aa r X i v : . [ m a t h . AG ] M a r ORDINARY REDUCTIONS OF ABELIAN VARIETIES
KIRTI JOSHI
Abstract.
I show that a conjecture of Joshi-Rajan on primes of Hodge-Witt reductionand in particular a conjecture of Jean-Pierre Serre on primes of good, ordinary reductionfor an abelian variety over a number field follows from a certain conjecture on Galois rep-resentations which may perhaps be easier to prove (and I prove this conjecture for abeliancompatible systems of a suitable type). This reduction (to a conjecture about certain sys-tems of Galois representations) is based on a new slope estimate for non Hodge-Witt abelianvarieties. In particular for any abelian variety over a number field with at least one primeof good ordinary or split toric reduction, I show that the conjecture of Joshi-Rajan and theconjecture of Serre on ordinary reductions can be reduced to proving that a certain rationaltrace of Frobenius is in fact an integer. The assertion that this trace is an integer is provedfor abelian systems of Galois representations (of suitable type).
It don’t mean a thing,if it ain’t got that swing...Louis Armstrong and Duke Ellington(and Irving Mills) Introduction
Let k be a perfect field. Let X/k be a smooth, projective variety over k . Let W = W ( k ) bethe ring of Witt vectors of k . Let us say, following [Illusie, 1979] that X is Hodge-Witt if thede Rham-Witt cohomology groups H i ( X, W Ω jX ) are finitely generated as W -modules for all i, j ≥
0. (see [Illusie, 1979]). Let us say that X is ordinary if and only if H i ( X, B Ω jX ) = 0for all i, j ≥
0, where B Ω jX = d (Ω j − X ) is the sheaf of locally exact j -forms on X (see[Illusie, 1979]). It is a theorem of [Illusie and Raynaud, 1983] that if X is ordinary then X is Hodge-Witt. For examples of ordinary varieties see [Illusie, 1990].Suppose now that X is an abelian variety. Then X is ordinary if and only if X has p -rankequal to dim( X ); and it is a less well-known theorem of Torsten Ekedahl (see [Illusie, 1983])that X is Hodge-Witt if and only if p -rank of X is at least dim( X ) −
1. In particular onesees from this that Hodge-Witt but non-ordinary abelian varieties exist.Readers unfamiliar with [Illusie, 1979], [Illusie and Raynaud, 1983], [Ekedahl, 1985, 1986]should consult Section 2 (especially Remark 2.1) for a “working definition” of Hodge-Wittvarieties adapted to abelian varieties which is more than adequate for reading this paper.Now suppose K/ Q is a number field and that X/K is a smooth, projective variety over K . Then Jean-Pierre Serre has conjectured that X has ordinary reduction modulo infinitelymany non-archimedean primes of K . In [Joshi and Rajan, 2000] it was conjectured thatthere exist infinitely many primes at which X has Hodge-Witt reduction. Clearly if X hasgood ordinary reduction at a finite prime p of K then X has good, Hodge-Witt reduction at p . Thus the set of primes of good ordinary reduction for X are contained in the set of primes of good, Hodge-Witt reduction for X . In [Joshi, 2014, Theorem 4.1.3] I give, amongst otherresults, several examples which show that the two sets of primes can have different densities.In [Joshi, 2014, Theorem 4.1.3] I also showed that Serre’s ordinarity conjecture for X is equivalent to the conjecture on Hodge-Witt reductions due to Joshi-Rajan (also see[Joshi and Rajan, 2000]) for X × K X (this uses an important result of [Ekedahl, 1985])and I also proved both the conjectures for abelian varieties with complex multiplication(CM case).In Theorem 4.2 of this note I prove, by using methods quite different from those pursuedin [Joshi, 2014, Theorem 4.1.3] for the CM case, that a certain conjecture on Galois rep-resentations (see Conjecture 3.6) implies that there exist infinitely many primes of good,Hodge-Witt reductions for any abelian variety. By [Joshi, 2014, Theorem 4.1.3] (stated hereas Theorem 4.1) this is enough to prove, assuming Conjecture 3.6, the result conjecturedby Serre that any abelian variety over a number field has infinitely many primes of goodordinary reduction (see Theorem 4.6).Let me remind the reader that for abelian surfaces the result was proved unconditionallyby [Ogus, 1982]. In [Joshi and Rajan, 2000], Rajan and I showed, again