Ordinary K3 surfaces over a finite field

Theorem A.

The endomorphism F preserves the Z-module M and satisfies

1. the 𝐙p⁢[F]-module M⊗𝐙p decomposes as M0⊕M1⊕M2 with

   1. F⁢Ms=qs⁢Ms for all s,

   2. M0, M1 and M2 are free 𝐙p-modules of rank 1, 20, 1 , respectively.

Definition (see [11]).

Let 𝒪K be a complete discrete valuation ring with fraction field K. We say that a K3 surface X over K satisfies (⋆) if there exists a finite extension K⊂L and algebraic space 𝔛/𝒪L satisfying

1. 𝔛L≅XL,

2. 𝔛 is regular,

3. the special fiber is a strict normal crossing divisor in 𝔛,

4. the relative dualizing sheaf of 𝔛/𝒪L is trivial.

Theorem B.

Fix an embedding ι:W⁢(Fq)→C. The resulting functor X↦(M,F,K) is a fully faithful functor between the groupoids of

1. ordinary K3 surfaces X over 𝐅q, and

2. triples (M,F,𝒦) consisting of

   1. an integral lattice M,

   2. an endomorphism F of the 𝐙-module M, and

   3. a convex subset 𝒦⊂M⊗𝐑,

   satisfying (M1)–(M5).

If every K3 surface over Frac⁡W⁢(Fq) satisfies (⋆), then the functor is essentially surjective.

Theorem C.

Let OK be a complete discrete valuation ring with perfect residue field k of characteristic p and fraction field K of characteristic 0. Let X be a projective K3 surface over OK and assume that Xk is ordinary. Then the following are equivalent:

1. 𝔛 is the base change from W⁢(k) to 𝒪K of the canonical lift of 𝔛k.

2. Hét2⁢(𝔛K¯,𝐙p)≅H0⊕H1⁢(-1)⊕H2⁢(-2) with Hi unramified 𝐙p⁢[GalK]-modules, free of rank 1, 20, 1 over 𝐙p, respectively.

Question.

Does there exist a triple (M,F,𝒦) satisfying (M1)–(M5) and the inequality 1+tr⁡F+q2<0?

Lemma 1.1.

One has H∙⁢(Xét,UA)=H∙⁢(X,UA) and H1⁢(Xét,UA)=0.

Proof.

The sheaf 𝒰A has a filtration whose graded pieces are 𝔪n/𝔪n+1⊗A/𝔪𝒪𝔛A/𝔪. Since these are coherent, we have

Moreover, since H1⁢(𝔛A/𝔪,𝒪𝔛A/𝔪) vanishes, we conclude that H1⁢(𝔛ét,𝒰A)=0. ∎

Lemma 1.2.

For every (A,m)∈ArtΛ there is a natural exact sequence

of abelian groups.

Proof.

Consider the complex 𝒰A⁢→pr⁢𝒰A on 𝔛ét in degrees 0 and 1. We have a short exact sequence

and thanks to Lemma 1.1 we obtain canonical isomorphisms

Now consider the short exact sequence

of length-two complexes concentrated in degrees 0 and 1. Using the Kummer sequence and the fact that 𝐆m is smooth, we see that

and similarly for 𝔛A/𝔪. It follows that the above short exact sequence of complexes induces a long exact sequence of (hyper-)cohomology groups

Since A/𝔪 is perfect and Pic⁡𝔛A/𝔪 is torsion-free, we have Hfl1⁢(𝔛A/𝔪,μpr)=0 and we conclude

which is what we had to show. ∎

Lemma 1.3.

For any quasi-compact quasi-separated scheme X there is a natural isomorphism

Proof.

This follows from [22, 0739], taking for ℬ the class of quasi-compact and quasi-separated schemes, and for Cov the fpqc covers consisting of finitely many affine schemes. ∎

Lemma 1.4 ([14, Corollary 1.4]).

The group Hfl2⁢(Xk¯,μp∞) is p-divisible. ∎

Lemma 1.5.

One has Hfl2⁢(Xk¯,μp∞)Gal⁡(k¯/k)=Hfl2⁢(Xk,μp∞).

Proof.

Since the p-th power map k¯×→k¯× is a bijection, and since Pic⁡𝔛k¯ is torsion-free, we have Hfli⁢(𝔛k¯,μp∞)=colimr⁡Hfli⁢(𝔛k¯,μpr)=0 for i∈{0,1}. It now follows from the Hochschild–Serre spectral sequence

that Hfl2⁢(𝔛k¯,μp∞)Gal⁡(k¯/k)=Hfl2⁢(𝔛k,μp∞). ∎

Theorem 1.6.

