# Derived isogenies and isogenies for abelian surfaces

Zhiyuan Li  SCMS, Fudan University, No,2005 Songhu Road, Shanghai, China
<EMAIL_ADDRESS>and  Haitao Zou  SCMS, Fudan University, No,2005
Songhu Road, Shanghai, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the twisted Fourier-Mukai partners of abelian
surfaces. Following the work of Huybrechts [31], we introduce the twisted
derived equivalences (also called derived isogenies) between abelian surfaces.
We show that there is a twisted derived Torelli theorem for abelian surfaces
over algebraically closed fields with characteristic $\neq 2,3$. Over the
complex numbers, the derived isogenies correspond to rational Hodge isometries
between the second cohomology groups, which is in analogy to the work of
Huybrechts and Fu–Vial on K3 surfaces. Their proof relies on the global
Torelli theorem over $\mathbb{C}$, which is missing in positive
characteristics. To overcome this issue, we firstly extend a trick given by
Shioda on integral Hodge structures, to rational Hodge structures, $\ell$-adic
Tate modules and $F$-crystals. Then we make use of Tate’s isogeny theorem to
give a characterization of the derived isogenies between abelian surfaces via
so called principal isogenies. As a consequence, we show the two abelian
surfaces are principally isogenous if and only if they are derived isogenous.

###### Key words and phrases:

abelian surface, isogenies, derived categories, twisted sheaves, Torelli
theorems

###### 2020 Mathematics Subject Classification:

Primary 14F08, 14K02; Secondary 14G17

The authors are supported by the NKRD Program of China (No. 2020YFA0713200),
NSFC General Program (No. 12171090) and Shanghai Pilot Program for Basic
Research (No. 21TQ00).

###### Contents

  1. 1 Introduction
  2. 2 Twisted abelian surface
  3. 3 Cohomological realizations of derived isogeny
  4. 4 Shioda’s Torelli theorem for abelian surfaces
  5. 5 Derived isogeny in characteristic zero
  6. 6 Derived isogeny in positive characteristic

##  1\. Introduction

###  1.1. Background

In the study of abelian varieties, a natural question is to classify the
Fourier-Mukai partners of abelian varieties. Due to Orlov and Polishchuk’s
_derived Torelli theorem_ for abelian varieties in (cf. [52, 54]), there is a
geometric/cohomological classification of derived equivalences between them.
More generally, one can consider the twisted derived equivalence or so called
derived isogeny between abelian varieties in the spirit of [31]: two abelian
varieties $X$ and $Y$ are derived isogenous if they can be connected by
derived equivalences between twisted abelian varieties, i.e. there exist
twisted abelian varieties $(X_{i},\alpha_{i})$ and $(X_{i},\beta_{i})$ such
that there is a zig-zag of derived equivalences

${\operatorname{D}^{b}(X,\alpha)}$${\operatorname{D}^{b}(X_{1},\beta_{1})}$${\operatorname{D}^{b}(X_{1},\alpha_{2})}$${\operatorname{D}^{b}(X_{2},\beta_{2})}$${\vdots}$${\operatorname{D}^{b}(X_{n},\alpha_{n+1})}$${\operatorname{D}^{b}(Y,\beta_{n})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$
(1.1.1)

where $\operatorname{D}^{b}(X,\alpha)$ is the bounded derived category of
$\alpha$-twisted coherent sheaves on $X$.

In [60], Stellari proved that derived isogenous complex abelian surfaces are
isogenous using the the Kuga–Satake varieties associated to their
transcendental lattices (cf. Theorem 1.2 in loc.cit.). However, the converse
is not true as there are isogenous abelian surfaces which are not derived
isogenous (cf. Remark 4.4 (ii) in loc.cit.). The main goal of this paper is to
give a cohomological and geometric classification of derived isogenies between
abelian surfaces over algebraically closed fields of arbitrary characteristic.

###  1.2. Twisted derived Torelli theorem for abelian surfaces in
characteristic zero

Let us first classify the derived isogenous between abelian surfaces in term
of isogenies. For this purpose, we need to introduce a new type of isogeny. We
say two abelian surfaces $X$ and $Y$ are _principally isogenous_ if there is a
isogeny $f$ from $X$ or $\widehat{X}$ to $Y$ of square degree. The first main
result is

######  Theorem 1.2.1.

Let $X$ and $Y$ be two abelian surfaces over $k=\bar{k}$ with
$\operatorname{char}(k)=0$. The following statements are equivalent.

