Frobenius splitting and ordinarity
Abstract.
We examine the relationship between the notion of Frobenius splitting and ordinarity for varieties. We show the following: a) The de RhamWitt cohomology groups of a smooth projective Frobenius split variety are finitely generated over . b) we provide counterexamples to a question of V. B. Mehta that Frobenius split varieties are ordinary or even HodgeWitt. c) a Kummer surface associated to an Abelian surface is split (ordinary) if and only if the associated Abelian surface is split (ordinary). d) for a surface defined over a number field, there is a set of primes of density one in some finite extension of the base field, over which the surface acquires ordinary reduction.
1991 Mathematics Subject Classification:
Primary 14XX; Secondary 11XX1. Introduction
Let be a perfect field of characteristic . An abelian variety over is said to be ordinary if the rank of is the maximum possible, namely equal to the dimension of . The notion of ordinarity was extended by Mazur [19], to a smooth projective variety over , using notions from crystalline cohomology. A more general definition was given by BlochKato [4] and IllusieRaynaud [14] using coherent cohomology. With this definition it is easier to see that ordinarity is an open condition. Ordinary varieties tend to have special properties, for example the existence of canonical SerreTate liftings for ordinary abelian varieties to characteristic , and the comparison theorems between crystalline cohomology and padic étale cohomology can be more easily established for such varieties. In brief, ordinary varieties play a key role in the study of varieties in characteristic .
One of the motivating questions in this paper is to study the relationship between the concept of Frobenius split varieties introduced by MehtaRamanathan [20], and ordinary varieties. Unlike ordinariness, the definition of Frobenius splitting can be extended to singular varieties, and this has proved to be of use in studying the cohomology and singularities of Schubert varieties. It was shown by Mehta and Srinivas [21], that smooth, projective varieties with trivial cotangent bundle, in particular abelian varieties, are ordinary if and only if they are Frobenius split. It can be seen from the theory of Cartier operator, that smooth, projective split surfaces are ordinary (see Theorem 2.4.1). Moreover any smooth, projective, ordinary variety with trivial canonical bundle is split (see Theorem 2.4.2).
We show (see Theorem 5.1.2) that if an abelian surface is split (hence ordinary), then the associated Kummer surface is also split (and hence ordinary). We recall that the surface is obtained by blowing up the singularities of the singular surface , obtained by identifying the points and in . Although this result can be proved by adic methods when the base field is finite, in this paper we prove this for a perfect field , by relating ordinarity to Frobenius splitting for such varieties. In the case of abelian and surfaces, it is possible to compare the notion of ordinarity with that of Frobenius splitting, and this allows us to handle the passage to the singular variety .
Ordinary varieties are HodgeWitt, in that the de RhamWitt cohomology groups are finitely generated over . A natural question that arises is whether Frobenius split varieties are HodgeWitt. We show (see Theorem 3.2.1) that for any smooth projective split variety over an algebraically closed field the cohomology groups are of finite type as modules. Using the work of IllusieRaynaud, we also see that the first differential is zero for all . During the course of writing of this paper, the first author refined these methods to control the nature of crystalline torsion for split varieties (see [15]) and has also shown that any smooth, projective and Frobenius split threefold is HodgeWitt.
The foregoing results raise the possibility that Frobenius split varieties should be HodgeWitt or even ordinary and indeed the question of whether or not Frobenius split varieties are ordinary was raised by V. B. Mehta. After the first draft of this paper and [15] were written and circulated, we found however that this general expectation, which had been further strengthened by low dimensional results like Theorem 2.4.1 for surfaces and [15] (for threefolds), turns out to be false in higher dimensions. In [15] it was shown that any Frobenius split, smooth projective threefold is HodgeWitt. It turns out that this is best possible. We found examples of Frobenius split varieties which are not ordinary (of dimension at least three) and are not even HodgeWitt (dimension at least four).
We also give an example of a smooth fibration of smooth, projective varieties and , where the base and fibers are ordinary, but the total space is not ordinary. This is in contrast to the fact that if is the projective bundle associated to a vector bundle on , then is ordinary if and only if is ordinary. A variant of our method also gives an example of a variety defined over a number field, whose reduction modulo all but a finite set of primes is HodgeWitt (and Frobenius split), but which has nonordinary reduction for infinitely many primes.