unconditionally, thatthere exist infinitely many primes of Hodge-Witt reduction for abelian threefolds. Serre’sconjecture on ordinary reductions for abelian varieties has also been established in a few casesunder restrictive assumptions on dimensions and endomorphism algebras or Mumford-Tategroups in [Pink, 1998, Tankeev, 1999, Noot, 2000] and the references therein for additionalresults. The methods of this paper have little overlap with these works.The proof given here proceeds in two steps. I use an observation of [Joshi, 2014, Theo-rem 4.1.3] (recalled here as Theorem 4.1) which allows one to reduce to proving that A × A has infinitely many primes of Hodge-Witt reduction (see Theorem 4.1). Then I prove, as-suming Conjecture 3.6, the existence of infinitely many primes of Hodge-Witt reduction forany abelian variety (see Theorem 4.2). Finally one gets from this, assuming Conjecture 3.6,the ordinarity result conjectured by Serre (see Theorem 4.6). The key new tool in the proofsof these two results, apart from reduction to the Hodge-Witt case provided by [Joshi, 2014,Theorem 4.1.3], is a (sharp) slope estimate (see Theorem 2.2 and Remark 2.24) for non-Hodge-Witt abelian varieties. In Theorem 3.8, I prove Conjecture 3.6 for abelian systems ofGalois representations.In Theorem 5.1 I show that any abelian variety with at least one prime of good ordinary orsplit toric reduction over a number field has infinitely many primes of Hodge-Witt reductionand hence any such abelian variety also has infinitely many primes of ordinary reductionprovided that Conjecture 3.10, which reduces the proof of Conjecture 3.6 to proving that acertain (rational) trace of Frobenius is in fact an integer, is true. In Theorem 3.11 I proveConjecture 3.10 for abelian systems of Galois representations.I thank Bryden Cais for many conversations and especially for patiently listening to severalof my unsuccessful attempts to prove Conjecture 3.6 in all generality. I also thank AdrianVasiu for a careful reading of an earlier version of this manuscript and a number of suggestionswhich led to several improvements and readability of this manuscript. Thanks are also dueto Brian Conrad for some conversations on abelian Galois representations.2. non Hodge-Witt abelian varieties Let X be a smooth, projective variety over a perfect field k of characteristic p >
0. Let W = W ( k ) be the ring of Witt vectors of k and let its quotient field be K . One says, following [Illusie and Raynaud, 1983], that X is Hodge-Witt if and only if the de Rham-Witt cohomology groups H i ( X, W Ω jX ) are of finite type W -modules for all i, j ≥
0. Supposenow that X is an abelian variety. It is a theorem of Torsten Ekedahl (see [Illusie, 1983]) that X is Hodge-Witt if and only if p -rank of X is ≥ dim( X ) − Remark 2.1.
Readers unfamiliar with Hodge-Witt varieties may take this p -rank conditionas an ad hoc definition of Hodge-Witt abelian varieties. In other words readers may adoptas a “working definition” the statement that X is Hodge-Witt if and only if p -rank of X is ≥ dim( X ) −
1. Since p -rank of an abelian variety is additive on taking products of abelianvarieties, it follows that if X, Y are abelian varieties over k then X × Y is Hodge-Witt if andonly if one of X, Y is Hodge-Witt and the other is ordinary (this is a very special case of abeautiful general theorem of [Ekedahl, 1985, 1986]). This observation for abelian varietiesis adequate for reading this paper. In [Chai et al., 2014] abelian varieties X which have p -rank ≥ dim( X ) − almost ordinary and p -divisible groups arising from themare said to be of extended Lubin-Tate type . It should be noted however that neither of theselabels reveal the most important property of these abelian varieties: the finiteness of the deRham-Witt cohomology equivalently the degeneration of the slope spectral sequence at E (see [Illusie and Raynaud, 1983]).In this section I prove the following slope estimate. As is conventional, a p -valuation v used for computing slopes is normalized so that v ( p ) = 1. Theorem 2.2.
Let X be an abelian variety over a perfect field k of characteristic p > .Assume that dim( X ) = g and that X is not Hodge-Witt. Then every slope ˜ λ of H g cris ( X/W ) satisfies ˜ λ ≥ . Proof.
