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On the etale cohomology of algebraic varieties

with totally degenerate reduction over p-adic

fields

to J. Tate

Wayne Raskind∗ Xavier Xarles∗∗

Introduction

Let K be a finite extension of Qp and X a smooth, projective, geometricallyconnected variety over K. Let K be an algebraic closure of K and X denotebase extension of X to K. In this paper, we analyze the etale cohomology of Xin the case when X has “totally degenerate reduction,” a notion defined below(see §1), and we generalize results for curves and abelian varieties due to Tate(unpublished), Raynaud [Ra] and Mumford [Mu1,2]. We will see that such vari-eties have reasonably simple but still very interesting etale cohomology that canbe analyzed with the help of integral structures that we define in §2. These arefinitely generated abelian groups that when tensored with a suitable coefficientring (a Tate twist of Z`), are isomorphic to the graded pieces of the `-adic etalecohomology of X for the monodromy filtration (see §§3-5 below). Using theseintegral structures, we go on to define in §6 what we call “p-adic intermediateJacobians,” J i(X), which we expect to play a similar role in the theory of alge-braic cycles on such an X as the intermediate Jacobians of Griffiths [Gr] do forcomplex varieties.

Our original motivation for studying these p-adic intermediate Jacobians wasto provide, at least in some cases, an algebro-geometric interpretation for theintermediate Jacobians of Griffiths. Recall that for X smooth and projectiveover C, two ways to define the J i(X) of Griffiths are as:

J i(X) = H2i−1(X,C)/(F i + Im(H2i−1(X,Z)))

orJ i(X) = F d−i+1H2d−2i+1(X,C)∗/Im(H2d−2i+1(X,Z)),

∗Partially supported by NSF grant 0070850, SFB 476 (Munster), CNRS France, and sab-batical leave from the University of Southern California∗∗Partially supported by grant BHA2000-0180 from DGI

where F · is the Hodge filtration, and Im means the image of the natural map oncohomology or homology induced by extending coefficients from Z to C. Thetwo definitions are the same by Poincare duality. The first is easy to remember,and the second is convenient for defining the Abel-Jacobi mapping:

CHi(X)hom → J i(X),

by integrating differential forms. Here the group on the left is that of cyclesthat are homologically equivalent to zero.

Now J i(X) is a (compact) complex torus, not necessarily an abelian variety,whose dimension is one half of B2i−1 = dimCH2i−1(X,C). We have that J1(X)is the Picard variety, Pic0(X), and Jd(X) is the Albanese variety, Alb(X). ForX a generic quintic 3-fold, Griffiths (loc. cit.) used the Abel-Jacobi map toJ2(X) to detect codimension two cycles that are homologically equivalent tozero, but not algebraically equivalent to zero. Tori of this type were first studiedby Weil (see [W1], Chapter X; [W2]), who defined J i(X) as

H2i−1(X,R)/Im(H2i−1(X,Z)),

with a different complex structure. In Weil’s theory, J i(X) is always an abelianvariety, but it is not, in general, functorial with respect to holomorphic maps.

The J i(X) of Griffiths have many good properties, but they have several short-comings from an algebro-geometric point of view. First, if X is an algebraicvariety defined over C, it has a model over some field that is finitely generatedover Q, and one can employ arithmetic techniques such as reduction modulo pin the study of algebraic cycles. But since J i(X) is not, in general, an algebraicvariety, one cannot apply such arithmetic techniques to it directly. Second, ifX is considered over an arbitrary field K, then there are Picard and Albanesevarieties defined over K, but there is presently no theory of J i(X) over K; thatis, there are, as yet, no “abstract intermediate Jacobians”. However, one canstill hope to construct such objects over a complete field by using analytic orformal techniques, and we begin to do so here.

Theorem 1 (see §§6,7) Let X be a smooth projective variety of dimension dhaving totally degenerate reduction over a p-adic field K (see §1 below for theprecise definition). Then for each integer i = 1, · · · , d, there exist rigid analytictori J i(X) of dimension equal to the Hodge number hi,i−1 together with an Abel-Jacobi mapping:

CHi(X)hom → J i(X)⊗Z Q.

We have that J1(X) is the Picard variety of X and Jd(X) the Albanese varietyof X.

Theorem 1 is really an application of Theorem 2 below, which is the main tech-nical result of this paper.

Theorem 2 (see §3, Corollary 1 and §5, Proposition 4) For X of dimensiond with totally degenerate reduction, there exist finitely generated, torsion freeabelian groups T i

j (i = −d, · · · , d; j = 0, · · · , d) with an unramified action of theabsolute Galois group G of K, maps N :T i

j → T i+2j−1, and a monodromy filtration

M• on H∗(X,Z`) for all ` such that

T ij (Y )⊗Q`(−j) ∼= GrM

−iHi+2j(X,Q`)

as G-modules. We have GrM−i+1H

i+2j(X,Q`) = 0. Moreover, for ` 6= p, thisfiltration coincides with the usual monodromy filtration on Hi(X,Q`), and themap N tensored with Q` is the `-adic monodromy map on the graded quotients.

The Z-structure on the graded quotients comes from the cohomology of a “Chowcomplex,” which is formed from the Chow groups of intersections of componentsof the special fibre of a strictly semi-stable regular proper model of X over thering of integers of K (we assume that this model exists). See §2 for a definitionof the Chow complexes. The idea of looking at these was suggested by analogieswith mixed Hodge theory, as described in the work of Deligne [De1], Rapoport-Zink [RZ], Steenbrink [St], and in work of Bloch-Gillet-Soule [BGS] and Consani[Con] on the monodromy filtration and Euler factors of L-functions. Such Z-structures have also been studied in some cases by Andre [A].

For ` 6= p, Theorem 2 is essentially known, as it follows without too much diffi-culty from the work of Rapoport-Zink [RZ1], although the result is not explicitlystated there. It is the p-adic cohomology part that requires a careful analysisof several different filtrations and their eventual coincidence. We note thatthere is no “simple” monodromy filtration on p-adic cohomology, in general, aswas first pointed out by Jannsen [Ja1], and we can only expect to get such afiltration on `-adic cohomology for all ` for the kind of varieties we consider here.

Examples of varieties to which the methods of this paper may be applied in-clude abelian varieties A such that the special fibre of the Neron model of Aover the ring of integers R of K is a torus, and products of Mumford curvesor other p-adically uniformizable varieties, such as Drinfeld modular varieties[Mus] and some unitary Shimura varieties (see e.g. [La], [Z], [RZ2] and [Va]).Unfortunately, these assumptions completely exclude the case where X has goodreduction, as in that case H2i−1(X,Q`) is pure of weight 2i − 1 by Deligne’stheorem (Riemann hypothesis [De2]). As pointed out to us by Fontaine, thereshould be many more examples if one considers motives with totally degeneratereduction.

To construct J i(X), we look for a subquotient of the etale cohomology groupH2i−1(X,Z`(i)) that is an extension of Z`’s by Z`(1)’s, as the `-Tate module

of an analytic torus must be. The monodromy filtration gives a natural way offinding such subquotients. Using Theorem 2, we see that the graded quotientsfor the monodromy filtration of this subquotient have Z-structures. The torsionfree quotients of these Z-structures give us our lattices M and M ′, and we thendefine a nondegenerate pairing

M ′ ×M∨ → K∗,

where M∨ = Hom(M,Z).

Then J i(X) is defined to be Hom(M∨, K∗)/M ′. To define the Abel-Jacobimapping, we use the groups H1

g (K,−) of Bloch-Kato ([BK2], §3), and the factthat

J i(X) ∼=∏

H1g (K, M1,`/M−3,`),

where M•,` is the monodromy filtration on H2i−1(X,Z`(i)) modulo torsion (see§§3 and 5), and the product is taken over all prime numbers ` . Then the Abel-Jacobi mapping is obtained from the `-adic Abel-Jacobi mapping ([Bl], [R]). SeeRemark 10 in §7 for why we have to tensor with Q to define the Abel-Jacobimapping. It would be helpful to define the Abel-Jacobi mapping by analyticmeans, so that it could be defined directly over a field such as Cp (the comple-tion of an algebraic closure of K). For example, Besser [Be] has interpreted thep-adic Abel-Jacobi mapping for varieties with good reduction over p-adic fieldsin terms of p-adic integration (see [Co]).

The dimension of our J i(X) is in general strictly less than half of B2i−1 forreasons of Galois equivariance. In fact, it is easy to see from the Hodge-Tatedecomposition [Fa1]:

H2i−1(X,Qp(i))⊗Qp Cp∼=

j+k=2i−1

Hj(XCp , ΩkXCp

)(i− k)

that the greatest dimension that J i(X) can have is hi,i−1, and it follows fromour theory that this is exactly the dimension when X has totally degeneratereduction. This points out a crucial difference from the complex case, wherethe dimension is always equal to half of B2i−1, but simply taking Hi,i−1 andquotienting out by the projection of the image of integral cohomology does not,in general, yield a (compact) complex torus. That there need not be an in-tegral structure on Hi,i−1 ⊕Hi−1,i for complex varieties was first pointed outby Grothendieck in his revision of the generalized Hodge conjecture [Gro]. Ina sequel to this paper, we will compare and contrast the p-adic case with thecomplex case in greater detail, and formulate a “p-adic generalized Hodge-Tateconjecture” for varieties of the type considered in this paper. See §8 for a specialcase of this conjecture and an example.

Note also that our J i(X) depend a priori on the special fibre of a regular propermodel of X over R. We expect that they can be defined directly from the mon-odromy filtration on X, but we have not managed to do this. We can show thatmodulo isogeny, they only depend on X (see the end of §6).

Many varieties with totally degenerate reduction have p-adic uniformizationsand vice-versa, but it is not clear to us that there is a precise connection be-tween these two classes of varieties. Some of what we do here in some sensereplaces arguments using the uniformization in the analysis of the etale coho-mology. However, the uniformization is very useful for some purposes for whichour analysis does not seem well-suited, for example an analytic definition of theAbel-Jacobi mapping along the lines of [MD].

The methods of this paper suggest going back to the complex case and tryingto reinterpret J i(X) in a similar way, using the idea that all complex varietiesshould have (totally degenerate) “semi-stable” reduction in the same way asthe varieties we study here over complete discretely valued fields (see [Ma] forwhat this means in the case of curves). This could help shed more light on theAbel-Jacobi mapping and the Hodge conjecture.

Remark 1 In papers of Ito [It] and de Shalit [dS], the monodromy and weightfiltrations are considered in `-adic and log-crystalline cohomology. The appro-priate versions of the “monodromy-weight conjecture” (see §3 below and thereferences mentioned there) are proved for varieties uniformized by the Drinfeldupper half space. We compare and contrast our results with theirs. Whereaswe assume the Hodge index theorem for the intersections of components of thespecial fibre of a regular proper model of our X, Ito proves this for the varietieshe considers. Thus Ito’s paper provides us with more examples to which themethods of this paper may be applied. de Shalit proves the p-adic version of themonodromy-weight conjecture by doing harmonic analysis and combinatorics onthe Bruhat-Tits building of PGLd+1(K). On the other hand, their results andmethods do not give a monodromy filtration on the p-adic cohomology of X, aswe do here.

