Finiteness of reductions of Hecke orbits

We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory and group theoretic techniques.


Introduction
Let Ā be an abelian variety over F p .When Ā is ordinary, then Ā admits a canonical (CM) lift, and every isogeny from Ā lifts to an isogeny in characteristic zero with source any fixed lift of Ā.The aim of this paper is to show that the situation is radically different for supersingular abelian varieties.In fact, we prove a general theorem for all Newton strata that interpolates between the ordinary case and the supersingluar case.
The first example of an abelian variety over F p without a CM lift was given by Oort in [Oor92].Further examples of such abelian varieties, including supersingular abelian varieties, were then constructed by Conrad, Chai and Oort in [CCO14].We prove that such examples are in fact quite abundant.
Theorem 1.1.Only finitely many supersingular abelian varieties of a given dimension admit CM lifts.
Combining this with a refined analysis of the Kuga-Satake construction, we are able to answer a question of Ito-Ito-Koshikawa [IIK18, Remark 1.3] Theorem 1.2.If p ≥ 5, then only finitely many supersingular K3-surfaces over F p admit CM lifts.
We remark that the theorem above makes no mention of polarizations, so does not simply follow from the Kuga-Satake map for integral models of Shimura varieties.It instead requires an analysis of a Kuga-Satake construction at the level of p-divisible groups due to Yang [Yan19].
Our results for general Newton strata use the notion of central leaves introduced by Oort in [Oor04,Theorem 5.3].The central leaf through Ā is a closed subvariety inside its Newton stratum, and essentially consists of all abelian varieties whose pdivisible group is geometrically isomorphic to that of Ā.In [Oor09, Section 5], Oort computes the dimensions of central leaves and shows that they are 0-dimensional in the supersingular stratum, and equal the entire Newton stratum in the ordinary case.
Theorem 1.3.Let W be a Newton stratum in the moduli space of principally polarized abelian varieties in characteristic p.The set of points of W admitting CM lifts is contained in a finite union of central leaves.
Let A be an abelian variety over a field K, with algebraic closure K.By the Hecke orbit of an abelian variety A, we mean the set of isomorphism classes of abelian varieties over K, which are isogenous to A K .If A is defined over a local field, and has good reduction Ā, then the image of the Hecke orbit of A in the Hecke orbit of Ā is called the reduction of the former.We prove the following result, which is in stark contrast to the ordinary case.This is the key input for proving the CM lifting theorems stated earlier.
Theorem 1.4.Let A be an abelian variety over a characteristic zero local field, and suppose A has good supersingular reduction.Then the reduction of its Hecke orbit is finite.
This theorem answers a question posed by Poonen in an unpublished preprint, and also makes progress towards understanding the p-adic distribution of Hecke orbits.The proof of Theorem 1.4 is entirely local, and we prove an analogous theorem in the setting of p-divisible groups first.Now suppose that K is a characteristic zero local field with ring of integers O K and residue field k.For a p-divisible group G over O K , we define the Hecke orbit of G and its reduction in the same way as above.In particular, the reduction is a collection of isomorphism classes of p-divisible groups over an algebraic closure of k.We also establish a finiteness theorem under a semisimplicity hypothesis on the p-adic Galois representation.
Theorem 1.5 (Theorem 2.13).Let G denote a p-divisible group over O K , such that the p-adic Galois representation associated to G is semisimple.Then the reduction of the Hecke orbit of G is finite.
In light of Theorem 1.4 and Theorem 1.5, we make the following conjecture.
Conjecture 1.6.For any p-divisible group G over O K , the reduction of its Hecke orbit is finite.
By Theorem 1.1 and Theorem 1.5, we know that Conjecture 1.6 holds in the case of supersingular reduction (without any semisimplicity conditions on the Galois representation), and in the case that the Galois representation is semisimple (without any condition on the Newton polygon of G ).We remark that unlike the situation in characteristic zero the Hecke orbit of Ā can contain positive dimensional families1 .This was first observed by Moret-Bailly [MB81], who constructed a complete family of supersingular abelian surfaces over P 1 Fp , such that all fibers are p-isogenous.
Outline of the paper: In § 2, we prove our results on finiteness of reductions of Hecke orbits.This is done using a Galois-theoretic result which relies on work of Sen and Serre.We apply this in § 3 to show the results on CM lifts of p-divisible groups and abelian varieties.Here we make crucial use of Oort's results on central leaves.Finally in § 4, we prove the finiteness result for CM lifts of supersingular K3 surfaces, by comparing the deformation theory of K3's with that of GSpin p-divisible groups.