One of the other motivating questions of this paper, is a conjecture of Serre (Conjecture 6.0.1) formulated originally in the context of abelian varieties. Let be a number field, and let be a smooth, projective variety defined over . Then the conjecture is that there should be a positive density of primes of , at which acquires ordinary reduction. We show that if is either an abelian variety or a surface defined over , (Theorem 6.6.2) then there is a finite extension of number fields, such that the set of primes of at which has ordinary reduction in the case of surfaces, or has rank at least two if is an abelian variety, is of density one. We note here that a proof of the result for a class of K3 surfaces was also given by Tankeev (see [31]) under somewhat restrictive hypotheses. Our proof mirrors closely the proof given by Ogus for abelian surfaces. The results of this section are essentially independent of the contents of the rest of the paper.
In the final section we study the relationship between ordinariness and the torsion in the de RhamWitt cohomology of varieties and discuss some questions and conjectures.
Acknowledgement: We would like to acknowledge our debt of gratitude to V. B. Mehta who has explained to us many of his ideas and insights on Frobenius splitting. This paper was motivated to a large extent by his questions about crystalline aspects of Frobenius splitting. We would also like to thank Luc Illusie, Minhyong Kim, Arthur Ogus, A. J. Parmeshwaran, Douglas Ulmer, Adrian Vasiu for correspondence, conversations and encouragement.
2. Preliminaries
2.1. Ordinary varieties
Let be a smooth projective variety over a perfect field of positive characteristic. Following BlochKato [4] and IllusieRaynaud [14], we say that is ordinary if for all , where
If is an abelian variety, then it is known that this definition coincides with the usual definition [4]. By [13, Proposition 1.2], ordinarity is an open condition in the following sense: if is a smooth, proper family of varieties parameterized by , then the set of points in , such that the fiber is ordinary is a Zariski open subset of . Although the following proposition is well known, we present it here as an illustration of the power of this fact.
Proposition 2.1.1.
Let denote the moduli space of principally polarized abelian varieties of dimension , equipped with a level structure of level , and . Then the set of ordinary points is open and dense in the moduli of principally polarized abelian varieties of dimension .
Proof.
Since the level , it is known that we obtain a fine moduli space over Further since the points of the moduli space over , are uniformized by the Siegel upper half plane, is an irreducible smooth variety over Since is coprime to , the moduli problem specializes, and we see that is irreducible. By [13, Proposition 1.2], the ordinary locus is open. To show it is dense it suffices to prove it is not empty. But this is easily done by choosing an ordinary elliptic curve (and there is always one in every characteristic) together with its principal polarization. Then we can take our ordinary abelian variety with principal polarization to be the product of this ordinary elliptic curve and we are done. ∎
2.2. Cartier Operator
Let be a smooth proper variety over a perfect field of characteristic , and let (or ) denote the absolute Frobenius of . We recall a few basic facts about Cartier operators from [11]. The first fact we need is that we have a fundamental exact sequence of locally free sheaves
(2.2.1) 
where is the sheaf of closed forms, where is the Cartier operator. The existence of this sequence is the fundamental theorem of Cartier (see [11]). Since the Cartier operator is also the trace map in Grothendieck duality theory for the finite flat map , we have a perfect pairing
(2.2.2) 
where , and the pairing is given by . This pairing is perfect and bilinear (see [21]).
In particular, on applying to the exact sequence
(2.2.3) 
we get
(2.2.4) 
2.3. split varieties
In this section we recall a few basic facts about ordinary and Frobenius split (split) varieties. In this section is a normal, projective variety over a perfect field . Let be the Frobenius morphism of .
Recall that is split if the canonical exact sequence of sheaves
(2.3.1) 
splits. Note that when is smooth this is an exact sequence of locally free modules.
Frobenius splitting was introduced by Mehta and Ramanathan in [20] and a number of remarkable properties were also investigated in that paper. It is known for instance that an abelian variety is split if and only if it is ordinary in the usual sense (see [21]).
In analogy with ordinary varieties, we now consider the openness of the Fsplit condition. Let be a smooth projective morphism. Let denote the fiber product . Then one has a morphism
(2.3.2) 
which is called the relative Frobenius morphism (see [11]). The restriction of to the fibers of induce the Frobenius morphism on the fibers.
The following proposition is the relative version of Proposition 9 of [20].
Proposition 2.3.1.
Let be a smooth projective morphism of schemes in characteristic . Let be the relative Frobenius morphism. Then for a point , the fiber is split if and only if the natural map
(2.3.3) 
is injective.
Proposition 2.3.2.