This paper is organized as follows: in §1, we discuss the assumptions we need onthe special fibre for our construction to go through. In §2, for a reduced projec-tive variety Y over a field, which satisfies the hypotheses of §1, we introduce ourChow complexes Ci

j(Y ), whose terms are direct sums of Chow groups of inter-sections of components of Y . The homology groups of these complexes will giveus the lattices mentioned above. In §3, we discuss the monodromy filtration onZ`-etale cohomology for ` 6= p, and in §4, we discuss the monodromy filtrationon the log-crystalline cohomology of the special fibre of a regular proper modelof X over A, as defined by Hyodo-Kato [HK] and Mokrane [Mok]. In §5, weuse work of Hyodo [H] and Tsuji [Tsu] to lift this filtration to a monodromyfiltration on the p-adic etale cohomology of X. This allows us to compare the

graded quotients for the monodromy filtration for the various cohomology the-ories with the cohomology of the Chow complexes mentioned above, tensoredwith the appropriate coefficient ring (a Tate twist of Z` for any prime number `).We then define the intermediate Jacobians in §6 and the Abel-Jacobi mappingin §7. Finally, in §8 we consider an example of the product of three Tate ellipticcurves, and we determine in some cases the dimension of the image in J2(X)of the Abel-Jacobi mapping restricted to cycles algebraically equivalent to zero.This dimension conjecturally depends on the rank of the space of multiplicativerelations between the Tate parameters of the curves. Our analysis shows thatJ2(X) is never equal to this image. This agrees with the complex case, sincethere we have that this image is contained in H1,2/[H1,2 ∩H3(X,Z)], and is allof this last group iff the three elliptic curves are isogenous and have complexmultiplication (see [Li], p. 1197). This cannot happen for Tate elliptic curves,since CM curves have everywhere potentially good reduction.

The authors would like to thank, respectively, the Universitat Autonoma deBarcelona and the University of Southern California for their hospitality. Thispaper was completed while the first author enjoyed the hospitality of Universitede Paris-Sud. We also thank J.-L. Colliot-Thelene, L. Clozel, L. Fargues, J.-M. Fontaine, O. Gabber, L. Illusie, K. Kunnemann, G. Laumon, A. Quiros, T.Tsuji and A. Werner for helpful comments and information.

1 Notation and Preliminaries

Let K be a finite extension of Qp, with valuation ring R and residue field F .We denote by F a separable closure of F , by W the ring of Witt vectors of F ,by W (F ) the ring of Witt vectors of F , and by Wn(F ) the ring of Witt vectorsof length n.

We denote by BDR the ring of p-adic periods of Fontaine. Fix a uniformizer πof K and the extension of the p-adic logarithm to K

∗with log(π) = 0. These

choices determine an embedding of Bst, the ring of periods of semi-stable vari-eties, in BDR . See ([I1], §1.2) for more details.

By an analytic torus over K, we mean the (rigid analytic) quotient of an alge-braic torus over K by a lattice.

If M is an abelian group, we denote by M/tors the torsion free quotient of M ,that is M/Mtors, where Mtors denotes the subgroup of torsion elements in M .We say that a map between abelian groups is an isomorphism modulo torsionif it induces an isomorphism between the free quotients. If M and M ′ are tor-sion free abelian groups of finite rank, we say that a map φ: M → M ′ is anisogeny if it is injective with finite cokernel. In this case, there exists a uniquemap ψ: M ′ → M , the dual isogeny, such that φ ψ = [e] and ψ φ = [e], wheree is the exponent of the cokernel of φ and [e] denotes the map multiplication by e.

We will use subindices for increasing filtrations and superindices for decreasingfiltrations. If M• is an increasing filtration on H (respectively F • a decreasingfiltration on H), we will denote by GrM

i (H) the quotient Mi(H)/Mi−1(H) (re-spectively, Gri

F (H) the quotient F i(H)/F i+1(H)). Observe that, if M• is anincreasing filtration, then F i(H) := M−i(H) is a decreasing filtration.

Let X be a smooth projective geometrically connected variety over K. We as-sume that X has a regular proper model X over R which is strictly semi-stable,which means that the following conditions hold:

(*) Let Y be the special fibre of X . Then Y is reduced; write

Y =n⋃

with each Yi irreducible. For each nonempty subset I = i1, . . . , ik of 1, . . . , n,we set

YI = Yi1 ∩ ... ∩ Yik,

scheme theoretically. Then YI is smooth and reduced over F of pure codimen-sion |I| in X if I 6= ∅. See ([dJ] 2.16 and [Ku], §1.9, 1.10) for a clear summaryof these conditions, as well as comparison with other notions of semi-stability.

Let F be an algebraic closure of F . We set Y I = YI ×F F . Note that theseneed not be connected.

Definition 1 We say that Y is totally degenerate over F if the following con-ditions are satisfied for each YI :

a) For every i = 0, ..., d, the Chow groups CHi(Y I) are finitely generatedabelian groups.

The groups CHiQ(Y I) satisfy the Hodge index theorem: let

ξ : CHi(Y I) → CHi+1(Y I)

be the map given by intersecting with the class of a hyperplane section.Suppose we have x ∈ CHi

Q(Y I) such that ξd−i(x) = 0. Then the form(−1)itr(xξd−2i(x)) is positive semi-definite.

b) For every prime number ` different from p, the etale cohomology groupsH2i+1(Y I ,Z`) are torsion, and the cycle map induces an isomorphism

CHi(Y I)/tors⊗ Z`∼= H2i(Y I ,Z`(i))/tors.

Note that this is compatible with the action of the Galois group.

c) If p > 0, let W be the ring of Witt vectors of F . Denote by H∗(Y I/W ) thecrystalline cohomology groups of Y I . Then H2i+1

crys (Y I/W ) are torsion, andCHi(Y I)⊗W (−i)/tors ∼= H2i

crys(Y I/W )/tors via the cycle map. HereW (−i) is W with the action of Frobenius multiplied by pi.

d) Y is ordinary, in that Hr(Y , Bωs) = 0 for all i, r and s. Here Bω is thesubcomplex of exact forms in the logarithmic de Rham complex on Y (seee.g. [I2], Definition 1.4). By [H] this is implied by the Y I being ordinaryin the usual sense, in that Hr(Y I , dΩs) = 0 for all I, r and s. In our case,if the groups Hr(Y , Wωs) are finitely generated and torsion-free and theslopes of Frobenius acting on Hi(Y /W ) are integers, then Y is ordinary.For more on the condition of ordinary, see [I2], Appendice and [BK1],Proposition 7.3)

If the natural maps

CHi(YI) → CHi(Y I)

are isomorphisms modulo torsion for all I, we shall say that Y is split totallydegenerate. After a finite extension, any totally degenerate variety becomes splittotally degenerate.

Remark 2 (i) For totally degenerate Y , the Chow groups CHi(Y I)Q :=CHi(Y I)⊗Q satisfy the hard Lefschetz Theorem. That is, if L is the classof a hyperplane section in CH1

Q(Y I) and ξ : CHiQ(Y I) → CHi+1

Q (Y I) de-notes the Lefschetz operator associated to L, then ξd−2i : CHi

Q(Y I) →CHd−i

Q (Y I) is an isomorphism for all i. This follows from the Hard Lef-schetz Theorem in `-adic etale cohomology, as proved by Deligne [De2]and the bijectivity of the cycle map modulo torsion.

(ii) Note that condition c) implies that for YI , we have

CHi(YI)⊗Z L → H2i(Y/W (F ))⊗W (F ) L

is an isomorphism, where L is the fraction field of W (F ). To see this,let G be the absolute Galois group of F . Then since L is a Q-algebra, wehave

CHi(YI)⊗Z L ∼= CHi(Y I)G ⊗Z L,

and H2i(YI/W (F )) ⊗W (F ) L ∼= [H2i(Y I/W ) ⊗W M ]G, where M is thefraction field of W . Combining these with c), we get the desired isomor-phism.

We will say that X has totally degenerate reduction if it has a regular propermodel X over R which is strictly semi-stable and whose reduction Y is totally de-generate over F . We find the name totally degenerate reduction a bit pejorative,because one of the main themes of this paper is that “bad reduction is good.”However, since this terminology is well-established, at least for dimension one,we decided to continue with it.

Remark 3 We expect that the condition that the cycle map be an isomorphismmodulo torsion can be weakened to say that this map is an isomorphism whentensored with Q` for all `, and the same with the crystalline cycle map. The endresult of this paper would be weakened to only give an intermediate Jacobian upto isogeny, using the following lemma.

Lemma 1 Let Y be a smooth, projective, irreducible variety over a separablyclosed field. Assume that the Chow groups are finitely generated abelian groupsand that the cycle map is an isomorphism when tensored with Q` for all `. Thenfor almost all `, the cycle map:

c` : CHi(Y )⊗Z Z` → H2i(Y,Z`(i))

is an isomorphism.

Proof: From the conditions, we know that the kernel of the cycle map is atorsion group. Since the Chow groups are assumed to be finitely generatedabelian groups, the `-torsion is zero for almost all `, and hence the map isinjective for almost all such `. As for the cokernel, by a theorem of Gabber [G],the etale cohomology groups Hj(Y,Z`) are torsion free for almost all `, and thusthe map is surjective on torsion for almost all `. Consider the pairing of finitelygenerated torsion free abelian groups:

CHi(Y )/tors× CHd−i(Y )/tors → Z

and the cohomological pairing of finitely generated Z`-modules given by cup-product:

H2i(Y,Z`(i))/tors×H2d−2i(Y,Z`(d− i))/tors → Z`.

By our assumptions on Y and Poincare duality, the first pairing is perfect whentensored with Q, and so its determinant is a non-zero integer, say d. By Poincareduality, the second pairing is perfect. Thus for primes ` not dividing d andsuch that the cohomology has no `-torsion, the cycle map is surjective. Thiscompletes the proof of the lemma.

Example 1 i) Assume that Y I are smooth projective toric varieties. ThenY is totally degenerate. Conditions a), b) and c) follow from ([FMSS],Corollary to Theorem 2) and ([Ku2], Proof of Theorem 6.13; see also [D],Theorems 10.8 and 12.11, or [F], Section 5.2, Theorem on p. 102 andthe argument on p. 103). Note that the odd integral cohomology of a toric

variety is trivial, and therefore the even integral cohomology is torsion free.Hence, by ([I2], Proposition 1.5(iii) and [I4]), we see that a toric varietyis ordinary.

ii) Let A be an abelian variety over K. Let A be the Neron model of Aover the ring of integers of K. Assume that the connected component ofidentity of the special fibre of A over F is a split torus. We will say that Ahas completely split toric reduction. By a theorem of Kunnemann ([Ku1],Theorem 4.6(iii)), after passing to a finite extension of K, if necessary, Ahas a regular proper semi-stable model A whose special fibre is a reduceddivisor with smooth components, which implies that it has strictly normalcrossings. In this case, the components of the special fibre are smoothprojective toric varieties. The intersection of any number of componentsis also a smooth projective toric variety. Thus A has totally degeneratereduction.

iii) Let ΩdK be the Drinfeld upper half space obtained by removing the K-

rational hyperplanes from PdK . This is a rigid analytic space, and its

quotient by a cocompact torsion free discrete subgroup Γ ⊂ PGLd+1(K)has the structure of a smooth projective variety (see [It] for a summary ofthe results in this direction, and [Mus], Theorem 4.1 and [Ku], Theorem2.2.5 for the original proofs). Ito proves the Hodge index theorem for thegroups CH∗(YI)Q (see [It], Conjecture 2.6 and Proposition 3.4). He alsoproves some of the results in sections §§3-5 of this paper. Note that in thisexample, the components of the special fibre and their intersections aresuccessive blow-ups of projective spaces along closed linear subschemes,and these are ordinary. Therefore, the results of this paper apply to theseDrinfeld modular varieties.

iv) There are Shimura varieties that can be p-adically uniformized by analyticspaces other than Drinfeld upper half planes (see e.g. [Rap], [RZ2] and[Va]). As pointed out in [La] and the introduction to [RZ2], the bad reduc-tion of a Shimura variety is only rarely totally degenerate. In any case,when they can be uniformized, Shimura varieties should have totally de-generate reduction. Unfortunately, it seems difficult to construct the goodmodels we need for this paper (but see the next remark).