Let X be a formal K3 surface over Λ with Xk ordinary. Then the enlarged formal Brauer group Ψ⁢(X) is representable by a p-divisible group over Λ. Its étale-local exact sequence

satisfies

1. Ψ∘⁢(𝔛) is a connected p-divisible group of height 1 , with

   Ψ∘⁢(𝔛)⁢[pr]⁢(A)=ker⁡(Hfl2⁢(𝔛A,μpr)→Hfl2⁢(𝔛A/𝔪,μpr))

   for all (A,𝔪)∈ArtΛ. It is canonically isomorphic to Br^⁢(𝔛),

2. Ψ⁢(𝔛)⁢[pr]⁢(A)=Hfl2⁢(𝔛A,μpr) for all (A,𝔪)∈ArtΛ,

3. Ψét⁢(𝔛)⁢[pr]⁢(A)=Hfl2⁢(𝔛A/𝔪,μpr) for all (A,𝔪)∈ArtΛ.

Proof.

The representability and (i) are shown in [1, Proposition IV.1.8].

To prove (ii), note that since H1⁢(𝔛A,𝐆m)=Pic⁡𝔛A is torsion-free, we have a natural isomorphism

Taking the colimit over r, we obtain a natural isomorphism

and in particular we see that Hfl1⁢(𝔛A,μp∞) is p-divisible. Now the long exact sequence associated to

induces an isomorphism

which proves (ii).

A similar argument shows (iii) for A with A/𝔪 algebraically closed, after which Lemma 1.5 implies the general case. ∎

Theorem 1.7 (Nygaard [14, Theorem 1.6]).

The map

is a bijection.

Proposition 1.8 ([14, Proposition 1.8]).

The map Pic⁡Xcan→Pic⁡X is a bijection.

Corollary 1.9 ([14, Proposition 1.8]).

The lift Xcan is algebraizable and projective.

Theorem 2.1.

Let X be a projective K3 surface over OK. Assume that Xk is ordinary. Then there is a natural injective map of GalK-modules

whose cokernel is a free Zp-module of rank 1.

Lemma 2.2.

For all i the natural map

is an isomorphism.

Proof.

As in the proof of Lemma 1.2, we have

and similarly

Let 𝒰n be the kernel of 𝒪𝔛n×→𝒪𝔛1× and 𝒰 the kernel of 𝒪𝔛×→𝒪𝔛1×. Then by the usual dévissage arguments the lemma reduces to showing that

is an isomorphism for all i.

Since the maps 𝒰n+1→𝒰n are surjective, we have Rlimn⁡𝒰n=𝒰. Since 𝒰 has a filtration with graded pieces isomorphic to 𝒪𝔛1, it has cohomology concentrated in degrees 0 and 2. These two facts imply

in 𝒟⁢(Ab). Similarly, we have

in 𝒟⁢(Ab). As R⁢Γét commutes with Rlim, the lemma follows. ∎

Corollary 2.3.

If Xk is ordinary, then Ψ⁢(X)⁢(K)⁢[pr]=Hfl2⁢(X,μpr).

Proof.

Indeed, we have

so the corollary follows from Lemma 2.2. ∎

Proposition 2.4.

If X is a projective K3 surface over OK, then for all r the natural map Hfl2⁢(X,μpr)→Hfl2⁢(XK,μpr) is injective.

Proof.

The Kummer sequence gives a commutative diagram with exact rows

Since 𝔛 is projective, we have Br=Br′ for 𝔛 and 𝔛K.

The left arrow in the diagram is an isomorphism since the special fiber 𝔛k is a principal divisor in 𝔛, so that Pic⁡𝔛→Pic⁡𝔛K is an isomorphism. By [8, Corollary 1.8] the natural maps of Br⁡𝔛 and Br⁡𝔛K to Br⁡K⁢(𝔛K) are injective, so that also the right arrow in the diagram is injective. We conclude that the middle map is injective. ∎

Proof of Theorem 2.1.

The proof is now formal. By Corollary 2.3 and Proposition 2.4 we have for every r and every finite extension K⊂L a canonical injection

Taking the colimit over all L, we obtain a GalK-equivariant injective map

and taking the limit over r, we obtain a GalK-equivariant injective map

Denote the cokernel of ρ by Q. Tensoring ρ with 𝐙/p⁢𝐙 yields an exact sequence

Since ρ1=ρ⊗𝐙/p⁢𝐙 is injective, we see that Tor⁡(Q,𝐙/p⁢𝐙) vanishes and that Q is torsion-free. ∎

Lemma 2.5.