With the notations of Proposition 2.3.1, there exists a Zariski open subset such that all the fibers of over points of are split.
2.4. Ordinary and split varieties
In this subsection we record a key lemma which we need and also record our proof that Frobenius split surfaces are ordinary. Our main tool here is the duality induced by the Cartier operator (see [21]).
We have the following lemma:
Lemma 2.4.1.
Let be any smooth projective variety over a perfect field. Assume is split. Then for all ,
Proof.
As is split, it follows that . Hence
But by the Leray spectral sequence applied to the Frobenius morphism and the projection formula we see that
and hence we see that
and so the lemma is proved. ∎
Theorem 2.4.1.
Let be any smooth projective, split surface over a perfect field. Then is ordinary.
Proof.
By Lemma 2.4.1 we know that for any split variety ,
for all . So when is a surface we need to check that the same vanishing is also valid for . But this is immediate from Serre duality and the following fact: Cartier operator induces a perfect pairing (we write it under assumption that is a surface)
given by and this induces a perfect pairing (see [21]). ∎
Theorem 2.4.2.
Let be any smooth projective, ordinary variety over a perfect field . If the canonical bundle of is trivial, then is Frobenius split.
Proof.
The obstruction to the splitting of the sequence
is an element of The duality pairing induced by the Cartier operator implies that
where denotes the canonical bundle. Since we have assumed that is trivial, it follows that
where the vanishing follows from the ordinarity assumption. Hence is Frobenius split. ∎
3. De RhamWitt cohomology of split varieties
3.1. de RhamWitt cohomology
The standard reference for de RhamWitt cohomology is [11]. Throughout this section, the following notations will be in force. Let be an algebraically closed field of characteristic , and a smooth, projective variety over . Let be the ring of Witt vectors of . Let be the quotient field of . Note that as is perfect, is a Noetherian local ring with a discrete valuation and with residue field . For any , let . comes equipped with a lift , of the Frobenius morphism of , which will be called the Frobenius of . We define a noncommutative ring , where are two indeterminate subject to the relations and and . The ring is called the Dieudonne ring of . The notation is borrowed from [14].
Let be the de RhamWitt procomplex constructed in [11]. It is standard that for each , are of finite type over . We define
(3.1.1) 
which are modules of finite type up to torsion. These cohomology groups are called HodgeWitt cohomology groups of .
Definition 3.1.1.
is HodgeWitt if for , the HodgeWitt cohomology groups are finite type modules.
The properties of the de RhamWitt procomplex are reflected in these cohomology modules and in particular we note that for each , the HodgeWitt groups are left modules over . The complex defined in a natural manner from the de RhamWitt procomplex computes the crystalline cohomology of and in particular there is a spectral sequence
(3.1.2) 
This spectral sequence induces a filtration on the crystalline cohomology of which is called the slope filtration and the spectral sequence above is called the slope spectral sequence (see [11]). It is standard (see [11] and [14]) that the slope spectral sequence degenerates at modulo torsion (i.e. the differentials are zero on tensoring with ) and at up to finite length (i.e. all the differential have images which are of finite length over W).
In dealing with the slope spectral sequence it is more convenient to work with a bigger ring than . This ring was introduced in [14]. Let be a graded algebra which is generated in degree by variables with the properties listed earlier (so is the the Dieudonne ring of ) and is a bimodule over generated in degree by with the properties and , and for any . The algebra is called the RaynaudDieudonne ring of (see [14]). The complex is a graded module over and is in fact a coherent, left module (in a suitable sense, see [14]).
3.2. A finiteness result
For a general variety , the de RhamWitt cohomology groups are not of finite type over , and the structure of these groups reflects the arithmetical properties of . For instance, in [4], [14] it is shown that for ordinary varieties are of finite type over . The following theorem lends more evidence towards the general expectation that Frobenius split varieties should be ordinary.
Theorem 3.2.1.
Let be any smooth, projective, split variety over an algebraically closed field of characteristic . Then for each , is a finite type module.
Remark 3.2.1.
When the formal Brauer group of (associated by Artin and Mazur; see [1]) is representable, is the Cartier module of this formal group. When this module is free of finite type over , the formal Brauer group is a divisible group of height equal to the dimension .
Corollary 3.2.1.
Remark 3.2.2.
Let be any smooth projective variety. In [14] it is has been shown that for all , are finite type modules.
Remark 3.2.3.