2. Finiteness for reductions of Hecke orbits 2.1.Let K be a field equipped with a rank 1 valuation.Throughout the paper, we will denote by O K the ring of integers of K. Fix an algebraic closure K of K. We denote by G K = Gal( K/K) the absolute Galois group of K, and by I K ⊂ G K the inertia subgroup.
In this section we suppose that K is a finite extension of Q p .We write Fp for the residue field of K. We denote by Qp the maximal unramified extension of Q p , and by K the compositum of K and Qp .
Proof.If ρ(G K ) = ∪ g∈S gρ(I K ) for some finite set S, then ∪ g∈S gH is a closed set containing ρ(G K ) and hence equals G.This shows that For the other direction, suppose that [G : H] < ∞.By a Theorem of Sen ([Sen73, Theorem 2]) and Serre ([Ser79, Theorem 1]), ρ( Definition 2.4.We say that two p-divisible groups over a finite field F q are equivalent if they become isomorphic over F p .
Lemma 2.5.For any h ≥ 1, the set of equivalence classes of p-divisible groups over F q of height h is finite.
Proof.By a result of Oort [Oor04, Corollary 1.7], there is an integer n = n(h) such that any two p-divisible groups over F p of height h, are isomorphic if and only if their p n -torsion subgroups are isomorphic.In particular, the equivalence class of a p-divisible group H over F q of height h is determined by its p n -torsion subgroup is a finite flat group scheme over F q of order p nh , there are only finitely many possibilities for H [p n ], and the lemma follows.
2.6.Let J(G ) denote the set of isomorphism classes of p-divisible groups over O K which are isogenous to G ⊗ OK O K .We set the set of isomorphism classes of reductions of elements of J(G ).
Analogously, we denote by I(A) the Hecke orbit of A, namely the set isomorphisms of abelian varieties over O K which are isogenous to A ⊗ OK O K .We then define }, the reduction of the Hecke orbit of A.
We now give a Galois-theoretic criterion for the finiteness of J(G , Proof.By Lemma 2.3, our hypothesis implies that ρ(I K ) has finite index in ρ(G K ).Therefore, after replacing K by a finite extension if necessary, we may assume that ρ(I K ) = ρ(G K ).Let K ρ the splitting field of ρ, namely the Galois extension defined by the subgroup Ker(ρ).Since ρ( Note that any isogeny of p-divisible groups with source G can be defined over is defined over the residue field of K, namely F q .The finiteness of J(G , Fp ) now follows from Proposition 2.5.
2.8.We will apply Proposition 2.7 in two cases.To explain the first of these, recall that for a p-divisible group H over F q , its Dieudonné module D(H ) is a finite free W (F q )-module equipped with a semi-linear Frobenius ϕ.If q = p r , then ϕ r acts linearly on D(H ), and we call this action the q-Frobenius Frob q on D(H ).Following Rappoport-Zink [RZ96], we say that H is decent if the action of Frob q on D(H ) is semisimple with eigenvalues which are all rational powers of q.That latter condition means that for each eigenvalue α, α m = q n for some integers m, n.
The semisimplicity condition is always satisfied if H = B[p ∞ ] is the p-divisible group arising from an abelian variety B over F q .Examples of a decent p-divisible groups include those of the form Proof.Let Rep G denote the Q p -linear Tannakian category of algebraic representations of G, and Isoc Qp the Tannakian category of isocrystals over K 0 , the maximal absolutely unramified subfield of K.As for D(H ), Frob q := ϕ r acting on Isoc Qp is linear.Using Fontaine's functor D cris , the representation ρ : G K → G(Q p ), gives rise to a functor

This functor sends
Now let W be a faithful representation of G/H, viewed as a representation of G, and write w : G → GL(W ).Then ρ • w is an unramified representation, so that the eigenvalues of Frob q acting on D cris (W ) are p-adic units.Since W is in the Tannakian category generated by V , it follows that D cris (W ) is in the Tannakian category generated by D[1/p].Hence Frob q acting on D cris (W ) is semi-simple and each of its eigenvalues have the form n i=1 α di i , where α 1 , • • • , α n are the eigenvalues of Frob q on D cris (V ).In particular any eigenvalue α of Frob q acting on D cris (W ) is a rational power of q.Since α is also a p-adic unit, it follows that α is a root of unity.
