2
r+
i(Y )(m −
r) .
Since
X and
Y are smooth, by duality we also get maps
H 2
n−2
r+
i(X)(n −
r).
Now suppose that
k is a finite field and
K an algebraic closure of
k. All the
cohomology groups discussed above, for varieties over
K which are base changedfrom varieties over
k, have a continuous action of
Gal(K/k). The maps
∗ are then compatible with the Galois action.
The following lemma implies that the maps ∗ and ∗ only depends on [ ] ∈
Lemma 2.2. Let
be as above. Suppose there exists a closed subvariety Z ⊂
X ×
Y such that   ⊂
Z, [ ] = 0
in CH∗
(Z) and pY 
Z :
Z →
Y is proper. Then
∗
and ∗
are the zero maps.
Proof. Suppose
Z is a closed subvariety of
X ×
Y such that the projection
pY 
Z :
Z →
Y is proper. It follows from the projection formula that ∗ can also bedefined as,
−→
Hi(Z) −→
H2
r−
i(Z)(−
r)
−→
H2
r−
i(Y )(−
r) ∼
=
H2
m−2
r+
i(Y )(m −
r),
where
cl( ) is considered as an element in
H2
r (Z)(−
r). The lemma followsfrom the fact that
cl( ) = 0 in
H2
r(Z)(−
r). The statement for ∗ follows byduality.
Congruences for rational points on varieties over finite fields
Let
X, Y, Z be smooth irreducible varieties over
K and let [ 1] (resp. [ 2]) be aproper correspondence from
X to
Y (resp.
Y to
Z). Analogous to the definitionof composition of correspondences [11, Chapter 16], we define [ 2] ◦ [ 1] ∈
P Corr(X, Z) by
2] ◦ [ 1] = [
pXZ ∗
(pXY ( 1
) ·
pY Z ( 2
))]
,
where the
p’s denote the projection morphisms from
X ×
Y ×
Z and the productis the refined intersection product. Note that
p
1
) ·
pY Z ( 2
) is a cycle which
is well defined upto rational equivalence on  1 ×
Y  2. Since  1 ×
Y  2 isproper over
Z, its image in
X ×
Z is also proper over
Z, so [ 2] ◦ [ 1] is a welldefined element of
P Corr(X, Z).
Lemma 2.3. Let X, Y, Z and 1
, 2
be as above. Then ( 2 ◦ 1
)∗ =
( 2
)∗ ◦
( 1
)∗
as maps from H ∗
(X) to H ∗
(Z) and ( 2 ◦ 1
)∗ =
( 1
)∗ ◦
( 2
)∗
as maps from
H ∗
(Z) to H ∗
(X).
Proof. The key point is the compatibility of the refined cycle class maps withrefined intersection products.
Let
a ∈
H ∗
(X). Then
( 2
)∗ ◦
( 1
)∗
(a) =
pYZ
Z ∗
(cl( 2
) ·
pY Z
Z ∗
(cl( 2
) ·
pXY Z
∗
(cl( 2 ◦ 1
) ·
pXZ
We use the projection formula several times along with compatibility of the cycle
class map with smooth pullbacks, products and proper pushforwards.
The key technical result which allows us to deduce congruences from cycle theoretic information is the following:
Proposition 2.4. Let X, Y,
be as above and assume that m =
n =
r. If
dim
(pX( 
)) < n then all the eigenvalues of (the geometric) Frobenius actingon
∗
(H i(Y )) ⊂
Hi(X) are algebraic integers which are divisible by
Proof. Replacing
k by a finite extension does not affect the conclusion of thelemma, so without loss of generality we may assume that
irreducible subvariety of
X ×
Y . Using the definition of ∗ as given in the proofof Lemma 2.2, ∗ is the composite of the following maps:
−→
Hi( )−→
H2
n−
i( )(−
n) −→
H2
n−
i(Y )(−
n) ∼
where we have used the hypothesis that
m =
n =
r.
→ is a proper dominant generically finite morphism with
smooth and geometrically irreducible. Then the projection formula shows that
and
pX (resp.
pY ) with
pXπ (resp.
pY π ) in the above
without changing the image of the composite.
