# Boundedness of the $p$-primary torsion of the Brauer groups of K3 surfaces

Christopher D. Lazda111Department of Mathematics, Harrison Building,
University of Exeter, EX4 4QF, United Kingdom. Email<EMAIL_ADDRESS>and Alexei N. Skorobogatov222Department of Mathematics, South Kensington
Campus, Imperial College London, SW7 2AZ, United Kingdom and Institute for the
Information Transmission Problems, Russian Academy of Sciences, Moscow,
127994, Russia. Email<EMAIL_ADDRESS>
###### Abstract

We prove that the transcendental Brauer group of a K3 surface $X$ over a
finitely generated field $k$ is finite, unless $k$ has positive characteristic
$p$ and $X$ is supersingular, in which case it is annihilated by $p$.

## Introduction

Let $k$ be a field, and $X$ a smooth, projective, geometrically connected
variety over $k$. If $\bar{k}$ is an algebraic closure of $k$, and
$\overline{X}$ is the base change of $X$ to $\bar{k}$, we let

${\rm{Br}}(\overline{X})^{k}:={\rm
Im}[{\rm{Br}}(X)\to{\rm{Br}}(\overline{X})]$

denote the transcendental Brauer group of $X$. If $k$ is a finitely generated
field of characteristic $0$, and $X$ is a K3 surface, it was proved in [SZ08]
that this group is finite. Similarly, if $k$ is finitely generated of
characteristic $p>0$, and $X$ is again a K3 surface, then it was proved in
[SZ15] (for odd $p$) and [Ito18] (for $p=2$) that the prime-to-$p$ torsion
subgroup
${\rm{Br}}(\overline{X})^{k}(p^{\prime})\subset{\rm{Br}}(\overline{X})^{k}$ is
finite.

In general, one cannot expect the $p$-primary torsion subgroup
${\rm{Br}}(\overline{X})^{k}\\{p\\}\subset{{\rm{Br}}}(\overline{X})^{k}$ to be
finite, see [Sko24, §4, Example 1]. However, one might still ask if this group
has finite exponent, that is, if it is annihiliated by some fixed power of
$p$. Moreover, the examples where ${\rm{Br}}(\overline{X})^{k}\\{p\\}$ is
infinite are all supersingular, so one might ask whether
${\rm{Br}}(\overline{X})^{k}\\{p\\}$ is finite whenever $X$ is of finite
height. Inspired by the analogous result for abelian varieties proved in
[D’Ad24], our main goal in this article is to give an affirmative answer to
these questions.

######  Theorem 1.

Suppose that $k$ is finitely generated of characteristic $p>0$, and that $X$
is a K3 surface.

  1. (1)

If $X$ is of finite height $h$, then ${\rm{Br}}(\overline{X})^{k}\\{p\\}$ is a
finite abelian group with at most $22-2h-\rho$ generators, where $\rho$ is the
Picard number of $\overline{X}$.

  2. (2)

If $X$ is supersingular, then ${\rm{Br}}(\overline{X})\cong\bar{k}$ is
annihilated by $p$.

In particular, ${\rm{Br}}(\overline{X})^{k}\\{p\\}$ is of finite exponent.

In fact, modulo the Tate conjecture, (2) is well-known, and essentially goes
back to Artin [Art74]. All we do here is observe that the Tate conjecture is
now known for K3 surfaces, thanks to work of Charles [Cha13], Madapusi Pera
[MP15], Kim–Madapusi Pera [KMP16] and Ito–Ito–Koshikawa [IIK21]. Our proof of
(1) is also based on the proof of the Tate conjecture for K3 surfaces given in
[MP15] and [KMP16], and in particular the arithmetic properties of the
Kuga–Satake construction proved there. Our main observation is that the
crystalline incarnation of Kuga–Satake descends from the perfection of $k$ to
give a map of $F$-isocrystals over $k$ itself (see Theorem 3.1 below).
Ultimately, this allows us to reduce Theorem 1 to the crystalline Tate
conjecture for abelian varieties proved by de Jong in [dJ98].

Combined with the above-mentioned results on the prime-to-$p$ torsion, we
obtain the following corollary. Recall that the characteristic exponent of $k$
is 1 if ${\rm char}(k)=0$, and $p$ if ${\rm char}(k)=p>0$.

Corollary A. Suppose that $k$ is finitely generated of characteristic exponent
$p$, and that $X$ is a K3 surface. Then ${\rm{Br}}(\overline{X})^{k}$ is
finite unless $p>1$ and $X$ is supersingular, in which case
${\rm{Br}}(\overline{X})^{k}$ is annihilated by $p$.

Question. Let $k$ be an infinite finitely generated field of characteristic
$p>0$, and $X$ a supersingular K3 surface. Is the group
${\rm{Br}}(\overline{X})^{k}$ infinite?

The answer to this question is positive when $X$ is the Kummer surface
attached to the self-product of a supersingular elliptic cuvre [Sko24, §4,
Example 1]. In general, for a supersingular K3 surface $X$ over an infinite
field $k$, the group ${\rm{Br}}(\overline{X})^{k}$ becomes infinite after
replacing $k$ by a finite extension.

Let $k^{\rm s}$ be the separable closure of $k$ in $\bar{k}$, and let
$\Gamma={\rm{Gal}}(k^{\rm s}/k)$. Write $X^{\rm s}=X\times_{k}k^{\rm s}$. The
natural map ${\rm{Br}}(X^{\rm s})\to{\rm{Br}}(\overline{X})$ is injective
because the Picard scheme of a K3 surface is smooth, see [CTS21, Theorem 5.2.5
(ii)], [D’Ad24, Corollary 3.4]. It is known that the cokernel of the natural
map ${\rm{Br}}(X)\to{\rm{Br}}(X^{\rm s})^{\Gamma}$ is the direct sum of a
finite group and a $p$-torsion group of finite exponent [CTS21, Theorem
5.4.12]. Thus we obtain the following statement.

Corollary B. Suppose that $k$ is finitely generated of characteristic exponent
$p$, and that $X$ is a K3 surface. Then ${\rm{Br}}(X^{\rm s})^{\Gamma}$ is the
direct sum of a finite group and a $p$-torsion group of finite exponent.

When the first version of this paper was written, the authors became aware of
the preprint of Z. Li and Y. Qin that gives a different proof of Corollary B.
Their result applies to arbitrary smooth projective varieties for which the
Tate conjecture for divisors is known, see [LQ24, Theorem 1.1, Corollary 1.2].

### Notation and conventions

  * •

For a given field $k$ of characteristic $p>0$, we will let $\bar{k}$ be an
algebraic closure of $k$, $k^{\rm s}$ the separable closure of $k$ in
$\bar{k}$, and $k^{\rm p}$ the perfect closure of $k$ in $\bar{k}$ (the
maximal inseparable extension of $k$ in $\bar{k}$). We write
$\Gamma={\rm{Gal}}(k^{\rm s}/k)$.

  * •

A variety over $k$ will mean a separated $k$-scheme of finite type. For a
variety $X$ over $k$ we will write $X^{\rm s}=X\times_{k}k^{\rm s}$, $X^{\rm
p}:=X\times_{k}k^{\rm p}$ and $\overline{X}:=X\times_{k}\bar{k}$.

