#  The Tate Conjecture for Motivic Endomorphisms of K3 Surfaces over Finite
Fields

Ziquan Yang

###### Abstract

The Tate conjecture for squares of K3 surfaces over finite fields was recently
proved by Ito–Ito–Koshikawa. We give a more geometric proof when the
characteristic is at least $5$. The main idea is to use twisted derived
equivalences between K3 surfaces to link the Tate conjecture to finiteness
results over finite fields, in the spirit of Tate.

##  1 Introduction

In this note, we give a proof of the Tate conjecture for squares of K3
surfaces over finite fields. We will phrase and study the problem from the
perspective of motives, so we first introduce some notation. Let $k$ be a
perfect field of characteristic $p$ with a chosen algebraic closure $\bar{k}$,
$\mathscr{M}_{\mathrm{Chow}}(k)$ be the category of Chow motives, which comes
equipped with fiber functors $\omega_{\ell}$ (resp. $\omega_{\mathrm{cris}}$)
given by $\ell$-adic cohomology for every prime $\ell\neq p$ (resp.
crystalline cohomology with $W[1/p]$ coefficients). Let
$\mathscr{M}_{\mathrm{hom}}(k)$ be the category
$\mathscr{M}_{\mathrm{Chow}}(k)$ but with $\mathrm{Hom}$ groups taken modulo
the kernel of $\omega:=\omega_{\ell}\times\omega_{\mathrm{cris}}$. Let
$\mathfrak{h}(-)$ denote the usual functor from the category of varieties over
$k$ to $\mathscr{M}_{\mathrm{hom}}(k)$ given by $X\mapsto(X,\mathrm{id},0)$.

######  Theorem 1.1.

Let $X$ be a K3 surface over a finite field $\mathbb{F}_{q}$ of charactersitic
$p\geq 5$. Then natural morphism

$\omega_{\ell}:\mathrm{End}(\mathfrak{h}(X)){\otimes}\mathbb{Q}_{\ell}\to\mathrm{End}_{F}(\mathrm{H}^{*}_{\mathrm{{\acute{e}}t}}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell}))$

is an isomorphism for every prime $\ell\neq p$, where $F$ denotes the
Frobenius action on
$\mathrm{H}^{*}_{\mathrm{{\acute{e}}t}}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell})$.

When $X$ is a K3 surface over $k$, $\mathfrak{h}(X)$ is known to admit a Chow-
Künneth decomposition
$\mathfrak{h}(X)=\mathfrak{h}^{0}(X)\oplus\mathfrak{h}^{2}(X)\oplus\mathfrak{h}^{4}(X)$.
For two K3 surfaces $X$ and $X^{\prime}$ over $k$, an isogeny from
$X^{\prime}$ to $X$ is an isomorphism
$f:\mathfrak{h}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X^{\prime})$
such that $\omega(f)$ preserves the Poincaré pairing. We consider the isogeny
category of K3 surfaces $\mathsf{IK3}(k)$, whose objects are K3 surfaces over
$k$ and whose morphisms are isogenies. For a give K3 surface $X$, we define
the automorphism group functor of $X$ in the isogeny category by

$\displaystyle
I_{X}(R):=\\{f\in(\mathrm{End}(\mathfrak{h}^{2}(X)){\otimes}R)^{\times}:\omega(f)\text{
preserves the Poincar\'{e} pairing}\\}$

for every $\mathbb{Q}$-algebra $R$. Then we have a group-theoretic version of
Thm 1.1:

######  Theorem 1.2.

Let $X$ be as in Thm 1.1. Assume that $\mathbb{F}_{q}$ is big enough so that
all line bundles on $X_{\bar{\mathbb{F}}_{q}}$ are defined over
$\mathbb{F}_{q}$ and
$F\in\mathrm{SO}(\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell}(1)))$.
Let $I_{X,\ell}$ denote the centralizer of the geometric Frobenius action in
$\mathrm{SO}(\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell}(1)))$.
Then $I_{X}$ is representable by a reductive group of finite type over
$\mathbb{Q}$ and $\omega_{\ell}$ induces an isomorphism
$I_{X}{\otimes}\mathbb{Q}_{\ell}\,{\cong}\,I_{X,\ell}$.

######  Theorem 1.3.

Let $X$ be a K3 surface over $\mathbb{F}_{q}$. Let $F$ denote the geometric
Frobenius action on $\mathrm{H}^{2}_{\ell}(X)$ or
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X)_{\mathbb{Q}}$. For any isometric
embedding $\iota:\Lambda_{\ell}\to\mathrm{H}^{2}_{\ell}(X)$, such that
$\iota(\Lambda_{\ell})$ is $F$-invariant, there exists another K3 surface
$X^{\prime}$ over $\mathbb{F}_{q}$, which is unique up to isomorphism and
comes with a derived $\ell$-isogeny
$f:\mathfrak{h}^{2}(X^{\prime})\to\mathfrak{h}^{2}(X)$ such that
$f(\mathrm{H}^{2}_{\ell}(X^{\prime})=\iota(\Lambda_{\ell})$.

Let $X,X^{\prime}$ be two K3 surfaces over $\mathbb{F}_{q}$. Suppose there
exists a derived $\ell$-isogeny
$f:\mathfrak{h}^{2}(X^{\prime})\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X)$
such that $f(\mathrm{H}^{2}_{\ell}(X^{\prime}))=\mathrm{H}^{2}_{\ell}(X)$,
then $X\cong X^{\prime}$.

We now explain the main ideas of proof. First, let $X$ and $\mathbb{F}_{q}$ be
as in the above theorem and $\mathfrak{t}$ be the transcendental part of
$\mathfrak{h}(X)$, i.e., the orthogonal complement to the span of submotives.
Similarly, we consider $I_{\mathfrak{t}(X)}$, which is defined just as $I_{X}$
but with $\mathfrak{h}^{2}(X)$ replaced by $\mathfrak{t}$ and the centralizer
$I_{\mathfrak{t},\ell}$ of the geometric Frobenius in
$\mathrm{SO}(\omega_{\ell}(\mathfrak{t}(1)))$. It suffices to show that
$\omega_{\ell}:\mathrm{End}(\mathfrak{t}(1)){\otimes}\mathbb{Q}_{\ell}\to\mathrm{End}_{F}(\omega_{\ell}(\mathfrak{t}(1)))$
and $\omega_{\ell}:I_{\mathfrak{t}}{\otimes}\mathbb{Q}_{\ell}\to
I_{\mathfrak{t},\ell}$ are isomorphisms.

Next, we geometrically construct lots of elements in
$\mathrm{End}(\mathfrak{t})$ using twisted derived equivalences. A twisted K3
surface is a pair $(X,\alpha)$ where $X$ is a K3 surface and $\alpha$ is a
Brauer class. The order of $(X,\alpha)$ is the order of $\alpha$. Up to
choices of B-fields, a derived equivalence
$\mathbb{D}(X,\alpha)\,{\cong}\,\mathbb{D}(X^{\prime},\alpha^{\prime})$
between two twisted K3 surfaces of $\ell^{\infty}$-order gives rise to an
isomorphism
$f:\mathfrak{t}\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{t}^{\prime}$,
where $\mathfrak{t}^{\prime}$ is the transcendental part of
$\mathfrak{h}(X^{\prime})$. We call compositions of such isomorphisms derived
transcendental $\ell$-isogenies. Let
$S_{\mathfrak{t}}^{\mathrm{der},\ell}\subset\mathrm{End}(\mathfrak{t})$ denote
the subgroup of derived transcendental $\ell$-isogenies from $X$ to itself.
After developing the structural theorems for derived transcendental
$\ell$-isogenies, we use the key fact that there are only finitely many K3
surfaces up to isomorphism over $\mathbb{F}_{q}$ to show the quotient
$S_{\mathfrak{t}}^{\mathrm{der},\ell}\backslash I_{\mathfrak{t}(X),\ell}$ is
compact (Thm 3.7).

Note that this compactness statement already implies that
$S_{\mathfrak{t}}^{\mathrm{der},\ell}$ is big. Our next step is to find a
reductive $\mathbb{Q}$-group $I^{\prime}$ which is equipped with a morphism
$\nu:I^{\prime}\to I_{\mathfrak{t}(X),\ell}$ and maps
$S_{\mathfrak{t}}^{\mathrm{der},\ell}\to
I^{\prime}(\mathbb{Q})\to\mathrm{End}(\mathfrak{t})$ compatible with $\nu$ in
the obvious sense. The compactness result and a theorem of Borel and Tis
implies that $v(I^{\prime}_{\mathbb{Q}_{\ell}})\subseteq
I_{\mathfrak{t},\ell}$ is a parabolic subgroup (see Thm 3.6). However,
$v(I^{\prime}_{\mathbb{Q}_{\ell}})$ is reductive and $I_{\mathfrak{t},\ell}$
is connected, so $v(I^{\prime}_{\mathbb{Q}_{\ell}})=I_{\mathfrak{t},\ell}$.
Then using the density of $I^{\prime}(\mathbb{Q})$ in
$I^{\prime}(\bar{\mathbb{Q}}_{\ell})$ we check that the image of
$I^{\prime}(\mathbb{Q})$ in $\mathrm{End}(\mathfrak{t})$ spans
$\mathrm{End}_{F}\omega_{\ell}(\mathfrak{t})$. This implies both Thm 1.1 and
1.2.