Let M = H ( X/W ) ⊗ W K , for a slope λ of M , let M λ be the slope λ part of M and in particular let M , M , M be respectively the slope zero, slope one and the slope halfsubmodules of M . Let m λ , m , m , m be their dimensions respectively. Further as X is notHodge-Witt so(2.3) m ≤ g − , and one has dim K ( M ) = 2 g . So one gets(2.4) 2 g = dim( M ) + dim( M ) + dim( M ) + X λ =0 , , dim( M λ ) . By definition of p -rank, the p -rank of X is m . By duality for abelian varieties m λ = m − λ .Therefore one can write this as(2.5) 2 g = 2 m + m + 2 X <λ< m λ . Hence one has, for every slope 0 < λ < of M ,(2.6) 2 m λ ≤ g − m − m. Observe that g − m − m cannot be negative by (2.5). Now the proof is split into two casesaccording to whether g − m − m = 0 or g − m − m = 0. KIRTI JOSHI
First suppose g − m − m = 0. So m λ ≤ g − m − m . Now for 0 < λ < λ = ab with a ≥ , b > a, b ) = 1. Then b | m λ so b ≤ m λ and hence b ≥ m λ . Hence in particularfor any 0 < λ < one has(2.7) λ ≥ m λ . As m λ ≤ g − m − m , from (2.6) and from (2.7), for any 0 < λ < , one has the fundamentalestimate:(2.8) λ ≥ m λ ≥ g − m − m . If 0 < λ < is a slope, then so is 1 − λ and 1 > − λ > and one has(2.9) 1 − λ > λ ≥ g − m − m , and hence one has for all slopes λ = 0 , , the estimate:(2.10) λ ≥ g − m − m . Now recall that H g cris ( X/W ) ⊗ K = ∧ g H ( X/W ) = ∧ g M and so any slope ˜ λ of Frobeniuson H g cris may be computed from slopes of M . Any slope ˜ λ of H g cris is of the form(2.11) ˜ λ = λ + · · · + λ g for some slopes λ , λ , . . . , λ g of M . If any of the λ j occurring in this expression is equalto one, then ˜ λ ≥ λ j < j = 1 , . . . , g . Let i be the number of times λ j = 0 and let i be the number of times occursin the above representation. Then˜ λ ≥ i g − i − i ) (cid:18) g − m − m (cid:19) (2.12) ≥ (cid:18) g − i g − m − m (cid:19) + i (cid:18) − g − m − m (cid:19) . (2.13)Note that − g − m − m >
0. This follows from (2.5) as(2.14) g − m − m X <λ< m λ and for slopes 0 < λ < one has m λ ≥ g − m − m ≥ , and hence(2.16) 1 g − m − m ≤ . So > ≥ g − m − m . Thus the second term on the right hand side of (2.12) is positive. Nowas(2.17) i ≤ m so(2.18) g − i ≥ g − m ≥ g − m − m . Thus in the case g − m − m > λ ≥ g − i g − m − m ≥ g − m g − m − m ≥ . Let me now address the case g − m − m = 0. One argues directly using (2.5) from whichone sees that m λ = 0 for all λ = 0 , ,
1. From(2.20) ˜ λ = λ + · · · + λ g assuming once again that none of the slopes in the above expression for ˜ λ are equal to one,one has the estimate(2.21) ˜ λ = 12 ( g − i ) ≥
12 ( g − m ) = m . So to prove our claim that ˜ λ ≥ g − m − m = 0 then m ≥
4. If m < | m this says m = 2 or m = 0. If m = 2 then(2.22) g = m + m m + 22 = m + 1and hence m = g − X is Hodge-Witt. This contradicts our assumption that X is non Hodge-Witt. Similarly if m = 0 then g = m hence X is ordinary but thiscontradicts our assumption that X is non Hodge-Witt. Thus if g − m − m = 0 then m ≥ ≤ m ≤ g − λ = m ≥
1. Thus one has in all cases(2.23) ˜ λ ≥ . This completes the proof of the theorem. (cid:3)
Remark 2.24.
Suppose X is ordinary. Then H gcris ( X/W ) has a slope zero part of rank one.So ˜ λ ≥ λ . Similarly if X is Hodge-Witt, but not ordinary, then X has p -rank g − λ = for H gcris ( X/W ). So the assumption that X is nonHodge-Witt cannot be relaxed to X is non-ordinary in Theorem 2.2. Moreover for every g ≥ k of characteristic p > { g , − g } each with multiplicity g and this abelianvariety is manifestly non Hodge-Witt with exactly one slope ˜ λ with ˜ λ = 1. So the estimateis the best possible in all dimensions g ≥ A conjecture about certain Galois representations
I make the following definition. Let K be a finite extension of Q , and fix an algebraicclosure of K and let G K be the Galois group of K (with respect to this algebraic closure).For any finite set of primes S of K , and a rational prime p , let S ℓ = S ∪ { l : l | ℓ } . For a p -adic representation of G K and prime p | p of K , I will write V p for the restriction of the G K -representation V p to the decomposition group D p at p . Let D cris ( V p ) be the (covariant)functor constructed in [Fontaine, 1994]. For any prime ℓ , let Q ℓ ( −
1) be the ℓ -adic Tatetwist. This is a G K -representation of Weil weight two and the unique slope of Frobenius on D cris ( Q p ( − Q ℓ (1) has Weil-weight minus two. KIRTI JOSHI
Definition 3.1.