  * •

A subset $\\{x_{i}\\}\subset k$ is called a $p$-basis of $k$ if the monomials
$\prod x_{i}^{e_{i}}$, where $0\leq e_{i}\leq p-1$, form a basis of $k$ as a
vector space over $k^{p}$, see [Stacks, Definition 07P1]. Any field of
characteristic $p$ admits a $p$-basis.

  * •

Unadorned tensor products will be over ${\mathbb{Z}}$.

  * •

For any scheme $S$ we set

${\rm H}^{i}_{\rm fppf}(S,{\mathbb{Z}}_{p}(1)):=\lim_{n}{\rm H}^{i}_{\rm
fppf}(S,\mu_{p^{n}}).$

We have followed the definition given in [Ill79, Chapitre II, §5] in the case
when $S$ is smooth and proper over an algebraically closed field. Outside this
case, the above definition is not really the ‘correct’ one,333The ‘correct’
definition is ${\rm H}^{i}({\bf R}\\!\lim_{n}{\bf R}\Gamma(S,\mu_{p^{n}}))$,
which always maps to the above group, and is isomorphic to it if
$\lim_{n}^{(1)}{\rm H}^{i-1}(S,\mu_{p^{n}})=0$. however, it will be good
enough for our purposes here.

  * •

For any abelian group $A$ we denote by $A\\{p\\}$ its $p$-primary torsion
subgroup, and by $A(p^{\prime})$ its prime-to-$p$ torsion subgroup.

##  1 The geometric Brauer group

Let $k$ be a field of characteristic $p$, and $X$ a smooth and proper variety
over $k$. We denote by $\rho={\rm{dim}}_{\mathbb{Q}}({\rm
NS\,}(\overline{X})\otimes{\mathbb{Q}})$ the (geometric) Picard number of $X$.
Let $r$ be the dimension of the ${\mathbb{Q}}_{p}$-vector space $({\rm
H}^{2}_{\rm cris}(\overline{X}/W({\bar{k}}))\otimes{\mathbb{Q}})^{F=p}$. Using
results of Illusie, it is shown in [Pet24, Theorem A.1] that there is an
isomorphism
${\rm{Br}}(\overline{X})\\{p\\}\cong({\mathbb{Q}}_{p}/{\mathbb{Z}}_{p})^{\oplus(r-\rho)}\oplus
B$, where $n\geq 0$ and $B$ is annihilated by a power of $p$. For K3 surfaces,
the following precise result is known.

######  Theorem 1.1.

Let $X$ be a K3 surface over $k$. If $X$ has finite height $h$, then
${\rm{Br}}(\overline{X})\\{p\\}\cong({\mathbb{Q}}_{p}/{\mathbb{Z}}_{p})^{\oplus(22-2h-\rho)}$.
If $X$ is supersingular, then ${\rm{Br}}(\overline{X})\cong\bar{k}$.

###### Proof.

We include a proof for the reader’s convenience. We will assume that
$k=\bar{k}$ is algebraically closed (thus $X=\overline{X}$), and we set
$W=W({\bar{k}})$.

For any smooth and proper variety $X$ over $k$ and any $i,j\geq 0$ we have a
short exact sequence [IR83, Corollaire IV.3.5 (a)]

$0\to{\rm H}^{j}(X,W\Omega^{i}_{X,{\rm log}})\to{\rm H}^{j}_{{\rm
Zar}}(X,W\Omega^{i}_{X})\xrightarrow{1-F}{\rm H}^{j}_{{\rm
Zar}}(X,W\Omega^{i}_{X})\to 0,$

where $W\Omega^{i}_{X}$ is the sheaf of de Rham–Witt differential forms with
its semi-linear Frobenius $F$ [Ill79, I.2.E]. By definition,

${\rm H}^{j}(X,W\Omega^{i}_{X,{\rm log}}):=\varprojlim{\rm H}^{j}_{\rm
Zar}(X,W_{n}\Omega^{i}_{X,{\rm log}}),$

where $W_{n}\Omega^{i}_{X,{\rm log}}$ is a subsheaf of ‘logarithmic’ forms in
$W_{n}\Omega^{i}_{X}$. In view of a canonical isomorphism ${\rm
H}^{i}(X,{\mathbb{Z}}_{p}(1))\cong{\rm H}^{i-1}(X,W\Omega^{1}_{X,{\rm log}})$
[Ill79, §II.5], we obtain an isomorphism

${\rm H}^{i}(X,{\mathbb{Z}}_{p}(1))\cong{\rm
H}^{i-1}(X,W\Omega^{1}_{X})^{F=1}.$

We also have an exact sequence

$0\to{\rm H}^{2}_{\rm fppf}(X,{\mathbb{Z}}_{p}(1))/p^{n}\to{\rm H}^{2}_{\rm
fppf}(X,\mu_{p^{n}})\to{\rm H}^{3}_{\rm fppf}(X,{\mathbb{Z}}_{p}(1))[p^{n}]\to
0$ (1)

together with the exact sequence

$0\to{\rm Pic}(X)/p^{n}\to{\rm H}^{2}_{\rm
fppf}(X,\mu_{p^{n}})\to{\rm{Br}}(X)[p^{n}]\to 0$ (2)

deduced from the Kummer exact sequence in the fppf topology.

Now let $X$ be a K3 surface. As explained in [Lie16, Theorem 4.8], $X$ is
Shioda supersingular (that is, $\rho=22$) if and only if it is Artin
supersingular (that is, the $F$-crystal ${\rm H}^{2}_{\rm cris}(X/W)$ is
purely of slope 1), thanks to the Tate conjecture for K3 surfaces [Cha13,
MP15, KMP16, IIK21]. (Note that while [Lie16, Theorem 4.8] is stated in odd
characteristic, this restriction comes from the fact that the Tate conjecture
for K3 surfaces was only known in odd characteristic when _loc. cit._ was
written. Given the Tate conjecture for K3 surfaces in characteristic $2$, the
proof of [Lie16, Theorem 4.8] works verbatim.)

If $X$ is supersingular, then the map ${\rm
Pic}(X)\otimes{\mathbb{Z}}_{p}\to{\rm H}^{2}_{\rm
fppf}(X,{\mathbb{Z}}_{p}(1))$ is an isomorphism by [Ogu79, Corollaries 1.6 and
1.7] (for the missing piece of the latter, see [Ill79, Chapitre II, Théorème
5.14]). We therefore deduce an isomorphism

${\rm{Br}}(X)[p^{n}]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm
H}^{3}_{\rm fppf}(X,{\mathbb{Z}}_{p}(1))[p^{n}]\cong\bar{k},$

see [Ill79, Chapitre II, §7.2(b)] for the second isomorphism.

An entirely similar argument using the $\ell$-adic Kummer sequence shows that
${\rm{Br}}(X)\\{\ell\\}=0$ for all $\ell\neq p$, and hence ${\rm{Br}}(X)$ is
$p$-torsion.

