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 – Springer-Verlag 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 0-cycles. 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 0-cycles of thegeneric fibre is trivial. In particular this leads to a vast generalization of the classical Chevalley-Warning 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 0-cycles.
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 0-cycles. 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 0-cycles of the generic fibre is trivial.
In particular this leads to a vast generalization of the classical Chevalley-Warningtheorem. 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 vari-eties. It is possible to extend Esnault’s theorem to a relative version for propervarieties using the classical theory of correspondences. To deal with non-propervarieties, 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(e-mail: naf@math.tifr.res.in, our proofs are the Grothendieck-Lefschetz trace formula and Deligne’s integralitytheorem [6] for the eigenvalues of Frobenius on l-adic 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 dom-inant 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 caseCH0(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 = Pn, 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 Chevalley-Warning theorem. In their work, they use the embedding of the hyper-surface in the smooth ambient variety Pn 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 diago-nal 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 corre-spondence 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 (geomet-ric) 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 Grothendieck-Lefschetz 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 decomposi-tion 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 con-sisting 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 p-adiques associ´ees aux repr´esen-tations 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 inP 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 VI-IX]; we note that the quasi-projectivityhypotheses 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 Borel-Moore) 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, H2n(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) ∈ H2k(|α|)(−k). This maps to an element, alsodenoted by cl(α), in H2k(X)(−k) which is zero if α = 0 ∈ CHk(X).
(4) For X smooth of pure dimension n there is a canonical isomorphism H2k(|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 4n−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 2m−2r+i(Y )(m−r) as the composite of the sequence of linear maps: −→ Hi(X × Y ) −→ Hi+2(n+m−r) H2r−i(| |)(−r) −→ H2r−i(Y )(−r) ∼ = H2m−2r+i(Y )(m − r) . Since X and Y are smooth, by duality we also get maps H 2n−2r+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) −→ H2r−i(Z)(−r) −→ H2r−i(Y )(−r) ∼ = H2m−2r+i(Y )(m − r), where cl( ) is considered as an element in H2r (Z)(−r). The lemma followsfrom the fact that cl( ) = 0 in H2r(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 the-oretic 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( )−→H2n−i( )(−n) −→ H2n−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 eigen-values of Frobenius on H ∗(W ) are algebraic integers. Since n − t > 0 and the geometric Frobenius acts on Ql(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 geo-metrically 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 Grothendieck-Lefschetz 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. Ql) 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 fieldsk(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 geomet-rically 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) = ∅. ThusX(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 non-normal.
Remark 3.4. The following example3 gives an example of a proper non-smooth
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 irre-ducible 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 the-ory 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 fory ∈ 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 projectionf : 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 Kollar-Miyaoka-Mori [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 Qp, O its ring of integers, k the resi-due 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 irre-ducible 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 thatCH0(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, Mad-hav 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 References
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