Before we give the proof of the above theorem, we need a few preparatory lemmas. Our proofs use the theory of higher Cartier operators as outlined in [11]. To set up conformity with notations from previous sections we recall Illusie’s notations and our definitions
(3.2.2) 
and
(3.2.3) 
The higher Cartier sheaves, are defined inductively in [11] (see page 519 of [11]). The formation of these sheaves is compatible with arbitrary base change (see [11], page 519).
Lemma 3.2.1.
Let be any smooth, projective, split variety over . Then for all and for all we have .
Proof.
The case is Lemma 2.4.1. We prove the result by induction on . We recall that we have an exact sequence (the arrow on the extreme right is the Cartier operator):
(see [11], page 519). This is essentially the definition of using . The result now follows trivially from the above exact sequence and the result for . ∎
Lemma 3.2.2.
Let be a smooth, projective, split variety over an algebraically closed field of characteristic . Then for all and for all , we have . In particular we have ,
Proof.
We recall the exact sequence (page 531, 2.5.1.2 of [11]):
Then proof follows from the vanishing of cohomology of . ∎
Proof.
[of Theorem 3.2.1] By ([11], page 613, Proposition 2.16), it suffices to prove that for all and for all , has bounded dimension. But and by definition is an isomorphism for all (the arrow in this isomorphism is the Cartier operator, ([11], see 2.5.1.2, page 531). Thus the required cohomology has dimension independent of . Next we need to check has bounded dimension for all . But by Lemma 3.2.1, this group is zero! Thus we can apply Proposition 2.16 of [11] to deduce that is a finite type module. ∎
4. Examples
4.1. Two Questions
One of the main motivations for this paper, are the following questions raised by V. B. Mehta.
Question 4.1.1.
Is any smooth projective, Frobenius split variety over a perfect field of characteristic of HodgeWitt type?
The above question is a weaker variant of the following.
Question 4.1.2.
Is any smooth projective, Frobenius split variety BlochKato ordinary?
We know by Theorem 2.4.1 that a smooth, projective Frobenius split surface is ordinary. In [15], it is shown that any Frobenius split smooth, projective three fold is HodgeWitt. Further it is known that for abelian varieties, the notions of Frobenius splitting and ordinarity coincide [21]. However in contrast to the expectation created by these results, we show in this section that the first question is false in dimensions greater than 3, and the second question is false in dimensions bigger than two.
Our examples also give examples of varieties which are HodgeWitt, but are not ordinary. These examples also provide examples of smooth, projective varieties , such that both and the the (smooth) fibers of are ordinary, but is not ordinary.
4.2.
Let be a smooth, projective variety with canonical bundle . We recall that by Cartier duality there is functorial isomorphism [20, Proposition 5], [22],
By means of this, we obtain a natural identification
Definition 4.2.1.
A section , such that under the above isomorphism, provides a splitting of , will be called as a splitting section.
The key result we need is a criteria on the relative embedding of a smooth subvariety in a smooth, Frobenius split variety, such that the blow up along the subvariety remains Frobenius split. We recall now some of the concepts and results regarding compatible Frobenius splitting of subvarieties. Let be a Frobenius split variety, and let
be a splitting of the Frobenius morphism. Suppose is a subvariety of , defined by a sheaf of ideals . In this case we have a notion of being compatibly Frobenius split in as follows:
Definition 4.2.2.
is said to be compatibly split by in , if
Let be a nonsingular variety and a nonsingular subvariety of codimension . Denote by the blow up of along , and by the exceptional divisor. The following result follows quite easily from [18, Proposition 2.1].
Proposition 4.2.1.
Let . Suppose that is a splitting section of , and that it vanishes to order or generically along . Then extends to a splitting of . Moreover if vanishes to order generically along , then is compatibly split in .
On the other hand, we have the following criterion for a blowup to be ordinary or HodgeWitt [13], [9]:
Proposition 4.2.2.
is ordinary (or HodgeWitt) if and only if both and are ordinary (resp. HodgeWitt).
The proof of this proposition follows from the decomposition of modules, compatible with the action of the Frobenius [9, IV 1.1.9],
and the fact that a smooth, proper variety is ordinary if and only if
is an isomorphism for all and .
Combining the above propositions, we obtain the following theorem:
Theorem 4.2.3.
Let be a smooth, projective Frobenius split variety, with a splitting section as above. Suppose that vanishes to order precisely generically along a smooth subvariety of codimension in . Further assume that is not ordinary (or not HodgeWitt). Then is Frobenius split but not ordinary (resp. not HodgeWitt).