Remark 4 Using the results of de Jong [dJ], it should be possible to prove thatfor any variety X which “looks like” it should have totally degenerate reduction,there is an alteration Y → X such that Y has strict semi-stable reduction anda motive for numerical equivalence on Y whose cohomology is isomorphic tothat of X. For example, in the theory of uniformization of Shimura varieties(see e.g. [RZ2], [Va]), one constructs models of Shimura varieties over p-adicinteger rings which are often not even regular, and it is not known at this timehow to provide models such as that required in this paper. But the special fibresof such models are often given by possibly singular toric varieties, and hence thegeneric fibre should have totally degenerate reduction.

2 Definition of the Chow complexes C ij(Y )

Let F be any field and Y a reduced projective variety of pure dimension d overF . We define Chow complexes Ci

j(Y ) for each j = 0, ..., d. The definition is in-spired by the work of Steenbrink [St] and Guillen/Navarro-Aznar[GN] on mixedHodge theory, Rapoport and Zink on the monodromy filtration [RZ], and Bloch,Gillet and Soule [BGS]. These will give us the Z-structures we require later.

As in §1, we write Y =⋃n

i=1 Yi with each Yi irreducible, and for a subsetI = i1, ..., im of 1, ..., n with i1 < ... < im, denote

YI = Yi1 ∩ ... ∩ Yim,

which by assumption is a reduced closed subscheme.We then write Y (m) :=

⊔YI , where the disjoint union is taken over all subsets

I of 1, ..., n with #I = m and YI 6= ∅.Now for each pair of integers (i, j) we define

Cij(Y ) =

k≥max0,iCHi+j−k(Y (2k−i+1)).

Note that there are only a finite number of summands here, because k runs frommaxi, 0 to i + j. Note also that Ci

j can be non-zero only if i = −d, ..., d andj = −i, ..., d− i.

Example 2 Ci0 = CH0(Y (i+1)), Ci

d = CHd+i(Y (1−i)), Ci−i = CH0(Y (1−i))

and Cid−i = CHd−i(Y (i+1)).

Denote by ρr : Yi1...im+1 → Yi1...ir...im+1

the inclusion maps. Here, as usual,Y

i1...ir...im+1denotes the intersection of the subvarieties Yij for j = 1, ..., r −

1, r + 1, ...m + 1 (delete Yir ).Now, for all i and m = 1, . . . , n we have maps

θi,m : CHi(Y (m)) → CHi(Y (m+1))

defined by θi,m =∑m+1

r=1 (−1)r−1ρ∗r , where ρ∗r is the restriction map. Let ρr∗ bethe Gysin map

CHi(Yi1...im+1) → CHi+1(Yi1...ir...im+1

and letδi,m : CHi(Y (m+1)) → CHi+1(Y (m))

be the map defined by δi,m =∑m+1

r=1 (−1)rρr∗. Let

d′ =⊕

k≥max0,iδi+j−k,2k−i

andd′′ =

k≥max0,iθi+j−k,2k−i+1.

Then define maps dij : Ci

j → Ci+1j by di

j = d′ + d′′.

Lemma 2 We have that dijd

i−1j = 0, and so we get a complex for each j =

0, ..., dim(Y ).

Proof The statement of the lemma is easily deduced from the following facts:For every i and m, δi+1,m−1δi,m = 0, θi,m−1θi,m = 0 and

θi−1,mδi,m + δi,m+1θi,m+1 = 0.

The first two equalities are easy and the last one is proved in [BGS], lemma 1.

Define T ij = Kerdi

j/Imdi−1j , the homology in degree i of the complex Ci

j definedabove.

We define pairings:

Cij × C−i

d−j → C0d

deg−→ Z

as the intersection pairing of each summand in Cij with the appropriate sum-

mand in C−id−j . Summands not of complementary dimension pair to zero. By the

projection formula, these pairings are compatible with the differentials, henceinduce pairings

T ij × T−i

d−j → Z,

which are nondegenerate on the torsion free quotients.

The monodromy operator N : Cij → Ci+2

j−1 is defined as the identity map on thesummands in common, and the zero map on different summands. N commuteswith the differentials, and so induces an operator on the T i

j , which we alsodenote by N . We have N i is the identity on Ci

j for i ≥ 0. This is proved byshowing that the summands in a given Ci

j persist throughout, and the othersoccurring in subsequent groups eventually disappear. The following result isa direct consequence of a result of Guillen and Navarro ([GN] Prop. 2.9 andTheoreme 5.2 or [BGS], Lemma 1.5 and Theorem 2), using the fact that theChow groups of the components of our Y satisfy the hard Lefschetz theoremand the Hodge index theorem. It is a crucial result for this paper.

Proposition 1 Suppose that Y satisfies the assumptions of §1. Then the mon-odromy operator N i induces an isogeny:

N i : T−ij+i → T i

for all i ≥ 0 and j.

3 The monodromy filtration in the case ` 6= p

Let K be a complete discretely valued field of characteristic zero with valuationring R and finite residue field F of characteristic p > 0. Let X be a smoothprojective variety over K. We assume that X satisfies the assumptions of §1.

In this section, we study the monodromy filtration on the etale cohomologyH∗(X,Q`) using the techniques developed in the last section. The goal is toestablish an isomorphism between the graded quotients for the monodromy fil-tration and an appropriate twist of T i

j ⊗Z Q` for any prime number l 6= p. Inthe next sections, we will consider the case ` = p.

Recall Grothendieck’s monodromy theorem in this situation (see [SGA VII, I]or [ST], Appendix): the restriction to the inertia group I of the `-adic repre-sentation associated to the etale cohomology Hi(X,Q`) of a smooth projectivevariety X with semistable reduction over a complete discretely valued field Kis unipotent.Thus we have the monodromy operator N : Hi(X,Q`)(1) → Hi(X,Q`), char-acterized by the fact that the restriction of the representation to the maximalpro-` quotient of the inertia group I is defined by the composition exp N t`,where t` : I → Z`(1) is the natural map. Using this monodromy operator,we can construct ([De2], 1.6.1, 1.6.13) the monodromy filtration, which is theunique increasing filtration

0 = M−i−1 ⊆ M−i ⊆ ... ⊆ Mi−1 ⊆ Mi = Hi(X,Q`),

such thatNMj(1) ⊆ Mj−2

andN j induces an isomorphism GrM

j (j) ∼= GrM−j .

The construction of the monodromy filtration in ibid. shows that I acts triviallyon the graded quotients GrM

j , and M I ⊆ M0. If the weights of Frobenius areintegers, then we also have the weight filtration ([De2], Proposition-Definition1.7.5): the unique increasing filtration on Hi(X,Q`) whose graded quotients arepure Gal(F/F )-modules. Recall also the monodromy-weight conjecture ([De1],Principe 8.1; [RZ], Introduction), which can be stated in two equivalent forms:

1. GrMj is a pure Gal(F/F )-module of weight i + j.

2. the weight filtration has the properties mentioned above that characterizethe monodromy filtration.

This conjecture in its first form was proved by Deligne in the equi-characteristiccase ([De2], Theoreme 1.8.4), and in its second form by Rapoport and Zinkin the case where X is of dimension 2 and has a model X with special fibreY =

∑niYi (as cycle), with ni prime to l for each i (see [RZ], Satz 2.13). We

will use the second form below.

Observe that when ` = p, there is in general no “simple” monodromy filtration,as is pointed out in ([Ja1], page 345). That is the reason for treating this caseseparately in the next two sections.

Consider the weight spectral sequence of Rapoport-Zink ([RZ], Satz 2.10; seealso [Ja2]), in its two forms:

Ei,j1 = Hi+j(Y , GrW

−i(RΨ(Z`)) =⇒ Hi+j(X,Z`).

Ei,j1 =

k≥max0,iHj+2i−2k(Y

(2k+1−i),Z`(i− k)) =⇒ Hi+j(X,Z`).

We denote by Ei,jr /tors the quotient of Ei,j

r by its torsion subgroup.We write the two forms of this spectral sequence because the first one has simpleindexing, and the groups in the second are very similar to the ones consideredin the last section, as we now see:

Proposition 2 Suppose X satisfies the conditions of §1. Then we have:

Ei,2j1 /tors ∼= Ci

j(Y )/tors⊗ Z`(−j)

Ei,2j+11 /tors = 0.

These are compatible with the differentials on both sides, the differentials on theleft hand side being those defined in section 2, tensored with Z`.

Proof: The two equalities are clear from our assumptions on Y . The compati-bility between differentials is deduced directly from the results of Rapoport andZink ([RZ], §2, especially Satz 2.10; see also [Ja2], §3).

Let us denote by W• the filtration on Hn(X,Z`) induced by the weight spectralsequence, and by M• the filtration defined by Mi(Hn) := Wi+n(Hn). Then thefiltration W• gives us the weight filtration on Hn(X,Q`), so the monodromyconjecture says that the filtration M• is the monodromy filtration.

Corollary 1 With assumptions as above, the spectral sequence of Rapoport-Zink degenerates at E2 modulo torsion. Thus

T ij (Y )/tors⊗ Z`(−j) ∼= GrM

−iHi+2j(X,Z`)/tors.

Proof: Since the E2/tors-terms are powers of Z`(r) for appropriate integers r,we have Hom(Er,s

2 , Er+t,s−t+12 ) = 0 modulo torsion for all r and s and t ≥ 2,

because these are of different weights and the residue field is finite. HenceEr,s

2 /tors = Er,st /tors for all r, s and t ≥ 2, and by an easy induction the

differentials are zero on the Er,st -terms modulo torsion for all r, s and t ≥ 2.

The second part of the assertion is clear because

GrM−iH

i+2j(X,Z`) = GrW2j Hi+2j(X,Z`) ∼= Ei,2j

Corollary 2 With assumptions as above, the filtration M• is the monodromyfiltration and the induced maps

GrM−iH

i+2j(X,Q`)N→ GrM

−i−2Hi+2j(X,Q`)(1)

coincide under the isomorphism above with the monodromy operator defined insection §2.

Proof: Using a weight argument as in the proof of the last corollary, it is easy tosee that NMi(1) ⊆ Mi−2. Thus one only has to show that the i-th power of themonodromy induces an isomorphism between GrM

i Hn ∼= GrM−iH

n(−i). Usingthe corollary above, this is reduced to showing that N i induces an isomorphism

T−ij+i ⊗Q`(−j − i) → T i

j ⊗Q`(−j − i).