  1. (i)

$X$ and $Y$ are derived isogenous.

  2. (ii)

$X$ and $Y$ are principally isogenous.

A notable fact for abelian surfaces is that besides their $1^{st}$ cohomology
groups, their $2^{nd}$ cohomology groups also carry rich structures. In the
untwisted case, Mukai and Orlov have showed [48, 52] that

$\mathrm{D}^{b}(X)\cong\mathrm{D}^{b}(Y)\Leftrightarrow\widetilde{\mathrm{H}}(X,\mathbb{Z})\cong_{\operatorname{Hdg}}\widetilde{\mathrm{H}}(Y,\mathbb{Z})\Leftrightarrow\mathrm{T}(X)\cong_{\operatorname{Hdg}}\mathrm{T}(Y),$

where $\widetilde{\mathrm{H}}(X,\mathbb{Z})$ and
$\widetilde{\mathrm{H}}(Y,\mathbb{Z})$ are the Mukai lattices,
$\mathrm{T}(X)\subseteq\mathrm{H}^{2}(X,\mathbb{Z})$ and
$\mathrm{T}(Y)\subseteq\mathrm{H}^{2}(Y,\mathbb{Z})$ denote the transcendental
lattices, $\cong_{\operatorname{Hdg}}$ means integral Hodge isometries (cf.
[12, Theorem 5.1]). The following result can be viewed as a generalization of
Mukai and Orlov’s result.

######  Corollary 1.2.2.

The statement (i) is also equivalent to the following equivalent conditions

  1. (iii)

the associated Kummer surfaces $\operatorname{Km}(X)$ and
$\operatorname{Km}(Y)$ are derived isogenous;

  2. (iv)

Chow motives $\mathfrak{h}(X)\cong\mathfrak{h}(Y)$ are isomorphic as Frobenius
exterior algebras;

  3. (v)

even degree Chow motives $\mathfrak{h}^{even}(X)\cong\mathfrak{h}^{even}(Y)$
are isomorphic as Frobenius algebra.

When $k=\mathbb{C}$, then the conditions above are also equivalent to

  1. (6)

$\mathrm{H}^{2}(X,\mathbb{Q})\cong\mathrm{H}^{2}(Y,\mathbb{Q})$ as a rational
Hodge isometry;

  2. (7)

$\widetilde{\mathrm{H}}(X,\mathbb{Q})\cong\widetilde{\mathrm{H}}(Y,\mathbb{Q})$
as a rational Hodge isometry;

  3. (8)

$\mathrm{T}(X)\otimes\mathbb{Q}\cong\mathrm{T}(Y)\otimes\mathbb{Q}$ as a
rational Hodge isometry.

Here, the motive $\mathfrak{h}(X)$ admits a canonical motivic decomposition
produced by Deninger–Murre [19]

$\mathfrak{h}(X)=\bigoplus_{i=0}^{4}\mathfrak{h}^{i}(X)$ (1.2.1)

such that $\mathrm{H}^{*}(\mathfrak{h}^{i}(X))\cong\mathrm{H}^{i}(X)$ for any
Weil cohomology $\mathrm{H}^{*}(-)$. It satisfies
$\mathfrak{h}^{i}(X)=\bigwedge^{i}\mathfrak{h}^{1}(X)$ for all $i$,
$\mathfrak{h}^{4}(X)\simeq\mathds{1}(-4)$ and
$\bigwedge^{i}\mathfrak{h}^{1}(X)=0$ for $i>4$ (cf. [37]). The motive
$\mathfrak{h}(X)$ is a Frobenius exterior algebra objects in the category of
Chow motives over $k$ and the even degree part

$\mathfrak{h}^{even}(X)=\bigoplus_{k\geq
0}^{2}\bigwedge^{2k}\mathfrak{h}^{1}(X)$ (1.2.2)

forms a Frobenius algebra object in the sense of [23].

The equivalences $(i)\Leftrightarrow(iv)\Leftrightarrow(v)$ are motivic
realizations of derived isogenies between abelian surfaces, which can be
viewed as an analogy of the motivic global Torelli theorem on K3 surfaces (cf.
[31, Conjecture 0.3] and [23, Theorem 1]). The equivalences
$(i)\Leftrightarrow(iii)\Leftrightarrow(viii)$ can be viewed as a
generalization of [60, Theorem 1.2]. The Hodge-theoretic realization
$(i)\Leftrightarrow(vi)$ follows a similar strategy of [31, Theorem 0.1],
which makes use of Shioda’s period map and Cartan–Dieudonné decomposition of a
rational isometry. The equivalences
$(vi)\Leftrightarrow(vii)\Leftrightarrow(viii)$ follow from the Witt
cancellation theorem (see §5.3).