Now suppose that $X$ has finite height $h$. Then ${\rm H}^{2}(X,W{\cal
O}_{X})$ is finitely generated over $W$, since it is the Dieudonné module of
the $p$-divisible group ${\widehat{{\rm{Br}}}}(X)$ [AM77, Corollary 4.3]. In
particular, the slope spectral sequence

$E_{1}^{i,j}={\rm H}^{j}(X,W\Omega^{i}_{X})\implies{\rm
H}^{i+j}(X,W\Omega^{\bullet}_{X})$

degenerates at $E_{1}$ by [Ill79, Chapitre II, Corollaire 3.14], and thus each
${\rm H}^{j}(X,W\Omega^{i}_{X})$ is finite free over $W$ by [IR83, Chapitre
IV, §4.6]. It then follows that all ${\rm H}^{i}_{\rm
fppf}(X,{\mathbb{Z}}_{p}(1))$ are finite free over ${\mathbb{Z}}_{p}$ by the
argument of [Pet24, Lemma A.2]. In particular, we deduce from (1) that

$\underset{n}{\rm colim}\;{\rm
H}^{2}(X,\mu_{p^{n}})\overset{\cong}{\longrightarrow}{\rm
H}^{2}(X,{\mathbb{Z}}_{p}(1))\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}/{\mathbb{Z}}_{p}$

is of the form $({\mathbb{Q}}_{p}/{\mathbb{Z}}_{p})^{\oplus n}$ for some
$n\geq 0$. By [Ill79, Théorème II.5.5 (5.5.3)] we have an isomorphism of
${\mathbb{Q}}_{p}$-vector spaces

${\rm H}^{i}(X,{\mathbb{Z}}_{p}(1))\otimes{\mathbb{Q}}\cong({\rm H}^{i}_{\rm
cris}(X/W)\otimes{\mathbb{Q}})^{F=p}.$

Thus $n$ is the multiplicity of the slope 1 part of ${\rm H}^{2}_{\rm
cris}(X/W)$, so that $n=22-2h$, cf. [Ill79, Chapitre II, §7.2(a)]. The Kummer
exact sequences (2) then fit together to give a surjection

$\underset{n}{\rm colim}\;{\rm
H}^{2}(X,\mu_{p^{n}})\twoheadrightarrow{\rm{Br}}(X)\\{p\\}$

with kernel $({\mathbb{Q}}_{p}/{\mathbb{Z}}_{p})^{\oplus\rho}$. This completes
the proof. ∎

This immediately proves Theorem 1 in the supersingular case. Moreover, to show
that Theorem 1 holds in the finite height case, it is enough to show that the
$p$-adic Tate module $T_{p}({\rm{Br}}(\overline{X})^{k})$ is zero. In doing
so, we are free to replace $k$ by a finite extension, so we can at any point
assume that $X(k)\neq\emptyset$, and that ${\rm Pic}(X)={\rm
Pic}(\overline{X})$.

The exactness of the Kummer sequence in the fppf topology gives rise to a
commutative diagram

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(3)

with exact rows. We let $T_{p}({\rm{Br}}(\overline{X}))^{k}$ denote the image
of the right hand vertical map of (3).

######  Lemma 1.2.

Suppose $X(k)\neq\emptyset$ and ${\rm{Pic}}(X)={\rm{Pic}}(\overline{X})$. Then
the natural map $T_{p}({\rm{Br}}(\overline{X}))^{k}\to
T_{p}({\rm{Br}}(\overline{X})^{k})$ is an isomorphism.

###### Proof.

The assumption implies that ${\rm H}^{1}(k,{\rm{Pic}}(X^{\rm s}))=0$. The map
${\rm{Br}}(X^{\rm s})\to{\rm{Br}}(\overline{X})$ is injective because the
Picard scheme of a K3 surface is smooth, see [CTS21, Theorem 5.2.5 (ii)],
[D’Ad24, Corollary 3.4]. Thus, the Hochschild–Serre spectral sequence gives
rise to an exact sequence

$0\to{\rm{Br}}(k)\to{\rm{Br}}(X)\to{\rm{Br}}(\overline{X})^{k}\to 0$

which is split by the choice of a $k$-point. Since the sequence is split, we
see that $T_{p}({\rm{Br}}(X))\to T_{p}({\rm{Br}}(\overline{X})^{k})$ is
surjective. Since $T_{p}({\rm{Br}}(\overline{X})^{k})\to
T_{p}({\rm{Br}}(\overline{X}))$ is injective, we deduce that
$T_{p}({\rm{Br}}(\overline{X}))^{k}=T_{p}({\rm{Br}}(\overline{X})^{k})$ as
required. ∎

######  Remark 1.3.

D’Addezio showed that $T_{p}({\rm{Br}}(\overline{A})^{k})=0$ for any abelian
variety over $k$ [D’Ad24, Theorem 5.1], by deducing this from
$T_{p}({\rm{Br}}(\overline{A}))^{k}=0$. See [Sko24, Theorem 3.2] for a
different proof.

We now let ${\rm H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}$
denote the image of

${\rm H}^{2}_{\rm fppf}(X,{\mathbb{Z}}_{p}(1))\to{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1)).$

If ${\rm Pic}(X)={\rm Pic}(\overline{X})$, we deduce from (3) an exact
sequence

$0\to{\rm{Pic}}(X)\otimes{\mathbb{Z}}_{p}\to{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}\to
T_{p}({\rm{Br}}(\overline{X}))^{k}\to 0$ (4)

and if in addition $X(k)\neq\emptyset$ we can replace the group appearing on
the right by $T_{p}({\rm{Br}}(\overline{X})^{k})$.

##  2 Crystalline cohomology

We continue to let $k$ be an arbitrary field of characteristic $p$. Choose a
Cohen ring $C$ for $k$, that is, a complete discrete valuation ring with
maximal ideal generated by $p$ and residue field isomorphic to $k$. Choose a
lifting $\sigma\colon C\to C$ of the Frobenius endomorphism of $k$. If we let
$\widehat{C}^{\rm p}$ denote the completed (direct) perfection of $C$ along
$\sigma$, then $\widehat{C}^{\rm p}$ is a Cohen ring for $k^{\rm p}$, and is
thus canonically isomorphic to $W(k^{\rm p})$. In particular, $\sigma$
determines an embedding $C\hookrightarrow W(k^{\rm p})$, such that $\sigma$ is
the restriction to $C$ of the canonical Witt vector Frobenius on $W(k^{\rm
p})$.