In ([RZ], §2), it is shown that the map

T ij ⊗Q`(−j)∼=GrM

−iHi+2j(X,Q`)

N→GrM−i−2H

i+2j(X,Q`)(1)∼=T i+2j−1 ⊗Q`(−j)

is the same as the map that we constructed in section 2, and hence we candeduce the result from Proposition 1.

Remark 5 Observe that, for almost all `, we have an isomorphism

T ij (Y )⊗ Z`(−j) ∼= GrM

−iHi+2j(X,Z`)

and hence the graded quotients of the monodromy filtration are torsion free foralmost all `.This is because by lemma 1 we have

Ei,2j1

∼= Cij(Y )⊗ Z`(−j)

Ei,2j+11 = 0

for almost all `, hence, because of the compatibility of the differentials,

Ei,2j2

∼= T ij (Y )⊗ Z`(−j)

Ei,2j+12 = 0.

so the E2-terms are torsion free for almost all ` because the groups T ij are finitely

generated abelian groups. For such `, the proof of Corollary 1 shows that theRapoport-Zink spectral sequence degenerates at E2.

Remark 6 For any variety X of finite type over a separably closed field k andprime number ` 6= char. k, there is a weight filtration defined on the `-adiccohomology group Hi(X,Q`). This is similar to the weight filtration in thetheory of mixed Hodge structures on complex varieties. See e.g. [Ja2] for a goodaccount of these filtrations and the various accompanying spectral sequences. In

these cases, the weight filtration is of length at most i. In the case of a varietyover a complete discretely valued field as we have studied in this section, theweight filtration is a priori of length 2i. The formalism is somewhat similar in allthree cases, so what accounts for the difference in the lengths of the filtrations?In the first two cases, X is not necessarily smooth or projective, and the field kcan be assumed to be a separable closure of a field that is finitely generated overthe prime subfield. In the discretely valued field case, the variety is smooth andprojective over the field, but need not have good reduction. In this case, it isultimately the presence of the monodromy operator that accounts for the longerfiltration.

4 The monodromy filtration on log-crystallinecohomology

Let F be a perfect field of characteristic p > 0, let W be the ring of Witt vec-tors with coefficients in F , and K0 be the field of fractions of W . Let L be alogarithmic structure on F , in the sense of Fontaine, Illusie and Kato (see [Ka]).Let (Y, M) be a smooth (F,L)-log-scheme, i.e. Y is a scheme over F and M isa fine log-structure with a smooth map of log schemes f : (Y, M) → (F,L). Weassume also that (Y, M) is semistable in the sense of [Mok], Definition 2.4.1.Hyodo and Kato [HK] have defined under these conditions the log-crystallinecohomology of Y , which we will denote by Hi(Y ×/W×). It is a W -module offinite type, with a Frobenius operator Φ that is bijective after tensoring by K0

and semilinear with respect to the Frobenius in W , and a monodromy operatorN that is nilpotent and satisfies NΦ = pΦN . This operator N then determinesa filtration on Hi(Y ×/W×) that we call the monodromy filtration. Recall thatthe log-crystalline cohomology can be computed as the inverse limit of the hy-percohomology of the de Rham-Witt complex Wnω•Y of level n.

Suppose now that Y is a projective variety. Then we have the spectral sequenceof Mokrane ([Mok], 3.23):

Ei,j1 =

k≥max0,iHj+2i−2k(Y (2k+1−i)/W )(i− k) =⇒ Hi+j(Y ×/W×),

where the twist by i− k means to multiply the Frobenius operator by pk−i. Wedenote by Ei,j

1 /tors the free quotient of the terms of this spectral sequence. Theassociated filtration on the abutment is called the weight filtration.

Proposition 3 With notation and assumptions as above, suppose further thatY satisfies the assumption c) of §1. Then if F = F , we have

Ei,2j1 /tors ∼= Ci

j(Y )/tors⊗W (−j)

Ei,2j+11 /tors = 0.

These are compatible with the differentials on both sides, the differentials on theleft hand side being those defined in section 1. In general, we have

Ei,2j1,K0

∼= Cij(Y )Gal(K/K0) ⊗K0(−j)

Ei,2j+11,K0

Proof: If F = F , the two equalities are clear by condition c) of §1, and thecompatibility between the differentials is clear by using the results of Mokrane([Mok], §3), who proves that the spectral sequence degenerates at E2 modulotorsion for any Y ([Mok], 3.32(2)). For the general case, apply Remark 1, (ii)of §1.

Corollary 3 With assumptions as in §1, the spectral sequence of Mokrane de-generates at E2, the weight filtration induces the monodromy filtration, and

T ij (Y )/tors⊗W (−j) ∼= GrM

−iHi+2j(Y

×/W×)/tors.

Moreover, the monodromy map on the graded quotients agrees with the mapdefined in §2. A similar statement holds for Y , if we tensor with K0.

Proof The degeneration of the spectral sequence is deduced using a slope ar-gument as in the proof of Corollary 1. The assertion about the coincidence ofthe weight filtration and the monodromy filtration is deduced from Proposition5 by using ([Mok], Proposition 3.18) as in the proof of Corollary 2.

Now, also assume that Y is ordinary (see condition d) of §1). Note that thiscondition is stable under base change to F . Then there exists a canonicaldecomposition as W -modules with a Frobenius (see [I3], Proposition 1.5. (b))

Hn(Y ×/W×) ∼=⊕

Hj(Y, Wωi)(−i).

We will identify the two sides of the isomorphism.The existence of this canonical decomposition allows us to define the filtrationby increasing slopes Uj as

Uj(Hn(Y ×/W×)) := ⊕r≤jHn−r(Y,Wωr)(−r),

which has the property that

GrUj Hn(Y ×/W×) ∼= Hn−j(Y,Wωj)(−j).

This filtration is opposed to the usual slope filtration, which is given by decreas-ing slopes. We can do the same thing for Y .

Corollary 4 With assumptions as in §1, we have that

M2j−n(Hn(Y×

/W×)/tors) ⊆ M2j−n+1(Hn(Y×

/W×)/tors)

⊆ Uj(Hn(Y×

/W×)/tors)

with finite index. If we tensor with K0 we have equalities, and this holds alsowhen Y is replaced by Y .

As a consequence, we have canonical isogenies

T i−jj (Y )/tors⊗W → Hi(Y , Wωj

Y)/tors.

Proof. Observe first that the filtration by increasing slopes has a splitting andhence the torsion free part of GrU

j Hi+2j(Y×

/W×) is equal to

(Hi+2j(Y

×/W×)/tors

Hence, to show the first assertion, we need only prove that the torsion-free partsof the E2-terms of the weight spectral sequence are of pure slope. Observe thatfor any i and any k, the cohomology groups

H2k(Y(i)

/W )(k)/tors

are of pure slope 0 because they are generated by algebraic cycles by our con-ditions in §1, c). So, the Ei,2j

1 -terms are of pure slope j modulo torsion, as areany of their subquotients, such as the Ei,2j

2 terms.From these facts, we have that

−iHi+2j(Y

×/W×)

)/tors ∼= T i

j (Y )/tors⊗W (−j)

is canonically isogenous to

GrUj Hi+2j(Y

×/W×)/tors ∼= Hi+j(Y , Wωj)/tors(−j).

Hence we have a canonical isogeny

T ij (Y )/tors⊗W → Hi+j(Y ,Wωj)/tors.

Now let’s consider the logarithmic Hodge-Witt pro-sheaves Wωj

Y ,logon Y , de-

fined as the kernel of 1− F on the pro-sheaves Wωj

Yfor the etale site.

The following result is clear by using Proposition 2.3 in [I3], which says that

Hi(Y , Wωj

Y ,log)⊗W (F ) ∼= Hi(Y ,Wωj

Corollary 5 With assumptions as in §1, we have a canonical isogeny

T i−jj (Y )/tors⊗Z Zp → Hi(Y , Wωj

Y ,log)/tors.

5 The monodromy filtration on p-adic cohomol-ogy

We return to the assumptions of §3. Thus K is a complete discretely valuedfield of characteristic zero with valuation ring R and finite residue field F ofcharacteristic p > 0. Let X be a smooth projective variety over K with a reg-ular proper model X over R whose special fibre satisfies the hypotheses of §1.The goal of this section is to show that the filtration on the p-adic cohomologyof X induced by the vanishing cycles spectral sequence has all of the propertiesthat a monodromy filtration should have.

First of all, recall that a p-adic representation V of GK is ordinary if there existsa filtration (FiliV )i∈Z of V , stable by the action of GK , such that the inertiasubgroup IK acts on FiliV/Fili+1V via χi, where χ denotes the cyclotomiccharacter. It is easy to see that this filtration is unique. The next theorem isthe main result in [H] (see also [I3], Theoreme 2.5 and Corollaire 2.7 ).

Theorem 3 (Hyodo) Let X be a proper scheme over R with semistable ordinaryreduction Y and generic fiber X. Then the p-adic representation Hm(X,Qp) isordinary. Moreover, the vanishing cycles spectral sequence

Ei,j2 = Hi(Y , RjΨ(Zp)) =⇒ Hi+j(X,Zp)

degenerates at E2 modulo torsion, and if F • denotes the corresponding filtrationon Hi+j(X,Zp), one has a canonical isomorphism as GK-modules

GriF Hi+j(X,Zp)/tors ∼= Hi(Y , Wωj

Y ,log)/tors(−j)

Suppose now that X has special fibre that satisfies the assumptions of §1. Then,as a consequence of Corollary 5 in §4, we get the following result.

Corollary 6 Let X be a proper scheme over R with semistable reduction Yand generic fiber X. Assume that Y satisfies the hypotheses of §1. Then, if F •

denotes the filtration obtained from the vanishing cycles spectral sequence, onehas canonical isogenies as GK-modules

T ij (Y )/tors⊗ Zp(−j) → Gri+j

F Hi+2j(X,Zp)/tors

We have then that the filtration F • has the same type of graded quotients asthe monodromy filtration in `-adic cohomology (modulo isogeny). In fact wewill prove that this filtration can be obtained from the monodromy filtration onthe log-crystalline cohomology by taking the functor Vst.

Recall that, a filtered (Φ, N)-module H is a K0-module H with a Frobenius Φand a monodromy map N verifying that NΦ = pΦN and an increasing filtrationFil• in H ⊗K0 K. Given a filtered (Φ, N)-module H, we define

Vst(H) := (Bst ⊗K0 H)N=0,Φ=1 ∩ Fil0(Bst ⊗K (H ⊗K0 K)).

It is a p-adic representation of GK , that is a Qp-vector space with a continuousaction of GK . Recall from §1 that we fixed a choice of uniformizer π of K, whichdetermined the embedding of Bst in BDR, and so this p-adic representation de-pends on that choice.

Consider the log-crystalline cohomology Hn(Y ×/W×) we used in the last sec-tion, where X has ordinary semistable reduction. Hyodo and Kato proved in[HK], §5, that we have an isomorphism

ρπ:Hn(Y ×/W×)⊗W K ∼= HndR(X/K)

depending on the uniformizer π that we have chosen. Using this isomorphism,we get a structure of filtered (Φ, N)-module on the log-crystalline cohomologyof the log-scheme Y .