Let
W be the Zariski closure of
pX( ) in
X and let dim
(W ) =
t. By the
as above,
σ :
W →
W with
σ
proper dominant generically finite and
W smooth, and
p :
pXπ =
iW σp, where
iW :
W →
X is the inclusion. Since
pXπ =
(iW σ )p andboth
and
W are smooth, it follows from the functoriality of pushforward maps
that
(pXπ )∗ =
(iW σ )∗
p∗. By the remarks of the previous paragraph we see thatthe image of ∗ is the same as that of the composite of the sequence:
H i(X)−→
H i(W )−→
H i( )−→
H i(Y ) .
and
W are smooth, by duality we get a factorisation of ∗ as a composite
H j (Y )−→
H j ( )−→
H 2
(t−
n)+
j (W )(t −
n)−→
H j (X) .
By Deligne’s integrality theorem [6], Expos´e XXI, Corollaire 5.5.3, all the eigenvalues of Frobenius on
H ∗
(W ) are algebraic integers. Since
n −
t > 0 and the
geometric Frobenius acts on Q
l(t −
n) by 
k
n−
t , the proposition follows.
3. Proofs of the main results
Using the results of the previous section, we now give the proofs of the resultsstated in the introduction.
Proof of Theorem 1.1. If
Y is not geometrically irreducible, then
Y (k) = ∅ sothere is nothing to prove. If
X2
(k) = ∅ then
X1
(k) = ∅ and the theorem is true.
Thus we can assume that
X2
(k) is nonempty and this implies that
X2 is geometrically irreducible. If
X1 is not geometrically irreducible, then after a basechange to a finite extension of
k,
X1 becomes a union of at least two connected
Congruences for rational points on varieties over finite fields
components, each of which maps dominantly to the base change of
X2. Hencethere is more than one connected component of
Z1 base changed to
k(X1
) whichmaps surjectively to each connected component of
Z2 base changed to
k(X1
).
This contradicts the injectivity of the map on Chow groups. Hence we can assumethat
X1 is also geometrically irreducible.
Let
W be an irreducible subvariety of
X1 which maps generically finitely and
dominantly to
X2 and let
d be the degree of
W over
X1. Let
g be the graph of
gin
X1 ×
k X2, let
W be the transpose of the graph of
g
W embedded in
X2 ×
k X1and let
g )/d . Since
W is a subvariety of
X1,
g is a proper correspondence since
g is proper. Hence
a proper correspondence of dimension
n1 = dim
X1. Since
f2 ◦
g =
f1, thecorrespondence 1 is naturally represented by a cycle supported in
X1 ×
Y X1.
Let
V2 be the open subset of
X2 over which
g
W is finite and let
V1 =
g−1
(V2
).
By the construction of
W ,
p∗
(
W ), which are subvarieties of
X1 ×
k
X2 ×
k X1, meet properly when pulled back to
V1 ×
k X2 ×
k X1. It follows thatone can write 1 =
P Corr(X1
, X2
) and
p1
(

) is a proper subvariety of
X
1 ×
k X1 and consider the proper correspon
X1 ×
Y X1. Hence we may consider its restriction (i.e. flat pullback)
γ2 to
Z1 ×
k(Y)Z1. Since
(I dX ×
g)
g , it follows from the previous paragraph that
Z1 ∗
(γ2
) can be represented by a cycle on
Z1 ×
k(Y ) Z2 which becomes
zero when restricted to
k(Z1
) ×
k(Y) Z2. Since the map
g∗ :
CH0
(Z1
Q is injective, it follows that
γ2 can be represented by a cycle on
Z1 ×
k(Y) Z1 whose support maps to a proper subvariety of
Z1 by the projection tothe first factor. By taking the Zariski closures in
X1 ×
Y X1, it follows that 2 canbe represented by a proper correspondence on
X1 ×
k X1 whose support maps toa proper subvariety of
X1 by the projection to the first factor.
By Proposition 2.4 all the eigenvalues of Frobenius acting on the image of
1
) are divisible by 
k. It follows from Lemma 2.3 and the definition
∗ is contained in the image of
g∗. By the definition of
2, we conclude that all the eigenvalues of Frobenius acting on the cokernel of
1
) are divisible by 
k. Note that since
g is dominant, the
The hypotheses of the theorem, and the above discussion, are not affected
if we replace
Y by an open subvariety
U and
Xi by
f −1
(U ),
i = 1
, 2, so we
may assume that
Y has only one rational point
y. Applying the GrothendieckLefschetz trace formula and the statement on eigenvalues above, we concludethat the 
X1
(k) ≡ 
X2
(k) mod 
k. Each rational point of
Xi must lie over theunique rational point
y of
Y , hence the theorem follows.
