OON PRIMES OF ORDINARY AND HODGE-WITT REDUCTION
KIRTI JOSHI
Abstract.
Jean-Pierre Serre has conjectured, in the context of abelian varieties, thatthere are infinitely primes of good ordinary reduction for a smooth, projective variety overa number field. We consider this conjecture and its natural variants. In particular weconjecture the existence of infinitely many primes of Hodge-Witt reduction (any prime ofordinary reduction is also a prime of Hodge-Witt reduction). The two conjectures are notequivalent but are related. We prove a precise relationship between the two; we prove severalresults which provide some evidence for these conjectures; we show that primes of ordinaryand Hodge-Witt reduction can have different densities. We prove our conjecture on Hodge-Witt and ordinary reduction for abelian varieties with complex multiplication. We includehere an unpublished joint result with C. S. Rajan (also independently established by FedorBogomolov and Yuri Zarhin by a different method) on the existence of primes of ordinaryreductions for K3 surfaces; our proof also shows that for an abelian threefold over a numberfield there is a set of primes of positive density at which it has Hodge-Witt reduction (thisis also a joint result with C. S. Rajan). We give a number of examples including those ofFermat hypersurfaces for which all the conjectures we make hold.
Contents
1. Introduction 21.1. Motivating question 2Acknowledgment 42. Preliminaries 42.1. Ordinary varieties 42.2. Cartier Operator 42.3. de Rham-Witt cohomology 53. Primes of ordinary reduction for K Mathematics Subject Classification.
Primary 14-XX; Secondary 11-XX. a r X i v : . [ m a t h . AG ] M a r On Ordinary and Hodge-Witt reduction
4. Primes of Hodge-Witt reduction 94.1. Hodge-Witt reduction 94.2. Hodge-Witt Abelian varieties 104.3. Primes of Hodge-Witt reduction for abelian threefolds 104.4. Hodge-Witt reduction of CM abelian varieties 125. Non Hodge-Witt reduction 155.1. Hodge-Witt torsion 166. Some explicit examples 186.1. Fermat varieties 18References 191.
Introduction
Motivating question.
Let k be a perfect field of characteristic p >
0. An abelianvariety A over k is said to be ordinary if the p -rank of A is the maximum possible, namelyequal to the dimension of A . The notion of ordinarity was extended by Mazur [30], to asmooth projective variety X over k , using notions from crystalline cohomology. A moregeneral definition was given by Bloch-Kato [5] and Illusie-Raynaud [20] using coherent coho-mology. With this definition it is easier to see that ordinarity is an open condition. Ordinaryvarieties tend to have special properties, for example the existence of canonical Serre-Tatelifting for ordinary abelian varieties to characteristic 0, and the comparison theorems be-tween crystalline cohomology and p-adic ´etale cohomology can be more easily establishedfor such varieties. In brief, ordinary varieties play a key role in the study of varieties incharacteristic p > K be a number field, and let X be a smooth, projective variety definedover K . Then there is a positive density of primes of K , at which X acquires ordinaryreduction. Ordinary varieties are Hodge-Witt , in that the de Rham-Witt cohomology groups H i ( X, W Ω jX ) are finitely generated over W . The class of Hodge-Witt varieties strictly con-tains the class of ordinary varieties. Varieties which are Hodge-Witt have distinct geometricalproperties from varieties which are non-Hodge-Witt. However we do not know how to readall the subtle information encoded in the de Rham-Witt cohomology yet. But an example ofthis behavior is implicit in the recent work of Chai-Conrad-Oort (see [8]). We note that theword ‘Hodge-Witt’ is not mentioned in loc. cit. but some of the lifting/non-lifting phenom-enon studied in [8] are dependent on whether the abelian variety in question is Hodge-Wittor non-Hodge-Witt. Hence for the reader’s convenience we provide a translation between thepoint of view of [8] and our point of view in Theorem 4.2.4. Following Serre, we ask moregenerally, if given a smooth, projective variety over a number field K , do there exist infinitelymany primes of Hodge-Witt reduction? This is our Conjecture 4.1.1. This conjecture doesnot appear to have studied before and we investigate it in section 4. We give some evidencefor this conjecture in this paper. We note that Conjecture 3.1.1 of Serre and our Conjec-ture 4.1.1 are not equivalent as there exists a smooth, projective variety over Q such that thesets of primes provided by Conjecture 3.1.1 and Conjecture 4.1.1 are of different densities(see Example 6.1.1). But the two Conjectures are almost equivalent: in Theorem 4.1.3 we irti Joshi 3 prove the precise relationship between the two conjectures. One of the main results is thatthis is true for abelian threefolds over number fields (this is joint work with C. S. Rajan–eventhough it is not stated in its present form in [21]).Some of the results of this note were included section 6,7 of my 2001 preprint [21] withC. S. Rajan. However when that preprint was accepted for publication in the InternationalMath. Res. Notices (see [22]), the referee felt that the result for K3 surfaces might be well-known to experts and at any rate the contents of sections 6,7 of that preprint be publishedindependently as the results contained in these two sections were independent of the mainresults of that paper (namely construction of examples of Frobenius split non-ordinary va-rieties). Subsequent to this the authors lost interest in separate publication of the contentsof these two sections. One of the important results of the aforementioned sections of [21] isabout ordinary reductions of K Q then any prime of Hodge-Wittreduction is a prime of ordinary reduction, but if the CM field is non-galois over Q , then theset of primes of Hodge-Witt reduction and the set of primes of ordinary reduction may notcoincide and we give an example of a CM field (and hence an abelian variety with CM bythis field) where the density of primes of Hodge-Witt reduction is while the set of primesof ordinary reduction is of density .In a different direction, Mustata-Srinivas and Mustata (see [35, 36]) have related theexistence of weak ordinary reductions to invariants of singularities (of subvarieties) andthis work has also led to some interest in ordinary varieties. We note, as an aside, thatCorollary 4.4.7 implies that the conjecture of Mustata-Srinivas (see [35, 36, Conjecture 1.1])is true for abelian varieties over number fields with CM (see Corollary 4.4.9). Since anyabelian variety with CM is definable over a number field, we see that the conjecture ofMustata-Srinivas is true for any abelian variety with CM over any field of characteristiczero. On Ordinary and Hodge-Witt reduction
In the final section we study the presence of infinite torsion in the de Rham-Witt coho-mology of varieties and discuss some questions and conjectures. In particular in Conjec-ture 5.1.5 we conjecture that for a smooth, projective variety X over a number field K ,with H n ( X, O X ) (cid:54) = 0, there exist infinitely many primes of K at which X has good, non-Hodge-Witt reduction. This conjecture includes, as a special case, the conjecture that thereexists infinitely many primes of good, supersingular reduction for any elliptic curve over K (for K = Q , this is the well-known theorem of [13]). Our conjecture also implies a strongerassertion: given any two elliptic curves over a number field K , there exist infinitely manyprimes of K at which both the curves have good supersingular reduction. It also includesthe assertion that given any g non-isogenous elliptic curves over a number field, then thereare infinitely many primes p such that at most g − Acknowledgment.