It remains to explain how to construct $I^{\prime}$. Ideally, if $X\times X$
satisfies some of the standard conjectures, then we can directly verify that
$I_{\mathfrak{t}}$ defines a reductive $\mathbb{Q}$-group and simply take
$I^{\prime}=I_{\mathfrak{t}}$. In fact, a little refinement of the argument
shows that $\mathrm{End}_{F}\omega_{\ell}(\mathfrak{t})$ is spanned by
$S^{\mathrm{der},\ell}_{\mathfrak{t}}$. We will carry out this strategy out
for K3 surfaces which arise from the Kummer construction (Ex. 3.10) and a
slight variant of this strategy to ordinary K3 surfaces (Ex. 3.11). To treat
the general case, we choose a polarization $\xi$ on $X$ and consider the Kuga-
Satake abelian variety $A$ attached to $(X,\xi)$. The abelian variety $A$
comes with a “CSpin-structure”. We consider the group scheme $\widetilde{I}$
which parametrizes the automorphisms of $A$ which preserve this CSpin-
structure and define $I^{\prime}:=\widetilde{I}/\mathbb{G}_{m}$. The main
point of considering $\widetilde{I}$ is that it fixes a polarization on $A$ up
to scalar, so that $I^{\prime}$ is reductive by the positivity of Rosati
involution. The general mechinery of the Kuga-Satake construction gives us a
morphism $I^{\prime}\to I_{\mathfrak{t},\ell}$ and we construct maps
$S_{\mathfrak{t}}^{\mathrm{der},\ell}\to
I^{\prime}(\mathbb{Q})\to\mathrm{End}(\mathfrak{t})$ by lifting-reduction
arguments.

######  Remark 1.4.

We remark that Thm 1.1 (including the case $p=2,3$) was previously proved in
[10] by Ito–Ito–Koshikawa, whose method relies more heavily on the Kuga-Satake
construction. The main idea there is to develop a refined CM lifting theory
for K3 surfaces over finite fields, and then deduce Thm 1.1 using Kisin’s
version of Tate’s theorem on the endomorphisms of abelian varieties, and the
fact that squares of K3 surfaces with CM satisfies the Hodge conjecture,
proved by Buskin. In comparison, our method relies more on the geometry of K3
surfaces themselves and Kuga-Satake abelian varieties are only used to bypass
the standard conjectures. In particular, we do not make use of any CM liftings
or Tate’s theorem111We do need to use the semisimplicity of Frobenius and the
divisorial Tate conjecture for individual K3’s, whose known proofs indeed
depend on Tate’s theorem via the Kuga-Satake construction. We will treat these
inputs as given.. There is a long standing connection between finiteness
results and the divisorial Tate conjecture, as shown in the work of Tate [17]
for abelian varieties, and works of Artin–Swinnerton-Dyer [2],
Lieblich–Maulik–Snowden [12], and Charles [7] for K3 surfaces. Our main
purpose of giving a new proof is to provide an example of such a connection
for codimension $2$ cycles, using theory of gerbes and twisted sheaves.

##  2 Derived $\ell$-Isogenies for K3 Surfaces

Let $k$ be an algebraically closed field. Let $Y$ be a K3 surface or a product
of two K3 surface over $k$ and let $\alpha$ be a Brauer class in
$\mathrm{Br}(Y)[\ell^{\infty}]$. The $\alpha$-twisted coherent sheaves form an
abelian category, so we can talk about its bounded derived category
$\mathbb{D}(Y,\alpha)$. To take Chern characters of $\alpha$-twisted sheaves,
we need the notion of B-field lifts:

######  Definition 2.1.

A $B$-field lift of $\alpha$ is an element in
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(Y,\mathbb{Q}_{\ell}(1))$ with the
following property: Suppose $\mathrm{ord}(\alpha)=\ell^{m}$ and
$B=\beta/\ell^{m}$. Then $\beta$ lies in
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(Y,\mathbb{Z}_{\ell}(1))$ and maps to
$\alpha$ under the composition
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(Y,\mathbb{Z}_{\ell}(1))\to\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(Y,\mu_{\ell^{m}})\to\mathrm{Br}(Y)$,
where the first map is the reduction map modulo $\ell^{m}$.

Given a complex of $\alpha$-twisted sheaf $\mathscr{E}$ on $Y$, and a B-field
lift $B$ of $\alpha$ as above, we can define its twisted Chern character $\rm
ch^{B}(\mathscr{E})$, which depends on the choice of $B$. The basic idea is
that, given a choice of a gerbe $\mathscr{X}$ representing $\beta$,
$\mathscr{E}^{{\otimes}\ell^{m}}$ becomes an untwisted sheaf on $X$, so we can
take the usual Chern character $\rm ch(\mathscr{E}^{{\otimes}\ell^{m}})$ and
define $\rm ch^{B}(\mathscr{E})$ as $\sqrt[\ell^{m}]{\rm
ch(\mathscr{E}^{{\otimes}\ell^{m}})}$, which depends only on $B$ and not on
$\mathscr{X}$. For details, see [4, App. A] and [5, §2]. The twisted Mukai
vector of $\mathscr{E}$ is defined by $v^{B}(\mathscr{E}):=\rm
ch^{B}(\mathscr{E})\sqrt{\mathrm{td}(Y)}$.

Now suppose we are given given two twisted K3 surfaces
$(X,\alpha),(X^{\prime},\alpha^{\prime})$ and a derived equialence
$\Phi:\mathbb{D}(X,\alpha)\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{D}(X^{\prime},\alpha^{\prime})$.
By Orlov’s theorem, $\Phi$ is induced by a complex $\mathscr{E}\in D(X\times
X^{\prime},\alpha^{-1}\boxtimes\alpha^{\prime})$, which is unique up to quasi-
isomorphism. To emphasize this dependence we denote $\Phi$ by
$\Phi_{\mathscr{E}}$. Given B-field lifts $B,B^{\prime}$ of
$\alpha,\alpha^{\prime}$, we have a twisted Mukai vector $v^{-B\boxtimes
B^{\prime}}(\mathscr{E})$. As an element of $\mathrm{Ch}^{*}(X\times
X^{\prime})_{\mathbb{Q}}$, it induces a map $\Phi^{-B\boxtimes
B^{\prime}}(\mathscr{E}):\mathrm{H}^{*}(X)_{\mathbb{Q}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{*}(X^{\prime})_{\mathbb{Q}}$.

######  Theorem 2.2.

The map $\Phi^{-B\boxtimes B^{\prime}}(\mathscr{E})$ as above restricts to a
$\mathbb{Z}_{\ell^{\prime}}$-integral isomorphism
$\mathrm{H}^{*}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Z}_{\ell^{\prime}})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{*}(X^{\prime},\mathbb{Z}_{\ell^{\prime}})$
for every prime $\ell^{\prime}\not\in\\{\ell,\mathrm{char\,}k\\}$. If
$\mathrm{char\,}k=p$, then $\Phi^{-B\boxtimes B^{\prime}}(\mathscr{E})$ in
addition restricts to a $W$-integral isomorphism
$\mathrm{H}^{*}_{\mathrm{cris}}(X/W)\to\mathrm{H}^{*}_{\mathrm{cris}}(X^{\prime}/W)$.

In general, $\Phi^{-B\boxtimes
B^{\prime}}(\mathscr{E}):\mathrm{H}^{*}(X)_{\mathbb{Q}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{*}(X^{\prime})_{\mathbb{Q}}$
does not need to preserve the codimension filtration, or restricts to an
isogeny
$\mathfrak{h}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X^{\prime})$.
Nonetheless, it always restricts to an isomorphism
$\mathfrak{t}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{t}(X^{\prime})$
which preserves the Poincaré pairing. One argues by Witt’s cancellation
theorem that $\mathrm{NS}(X)_{\mathbb{Q}}$ is isomorphic to
$\mathrm{NS}(X^{\prime})_{\mathbb{Q}}$ as quadratic forms, so that given such
$\mathfrak{t}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{t}(X^{\prime})$
we can always complete to an isogeny
$\mathfrak{h}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X^{\prime})$
by adding an isomorphism
$\mathfrak{a}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{a}^{2}(X^{\prime})$
which preserves the Poincaré pairing.

######  Definition 2.3.

  1. (a)

A primitive derived transcendental $\ell$-isogeny is an isomorphism
$\mathfrak{t}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{t}(X^{\prime})$
is given by $v^{-B\boxtimes B^{\prime}}(\mathscr{E})$ for some choices of
$B,B^{\prime},\alpha,\alpha^{\prime}$ and
$\Phi_{\mathscr{E}}:\mathbb{D}(X,\alpha)\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{D}(X^{\prime},\alpha^{\prime})$.

  2. (b)

A primitive derived $\ell$-isogeny
$f:\mathfrak{h}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X^{\prime})$
is an isogeny whose restriction to the transcendental parts is a primitive
derived transcendental $\ell$-isogeny.

  3. (c)

A (transcendental) derived $\ell$-isogeny is a finite composition of the
primitive ones.