It follows that Frob q acting on D cris (W ) has finite order.This implies that for some power q ′ of q, W, viewed as a representation of G K /I K , can be identified with (D cris (W )⊗W (F q ′ )) ϕ=1 , with G K /I K acting via its action on W (F q ′ ).In particular, we see that G K /I K acts on W through a finite quotient.Since ρ(G K ) is dense in G, it is dense in the image of G in GL(W ), and hence this image is finite.As W was a faithful representation of G/H, this proves the proposition.
Proof.The first statement follows immediately Proposition 2.9 and Proposition 2.7.
For the second statement, as the special fiber of A is supersingular, we may replace K by a finite extension so that the Galois action on the prime-to-p torsion of A is through scalars (with Frobenius mapping to the scalar q 1/2 where q is the size of the residue field of K).Further, we observe that A[p ∞ ] ⊗ OK F q is decent as A has supersingular reduction, and hence the hypothesis of Proposition 2.7 holds.Let K ρ be as in the proof of Proposition 2.7.We have that every isogeny from A is defined over K ρ and hence all A ′ isogenous to A have reductions defined over F q , the residue field of K.By Zarhin's trick [Zar77, Theorem 4.1] there are only finitely many isomorphism classes of abelian varieties over F q , and hence I(A, F p ) is a finite set.
2.11.Our second application of Proposition 2.7 is more indirect, and proceeds by showing that even if G does not satisfy the hypothesis of Proposition 2.7 one can sometimes construct an auxiliary p-divisible group which does.
Lemma 2.12.Suppose that the connected component of the identity in G is reductive.Then after replacing K by a finite extension, there exists a p-divisible group Proof.After replacing K by a finite extension, we may assume that G is reductive, and we set T = G/H.As T is abelian, it is a torus.Let Z G denote the center of G, and G der its derived subgroup.The map Z G × G der → G is surjective with finite kernel, so we obtain a surjective map Z G → T. This implies that there is a subtorus T sub ⊂ Z G , such that the map T sub → T is an isogeny.Let χ : G K → T (Q p ) be the map induced by ρ, and let σ ∈ G K be a lift of the q-Frobenius.For some positive integer m, τ = χ(σ m ) lifts to an element of T sub (Q p ). Thus after replacing K by a finite extension, we may assume that τ lifts to an element τ ′ ∈ T sub (Q p ).Since T sub → T is an isogeny, the subgroup generated by τ ′ is bounded, so there is an unramified Galois character of the form By construction, the underlying Z p -modules of T p G and T p G ′ are canonically identified, and the action of G K on G ′ is obtained by multiplying its action on T p G by ψ(σ) −1 .Hence we have Corollary 2.13.If the p-adic Galois representation associated to G is semisimple, then the set J(G , F p ) is finite.
Proof.By Lemma 2.12 we can find G ′ such that H ′ = G ′ .By Proposition 2.7 J(G ′ , F p ) is a finite set.On the other hand the two p-divisible groups G and G ′ are isomorphic over K and therefore J(G , F p ) = J(G ′ , F p ).

Finiteness of p-divisible groups admitting a CM lift
3.1.In this section we assume that K is a finite extension of K 0 = W ( Fp )[1/p].
A p-divisible group G of (constant) height h, over any base, is said to have CM by a commutative semisimple Q p -algebra F if there is an injective homomorphism We say that G is CM, or has CM if G has CM by some F as above.