Some parts of this paper were mostly written while the author was at,or visiting, the Tata Institute of Fundamental Research and support and hospitality of theinstitute is gratefully acknowledged. We also thank Vikram Mehta for conversations. Wethank C. S. Rajan for many conversations and also for allowing us to include our previouslyunpublished joint results here. We would also like to thank Karl Schwede and Kevin Tucker,organizers of the 2011 AIMS workshop on “Relating test ideals and multiplier ideals” forgiving me an opportunity to speak on results of [21]. We decided to revise the unpublishedsections of [21] because of this workshop and the support and hospitality of AIMS is gratefullyacknowledged. I thank the anonymous referee for suggesting a number of improvements andalso for pointing out a mistake in the earlier version of Proposition 4.4.2 (I had claimed astronger assertion than what is now proved–but this is quite adequate for what is neededhere) and for a careful reading of the manuscript.2.
Preliminaries
Ordinary varieties.
Let X be a smooth projective variety over a perfect field k ofpositive characteristic. Following Bloch-Kato [5] and Illusie-Raynaud [20], we say that X isordinary if H i ( X, B jX ) = 0 for all i ≥ , j >
0, where B jX = image (cid:0) d : Ω j − X → Ω jX (cid:1) . If X is an abelian variety, then it is known that this definition coincides with the usualdefinition [5]. By [19, Proposition 1.2], ordinarity is an open condition in the followingsense: if X → S is a smooth, proper family of varieties parameterized by S , then the set ofpoints s in S , such that the fiber X s is ordinary is a Zariski open subset of S . Although thefollowing proposition is well known, we present it here as an illustration of the power of thisfact.2.2. Cartier Operator.
Let X be a smooth proper variety over a perfect field of charac-teristic p >
0, and let F X (or F ) denote the absolute Frobenius of X . We recall a few basicfacts about Cartier operators from [17]. The first fact we need is that we have a fundamentalexact sequence of locally free sheaves(2.2.1) 0 → B iX → Z iX C −→ Ω iX → , where Z iX is the sheaf of closed i -forms, where C is the Cartier operator. The existence ofthis sequence is the fundamental theorem of Cartier (see [17]). Since the Cartier operator irti Joshi 5 is also the trace map in Grothendieck duality theory for the finite flat map F , we have aperfect pairing(2.2.2) F ∗ (Ω jX ) ⊗ F ∗ (Ω n − jX ) → Ω nX where n = dim( X ), and the pairing is given by ( ω , ω ) (cid:55)−→ C ( w ∧ w ). This pairing isperfect and O X -bilinear (see [32]).In particular, on applying Hom( − , Ω nX ) to the exact sequence(2.2.3) 0 → B nX → Z nX → Ω nX → → O X → F ∗ ( O X ) → B X → de Rham-Witt cohomology. The standard reference for de Rham-Witt cohomologyis [17]. Throughout this section, the following notations will be in force. Let k be analgebraically closed field of characteristic p >
0, and X a smooth, projective variety over k .Let W = W ( k ) be the ring of Witt vectors of k . Let K = W [1 /p ] be the quotient field of W . Note that as k is perfect, W is a Noetherian local ring with a discrete valuation andwith residue field k . For any n ≥
1, let W n = W ( k ) /p n . W comes equipped with a lift σ : W → W , of the Frobenius morphism of k , which will be called the Frobenius of W . Wedefine a non-commutative ring R = W σ [ V, F ], where
F, V are two indeterminate subject tothe relations
F V = V F = p and F a = σ ( a ) F and aV = V σ ( a ). The ring R is called theDieudonne ring of k . The notation is borrowed from [20].Let { W n Ω ∗ X } n ≥ be the de Rham-Witt pro-complex constructed in [17]. It is standardthat for each n ≥ , i, j ≥ H i ( X, W n Ω jX ) are of finite type over W n . We define(2.3.1) H i ( X, W Ω jX ) = lim ←− n H i ( X, W n Ω jX ) , which are W -modules of finite type up to torsion. These cohomology groups are calledHodge-Witt cohomology groups of X . Definition 2.3.1. X is Hodge-Witt if for i, j ≥
0, the Hodge-Witt cohomology groups H i ( X, W Ω jX ) are finite type W -modules.The properties of the de Rham-Witt pro-complex are reflected in these cohomology mod-ules and in particular we note that for each i, j , the Hodge-Witt groups H i ( X, W Ω jX ) are leftmodules over R . The complex W Ω ∗ X defined in a natural manner from the de Rham-Wittpro-complex computes the crystalline cohomology of X and in particular there is a spectralsequence(2.3.2) E i,j = H i ( X, W Ω jX ) ⇒ H ∗ cris ( X/W )This spectral sequence induces a filtration on the crystalline cohomology of X which iscalled the slope filtration and the spectral sequence above is called the slope spectral sequence(see [17]). It is standard (see [17] and [20]) that the slope spectral sequence degenerates at E modulo torsion (i.e. the differentials are zero on tensoring with K ) and at E up to finitelength (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 biggerring than R . This ring was introduced in [20]. Let R = R ⊕ R be a graded W -algebrawhich is generated in degree 0 by variables F, V with the properties listed earlier (so R isthe Dieudonne ring of k ) and R is a bimodule over R generated in degree 1 by d with the On Ordinary and Hodge-Witt reduction properties d = 0 and F dV = d , and da = ad for any a ∈ W . The algebra R is called theRaynaud-Dieudonne ring of k (see [20]). The complex ( E ∗ ,i , d ) is a graded module over R and is in fact a coherent, left R -module (in a suitable sense, see [20]).For a general variety X , the de Rham-Witt cohomology groups are not of finite type over W , and the structure of these groups reflects the arithmetical properties of X . For instance,in [5], [20] it is shown that for ordinary varieties H i ( X, W Ω j ) are of finite type over W .3. Primes of ordinary reduction for K surfaces Serre’s conjecture.
The following more general question, which is one of the motivat-ing questions for this paper, is the following conjecture which is well-known and was raisedinitially for abelian varieties by Serre:
Conjecture 3.1.1.