Set $C_{n}:=C/p^{n}$. If $\\{x_{i}\\}_{i\in I}\subset C$ is a lift of a
$p$-basis for $k$, then it follows from a more or less direct calculation that
the module

$\widehat{\Omega}_{C}:=\lim_{n}\Omega_{C_{n}/{\mathbb{Z}}}$

of $p$-adically complete differentials on $C$ is given by
$\widehat{\oplus}_{i\in I}\,C\,{\rm d}x_{i}$. For any smooth and proper scheme
$X/k$ we have the crystalline cohomology groups ${\rm H}^{q}_{\rm
cris}(X/C)=\lim_{n}{\rm H}^{q}_{\rm cris}(X/C_{n})$. By [BO78, Summary 7.26]
these are finitely generated $C$-modules. They are equipped with an integrable
connection

$\nabla\colon{\rm H}^{q}_{\rm cris}(X/C)\to\widehat{\Omega}_{C}\otimes_{C}{\rm
H}^{q}_{\rm cris}(X/C)$

by the construction explained in [BO78, page 7.16], and a horizontal Frobenius

$F\colon\sigma^{*}{\rm H}^{q}_{\rm cris}(X/C)\to{\rm H}^{q}_{\rm cris}(X/C)$

by functoriality. The construction of [BI70] (explained in [Pet03, §5.2])
provides a cycle class map

$c_{1}\colon{\rm Pic}(X)\to{\rm H}^{2}_{\rm cris}(X/C)(1)$

which lands inside the subspace of horizontal sections (because it factors
through absolute crystalline cohomology ${\rm H}^{2}_{\rm
cris}(X/{\mathbb{Z}}_{p})(1)$) on which $F=1$ (because $F^{*}$ acts as
multiplication by $p$ on ${\rm{Pic}}(X)$). For any $0\leq r\leq d$ there is a
horizontal and Frobenius-equivariant cup product pairing

${\rm H}^{q}_{\rm cris}(X/C)(r)\times{\rm H}^{2d-q}_{\rm
cris}(X/C)(d-r)\to{\rm H}^{2d}_{\rm cris}(X/C)(d)\cong C,$ (5)

where the target here has the unit connection and Frobenius structure.

In the case that $k$ is perfect, Illusie constructs in [Ill79, Chapitre II,
Théorème 1.4] a functorial isomorphism between the crystalline cohomology of
$X$ and the (Zariski or étale) hypercohomology of the de Rham–Witt complex of
$X$. This is deduced from a similar isomorphism

${\rm H}^{q}_{\rm cris}(X/W_{n}(k))\overset{\cong}{\longrightarrow}{\rm
H}^{q}(X,W_{n}\Omega^{\bullet}_{X})$

for each $n$, which in fact holds for any smooth variety $X/k$ (not
necessarily proper). We will need the following marginal extension of this
result.

######  Lemma 2.1.

Let $X$ be a smooth variety over a finitely generated field $k$ of
characteristic $p$, and let $k_{0}$ denote the algebraic closure of
${\mathbb{F}}_{p}$ inside $k$. Then for each $n$ there is an isomorphism

${\rm H}^{q}_{\rm cris}(X/W_{n}(k_{0}))\overset{\cong}{\longrightarrow}{\rm
H}^{q}(X,W_{n}\Omega^{\bullet}_{X})$

functorial in $X$ and $k$.

###### Proof.

This is surely well-known to the experts, but we include a proof here for lack
of a precise reference. By choosing a separating basis for $k/k_{0}$, we may
write $k={\rm colim}_{i\in I}R_{i}$ as a filtered colimit of smooth
$k_{0}$-algebras, with all transition maps $R_{i}\to R_{j}$ localisations.
Moreover, we may assume that there exists a cofiltered diagram
$\\{X_{i}\\}_{i\in I}$, with each $X_{i}$ a smooth $R_{i}$-scheme, such that
$X_{j}=X_{i}\times_{R_{i}}R_{j}$ whenever $j\geq i$, and
$X=X_{i}\times_{R_{i}}k$. In particular, $X={\rm lim}_{i}X_{i}$.

Then thanks to [Ill79, Chapitre II, Théorème 1.4] there is a natural quasi-
isomorphism ${\bf R}\Gamma_{\rm
cris}(X_{i}/W_{n}(k_{0}))\overset{\cong}{\longrightarrow}{\bf
R}\Gamma(X_{i},W_{n}\Omega^{\bullet}_{X_{i}})$ for each $i$, and we can see
that

${\rm colim}_{i\in I}{\bf
R}\Gamma(X_{i},W_{n}\Omega^{\bullet}_{X_{i}})\overset{\cong}{\longrightarrow}{\bf
R}\Gamma(X,W_{n}\Omega^{\bullet}_{X})$

by localising on $X$ and using the definition of $W_{n}\Omega^{\bullet}$ for
smooth ${\mathbb{F}}_{p}$-algebras. It is therefore enough to show that the
natural map

${\rm colim}_{i\in I}{\bf R}\Gamma_{\rm cris}(X_{i}/W_{n}(k_{0}))\to{\bf
R}\Gamma_{\rm cris}(X/W_{n}(k_{0}))$

is an isomorphism. This question is local on $X$, so we can assume that
$X={\rm Spec}(A)$ is affine, and so are all $X_{i}={\rm Spec}(A_{i})$. In this
case, we lift some fixed $R_{i_{0}}$ to a smooth $W_{n}(k_{0})$-algebra
$\widetilde{R}_{i_{0}}$, and $A_{i_{0}}$ to a smooth
$\widetilde{R}_{i_{0}}$-algebra $\widetilde{A}_{i_{0}}$. Since the transition
maps in the system $\\{R_{i}\\}_{i\geq i_{0}}$ are all localisations, we can
lift each $R_{i}$ uniquely to a localisation $\widetilde{R}_{i}$ of
$\widetilde{R}_{i_{0}}$, and then set
$\widetilde{A}_{i}:=\widetilde{A}_{i_{0}}\otimes_{\widetilde{R}_{i_{0}}}\widetilde{R}_{i}$.
Thus $\widetilde{A}:={\rm colim}_{i\in I}\widetilde{A}_{i}$ is a lifting of
$A$ to a quasi-smooth $W_{n}(k_{0})$-algebra in the sense of [Ber74, Chapitre
IV, Définition 1.5.1]. We therefore have

${\bf R}\Gamma_{\rm cris}({\rm Spec}(A)/W_{n}(k_{0}))={\bf R}\Gamma_{\rm
dR}(\widetilde{A}/W_{n}(k_{0}))$

and

${\bf R}\Gamma_{\rm cris}({\rm Spec}(A_{i})/W_{n}(k_{0}))={\bf R}\Gamma_{\rm
dR}(\widetilde{A}_{i}/W_{n}(k_{0}))$

for each $i$ by [Ber74, Chapitre V, Théorème 2.3.2]. Hence the claim reduces
to the fact that algebraic de Rham cohomology commutes with filtered colimits
of rings. ∎

More generally, if $S$ is an ${\mathbb{F}}_{p}$-scheme, and $f\colon X\to S$
is smooth and proper, then we may consider the relative crystalline cohomology
groups ${\bf R}^{q}f_{{\rm cris}*}{\cal O}^{\rm cris}_{X/{\mathbb{Z}}_{p}}$,
which are sheaves of ${\cal O}^{\rm cris}_{S/{\mathbb{Z}}_{p}}$-modules on
$(S/{\mathbb{Z}}_{p})_{\rm cris}$. In the special case when $S={\rm Spec}(k)$,
the group ${\rm H}^{q}_{\rm cris}(X/C)$ is simply the realisation of ${\bf
R}^{q}f_{{\rm cris}*}{\cal O}^{\rm cris}_{X/{\mathbb{Z}}_{p}}$ on the pro-PD-
thickening ${\rm Spec}(k)\hookrightarrow{\rm Spf}(C)$.