Let g ≥ S be a finite set of primes of K . A pair ofcontinuous, ℓ -adic Galois representations (one for each rational prime ℓ ), { V ℓ } ℓ , { V ′ ℓ } ℓ of G K are said to be a twist-coupled system of Galois representations of type ( g, g −
2) if they satisfythe following conditions:( G .1) Each V ℓ is unramified outside S ∪ { ℓ } .( G .2) Each V ℓ is pure of Weil weight g .( G .3) For any prime p S ℓ the characteristic polynomial of Frobenius of V ℓ at p is apolynomial f p ,ℓ ( t ) ∈ Q [ t ], and if ℓ, ℓ ′ are primes and p S ℓ ∪ S ℓ ′ then f p ,ℓ ( t ) = f p ,ℓ ′ ( t ).( L .1) For all primes p , V p is potentially semistable.( L .2) For all primes p S , V p is crystalline.( L .3) For all primes p S the characteristic polynomial of the linearized Frobenius of D cris ( V p ) is f p ( t ) ∈ Q [ t ] and if p / ∈ S ℓ one has f p ,ℓ ( t ) = f p ( t ).( E ) Except for a set of finitely many primes p including those in S , the characteristicpolynomial f p ( t ) ∈ Z [ t ].( S ) For all primes outside the exceptional set of primes of ( E ) the slopes of Frobenius on D cris ( V p ) are in the interval [1 , g − H ) For all primes p , V p has Hodge-Tate weights in [0 , g ], and one has gr ( D HT ( V p )) = 0.( T ) For all ℓ there exists a continuous isomorphism V ℓ ≃ / / V ′ ℓ ( −
1) of G K -modules (here Q ℓ ( −
1) is the Tate twist).The following simple lemma, while not essential in my proof, explicates the condition ( S ). Lemma 3.2.
Let f ( t ) ∈ Z [ t ] be a non-zero polynomial whose roots are p -Weil numbersof weight m ≥ . Suppose that L/ Q is a finite extension containing all the roots of f ( t ) and suppose for any p -adic valuation v p of L , normalized so that v p ( p ) = 1 , extending thestandard p -adic valuation of Q p , one has v p ( α ) ≥ for any root α of f ( t ) . Then there is analgebraic integer β such that α = pβ .Proof. Let ( p ) = p a · · · p a r r be the prime factorization of p in L . Then the estimate v p i ( α ) ≥ v p i ( p ) = 1, says that α ∈ p a i i . Hence α ∈ p a i i for all i = 1 , . . . , r . Thus α ∈ ∩ p a i i = p a · · · p a r r = ( p ). Therefore α = pβ for some algebraic integer β as asserted. (cid:3) I will write the quantities corresponding to V ′ p , such as characteristic polynomials of Frobe-nius, traces of Frobenius etc. as primed quantities: f ′ p ( t ) , f ′ p ,ℓ ( t ) etc. Hopefully there will beno confusion with derivatives (which will not be used in this paper).The contents of the following remarks will clarify this list of properties of twist-coupledsystems and will be used in the rest of the paper. The remarks are immediate from definitionsor elementary considerations. Remark 3.3.
Note that the properties are indexed by global conditions ( G .1)–( G .3) andby local conditions ( L .1)–( L .3) which members of { V ℓ } ℓ satisfy. The assumptions ( G .1)–( G .3) and ( L .1)–( L .3) are satisfied by all geometric Galois representations. That ( G .1)–( G .3), ( L .1)–( L .3) hold in geometric contexts is the work of many mathematicians: prop-erties ( G .1)–( G .3) hold by [Deligne, 1974], properties ( L .1)–( L .3) first arose in the workof [Fontaine, 1982] and have now been established in geometric contexts (see [Colmez,2001-2002] for a detailed bibliography). That the property ( L .3) holds in geometric con-text is due to [Katz and Messing, 1974].(1) Through ( T ) the representation { V ′ ℓ } ℓ also satisfies ( G .1)–( G .3) and ( L .1)–( L .3). (2) Properties ( E ), ( S ), ( H ) are not invariant under Tate twists and hence are not in-herited by { V ′ ℓ } through ( T ).(3) Property ( E ) of V p says that in some sense V p is “effective.”(4) Note that properties ( S ), ( E ) and ( H ) only involve { V ℓ } ℓ . Proposition 3.4.