Let
X/K be a smooth projective variety over a number field. Thenthere is a positive density of primes v of K for which X has good ordinary reduction at v .Let K be a number field, and let X denote either an abelian variety or a K K . Our aim in this section is to show that there is a finite extension L/K ofnumber fields, such that the set of primes of L at which X has ordinary reduction in the caseof K p -rank at least two if X is an abelian variety of dimension at leasttwo, is of density one. Our proof closely follows the method of Ogus for abelian surfaces (see[40, page 372]).We note here that a proof of the result for a class of K3 surfaces was also given byTankeev (see [46]) under some what restrictive hypothesis. The question of primes of ordinaryreduction for abelian varieties has also been treated recently by R. Noot (see [37]), R. Pink(see [41]) and more recently A. Vasiu (see [48]) has studied the question for a wider classof varieties. The approach adopted by these authors is through the study of Mumford-Tategroups.Let O K be the ring of integers of K ; for a finite place v of K lying above a rational prime p , let O v be the completion of O K with respect to v and let k v be the residue field at v ofcardinality q v = p c v . Assume that v is a place of good reduction for X as above and write X v for the reduction of X at v . We recall here the following facts:3.2. Trace of Frobenius.
The Frobenius endomorphism F v is an endomorphism of the l -adic cohomology groups H il := H iet ( X ⊗ K, Q l ) for a prime l (cid:54) = p . The l -adic character-istic polynomial P i,v ( t ) = det(1 − tF v | H iet ( X ⊗ K, Q l )) is an integral polynomial and isindependent of l . Let a v = Tr( F v | H et ( X ⊗ K, Q l )denote the trace of the l -adic Frobenius acting on the second ´etale cohomology group. Then a v is a rational integer (see [40]).3.3. Deligne Weil estimate. (Deligne-Weil estimates) [10]: It follows from Weil estimatesproved by Weil for abelian varieties and by Deligne in general that | a v | ≤ dp where d = dim H l is a constant independent of the place v . irti Joshi 7 Katz-Messing theorem.
Let φ v denote the crystalline Frobenius on H icris ( X/W ( k v )). φ c v v is linear over W ( k v ), and the characteristic polynomials of the crystalline Frobenius andthe l -adic Frobenius are equal: P i,v ( t ) = det(1 − tφ c v v | H icris ( X/W ( k v )) ⊗ K v ) , (see [25]).3.5. Semi-simplicity of the crystalline Frobenius. If X is a K φ c v v is semi-simple. We remarkthat although this result is not essential in the proof of the theorem, it simplifies the proof.3.6. Mazur’s theorem.
We recall from [30], [11], [4] the following theorem of B. Mazur.There are two parts to the theorem of Mazur that we require and we record them separatelyfor convenience. After inverting finitely many primes v ∈ S in K , we can assume that X has good reduction outside S . Using Proposition 3.7.1 (see below) we can assume that H i ( X, Ω jX ) and H i ( X/W ( k v )) are torsion-free outside a finite set of primes of K . As X isdefined over characteristic zero, the Hodge to de Rham spectral sequence degenerates at E stage. Thus all the hypothesis of Mazur’s theorem are satisfied. The two parts of Mazur’stheorem that we require are the following:3.6.1. Hodge and Newton polygons: Mazur’s proof of Katz’s conjecture.
Let L v be a finiteextension of field of fractions of W ( k v ), over which the polynomial P i,v ( t ) splits into linearfactors. Let w denote a valuation on L v , such that w ( p ) = 1. The Newton polygon of thepolynomial P i,v ( t ) lies above the Hodge polygon in degree i , defined by the Hodge numbers h j,i − j of degree i . Moreover they have the same endpoints.3.6.2. Divisibility.
The crystalline Frobenius φ v is divisible by p i when restricted to F i H jdR ( X/W ( k v )) := H ( X/W ( k v )) , Ω ≥ iX ) . Crystalline torsion.
We will also need the following proposition (from [21]) which iscertainly well-known but as we use it in the sequel, we record it here for convenience.
Proposition 3.7.1.
Let
X/K be a smooth projective variety. Then for all but finitely manynonarchimedean places v , the crystalline cohomology H i cris ( X v /W ( k v )) is torsion free for all i .Proof. We choose a smooth model
X →
Spec( O K ) − V ( I ) for some non-zero proper ideal I ⊂ O K . The relative de Rham cohomology of the smooth model X is a finitely generated O K -module and has bounded torsion. After inverting a finite set S of primes, we can assumethat H idR ( X , O S ) is a torsion-free O S module, where O S is the ring of S -integers in K . By thecomparison theorem of Berthelot (see [3]), there is a natural isomorphism of the crystallinecohomology of X p to that of the de Rham cohomology of the generic fiber of a lifting to Z p . H i cris ( X v /W ( k v )) ≡ H idR ( X , O S ) ⊗ W ( k v ) . This proves our proposition. Moreover the proof shows that we can assume after invertingsome more primes, that the Hodge filtrations F j H i dR ( X , O S ) are also locally free over O S ,such that the sub-quotients are also locally free modules. (cid:3) We first note the following lemma which is fundamental to the proof.
Lemma 3.7.2 (Joshi-Rajan) . With notation as above, assume the following:
On Ordinary and Hodge-Witt reduction (1) if X is a K -surface, then X does not have ordinary reduction at v . (2) if X is an abelian variety, then the p -rank of the reduction of X at v , is at most 1.Then p | a v . Proof.
Let w be a valuation as in 3.6.1 above. If X is an abelian variety defined over thefinite field k v , then the p -rank of X is precisely the number of eigenvalues of the correct powerof the crystalline Frobenius acting on H ( X/W ( k v )) ⊗ Q p , which are p -adic units. Supposenow α is an eigenvalue of the crystalline Frobenius φ c v v acting on H ( X/W ( k v )) ⊗ Q p . Incase (2), the hypothesis implies that w ( α ) is positive, and hence w ( a v ) is strictly positive.As a v is a rational integer, the lemma follows.When X is a K φ a v v acting on H ( X/W ( k v )) ⊗ Q p is a p -adic unit. Hence if X is not ordinary, then for any α as above, wehave w ( α v ) is positive. Again since a v is a rational integer, the lemma follows. (cid:3) The following theorem was proved in [21]. We note that the first assertion of Theorem 3.7.3was also independently established by Fedor Bogomolov and Yuri Zarhin (see [6]). Theirmethod is different method from the one adopted here.