In general, the ${\bf R}^{q}f_{{\rm cris}*}{\cal O}^{\rm
cris}_{X/{\mathbb{Z}}_{p}}$ are not crystals, however, it follows from [BO78,
Theorem 7.8, Corollary 7.11] that they are ‘derived crystals’. Thus, if ${\bf
R}^{q^{\prime}}f_{{\rm cris}*}{\cal O}^{\rm cris}_{X/{\mathbb{Z}}_{p}}$ is
flat for each $q^{\prime}>q$, then ${\bf R}^{q}f_{{\rm cris}*}{\cal O}^{\rm
cris}_{X/{\mathbb{Z}}_{p}}$ is indeed a crystal, and this will always be the
case if $X/S$ is an abelian scheme or relative K3 surface, since the
cohomology of an abelian variety or K3 surface is torsion free. Another
application of [BO78, Theorem 7.8, Corollary 7.12] shows that, under the same
flatness hypothesis, the formation of ${\bf R}^{q}f_{{\rm cris}*}{\cal O}^{\rm
cris}_{X/{\mathbb{Z}}_{p}}$ commutes with base change. There is also a
relative version of the cup product pairing (5).

If $X/k$ is a K3 surface, then ${\rm H}^{2}_{\rm cris}(X/C)$ is free of rank
$22$. After base changing to $\bar{k}$, it follows from [Ill79, Chapitre II,
Theorem 5.14] that the cycle class map $c_{1}$ factors as

${\rm Pic}(\overline{X})\to{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))\overset{\cong}{\longrightarrow}{\rm
H}^{2}_{\rm cris}(\overline{X}/W(\bar{k}))(1)^{F=1}$

where the first map comes from the Kummer exact sequence and the second map is
an isomorphism.

If $\xi$ is a primitive polarisation of $X$, of degree $\xi^{2}=2d$, we write
${\rm{Pic}}(X)^{\xi}$ for the orthogonal complement to $\xi$, and

$P{\rm H}^{2}_{\rm cris}(X/C):=\langle c_{1}(\xi)\rangle^{\perp}\subset{\rm
H}^{2}_{\rm cris}(X/C)$

for the analogous primitive crystalline cohomology of $(X,\xi)$. This is
stable under both $\nabla$ and $F$. We also have the versions
${\rm{Pic}}(\overline{X})^{\xi}$ and $P{\rm H}^{2}_{\rm
cris}(\overline{X}/W(\bar{k}))$ of these groups after base changing to
$\bar{k}$.

######  Definition 2.2.

Define the primitive subgroup of ${\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))$ by

$\displaystyle P{\rm H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))$
$\displaystyle:=P{\rm H}^{2}_{\rm cris}(\overline{X}/W(\bar{k}))(1)^{F=1}$
$\displaystyle={\rm H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))\cap
P{\rm H}^{2}_{\rm cris}(\overline{X}/W(\bar{k}))(1).$

Similarly, define the primitive subgroup

$P{\rm H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}:={\rm
H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}\cap P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1)).$

######  Lemma 2.3.

Suppose that $(X,\xi)$ is a primitively polarised K3 surface with
$X(k)\neq\emptyset$ and ${\rm Pic}(X)={\rm Pic}(\overline{X})$. Then (4)
induces an exact sequence

$0\to{\rm{Pic}}(X)^{\xi}\otimes{\mathbb{Q}}_{p}\to P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}\otimes{\mathbb{Q}}\to
T_{p}({\rm{Br}}(\overline{X})^{k})\otimes{\mathbb{Q}}\to 0$

of ${\mathbb{Q}}_{p}$-vector spaces.

###### Proof.

The only part that is not clear is the claim that the last map is surjective,
which is equivalent to showing that ${\rm Pic}(X)+P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))$ is of finite index in ${\rm
H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))$. To see this, take
$x\in{\rm H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))$. By Frobenius
compatibility of the cup product pairing, we see that $x\cup\xi\in
H^{4}(\overline{X}/W(\bar{k})(2))^{F=1}={\mathbb{Z}}_{p}$. Hence $\xi^{2}\cdot
x-(x\cup\xi)\xi$ lies in $P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))$, and we are done. ∎

If $A/k$ is an abelian variety, then ${\rm H}^{1}_{\rm cris}(A/C)$ is a free
$C$-module of rank $2\,{\rm{dim}}A$, and can be identified with the Dieudonné
module of the $p$-divisible group $A[p^{\infty}]$ by [BBM82, (3.3.7.2)]. In
particular, it follows from [BM90, Théorème 4.1.1] that there is a commutative
diagram

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
27.69067pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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End}(A[p^{\infty}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-31.99889pt\hbox{\hbox{\kern 0.0pt\raise
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53.5802pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm End}_{C}({\rm H}^{1}_{\rm
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End}(\overline{A}[p^{\infty}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
33.44067pt\raise-36.34889pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
51.69067pt\raise-42.49889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
51.69067pt\raise-42.49889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
End}_{W(\bar{k})}({\rm H}^{1}_{\rm
cris}(\overline{A}/W(\bar{k}))^{F=1}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(6)

in which both horizontal arrows are isomorphisms.

##  3 The Kuga–Satake construction

Recall that a K3 surface $X$ is called superspecial if the Hodge and the
conjugate filtrations on ${\rm H}^{2}_{\rm dR}(X/k)$ coincide. Thanks to the
Tate conjecture this is equivalent to $X$ being supersingular of Artin
invariant $1$.

Let $X$ be a non-superspecial K3 surface over a field $k$ of characteristic
$p$, together with a primitive polarisation $\xi$ of degree $\xi^{2}=2d$. We
consider the lattice

$L_{d}:={\rm E}_{8}(-1)^{\oplus 2}\oplus{\rm U}^{\oplus
2}\oplus{\mathbb{Z}}(-2d),$

where ${\rm U}$ is the hyperbolic plane, and $(m)$ denotes multiplying the
bilinear form by $m$.

######  Theorem 3.1.

Up to replacing $k$ by a finite separable extension, there exists an abelian
variety $A/k$, equipped with an action of the Clifford algebra ${\rm
Cl}(L_{d})$, and:

  * •

a horizontal, Frobenius equivariant embedding

$P{\rm H}^{2}_{\rm
cris}(X/C)(1)\otimes{\mathbb{Q}}\hookrightarrow{\rm{End}}_{{\rm
Cl}(L_{d})}({\rm H}^{1}_{\rm cris}(A/C))\otimes{\mathbb{Q}};$ (7)

  * •

for all $\ell\neq p$, a $\Gamma$-equivariant embedding

$P{\rm H}^{2}_{\rm{\acute{e}t}}(X_{k^{\rm
s}},{\mathbb{Q}}_{\ell}(1))\hookrightarrow{\rm{End}}_{{\rm Cl}(L_{d})}({\rm
H}^{1}_{\rm{\acute{e}t}}(A_{k^{\rm s}},{\mathbb{Q}}_{\ell})).$

###### Proof.

If $p\neq 2$, then the integral version of this statement is almost [MP15,
Theorem 5.17]. The missing point is to descend the $p$-adic cohomological
comparison from $W(k^{\rm p})$ to $C$, but this is a simple consequence of
[MP15, Theorem 5.14]. Indeed, it follows from _loc. cit._ that, after taking a
suitable étale cover $T\rightarrow{\sf M}^{\circ}_{2d}$ of the moduli space of
primitively polarised K3 surfaces, there is an embeding

${\bm{P}}_{\rm cris}(1)\hookrightarrow{\bm{H}}_{\rm
cris}\otimes{\bm{H}}^{\vee}_{\rm cris}$ (8)

of $F$-crystals on $T_{{\mathbb{F}}_{p}}$, where ${\bm{P}}_{\rm cris}$ is the
relative primitive (crystalline) cohomology of the universal K3 surface, and
${\bm{H}}_{\rm cris}$ is the relative (crystalline) cohomology of the
Kuga–Satake abelian scheme. The embedding (7) is then obtained by simply
pulling back via ${\rm Spec}(k)\to T$ and inverting $p$. That it lands inside
the subspace of ${\rm Cl}(L_{d})$-equivariant endomorphisms can be checked
after base changing to $k^{\rm p}$. In fact, since we only need a comparison
for _rational_ crystalline cohomology, and $X$ is non-superspecial, it would
suffice to apply [MP15, Proposition 5.6, Lemma 5.9] (see the argument outlined
below in the case $p=2$).