Corollary 3.1. Let f :
X →
Y be a proper dominant generically finite mor
phism of smooth irreducible varieties over a finite field k such that the extension
of function fields k(Y ) →
k(X) is purely inseparable. Then for any y ∈
Y (k),

f −1
(y)(k) ≡ 1 mod 
k
.
k(X) red is isomorphic to Spec
(k(X)), the hypothesis on
CH0 is
Remark 3.2. Since we only use intersection theory (resp. cohomology groups)with rational (resp. Q
l) coefficients, the above results hold even when
X and
Yare quotients of smooth varieties by finite groups.
Proof of Corollary 1.4. As in the proof of Theorem 1.1, we may assume that
Xand
Y are geometrically irreducible and that
Y has a unique rational point.
By a result of De Jong [5, 5.15] there exists an irreducible variety
X over
k
which is the quotient of a smooth variety by a finite group and a proper dominantgenerically finite morphism
π :
X →
X such that the extension of function fields
k(X) →
k(X ) is purely inseparable. Let
f =
f π :
X →
Y and let
Z be thegeneric fibre of
f . Since
X is irreducible over
k,
Z is irreducible over
k(Y ).
Zis geometrically irreducible since it is smooth over
k(Y ); since the extension offunction fields induced by the map
Z →
Z is
k(X) →
k(X ),
Z is also geometrically irreducible. The induced morphism
(Z
assumptions of Lemma 3.6 below, so
CH0
(Z
1.2 (cf. Remark 3.2) to the morphism
f :
X →
Y we see that
X (k) = ∅. Thus
X(k) = ∅.
Remark 3.3. For singular
X it is not always true that 
f −1
(y)(k) ≡ 1 mod 
k,however the only examples we know where this fails are nonnormal.
Remark 3.4. The following example3 gives an example of a proper nonsmooth
variety over a finite field which has no rational point and such that the Chow group
of zero cycles of degree 0 is trivial: let
C be a smooth projective geometrically
ireducible curve over
k, a finite field, with
C(k) = ∅ and let
p be any closed point
of
C. Let
t be a closed point of
P1 of degree greater than one, and let
X be the
blowup of
C ×
k P1 with centre the closed subscheme
p ×
variety obtained by blowing down the strict transform of
C ×
k t in
X . Then
X isrationally chain connected but
X(k) = ∅.
Remark 3.5. X(k) = ∅ if
X is a proper variety over a finite field
k which isrationally connected i.e. any two general points of
X( ) are contained in an irreducible rational curve in
X defined over
3 We learnt of such an example, due to J. Koll´ar (unpublished), from a talk by J. Iyer at the
Congruences for rational points on varieties over finite fields
(To prove this we may replace
X by
X as in the proof of Corollary 1.4. Sincethe extension of function fields
k(X) →
k(X ) is purely inseparable, it followsthat
X is also rationally connected, so
CH0
(X
follows that
X (k) ≡ 1 mod 
k, hence
X (k) = ∅. Thus
X(k) = ∅.) However, adegeneration of a rationally connected variety is in general only rationally chainconnected, so one cannot use this to prove Corollary 1.4 even if
Z is rationallyconnected.
Lemma 3.6. Let f :
X →
Y be a proper dominant morphism of irreducible
varieties over an algebraically closed field K. Assume that Y is smooth, f is
generically finite and the extension of function fields K(Y ) →
K(X) is purely
inseparable. Then f∗ :
CH0
(X)Q →
CH0
(Y )Q
is an isomorphism.
Proof. f∗ is surjective because
f is surjective. Using the refined intersection theory of [11] and the hypothesis on
Y , we see that there is a natural map
f ∗ :
CH0
(Y )Q →
CH0
(X)Q. By the assumptions on
f there exists an open subset
Uof
Y such that for
V =
f −1
(U ),
f 
V :
V →
U is a bijection. By the movinglemma (which is trivial for zero cycles)
CH0
(X) (resp.
CH0
(Y )) is generatedby the closed points in
V (resp.