Theorem 3.7.3 (Joshi-Rajan) . Let X be a K surface or an abelian variety of dimensionat least two defined over a number field K . Then there is a finite extension L/K of numberfields, such that (1) if X is a K -surface, then X × K L has ordinary reduction at a set of primes ofdensity one in L . (2) if X is an abelian variety, then there is set of primes O of density one in L , suchthat the reduction of X × K L at a prime p ∈ O has p -rank at least two.Proof. Our proof follows closely the method of Serre and Ogus (see [40]). Fix a prime l , andlet ρ l denote the corresponding galois representation on H l . The galois group G K leaves alattice V l fixed, and let ρ l denote the representation of G L on V l ⊗ Z /l Z . Let L be a galoisextension of Q , containing K and the l th roots of unity, and such that for σ ∈ G L , ρ l ( σ ) = 1 . We have a v ≡ d (mod l ) , where d = dim H l .Let v be a prime of L of degree 1 over Q , lying over the rational prime p. Since p splitscompletely in L , and L contains the l th roots of unity, we have p ≡ l ) . Now choose l > d . Since | a v | ≤ dp and is a rational integer divisible by p from the above lemma, itfollows on taking congruences modulo l that a v = ± dp. Now a v is the sum of d algebraic integers each of which is of absolute value p with respectto any embedding. It follows that all these eigenvalues must be equal, and equals ± p. Hencewe have that F v = ± pI as an operator on H l . By the semi-simplicity of the crystalline Frobenius for abelian vari-eties and K φ v = ± pI . But this contradicts the divisibilityproperty of the crystalline Frobenius 3.6.2, that the crystalline Frobenius is divisible by p irti Joshi 9 on F H dR ( X × k v ). Hence v has to be a prime of ordinary reduction, and this completes theproof of our theorem. (cid:3) Ogus’ method can in fact be axiomatized to give positive density results whenever certaincohomological conditions are satisfied. We present this formulation for the sake of complete-ness.
Proposition 3.7.4 (Joshi-Rajan) . Let X be a smooth projective variety over a number field K . Assume the following conditions are satisfied: (1) dim H ( X, O X ) = 1 , (2) The action of the crystalline Frobenius of the reduction X p of X at a prime p issemi-simple for all but finite number of primes p of K .Then the galois representation H et ( X, Q (cid:96) ) is ordinary at a set of primes of positive density in K and the F -crystal H cris ( X p /W ) is ordinary for these primes. In other words, the motive H ( X ) has ordinary reduction for a positive density of primes of K . Primes of Hodge-Witt reduction
Hodge-Witt reduction.
Let X be a smooth projective variety over a number field K .We fix a model X →
Spec( O K ) which is regular, proper and flat and which is smooth over asuitable non-empty subset of Spec( O K ). All our results are independent of the choice of themodel. In what follows we will be interested in the smooth fibers of the map X →
Spec( O K ),in other words we will always consider primes of good reduction. Henceforth p will alwaysdenote such a prime and the fiber over this prime will be denoted by X p .Since ordinary varieties are Hodge-Witt, we can formulate a weaker version of Conjec-ture 3.1.1. Conjecture 4.1.1 (Joshi-Rajan) . Let
X/K be a smooth projective variety over a numberfield then X has Hodge-Witt reduction modulo a set of primes of K of positive density. Remark 4.1.2.
Let us understand what our conjecture means in the context of a K3 surface.Let
X/K be a smooth, projective K3 surface over a number field K . So Conjecture 4.1.1predicts in this case that there are infinitely many primes of Hodge-Witt reduction for X .Now X has Hodge-Witt reduction at a prime p of K if and only if X is of finite height (i.e.the formal group attached to X by [1] is of finite height).As we remarked in the Introduction, Conjecture 4.1.1 and Conjecture 3.1.1 are not equiv-alent (see Example 6.1.1). The two however are related and the following clarifies the rela-tionship between Conjecture 3.1.1 and Conjecture 4.1.1. Our theorem is the following: Theorem 4.1.3.
Let X be a smooth, projective variety over a number field K . Then thefollowing are equivalent: (1) There are infinitely many primes of ordinary reduction for X . (2) There are infinitely many primes of ordinary reduction for X × K X (3) There are infinitely many primes of Hodge-Witt reduction for X × K X .Proof. Let us equip ourselves with flat, regular, proper models for X and X × K X over aZariski-open subscheme of U ⊂ Spec( O K ) with smooth, proper, fibres over U . Let p ∈ U bea prime of good, ordinary reduction for X . As a product of ordinary varieties is ordinary(see [19]), so X × K X is also ordinary at p . Thus (1) = ⇒ (2). Since any prime p of good, ordinary reduction for X × K X is also a prime of good, Hodge-Witt reduction for X × K X ,so we have (2) = ⇒ (3). Let us prove that (3) = ⇒ (1). Assume p is a prime of Hodge-Wittreduction for X × K X . By [12, III, Prop 2.1(ii) and Prop 7.2(ii)] we see that a product isHodge-Witt if and only if at least one factor is ordinary and other is Hodge-Witt. Thus if X × K X is Hodge-Witt at p then X must be ordinary at p . Thus X is ordinary at p . (cid:3) Hodge-Witt Abelian varieties.
Let X be a p -divisible group over a perfect field k of characteristic p >
0. Let X t denote the Cartier dual of X and let X (0 , be the local-localpart of X . Following [8, Def 3.4.2] we say that a p -divisible group X is of extended Lubin-Tate type if dim X (0 , ≤ X t (0 , ≤
1. Equivalently X is of extended Lubin-Tate typeif X isogenous to an ordinary p -divisible group or X is isogenous to a product of an ordinary p -divisible group and one copy of the p -divisible group of a supersingular elliptic curve. Webegin with the following proposition: Proposition 4.2.1.
Let k be a perfect field of characteristic p > . Let A be an abelianvariety over k . Then the following are equivalent: (1) A is Hodge-Witt. (2) A has p -rank ≥ dim( A ) − . (3) The slopes of Frobenius of A are in { , , } and multiplicity of is at most two. (4) The p -divisible group A [ p ∞ ] is of extended Lubin-Tate type.Proof. The equivalence (1) ⇐⇒ (2) is [18]. The equivalence (2) ⇐⇒ (3) ⇐⇒ (4) is immedi-ate from the definition. (cid:3) Remark 4.2.2.
In [8] an abelian variety A with p -rank equal to dim( A ) − almost ordinary abelian variety . Thus we see that A is Hodge-Witt if and only if A is ordinaryor A is almost ordinary. Remark 4.2.3.
Properties of Hodge-Witt varieties and non-Hodge-Witt varieties can berather distinct. A classic example of this phenomenon is implicit in [8]. Since the phrase
Hodge-Witt is never mentioned in [8] we provide the following transliteration of the relevantresults.
Theorem 4.2.4 (Reformulation of [8, Theorem 3.5.1, Corollary 3.5.6 and Remark 3.5.7]) . Let k be a perfect field of characteristic p > , let A be an abelian variety with a CM algebra K with dim( K ) = 2 dim( A ) and an embedding K (cid:44) → End (A) = End (A) . If A is Hodge-Witt, then there exists an isogeny A (cid:48) → A such that A (cid:48) admits a CM-lifting to characteristiczero. On the other hand if A is not Hodge-Witt, there exists an isogeny A (cid:48) → A , such that A (cid:48) does not admit any CM-lifting to characteristic zero. Primes of Hodge-Witt reduction for abelian threefolds.
Our next theorem showsthat the Conjecture 4.1.1 is true for abelian varieties of dimension at most three. This resultis implicit in [21].