######  Remark 2.4.

We remark that isomorphisms between K3 surfaces and “reflection in
$(-2)$-curves” on a single K3 (see [18, §6]) give rise to primitive derived
$\ell$-isogenies with $B,B^{\prime},\alpha,\alpha^{\prime}$ all being zero.

### 2.1 Existence Theorems

######  Theorem 2.5.

Let $X$ be a K3 surface over $k$. Denote by $\Lambda_{\ell}$ the quadratic
lattice given by
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Z}_{\ell})$. For every
isometric embedding
$\iota:\Lambda_{\ell}\hookrightarrow\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Q}_{\ell})$,
there exists another K3 surface $X^{\prime}$ together with a derived
$\ell$-isogeny $f:\mathfrak{h}^{2}(X^{\prime})\to\mathfrak{h}^{2}(X)$ such
that
$f(\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Z}_{\ell}))=\iota(\Lambda_{\ell})$.

###### Proof.

This is part of [5, Thm 1.3]. We explain the main steps for readers’
convenience: First, by Cartan-Dieudonné theorem, every element of
$\mathrm{O}(\Lambda_{\ell}{\otimes}\mathbb{Q}_{\ell})$ can be written as a
composition of relfections, so it suffices to treat the case when
$\iota(\Lambda_{\ell})=s_{b}(\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Z}_{\ell}))$,
where $s_{b}$ is the reflection in
$b\in\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Q}_{\ell})$. We may
assume $b$ is a primitive element in
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Z}_{\ell})$.

Now set $n=b^{2}/2$ and $B:=b/n$. Let $\alpha$ be the Brauer class defined by
$B$. Let $X^{\prime}$ be the moduli space of stable $\alpha$-twisted sheaves
with Mukai vector $(n,0,0)$. Then there is an equivalence
$\Phi_{\mathscr{E}}:\mathbb{D}(X^{\prime},\alpha^{\prime})\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{D}(X,\alpha)$
for some $\alpha^{\prime}\in\mathrm{Br}(X^{\prime})$. For a suitable B-field
lift $B^{\prime}$ of $\alpha^{\prime}$, $v^{-B^{\prime}\boxtimes
B}(\mathscr{E})$ restricts to the desired isogeny $f$. For details of this
argument, see [5, Lem. 6.2(1)]. ∎

When $k=\mathbb{C}$, this gives an adelic proof of [9, Thm 0.1], which refines
[6, Thm 1.1]:

######  Theorem 2.6.

Let $X,X^{\prime}$ be algebraic K3 surfaces over $\mathbb{C}$. Every Hodge
isometry
$g:\mathrm{H}^{2}(X^{\prime},\mathbb{Q})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{2}(X,\mathbb{Q})$
is given by a derived isogeny.

###### Proof.

By applying Thm 2.5 for different $\ell$’s, we find a derived isogeny
$f:\mathfrak{h}^{2}(X^{\prime\prime})\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X)$
such that
$f^{*}(\mathrm{H}^{2}(X^{\prime\prime},\mathbb{Z}))=g(\mathrm{H}^{2}(X^{\prime},\mathbb{Z}))$
for some other K3 surface $X^{\prime\prime}$. The composition $g\circ f^{*}$
is an integral Hodge isometry
$\mathrm{H}^{2}(X^{\prime\prime},\mathbb{Z})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{2}(X^{\prime},\mathbb{Z})$.
By [18, Lem. 6.2], up to precomposing $\pm f$ by a composition of reflections
in $(-2)$-curves, we may assume $g\circ f^{*}$ preserves the ample cones.
Therefore, $g\circ f^{*}$ is induced by an isomorphism by the global Torelli
theorem. ∎

### 2.2 Torelli Theorem

If a lifting $X_{W}$ of $X$ over $W$ carries a lifting of all line bundles on
$X$, i.e., the natural map $\mathrm{Pic}\,(X_{W})\to\mathrm{Pic}\,(X)$ is an
isomorphism, then we say that $X_{W}$ is a perfect lifting. By [11, Prop.
4.10], perfect liftings always exist for non-supersingular K3 surfaces.

######  Theorem 2.7.

Let
$f:\mathfrak{h}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X^{\prime})$
be a derived $\ell$-isogeny between non-supersingular K3 surfaces
$X,X^{\prime}$. For every perfect lifting $X_{W}$ of $X$ over $W$, there
exists a perfect lifting $X^{\prime}_{W}$ such that $f$ lifts to an isogeny
between their generic fibers of $X_{W}$ and $X_{W}^{\prime}$.

###### Proof.

It suffices to treat the case when $f$ is primitive, so that the restriction
of $f$ to the transcendental parts is given by a twisted Fourier-Mukai
equivalence
$\Phi_{\mathscr{E}}:\mathbb{D}(X,\alpha)\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{D}(X^{\prime},\alpha^{\prime})$
for some Brauer classes $\alpha,\alpha^{\prime}$ of $\ell$-power order.
Moreover, since we are considering perfect liftings, which in particular carry
all liftings of $(-2)$-curves, using [18, Lem. 6.2] we reduce to the case when
$f$ is polarizable. Any lifting $X_{W}$ carries a unique lift $\alpha_{W}$ of
the Brauer class $\alpha$, so we may now conclude by [5, Thm 4.6], which
states that for some lifting $X^{\prime}_{W}$ of $X$, $\mathscr{E}$ lifts up
to quasi-isomorphism to a complex in $\mathbb{D}(X_{W}\times
X_{W}^{\prime},-\alpha_{W}\boxtimes\alpha_{W}^{\prime})$, where
$\alpha^{\prime}_{W}$ is the lift of $\alpha^{\prime}$ on $X_{W}^{\prime}$.
Moreover, if $X$ is ordinary and $X_{W}$ is the canonical lifting, then
$X^{\prime}$ is also ordinary and $X^{\prime}_{W}$ is the canonical lifting of
$X^{\prime}$.

For the second statement, note that since
$\mathrm{H}^{2}_{\mathrm{cris}}(X/W)\,{\cong}\,\mathrm{H}^{2}_{\mathrm{cris}}(X^{\prime}/W)$,
$X$ and $X^{\prime}$ have the same height. Moreover,
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X_{\bar{K}},\mathbb{Z}_{p})\,{\cong}\,\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(X^{\prime}_{\bar{K}},\mathbb{Z}_{p})$
as $\mathrm{Gal}_{K}$-modules. It follows from [16, Thm C] that if $X_{W}$ is
the canonical lifting, then so is $X^{\prime}_{W}$. ∎

######  Theorem 2.8.

Let
$f:\mathfrak{h}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X^{\prime})$
be derived $\ell$-isogeny. If $f$ is polarizable and $\omega(f)$ restricts to
an isomorphism
$\mathrm{H}^{2}(X)\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{2}(X^{\prime})$,
then $f$ is induced by a unique isomorphism
$X^{\prime}\stackrel{{\scriptstyle\sim}}{{\to}}X$.

######  Remark 2.9.

Thm 2.8 is proved in [5] in more generality, which also deals with derived
equivlences between twisted K3 surfaces whose Brauer classes are not of prime-
to-$p$ order. The proof there also becomes significantly more involved because
when the order of $\alpha\in\mathrm{Br}(X)$ is not prime-to-$p$, it is not
true that every lifting $X_{W}$ of $X$ over $W$ carries a lifting of $\alpha$,
so the proof of Thm 2.7 breaks down.

### 2.3 Galois Descent

Using Thm 2.8 we give a Galois descent theorem for derived $\ell$-isogenies:

######  Theorem 2.10.

Let $k$ be a perfect field with algebraic closure $\bar{k}$. Let
$f:\bar{X}^{\prime}\stackrel{{\scriptstyle\sim}}{{\to}}\bar{X}$ be a derived
$\ell$-isogeny bewteen two K3 surface $\bar{X},\bar{X}^{\prime}$ over
$\bar{k}$. Suppose $\bar{X}$ admits a model $X$ over $k$ and the
$\mathbb{Z}_{\ell}$-lattice
$f^{*}(\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X}^{\prime},\mathbb{Z}_{\ell}))$
in $\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X},\mathbb{Z}_{\ell})$ is
invariant under the induced $\mathrm{Gal}_{k}$-action, then $\bar{X}^{\prime}$
admits a model $X^{\prime}$ over $k$ such that $f$ decends to a derived
$\ell$-isogeny
$\mathfrak{h}^{2}(X^{\prime})\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X)$.

###### Proof.

For each $\sigma\in\mathrm{Gal}_{k}$, we use supscript $\sigma$ to denote base
change via $\sigma:\bar{k}\to\bar{k}$. Let us write $b_{\sigma}:\bar{\to}X$
and $b^{\prime}_{\sigma}:(X^{\prime})^{\sigma}\to X^{\prime}$ for the
canonical base change morphisms. Suppose
$\\{\varphi_{\sigma}:\bar{X}^{\sigma}\stackrel{{\scriptstyle\sim}}{{\to}}\bar{X}\\}_{\sigma\in\mathrm{Gal}_{k}}$
is the descent datum for $\bar{X}$ defined by the model $X$. Our goal is to
transport $\\{\varphi_{\sigma}\\}$ to a descent datum for $X^{\prime}$. Let
$\varphi^{\prime}_{\sigma}:\mathfrak{h}^{2}((X^{\prime})^{\sigma})\rightsquigarrow\mathfrak{h}^{2}(X^{\prime})$
be the defined by the composition

$\mathfrak{h}^{2}((\bar{X}^{\prime})^{\sigma})\stackrel{{\scriptstyle
f^{\sigma}}}{{\to}}\mathfrak{h}^{2}(\bar{X}^{\sigma})\stackrel{{\scriptstyle\varphi_{\sigma}}}{{\to}}\mathfrak{h}^{2}(\bar{X})\stackrel{{\scriptstyle
f^{-1}}}{{\to}}\mathfrak{h}^{2}(\bar{X}^{\prime}).$

It is clear that $\varphi^{\prime}_{\sigma}$ is a derived $\ell$-isogeny.