###  1.3. Shioda’s trick

The proof of Theorem 1.2.1 is concluded by a new ingredient so called rational
Shioda’s trick on abelian surfaces. The original Shioda’s trick in [58] plays
a key role in the proof of Shioda’s global Torelli theorem for abelian
surfaces, which links the weight-1 integral Hodge structure to the weight-2
integral Hodge structure of an abelian surface. We generalize it in the
following form.

######  Theorem 1.3.1 (Shioda’s trick, see §4).

Let $X$ and $Y$ be two complex abelian surfaces. Then for any admissible Hodge
isometry

$\psi\colon\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})$

we can find an isogeny $f\colon Y\to X$ of degree $d^{2}$ such that
$\psi=\frac{f^{*}}{d}$.

As an application, the generalized Shioda’s trick gives the algebraicity of
some cohomological cycles. For any integer $d$, one can consider a Hodge
similitude of degree $d$

$\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q}),$

called a Hodge isogeny of degree $d$. Due to the Hodge conjecture on products
of abelian surfaces, we know that every Hodge isogeny is algebraic. Our
generalized Shioda’s trick actually shows that it is induced by certain
isogenies. Similarly, we prove the $\ell$-adic and $p$-adic Shioda’s trick,
which gives a proof of Tate conjecture for isometries between the
$2^{nd}$-cohomology groups (as either Galois-modules or crystals) of abelian
surfaces over finitely generated fields. See Corollary 4.6.4 for more details.

###  1.4. Results in positive characteristic

The second part of this paper is to investigate the twisted derived Torelli
theorem over positive characteristic fields. Due to the absence of a
satisfactory global Torelli theorem, one can not follow the argument in
characteristic zero directly. Instead, we need some input from $p$-adic Hodge
theory. Our formulation is the following.

######  Theorem 1.4.1.

Let $X$ and $Y$ be two abelian surfaces over $k=\bar{k}$ with
$\operatorname{char}(k)=p>3$. Then the following statements are equivalent.

  1. (i′)

$X$ and $Y$ are prime-to-$p$ derived isogenous.

  2. (ii′)

$X$ and $Y$ are prime-to-$p$ principally isogenous.

Moreover, in case that $X$ is supersingular, then $Y$ is derived isogenous to
$X$ if and only if $Y$ is supersingular.

Here, we say a derived isogeny as (1.1.1) is prime-to-$p$ if its crystalline
realization is integral (see Definition 3.1.3 for details), which is a
condition somewhat technical. The main ingredients in the proof of Theorem
1.4.1 are the lifting-specialization technique, which works well for prime-
to-$p$ derived isogenies. Actually, our method shows that there is an
implication $(i^{\prime})\Rightarrow(ii^{\prime})$ for derived isogenies which
are not necessarily being prime-to-$p$ (see Theorem 6.5.1). Conversely, we
believe that the existence of quasi-liftable isogenies will imply the
existence of derived isogeny (see Conjecture 6.5.2). The only obstruction is
the existence of the specialization of non-prime-to-$p$ derived isogenies
between abelian surfaces. See Remark 6.3.3.

Another natural question is whether two abelian surfaces are derived isogenous
if and only if their associated Kummer surfaces are derived isogenous over
positive characteristic fields. Unfortunately, we can not fully prove the
equivalence. Instead, we provide a partial solution of this question. See
Theorem 6.5.3 for more details.