Theorem 4.3.1 (Joshi-Rajan) . Let
A/K be an abelian variety of dimension at most threeover a number field K . Then there exists a finite extension L/K and a set of primes ofdensity one in L at which A has Hodge-Witt reduction.Proof. The case dim( A ) = 1 is immediate as smooth, projective curves are always Hodge-Witt. Next assume that dim( A ) = 2, then we know by Ogus’s result that an abelian surface irti Joshi 11 defined over a number field has a positive density of primes of ordinary reduction. So we aredone in this case.Thus we need to address dim( A ) = 3. Recall that an abelian variety A over a perfect fieldis Hodge-Witt if and only if the p -rank of A is at least dim( A ) − (cid:3) For surfaces the geometric genus appears to detect the size of the set of primes which ispredicted in Conjecture 4.1.1.
Theorem 4.3.2 (Joshi-Rajan) . Let X be a smooth, projective surface with p g ( X ) = 0 ,defined over a number field K . Then for all but finitely many primes p , X has Hodge-Wittreduction at p .Proof. By the results of [38], [17], [20], it suffices to verify that H ( X p , W ( O X p )) = 0for all but a finite number of primes p of K . But the assumption that p g ( X ) = 0 entailsthat H ( X, O X ) = H ( X, K X ) = 0. Hence by the semicontinuity theorem, for all but finitenumber of primes p of K , the reduction X p also has p g ( X p ) = 0. Then by [17], [38] one seesthat H ( X p , W ( O X p )) = 0 and so X p is Hodge-Witt at any such prime. (cid:3) Corollary 4.3.3 (Joshi-Rajan) . Let
X/K be an Enriques surface over a number field K .Then X has Hodge-Witt reduction modulo all but finite number of primes of K .Proof. This is immediate from the fact that for an Enriques surface over K , p g ( X ) = 0. (cid:3) When X is a smooth Fano surface over a number field, one can prove a little more: Theorem 4.3.4 (Joshi-Rajan) . Let X be a smooth, projective Fano surface, defined over anumber field K . Then for all but finitely many primes p , X has ordinary reduction at p andmoreover the de Rham-Witt cohomology of X p is torsion free.Proof. It follows from the results of [34] and [16], that if X is a smooth, projective andFano variety X over a number field, for all but finitely many primes p , the reduction X p isFrobenius split. By [22, Proposition 3.1] we see the reduction modulo all but finitely manyprimes p of K gives an ordinary surface. Then by Lemma 9.5 of [5] and Proposition 3.7.1,the result follows. (cid:3) Example 4.3.5 (Joshi-Rajan) . Let K = Q and X ⊂ P n be any Fermat hypersurfaceof degree m and n ≥
6. If m < n + 1 then this hypersurface is Fano but by [47] thishypersurface does not have Hodge-Witt reduction at primes p satisfying p (cid:54)≡ m . Thisgives examples of Fano varieties which are ( F -split but are) not Hodge-Witt or ordinary. Remark 4.3.6 (Joshi-Rajan) . It is clear from Example 4.3.5 that there exist Fano varietiesover number fields which have non-Hodge-Witt reduction modulo an infinite set of primesand thus this indicates that in higher dimension p g ( X ) is not a good invariant for measuringthis behavior. The following question and subsequent examples suggests that the Hodgelevel may intervene in higher dimensions (see [9, 42] for the definition of Hodge level). Question 4.3.7 (Joshi-Rajan) . Let
X/K be a smooth, projective Fano variety over a numberfield. Assume that X has Hodge level ≤ X have Hodge-Witt reduction modulo all but a finite number of primes of K ? Remark 4.3.8 (Joshi-Rajan) . A list of all the smooth complete intersection in P n whichare of Hodge level ≤ Remark 4.3.9.
In [23] we have answered the Question 4.3.7 affirmatively for dim( X ) = 3where the Hodge level condition is automatic.4.4. Hodge-Witt reduction of CM abelian varieties.
We will recall a few facts aboutCM abelian varieties. For a modern reference for CM abelian varieties reader may consult[8]. Let K be a CM field. Let L be a number field and let A be an abelian variety over L with End (A) = K. In other words L is an abelian variety with complex multiplication(“CM”) by K . We will assume that , by passing to a finite extension i if required that L contains(1) the galois closure N of K (2) and that A has good reduction at all finite primes of L .Assumption (2) is guaranteed by the theory of Complex multiplication (see [8]). Let Φ bethe CM-type of A . Let E ( K, Φ) be the reflex field. In this situation by [49] we know thatthe Dieudonne module of the reduction ¯ A at any finite prime p of L is determined by Φ andthe splitting of rational primes in K . In particular it is possible to determine completely thereduction type of A from the decomposition of p in O K . But in general this is a complicatedcombinatorial problem (see [14, 44, 50]) depending on the galois closure of K and also onthe CM-type. In the present section we give sufficient conditions on prime decomposition of p which allows us to conclude the infinitude of primes of Hodge-Witt and ordinary reductionfor A . Before proceeding we make some elementary observations. Definition 4.4.1.
Let K/ Q be a finite extension. Let p be a prime number which is unram-ified in K . We say that p splits almost completely in K and we have a prime factorization:( p ) = n (cid:89) i =1 p i , with n ≥ [ K : Q ] − K : Q ] > n = [ K : Q ] if [ K : Q ] ≤
2. We say that p splitscompletely in K if we have a prime factorization:( p ) = n (cid:89) i =1 p i , with n = [ K : Q ]. Let f i be the degree of the residue field extension of p i over Z /p . Thenthe residue field degree sequence of p in K is the tuple ( f , f , . . . ).It is clear from Definition 4.4.1 that if p splits completely in K then p splits almostcompletely in K . Also from our definition p splits almost completely in a quadratic extensionif and only if p splits completely. The following proposition is an elementary consequence ofDefinition 4.4.1 and standard facts about splitting of primes in K but we give a proof forcompleteness. Proposition 4.4.2.