We claim that $\omega(\varphi^{\prime}_{\sigma})$ restricts to an isomorphism
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}((\bar{X}^{\prime})^{\sigma},\mathbb{Z}_{\ell})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X},\mathbb{Z}_{\ell})$.
Since $f$ is an $\ell$-isogeny, it suffices to check that
$\varphi^{\prime}_{\sigma}$ is $\ell$-integral. Recall that $b_{\sigma}$ and
$b_{\sigma}^{\prime}$ induce canonical identifications
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X}^{\sigma},\mathbb{Z}_{\ell})=\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X},\mathbb{Z}_{\ell})$
and
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}((X^{\prime})^{\sigma},\mathbb{Z}_{\ell})=\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X}^{\prime},\mathbb{Z}_{\ell})$,
the action of $\sigma$ on
$\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X},\mathbb{Z}_{\ell})$ comes
exactly from $\varphi_{\sigma}$. Therefore, the claim follows from the
assumption that
$f^{*}(\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X}^{\prime},\mathbb{Z}_{\ell}))$
is ${\mathrm{Gal}_{k}}$-stable.

By Thm 2.8, $\varphi^{\prime}_{\sigma}$ is induced by an isomorphism, which we
still denote by $\varphi^{\prime}_{\sigma}$. Since ${\varphi_{\sigma}}$
satisfies the cocycle condition, one easily checks using the cohomological
rigidity of K3 surfaces ([15, Prop. 3.4.2]) that
$\\{\varphi^{\prime}_{\sigma}\\}$ also satisfies the cocyle condition.
Therefore, $\\{\varphi^{\prime}_{\sigma}\\}$ defines a descent datum for
$X^{\prime}$, which gives rise to a model $Y^{\prime}$ over $k$. By
construction, the map
$f^{*}:\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X}^{\prime},\mathbb{Q}_{\ell})\to\mathrm{H}^{2}_{\mathrm{{\acute{e}}t}}(\bar{X},\mathbb{Q}_{\ell})$
is ${\mathrm{Gal}_{k}}$-equivariant for the ${\mathrm{Gal}_{k}}$-actions
induced by $X,X^{\prime}$. By a standard averaging argument, $f$ descends to
an isogeny
$\mathfrak{h}^{2}(X^{\prime})\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{h}^{2}(X)$.
∎

##  3 Relations between cycle conjectures

### 3.1 Linear Algebra

Let $V_{\ell}$ be a $\mathbb{Q}_{\ell}$ vector space of dimension $2r$
equipped with a non-degenerate bilinear pairing $q:V_{\ell}\times
V_{\ell}\to\mathbb{Q}_{\ell}$. Note that $q$ provides an identification of
$V_{\ell}$ with $V_{\ell}^{\vee}$, through which induces a natural pairing
$q^{{\otimes}(1,1)}$ on
$\mathrm{End}(V_{\ell})\,{\cong}\,V_{\ell}{\otimes}V_{\ell}$ given by the
formula

$\displaystyle
q^{{\otimes}(1,1)}(\alpha,\beta)=\mathrm{tr}(\alpha\beta^{\dagger})$ (3.1)

where $\beta^{\dagger}$ denotes the adjoint of $\beta$ under $q$.

Let $\mathrm{U}(E)$ denote the group scheme defined by units of $E$: For each
$\mathbb{Q}$-algebra $R$, set
$\mathrm{U}(E):=(E{\otimes}_{\mathbb{Q}}R)^{\times}$. In particular, if $E$ is
a field extension of $\mathbb{Q}$, then $\mathrm{U}(E)$ is nothing but the
usual Weil restriction $\mathrm{Res}\,_{E/\mathbb{Q}}\mathbb{G}_{m}$.

######  Lemma 3.2.

If the natural map $E{\otimes}\mathbb{Q}_{\ell}\to\mathrm{End}(V_{\ell})$ is
injecture, or equivalently,
$\dim_{\mathbb{Q}}E=\dim_{\mathbb{Q}_{\ell}}\mathbb{Q}_{\ell}\langle
E\rangle$, then the natural map
$\mathrm{U}(E){\otimes}\mathbb{Q}_{\ell}\to\mathrm{GL}(V_{\ell})$ is
injective.

###### Proof.

When evaluated on a $\mathbb{Q}_{\ell}$-algebra $R$,
$\mathrm{U}(E){\otimes}\mathbb{Q}_{\ell}\to\mathrm{GL}(V_{\ell})$ becomes the
map $(E{\otimes}_{\mathbb{Q}}R)^{\times}\to\mathrm{GL}(V_{\ell}{\otimes}R)$.
This map is a restriction of the $R$-linear map
$E{\otimes}_{\mathbb{Q}}R\to\mathrm{End}(V_{\ell}{\otimes}R)$ which is
obtained from $E{\otimes}\mathbb{Q}_{\ell}\to\mathrm{End}(V_{\ell})$ via base
change. Then we use the fact that algebras over a field are flat. ∎

For a quick non-example, note that if $\dim V_{\ell}=1$,
$E=\mathbb{Q}(\alpha)$ for some $\alpha\in\mathbb{Q}_{\ell}$ which is a root
to some irreducible polynomial over $\mathbb{Q}$ of degree $\geq 2$, then
$\mathrm{U}(E){\otimes}\mathbb{Q}_{\ell}$ cannot inject into
$\mathrm{GL}(V_{\ell})$ as $\dim\mathrm{U}(E)\geq 2$.

######  Lemma 3.3.

Let $W_{\ell}$ be any finite dimensional $\mathbb{Q}_{\ell}$ vector space
equipped with a non-degenerate pairing $W_{\ell}\times
W_{\ell}\to\mathbb{Q}_{\ell}$. Let $W$ be a $\mathbb{Q}$-vector space embedded
in $W_{\ell}$ such that the natural map
$j_{\ell}:W{\otimes}\mathbb{Q}_{\ell}\to W_{\ell}$ is surjective. If the
pairing on $W_{\ell}$ restricts to a pairing $W\times W\to\mathbb{Q}$ on $W$,
then $j_{\ell}$ must also be injective.

###### Proof.

Choose a section of the projection $W\to W/(W\cap\ker(j_{\ell}))$. Then we
obtain a finite dimensional subspace $N\subset W$ such that the restriction of
$j_{\ell}$ to $N{\otimes}\mathbb{Q}_{\ell}$ is an isomorphism. In other words,
$N$ provides a $\mathbb{Q}$-linear structure on $W$. Note that the pairing on
$N$ which is inherited from $W$ is necessarily non-degenerate. Since
$j_{\ell}$ is surjective, the map $W^{\vee}\to
W_{\ell}^{\vee}=(N{\otimes}\mathbb{Q}_{\ell})^{\vee}$ is injective. As this
map factors as $W^{\vee}\to N^{\vee}\to(N{\otimes}\mathbb{Q}_{\ell})^{\vee}$,
$W^{\vee}\to N^{\vee}$ must also be injective, so that $N=W$. ∎

######  Set-up 3.4.

Let $F\in\mathrm{SO}(V_{\ell})$ be a semisimple element such that the set of
eigenvalues of $F$ is of the form
$\\{\alpha_{1}^{\pm},\alpha_{2}^{\pm},\cdots,\alpha_{r}^{\pm}\\}$ for distinct
$\alpha_{i}$’s in $\bar{\mathbb{Q}}_{\ell}$ and none of $\alpha_{i}$’s is a
root of unity. Let $E$ be a vector space over $\mathbb{Q}$ embedded in
$\mathrm{End}(V_{\ell})$ such that $E$ is closed under multiplication and
$q^{{\otimes}(1,1)}$ restricts to a $\mathbb{Q}$-bilinear pairing $E\times
E\to\mathbb{Q}$ on $E$. Let $I_{\ell}$ be the centralizer of $F$ in the
reductive group $\mathrm{SO}(V_{\ell})$ over $\mathbb{Q}_{\ell}$.