If $p=2$ then we may obtain the required map in rational crystalline
cohomology by noting that the proofs of [MP15, Proposition 5.6, Lemma 5.9]
work verbatim in this case. We explain them here for the reader’s convenience.
Since $X$ is non-superspecial, it corresponds to a smooth point of the moduli
space ${\sf M}_{2d,{\mathbb{Z}}_{(2)}}$, which then lifts to a point $x$ in
the domain $\tilde{{\sf M}}_{2d,{\mathbb{Z}}_{(2)}}^{\rm sm}$ of the $2$-adic
Kuga–Satake map $\iota^{\rm KS}_{{\mathbb{Z}}_{(2)}}$ constructed in [KMP16,
Appendix A]. In particular, we may choose an étale map $T\to\tilde{{\sf
M}}^{\rm sm}_{2d,{\mathbb{Z}}_{(2)}}$, with $T$ a scheme smooth over
${\mathbb{Z}}_{(2)}$, such that the Kuga–Satake abelian scheme is defined over
$T$ and $x$ is in the image of $T$. Let $\bm{P}_{{\rm dR},{\mathbb{Q}}}(1)$
and $\bm{H}_{{\rm dR},{\mathbb{Q}}}$ denote the (primitive) relative de Rham
cohomology of the universal K3 surface and Kuga–Satake abelian scheme over
$T_{{\mathbb{Q}}}$ respectively. Then, thanks to [MP15, Proposition 5.6],
there is an injection

$\alpha_{{\rm dR},{\mathbb{Q}}}\colon{\bm{P}}_{{\rm
dR},{\mathbb{Q}}}(1)\hookrightarrow{\bm{H}}_{{\rm
dR},{\mathbb{Q}}}\otimes{\bm{H}}^{\vee}_{{\rm dR},{\mathbb{Q}}}$

of modules with connection on $T_{{\mathbb{Q}}}$ whose pullback via any point
$s\colon{\rm Spec}(F)\to T_{{\mathbb{Q}}}$ comes from an injection

$\alpha_{s}\colon{\bm{P}}_{s}(1)\hookrightarrow{\bm{H}}_{s}\otimes{\bm{H}}_{s}^{\vee}$
(9)

of motives for absolute Hodge cycles over $F$. Let $\widehat{T}$ denote the
$2$-adic completion of $T$ as a formal scheme over ${\mathbb{Z}}_{2}$. Then,
via the usual comparison between crystalline and de Rham cohomology, we may
view (the restriction of) $\alpha_{{\rm dR},{\mathbb{Q}}}$ (to the rigid
analytic variety $\widehat{T}_{{\mathbb{Q}}_{2}}$) as an injection

$\alpha_{{\rm cris},{\mathbb{Q}}}\colon{\bm{P}}_{{\rm
cris}}(1)\otimes{\mathbb{Q}}\hookrightarrow{\bm{H}}_{{\rm
cris}}\otimes{\bm{H}}^{\vee}_{{\rm cris}}\otimes{\mathbb{Q}}$

of isocrystals on $T_{{\mathbb{F}}_{2}}$. To see that it is an injection of
$F$-isocrystals, it is enough to prove compatibility with Frobenius, and it is
enough to show this after pullback to any field-valued point $s_{0}\colon{\rm
Spec}({\mathbb{F}})\to T_{{\mathbb{F}}_{2}}$. In fact, because the category of
isocrystals on a connected scheme is Tannakian, it suffices to do this at one
point on each connected component of $T_{{\mathbb{F}}_{2}}$, so we can assume
that ${\mathbb{F}}$ is finite. If we take a lift $s\colon{\rm
Spec}(W({\mathbb{F}}))\to T$ of $s_{0}$ to the Witt vectors of ${\mathbb{F}}$,
we therefore need to show that the de Rham realisation

$\alpha_{s_{\mathbb{Q}},{\rm dR}}\colon{\bm{P}}_{s_{{\mathbb{Q}}},{\rm
dR}}(1)\hookrightarrow{\bm{H}}_{s_{{\mathbb{Q}}},{\rm
dR}}\otimes{\bm{H}}_{s_{{\mathbb{Q}}},{\rm dR}}^{\vee}$

of (9) is compatible with Frobenius (via the identifications of both sides
with an appropriate crystalline cohomology group). Since
$\alpha_{s_{\mathbb{Q}},{\rm dR}}$ arises from an absolute Hodge cycle on an
abelian variety with good reduction, this follows from [MP15, Theorem 2.8 and
Lemma 2.11].

The upshot, then, is that we get the $2$-adic, rational analogue of (8), which
we may again pullback via the map $x\colon{\rm Spec}(k)\to T$ to obtain (7). ∎

After base changing to $\bar{k}$, we thus obtain from (7) an $F$-equivariant
embedding

$P{\rm H}^{2}_{\rm
cris}(\overline{X}/W(\bar{k}))(1)\otimes{\mathbb{Q}}\hookrightarrow{\rm{End}}_{{\rm
Cl}(L_{d})}({\rm H}^{1}_{\rm
cris}(\overline{A}/W(\bar{k})))\otimes{\mathbb{Q}}.$ (10)

After replacing $k$ by a finite separable extension, we can assume that the
natural map ${\rm{End}}(A)\to{\rm{End}}(\overline{A})$ is an isomorphism
(using a theorem of Chow [Con06, Theorem 3.19] which says that
${\rm{End}}(A^{\rm s})\to{\rm{End}}(\overline{A})$ is an isomorphism). In this
case, let $L(A)\subset{\rm{End}}_{{\rm Cl}(L_{d})}(A)$ be the subgroup of
special endomorphisms, that is, the endomorphisms whose crystalline
realisation comes from the cohomology of $X$ via (10).

######  Theorem 3.2.

Suppose $X$ is non-superspecial, and $k$ is large enough that
${\rm{End}}(A)={\rm{End}}(\overline{A})$ and ${\rm Pic}(X)={\rm
Pic}(\overline{X})$. Then there is an injection

$L(A)\hookrightarrow{\rm{Pic}}(X)^{\xi}$

such that the diagram

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
49.1379pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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(11)

commutes.