Now assume only that X has ordinary reduction, and consider the filtrationby increasing slopes U• of the last section. Then the induced (Φ, N)-modulestructure on the Ui via the Hyodo-Kato isomorphism ρπ gives a filtration byfiltered (Φ, N)-modules. Applying the functor Vst, and using the main result ofTsuji ([Ts], p. 235) on the conjecture Cst

Hn(X,Qp) ∼= Vst(Hn(Y ×/W×)),

we get a filtration on the p-adic cohomology Hn(X,Qp).

Theorem 4 Assume only that X has ordinary reduction. Then the filtrationsVst(Ui) and Fn−i on Hn(X,Qp) are the same.

Proof. First of all, note that the filtered (Φ, N)-module Hn(Y ×/W×) is ordi-nary in the sense of ([P], p. 186). This is because Hn(Y ×/W×) is isomorphic,by Tsuji’s theorem, to Dst(Hn(X,Qp)), and the functor Dst takes ordinaryp-adic representations to ordinary filtered (Φ, N)-modules (see [P], Theoreme1.5).Now observe that the graded quotients D with respect to the U• filtration havethe property that there exists i such that Fili(DK) = DK and Fili+1(DK) = 0,p−iΦ acts as an automorphism on a lattice in D and N = 0. This is due tothe fact that the U• filtration is opposed to the Hodge filtration, because ourfiltered (Φ, N)-module is ordinary (see [P], middle of p. 187 and [I3] 2.6 c),middle of p. 217). But by ([P], Lemme 2.3), the inertia group acts on Vst(D) asχi. Moreover, the functor Vst is exact for the ordinary (Φ, N)-filtered modules(see [P], 2.7). So the Vst(U•) filtration has the same graded quotients as thefiltration Fn−•, and hence these filtrations must be the same.

We summarize our discussion in the following result.

Proposition 4 For X with totally degenerate reduction, consider the filtrationM• on Hn = Hn(X,Qp) defined as Mi(Hn) := F j(Hn), where j = n−i

j = n−i+12 depending on the parity of n− i. Then we have

T ij (Y )⊗Qp(−j) ∼= GrM

−iHi+2j(X,Qp),

and the maps on these graded quotients induced by N on the T ij coincide after

applying the functor Vst with the maps N defined in the section §2. Moreover,they verify that N i induces an isomorphism GrM

i∼= GrM

−i(−i). The other gradedquotients are zero.

6 p-adic Intermediate Jacobians

Let K be a complete discretely valued field of characteristic zero with finiteresidue field F of characteristic p > 0. Denote by G the absolute Galois groupof K. Let X be a smooth projective variety over K of dimension d with a reg-ular proper model X over R whose special fibre satisfies the assumptions of §1.

The goal of this section is to define, for every j = 1, . . . d, rigid analytic toriJj(X) associated to X . These are constructed from the monodromy filtrationintroduced in §§3-5.

To shorten notation, we will denote T ij (Y ) by T i

j . Suppose first of all that theabsolute Galois group of F acts trivially on T−1

j /tors and on T 1j−1/tors for any

j; this will happen after a finite unramified extension of K. We define a pairingfor every j:

−,−l : T−1j /tors× T 1∨

j−1/tors → K∗(`) := lim←−n

K∗/K∗`n

where K∗(`) is the `-completion of the multiplicative group of K. This is done byusing properties of the monodromy filtration Mi(H) on H := H2j−1(X,Z`(j)).The idea is to look at the quotient

M := (M1(H)/M−3(H))/tors,

which is a subquotient of H2j−1(X,Z`(j)), and it is, modulo torsion, an exten-sion of Z`’s by Z`(1)’s. But due to the possible torsion we have to do a slightmodification of M .

Suppose first that ` 6= p. We have then a natural epimorphism from M toW1 := GrM

1 H/tors. Define then W−1 as the kernel of this map. Observe thatW−1 is isogenous to GrM

−1H/tors, and the natural isogeny ψ: GrM−1H/tors →

W−1 has cokernel isomorphic to the cokernel of M1(H)/M−3(H)→GrM1 H =

M1(H)/M−1(H) and with exponent e`. Considering the dual isogeny ϕ fromW−1 to GrM

−1H/tors such that ψ ϕ = [e`] (see §1), we can push out theextension

0 → W−1 → M → W1 → 0,

with respect to this isogeny, and, recalling that we have isomorphisms

GrM−1H/tors ∼= T 1

j−1/tors⊗ Z`(1) and GrM1 H/tors ∼= T−1

j /tors⊗ Z`,

we get an extension

0 → T 1j−1/tors⊗ Z`(1) → E′

` → T−1j /tors⊗ Z` → 0.

For ` = p we can do a similar argument, but because we have only isogenies

ψ1:T 1j−1/tors⊗ Zp(1) → GrM

−1H/tors

andψ2: T−1

j /tors⊗ Zp → GrM1 H/tors,

we have to push out with respect to ψ1 and pull back with respect to the dualisogeny associated to ψ2 to get an extension E′

p as above.Denote by e :=

∏` e`; it is well defined because e` = 1 for almost all ` since the

graded quotients for the monodromy filtration are torsion-free for almost all `(see remark 5). We use these numbers to normalize this extensions: for any `consider the extension E` obtained from E′

` by pushing out with respect to themap

[e/e`] : T 1j−1/tors⊗ Z`(1) → T 1

j−1/tors⊗ Z`(1).

Lemma 3 Consider the Q`-vector space E` ⊗Z`Q` constructed above. Then,

for ` 6= p, we have that the monodromy map is given by the composition

E` ⊗Z`Q` → T−1

j ⊗Q`N→ T 1

j−1 ⊗Q` → E` ⊗Z`Q`

where the left and right hand side maps are the natural ones and the map Nis the composition of the map N defined in §2 and the multiplication-by-e map.For ` = p, applying the functor Dst to this composition we get the monodromymap on the corresponding filtered (Φ, N)-module.

Proof: Consider first the case ` 6= p. Observe that, for a Q`-module V withaction of the absolute Galois group such that monodromy filtration verify thatMi−1 = 0 ⊆ Mi ⊆ Mi+1 ⊆ Mi+2 = V , then the monodromy map N is thecomposition of

V → V/MiN ′→ Mi(1) → V,

where N ′: V/Mi → Mi is the induced map by N .Now, recall that, by corollary 2, and if X has totally degenerate reduction, andM• is the monodromy filtration on the `-adic cohomology Hi+2j(X,Q`), theinduced map

GrM−iH

i+2j(X,Q`)N ′→ GrM

−i−2Hi+2j(X,Q`)(1)

coincides with the map N :T ij ⊗ Q`(−j) → T i+2

j−1 ⊗Q`(−j) that we defined insection §2, by composing with the isomorphism

T ij (Y )⊗Q`(−j) ∼= GrM

−iHi+2j(X,Q`)

deduced from the weight spectral sequence and the cycle maps on the compo-nents of the special fiber Y

(k). In particular we have that, by using the notation

in the construction above, the composition

M ⊗Z`Q` → W1⊗Z`

Q`∼= T−1

j ⊗Q`N→ T 1

j−1⊗Q`∼= W−1⊗Z`

Q` → M ⊗Z`Q`

is the monodromy map on M .To get the result observe that the Q`-vector space E⊗Z`

Q` was obtained fromM ⊗Z`

Q` by pushing out with respect to the map multiplication by e on theT 1

j−1 ⊗Z`Q`, so we have a commutative diagram

0 → T 1j−1 ⊗Q`(1) → E` ⊗Z`

Q` → T−1j ⊗Q` → 0

↑ [e] ↑ g ‖0 → T 1

j−1 ⊗Q`(1) → M ⊗Z`Q` → T−1

j ⊗Q` → 0,

and the monodromy operator N on E` is obtained from the monodromy operatoron M by composing with the isomorphism g and its inverse:

E`g−1

→ MN→ M(−1)

g→ E`(−1).

Finally, to obtain the result in the case ` = p, one uses first a similar argumentfor the log-crystalline cohomology by applying Corollary 3 and then Proposition4.

Now we define the pairing as follows: let α ∈ T−1j /tors, β ∈ T 1∨

j−1/tors. Ten-soring the subgroups these elements generate with, respectively Z` and Z`(1),pulling back the extension E` by the former and pushing out by the latter, weget an extension of Z` by Z`(1), hence an element of

Ext1G(Z`,Z`(1)) = K∗(`),

the `-completion of the multiplicative group of K. We denote this element byα, β` and the product of the pairings over all ` by α, β. This pairing apriori takes values in K∗.

Proposition 5 The image of the pairing −,− in K∗ is contained in theimage of the natural map K∗ → K∗.

In the proof, we shall use the following simple lemma:

Lemma 4 Let K be a finite extension of Qp with normalized valuation v. De-note by K∗ the completion of the multiplicative group of K with respect to itssubgroups of finite index, and by v the “completed valuation”

K∗ → Z.

Then the completed valuation of an element α ∈ K∗ lies in the image of Z in Ziff α lies in the image of K∗ in K∗.

Proof of lemma: Let R be the ring of integers in K. Then since K is finiteover Qp, the groups R∗/R∗n are finite for every integer n, and R∗ is completewith respect to the topology induced by its subgroups of finite index. Thus thenatural map R∗ → R∗ is an isomorphism and the map K∗ → K∗ is injective.Hence the induced map

K∗/K∗ → Z/Z

is an isomorphism, which proves the lemma.

Remark 7 The use of this lemma is the one place where our construction willnot work over a larger field such as Cp. What seems to be needed is to defineour extensions in some sort of category C of “analytic mixed motives” over Cp,in which we would have Ext1C(Z,Z(1)) = C∗

p. As is well-known, extensions ofCp by Cp(1) in the category of continuous Gal(Cp/K)-modules are trivial, sothis category won’t do.

Proof of Proposition: It is sufficient to show that for each `, the diagram ofpairings:

T−1j × T 1∨

j−1 → K∗(`) → Z`

‖ ‖ ↑T−1

j × T 1∨j−1 → Z.

is commutative. The pairing in the top row is that just defined, and the pairingin the bottom row is constructed from the isogeny

N : T−1j → T 1

defined in the lemma 3. This shows that the top pairing takes values in Z, hencefactors through the image of K∗ in K∗(`) for every `. Clearly, we only need toprove that this diagram is commutative after tensoring by Q.Now, the commutativity of the diagram is deduced from Lemma 3 by using thesame argument as in the paper of Raynaud ([Ray2], section 4.6). Concretely,given an exact sequence 0 → E′ → E → E/E′ → 0 as Galois Q`-modules suchthat E′(−1) and E/E′ are Q`-vector spaces of finite dimension with trivialGalois action, one can construct as before a pairing

−,− : E/E′ × E′(−1)∨ → K∗(`) ⊗Z`Q`

v`→ Q`.

Then the map N ′:E/E′ → E′(−1) constructed from this pairing coincides withthe map induced from the monodromy on E.

Definition 2 If the Galois action on T−1j (Y ) and T 1

j−1(Y ) is trivial, we definethe i-th intermediate Jacobian J i(X) as follows: consider the pairing

−,− : T−1j /tors× T 1∨

j−1/tors →∏

K∗(`).

Its image is contained in the image of K∗ via the diagonal morphism, which isan injection. Now define:

Jj(X) := Hom(T 1∨j−1/tors,Gm)/T−1

j /tors,

where we are viewing T−1j /tors into Hom(T 1∨

j−1/tors,Gm) via the pairing −,−.This quotient has sense in a rigid-analytic setting due to Proposition 2, whichtells us that T−1

j /tors is a lattice in the algebraic torus Hom(T 1∨j−1/tors,Gm).