We first make some elementary observations: Let $W_{i}^{\pm}$ be the
eigenspace of the eigenvalue $\alpha_{i}^{\pm}$. Then
$\bar{V}_{\ell}:=V_{\ell}{\otimes}\bar{\mathbb{Q}}_{\ell}$ admits a
decomposition $V_{\ell}=\oplus_{i=1}^{r}(W_{i}^{+}\oplus W_{i}^{-})$. In
particular, every $W_{i}$ (resp. $W_{i}^{-}$) is isotropic and only pairs
nontrivially with $W_{i}^{-}$ (resp. $W_{i}$). More precisely, $q$ restricts
to a perfect pairing $W_{i}^{+}\times W_{i}^{-}\to\bar{\mathbb{Q}}_{\ell}$
with which we can identify $W_{i}^{+}$ with $(W_{i}^{-})^{\vee}$. There is a
natural isomorphism

$\displaystyle\mathrm{GL}(W_{i}^{+})\,{\cong}\,\mathrm{SO}_{F}(W_{i}^{+}\oplus
W_{i}^{-})\subset\mathrm{End}(W_{i}^{+})\times\mathrm{End}(W_{i}^{-})$ (3.5)

given by $g\mapsto(g,(g^{\vee})^{-1})$.

######  Theorem 3.6.

Suppose $I$ is a group over $\mathbb{Q}$ together with a morphism
$i:I(\mathbb{Q})\to E$ of sets and a morphism
$i_{\ell}:I{\otimes}\mathbb{Q}_{\ell}\to I_{\ell}$ of algebraic groups such
that the diagram

${I(\mathbb{Q})}$${I(\mathbb{Q}_{\ell})}$${E}$${\mathrm{End}V_{\ell}}$$\scriptstyle{i}$$\scriptstyle{i_{\ell}(\mathbb{Q}_{\ell})}$

commutes. If $i(I(\mathbb{Q}))\backslash I(\mathbb{Q}_{\ell})$ is compact,
then the following are equivalent: (a) $I$ is reductive. (b) The image of
$i_{\ell}$ is $I_{\ell}$. (c) The image of $I(\mathbb{Q})$ in
$\mathrm{End}V_{\ell}$ span $\mathrm{End}_{F}(V_{\ell})$.

###### Proof.

We first remark that by [3, Prop. 9.3], the compactness of
$I(\mathbb{Q})\backslash I(\mathbb{Q}_{\ell})$ ensures that
$I{\otimes}\mathbb{Q}_{\ell}$ is a parabolic subgroup of $I_{\ell}$.

(a) $\Rightarrow$ (b) The quotient $Q:=I{\otimes}\mathbb{Q}_{\ell}\backslash
I_{\ell}$ is proper. Since $I{\otimes}\mathbb{Q}_{\ell}$ is reductive, $Q$
must also be affine. The connectedness of $I_{\ell}$ implies that $Q$ is just
a point.

(b) $\Rightarrow$ (c): Since $I$ is reductive, $I(\mathbb{Q})$ is Zariski
dense in $I_{\ell}$. Therefore, it suffices to show that
$I(\bar{\mathbb{Q}}_{\ell})$ spans $\mathrm{End}_{F}(V_{\ell})$. This follows
easily from the observation that
$I_{\ell}{\otimes}\bar{\mathbb{Q}}_{\ell}=\prod_{i=1}^{r}\mathrm{GL}(W_{i}^{+})$
under (3.5). Indeed, it suffices to check that for any vector space $W$ over
any field, the image of $\mathrm{GL}(W)$ in
$\mathrm{End}(W)\times\mathrm{End}(W)$ under the embedding
$g\mapsto(g,g^{-1})$ spans the entire space.

(c) $\Rightarrow$ (a): Under the isomorphism
$I_{\ell}{\otimes}\bar{\mathbb{Q}}_{\ell}=\prod_{i=1}^{r}\mathrm{GL}(W_{i}^{+})$,
$I{\otimes}\bar{\mathbb{Q}}_{\ell}$ takes the form $\prod_{i=1}^{r}G_{i}$,
where $G_{i}$ is a parabolic subgroup of $\mathrm{GL}(W_{i}^{+})$. Suppose for
some $i$, $G_{i}$ is a proper subgroup of $\mathrm{GL}(W_{i})$. Then $G_{i}$
has to fix some nontrivial flag in $W_{i}$, so that
$G_{i}(\bar{\mathbb{Q}}_{\ell})\subset\mathrm{End}(W_{i})$ cannot possibly
span $\mathrm{End}(W_{i})$. This contradicts (c). Therefore, we must have (b),
which implies (a). ∎

### 3.2 Finiteness and Tate Conjecture

Let $X$ be a polarized K3 surface over $\mathbb{F}_{q}$. Let $F$ be the
Frobenius action on $\mathrm{H}^{2}_{\ell}(X)(1)$ and assume that
$F\in\mathrm{SO}(\mathrm{H}^{2}_{\ell}(X)_{\mathbb{Q}}(1))$, and the
$F$-invariants of $\mathrm{H}^{2}_{\ell}(X)_{\mathbb{Q}}(1)$ are the same as
$F^{2}$-invariants. Let $\mathfrak{a}$ and $\mathfrak{t}$ denote the algebraic
part and the transcendental part of $\mathfrak{h}^{2}(X)$ respectively. Recall
that by the Tate conjecture for K3 surfaces ([13, Thm 1]),
$\mathrm{H}^{2}_{\ell}(X)_{\mathbb{Q}}(1)^{F}$ can be identified with
$\mathrm{NS}(X){\otimes}\mathbb{Q}_{\ell}$. Let $T_{\ell}(X)$ denote the
orthogonal complemenet of $\mathrm{NS}(X){\otimes}\mathbb{Q}_{\ell}$. Let
$I_{\ell}$ (resp. $I_{\mathfrak{t},\ell}$) denote the centralizer of $F$ in
$\mathrm{SO}(\mathrm{H}^{2}_{\ell}(X)_{\mathbb{Q}}(1))$ (resp.
$\mathrm{SO}(T_{\ell}(X)_{\mathbb{Q}})$). Note that $I_{\ell}$ canonically
decomposes as $I_{\mathfrak{a},\ell}\times I_{\mathfrak{t},\ell}$, where
$I_{\mathfrak{a},\ell}=\mathrm{SO}(\mathrm{H}^{2}_{\ell}(X)_{\mathbb{Q}}(1)^{F})$.
Define $I_{\mathfrak{t},\ell}$ to be the functor which sends every
$\mathbb{Q}$-algebra $R$ to the subgroup of
$(\mathrm{End}(\mathfrak{t}){\otimes}R)^{\times}$ which preserves the Poincaré
pairing $\mathfrak{t}(1){\otimes}\mathfrak{t}(1)\to\mathbf{1}$. Let
$S^{\mathrm{der},\ell}\subset\mathrm{End}(\mathfrak{h}^{2}(X))$ (resp.
$S^{\mathrm{der},\ell}_{\mathfrak{t}}\subset\mathrm{End}(\mathfrak{t})$) be
the subgroup of derived (resp. derived transcendental) $\ell$-isogenies.

######  Theorem 3.7.

The quotient $S^{\mathrm{der},\ell}_{\mathfrak{t}}\backslash
I_{\mathfrak{t},\ell}(\mathbb{Q}_{\ell})$ is compact.

###### Proof.

Note that $I_{\ell}=I_{\mathfrak{a},\ell}\times I_{\mathfrak{t},\ell}$. Let
$p_{\mathfrak{a}}(\mathsf{K})$ and $p_{\mathfrak{t}}(\mathsf{K})$ be the
projections of the $I_{\mathfrak{a},\ell}(\mathbb{Q}_{\ell})\times
I_{\mathfrak{t},\ell}(\mathbb{Q}_{\ell})$ repsectively. Then
$\mathsf{K}\subseteq p_{\mathfrak{a}}(\mathsf{K})\times
p_{\mathfrak{t}}(\mathsf{K})$ is a subgroup of finite index. Therefore, it
suffices to show that the double quotient $S^{\mathrm{der},\ell}\backslash
I_{\ell}(\mathbb{Q}_{\ell})/\mathsf{K}$ is finite.

By combining Thm 2.5 and Thm 2.10, one quickly sees that the elements in the
double quotient $S^{\mathrm{der},\ell}\backslash
I_{\ell}(\mathbb{Q}_{\ell})/\mathsf{K}$ corresponds bijectively to the set of
isomorphism class of K3 surfaces $X^{\prime}$ over $\mathbb{F}_{q}$ which
admits a derived $\ell$-isogeny
$f:\mathfrak{h}^{2}(X^{\prime})\to\mathfrak{h}^{2}(X)$. Thanks to [13, Cor.
2], which is a corollary to [12, Main Theorem] and the Tate conjecture for
K3’s [13, Thm 1], there are only finitely isomorphism classes of K3 surfaces
over $\mathbb{F}_{q}$. Therefore, $S^{\mathrm{der},\ell}\backslash
I_{\ell}(\mathbb{Q}_{\ell})/\mathsf{K}$ is finite. ∎

######  Corollary 3.8.

Suppose there is a reductive group $I$ over $\mathbb{Q}$ which is equipped
with (set-theoretic) maps
$S^{\mathrm{der},\ell}_{\mathfrak{t}}\stackrel{{\scriptstyle
j}}{{\to}}I(\mathbb{Q})\stackrel{{\scriptstyle
i}}{{\to}}\mathrm{End}(\mathfrak{t})$ and a morphism $i_{\ell}:I\to
I_{\mathfrak{t},\ell}$ such that $i_{\ell}\circ j=\omega_{\ell}$ and
$\omega_{\ell}\circ i=i_{\ell}$. Then
$\mathrm{End}_{F}T_{\ell}(X)_{\mathbb{Q}}$ is spanned by
$\omega_{\ell}(i(I(\mathbb{Q}))$, and
$\omega_{\ell}:I_{\mathfrak{t}}{\otimes}\mathbb{Q}_{\ell}\to
I_{\mathfrak{t},\ell}$ is an isomorphism. In particular, Thm 1.1 and 1.2 hold
for $X$.