If G is a p-divisible group over O K , we can form its formal group G .If G has CM by F, then Lie G ⊗ OK K ≃ ⊕ σ V σ where σ runs over Q p -algebra maps F → K, and for a ∈ F, we have aV σ = σ(a)V σ .For each σ, the summand V σ is either trivial, or one dimensional over K [CCO14, Lemma 3.7.1.3].We denote by Φ the set of σ for which V σ is one dimensional, and we call Φ the CM type of G .Lemma 3.2.Let G be a p-divisible group over O K with CM by F. Then there exists a finite extension K ′ /Q p contained in K, and a p-divisible group Proof.This is well known.Let D be the weakly admissible module D cris (T p G [1/p]), so that D is an F ⊗ Qp K 0 -module equipped with an injective, semi-linear Frobenius and a one step filtration Fil The filtration on D K is induced by a cocharacter µ ∈ X * (Res F/Qp G m ).Choose K ′ /Q p finite and contained in K, such that µ is defined over K ′ , and let K ′ 0 denote the maximal unramified subfield of K ′ .Then there exists a free Then we can identify D with F ⊗ Qp K 0 via ι, and the Frobenius on D given by δσ, where δ ∈ (F ⊗ Qp K 0 ) × and σ denotes the Frobenius on K 0 .After possibly replacing K ′ by a larger field, there exists Then c • ι respects Frobenius, and also respects filtrations as ι does.D ′ along with the Frobenius δ ′ σ and filtration Fil 1 D ′ K ′ ⊂ D ′ K ′ is a weakly admissible module as D is.This weakly admissible module equals D cris (T p G ′ [1/p]) where G ′ /O K ′ is a pdivisible group, for example by [Kis06].The isomorphism D ′ K0 ≃ D K0 induces a quasi-isogeny between the Tate-modules of G ′ and G as Gal( K/K)-representations, and hence (after multiplication by a power of p) an isogeny G ′ OK → G .After replacing K ′ by a finite extension, we may assume that the kernel of this isogeny is defined over O K ′ and the theorem follows.
3.3.Let H be a p-divisible group over Fp .We say that H admits a CM lift, if there exists a finite extension K/W ( Fp )[1/p], and a CM p-divisible group G over We remark that there is essentially no extra generality gained by considering CM lifts to more general base rings.More precisely, if R is an integral, normal, flat W ( Fp )-algebra, and G is a CM deformation of H to R, then there is a finite extension K/W ( Fp )[1/p], and an inclusion O K → R, such that G arises from a CM deformation of H over O K .This can be deduced from the fact that the rigid analytic period morphism in [RZ96, §5] is étale, together with the fact that any Theorem 3.4.Let H / Fp be a p-divisible group.Then, the isogeny class of H contains only finitely many isomorphism classes of p-divisible groups which admit a CM lift.
Proof.Since the algebra End(H ) ⊗ Q has finite dimension over Q p , there are only finitely many choices for the CM algebra F. Given F, there are only finitely many choices for the CM type Φ.Thus, we may fix F and Φ, and consider only p-divisible groups in the isogeny class of H , which admit a CM lift having CM by F and CM type Φ.
Let G , G 1 be such lifts, defined over some finite extension K/W ( Fp )[1/p].By Lemma 3.2, there exists a p-divisible group G ′ with CM by F, defined over a finite extension Now the Zariski closure of the image of G K ′ acting on T p G ′ is a closed subgroup of the torus Res F/Qp G m , hence is reductive.Hence J(G ′ , Fp ) is finite by Theorem 2.13, and the theorem follows.
3.5.Let A g denote the moduli space of principally polarized abelian varieties of dimension g.For a given Newton polygon ν we denote by W ν ⊂ A g, Fp the corresponding Newton stratum.It is a locally closed subscheme.
For any x ∈ A g ( Fp ), the associated p-divisible group G x carries a principal polarization, an isomorphism, ψ x , of G x and its Cartier dual.The polarized central leaf through a point x ∈ A g is the locus of points where the associated polarized Note that these conditions imply that Fil 1 = Fil 2 ⊥ , that ϕ(Fil 2 L) ⊂ p 2 L, and that for i = 0, 1, 2, gr i L has rank 1, 20, 1 respectively.4.2.Now suppose that R = W (k) with k a perfect field, which will be either Fp or a finite field in application.
A K3-crystal L over k is said to be supersingular, if the slopes of ϕ are all 1.If L is supersingular, and k is algebraically closed, then L ϕ=p is a free Z p -module of rank 22 ([Ogu79, Theorem 3.3]), which also admits a bilinear form (which will no longer be perfect).
If k = F q is a finite field, then we say a K3-crystal L over k is decent if the q-Frobenius on L has eigenvalues which are rational powers of q.We note that this implies the Z p -module L ϕ=p ⊂ L has rank 22.Note that every K3-crystal over F p admits a decent model over F q for some q, see [Kot85, §4.3] Lemma 4.3.Let L be a filtered K3-crystal over R, as above.Then the filtration on L is induced by a cocharacter µ : G m → GO(L, •, • ).In particular, the subgroup P ⊂ GO(L, •, • ) preserving the filtration is parabolic.