Now we consider the case that there is some non-trivial action of GF on T−1j /tors

and on T 1j−1/tors, necessarily through a finite quotient. After a finite unramified

Galois extension L/K, we have a pairing

T−1j /tors× T 1∨

j−1/tors → L∗.

If we show that this pairing respects the action of the Galois group of L/K onboth sides, we will have an identification of T−1

j /tors and a lattice in the toruswith character group T 1

j−1/tors. We can define hence the Jj(X) as the non-split analytic torus defined as the quotient. But this pairing has been definedby using an extension in Ext1G(T−1

j /tors⊗ Z`, T1j−1/tors⊗ Z`(1)) for any `, so

it is clear that pairing we get

−,− : T−1j /tors× T 1∨

j−1/tors →∏

L∗(`)

respects the Galois action of Gal(L/K).

Remark 8 Observe that the dimension of Jj(X) is equal to the rank of T 1j−1,

which is equal to dimQp(Hj(Y ,Wωj−1

Y ,log)⊗Zp Qp) by Corollary 5, which is equal

to hj,j−1 := dimK(Hj(X, Ωj−1X )) by [I2].

Note that this intermediate Jacobian, which a priori depends of a regular propermodel of X over R, depends modulo isogeny only on the variety X. To showthis, observe first that the monodromy filtration we constructed in the p-adiccohomology Hn(X,Qp) is uniquely defined, because there are no maps betweenQp(j) and Qp(i) if i 6= j. On the other hand, the Tate module Vp(Jj(X)) isisomorphic to M1(H)/M−3(H), where H = H2j−1(X,Qp(j)) and M• is themonodromy filtration. But an analytic torus over K is determined moduloisogeny by its Tate module (see for example [Se], Ch. IV, §A.1.2).

7 The Abel-Jacobi mapping

Let K be a finite extension of Qp, and let X be a smooth projective variety overK. We assume that X has a regular proper model X over the ring of integersR of K, with special fibre satisfying the assumptions in §1. In the last section,

we defined intermediate Jacobians Jj(X), which we will denote by Jj when Xis fixed throughout the discussion. In this section, we define the Abel-Jacobimapping:

CHj(X)hom → Jj(X)⊗Z Q,

where CHj(X)hom is the subgroup of the Chow group CHj(X) consisting ofcycles in the kernel of the cycle map:

CHj(X) →∏

H2j(X,Z`(j)),

where the product is taken over all prime numbers `.Let T` = T`(Jj(X)), the Tate module of Jj(X), and let T =

∏` T`. Put

V` = T` ⊗Z`Q`, and let V =

∏` V`. Recall the groups H1

g (K, T ),H1g (K, V ) of

Bloch-Kato ([BK], 3.7). We have

H1g (K, Vp) = Ker[H1(K, Vp) → H1(K,Vp ⊗Qp

BDR)],

where BDR is the field of “p-adic periods” of Fontaine. For ` 6= p, we have:

H1g (K, V`) = H1(K, V`).

H1g (K,T ) is defined as the inverse image of H1

g (K, V ) under the natural map

H1(K, T ) → H1(K,V ).

The following Lemma is proved as in the case of an abelian variety in [BK],section 3.

Lemma 5 We have an isomorphism:

J i(K) → H1g (K, T ),

compatible with passage to a finite extension of K and with norms.

Remark 9 The content of the lemma is that the extensions of Zp by Zp(1) thatgive pairings with values in K∗ (as opposed to K∗) are precisely the de Rhamextensions.

Now, the following result is essentially due to Faltings [Fa2]: it is a consequenceof his results that the etale cohomology of a smooth variety (not necessarilyprojective) is de Rham in the sense defined by Fontaine (see for example [I2]).

Theorem 5 (Faltings) Let X be a smooth projective variety. Let H be equalto

∏` H2j−1(X,Z`(j)) and let cl′ : CHj(X)hom → H1(K,H) be the canonical

etale Abel-Jacobi mapping (see [Bl] and [R]). Then the image of cl′ is containedin H1

g (K, H).

Corollary 7 Let X be a smooth projective variety over K with reduction sat-isfying the assumptions of §1. Let M· be the monodromy filtration on the coho-mology H2j−1(X,Q`(j)), and let V =

∏` V`. Then the `-adic Abel-Jacobi map

cl′ factorizes through a map cl′ : CHj(X)hom → H1g (K,M1).

Proof: Observe first that for every ` we have that V`/M1 is a successive exten-sion of Q`-vector spaces of the form Q`(i), with i < 0. Then the result followsfrom the fact that [V`/M1]G = 0 and H1

g (K,Q`(i)) = 0 for i < 0 ([BK], Example3.5).

Now we define the rigid-analytic Abel-Jacobi mapping as the composition

CHj(X)hom → H1g (K, M1) → H1

g (K, V ),

using Lemma 5 and Corollary 7.

Example 3 Let X be a regular proper scheme of relative dimension d over R,whose special fibre satisfies the assumptions of §1. Then J1(X) is the Jacobianvariety of X, Jd(X) is the Albanese variety of X and the maps defined aboveare the classical Abel-Jacobi mappings, tensored with Q.

Remark 10 (i) Recall that by definition, H1g (K, T ) contains the torsion sub-

group of H1(K,T ) for any finitely generated Z`-module T . Now for i < 0,H1(K,Z`(i)) has nontrivial torsion, in general, and so it is not clear thatwe can define the Abel-Jacobi mapping to Jj(X) integrally. Of course,we can define the map on the inverse image of H1(K,T ∩M1) under theAbel-Jacobi mapping, but it is not clear how this subgroup changes as weextend the base field K.

(ii) As mentioned above, the dimension of Jj(X) is the Hodge number hj,j−1.This differs in general from the dimension of the complex intermediateJacobian of Griffiths, which is always equal to B2j−1

2 . Using higher oddK-theory, one can in some sense “recover” the “rest” of the classical in-termediate Jacobian. For example, for j = 2, we have the exact sequence:

0 → M−3 → M1 → M1/M−3 → 0.

The term on the right is the Tate module of our J2(X). Using Proposition7, we get (at least after tensoring with Q; see the last remark) a naturalmap:

kerα2 → H1(K, M−3),

where α2 is the Abel-Jacobi mapping we have just defined. Now M−3

modulo torsion is a direct sum of Z`(2)’s, and we have

lim←−n

K3(K,Z/`n) ∼= H1(K,Z`(2)).

Thus we can relate kerα2 to K3(K). Using work of Hesselholt-Madsen(see [HM], Theorem A), we can relate the kernel of αj successively tohigher odd algebraic K-groups K2r−1(K), for r = 2, · · · , j. We hope topursue this point in another paper.

8 An Example

In this section we describe in detail an example that motivated our theory.

To warm up, we analyze the case of the product X of two Tate elliptic curvesE1, E2. The results here are due to Serre and Tate (see [Se], Ch. IV §A.1.4), butwe interpret them in terms of our monodromy filtration, which will put themin a form suitable for generalization to higher cohomology groups.

Let V = H1(E1,Qp(1)) and W = H1(E2,Qp). By the Kunneth formula, wehave that H2(X,Qp(1)) is the direct sum of V ⊗W and two copies of the trivialrepresentation Qp. The monodromy filtration on M = V ⊗ W is the tensorproduct of the monodromy filtrations on V and W . We thus have

M0 =∑

Vi ⊗Wj .

We have exact sequences:

0 → Qp(1) → V → Qp → 0

0 → Qp → W → Qp(−1) → 0,

which encode all of the information about the monodromy filtrations. Let qi (i =1, 2) be the class of the extension V (resp. W ) in

Ext1G(Qp,Qp(1)) = K∗ (p) ⊗Zp Qp.

Tensoring the first sequence with W and the second by V , we get exact sequences

0 → W (1) → V ⊗W → W → 0.

0 → V → V ⊗W → V (−1) → 0.

This gives us a map:

V ⊕W (1) → V ⊗W.

and since V = V1,W = W1, we get a surjection

V ⊕W (1) → M0.

Note that the dimension of the space on the left is 4 and the dimension of thaton the right is 3. Thus the kernel is of dimension 1, and is easily seen to be

V ∩W (1) = Qp(1), where these two spaces are viewed as subspaces of M0 viathe exact sequences above. Now M−2 = V−1⊗W−1 = Qp(1), and summarizing,we have an exact sequence:

0 → Qp(1) → M0 → M0/M−2 → 0,

where the Galois group acts on the right hand space through a finite quotient.Passing to an open subgroup H of finite index that acts trivially on this spaceand taking the standard basis of Q2

p, we see that the map

M0/M−2 → H1(H,Qp(1))

sends the vector (a1, a2) to the class of qa11 qa2

2 in K∗ (p) ⊗ZpQp. Thus MH

0 6= 0iff there exist nonzero p-adic numbers a1, a2 such that qa1

1 qa22 = 1. In fact, we

Theorem 6 (see [Se], Ch. IV, §A.1.2) E1 and E2 are isogenous iff there existnonzero integers a1, a2 such that qa1

1 qa22 = 1. The map

Hom(E1, E2)⊗Z Qp → HomG(Vp(E1), Vp(E2))

is an isomorphism.

Taking into account the other contributions to H2(X,Qp(1)), we get

Corollary 8 For X the product of two Tate elliptic curves, the cycle map:

Pic(X)⊗Qp → H2(X,Qp(1))G

is surjective.

This last statement is the “p-adic Tate conjecture” for Tate elliptic curves andtheir product (see Remark 12(ii) below) .

Now we generalize this to H3 of the product X of three Tate elliptic curves,Ei, i = 1, 2, 3 with parameters qi. We study J2(X). Now X is a 3-dimensionalabelian variety, so we have:

dim H3(X,Q`) ∼= dim3∧

H1(X,Q`) =(

)= 20.

Thus the complex intermediate Jacobian would be of dimension 10. Our J2(X)will be of dimension 9, and we explain below where the “lost” dimension went.We calculate the monodromy filtration on H3(X,Q`(2)). As above, the mon-odromy filtration on each Vi = H1(Ei,Q`(1)) is given by:

Vi,−1 = Q`(1), Vi,0 = Q`(1), Vi,1 = V,

and Vi/Vi,−1 = Q`. The class of the extension

0 → Vi,−1 → Vi → Vi/Vi,−1 → 0

in H1(K,Q`(1)) = K∗ (l) ⊗Z`Q` is given by the class of the Tate parameter qi

of Ei in this group. The monodromy filtration on M = V1 ⊗ V2 ⊗ V3(−1) is thetensor product of the monodromy filtrations on the Vi. Thus

Mn =∑

r+s+t=n

V1,r ⊗ V2,s ⊗ V3,t(−1);

note that this sum is not direct. We calculate easily the following table, in whichthe top row is the level of the filtration and the bottom row is the dimension:

−3 −2 −1 0 1 2 31 1 4 4 7 7 8 .

If we include the other contributions to H3(X,Q`(2)) from the Kunneth for-mula, we find the following table, which gives the dimension of the spaces in thevarious steps of the monodromy filtration on H3(X,Q`(2)):

−3 −2 −1 0 1 2 31 1 10 10 19 19 20 .