Suppose { V ℓ , V ′ ℓ } ℓ is a twist-coupled family of Galois representations.Then(1) Properties ( G .1)–( G .3), ( L .1)–( L .3) are hold for { V ′ ℓ } .(2) The property ( E ) is inherited by { V ′ ℓ } .(3) For all primes p S , V ′ p is crystalline.(4) For all but a finite number of primes, the slopes of Frobenius on D cris ( V ′ p ) are in [0 , g − .(5) The family { V ssℓ , V ′ ssℓ } ℓ of semisimplifications of { V ℓ , V ′ ℓ } ℓ is also a twist coupled sys-tem.Proof. (1), (3) and (4) are immediate from the fact that ( G .1)–( G .3), ( L .1)–( L .3), ( S ), ( E ),( H ) and ( T ) hold for { V ℓ } and as V ′ ℓ = V ℓ (1) by ( T ). Claim (2) is now immediate from ( T ),( E ) and ( S ) for { V ℓ } . Clearly (5) is entirely formal. (cid:3) Remark 3.5.
In particular one sees from Proposition 3.4(2) that if ( G .1)–( G .3), ( L .1)–( L .3), ( S ), ( E ), ( H ) and ( T ) hold then ( E ) also holds for { V ′ ℓ } ℓ . As V ℓ = V ′ ℓ ( −
1) by ( T ),and as ( E ) holds for V ′ ℓ , one has divisibility of traces of Frobenius elements acting on V ℓ .Thus the conditions ( G .1)–( G .3), ( L .1)–( L .3), ( S ), ( E ), ( H ) and ( T ) provide a natural wayof encoding the divisibility by p of trace of Frobenius element at p acting on V ℓ .I propose the following conjecture. Conjecture 3.6.
Let
X/K be a smooth, projective variety over a number field K of dimen-sion g ≥ H g ( X, O X ) = 0 . Then the family of ℓ -adic Galois representations:(3.7) { V ℓ = H get ( X, Q ℓ ) , V ′ ℓ = V ℓ (1) } ℓ is not a twist coupled system of Galois representations of type ( g, g − g = 2 and any smooth projective variety X (with H ( X, O X ) = 0), Conjecture 3.6 isimmediate by using methods of [Ogus, 1982, Joshi and Rajan, 2000, Bogomolov and Zarhin,2009]. To see this it is sufficient to note that the methods of loc. cit. show that if thetraces of Frobenius at all but finite number of primes p (for ℓ -adic ´etale cohomology, with anappropriate ℓ for X ) are divisible by p , then the semisimplification of H et ( X, Q ℓ ) is isomorphicto ⊕ Q ℓ ( −
1) (as a G K -module) (and by compatibility (( G .1)–( G .3)) this must hold for all ℓ )which obviously contradicts ( H ) (by the Hodge-Tate decomposition Theorem furnished by p -adic Hodge Theory). Thus { V ℓ = H et ( X, Q ℓ ) , V ′ ℓ = V ℓ (1) } ℓ is not a twist-coupled system.Note that the hypothesis in loc. cit. for g = 2, that X is a K3 surface is used only to provethat there is exactly one unit eigenvalue of Frobenius. Here is some additional evidence forConjecture 3.6. Theorem 3.8.