4.3.
We can now give the examples of Frobenius split varieties which are not ordinary or HodgeWitt.
Example 4.3.1.
Let be a supersingular elliptic curve in . is contained in the zero locus of nondegenerate quadric . Let and be linear polynomials such that the ideals generated by choosing any combination of define complete intersection subvarieties in . Then it can be checked using [20, Proposition 7] that the section gives rise to a splitting of , vanishing to order along . Hence the blow up of is Frobenius split (and is HodgeWitt) but is not ordinary.
Example 4.3.2.
A natural question that arises in the study of the geometry of ordinary varieties, is whether a variety is ordinary, if it is fibered over an ordinary variety, such that the smooth fibres are ordinary. The above example also provides an example of a variety fibered over , such that the (smooth) fibers are ordinary but the variety itself is not ordinary. The elliptic curve is defined as the complete intersection of two nondegenerate quadrics which generates a pencil of quadrics. The strict transform of these quadrics in the blowup gives a fibration of over by Frobenius split varieties. However Frobenius split (smooth) surfaces are ordinary, and this gives us the desired example.
Example 4.3.3.
The above example can be generalized. Let be a smooth hypersurface in , for example a Fermat hypersurface of degree . Choose a system of coordinates on , where is given by . Then is a splitting section vanishing precisely to order generically along . The blowup of along is then Frobenius split. Recall that results of [32] give explicit conditions on under which the Fermat hypersurface is not HodgeWitt. For instance assume that , does not divide and . Then this hypersurface is not HodgeWitt and so the blowup is not HodgeWitt. But the blowup is Frobenius split but neither HodgeWitt nor ordinary. The results of [32] can also be used to give examples in dimensions four and five as well.
Example 4.3.4.
Let be an elliptic curve. We embed in by using the embedding given by the linear system . The blowup of along has split and HodgeWitt reduction at all but finite number of primes. In fact, by the blowup formula for HodgeWitt cohomology, blowup of along any smooth projective embedded curve is HodgeWitt. By [8] we know that the reduction of is supersingular at infinitely many primes, and if has CM, then it has supersingular reduction at a set of primes of density . Hence there are infinitely many primes where the blowup has split (and HodgeWitt) but nonordinary reduction.
Example 4.3.5.
The above examples might lead one to raise the question whether Fano split varieties are ordinary or HodgeWitt. But even this turns out to be false. See Example 7.1.1.
5. Kummer surfaces
In this section we explore further the relationship between Frobenius split and ordinary varieties, especially in the context of Kummer surfaces.
5.1. Kummer Surfaces over perfect fields
Let be an abelian surface over a perfect field of odd, positive characteristic . Denote by , the involution on the abelian surface. Let denote the quotient variety of with respect to this involution. It is known that is Gorenstein having only quotient singularities. Denote by the dualizing sheaf. It is known that the structure sheaf on .
There exists a smooth, projective variety , which is a blow up of , at the sixteen singular points of is a surface, in that it is simply connected and is one dimensional. is the Kummer surface associated to the abelian surface
Theorem 5.1.1.
With notation as in the above theorem, is Frobenius split if and only if is Frobenius split.
Since the canonical bundles of and are trivial, we see by Theorems 2.4.1 and 2.4.2, that the above theorem is equivalent to proving the following:
Theorem 5.1.2.
Let be an abelian surface over a perfect field of characteristic . Let be the associated Kummer surface. Then is ordinary if and only if is ordinary.
Proof.
Let be a scheme over . In order to split , we need a map such that the composite with the is the identity. Suppose now that is a reduced equidimensional Gorenstein scheme. By applying duality for the Frobenius morphism, we obtain a canonical isomorphism of sheaves on , as in [22, Lemma 1],
(5.1.1) 
where denotes the dualizing sheaf of . In particular, Frobenius splittings of are induced by sections of .
Let denote any one of the varieties . By definition and are smooth, and it is known that is Gorenstein. Further the dualizing sheaf of is the structure sheaf in each of the above cases. Moreover the action is trivial on , we have a natural isomorphism
Let denote a section in any one of the above cohomology groups, and we continue to denote by , its image in the other cohomology groups. By the isomorphism 5.1.1, gives rise to a morphism To check that gives a splitting section, that the composite is the identity, it is enough to check at a point on , since is projective and any global map is a constant. By the local nature of duality, the morphism 5.1.1, is an isomorphism of sheaves, and it is enough to check the splitting condition in the formal neighborhood of a smooth point on .