Similarly, one may ask whether such results also hold for K3 surfaces. Recall
that two K3 surfaces $S$ and $S^{\prime}$ over a finite field $\mathbb{F}_{q}$
are (geometrically) isogenous in the sense of [65] if there exists an
algebraic correspondence $\Gamma$ which induces an isometry of
$\operatorname{Gal}(\bar{\mathbb{F}}_{p}/k)$-modules

$\Gamma^{\ast}_{\ell}\colon\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(S_{\bar{\mathbb{F}}_{p}},\mathbb{Q}_{\ell})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(S^{\prime}_{\bar{\mathbb{F}}_{p}},\mathbb{Q}_{\ell}),$

for all $\ell\nmid p$ and an isometry of isocrystals

$\Gamma^{\ast}_{p}\colon\mathrm{H}^{2}_{\operatorname{crys}}(S_{k}/K)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(S^{\prime}_{k}/K),$

for some finite extension $k/\mathbb{F}_{q}$ and the fraction field $K$ of
$W=W(k)$. Then we say the isogeny is prime-to-$p$ if the isometry
$\Gamma^{\ast}_{p}$ is integral, i.e.,
$\Gamma^{\ast}_{p}\left(\mathrm{H}^{2}_{\operatorname{crys}}(S_{k}/W)\right)=\mathrm{H}^{2}_{\operatorname{crys}}(S^{\prime}_{k}/W)$.
Then we have a formulation of the twisted derived Torelli conjecture for K3
surfaces.

######  Conjecture 1.4.2.

For two K3 surfaces $S$ and $S^{\prime}$ over a finite field $k$ with
$\mathrm{char}(k)=p>0$, then the following are equivalent.

  1. (a)

There exists a prime-to-$p$ derived isogeny
$\operatorname{D}^{b}(S)\sim\operatorname{D}^{b}(Y)$.

  2. (b)

There exists a prime-to-$p$ isogeny between $S$ and $S^{\prime}$.

The implication $(a)\Rightarrow(b)$ is clear, while the converse remains open.
In the case of Kummer surfaces, our results provide some evidence of
Conjecture 1.4.2. We shall mention that recently Bragg and Yang have studied
the derived isogenies between K3 surfaces over positive characteristic fields
and they provided a weaker version of the statement in Conjecture 1.4.2 (cf.
[9, Theorem 1.2]).

### Organization of the paper.

We will start with two preliminary sections, in which we include some well-
known constructions and facts: In Section 2, we perform the computations of
the Brauer group of abelian surfaces via the Kummer construction. This allows
us to prove the lifting lemma for twisted abelian surfaces of finite height.
In Section 3, we collect the knowledge on derived isogenies between abelian
surfaces and their cohomological realizations, which include the motivic
realization, the $\mathbf{B}$-field theory, the twisted Mukai lattices, a
filtered Torelli theorem and its relation to the moduli space of twisted
sheaves.

In Section 4, we revise Shioda’s work and extend it to rational Hodge
isogenies. This is the key ingredient for proving Theorem 1.2.1. Furthermore,
after introducing the admissible $\ell$-adic and $p$-adic bases, we prove the
$\ell$-adic and $p$-adic Shioda’s trick for admissible isometries on abelian
surfaces. As an application, we prove the algebracity of these isometries on
abelian surfaces over finitely generated fields.

Section 5 and 6 are devoted to proving Theorem 1.2.1 and Theorem 1.4.1.
Theorem 1.2.1 is essentially Theorem 5.1.3 and Theorem 5.2.5. The proof of
Theorem 1.4.1 is much more subtle. We establish the lifting and the
specialization theorem for prime-to-$p$ derived isogeny. Then one can conclude
$(i^{\prime})\Leftrightarrow(ii^{\prime})$ from Theorem 1.2.1 for abelian
surfaces of finite heights. At the end of Section 6, we follow Bragg and
Lieblich’s twistor line argument in [7] to conclude the supersingular case of
Theorem 1.4.1.

### Acknowledgement

The authors are grateful to the useful comments by Ziquan Yang.

### Notations and Conventions

Throughout this paper, we will use the symbol $k$ to denote a field. If $k$ is
a perfect field and $\operatorname{char}{k}=p>0$, we denote $W\coloneqq W(k)$
for the ring of Witt vectors in $k$, which is equipped with a morphism
$\sigma\colon W\rightarrow W$ induced by the Frobenius map on $k$. If $k$ is
not perfect, we choose a Cohen ring $W\subset W(\bar{k})$ with $W/pW=k$,
inside the ring of Witt vectors in a fixed algebraic closure $\bar{k}$ of $k$.