Let X be a smooth, projective variety of dimension g over a number field andsuppose H g ( X, O X ) = 0 . Assume that { ( ρ ℓ , H get ( X, Q ℓ )) } ℓ is a family of abelian Galois repre-sentations (i.e. factors as a representation ρ ℓ : G abK → GL( H get ( X, Q ℓ )) . Then Conjecture 3.6holds for X . KIRTI JOSHI
Proof. As { H get ( X, Q ℓ ) } ℓ is a family of abelian representations, so is its Tate-twist { V ′ ℓ = H get ( X, Q ℓ )(1) } ℓ . Replacing these representations by their semi-simplifications one can as-sume that one has a family of semi-simple abelian representations. Assume the Conjecture 3.6is false for X . This means that(3.9) { V ℓ = H get ( X, Q ℓ ) , H get ( X, Q ℓ )(1) } ℓ forms a twist-coupled system of G K -representations (so properties ( G .1)–( G .3), ( L .1)–( L .3),( S ), ( E ), ( H ) and ( T ) hold). In particular by Proposition 3.4(2), { V ′ ℓ } satisfies ( E ). By( G .1)–( G .3), ( L .1)–( L .3) and [Serre, 1968, Theorem of Tate, page III-7] V ′ p is locally algebraicand hence by [Serre, 1968, Proposition, page III-9] V ′ p arises from a Q -rational representationof an algebraic torus. It is possible to choose a prime p such that for any prime p of K lyingover p such that V ′ p is crystalline at p (and hence Hodge-Tate at p | p ) and the algebraic torusgiving rise to this representation is split at p . Pick such a prime p . Then V ′ p is a directsum of crystalline characters of D p . As { V ′ ℓ } satisfies ( E ), the characteristic polynomialsof Frobenius elements at all but finite number of primes of K (acting on V ′ p ) have integercoefficients. Then the integrality of the characteristic polynomials of Frobenius elements in G K (over all but finitely many primes of K ) shows by [Serre, 1968, Chapter II, Corollary2, page II-36] that the Hodge-Tate weights of V ′ p are non-negative (note that loc. cit. theHodge-Tate weights are determined by the algebraic characters of the aforementioned toruswhich appear in rational representation of this torus provided by [Serre, 1968, Proposition,page III-9]). So V ′ p has non-negative Hodge-Tate weights and hence V p = V ′ p ( −
1) has strictlypositive Hodge-Tate weights. This contradicts ( H ) as a Theorem of Tate (see [Tate, 1967])says that if V p is a p -adic representation with strictly positive Hodge-Tate weights (at primesover p ) then ( V p ⊗ C p ) D p = 0 for any prime p lying over p . On the other hand one has H g ( X, O X ) = 0 and hence ( V p ⊗ C p ) D p = 0. Hence one has arrived at a contradiction. (cid:3) In some situations one can replace Conjecture 3.6 by the somewhat simpler Conjecture 3.10given below.
Conjecture 3.10.
Let K be a number field. Let S be a finite set of primes of K . Suppose { V ′ ℓ } ℓ is a family of continuous Galois representations satisfying the properties ( G .1)–( G .3),( L .1)–( L .3), ( E ) and suppose that for a prime p (lying over a rational prime p ) the followinghypothesis hold (here D p ⊂ G K is the decomposition group at p and I p is the inertia subgroupat p , and V ′ p is the restriction of V ′ p to D p ):( O .1) The vector of Hodge-Tate weights of V ′ q is constant as q varies over primes of K .( O .2) For any prime ℓ not lying below p , the representation V ′ ℓ of D p is unramified (resp.unipotent).( O .3) The representation V ′ p of D p is crystalline ordinary (resp. semi-stable ordinary, i.e.,equipped with a D p -invariant filtration whose graded pieces are isomorphic to Q p ( − i )for i ∈ Z ).( O .4) For any prime ℓ not lying below p , the characteristic polynomial of Frobenius at p acting on V ′ ℓ coincides with the characteristic polynomial of Frobenius on D st ( V ′ p ).( O .5) The trace a ′ p of Frobenius φ ′ p on D st ( V ′ p ) is rational (i.e. a ′ p ∈ Q ).Then the trace a ′ p of Frobenius φ ′ p on D st ( V ′ p ) is an integer.Here is some partial evidence for Conjecture 3.10. Theorem 3.11.
Suppose { ρ ′ ℓ : G abK → GL( V ′ ℓ ) } ℓ is an family of continuous, abelian Galoisrepresentations which satisfies all the hypothesis of Conjecture 3.10. Then Conjecture 3.10is true for { V ′ ℓ } ℓ .Proof. This is proved in a manner similar to Theorem 3.8. Replacing our system by itssemisimplification one can assume that one has a semisimple system of representations of G abK . First choose a prime q = p and a prime q of K lying over q such that V ′ q is crystalline at q . This means, by ( G .1)–( G .3), ( L .1)–( L .3) and [Serre, 1968, Theorem of Tate, page III-7],that V ′ q is locally algebraic and hence by [Serre, 1968, Proposition, page III-9] V ′ q arises froma Q -rational representation of an algebraic torus. As { V ′ ℓ } satisfies ( E ), the characteristicpolynomials of Frobenius elements at all but finite number of primes (acting on V ′ q ) areintegers. The integrality of the characteristic polynomials of Frobenius elements in G K (overall but finitely many primes of K ) shows, by [Serre, 1968, Chapter II, Corollary 2, page II-36],that the Hodge-Tate weights of V ′ q are non-negative. By ( O .1) the Hodge-Tate weights of V ′ p are also non-negative. Since V ′ p is semistable and ordinary (by ( O .3)), the non-negativityof Hodge-Tate weights of V ′ p says that the characteristic polynomial of φ ′ p on D st ( V ′ p ) has p -adic integer coefficients. Thus by ( O .5) one sees that the trace of Frobenius φ ′ p is a rationalinteger. This proves the assertion. (cid:3) Hodge-Witt and Ordinary reductions
Before proceeding further let me recall the following observation of [Joshi, 2014, Theo-rem 4.1.3]. The only case Theorem 4.1 of interest for this paper is the case when X is anabelian variety and as remarked in Remark 2.1 if one uses the “working definition” that anabelian variety X is Hodge-Witt if and only if X has p -rank ≥ dim( X ) −
1, then Theorem 4.1given below is quite elementary to prove.