###### Proof.

The first statement is a direct consequence of Thm 3.6 and Thm 3.7. For the
second, we first apply Lem. 3.3 to
$W_{\ell}=\mathrm{End}_{F}T_{\ell}(X)_{\mathbb{Q}}$ and $W=E$ to obtain the
finite-dimensionality of $\mathrm{End}(\mathfrak{t})$ and the injectivity of
$E{\otimes}\mathbb{Q}_{\ell}\to\mathrm{End}_{F}T_{\ell}(X)_{\mathbb{Q}}$. Then
we apply Lem. 3.2 to $V_{\ell}=T_{\ell}(X)_{\mathbb{Q}}$ and
$E=\mathrm{End}(\mathfrak{t})$ to conclude that the morphism
$I_{\mathfrak{t}}{\otimes}\mathbb{Q}_{\ell}\to I_{\mathfrak{t},\ell}$ is
injective. ∎

######  Remark 3.9.

If we know that (A) the span $\mathbb{Q}\langle S^{\mathrm{der},\ell}\rangle$
in $E$ is finite dimensional, and (B) the Poincaré pairing on
$p_{\mathfrak{t}}(\mathbb{Q}\langle S^{\mathrm{der},\ell}\rangle)$ is negative
definite, as predicted by the standard conjectures, then we can simply take
$I$ to be the subgroup
$I^{\prime}_{\mathfrak{t}}:=\mathrm{U}(\mathbb{Q}\langle
S^{\mathrm{der},\ell}\rangle)\cap I_{\mathfrak{t}}$ of $I_{\mathfrak{t}}$.
Indeed, using (3.1) one quickly checks that $I^{\prime}_{\mathfrak{t}}$ embeds
into $\mathrm{O}(p_{\mathfrak{t}}(\mathbb{Q}\langle
S^{\mathrm{der},\ell}\rangle))$ via right multiplication. This implies that
$I_{\mathfrak{t}}(\mathbb{R})$ is compact, so that $I^{\prime}_{\mathfrak{t}}$
is reductive. In fact, in this case, we know that
$\mathrm{End}_{F}T_{\ell}(X)_{\mathbb{Q}}$ is spanned by
$S^{\mathrm{der},\ell}_{\mathfrak{t}}$.

######  Example 3.10.

Suppose $X$ comes from the Kummer construction, i.e., $X$ is the
desingularization of the quotient $A/A[2]$ for some abelian surface $A$. Set
$\widetilde{A}:=X\times_{A/A[2]}A$. Then we have a diagram

${\widetilde{A}}$${X}$${A}$

.

This correspondence induces an isomorphism
$\Psi:\mathfrak{h}^{2}(A)\oplus(\mathbf{L}^{\oplus
16})\,{\cong}\,\mathfrak{h}^{2}(X)$, where each $\mathbf{L}$ corresponds to an
exceptional divisor, which has self-intersection number $-2$. If
$q_{X}:\mathfrak{h}^{2}(X)(1)^{{\otimes}2}\to\mathbf{1}$ and
$q_{A}:\mathfrak{h}^{2}(A)(1)^{{\otimes}2}\to\mathbf{1}$ denote the Poincaré
pairings, then $q_{X}\circ\Psi^{{\otimes}2}=2q_{A}$. It is easy to see, using
[1, Thm 1.1] and [8, Thm 1], that $X$ satisfies (A) and (B) in Rmk 3.9.

######  Example 3.11.

Suppose $X$ is an ordinary K3 surface. Then by Thm 2.7 elements in
$S^{\mathrm{der},\ell}$ all lift to the canonical lifting of $X$. Therefore,
it follows from Hodge theory that $X$ satisfies (A) and (B) in Rmk 3.9. We
remark that the Tate conjecture for any power of an ordinary K3 surface was
proved by Zarhin in [19]. The main point is that the transcendental Tate
classes are all spanned by graphs of powers of the Frobenius endomorphism. The
$q^{-1}$-multiple of the graph of the $\mathbb{F}_{q}$-linear Frobenius
endomorphism is indeed an isogeny. However, it cannot be a derived
$\ell$-isogeny as long as $X$ is not supersingular. Therefore, for squares of
ordinary K3’s, our proof yields a different explanation for algebraic cycles
spanning the transcendental classes such that the method of constructing these
cycles works for arbitrary fields.

##  4 Uncondtional Proof

### 4.1 Kuga-Satake Construction

Let $\mathscr{M}_{2d}^{\circ}$ denote the moduli stack of polarized K3 surface
of degree $2d$ over $\mathbb{Z}_{(p)}$. Let $L_{d}$ denote the lattice defined
in [13, (3.10)]. Let $\mathcal{S}(L_{d})$ (resp.
$\widetilde{\mathcal{S}}(L_{d})$) denote the canonical integral model of the
associated orthogonal (resp. spinor) Shimura variety. There are sheaves
$\mathbf{L}_{B},\mathbf{L}_{\mathrm{dR}},\mathbf{L}_{\ell},\mathbf{L}_{p},\mathbf{L}_{\mathrm{cris}}$
defined on
$\mathcal{S}(L_{d})(\mathbb{C}),\mathcal{S}(L_{d}),\mathcal{S}(L_{d}),\mathcal{S}(L_{d})_{\mathbb{Q}},\mathcal{S}(L_{d})_{\mathbb{F}_{p}}$
respectively, with respect to the understood Grothendieck topology ([13,
§4.1]).

######  Theorem 4.1.

There exists an étale morphism
$\rho:\widetilde{\mathscr{M}}_{2d}^{\circ}\to\mathcal{S}(L_{d})$ such that for
every algebraically closed field $\kappa$,
$u\in\widetilde{\mathscr{M}}_{2d}^{\circ}(\kappa)$ and $t:=\rho(u)$, the
following holds: If $(X,\xi)$ is the fiber of the universal family over $u$,
then there are canonical isometry
$\alpha_{*}:\mathbf{L}_{*,t}(-1)\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{P}^{2}_{*}(X)$
for $*=B,\mathrm{dR},\ell,p,\mathrm{cris}$ whenever the cohomology theory $*$
is applicable.

If $p\mid d$, then $L_{d}$ is not self-dual at $p$. Nonetheless, by [14, §6.8]
we can find a quadratic lattice $\widetilde{L}$ which is self-dual at $p$ and
of signature $(r,2)$ for some $r>19$, together with an isometric embedding
$L_{d}\hookrightarrow\widetilde{L}$ onto an orthogonal direct summand. This
gives rise to maps $\mathcal{S}(L_{d})\to\mathcal{S}(\widetilde{L})$. Set
$N:=(L_{d})^{\perp}\subset\widetilde{L}$. If $p\nmid d$, then we just set
$\widetilde{L}=L_{d}$. We additionally consider
$\widetilde{\mathcal{S}}(\widetilde{L})$, which is equipped with a polarized
abelian scheme $\widetilde{\mathscr{A}}$, and sheaves
$\widetilde{\mathbf{L}}_{B},\widetilde{\mathbf{L}}_{\mathrm{dR}},\widetilde{\mathbf{L}}_{\ell},\widetilde{\mathbf{L}}_{p},\widetilde{\mathbf{L}}_{\mathrm{cris}}$.
To simplify notation, we view each scheme $T$ over $\mathcal{S}(L_{d})$
simultaneously as a scheme over $\mathcal{S}(\widetilde{L})$. Then there is a
canonical embedding $N\subset\mathrm{LEnd}(\widetilde{\mathscr{A}}_{T})$
together with compatible isometries
$\mathbf{L}_{*,T}\stackrel{{\scriptstyle\sim}}{{\to}}N^{\perp}\subset\widetilde{\mathbf{L}}_{*,T}$
($*=B,\ell,\mathrm{dR},p,\mathrm{cris}$ whenever applicable) [13, Prop. 4.9].