Proof.Let L 2 = Fil 2 L, and choose a submodule Then (L 1 ) ⊥ is free of rank 2 and surjects on gr 0 L, as •, • is perfect and strict for filtrations.Thus, we can choose a rank 1 direct summand L 0 ⊂ (L 1 ) ⊥ which maps isomorphically to gr 0 L. Then •, • induces a perfect pairing between the rank 1 subspaces L 0 , L 2 ⊂ (L 1 ) ⊥ .Thus, modifying L 0 ⊂ (L 1 ) ⊥ we may assume that L 0 is isotropic for •, • .Now L = L 2 ⊕ L 1 ⊕ L 0 , and we define µ by requiring that µ(z) by z i on L i .Since L i and L j are orthogonal for i + j = 2, µ(z) acts on •, • by z 2 .In particular µ factors through GO(L, •, • ).
4.4.Now let k be a perfect field of characteristic p, which is either algebraically closed or an algebraic extension of F p .Let X/k denote a K3-surface.It is well known that the deformation functor of X is smooth and pro-representable and formally smooth of dimension 20 over W = W (k). Let Spf RX denote the universal deformation space of X, and let X u π − → Spf R denote the universal deformation of X. Choose a set {p, x 1 . . .x 20 } of elements that generate the maximal ideal of RX ; define σ to be the lift of the Frobenius endomorphism of RX mod p such that σ is the usual Frobenius on W (F q ), and σ(x i ) = (x i ) p .
Then L u = R 2 f * π is a filtered K3-crystal over RX .To give a more explicit description of this filtered K3-crystal, we need the following Lemma 4.5.Let L be the K3-crystal attached to X.There exists a free Z p -module with quadratic form (T, •, • ′ ) and an isomorphism ι : (T, Proof.The bilinear form on L is self dual.It follows from the theory of nondegenerate bilinear forms over finite fields that after replacing k by at most a quadratic extension, (L, •, • ) has determinant a square and also admits a rank-11 isotropic subspace.Indeed, there is a unique quadratic form on L (up to isomorphism) that satisfies these conditions, whence it follows that (T, •, • ′ ) Zp ⊗ W is isomorphic to (L, •, • ) where T is a rank-22 free Z p -module, and •, • ′ is the unique self-dual quadratic form on T that has square determinant and admits a rank-11 It follows that •, • is perfect on L ′ u , and hence L ′ u = L u .This proves the claim.It follows that w = b u b −1 ∈ SO( RX ).Since the map U opp → P 0 \ GO(L 0 , •, • ) is an open immersion, and w is the identity mod n, we can write w = λ • u with u ∈ U opp ( RX ) and λ ∈ P ( RX ) both reducing to 1 mod n.Conjugating j by an element of P ( RX ) has the effect of replacing b u = λub by its σ-conjugate by the same element.Now let m ≥ 1, and suppose that j can be chosen so that b u = b u (m) = ub modulo n m , so that λ = λ(m) ≡ 1 mod n m .Then, Since λ ≡ 1 mod n n , σ(λ) ≡ 1 mod n pm , and hence b u (m + 1) = λ −1 b u (m)σ(λ) ≡ ub mod n pm .This shows that ub is the σ-conjugate of b u (1) by the convergent product . . .λ(2)λ(1), so j can be chosen with b u = ub.
It remains to show that the map u : Spf RX u − → Û opp is an isomorphism.Given that these are both smooth 20-dimensional formal schemes over W , it suffices to prove this modulo the ideal (p, m 2 ).Now let S = RX /(p, m 2 ), and let L S denote filtered K3-crystal over S given by L u | S .After, modifying our chosen isomorphism j by u, L S may be identified with where we have used that σ(u) = 1 in S.
Work of Nygaard-Ogus [NO85, Theorem 5.2, 5.3] implies that deformations of X to S correspond bijectively to isotropic lifts of Fil 2 L mod m.These lifts are in bijection with points of U opp (S) which are 1 modulo m, and hence with Û opp (S).This implies that u induces an isomorphism on S points, and hence is an isomorphism.Definition 4.9.A GSpin-structure on a p-divisible group H /k is the data of an isomorphism ι : H ⊗ W (k) → D(H ), such that ι(s α,p ) ∈ D(H ) ⊗ are Frobeniusinvariant.We say that ι 1 and ι 2 are isomorphic GSpin-structures if ι 1 (s α,p ) = ι 2 (s α,p ).