Suppose [ K : Q ] = 2 g be a CM field and let K ⊂ K be the totally realsubfield of K . Then we have the following assertions: (1) If p splits completely in K then p splits completely in K . irti Joshi 13 (2) If p splits almost completely, but not completely in K then p splits completely in K and exactly one prime lying over p in K is inert in K and the rest split in K . (3) If K/ Q is galois then p splits almost completely in K if and only if p splits completelyin K .Proof. Let [ K : Q ] = 2 g . Let(4.4.1) ( p ) = p · · · p n be the prime factorization of p in K (recall that by definition p splits completely means p isunramified) and let(4.4.2) ( p ) = q · · · q (cid:96) , be the prime factorization of p in K . Let m be the number of primes in q , . . . , q (cid:96) whichsplit in K . By renumbering the p , . . . , p n we can assume that p , . . . , p m are the primeslying over the split primes q , . . . , q m . The rest lie over primes of K (over p ) which are inertin K . Thus we have(4.4.3) 2 g = 2 m + ( (cid:96) − m )equivalently(4.4.4) 2 g = (cid:96) + m. Further if f ( p | p ) denotes the degree of the residue field of a prime p lying over p then wehave for K the equation(4.4.5) (cid:96) ≤ (cid:96) (cid:88) i =1 f ( q i | p ) = g. Thus we see that 2 g = (cid:96) + m ≤ (cid:96) ≤ g as m ≤ (cid:96) . Thus we have 2 g ≤ (cid:96) ≤ g so (cid:96) = g .Thus p splits completely in K . This proves the first assertion.Now suppose p splits almost completely in K but not completely in K . Then [ K : Q ] =2 g > p splits almost completely for [ K : Q ] = 2 if and only if p splits in K . So if p is a such a prime then n = 2 g − > g >
1. Further, usingthe notation established in the previous case we have(4.4.6) 2 m + ( (cid:96) − m ) = (cid:96) + m = 2 g − . Thus (cid:96) + m = 2 g −
1. We claim that m ≥ g −
1. Suppose m < g − g − (cid:96) + m < (cid:96) + g − , so g < (cid:96) . But as (cid:96) ≤ (cid:80) (cid:96)i =1 f ( q i | p ) = g one gets (cid:96) ≤ g . So g < (cid:96) and (cid:96) ≤ g which is acontradiction. This gives m ≥ g −
1. If m = g then we are in the completely split case.So m = g −
1. Then 2 g − (cid:96) + m = (cid:96) + g − (cid:96) = g . Further as g > m = g − > q , . . . , q g is inert in K as claimed. Thisproves the second assertion.Finally suppose K/ Q is galois and p splits almost completely then p is completely splitas all the residue field degrees are equal, while from the preceding discussion exactly oneresidue field degree of prime lying over p in K is two if p splits almost completely but notcompletely split, so we are done by the second assertion. (cid:3) The following is also immediate from this proof.
Corollary 4.4.3.
Let K be a CM field. Let p be a rational prime which is unramified in K .Then the following are equivalent: (1) p splits almost completely in K . (2) the residue field degree sequence of p in K is (1 , , , · · · ) or (2 , , , · · · ) . We are now ready to prove our theorem.
Theorem 4.4.4.
Let
A, L, K be as above with K a CM field. Let p L be a prime of L lyingover an unramified prime p of Q . If p splits almost completely in K , then A has Hodge-Wittreduction at p L .Proof. We follow a method due to [14, 44, 50]–especially [44] who proved the result when p is completely split (in which case A has ordinary reduction at p L )–also see [14, 50]. We notethat [14] spells out the details quite well (for abelian surfaces–and is also enough to provewhat we need here). Suppose p L is a prime lying over p in L . Let κ be the residue field of p L .Then the reduction of A at p L is a abelian scheme over κ . It will be convenient to extendthe base field of the reduction from κ to an algebraic closure ¯ κ of κ . Let X be the p -divisiblegroup of this abelian scheme over ¯ κ . Let O K be the ring of integers of K , let K ⊂ K be thetotally really subfield and O K be the ring of integers of K . Let σ : O K → O K be complexconjugation. Then O K operates on the p -divisible group X .Thus we are in the following situation: we have a p -divisible group X over ¯ κ with CMby K and we have to show that if p splits almost completely in K then X is of extendedLubin-Tate type.Let M = M ( X ) be the Dieudonne module of X . Then M is a W = W (¯ κ )-module and letus note, by the standard theory of complex multiplication (see [8]), that M is an O K -moduleof rank one and hence, or at any rate, M is an O K ⊗ Z p -module. We may decompose O K ⊗ Z p using the factorization of ( p ). This also gives us a decomposition of the Dieudonne module M = M ( X ) of X .Assume that [ K : Q ] = 2 g and ( p ) = (cid:81) ni =1 p i . Then O K ⊗ Z p = n (cid:89) i =1 W ( k ( p i ))where k ( p i ) is the residue field of p i . Then we have to show that if n ≥ g − X is ofextended Lubin-Tate type.If n = 2 g then p splits completely and we are done by [44], but we recall his argumenthere as it is also needed in case p splits almost completely. If p is completely split, then byProposition 4.4.2(1) every p i pairs uniquely with a prime p j such that p j = σ ( p i ) (i.e p i , p j live over a prime of O K which splits completely in O K ). The factor of M , correspondingto p i , p j provides a Dieudonne module of a p -divisible group, which descends to the residuefield of the unique prime of O K which splits into p i , p j in O K , gives a p -divisible group oftype G , × G , (in the standard notation of p -divisible groups) with slopes 0 ,
1. If p iscompletely split, then there are exactly g such factors and hence X is ordinary p -divisiblegroup and hence A has ordinary reduction at p L . If p splits almost completely but does notsplit completely then by Proposition 4.4.2(2) all but one of p i can be paired with exactlyone p j as above and there are exactly 2 g − p g − (after renumbering if required), lives over the unique prime of O K lying over p which is inertin K . As is shown in [44], the factor of M corresponding to p g − gives an indecomposableDieudonne module of rank two over the residue field of p g − with slope , and hence a irti Joshi 15 p -divisible group G , . Thus in this case X is extended Lubin-Tate group of height 2 g . Thiscompletes the proof of the theorem. (cid:3) Theorem 4.4.5.
Let
A, L, K be as in 4.4. Then there exists infinitely many primes of Hodge-Witt reduction. In particular Conjecture 4.1.1 is true for abelian varieties with complexmultiplication and in general the density of such primes is greater than the density of primesof ordinary reduction.Proof.
This is a consequence of Theorem 4.4.4 and the Chebotarev density theorem. Let N be the galois closure of K in ¯ Q , let G = Gal( N/ Q ). Let H ⊂ G be the subgroup fixing H . Then action of G on the coset space G/H embeds
G (cid:44) → S n where n = [ G : H ] andin particular Frob p ∈ G acts on G/H (by permutation). This action depends only on theconjugacy class of Frob p . It is well-known, see for instance [28] and [27] that the splittingtype is determined by the cycle decomposition of Frob p on G/H and that the splitting typeof p in K is determined by the sequence of residue field degrees. In particular we can “readoff” all the information we need from the table of conjugacy classes of G . By Corollary 4.4.3we see that p splits completely if and only if the cycle decomposition is identity, and the p splits almost completely if the cycle decomposition is identity or exactly one transposition.Let G tr be the union of the conjugacy classes of these two types. Then G tr ⊇ { } andhence G tr (cid:54) = ∅ . In particular we see that the density of primes of Hodge-Witt reduction is | G tr || G | ≥ | G | > (cid:3) Let us first state some obvious corollaries.
Corollary 4.4.6.