#### CSpin-structures

Let $\widetilde{H}$ denote $\mathrm{Cl}(\widetilde{L})$, viewed as a
$\mathrm{Cl}(\widetilde{L})$-bimodule. Equip
$\mathrm{End}(\widetilde{H}_{\mathbb{Q}})$ with a pairing by
$(\alpha,\beta)\mapsto 2^{r+2}\mathrm{tr}(\alpha\circ\beta)$. Then the map
$\widetilde{L}_{\mathbb{Q}}\to\mathrm{End}(\widetilde{H}_{\mathbb{Q}})$ given
by left multiplication is an isometric embedding. Denote by $\pi^{\prime}$ the
idempotent projector from $\mathrm{End}(H^{\prime}_{\mathbb{Q}})$ onto
$\widetilde{L}_{\mathbb{Q}}$. The group
$\mathrm{CSpin}(\widetilde{L}_{\mathbb{Q}})$ is by definition the stabilizer
in $\mathrm{GL}(\widetilde{H}_{\mathbb{Q}})$ of the
$\mathbb{Z}/2\mathbb{Z}$-grading, right $\mathrm{Cl}(\widetilde{L})$-action,
and the tensor $\pi^{\prime}$. We can endow a nondegenerate symplectic pairing
$\Psi:\widetilde{H}\times\widetilde{H}\to\mathbb{Z}$ such that the morphism
$\mathrm{CSpin}(\widetilde{L}_{\mathbb{Q}})\to\mathrm{GL}(\widetilde{H}_{\mathbb{Q}})$
factors through $\mathrm{GSp}(\widetilde{H}_{\mathbb{Q}},\Psi)$. This gives an
embedding of $\widetilde{\mathcal{S}}(L_{d})$ into a Siegel Shimura variety,
from which it inherits a weakly polarized universal abelian scheme
$(\widetilde{\mathscr{A}},\lambda^{\prime})$. Moreover,
$\widetilde{\mathscr{A}}$ is equipped with a “CSpin-structure”: a
$\mathbb{Z}/2\mathbb{Z}$-grading, a left $\mathrm{Cl}(\widetilde{L})$-action
and various realizations of $\pi$. More precisely, for each algebraically
closed field$\kappa$, and $\kappa$-point $t$ on $\mathcal{S}(\widetilde{L})$
with lift $s$ on $\widetilde{\mathcal{S}}(\widetilde{L})$, there exists an
idempotent projector $\widetilde{\boldsymbol{\pi}}_{*,s}$ on
$\mathrm{End}(\mathrm{H}^{1}_{*}(\widetilde{\mathscr{A}}_{s}))$ whose image is
canonically identified with $\widetilde{\mathbf{L}}_{*,t}$ for
$*=B,\mathrm{dR},\ell,p,\mathrm{cris}$ whenever the cohomology $*$ is
applicable. These tensors are compatible in a natural way: If
$\mathrm{char\,}\kappa=0$, the tuple
$(\widetilde{\boldsymbol{\pi}}_{\mathrm{dR},s},\widetilde{\boldsymbol{\pi}}_{\widehat{\mathbb{Z}},s})$
is absolute Hodge. In particular, if $\kappa=\mathbb{C}$,
$(\widetilde{\boldsymbol{\pi}}_{\mathrm{dR},s},\widetilde{\boldsymbol{\pi}}_{\widehat{\mathbb{Z}},s})$
comes from the Betti tensor $\pi_{B,s}$. If $\mathrm{char\,}\kappa=p$, $K$ is
a finite extension of $W(\kappa)[1/p]$, and $s_{K},t_{K}$ are $K$-points which
specialize to $s,t$ respectively,
$\widetilde{\boldsymbol{\pi}}_{\widehat{\mathbb{Z}}^{p},s_{\bar{K}}}$ and
$\widetilde{\boldsymbol{\pi}}_{\widehat{\mathbb{Z}}^{p},s}$ are compatible via
the smooth and proper base change theorem and
$\widetilde{\boldsymbol{\pi}}_{\mathrm{cris},s}$ and
$\widetilde{\boldsymbol{\pi}}_{\mathrm{dR},s_{K}}$ are compatible via the
Berthelot-Ogus isomorphism
$\mathrm{H}^{1}_{\mathrm{cris}}(\widetilde{\mathscr{A}}_{s_{K}}){\otimes}K\,{\cong}\,\mathrm{H}^{1}_{\mathrm{dR}}(\widetilde{\mathscr{A}}_{s_{K}})$.

Denote the composition
$\widetilde{\mathscr{M}}_{2d}^{\circ}\stackrel{{\scriptstyle\rho}}{{\to}}\mathcal{S}(L_{d})\to\mathcal{S}(\widetilde{L})$
by $\rho^{\prime}$.

######  Lemma 4.2.

Let $\kappa$ be a field. Let $t\in\widetilde{\mathcal{S}}(\widetilde{L})(k)$
be a point and $s\in\mathcal{S}(\widetilde{L})(k)$ be a lift of the image of
$t$.

  1. (a)

$\mathrm{Fil}^{1}\mathrm{H}^{1}_{\mathrm{dR}}(\mathscr{A}_{s})=\ker(x)$ for
any generator $x$ of the one-dimensional subspace
$\mathrm{Fil}^{1}\widetilde{\mathbf{L}}_{\mathrm{dR},s}$. Under the inclusion
$\mathbf{L}_{\mathrm{dR}}\subset\widetilde{\mathbf{L}}_{\mathrm{dR}}$ we have
that
$\mathrm{Fil}^{1}\mathbf{L}_{\mathrm{dR}}=\mathrm{Fil}^{1}\widetilde{\mathbf{L}}_{\mathrm{dR}}$.

  2. (b)

Let $R$ be any $\mathbb{Q}_{\ell}$-algebra. If
$\psi_{\ell}\in\mathrm{GL}(\mathrm{H}^{1}_{\ell}(\widetilde{\mathscr{A}}_{s}){\otimes}R)$
is an element which preserves the $\mathbb{Z}/2\mathbb{Z}$-grading,
$\mathrm{Cl}(\widetilde{L})$-action and $\widetilde{\boldsymbol{\pi}}_{\ell}$,
then $\psi_{\ell}$ preserves the symplectic pairing on
$\mathrm{H}^{1}_{\ell}(\widetilde{\mathscr{A}}_{s}){\otimes}R$ induced by the
weak polarization $\lambda^{\prime}_{s}$, which is well defined up to a
multiple of $R^{\times}$.

###### Proof.

(a) The first statement follows from [14, Prop. 4.7(v)]. The second statement
follows from the fact that
$N\subseteq\mathrm{Fil}^{0}\widetilde{\mathbf{L}}_{\mathrm{dR}}$ is
perpendicular to $\mathrm{Fil}^{1}\widetilde{\mathbf{L}}_{\mathrm{dR}}$. (b)
By [14, Prop. 3.14], there exists an isomorphism
$\eta_{\ell}:H^{\prime}_{\mathbb{Q}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{H}^{1}_{\ell}(\mathscr{A}_{s})_{\mathbb{Q}}$
which preserves the $\mathbb{Z}/2\mathbb{Z}$-grading,
$\mathrm{Cl}(\widetilde{L})$-action and sends $\pi$ to
$\widetilde{\boldsymbol{\pi}}_{\ell}$ such that $\eta_{\ell}$ sends $\Psi$ to
the pairing induced by the weak polarization $\lambda^{\prime}_{s}$. Then we
use the fact that the stablizer of the $\mathbb{Z}/2\mathbb{Z}$-grading,
$\mathrm{Cl}(\widetilde{L})$-action and sends $\pi$ in
$\mathrm{GL}(H^{\prime}_{\mathbb{Q}_{\ell}})$ is
$\mathrm{CSpin}(\widetilde{L}_{\mathbb{Q}_{\ell}})$, which lies in
$\mathrm{GSp}(H^{\prime}_{\mathbb{Q}_{\ell}},\Psi)$. ∎

### 4.2 Unconditional Proof

Let $k$ be a finite field over $\mathbb{F}_{p}$. Let
$u\in\widetilde{\mathscr{M}}^{\circ}_{2d}(k)$ be a point given by the tuple
$(X,\xi,\epsilon)$ for some choice of
$\epsilon:\det(L_{d}{\otimes}\mathbb{Z}_{2})\stackrel{{\scriptstyle\sim}}{{\to}}\det(\mathrm{P}^{2}_{2}(X))$.
Set $t=\rho(u)$ and choose a lift
$s\in\widetilde{\mathcal{S}}(\widetilde{L})(k)$ of $t$. Up to replacing $k$ by
a finite extension, we assume that all line bundles on $X_{\bar{k}}$ and all
endomorphisms of $\widetilde{\mathscr{A}}_{s,\bar{k}}$ are defined over $k$.
Let $S_{0}^{\mathrm{der},\ell}$ be the stabilizer of $\xi$ and
$\det(\mathrm{P}^{2}_{2}(X))$ in $S^{\mathrm{der},\ell}$. Since
$\mathrm{rank\,}\mathrm{Pic}\,(X)\geq 2$, the restriction of
$p_{\mathfrak{t}}$ to $S_{0}^{\mathrm{der},\ell}$ has the same image as
$S^{\mathrm{der},\ell}$. Suppose that $X$ has finite height.

######  Lemma 4.3.

For every $f\in S_{0}^{\mathrm{der},\ell}$, there exists a CSpin-isogeny
$\psi:\widetilde{\mathscr{A}}_{s}\to\widetilde{\mathscr{A}}_{s}$ which fixes
$N_{s}$ and induces the same action on $\mathbf{L}_{*,t}$ for
$*=\mathrm{cris},\widehat{\mathbb{Z}}^{p}$ as $f$ via $\alpha_{*}$. Moreover,
$\psi$ is unique up to a $\mathbb{Q}^{\times}$-multiple.