Let $X$ be a smooth projective variety over $k$. We denote by
$\mathrm{H}^{\bullet}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Z}_{\ell})$
the $\ell$-adic étale cohomology group of $X_{\bar{k}}$. The
$\mathbb{Z}_{\ell}$-module
$\mathrm{H}^{\bullet}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Z}_{\ell})$
has been endowed with a canonical
$G_{k}=\operatorname{Gal}(\bar{k}/k)$-action. We use
$\mathrm{H}^{i}_{\operatorname{crys}}(X/W)$ to denote the $i$-th crystalline
cohomology group of $X$ over the $p$-adic base $W\twoheadrightarrow k$, which
is a $W$-module. It is endowed with a natural $\sigma$-linear map

$\varphi\colon\mathrm{H}^{i}_{\operatorname{crys}}(X/W)\rightarrow\mathrm{H}^{i}_{\operatorname{crys}}(X/W)$

induced from the absolute Frobenius morphism $F_{X}\colon X\to X$.

We denote by $\operatorname{D}^{b}(X)$ the bounded derived category of
coherent sheaves $X$. A derived equivalence means a $k$-linear exact
equivalence between triangulated categories in the form

$\Psi\colon\operatorname{D}^{b}(X)\xrightarrow{\sim}\operatorname{D}^{b}(Y).$

If $\Psi$ is of the form

$\Psi^{P}(E)=\mathbf{R}\\!q_{*}(p^{*}E\otimes\mathcal{P}),$

then we call it a Fourier–Mukai transform with a kernel
$P\in\operatorname{D}^{b}(X\times Y)$ and the projections $p\colon X\times
Y\to X$, $q\colon X\times Y\to Y$, and $X,Y$ are called a pair of
Fourier–Mukai partners.

When $X$ is an abelian variety over $k$, we denote $\widehat{X}$ for its dual
abelian variety and $X[p^{\infty}]$ for the associated $p$-divisible group.
There is a natural identification of its contravariant Dieudonné module with
its first crystalline cohomology:

$\mathbb{D}(X[p^{\infty}])\coloneqq
M(X[p^{\infty}]^{\vee})\cong\mathrm{H}^{1}_{\operatorname{crys}}(X/W),$

where $M(-)$ is the Dieudonné module functor on $p$-divisible groups defined
in [44, 17].

For any abelian group $G$ and an integer $n$, we denote $G[n]$ for the
subgroup of $n$-torsions in $G$ and $G\\{n\\}$ for the union of all $n$-power
torsions.

For a lattice $L$ in $\mathbb{Z}$ or $\mathbb{Q}$ and an integer $n$, we use
$L(n)$ for the lattice twisted by $n$, i.e., $L=L(n)$ as $\mathbb{Z}$ or
$\mathbb{Q}$-module, but

$\langle x,y\rangle_{L(n)}=n\langle x,y\rangle_{L}.$

The reader shall not confuse it with the Tate twist.

##  2\. Twisted abelian surface

In this section, we give some preliminary results in the theory of twisted
abelian surfaces, especially for those in positive characteristic. As most of
these results are well-known in characteristic zero, the readers who are only
interested in this case may skip this part.

We will frequently use the terminology “gerbe”, on which the readers may refer
[25] or [39] for more details.

###  2.1. Gerbes on abelian surfaces and associated Kummer surfaces

Let $X$ be an abelian surface over a field $k$ and let $\mathscr{X}\rightarrow
X$ be a $\mu_{n}$-gerbe over $X$. This corresponds to a pair $(X,\alpha)$ for
some $\alpha\in\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{n})$, where the
cohomology group is with respect to the flat topology. Since $\mu_{n}$ is
commutative, there is a bijection of sets

$\left\\{\text{$\mu_{n}$-gerbes on
$X$}\right\\}\\!/\simeq\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{n}),$

where $\simeq$ is the $\mu_{n}$-equivalence defined as in [25, IV.3.1.1]. We
may write $\alpha=[\mathscr{X}]$. The Kummer exact sequence induces a
surjective map

$\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{n})\rightarrow\operatorname{Br}(X)[n]$
(2.1.1)

where the right-hand side is the _cohomological Brauer group_
$\operatorname{Br}(X)\coloneqq\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{G}_{m})$.
For any $\mu_{n}$ gerbe $\mathscr{X}$ on $X$, there is an associated
$\mathbb{G}_{m}$-gerbe on $X$ via (2.1.1), denoted by
$\mathscr{X}_{\mathbb{G}_{m}}$. Let $\mathscr{X}^{(m)}$ be the gerbe
corresponding to cohomological class
$m[\mathscr{X}]\in\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{n})$. If
$[\mathscr{X}_{\mathbb{G}_{m}}]=0$, then we will call $\mathscr{X}$ an
essentially-trivial $\mu_{n}$-gerbe.