Let U be a free Zp-module of rank 2 and b:U×U→Zp a non-degenerate symmetric bilinear form. Let L⊂U be a totally isotropic rank-1 submodule. If L is saturated in

then U∨=U.

Proof.

Since L is saturated in U∨, it is also saturated in U⊂U∨ and we may choose a basis (e,f) for U with L=〈e〉. Set d:=b⁢(e,f). Since b⁢(e,e)=0, the determinant of b is -d2. Since e/d lies in U∨, we must have that d is a unit and therefore U∨=U. ∎

Proof of Theorem C.

Assume that (ii) holds. Then we have

with the Hi unramified. Since the Tate module of a p-divisible group is Hodge–Tate of weights 0 and -1, we have Hom⁡(Tp⁢Ψ⁢(𝔛)K¯,H2⁢(-1))=0, and by Theorem 2.1 we see that

By Tate’s theorem [19, Theorem 4] this implies that

with Tp⁢Ψ0⁢(𝔛)K¯=H0⁢(1) and Tp⁢Ψét⁢(𝔛)K¯=H1. It follows that 𝔛 is the base change of the canonical lift of 𝔛k to 𝒪K.

Conversely, assume that 𝔛 is the base change of the canonical lift of 𝔛k to 𝒪K. Let H1 be the image of the direct summand Tp⁢Ψét⁢(𝔛)K¯ under the embedding

of Theorem 2.1. It is a primitive submodule, and considering Hodge–Tate weights, we see that the restriction of the bilinear form on H2 to H1 is non-degenerate. Let U⊂Hét2⁢(𝔛K¯,𝐙p⁢(1)) be its orthogonal complement. Then U is a rank 2 lattice over 𝐙p. The inclusions of H1 and U as mutual orthogonal complements inside the self-dual lattice Hét2⁢(𝔛K¯,𝐙p⁢(1)) induce an isomorphism

and an identification

Consider the unramified GalK-module H0:=Tp⁢Ψ∘⁢(𝔛)K¯⁢(-1). We have that H0⁢(1) is a totally isotropic line in U. We claim that it is saturated in U∨. Indeed, if x∈U∨ satisfies p⁢x∈H0⁢(1) then (x,α⁢(x)) defines a p-torsion element in the cokernel of the embedding Tp⁢Ψ⁢(𝔛)K¯→Hét2⁢(𝔛K¯,𝐙p⁢(1)), which must be trivial by Theorem 2.1. By Lemma 2.5 we conclude that U=U∨ and that Hét2⁢(𝔛K¯,𝐙p⁢(1))=U⊕H1.

Now U is a unimodular 𝐙p-lattice of rank 2 containing an isotropic line. Moreover, since the intersection pairing on Hét2⁢(𝔛K¯,𝐙p⁢(1)) is even, so is the lattice U. It follows that there is a unique isotropic line H2⁢(-1)⊂U with U=H0⁢(1)⊕H2⁢(-1) and with H0 and H2 dual unramified representations. We find

as claimed. ∎

Remark 2.6.

Using the results on integral p-adic Hodge theory by Bhatt, Morrow, and Scholze [2], one can show that the splitting of Hét2⁢(𝔛K¯,𝐙p) as in Theorem C implies an analogous splitting of the filtered crystal Hcrys2⁢(𝔛/W). If p>2, then the splitting of Hcrys2⁢(𝔛/W) implies that 𝔛 is the canonical lift of 𝔛k (see [7] and [14, Lemma 1.11, Theorem 1.12]). For p=2, however, the splitting of Hcrys2⁢(𝔛/W) is a weaker condition than the splitting of Hét2⁢(𝔛K¯,𝐙p), see also [7, Section 2.1.16.b].

Theorem 3.1 (Nygaard [14, Section 3], Yu [20, Lemma 2.3]).

There is a unique endomorphism F of H2⁢(Xcanι,Z⁢[1p]) such that

1. for every ℓ≠p the map F corresponds under the comparison isomorphism

   H2⁢(Xcanι,𝐙⁢[1p])⊗𝐙ℓ→∼Hét2⁢(X𝐅¯q,𝐙ℓ)

   to the geometric Frobenius Frob on étale cohomology,

2. the map F corresponds under the comparison isomorphism

   H2⁢(Xcanι,𝐙⁢[1p])⊗Bcrys→∼Hcrys2⁢(X/W)⊗WBcrys

   to the endomorphism ϕa⊗id, where ϕ denotes the crystalline Frobenius.