We now choose to be a non torsion point on . We continue to denote by , the image of in and . We then have an isomorphism of the formal completions,
compatible with the isomorphism 5.1.1. Hence a section gives a splitting section for if and only if it gives a splitting section for , or equivalently for . Hence is split is equivalent to being split, and this is equivalent to being split. ∎
Remark 5.1.1.
Over finite fields, it is possible to give a different proof using adic methods. For a smooth, projective surface with trivial canonical bundle, the condition for being Frobenius split is that the Frobenius is an isomorphism. When the surfaces are defined over finite fields, it follows from the Katz congruence formula for the zeta function [16], applied to a surface, that there is precisely one eigenvalue of the crystalline Frobenius acting on which is a adic unit. From the shape of the Hodge polygon and duality in the case of abelian and surfaces, we then conclude that this is equivalent to surface being ordinary. By comparing and , we obtain a different proof over finite fields, that the Kummer surface is ordinary if the abelian surface is ordinary. However these adic methods do not seem to generalize to an arbitrary perfect base field.
Remark 5.1.2.
In the course of the proof of the Tate conjecture for ordinary surfaces over a finite field [26], it is shown that the KugaSatake abelian variety associated to is ordinary, provided is ordinary. If is a Kummer surface associated to an abelian surface , then it is known that is isogeneous to a sum of copies of . It follows that if is ordinary, then is ordinary.
Remark 5.1.3.
Combining the above theorem with a theorem of Ogus [27, page 372], it follows that for a Kummer surface defined over a number field, there is a finite extension over which the variety acquires ordinary reduction at a set of primes of density one. In the next section, we will show that Ogus’ proof extends to prove this result for any surface defined over a number field.
6. Primes of ordinary reduction for surfaces
The following more general question, which is one of the motivating questions for this paper, is the following conjecture which is wellknown and was raised initially for abelian varieties by Serre:
Conjecture 6.0.1.
Let be a smooth projective variety over a number field. Then there is a positive density of primes of for which has good ordinary reduction at .
Let be a number field, and let denote either an abelian variety or a surface defined over . Our aim in this section is to show that there is a finite extension of number fields, such that the set of primes of at which has ordinary reduction in the case of surfaces, or has rank at least two if is an abelian variety of dimension at least two, is of density one. Our proof closely follows the method of Ogus for abelian surfaces (see [27, page 372]).
We note here that a proof of the result for a class of K3 surfaces was also given by Tankeev (see [31]) under some what restrictive hypothesis. The question of primes of ordinary reduction for abelian varieties has also been treated recently by R. Noot (see [24]), R. Pink (see [28]) and more recently A. Vasiu (see [33]) has studied the question for a wider class of varieties. The approach adopted by these authors is through the study of MumfordTate groups.
Let be the ring of integers of ; for a finite place of lying above a rational prime , let be the completion of with respect to and let be the residue field at of cardinality . Assume that is a place of good reduction for as above and write for the reduction of at . We recall here the following facts:
6.1.
(Weil, Deligne, Ogus)[27]: The Frobenius endomorphism is a semisimple endomorphism of the adic cohomology groups for a prime . The adic characteristic polynomial is an integral polynomial and is independent of . Let
denote the trace of the adic Frobenius acting on the second étale cohomology group. is a rational integer.
6.2.
(DeligneWeil estimates) [6]: It follows from Weil estimates proved by Weil for abelian varieties and by Deligne in general that
where is a constant independent of the place .
6.3.
(KatzMessing theorem) [17]: Let denote the crystalline Frobenius on . is linear over , and the characteristic polynomials of the crystalline Frobenius and the adic Frobenius are equal:
6.4.
6.5.
(Mazur’s theorem) [19], [7], [3]: There are two parts to the theorem of Mazur that we require. After inverting finitely many primes in , we can assume that has good reduction outside . Using Proposition 6.6.1 (see below) we can assume that and are torsionfree outside a finite set of primes of . As is defined over characteristic zero, the Hodge to de Rham spectral sequence degenerates at stage. Thus all the hypothesis of Mazur’s theorem are satisfied. The two parts of Mazur’s theorem that we require are the following:
6.5.1.
(Mazur’s proof of Katz’s conjecture): Let be a finite extension of field of fractions of , over which the polynomial splits into linear factors. Let denote a valuation on such that . The Newton polygon of the polynomial lies above the Hodge polygon in degree , defined by the Hodge numbers of degree . Moreover they have the same endpoints.