If $k$ has characteristic $p\neq 2$, there is an associated Kummer surface
$\widetilde{X}$ constructed as follows:

${\widetilde{X}}$${X}$${\operatorname{Km}(X)}$${X/{\iota}}$$\scriptstyle{\widetilde{\sigma}}$$\scriptstyle{\pi}$$\scriptstyle{\sigma}$
(2.1.2)

where

  * •

$\iota$ is the involution of $X$;

  * •

$\sigma$ is the crepant resolution of quotient singularities;

  * •

$\widetilde{\sigma}$ is the blow-up of $X$ along the closed subscheme
$X[2]\subset X$. Its birational inverse is denoted by
$\widetilde{\sigma}^{-1}$.

Let $E\subset\widetilde{X}$ be the exceptional locus of $\widetilde{\sigma}$.
Then we have a composition of the sequence of morphisms

$(\widetilde{\sigma}^{-1})^{*}:\operatorname{Br}(\widetilde{X})\to\operatorname{Br}(\widetilde{X}\setminus
E)\cong\operatorname{Br}(X\setminus X[2])\cong\operatorname{Br}(X).$

Here, the last isomorphism
$\operatorname{Br}(X)\to\operatorname{Br}(X\setminus X[2])$ is due to
Grothendieck’s purity theorem (cf. [27, 63]).

######  Proposition 2.1.1.

When $k=\bar{k}$ and $\operatorname{char}(k)\neq 2$, the
$(\widetilde{\sigma}^{-1})^{*}\pi^{*}$ induces an isomorphism between
cohomological Brauer groups

$\Theta\colon\operatorname{Br}(\operatorname{Km}(X))\to\operatorname{Br}(X).$
(2.1.3)

In particular, when $X$ is supersingular over $\bar{k}$, then
$\operatorname{Br}(X)$ is isomorphic to the additive group $\bar{k}$.

###### Proof.

For torsions of (2.1.3) whose orders are coprime to $p$, the proof is
essentially the same as [59, Proposition 1.3] by the Hochschild–Serre spectral
sequence and the fact that $\mathrm{H}^{2}(\mathbb{Z}/2\mathbb{Z},k^{*})=0$
(cf. [64, Proposition 6.1.10]) as $\operatorname{char}(k)>2$. See also [60,
Lemma 4.1] for the case $k=\mathbb{C}$. For $p$-primary torsion part, we have

$\operatorname{Br}(\operatorname{Km}(X))\\{p\\}\cong\operatorname{Br}(X)^{\iota}\\{p\\}$

from the Hochschild–Serre spectral sequence, where
$\operatorname{Br}(X)^{\iota}$ is the $\iota$-invariant subgroup. Hence it
suffices to prove that $\iota$ acts trivially on $\operatorname{Br}(X)$.

In fact, $\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{p})$ can be
$\iota$-equivariantly embedded to $\mathrm{H}^{2}_{\operatorname{dR}}(X/k)$ by
de Rham–Witt theory (cf. [50, Proposition 1.2]). The action of $\iota$ on
$\mathrm{H}^{2}_{\operatorname{dR}}(X/k)=\wedge^{2}\mathrm{H}^{1}_{\operatorname{dR}}(X/k)$
is the identity, as its action on $\mathrm{H}^{1}_{\operatorname{dR}}(X/k)$ is
given by $x\mapsto-x$. Thus the involution on
$\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{p})$ is trivial. Then by the exact
sequence

$0\to{\rm
NS}(X)\otimes\mathbb{Z}/p\to\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{p})\to\operatorname{Br}(X)[p]\to
0,$

we can deduce that $\operatorname{Br}(X)[p]$ is invariant under the
involution. Furthermore, for $p^{n}$-torsions with $n\geq 2$, we can proceed
by induction on $n$. Assume that all elements in $\operatorname{Br}(X)[p^{d}]$
are $\iota$-invariant if $1\leq d<n$. By abuse of notation, we still use
$\iota$ to denote the induced map
$\operatorname{Br}(X)\to\operatorname{Br}(X)$. For
$\alpha\in\operatorname{Br}(X)[p^{n}]$,
$p\alpha\in\operatorname{Br}(X)[p^{n-1}]$ is $\iota$-invariant. This gives

$p\alpha=\iota(p\alpha)=p\iota(\alpha),$

which implies $\alpha-\iota(\alpha)\in\operatorname{Br}(X)[p]$. Applying
$\iota$ on $\alpha-\iota(\alpha)$, we can obtain

$\alpha-\iota(\alpha)=\iota(\alpha)-\alpha.$

It implies $\alpha-\iota(\alpha)$ is also a $2$-torsion element. Since $p$ is
coprime to $2$, we can conclude that $\alpha=\iota(\alpha)$.