###### Proof.

By Thm 2.7, we can find perfect liftings $X_{W}$ and $X^{\prime}_{W}$ of $X$
such that $f$ lifts to an isogeny
$f_{K}:\mathfrak{h}^{2}(X_{K})\to\mathfrak{h}^{2}(X^{\prime}_{K})$ between the
generic fibers. Choose an isomorphism $\bar{K}\,{\cong}\,\mathbb{C}$. Let
$t_{W},t^{\prime}_{W}$ be the liftings of $t$ given by $X_{W},X^{\prime}_{W}$
and $t_{\mathbb{C}},t_{\mathbb{C}}^{\prime}$ be
$t_{W}{\otimes}\mathbb{C},t^{\prime}_{W}{\otimes}\mathbb{C}$. Then
$f_{K}{\otimes}\mathbb{C}$ gives a rational Hodge isometry
$\mathbf{L}_{B,t_{\mathbb{C}}}{\otimes}\mathbb{Q}\,{\cong}\,\mathbf{L}_{B,t^{\prime}_{\mathbb{C}}}{\otimes}\mathbb{Q}$.
Recall that $N_{t_{\mathbb{C}}}$ is canonically identified with
$N_{t^{\prime}_{\mathbb{C}}}$, so that can extend
$\mathbf{L}_{B,t_{\mathbb{C}}}{\otimes}\mathbb{Q}\,{\cong}\,\mathbf{L}_{B,t^{\prime}_{\mathbb{C}}}{\otimes}\mathbb{Q}$
to a Hodge isometry
$\widetilde{\mathbf{L}}_{B,t_{\mathbb{C}}}{\otimes}\mathbb{Q}\,{\cong}\,\widetilde{\mathbf{L}}_{B,t^{\prime}_{\mathbb{C}}}{\otimes}\mathbb{Q}$.
Since the map $\widetilde{\mathcal{S}}(\widetilde{L})\to\mathcal{S}(L)$ is
étale, the liftings $t_{W},t^{\prime}_{W}$ of $t$ can be lifted to liftings
$s_{W},s^{\prime}_{W}$ of $s$ over $W$. Set
$s_{\mathbb{C}},s^{\prime}_{\mathbb{C}}$ to be the $\mathbb{C}$-fibers of
$s_{W},s^{\prime}_{W}$. By [5, Lem. 5.2], the rational Hodge isometry
$\widetilde{\mathbf{L}}_{B,t_{\mathbb{C}}}{\otimes}\mathbb{Q}\,{\cong}\,\widetilde{\mathbf{L}}_{B,t^{\prime}_{\mathbb{C}}}{\otimes}\mathbb{Q}$
can be lifted to a CSpin-isogeny
$\widetilde{\mathscr{A}}_{s_{\mathbb{C}}}\to\widetilde{\mathscr{A}}_{s^{\prime}_{\mathbb{C}}}$.
We get the desired $\psi$ by specialzation. ∎

######  Lemma 4.4.

For each CSpin-isogeny
$\psi:\widetilde{\mathscr{A}}_{s}\to\widetilde{\mathscr{A}}_{s}$ which
preserves $N_{s}\subset\mathrm{LEnd}(\widetilde{\mathscr{A}}_{s})$ there
exists an isogeny $f:\mathfrak{h}^{2}(X)\to\mathfrak{h}^{2}(X^{\prime})$ which
preserves $\xi$ and $\epsilon$ such that $\psi$ and $f$ induce the same
actions on $\mathbf{L}_{*,t}{\otimes}\mathbb{Q}$ for
$*=\mathrm{cris},\widehat{\mathbb{Z}}^{p}$.

###### Proof.

This follows from a slight refinement of the argument as [18, Prop.
5.2]222There is a typo in loc. cit. $p\geq 13$ should be $p\geq 5$.. For
reader’s convenience we sketch the argument: First, note that $\psi$ induces
an isomorphism $\phi\in\mathrm{O}(\mathrm{H}^{2}_{\mathrm{cris}}(X/W)[1/p])$
which preserves the class of $\xi$. By [18, Lem. 4.5], there exists a finite
flat extension $V$ of $W$, and a lifting $X_{V}$ of $X$ such that
$\mathrm{Pic}\,(X_{V})\to\mathrm{Pic}\,(X)$ is an isomorphism, and the
automorphism of $\mathrm{H}^{2}_{\mathrm{dR}}(X_{V}[1/p])$ induced by $\phi$
via the Berthelot-Ogus isomorphism preserves the Hodge filtration. This
$X_{V}$ gives rise to a $V$-valued point $u_{V}$ which lifts $u$. Set
$t_{V}=\rho^{\prime}(u_{V})$. Since the map
$\widetilde{\mathcal{S}}(L_{d})\to\mathcal{S}(L_{d})$ are étale, $u_{V}$
induces liftings $t_{V}$ and $s_{V}$ of $t$ and $s$ over $V$. Next, set
$K:=V[1/p]$ and recall that
$\mathrm{Fil}^{1}\mathrm{H}^{1}_{\mathrm{dR}}(\mathscr{A}_{s_{K}})=\ker(x)$
for any generator $x$ of the one-dimensional subspace
$\mathrm{Fil}^{1}\widetilde{\mathbf{L}}_{\mathrm{dR},t_{K}}$. By Hodge theory,
the cycle classes of $N$ in $\widetilde{\mathbf{L}}_{\mathrm{dR},t_{K}}$
cannot lie in $\mathrm{Fil}^{1}\widetilde{\mathbf{L}}_{\mathrm{dR},t_{K}}$, so
we have
$\mathrm{Fil}^{1}\mathbf{L}_{\mathrm{dR},t_{K}}=\mathrm{Fil}^{1}\widetilde{\mathbf{L}}_{\mathrm{dR},t_{K}}$
under the inclusion
$\mathbf{L}_{\mathrm{dR},t_{K}}\subseteq\widetilde{\mathbf{L}}_{\mathrm{dR},t_{K}}$.
By our construction of $s_{V}$, $\psi$ preserves
$\mathrm{Fil}^{1}\mathrm{H}^{1}_{\mathrm{dR}}(\mathscr{A}_{s_{K}})$ via the
Berthelot-Ogus isomorphism. This implies that, up to replacing $\psi$ by a
$p$-power multiple, $\psi$ lifts to a CSpin-isogeny
$\psi_{K}:\mathscr{A}_{s_{K}}\to\mathscr{A}_{s_{K}}$, which necessarily
preserves $N\subset\mathrm{LEnd}(\mathscr{A}_{s_{K}})$. Now choose an
isomorphism $\bar{K}\,{\cong}\,\mathbb{C}$. Then $\psi_{K}{\otimes}\mathbb{C}$
induces an isometry in
$\mathrm{O}(\mathrm{H}^{2}(X_{s_{K}}(\mathbb{C}),\mathbb{Q}))$ which preserves
$\xi_{K}(\mathbb{C})$ and the Hodge structure. Finally, apply Thm 2.6 to
obtain an auto-isogeny of $X_{K}{\otimes}\mathbb{C}$. Specialize this isogeny
to $X$ and we are done. ∎

Proofs of Thm 1.1 and 1.2. Consider group scheme $\widetilde{I}$ defined by

$\displaystyle\widetilde{I}(R):=\\{\psi\in(\mathrm{End}(\widetilde{\mathscr{A}}_{s}){\otimes}R)^{\times}:\,$
$\displaystyle\psi\text{ respects the $\mathbb{Z}/2\mathbb{Z}$-grading
$\mathrm{Cl}(\widetilde{L})$-action and $\pi_{*}{\otimes}_{\mathbb{Z}}R$ for
$*=\mathrm{cris},\widehat{\mathbb{Z}}^{p}$}\\}$

for each $\mathbb{Q}$-algebra $R$. Set
$I^{\prime}:=\widetilde{I}/\mathbb{G}_{m}$. It follows from Lem. 4.2(b) and
the positivity of Rosati involution that $I^{\prime}(\mathbb{R})$ is compact,
so that $I^{\prime}$ is reductive. Note that
$I^{\prime}{\otimes}\mathbb{Q}_{\ell}$ admits a natural morphism into the
centralizer of $F$ in
$\mathrm{SO}(\mathbf{L}_{\bar{s},\ell})\stackrel{{\scriptstyle\alpha_{\ell}}}{{=}}\mathrm{SO}(\mathrm{P}^{2}_{\mathrm{{\acute{e}}t}}(X,\mathbb{Q}_{\ell}(1)))$.
Let $\mathfrak{t}$ denote $\mathfrak{t}(X)$ and let $I_{\mathfrak{t}}$ be as
defined in § 3.2. Since
$I^{\prime}(\mathbb{Q})=\widetilde{I}(\mathbb{Q})/\mathbb{G}_{m}(\mathbb{Q})$
by Hilbert theorem 90, Lem. 4.3 and 4.4 give us maps
$S_{0}^{\mathrm{der},\ell}\to I^{\prime}(\mathbb{Q})\to
I_{\mathfrak{t}}(\mathbb{Q})$ which are compatible with the morphisms to
$I_{\mathfrak{t}}(\mathbb{Q}_{\ell})$. It is easy to check that
$p_{\mathfrak{t}}(S_{0}^{\mathrm{der},\ell})=S_{\mathfrak{t}}^{\mathrm{der},\ell}$
is surjective, so there exists at least a set-theoretic section
$S_{\mathfrak{t}}^{\mathrm{der},\ell}\to S_{0}^{\mathrm{der},\ell}$. Then we
conclude by Cor. 3.8. ∎

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