###### Proof.

If $p\neq 2$, this is just the rational version of [MP15, Theorem 5.17 (4)],
with a stonger hypothesis and weaker conclusion. (As noted in [MP15, p.645],
the $\ell$-adic realisations of any special endomorphism also come from the
cohomology of $X$, hence the definition of $L(A)$ given in [MP15, Theorem 5.17
(4)] coincides with the notion of special endomorphism defined earlier in the
same paper.)

If $p=2$, then the required injection $L(A)\hookrightarrow{\rm{Pic}}(X)^{\xi}$
is constructed in [KMP16, Proposition A.15], and it is shown there that it is
compatible with all $\ell$-adic realisations, in the sense that the diagrams

$\textstyle{P{\rm
H}^{2}_{\rm{\acute{e}t}}(\overline{X},{\mathbb{Q}}_{\ell}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm{End}}_{{\rm
Cl}(L_{d})}({\rm
H}^{1}_{\rm{\acute{e}t}}(\overline{A},{\mathbb{Q}}_{\ell}))}$$\textstyle{{\rm{Pic}}(X)^{\xi}\otimes{\mathbb{Q}}_{\ell}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L(A)\otimes{\mathbb{Q}}_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

for $\ell\neq 2$ all commute.

To see that the $2$-adic analogue (11) of this square commutes, we follow the
proof of [KMP16, Proposition A.15]. Indeed, we can assume that $k={\bar{k}}$,
and we let $W=W({\bar{k}})$ denote the ring of Witt vectors. Given $f\in
L(A)$, it is shown in [KMP16, Lemma A.13] that there exists a finite extension
$K$ of $W[1/2]$ such that the triple $(X,\xi,f)$ lifts to ${\cal O}_{K}$. In
other words, there exists a polarised K3 surface $(Y,\xi)$ over $K$ lifting
$(X,\xi)$, whose Kuga–Stake variety $B$ is a lift of $A$, in such a way that
$f$ lifts to a special endomorphism of $B$. After making a further finite
extension of $K$ we will assume that
${\rm{Pic}}(Y)={\rm{Pic}}(Y_{\overline{K}})$ and ${\rm End}(B)={\rm
End}(B_{\overline{K}})$. We let $L^{\prime}(B)$ denote the set of special
endomorphisms of $B$ which reduce modulo $2$ to a special endomorphism of $A$.
We then have a diagram

$\textstyle{{\rm{Pic}}(X)^{\xi}}$$\textstyle{L(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm{Pic}}(Y)^{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which can be seen to commute by, for example, considering the maps into
$\ell$-adic cohomology for any prime $\ell\neq 2$. By construction $f$ is in
the image of the right hand map.

We now consider the following cube:

---
$\textstyle{P{\rm H}^{2}_{\rm
cris}(X/W)\otimes_{W}K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\rm
cris}}$$\textstyle{{\rm End}_{{\rm Cl}(L_{d})}({\rm H}^{1}_{\rm
cris}(A/W))\otimes_{W}K}$$\textstyle{P{\rm H}^{2}_{\rm
dR}(Y/K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\rm
dR}}$$\textstyle{{\rm End}_{{\rm Cl}(L_{d})}({\rm H}^{1}_{\rm
dR}(B/K))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm{Pic}}(X)^{\xi}\otimes{\mathbb{Q}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\textstyle{L(A)\otimes{\mathbb{Q}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm{Pic}}(Y)^{\xi}\otimes{\mathbb{Q}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}(B)\otimes{\mathbb{Q}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By definition, the top face of this cube commutes. Moreover, by compatibility
of the cycle class map with the comparison between crystalline and de Rham
cohomology, the left hand face commutes. We have just shown that the bottom
face commutes, and it is a straightforward consequence of functoriality of
cohomology that the right hand face commutes. To see that the front face
commutes, we note that by [KMP16, Proposition A.15] the diagram

$\textstyle{P{\rm
H}^{2}_{\rm{\acute{e}t}}(Y_{\overline{K}},{\mathbb{Q}}_{2}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{2}}$$\textstyle{{\rm{End}}_{{\rm
Cl}(L_{d})}({\rm
H}^{1}_{\rm{\acute{e}t}}(B_{\overline{K}},{\mathbb{Q}}_{2}))}$$\textstyle{{\rm{Pic}}(Y)^{\xi}\otimes{\mathbb{Q}}_{2}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L(B)\otimes{\mathbb{Q}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

commutes, so we may apply Fontaine’s functor ${\bf D}_{\rm
dR}:=(-\otimes_{{\mathbb{Q}}_{2}}B_{\rm dR})^{{\rm Gal}(\overline{K}/K)}$ and
use [MP15, Lemma 2.11] to identify ${\bf D}_{\rm dR}(\alpha_{2})$ with
$\alpha_{{\rm dR}}$.

By construction, $f\in L(A)$ is in the image of $L^{\prime}(B)\to L(A)$. Thus,
by the commutativity of the ‘other’ 5 faces of the above cube, it follows that
the images of $f$ under the two possible ways

$L(A)\otimes{\mathbb{Q}}_{2}\rightrightarrows{\rm{End}}_{{\rm Cl}(L_{d})}({\rm
H}^{1}_{\rm cris}(A/W))\otimes{\mathbb{Q}}$

of following round the square (11) coincide. Since $f$ was arbitrary, we see
that these two maps are in fact equal. ∎

In particular, passing to $F$-invariants in the top row of (11), we obtain the
commutative diagram

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
39.57443pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))\otimes{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.57443pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
63.57443pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
63.57443pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm{End}}_{{\rm
Cl}(L_{d})}(\overline{A}[p^{\infty}])\otimes{\mathbb{Q}}}$}}}}}}}{\hbox{\kern-31.34906pt\raise-42.04109pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm{Pic}}(X)^{\xi}\otimes{\mathbb{Q}}_{p}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-29.59665pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-6.03333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
82.28079pt\raise-42.04109pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L(A)\otimes{\mathbb{Q}}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.28079pt\raise-42.04109pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
31.34908pt\raise-42.04109pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
106.62012pt\raise-31.54109pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.62012pt\raise-6.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{split}$
(12)

##  4 Completion of the proof

In the previous section, we constructed, for any non-superspecial, primitively
polarised K3 surface $(X,\xi)$ over a field $k$ of charactersitic $p>0$, the
commutative diagram (12). The last key step in the proof of Theorem 1 is now
the following.

######  Proposition 4.1.

Suppose that $k$ is finitely generated, and $(X,\xi)$ is non-superspecial.
Then the image of the map

$P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}\otimes{\mathbb{Q}}\to{\rm{End}}_{{\rm
Cl}(L_{d})}(\overline{A}[p^{\infty}])\otimes{\mathbb{Q}}$

is contained within ${\rm{End}}_{{\rm
Cl}(L_{d})}(A[p^{\infty}])\otimes{\mathbb{Q}}$.