Let K be a CM field with galois closure N/ Q and G = Gal( N/ Q ) . Let G tr be as above. Then the density of primes of Hodge-Witt reduction for A is at least | G tr || G | and the density of primes of ordinary reduction is at least | G | . Corollary 4.4.7. If A, L, K, p L are as in Theorem 4.4.4. Then Conjecture 3.1.1 is true for A .Proof. This is clear from the preceding results. (cid:3)
Corollary 4.4.8. If A, L, K, p L are as above, then Question 5.1.2 has an affirmative answerfor CM abelian varieties. Corollary 4.4.9.
Let
A/L be an abelian variety over a field of characteristic zero. Assume A has complex multiplication by a CM field K . Then the conjecture of Mustata-Srinivas (see [35, Conjecture 1.1] ) is true for A . Non Hodge-Witt reduction
Let us briefly discuss non Hodge-Witt reduction. By definition a smooth, projective varietyis non Hodge-Witt if and only if some de Rham-Witt cohomology group H i ( X, W Ω jX ) is notfinitely generated as a W -module. In other words, for some i, j ≥ H i ( X, W Ω jX ) has infinitetorsion. In this section we briefly describe what we know about primes of non Hodge-Wittreduction and what we expect to be true and how this relates to some of other well-knownconjectures. Hodge-Witt torsion.
We include here some observations probably well-known to theexperts, but we have not found them in print. We assume as in the previous section that
X/K 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 K and whosegeneric fiber is X .Before we proceed we record the following: Proposition 5.1.1 (Joshi-Rajan) . Let
X/K be a smooth projective variety over a numberfield K . Then there exists an integer N such that for all primes p in K lying over anyrational prime p ≥ N , the following dichotomy holds (1) either for all i, j ≥ , the Hodge-Witt groups are free W -modules (of finite type), or (2) there is some pair i, j such that H i ( X p , W Ω X p ) has infinite torsion.Proof. Choose a finite set of primes S of K , such that X has a proper, regular model overSpec( O K,S ), where O K,S denotes the ring of S -integers in K . Choose N large enough so thatfor any prime p lying over a rational prime p > N , we have p (cid:54)∈ S , and all the crystallinecohomology groups of X p are torsion-free. We note that this choice of N may depend on thechoice of a regular proper model for X over Spec( O K,S ). If p is such that H i ( X p , W Ω jX p )are all finite type, then by the degeneration of the slope spectral sequence at the E -stageby Bloch-Nygaard (see Theorem 3.7 of [17]), and the fact that the crystalline cohomologygroups are torsion free, it follows that the Hodge-Witt groups are free as well. If, on theother hand, some Hodge-Witt group of X p is not of finite type over W , then we are in thesecond case. (cid:3) Question 5.1.2 (Joshi-Rajan) . Let
X/K be a smooth projective variety over a number field.When does there exist an infinite set of primes of K such that the Hodge-Witt cohomologygroups of the reduction X p at p are not Hodge-Witt?We would like to explicate the information encoded in such a set of primes (when it exists). Proposition 5.1.3 (Joshi-Rajan) . Let
A/K be an abelian surface over a number field K .Then there exists infinitely many primes p such that H ( X p , W ( O X p )) has infinite torsionif and only if there exists infinitely many primes p of supersingular reduction for X . Inparticular, let E be an elliptic curve over Q and let X = E × Q E . Then for an infiniteset of primes of Q , the Hodge-Witt groups H i ( X p , W Ω jX p ) are not torsion free for ( i, j ) ∈{ (2 , , (2 , } .Proof. The first part follows from the results of [17, Section 7.1(a)]. The second part followsfrom combining the first part with Elkies’s theorem (see [13]), that given an elliptic curve E over Q , there are infinitely many primes p of Q such that E has supersingular reduction. (cid:3) Remark 5.1.4.
Question 5.1.2 which was raised by us in [21] can now formulated as a moreprecise conjecture (in the light of Corollary 4.4.8). Our formulation is the following.
Conjecture 5.1.5.
Let
X/K be any smooth, projective variety of dimension n over a numberfield K . Assume that H n ( X, O X ) (cid:54) = 0. Then there exists infinitely may primes p of K suchthat the domino associated to the differential H n ( X p , W ( O X p )) → H n ( X p , W Ω X p ) is non-zero (here X p is the “reduction” of X at p ). In particular X has non-Hodge-Witt reductionat p . irti Joshi 17 Remark 5.1.6.
Let us remind the reader that the domino (see [20]) associated to a deRham-Witt differential such as H n ( X p , W ( O X p )) → H n ( X p , W Ω X p ) is a measure of infinitetorsion in the slope spectral sequence. In particular the conjecture says that for the reduction, H n ( X p , W ( O X p )) has infinite torsion for infinitely many primes p . Let us understand whatConjecture 5.1.5 means in the context of a K3 surface. Let X/K be a smooth, projectiveK3 surface over a number field K . As H ( X, O X ) (cid:54) = 0, the hypothesis of Conjecture 5.1.5is valid. Now X has Hodge-Witt reduction at a prime p of K if and only if X p is of finiteheight (i.e. the formal group attached to X p is of finite height). So X p has non-Hodge-Wittreduction if and only if the formal group attached to X p is of infinite height. In this casefor the reduction one has H ( X p , W ( O X p )) = k [[ V ]] with F = 0 (as a module over theCartier-Dieduonne algebra–see [17]). Remark 5.1.7.
Let us remark that Conjecture 5.1.5 implies Elkies’ Theorem on infinitudeof primes of supersingular reduction valid for any number field (and not just Q ). To seethis let E/K be an elliptic curve over any number field K . Let A = E × K E . Then A isan abelian surface and hence H ( A, O A ) (cid:54) = 0 and so Conjecture 5.1.5 predicts that there areinfinitely may primes p of K such that A has non-Hodge-Witt reduction at p . But A hasnon-Hodge-Witt reduction if and only if the p -rank of A is zero. This means E has p -rankzero at all such p . Thus E has supersingular reduction at p . Hence E has supersingularreduction at infinitely many primes p of K . Remark 5.1.8.
Let us also remark that Conjecture 5.1.5 also implies that if E , E are twoelliptic curves over number fields, then there are infinitely many primes p (of the numberfield) where both E , E have supersingular reduction. The set of such primes is, of course,expected to be very thin. Remark 5.1.9.