If $X$ is supersingular, then $\operatorname{Km}(X)$ is also supersingular. We
have already known that the Brauer group of a supersingular K3 surface is
isomorphic to $k$ by [2]. Thus $\operatorname{Br}(X)\cong k$. ∎

######  Remark 2.1.2.

In the case $A$ being supersingular, the method of [2] can not be directly
applied to show that $\operatorname{Br}(X)=k$ as
$\mathrm{H}^{1}_{\operatorname{fl}}(X,\mu_{p^{n}})$ is not trivial in general
for an abelian surface $X$.

######  Remark 2.1.3.

For abelian surfaces over a non-algebraically closed field or more general
ring, we still have the canonical map (2.1.3), but it is not necessarily an
isomorphism.

######  Remark 2.1.4.

For a cohomology theory $\mathrm{H}^{\bullet}(-)$ with nice properties e.g.
satisfying the blow-up formula, we have a canonical decomposition

$\mathrm{H}^{2}(\operatorname{Km}(X))\cong\mathrm{H}^{2}(X)\oplus\pi_{*}\Sigma,$

where $\Sigma$ is the summand in $\mathrm{H}^{2}(\widetilde{X})$ generated by
the exceptional divisors of $\widetilde{\sigma}$.

###  2.2. A lifting lemma

In [5], Bragg has shown that a twisted K3 surface can be lifted to
characteristic $0$. Though his method can not be directly applied to twisted
abelian surfaces, one can still obtain a lifting result for twisted abelian
surfaces via using the Kummer construction. The following result will be
frequently used in this paper.

######  Lemma 2.2.1.

Let $\mathscr{X}\to X$ be a $\mathbb{G}_{m}$-gerbe on an abelian surface $X$
over $k=\bar{k}$. Suppose $\mathrm{char}(k)>2$ and $X$ has finite height. Then
there exists a lifting $\mathfrak{X}\to\mathcal{X}$ of $\mathscr{X}\to X$ over
some discrete valuation ring $W^{\prime}$ whose residue field is $k$ such that
the specialization map

${\rm NS}(\mathcal{X}_{K^{\prime}})\to{\rm NS}(X)$

on Néron-Severi groups is an isomorphism. Here, $K^{\prime}$ is the fraction
field of $W^{\prime}$ and $\mathcal{X}_{K^{\prime}}$ is the generic fiber of
$\mathcal{X}\to\operatorname{Spec}W^{\prime}$.

###### Proof.

The existence of such lifting is ensured by [5, Theorem 7.10], [38, Lemma 3.9]
and Proposition 2.1.1. Roughly speaking, let
$\mathscr{S}\to\operatorname{Km}(X)$ be the associated twisted Kummer surface
via the isomorphism (2.1.3) in Proposition 2.1.1. Then [5, Theorem 7.10]
asserts that there exists a lifting $\mathfrak{S}\to\mathcal{S}$ of
$\mathscr{S}\to\operatorname{Km}(X)$ such that the specialization map of
Néron-Severi groups is an isomorphism

${\rm NS}(\mathcal{X}_{K^{\prime}})\xrightarrow{\sim}{\rm NS}(X).$ (2.2.1)

Then [38, Lemma 3.9] says that one can find a lifting $\mathcal{X}/W^{\prime}$
of $X$ such that $\operatorname{Km}(\mathcal{X})\cong\mathcal{S}$ over
$W^{\prime}$. According to Remark 2.1.3, there is a canonical map

$\Theta:\operatorname{Br}(\operatorname{Km}(\mathcal{X}))\to\operatorname{Br}(\mathcal{X})$

as in (2.1.3). Consider the image
$\Theta([\mathfrak{S}])\in\operatorname{Br}(\mathcal{X})$, one can take
$\mathfrak{X}\to\mathcal{X}$ to be the associated twisted abelian surface.
Then $\mathfrak{X}\to\mathcal{X}$ will be a lifting of $\mathscr{X}\to X$ as
the restriction of the Brauer class $[\mathfrak{X}]$ to $X$ is
$[\mathscr{X}]$. ∎

##  3\. Cohomological realizations of derived isogeny

In this section, we briefly recall the action of derived isogenies on the
cohomology groups of abelian surfaces and define the prime-to-$\ell$ derived
isogenies. This action has the following two forms

  1. (1)

the motivic realization, which induces rational isomorphisms on the cohomology
groups;

  2. (2)

the realization on the integral twisted Mukai lattices.