U ). This shows that
f ∗ is a surjection since for
y ∈
U ,
f ∗
([
y]
) = [
f −1
(y)]. Since
f∗
f ∗ is multiplication by deg
(f ), it followsthat
f∗ is an isomorphism.
Proof of Corollary 1.2. Let
Y =
i ) and let
X ⊂
Y ×
P be the zero
i . Since the
Li are basepoint free, the
map
X →
P is smooth, hence
X is also smooth. Since the
Li are assumed to bevery ample, Bertini’s theorem implies that the generic fibre
Z of the projection
f :
X →
Y is smooth. The assumptions on the
Li and the adjunction formulaimply that
Z is a Fano variety. By a theorem of Campana [2] and KollarMiyaokaMori [15]
Z is rationally chain connected, so
CH0
(Z
concluded by applying Corollary 1.2 to
f .
Remark 3.7. The proof shows that instead of assuming that the
Li are very ampleit suffices to assume that they are basepoint free and that
Z is smooth.
4. Further questions
It seems likely that the following mixed characteristic analogue of Corollary 1.4has a positive answer:
Question 4.1. Let
K be a finite extension of Q
p,
O its ring of integers,
k the residue field and
X a smooth proper variety over
K such that
CH0
(X
X →
Spec(O) is a proper scheme with generic fibre isomorphic to
X and closedfibre
X0, then is
X0
(k) = ∅?
From our proofs we do not obtain any information about the valuations of the
eigenvalues of Frobenius acting on the ´etale cohomology of the fibres of
f . Oneis thus lead to ask the following:
Question 4.2. Let
f :
X →
Y be a proper dominant morphism of smooth irreducible varieties over a finite field
k. Let
Z be the generic fibre of
f and assumethat
CH0
(Z
k(X) Q = Q. Then for all
y ∈
Y (k) and
i > 0, does 
k divide all the
eigenvalues of Frobenius acting on
H i((f −1
(y)) , Q
Theorem 1.1 has the following Hodge theoretic analogue:
Theorem 4.3. Let fi :
Xi →
Y , i = 1
, 2
, be proper dominant morphisms of
smooth irreducible varieties over C
and let g :
X1 →
X2
be a dominant morphismover Y . Let Zi be the generic fibre of fi and assume that g∗ :
CH0
(Z1C
(X2
, C
))) = 0
for all i, where F denotes the Hodge filtration of Deligne.
The proof is essentially the same as that of Theorem 1.1 so we omit the details:one only needs to replace Proposition 2.4 by its Hodge theoretic counterpart.
Question 4.4. Let
f :
X →
Y be a proper dominant morphism of smoothirreducible varieties over C. Let
Z be the generic fibre of
f and assume that
CH0
(ZC
)
(H i(f −1
(y), C
)) =
(X) Q = Q. Then for all
y ∈
Y (C
) and
i > 0, is
gr F
It seems likely, as was suggested to us by P. Brosnan, that there should be
a purely motivic statement from which Corollary 1.2 and Theorem 4.3 shouldfollow after taking realisations. The category of motives over a base of Corti andHanamura [3] would seem to be a natural choice in which to formulate such astatement.
Remark 4.5. After this paper was submitted both Question 4.2 and (a strongerform of) Question 4.4 were shown to have positive answers by H. Esnault; herresults appear in the appendix to this article [8]. A positive answer to Question4.1 in case
X is regular was also obtained by her in [9]; the results of that paperinclude a stronger version of Corollary 1.2 in the case that
Y is a curve.
Acknowledgements. N.F. thanks Arvind Nair for a remark which convinced him that Corollary1.2 should be true (before Theorem 1.1 was proved) and Patrick Brosnan, H´el`ene Esnault, Madhav Nori and V. Srinivas for their comments on the results of this paper. He also thanks H´el`eneEsnault and Marc Levine for an invitation to visit the University of Essen in May–June 2003
Congruences for rational points on varieties over finite fields
where he was supported by the DFG and via the Wolfgang Paul prize program of the HumboldtStiftung; lectures of H´el`ene Esnault that he attended at that time provided part of the motivationfor the results and questions in this paper.
We also thank Bruno Kahn for his comments on the first version of this paper and for
suggesting Theorem 1.1 as a common generalisation of Corollaries 1.2 and 1.3.
We thank the referee for his suggestions which helped in improving the exposition of the
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