Proposition 4.11 ([Yan19, Lemma A.7]
).There exists a p-divisible group H over k whose Dieudonné module D(H ) is given by H, with Frobenius acting as bσ.
We will now use description of the universal deformation space of X to prove that H admits a CM lift if X does.Write R = RX .The opposite unipotent of μ in GSpin is canonically isomorphic to U opp , the opposite unipotent of µ SO in SO.Thus, we may regard u ∈ U opp ( R) as an endomorphism of H ⊗ R.
Let Proof.The weakly admissible module D(X y ) is constructed using the crystalline and de Rham cohomology of X y , and the isomorphism between them as in [BO83, §2], and similarly for D(H y ).Let us briefly recall the construction.
Let Ry denote the P D-completion of R with respect to Ker(y) + p R. Then L Ry = L u ⊗ R Ry is equipped with a Frobenius and filtration.Moreover, there is a unique Frobenius equivariant map L = L 0 → L Ry [1/p] which lifts the identity over n.It may be constructed by choosing any lift of the identity s 0 , and taking the limit s = lim i ϕ i (s 0 ) = lim i ϕ i • s 0 • ϕ −i , which converges.This allows us to identify L ⊗ W (k) K with L y [1/p], where L y = L u ⊗ R,y O K .If K 0 ⊂ K denotes the maximal unramified subfield, this gives D(X y ) ∼ = L ⊗ W (k) K 0 the structure of a weakly admissible module over K.Note that this identification is not in general given by the identity of L. There is an analogous construction starting with H R in place on L u .Now let L u (1) denote the K3-crystal with underlying module L u , equipped with the Frobenius given by p −1 b u σ, and the filtration given by Fil i L u (1) = Fil i+1 L u .By construction, there is an inclusion L u (1) ⊂ H ⊗ R .Applying the above construction to both sides, we obtain D(X y )(1) ⊂ D(H y ) ⊗ , as required.Proposition 4.14.Suppose that the K3-surface X admits a CM lift.Then so does the associated Kuga-Satake p-divisible group H . Proof.Let K, K 0 and y : R → O K , be as in the previous lemma, and denote by G(D(X y )(1)) and G(D(H y )) the Tannakian groups of the weakly admissible modules D(X y )(1) and D(H y ) ⊗ , associated to fiber functor which takes a weakly admissible module to its underlying K 0 -vector space.By construction, we have the ι q : H ⊗ W (F q ) → H that respects tensors.It follows that the GSpin-structure is indeed defined over F q , as claimed4 .

2. 2 .
Let G /O K denote a p-divisible group.Denote by T p G the p-adic Tate module of G , and let ρ = ρ G : G K → GL(T p G ) be the Galois representation associated to G .We denote by G (resp.H) the Zariski closure of ρ denote the subgroup preserving T p G .After replacing K by a finite extension, and so τ ′ by a power, we may assume that τ ′ ∈ C.The action of C on T p G commutes with the action of G K .Hence by Tate's theorem, it induces a map C → Aut G .Since G is defined over O K , for any γ ∈ G K /I K , we have a canonical isomorphism γ * G ≃ G over O K .Denote by c γ the composite of this isomorphism and the automorphism ψ(γ) −1 ∈ C viewed as an automorphism of G (here we view ψ as a character on G K /I K ).Then c γ defines a descent datum on G [p n ]| O K , for each n.By étale descent, c γ arises from a unique p-divisible group G ′ over O K .
Fil ⊂ H denote the filtration of H induced by μ.By [Kis10, Section 1.5], the data(H ⊗ R, Fil ⊗ R, u • ( bσ)) arises from the Dieudonné module of a p-divisible group H R over Spf R, which deforms H .Note that the tensors s α,0 ∈ D(H R) ⊗ are Frobenius invariant and in Fil 0 .Lemma 4.13.Let K/W (k)[1/p] be a finite extension, y : R → O K , a map of W (k)-algebras, and X y (resp.H y ) the K3 surface (resp.p-divisible group) over O K corresponding to y.Let D(X y ) (resp.D(H y )) denote the weakly admissible module over K associated to X y (resp.H y ).Then there is a canonical inclusion D(X y )(1) ⊂ D(H y ) ⊗ compatible with filtrations and Frobenius.