Theorem 4.1.
Let X be a smooth, projective variety over a number field K . Then thefollowing are equivalent:(1) There exist infinitely many primes of ordinary reduction for X .(2) There exist infinitely many primes of ordinary reduction for X × K X (3) There exist infinitely many primes of Hodge-Witt reduction for X × K X . For dim( X ) = 3 the existence of infinitely many primes of Hodge-Witt reductions, (withoutassuming Conjecture 3.6), is due to [Joshi and Rajan, 2000] (also see [Joshi, 2014]). Theorem 4.2.
Let K be a number field. Let X/K be an abelian variety over K of dimension g ≥ . Assume Conjecture 3.6 is true for X if g ≥ . Then there exist infinitely many primesof Hodge-Witt reduction for X .Proof. Since any abelian variety of dimension g = 1 has Hodge-Witt for an infinite set ofprimes, one can assume that g ≥
2. Suppose the assertion is not true. Then I show that(4.3) { V ℓ = H get ( X, Q ℓ ) , V ′ ℓ = V ℓ (1) } ℓ is a system of twist-coupled Galois representations of type ( g, g − T ) holds asone has the tautological isomorphism V ℓ = V ℓ (1)( −
1) = V ′ ℓ ( − { V ℓ , V ′ ℓ } ℓ satisfiesall the properties ( G .1)–( G .3), ( L .1)–( L .3), ( S ), ( E ), ( H ) and ( T ) except possibly ( S ). Byour assumption X does not have Hodge-Witt reduction at all but a finite number of primes p of K . Suppose p is a prime of good, non Hodge-Witt reduction. Let X p be the reductionmodulo p of X (in a regular, smooth, proper model of X over a suitable localization of the ring of integers of K ). Let κ ( p ) be the residue field of p . Then by Theorem 2.2 one seesthat if α is any eigenvalue of Frobenius on H get ( X, Q ℓ ) (equivalently on H gcris ( X p /W ( κ ( p ))),by [Katz and Messing, 1974]) then α satisfies v p ( α ) ≥
1. As X p has dimension g , one has byPoincar´e duality for crystalline cohomology of X p that v p ( α ) ≤ g −
1. Hence for any prime p of non Hodge-Witt reduction ( S ) holds for D cris ( V p ). Thus one has arrived at a twist coupledsystem of Galois representations of type ( g, g − X thenone has arrived at a contradiction. (cid:3) Corollary 4.4.
Let
X/K be an abelian variety over a number field K . If Conjecture 3.6 istrue for X then there exists a set of primes of positive density at which X has Hodge-Wittreduction.Proof. If the assertion is not true then the set of primes of Hodge-Witt reduction has densityzero and examining the proof of Theorem 4.2 one is again led to a contradiction. (cid:3)
Remark 4.5.
Let me point out that if one assumes that X has non-ordinary reduction at p in Theorem 2.2 then it follows that the traces of Frobenius of p on H get ( X, Q ℓ ) are divisible by p . However this is not enough to conclude that all eigenvalues of Frobenius are divisible by p .Thus the assumption in Theorem 2.2 that X is non Hodge-Witt places stronger constraintson the crystalline cohomology of X than non-ordinarity does. Theorem 4.6 (Serre’s Ordinarity Conjecture) . Let
A/K be any abelian variety over K . IfConjecture 3.6 is true for A × K A then there exists a set of primes of positive density of goodordinary reduction for A .Proof. It is sufficient, by [Joshi, 2014, Theorem 4.1.3] (also see Remark 2.1), to prove that X = A × K A has infinitely many primes of Hodge-Witt reduction. This is immediatefrom Theorem 4.2 if A × K A satisfies Conjecture 3.6. The density assertion follows fromTheorem 4.2 and Corollary 4.4. (cid:3) Corollary 4.7.