The `-Tate vector space V`(J2(X)) of our J2(X) is M1/M−3, which is of di-mension 18, and thus the dimension of J2(X) is 9. The “lost” dimension comesfrom the fact that Gr2M ∼= Q`(−1) as Galois modules, and hence this cannotcontribute to the Tate module of a p-adic analytic torus. As mentioned abovein Remark 10, the lost part can be directly related to K3, via extensions of Q`

by Q`(2).

In some cases, we can also determine the image of the restriction of the Abel-Jacobi mapping to CH2(X)alg; this is an abelian variety, which is the universalabelian variety into which CH2(X)alg maps via a regular homomorphism (see[Mu]); we denote it by J2

a(X). Its dimension should depend on the multiplicativerelations between the qi with integer exponents, as we now explain. To simplifythe exposition and not lose the reader (and authors) in a maze of indices, wedescribe the case of H3, and leave the general case to a future paper. We alsoassume that the Galois group acts trivially on the T i

j (Y ), where Y isthe geometric special fibre of a strictly semi-stable model X of X over the ringof integers of K.

With notation as in the previous sections, we have an extension:

0 → T 30 ⊗Q`(1) → M−1 → T 1

1 ⊗Q` → 0.

As in the definition of the intermediate Jacobian, we can define a pairing:

(T 11 ⊗Z Q)× (T 3∨

0 ⊗Z Q) → K∗(`) ⊗Z Q

by taking elements α ∈ T 11 ⊗Z Q, β ∈ T 3∨

0 ⊗Z Q, tensoring the subgroups theygenerate by Q` and Q`(1), respectively, pulling back the extension above by thefirst and pushing it out by the second. Taking the product for all ` and usingProposition 5 above, we see that we get a K∗ ⊗Z Q-valued pairing and hence amap:

N1 : T 11 ⊗Z Q → Hom(T 3∨

0 ⊗Z Q,K∗ ⊗Z Q),

that we call the enriched monodromy map. We denote also by

N ′1 : T−1

2 ⊗Z Q → Hom(T 3∨0 ⊗Z Q,K∗ ⊗Z Q),

the composition of the map N : T−12 → T 1

1 (tensored with Q) with the map N1

just defined.Recall that if Z and X are two smooth projective varieties over a field, thena correspondence between them is an element η of CHr(Z ×X); such a cycleinduces a Galois equivariant map:

η∗ : Hi(Z,Q`(1)) → Hi+2r(X,Q`(1 + r))

defined by η∗(α) = pr2∗(pr∗1(α) ∪ [η]). Here [η] is the cohomology class of η inH2r(Z ×X,Q`(r)).

Lemma 6 Let A be an abelian variety contained in J2a(X). Write Vl(A) as an

extension

(∗) 0 → S10 ⊗Q`(1) → Vl(A) → S−1

1 ⊗Q` → 0,

where Sij are finitely generated abelian groups (see the proof below for why this

is possible). Then for some finite extension L/K, we have:

S10 ⊗Z Q ⊆ kerN1,L : T 1

1 ⊗Z Q → Hom(T 3∨0 ⊗Z Q, L∗ ⊗Z Q)

S−11 ⊗Z Q ⊆ kerN ′

1,L : T−12 ⊗Z Q → Hom(T 3∨

0 ⊗Z Q, L∗ ⊗Z Q).

Proof: Since A is contained in J2(X), it is an analytic torus. Denote by S10 and

by S−11 the corresponding lattices of A, such that A(L) ∼= Hom(S1

0 , L∗)/S−11 .

Now the inclusion map from A to J2(X) gives us monomorphisms i: S10 → T 1

and i′:S−11 → T−1

2 . We must show that N1 (i⊗ZQ) = 0 and N ′1 (i′⊗ZQ) = 0

for some finite extension L/K. To prove this it is sufficient to show, by applyingLemma 4, that, for any prime `, the composition

S10 ⊗Z Q` → T 1

1 ⊗Z Q`NL,`−→ Hom(T 3∨

0 ⊗Z Q`, L∗(`) ⊗Z`

is trivial.There exists a divisor D on X with desingularization D′ and a correspondenceη ∈ CH2(X × D′) such that V`(A) is the image of η∗ in H3(X,Q`(2)). Take

L a finite extension of K such that D′ has semi-stable reduction. Then thecohomology H1(D′,Qp(1)) is semi-stable.First consider the case ` 6= p. Denoting by M• the monodromy filtration onthe cohomology, we have that the correspondence η gives a map respecting thisfiltration. This gives us a map

GrM−1(H

1(D′,Qp(1))) → S10 ⊗Z Q` → T 1

1 ⊗Z Q`.

We have, on the other hand, a commutative diagram of monodromy operators

T 11 ⊗Z Q`

N1,L→ Hom(T 3∨0 ⊗Z Q`, L

∗(`) ⊗Z`Q`)

↑ ↑GrM

−1(H1(D′,Qp(1)))

N1,L→ Hom(GrM−3(H

1(D′,Qp(1)))∨(1), L∗(`) ⊗Z`Q`).

But GrM−3(H

1(D′,Qp(1))) = 0, since there is no graded piece of weight ≤ −3 inthe monodromy filtration on H1(D′,Qp(1)), and hence the image of S1

−1⊗Z Q`

in T 11 ⊗Z Q` is contained in kerNL, as claimed.

Now, to show the case ` = p, we use instead of the p-adic cohomology of D′

the log-crystalline cohomology of a suitable model of D′. We have then, byapplying Tsuji’s theorem if necessary, that the correspondence η gives us a mapbetween the respective log-crystalline cohomologies, respecting the monodromyfiltration. Then the same argument applies as we have just given for ` 6= p. Asimilar argument applies also to the other claimed inclusion.

Conversely, we have the

Conjecture 1 (p-adic Generalized Hodge conjecture for H3) For X with totallydegenerate reduction, we have

Gr−1N1H3(X,Qp)=⋃

[L:K]<∞[kerN1,L : T 1

1 ⊗Z Q → Hom(T 3∨0 ⊗Z Q, L∗ ⊗Z Q)]⊗Q Qp(−1).

Gr1N1H3(X,Qp)=⋃

[L:K]<∞[kerN ′

1,L : T−12 ⊗Z Q → Hom(T 3∨

0 ⊗Z Q, L∗ ⊗Z Q)]⊗Q Qp(−2).

Note that these spaces are of the same dimension, since the monodromy operatorN : T−1

2 → T 11 is an isogeny. This is as it should be, since N1H3(X,Qp(2))

is analogous to a pure Hodge structure of odd weight, and so should be evendimensional.

Remark 11 To explain the motivation for phrasing this conjecture as we have,consider the extension:

0 → M−3 → M−1 → M−1/M−3 → 0,

where M = Vp(E1)⊗ Vp(E2)⊗ Vp(E3)(−1).Taking H-cohomology for an open subgroup H of finite index in G, we get anexact sequence:

0 → MH−1 → [M−1/M−3]H

∂H→ H1(H, M−3).

The space in the middle is isomorphic to (T 11 ⊗Qp)H . As we have seen above in

the case of the product of two Tate elliptic curves, for the product of three Tateelliptic curves Ei with parameters qi, the space M−1/M−3 is three dimensional,and for sufficiently small H, the map ∂H takes a vector (a1, a2, a3) to

∏qaii ,

viewed as an element of L∗ (p). Thus ker ∂H corresponds to relations betweenthe qi in L∗ (p) with p-adic exponents. There is no reason why these have tobe relations with integral exponents, as they are for the case of two q′is. Forexample, suppose we choose a uniformizing parameter π of K and a logarithmwith log(π) = 0. Then finding p-adic numbers a1, a2, a3 with

qaii = 1

is equivalent to solving the simultaneous equations:

ai log(qi) = 0

aivi = 0

in Qp, where vi = v(qi). This is a system of two equations in three unknowns,so there should always be a solution other than (0, 0, 0). But there is no reasonto believe that there is always an integral solution. In other words, the kernel ofthe monodromy map:

T 11 → Hom(T 3∨

0 ,K∗)

tensored with Zp, may be strictly smaller than the kernel of the map:

T 11 ⊗ Zp → Hom(T 3∨

0 ,K∗ (p)).

Note that this map is the completion of the first map with respect to subgroups offinite index, not the tensor product of this map with Zp. This is why its kernelcan be bigger, since the natural map

K∗ ⊗Z Zp → K∗ (p)

is surjective but not injective.

In the following, we shall prove this conjecture in some cases when X is theproduct of three Tate elliptic curves. Complex analogues of these results arewell-known (see e.g. [L], p. 1197). The strategy is simple: by Lemma 5, forH a sufficiently small open subgroup of G with fixed field L, the dimensionof kerN1,L provides an upper bound for the dimension of Gr−1N1H3(X,Qp).For X as above, we can easily compute this bound, and in some cases we canexplicitly produce divisorial correspondences η such that the sum of the imagesof η∗ is of the required dimension.

Conjecture 2 (Compare 7.5 on p. 1197 of [L]) Let X = E1 × E2 × E3 be aproduct of three Tate elliptic curves with parameters qi (i = 1, 2, 3), and let r bethe rank of the space of triples of integers (n1, n2, n3) with

qnii = 1.

dimJ2a(X) = 6 + r.

For example, for “generic” qi (no multiplicative relations), the dimension is6, and if all of the qi are equal, the dimension is 8. In particular, for X theproduct of three Tate elliptic curves, the restriction of the Abel-Jacobi map toCH2(X)alg is never surjective.

Proposition 6 The conjecture is true if there are two independent multiplica-tive relations between the qi, if there is one relation and two of the three areisogenous, or if there are no nontrivial multiplicative relations between the qi.

Proof: The dimension is at least 6 since J2(X) contains two copies of X, viaalgebraic cohomology classes of the type

α⊗ β ⊗ γ ∈ H0(Ei)⊗H1(Ej)⊗H2(Ek),

for each permutation (i, j, k) of (1, 2, 3).

To determine the dimension of J2a(X), assume first that the rank of the space

of relations is two. Since the valuations of the qi are all positive, all of the 2by 2 minors of the two vectors must be nonzero. We can row reduce chosenbasis vector for the relation exponents so that each contains exactly one 0, andthese are in different coordinate positions. But then by Theorem 6, we have tworelations between two of the qi’s, which gives isogenies between the respectiveEi. Then it is easy to see that the dimension of J2

a(X) is 8, because [M−1/M−3]G

is then spanned by three cohomology classes of the type:

αij ⊗ β ∈ H1(Ei)⊗H1(Ej)⊗H1(Ek),

where αij is the class of the graph of a nonzero isogeny between Ei and Ej .These classes cannot all lift to MG

−1, as then the map:

[M−1/M−3]G → H1(G,M−3)

would be zero, which can’t be, since there are linear combinations of the valua-tions of the qi that are nonzero.

If there is say a multiplicative relation between q1 and q2, and these have nomultiplicative relations with q3, then we can produce another cohomology classsupported on the graph of the isogeny, as above.

If there are no relations, then the dimension is at most six, and so is exactlysix. This completes the proof of the proposition.