6.5.2.
(Divisiblity property): The crystalline Frobenius is divisible by when restricted to .
6.6.
(Crystalline torsion): We will also need the following proposition which is certainly wellknown but as we use it in the sequel, we record it here for convenience.
Proposition 6.6.1.
Let be a smooth projective variety. Then for all but finitely many nonarchimedean places , the crystalline cohomology is torsion free for all .
Proof.
We choose a smooth model for some nonzero proper ideal . The relative de Rham cohomology of the smooth model is a finitely generated module and has bounded torsion. After inverting a finite set of primes, we can assume that is a torsionfree module, where is the ring of integers in . By the comparison theorem of Berthelot (see [2]), there is a natural isomorphism of the crystalline cohomology of to that of the de Rham cohomology of the generic fiber of a lifting to .
This proves our proposition. Moreover the proof shows that we can assume after inverting some more primes, that the Hodge filtrations are also locally free over , such that the subquotients are also locally free modules. ∎
We first note the following lemma which is fundamental to the proof.
Lemma 6.6.1.
With notation as above, assume the following:
a) if is a surface, then does not have ordinary reduction
at .
b) if is an abelian variety, then the rank of the
reduction of at , is at most 1.
Then
Proof.
Let be a valuation as in 6.5.1 above. If is an abelian variety defined over the finite field , then the rank of is precisely the number of eigenvalues of the correct power of the crystalline Frobenius acting on , which are adic units. Suppose now is an eigenvalue of the crystalline Frobenius acting on In case b), the hypothesis implies that is positive, and hence is strictly positive. As is a rational integer, the lemma follows.
When is a surface, it follows from the shapes of the Newton and Hodge polygons, that ordinarity is equivalent to the fact that precisely one eigenvalue of acting on is a adic unit. Hence if is not ordinary, then for any as above, we have is positive. Again since is a rational integer, the lemma follows. ∎
Theorem 6.6.2.
Let be a surface or an abelian variety of dimension at least two defined over a number field . Then there is a finite extension of number fields, such that

if is a surface, then has ordinary reduction at a set of primes of density one in .

if is an abelian variety, then there is set of primes of density one in , such that the reduction of at a prime has rank at least two.
Proof.
Our proof follows closely the method of Serre and Ogus (see [27]). Fix a prime , and let denote the corresponding Galois representation on . The Galois group leaves a lattice fixed, and let denote the representation of on Let be a Galois extension of containing and the roots of unity, and such that for , We have
where .
Let be a prime of of degree over lying over the rational prime Since splits completely in , and contains the roots of unity, we have Now choose . Since and is a rational integer divisible by from the above lemma, it follows on taking congruences modulo that
Now is the sum of algebraic integers each of which is of absolute value with respect to any embedding. It follows that all these eigenvalues must be equal, and equals Hence we have that
as an operator on . By the semisimplicity of the crystalline Frobenius for abelian varieties and surfaces [27], it follows that . But this contradicts the divisiblity property of the crystalline Frobenius 6.5.2, that the crystalline Frobenius is divisible by on . Hence has to be a prime of ordinary reduction, and this completes the proof of our theorem. ∎
Ogus’ method can in fact be axiomatized to give positive density results whenever certain cohomological conditions are satisfied. We present this formulation for the sake of completeness.
Proposition 6.6.3.
Let be a smooth projective variety over a number field . Assume the following conditions are satisfied:

,

The action of the crystalline Frobenius of the reduction of at a prime is semisimple for all but finite number of primes of .
Then the Galois representation is ordinary at a set of primes of positive density in and the crystal is ordinary for these primes. In other words, the motive has ordinary reduction for a positive density of primes of .
7. Primes of HodgeWitt reduction
7.1. HodgeWitt reduction
Let be a smooth projective variety over a number field . We fix a model which is regular, proper and flat and which is smooth over a suitable nonempty subset of . All our results are independent of the choice of the model. In what follows we will be interested in the smooth fibers of the map , in other words we will always consider primes of good reduction. Henceforth will always denote such a prime and the fiber over this prime will be denoted by .
Since ordinary varieties are HodgeWitt, we can formulate a weaker version of Conjecture 6.0.1.
Conjecture 7.1.1.
Let be a smooth projective variety over a number field then has HodgeWitt reduction modulo a set of primes of of positive density.