Let K be any field of characteristic zero. If Conjecture 3.6 is true for anabelian variety and also for the self-product of this variety over K then this abelian varietyhas infinitely many primes of Hodge-Witt and ordinary reductions.Proof. This is immediate from Theorem 4.6. (cid:3)
Remark 4.8.
In [Joshi, 2014] it is conjectured that for a class of smooth, projective varietiesincluding abelian varieties, the set of primes of non Hodge-Witt reductions in dimensiondim( X ) is also infinite. This, I believe, is the correct analog in higher dimensions, of Elkies’Theorem (see [Elkies, 1987]) on the infinitude of primes of supersingular reductions for ellipticcurves over Q (in [Joshi, 2014] it is noted that Elkies’s result is a very special case of thisconjecture). 5. The Toric Case
In case of abelian varieties over a number field with at least one prime of ordinary orsplit-toric reduction one may replace Conjecture 3.6 by Conjecture 3.10 in the proof ofTheorem 4.2. This is proved in Theorem 5.1 below.
Theorem 5.1.
Let
X/K be an abelian variety of dimension g ≥ over a number field K and suppose that p is a prime of K at which X has either ordinary or split toric reduction.If Conjecture 3.10 is true then X has infinitely many primes of Hodge-Witt reduction. Proof.
Let us suppose the theorem is false. Then the proof of Theorem 4.2 shows that { V ℓ = H get ( X, Q ℓ ) , V ′ ℓ = V ℓ (1) } ℓ forms a twist-coupled system of Galois representations. SoProperties ( G .1)–( G .3), ( L .1)–( L .3), ( S ), ( E ), ( H ) and ( T ) hold.Let a p (resp. a ′ p ) be the trace of Frobenius on D st ( V p ) (resp. D st ( V ′ p )). Suppose for themoment that a p is an integer. Then the property ( T ) says that a ′ p is rational. Since V p isordinary and H g ( X, O X ) = 0 one has(5.2) a p p. On the other hand the isomorphism V ℓ ≃ V ′ ℓ ( −
1) of G K -modules says that one has(5.3) a p = a ′ p · p. Now it is clear that (5.2) and (5.3) cannot hold simultaneously provided one knows that a ′ p ∈ Z .First suppose that X has good ordinary reduction at p . Suppose ℓ is not a prime lyingbelow p . Then by the hypothesis, that X has good reduction at p , the trace of Frobenius at p acting on V ℓ is an integer. By ( L .3) this integer is also the trace of Frobenius acting on D st ( V p ) = D crys ( V p ). As V ′ p = V p (1) this says that the trace of Frobenius at p acting on V ′ ℓ is rational. Hence all the hypothesis of Conjecture 3.10 hold for { V ′ ℓ } ℓ . Thus one can invokeConjecture 3.10 and the assertion follows.Now suppose that p is a prime of split toric reduction lying over a rational prime p . Letus consider the restriction of the G K -representation of V ℓ (for a rational prime ℓ not lyingbelow p ) to the decomposition group D p at p . Let K p be the completion of K at p .I have to prove that the traces a p (resp. a ′ p ) are rational. This is certainly well-known butI recall this for convenience.Let σ be any element of the Weil group of K p (over its algebraic closure) lifting Frobeniusof the residue field at p and acting on V ℓ (through the action of D p on V ℓ ). By abuse ofterminology I will call such an element a Frobenius at p . Then it follows from [Grothendieck,1972] and [Coleman and Iovita, 2009] that a p = Tr( ρ ℓ (Frob p )) and it is immediate from[Grothendieck, 1972, Theorem 4.3(b), page 359]) and the fact that as D p -modules one hasan isomorphism V ℓ = H get ( X, Q ℓ ) = ∧ g H et ( X, Q ℓ ) that a p is an integer and so a ′ p is rational.Hence all the hypothesis of Conjecture 3.10 hold for { V ′ ℓ } ℓ . Now one invokes Conjecture 3.10to see that a ′ p is an integer. This together with (5.3) and (5.2) completes the proof. (cid:3) This has the following corollary.
Theorem 5.4.
Let
X/K be an abelian variety over a number field K and suppose that p is a prime of K at which X has good ordinary or split toric reduction. Assume thatConjecture 3.10 is true. Then X has infinitely many primes of ordinary reduction.Proof. This follows from Theorem 5.1 and Theorem 4.1 because if X has a prime of splittoric reduction then so does X × X . Similarly if X has one prime of good ordinary reductionthen so does X × K X . Now the assertion follows from Conjecture 3.10. (cid:3) References
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Math. department, University of Arizona, 617 N Santa Rita, Tucson 85721-0089, USA.
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