Let us note that even for pairs of elliptic curves over Q with small con-ductors, the first prime of common supersingular reduction can be very large. Here is aparticularly interesting example with “modular flavor.” Consider the modular curve X (37),which has genus two and its Jacobian, J (37), is isogenous to a product of two elliptic curvesof conductor N = 37. The curves were described by Mazur and Swinnerton-Dyer in theirclassic paper on Weil curves and the two elliptic curve factors of J (37) are not isogenousto each other. The two curves appear as 37a and 37b1 in John Cremona’s tables. So tofind a prime of non-Hodge-Witt reduction for J (37) is the same as finding a prime of com-mon supersingular reduction for the elliptic curve factors of J (37). The first such a prime is p = 18489743 (a number several order of magnitudes larger than the conductor). This makessearching for such primes for pairs of elliptic curves rather tedious, at least with the minisculecomputational resources available to us, but the evidence for the conjecture, gathered using[43], is quite convincing despite this.In the direction of the Conjecture 5.1.5 we have the following: Theorem 5.1.10.
Let X be one of the following: (1) a Fermat hypersurface over Q , or (2) an abelian variety with complex multiplication over some number field K .Then Conjecture 5.1.5 holds for X .Proof. The assertion (1) is due to Joshi-Rajan (see [21] and uses results of [47]). The secondassertion, after [44, Theorem 1.1], is reduced to a Chebotarev argument similar to the one used in the proof of Theorem 4.4.5. To be precise one needs to show that there are infinitelymany primes p such that at least two of the primes p , . . . , p r lying over p in K remainsinert in K (which is clear by the method of proof of Theorem 4.4.5). (cid:3) Remark 5.1.11.
These are the only examples of this phenomena we know so far related tothe above question. 6.
Some explicit examples
In this subsection we describe some explicit examples of the diverse phenomena which thereader may find instructive or insightful.6.1.
Fermat varieties.
Let us begin with Fermat varieties and explicitly compute densitiesof primes of various types of reduction. The examples we consider here are general type. Let F m,n be the closed subscheme of P n +1 defined by the homogeneous equation F n,m : X m + · · · X mn +1 = 0 . The notation is such that F n,m has dimension n . Let δ ord n,m , δ hw m,n , and δ non − hw n,m denote theasymptotic density of primes of ordinary, Hodge-Witt and non-Hodge-Witt reduction for F n,m / Q (we ignore primes of bad reduction). Then the densities may be calculated byusing the results of [47]. Except for a finite list of exceptional F n,m , which are given in thetable below, the densities are given by the following (we write δ ? n,m for δ ? ( F n,m , Q ), with? = ord , hw , non − hw): δ ord n,m = 1 φ ( m )(6.1.1) δ hw n,m = 1 φ ( m )(6.1.2) δ non − hw n,m = 1 − φ ( m )(6.1.3)In the table we list the density of primes of different types of reduction for F n,m , we also listthe condition for a prime so that F n,m has the reduction of the given sort. The table alsoshows that the density of ordinary and Hodge-Witt reductions can be different. In particularfor the septic Fermat surface F , ⊂ P the densities are δ ord2 , = < δ hw2 , = = δ non − hw2 , . So theseptic Fermat surface shows rather interesting behavior. We do not know why this surfaceshows this exceptional behavior.( n, m ) ordinary δ ord n,m Hodge-Witt δ hw n,m non Hodge-Witt δ non − hw n,m (1 , m ) ? 1 0( n,
1) 1 1 0( n,
2) 1 1 0(2 , , , ( p ≡ , , ( p ≡ , , ( p ≡ , , ( p ≡ , ( p ≡ , , , ( p ≡ , , n, m ) φ ( m ) 1 φ ( m ) − φ ( m ) irti Joshi 19 Example 6.1.1. [Product of two elliptic curves] Here is another example of a similar sort.Let E/ Q : y = x − x and E (cid:48) / Q : y = x + 1 be elliptic curves over Q with complexmultiplication by Q ( i ) and Q ( ζ ) respectively. Then E has ordinary reduction if and onlyif p ≡ E (cid:48) has ordinary reduction if and only if p ≡ A = E × E (cid:48) .This is an abelian surface, and we want to calculate the densities δ ord A and δ hw A or primesof ordinary reduction and Hodge-Witt reduction for A . We consider primes p for which A has good reduction. Now suppose p is a prime of good reduction for A . Then we see byconsidering Newton and Hodge polygons of A = E × E (cid:48) that A has ordinary reduction at p ifand only if both E and E (cid:48) have ordinary reductions at p (so p ≡ p ≡ δ ord A = . While A has Hodge-Witt reduction if and only if one of E, E (cid:48) isordinary (curves always have Hodge-Witt reduction, so
E, E (cid:48) are always Hodge-Witt). Thusthis happens if and only if p is 1 mod 4 or 1 mod 3. Thus we see that δ hw A = + − = .Thus we see that for Abelian surfaces it is possible to have δ ord A < δ hw A < Example 6.1.2. [CM by a field galois (over Q )] Let C : y = x −
1. Here the genus g = 2 and the Jacobian of C has CM by Q ( ζ ) which is cyclic and galois. Then [14] itsJacobian is ordinary if and only if p ≡ p >
5, the reductionis non-Hodge-Witt. In particular the set of prime of ordinary (=Hodge-Witt) reduction hasdensity φ (5) (where φ (5) = | Gal( Q ( ζ ) / Q ) | = 4) and the set of primes of non-Hodge-Wittreduction is 1 − φ (5) = . Example 6.1.3. [CM by a non galois CM field] Consider the field K = Q [ x ] / ( x +134 x +89).This is a quartic non-galois CM field, its normal closure N has G = Gal( N/ Q ) = D the dihedral group of order 8. This field and abelian surfaces with CM by this field arediscussed in [14]. Examining the conjugacy classes of G we see that the conjugacy classof transpositions has two elements, and the conjugacy class of the identity is of course oneelement. Now there exists an abelian variety A over some number field L , with CM by K .For A , by Theorem 4.4.5 the set of primes of Hodge-Witt reduction has density ≥ δ hw ≥ while the density of primes of ordinary reduction is ≥ δ ord ≥ . We note that some primesother than the almost split primes contribute to the densities, and one can, using [14],describe them explicitly, but we do not work this out here–as this sort of calculation shouldbe worked out in some generality and this will require some additional combinatorics whichbelongs to a separate paper by itself. For all other sufficiently large primes, the reduction isnon-Hodge-Witt and there are infinitely many of these as well (from the above bounds). Example 6.1.4 (Surfaces of general type) . Suppose
C, C (cid:48) are smooth projective curvesover Q . Assume that C is hyperelliptic curve of genus g ≥ C (cid:48) is at least two, the Jacobian of C (resp. C (cid:48) ) is notisogenous to a factor of the Jacobian of C (cid:48) (resp. C ). Then C has good ordinary reductionat a positive density of primes. Now consider the surface S = C × Q C (cid:48) , then S is a surfacewith good Hodge-Witt reduction at positive density set of primes; and S has good ordinaryreduction precisely when both C, C (cid:48) have good ordinary reductions. Thus we have examplesof surfaces of general type where Conjecture 4.1.1 and Conjecture 3.1.1 hold and providedifferent sets of primes.
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