###### Proof.

Since ${\rm{End}}_{{\rm
Cl}(L_{d})}(A[p^{\infty}])={\rm{End}}(A[p^{\infty}])\cap{\rm{End}}_{{\rm
Cl}(L_{d})}(\overline{A}[p^{\infty}])$, it suffices to show that the image is
contained within ${\rm{End}}(A[p^{\infty}])\otimes{\mathbb{Q}}$. Thanks to
(6), it is enough to show that the image of the map

$P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}\otimes{\mathbb{Q}}\to{\rm{End}}_{W(\bar{k})}({\rm
H}^{1}_{\rm cris}(\overline{A}/W(\bar{k}))\otimes{\mathbb{Q}}$

is contained within ${\rm{End}}_{C}({\rm H}^{1}_{\rm
cris}(A/C))^{\nabla=0}\otimes{\mathbb{Q}}$. By (7), it is enough to show that
the image of

$P{\rm H}^{2}_{\rm fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}\to P{\rm
H}^{2}_{\rm cris}(\overline{X}/W(\bar{k}))$

is contained within $P{\rm H}^{2}_{\rm cris}(X/C)^{\nabla=0}$, and hence it is
enough to show that the composite map

${\rm H}^{2}_{\rm fppf}(X,{\mathbb{Z}}_{p}(1))\to{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))\to{\rm H}^{2}_{\rm
cris}(\overline{X}/W(\bar{k}))$

factors through ${\rm H}^{2}_{\rm cris}(X/C)^{\nabla=0}$. In fact, we may work
modulo $p^{n}$ and show that the composite map

${\rm H}^{2}_{\rm fppf}(X,\mu_{p^{n}})\to{\rm H}^{2}_{\rm
fppf}(\overline{X},\mu_{p^{n}})\to{\rm H}^{2}_{\rm
cris}(\overline{X}/W_{n}(\bar{k}))$

factors through ${\rm H}^{2}_{\rm cris}(X/C_{n})^{\nabla=0}$, for all $n$.

Let $k_{0}$ denote the algebraic closure of ${\mathbb{F}}_{p}$ inside $k$, and
consider the absolute crystalline cohomology group ${\rm H}^{2}_{\rm
cris}(X/W_{n}(k_{0}))$. Then the map

${\rm H}^{2}_{\rm cris}(X/W_{n}(k_{0}))\to{\rm H}^{2}_{\rm
cris}(\overline{X}/W_{n}(\bar{k}))$

factors through ${\rm H}^{2}_{\rm cris}(X/C_{n})^{\nabla=0}$, so it is enough
to produce a commutative diagram

$\textstyle{{\rm H}^{2}_{\rm
fppf}(X,\mu_{p^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{2}_{\rm
cris}(X/W_{n}(k_{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{2}_{\rm
fppf}(\overline{X},\mu_{p^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{2}_{\rm cris}(\overline{X}/W_{n}(\bar{k})).}$

To do this, we use [Ill79, Chapitre II, Théorème 1.4] and Lemma 2.1 to get
identifications

${\rm H}^{2}_{\rm cris}(X/W_{n}(k_{0}))\overset{\cong}{\longrightarrow}{\rm
H}^{2}(X,W_{n}\Omega^{\bullet}_{X}),\quad{\rm H}^{2}_{\rm
cris}(\overline{X}/W_{n}(\bar{k}))\overset{\cong}{\longrightarrow}{\rm
H}^{2}(\overline{X},W_{n}\Omega^{\bullet}_{\overline{X}})$

(note here the use of the hypothesis that $k$ is finitely generated). In fact,
since each $W_{n}\Omega^{i}$ is a coherent $W_{n}({\cal O})$-module, we may
replace the Zariski topology here by the étale topology. Moreover, by pushing
forward from the flat to the étale site, we obtain identifications

$\displaystyle{\rm H}^{2}_{\rm fppf}(X,\mu_{p^{n}})$
$\displaystyle\overset{\cong}{\longrightarrow}{\rm
H}^{1}_{\rm{\acute{e}t}}(X,{\cal O}_{X}^{\times}/({\cal
O}_{X}^{\times})^{p^{n}})$ $\displaystyle{\rm H}^{2}_{\rm
fppf}(\overline{X},\mu_{p^{n}})$
$\displaystyle\overset{\cong}{\longrightarrow}{\rm
H}^{1}_{\rm{\acute{e}t}}(\overline{X},{\cal O}_{\overline{X}}^{\times}/({\cal
O}_{\overline{X}}^{\times})^{p^{n}})$

via which the map ${\rm H}^{2}_{\rm fppf}(\overline{X},\mu_{p^{n}})\to{\rm
H}^{2}_{\rm{\acute{e}t}}(\overline{X},W_{n}\Omega^{\bullet}_{\overline{X}})$
is induced by

${\cal O}_{\overline{X}}^{\times}/({\cal
O}_{\overline{X}}^{\times})^{p^{n}}\overset{{\rm
d}\\!\log}{\longrightarrow}W_{n}\Omega^{1}_{\overline{X}}$

It is now enough to observe that there is an analogous map

${\cal O}_{X}^{\times}/({\cal O}_{X}^{\times})^{p^{n}}\overset{{\rm
d}\\!\log}{\longrightarrow}W_{n}\Omega^{1}_{X}.$

on $X_{\rm{\acute{e}t}}$, and that the commutativity of the diagram

$\textstyle{{\rm H}^{1}_{\rm{\acute{e}t}}(X,{\cal O}_{X}^{\times}/({\cal
O}_{X}^{\times})^{p^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{2}_{\rm{\acute{e}t}}(X,W_{n}\Omega^{\bullet}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{1}_{\rm{\acute{e}t}}(\overline{X},{\cal O}_{\overline{X}}^{\times}/({\cal
O}_{\overline{X}}^{\times})^{p^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{2}_{\rm{\acute{e}t}}(\overline{X},W_{n}\Omega^{\bullet}_{\overline{X}})}$

is clear. ∎

###### Proof of Theorem 1.

We may assume that $X$ is of finite height, and admits a primitive
polarisation $\xi$. Set $2d=\xi^{2}$. Extending $k$ if necessary, we may
assume that $X(k)\neq\emptyset$, that ${\rm Pic}(X)={\rm Pic}(\overline{X})$,
that the Kuga–Satake variety $A$ is defined over $k$, and that ${\rm
End}(A)={\rm End}(\overline{A})$. We therefore have the following commutative
diagram:

$\textstyle{P{\rm H}^{2}_{\rm
fppf}(\overline{X},{\mathbb{Z}}_{p}(1))^{k}\otimes{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm{End}}_{{\rm
Cl}(L_{d})}(A[p^{\infty}])\otimes{\mathbb{Q}}}$$\textstyle{{\rm{Pic}}(X)^{\xi}\otimes{\mathbb{Q}}_{p}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L(A)\otimes{\mathbb{Q}}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Thanks to [dJ98, Theorem 2.6] we can identify the group ${\rm{End}}_{{\rm
Cl}(L_{d})}(A[p^{\infty}])\otimes{\mathbb{Q}}$ appearing in the top right hand
corner of this diagram by ${\rm{End}}_{{\rm
Cl}(L_{d})}(A)\otimes{\mathbb{Q}}_{p}$ (again, here we are using the
hypothesis that $k$ is finitely generated). By definition, the image of the
upper horizontal map is contained within the image of the right hand vertical
map. It follows that the lower horizontal and left hand vertical maps are
isomorphisms. Lemma 2.3 now shows that
$T_{p}({\rm{Br}}(\overline{X})^{k})\otimes{\mathbb{Q}}=0$, and hence that
$T_{p}({\rm{Br}}(\overline{X})^{k})=0$, completing the proof. ∎

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