Moreover, F preserves the Hodge structure on H2⁢(Xcanι,Q). ∎

Lemma 3.2.

Let V be an unramified GalK-representation over Qp, and let n be an integer. Let Frob be the geometric Frobenius endomorphism of V, relative to Fq. Consider the GalK-module V⁢(-n):=V⊗Qp⁢(-n).

1. The map

   K×→GL(V(-n)),x↦Frobv⁢(x)⊗qn⁢v⁢(x)NmK/𝐐p(x)-n

   factors over the reciprocity map K×→GalKab and induces the action of GalK on V⁢(-n).

2. Dcrys⁢(qn⁢Frob)=ϕa as endomorphisms of Dcrys⁢(V⁢(-n)).

Proof.

The first statement follows from Lubin–Tate theory, see for example [17, Section 3.1, Theorem 2]. For the second, one uses that the functor Dcrys commutes with Tate twists to reduce to the case n=0. In this case, the statement follows from the observation that Frob⊗1=1⊗ϕa as endomorphisms of (V⊗𝐙pW⁢(𝐅¯q))Gal𝐅q, where the action of Gal𝐅q is the diagonal one. ∎

Proof of Theorem A.

By its definition (in Theorem 3.1), it is clear that the endomorphism F of H2⁢(Xcanι,𝐐)⊗𝐐p=Hét2⁢(XK¯,𝐐p) satisfies

on Hcrys2⁢(X/W)⁢[1p].

By Theorem C we have a decomposition

with Hi unramified and by Lemma 3.2, also the endomorphism

of Hét2⁢(XK¯,𝐐p) satisfies

Since Dcrys is fully faithful, we must have F=F′. But it then follows immediately that F preserves the 𝐙p-lattice Hét2⁢(XK¯,𝐙p), and that Hét2⁢(XK¯,𝐙p) decomposes as described in the theorem. ∎

Lemma 3.3.

Let (M,F) be a pair satisfying (M1)–(M4). Then complex conjugation on M⊗C maps the subspace Ms⊗Zp,ιC to M2-s⊗Zp,ιC.

Proof.

By property (M4), the one-dimensional subspace M0⊗𝐙p,ι𝐂 of M⊗𝐂 is the unique eigenspace for the endomorphism F corresponding to an eigenvalue u∈𝐐p⊂𝐂 with vp⁢(u)=0. By (M3) we have u⁢u¯=q2, and hence also the eigenvalue u¯=q2u lies in 𝐐p⊂𝐂. Since vp⁢(u¯)=vp⁢(q2), we see that the corresponding eigenspace is M2⊗𝐙p,ι𝐂. Similarly, complex conjugation maps M2 to M0. By (M2), the subspace M1 is the orthogonal complement of M0⊕M2, and hence is preserved by complex conjugation. ∎

Proposition 3.4.

Let (M,F) be the pair associated to an ordinary K3 surface X over Fq. Then we have Ms⊗Zp,ιC=Hs,2-s⁢(Xcanι) as subspaces of M⊗C=H2⁢(Xcanι,C).

Proof.

Indeed, under the “Hodge–Tate” comparison isomorphism

the subspace (Fili⁡HdR2⁢(Xcan,K/K))⊗K𝐂p is mapped to ⊕s≥iMs⊗𝐙p𝐂p. Extending ι to an embedding 𝐂p→𝐂, we see that the Hodge filtration on H2⁢(Xcanι,𝐂) agrees with the filtration on M⊗𝐂 induced by the decomposition on M⊗𝐙p, and hence by Lemma 3.3 we have Ms⊗𝐙p𝐂=Hs,2-s⁢(Xcanι). ∎

Proposition 3.5.

We have natural isomorphisms

and a class λ∈Pic⁡XF¯q is ample if and only if its image in Pic⁡Xcan,K¯ is ample.

Proof.

The first isomorphism follows from the fact that canonical lifts commute with finite unramified extensions. The second isomorphism follows from the triviality of the action of Gal⁡(K/Knr) on Pic⁡Xcan,K¯⊂Hét2⁢(Xcan,K¯,𝐐ℓ⁢(1)) and the vanishing of Br⁡Knr.