For surfaces the geometric genus appears to detect the size of the set of primes which is predicted in Conjecture 7.1.1.
Theorem 7.1.1.
Let be a smooth, projective surface with , defined over a number field . Then for all but finitely many primes , has HodgeWitt reduction at .
Proof.
By the results of [25], [11], [14], it suffices to verify that is zero for all but finite number of primes of . But the assumption that entails that . Hence by the semicontinuity theorem, for all but finite number of primes of , the reduction also has . Then by [11], [25] one sees that and so is HodgeWitt at any such prime. ∎
Corollary 7.1.1.
Let be an Enriques surface over a number field . Then has HodgeWitt reduction modulo all but finite number of primes of .
Proof.
This is immediate from the fact that for an Enriques surface over , . ∎
When is a smooth Fano surface over a number field, one can prove a little more:
Theorem 7.1.2.
Let be a smooth, projective Fano surface, defined over a number field . Then for all but finitely many primes , has ordinary reduction at and moreover the de RhamWitt cohomology of is torsion free.
Proof.
It follows from the results of [23] and [10], that if is a smooth, projective and Fano variety over a number field, for all but finitely many primes , the reduction is split. By Theorem 2.4.1 we see the reduction modulo all but finitely many primes of gives an ordinary surface. Then by Lemma 9.5 of [4] and Proposition 6.6.1, the result follows. ∎
Example 7.1.1.
Let and be any Fermat hypersurface of degree and . If then this hypersurface is Fano but by [32] this hypersurface does not have HodgeWitt reduction at primes satisfying . This gives examples of Fano varieties which are (split but are) not HodgeWitt or ordinary.
Remark 7.1.1.
It is clear from Example 7.1.1 that there exist Fano varieties over number fields which do not have HodgeWitt reduction modulo an infinite set of primes and thus this indicates that in higher dimension is not a good invariant for measuring this behavior. The following question and subsequent examples suggests that the Hodge level may intervene in higher dimensions.
Question 7.1.1.
Let be a smooth, projective Fano variety over a number field. Assume that has Hodge level in the sense of [5]. Then does have HodgeWitt reduction modulo all but a finite number of primes of ?
Remark 7.1.2.
Remark 7.1.3.
Recall that an abelian variety over a perfect field is HodgeWitt if and only if the rank of is at least (see [12]). This together with Theorem 6.6.2 gives
Theorem 7.1.3.
Let be an abelian threefold over a number field . Then there exists a set of primes of positive density in such that has HodgeWitt reduction at these primes.
7.2. HodgeWitt torsion
We include here some observations probably wellknown to the experts, but we have not found them in print. We assume as in the previous section that is smooth projective variety over a number field and that we have fixed a regular, proper model smooth over some open subscheme of the ring of integers of and whose generic fiber is .
Before we proceed we record the following:
Proposition 7.2.1.
Let be a smooth projective variety over a number field . Then there exists an integer such that for all primes in lying over any rational prime , the following dichotomy holds

either for all , the HodgeWitt groups are free modules (of finite type), or

there is some pair such that has infinite torsion.
Proof.
Choose a finite set of primes of , such that has a proper, regular model over , where denotes the ring of integers in . Choose large enough so that for any prime lying over a rational prime , we have , and all the crystalline cohomology groups of are torsionfree. We note that this choice of may depend on the choice of a regular proper model for over . If is such that are all finite type, then by the degeneration of the slope spectral sequence at the stage by BlochNygaard (see Theorem 3.7 of [11]), and the fact that the crystalline cohomology groups are torsion free, it follows that the HodgeWitt groups are free as well. If, on the other hand, some HodgeWitt group of is not of finite type over , then we are in the second case. ∎
Question 7.2.1.
Let be a smooth projective variety over a number field. When does there exist an infinite set of primes of such that the HodgeWitt cohomology groups of the reduction at are not HodgeWitt?
We would like to explicate the information encoded in such a set of primes (when it exists).
Proposition 7.2.2.
Let be an abelian surface over a number field . Then there exists infinitely many primes such that has infinite torsion if and only if there exists infinitely many primes of supersingular reduction for . In particular, let be an elliptic curve over and let . Then for an infinite set of primes of , the HodgeWitt groups are not torsion free for .
Proof.
Example 7.2.1.
Results of [32] on Fermat hypersurfaces together with foregoing discussion indicate similar examples as above. These are the only examples of this phenomena we know so far related to the above question.
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