Remark 12 i) When there is just one multiplicative relation which doesnot arise from an isogeny between two of the curves, there is no obviousway to produce the required abelian subvariety, but the generalized Hodgeconjecture predicts that it should exist.

ii) In a sequel to this paper [R3], we will formulate a p-adic generalized Hodge-Tate conjecture, which predicts the dimension of the coniveau filtrationN iHj(X,Qp) for any i, j in terms of the kernels of monodromy operators.

iii) Since the image of the Abel-Jacobi mapping restricted to CH2(X)alg isnever all of J2(X) when X is the product of three Tate elliptic curves,we hope to be able to use the quotient of J2(X) by this image to detectcodimension two cycles that are homologically equivalent to zero, but notalgebraically equivalent to zero. A regular proper model for such productshas been constructed by Gross-Schoen ([GS], §6, especially Props. 6.1.1and 6.3, and Cor. 6.4), but we do not believe it is strictly semi-stable.Further blowing-up should make it such, however. See also the paper ofHartl [Ha], where he constructs strictly semi-stable regular models of ram-ified base changes of varieties with strictly semi-stable reduction, and ofproducts of such.

References

[A] Y. Andre, p-adic Betti lattices, in p-adic analysis, Trento 1989, 23-63,Lecture Notes in Mathematics 1454, 1990

[Be] A. Besser A generalization of Coleman’s p-adic integration, InventionesMath. 142 (2000) 397-434

[Bl] S. Bloch Algebraic cycles and values of L-functions I, J. f. die reine undangewandte Math. 350 (1984) 94-108

[BK1] S. Bloch and K. Kato, p-adic etale cohomology, Publ. Math. I.H.E.S. 63(1986) 107-152

[BK2] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, inThe Grothendieck Festschrift, Vol I, Progr.Math. 87, Birkhauser, Boston,1990.

[BGS] S. Bloch, H. Gillet and C. Soule, Algebraic cycles on degenerate fibers,in Arithmetic Aspects of Algebraic Geometry, Cortona 1994, F. Cataneseeditor, 45-69.

[Co] P. Colmez, Integration sur les varietes p-adiques, Asterisque 248, SocieteMathematique de France, 1998

[Con] C. Consani, Double complexes and Euler facotrs, Comp. Math. 111 (1998)323-358

[D] V.I. Danilov, The geometry of toric varieties (in Russian), Uspekhi Mat.Nauk 33:2 (1978), 85-134; English translation in Russian MathematicalSurveys 33:2 (1978) 97-154.

[DeJ] A.J. de Jong, Smoothness, alterations and semi-stability, Publ. Math.I.H.E.S. 83 (1996) 52-96.

[De1] P. Deligne, Theorie de Hodge I, Actes du Congres International des Ma-thematiciens, Nice, 1970, Tome I, 425-430.

[De2] P. Deligne, La conjecture de Weil II, Publ. Math. I.H.E.S. 52 (1980)137-252.

[dS] E. de Shalit, The p-adic monodromy-weight conjecture for p-adically uni-formized varieties, preprint 2003

[Fa1] G. Faltings, p-adic Hodge theory, Journal of the American MathematicalSociety, 1 (1988)

[Fa2] G. Faltings, Crystalline cohomology and p-adic Galois representations, inAlgebraic Analysis, Geometry and Number Theory, Baltimore MD 1988,25-80, Johns Hopkins University Press, 1989

[F] W. Fulton, Introduction to Toric Varieties, Annals of Math. Studies,Volume 131, Princeton University Press, 1993.

[FMSS] W. Fulton, R. MacPherson, F. Sottile and B. Sturmfels, Intersection the-ory on spherical varieties, J. Alg. Geometry 4 (1995) 181-193

[G] O. Gabber, Sur la torsion dans la cohomologie `-adique d’une variete,C.R.A.S. 297 (1983) 179-182.

[GN] F. Guillen and V. Navarro-Aznar, Sur le theoreme local des cycles invari-ants, Duke Math. J. 61 (1990), 133-155.

[Gr] P. Griffiths, On the periods of certain rational integrals: I and II, Ann.Math. 90 (1969), 460-541.

[GS] B. Gross and C. Schoen, The modified diagonal cycle on a triple productof curves, Ann. de l’Institut Fourier 45 (1995) 649-679.

[Gro] A. Grothendieck, Hodge’s general conjecture is false for trivial reasons,Topology 8 (1969) 299-303.

[Ha] U. Hartl, Semi-stability and base change, Archiv der Mathematik 77(2001), 215-221

[HM] L. Hesselholt and I. Madsen, On the K-theory of local fields, preprint 2002,available on the K-Theory preprint archive

[H] O. Hyodo, A note on p-adic etale cohomology in the semi-stable reductioncase, Inv. Math. 91 (1988), 543-557.

[HK] O. Hyodo and K. Kato, Semi-stable reduction and crystalline cohomologywith logarithmic poles, in Periodes p-adiques, Seminarie de Bures, 1988(J.-M. Fontaine ed.), Asterisque 223 (1994), 221-268.

[I1] L. Illusie, Cohomologie de de Rham et cohomologie etale p-adique (D’apresG. Faltings, J.-M. Fontaine et al.) in Seminaire Bourbaki, 1989-1990,Asterisque 189-190 (1990), 325-374

[I2] L. Illusie, Ordinarite des intersections completes generales, in The Gro-thendieck Festschrift, Vol II, Progr. Math, Vol. 87, Birkhauser, Boston,1990

[I3] L. Illusie, Reduction semi-stable ordinaire, cohomologie etale p-adique etcohomologie de de Rham, d’apres Bloch-Kato et Hyodo, in Periodes p-adiques, Seminaire de Bures, 1988 (J.-M. Fontaine ed.), Asterisque 223(1994), 209-220.

[I4] L. Illusie, Autour du theoreme de monodromie locale, in Periodes p-adiques, 9-57.

[It] T. Ito, Weight-Monodromy conjecture for p-adically uniformized varieties,preprint 2003 (ArXiv:math.NT/0301201)

[Ja1] U. Jannsen, On the `-adic cohomology of varieties over number fields andits Galois cohomology, in Galois Groups over Q, Y.Ihara, K. Ribet andJ.-P. Serre ed., Math. Sci. Res. Inst. Publ. 16, Springer-Verlag, NewYork (1989), 315-360.

[Ja2] U. Jannsen, Mixed Motives and Algebraic K-Theory, Lecture Notes inMathematics, Volume 1400, Springer-Verlag 1990

[Ka] K. Kato, Logarithmic structures of Fontaine-Illusie, in Algebraic Analy-sis, Geometry and Number Theory, Proceedings of the JAMI InauguralConference, Jun-Ichi Igusa, editor, Special Issue of the American Journalof Mathematics, Johns Hopkins Press, 1989

[Ku1] K. Kunnemann, Projective regular models for abelian varieties, semi-stable reduction, and the height pairing, Duke Math. J. 98 (1998) 161-212.

[Ku2] K. Kunnemann, Algebraic cycles on toric fibrations over abelian varieties,Math. Zeitschrift 232 (1999) 427-435

[La] R. Langlands, Sur la mauvaise reduction d’une variete de Shimura, inJournees de Geometrie Algebrique de Rennes, Asterisque 65 (1979)

[Li] D. Lieberman, On higher Picard varieties, American J. of Mathematics,90 (1968) 1165-1199

[Ma] Y. Manin, Three-dimensional hyperbolic geometry as ∞-adic Arakelovgeometry, Invent. Math. 104 (1991) 223-244

[MD] Y. Manin and V. Drinfeld, Periods of p-adic Schottky groups, J. reine undang. Math 262/263 (1973) 239-247

[Mok] A. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato,Duke Math. J. 72 (1993) 301-337.

[Mu1] D. Mumford, An analytic construction of degenerating curves over a com-plete local ring, Compositio Math. 24 (1972) 129-174.

[Mu2] D. Mumford, An analytic construction of degenerating abelian varietiesover complete local rings, Compositio Math. 24 (1972) 239-272

[Mur] J. Murre, Un resultat en theorie de cycles algebriques de codimensiondeux, Comptes Rendus de l’Academie des Sciences 296 (1983) 981-984

[Mus] G.A. Mustafin, Non archimedean uniformization (in Russian), Mat. Sb.105 (147) (1978) 207-237, 287; English translation in Mathematics of theUSSR: Sbornik 34 (1978) 187-214

[P] B. Perrin-Riou, Representations p-adiques ordinaires, in Periodes p-adi-ques, Seminaire de Bures, 1988 (J.-M. Fontaine ed.), Asterisque 223 (1994),185-208.

[Rap] M. Rapoport, On the bad reduction of Shimura varieties, in Automor-phic Forms, Shimura Varieties and L-Functions, L. Clozel and J. S. Milneeditors, Perspectives in Mathematics, Vol. 11, Academic Press 1990

[RZ1] M. Rapoport and Th. Zink, Uber die locale Zetafunktion von Shimurava-rietaten, Monodromie-filtration und verschwindende Zyklen in ungleicherCharacteristic, Invent. Math. 68 (1982), 21-101.

[RZ2] M. Rapoport and Th. Zink, Period spaces for p-divisible groups, Annalsof Mathematics Studies, Volume 141, Princeton University Press, 1996

[R1] W. Raskind, Higher `-adic Abel-Jacobi mappings and filtrations on Chowgroups, Duke Math. J. 78 (1995) 33-57.

[R2] W. Raskind, A p-adic Tate conjecture for varieties with totally degeneratereduction, in preparation 2003

[R3] W. Raskind, On a generalized p-adic Hodge-Tate conjecture for varietieswith totally degenerate reduction, in preparation 2003

[Ray1] M. Raynaud, Varietes abeliennes et geometrie rigide, Actes du ICM, Nice,1970, Tome I, 473-477

[Ray2] M. Raynaud, 1-motifs et monodromie geometrique, in Periodes p-adiques,Seminaire de Bures, 1988 (J.-M. Fontaine ed.), Asterisque 223 (1994),295-319.

[Se] J.-P. Serre, Abelian `-adic representations and elliptic curves, Benjamin1967

[ST] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Annals ofMath. 88 (1968) 492-517

[St] J.H. Steenbrink, Limits of Hodge Structures, Invent. Math. 31 (1976),223-257.

[Tsu] T. Tsuji, p-adic etale cohomology and crystalline cohomology in the semi-stable reduction case, Inventiones Math 137 (1999) 233-411.

[Va] Y. Varshavsky, p-adic uniformization of unitary Shimura varieties, Publ.Math. I.H.E.S. 87 (1998) 57-119.

[W1] A. Weil, Foundations of Algebraic Geometry, American Mathematical So-ciety Colloquium Publications Series, Volume 29, Revised Edition, 1962.

[W2] A. Weil, On Picard varieties, American Journal of Math., 74 (1952) 865-894; also in Oeuvres Scientifiques, Volume II, Springer-Verlag Berlin 1978.

[Z] T. Zink, Uber die schlechte Reduktion einiger Shimuramannigfaltigkeiten,Comp. Math 45 (1981) 15-107.

Authors’ addresses

Wayne Raskind: Dept. of MathematicsUniversity of Southern CaliforniaLos Angeles, CA 90089-1113, USAemail: raskind@math.usc.edu

Xavier Xarles: Departament de MatematiquesUniversitat Autonoma de Barcelona08193 Bellaterra, Barcelona, Spainemail: xarles@mat.uab.es

on the ´etale cohomology of algebraic varieties with ...mat.uab.es/~xarles/scohomo.pdf · on the...

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