The story over $\mathbb{C}$ comes back to [66, 33, 31]. Over a general field,
we refer [41] for the non-twisted Mukai realization, [40, 6] for the
definition of twisted Mukai lattices, and [30, 23] for the motivic
realization.

Following the works in [41, 28], we extend the filtered Torelli theorem to
twisted abelian surfaces over an algebraically closed field $k$ with
$\operatorname{char}(k)\neq 2$. As a corollary, we show that any Fourier–Mukai
partner of a twisted abelian surface is isomorphic to a moduli space of stable
twisted sheaves on itself or its dual (cf. Theorem 3.4.5).

###  3.1. Motivic realization of derived isogeny on cohomology groups

In [30, 31], Huybrechts shows that (twisted) derived equivalent K3 surfaces
over a field $k$ have isomorphic Chow motives, which also holds for general
algebraic surfaces over $k$ (as remarked in §2.4 of loc.cit.). Moreover, Lie
and Vial proved that any twisted derived equivalence induces an isomorphism
between the second component of Chows motives by a weight-argument (cf. [23,
§1.2]). In this part, we record their results for the convenience of the
reader. We will focus on abelian surfaces over $k$ as a typical type of
examples.

For any abelian surface $X$ over a field $k$, one may consider idempotent
correspondences $\pi^{2}_{\operatorname{alg},X}$ and
$\pi^{2}_{\operatorname{tr},X}$ in $\operatorname{CH}^{2}(X\times
X)_{\mathbb{Q}}$ defined as

$\pi_{\operatorname{alg},X}^{2}\coloneqq\sum_{i=1}^{\rho}\frac{1}{\deg(E_{i}\cdot
E_{i})}E_{i}\times
E_{i},\quad\pi^{2}_{\operatorname{tr},X}=\pi^{2}_{X}-\pi^{2}_{\operatorname{alg},X},$

where $\pi^{2}_{X}$ is the idempotent correspondence given by the Chow–Künneth
decomposition (1.2.1) and $E_{i}$ are non-isotropic divisors generating the
Néron–Severi group ${\rm NS}(X_{k^{s}})$ as a orthogonal basis. Consider the
decomposition of $\mathfrak{h}^{2}(X)$:

$\mathfrak{h}^{2}(X)=\mathfrak{h}_{\operatorname{alg}}^{2}(X)\oplus\mathfrak{h}_{\operatorname{tr}}^{2}(X)$

given by $\pi^{2}_{\operatorname{alg},X}$ and $\pi^{2}_{\operatorname{tr},X}$.
It is not hard to see $\mathfrak{h}^{2}_{\operatorname{alg}}(X)$ is a Tate
motive after base change to the separable closure $k^{s}$, whose Chow
realization is

$\operatorname{CH}_{\mathbb{Q}}^{*}(\mathfrak{h}^{2}_{\operatorname{alg}}(X_{k^{s}}))\cong{\rm
NS}(X_{k^{s}})_{\mathbb{Q}}.$

According to the main result in [14], any derived equivalence
$\operatorname{D}^{b}(X,\alpha)\xrightarrow{\sim}\operatorname{D}^{b}(Y,\beta)$
can be uniquely (up to isomorphism) written as a Fourier-Mukai transform with
kernel $\mathcal{P}\in\operatorname{D}^{b}(X\times
Y,\alpha^{-1}\boxtimes\beta)$

$\Phi^{\mathcal{P}}\colon\operatorname{D}^{b}(X,\alpha)\xrightarrow{\sim}\operatorname{D}^{b}(Y,\beta).$

Consider the cycle class

$[\Gamma_{\operatorname{tr}}]=v_{2}(\mathcal{P})\in\operatorname{CH}^{2}(\mathscr{X}\times\mathscr{Y})_{\mathbb{Q}}\cong\operatorname{CH}^{2}(X\times
Y)_{\mathbb{Q}},$

where $v_{2}(\mathcal{P})$ is the dimension two component of the Mukai vector
of $\mathcal{P}$. It will induce an isomorphism of motives by a weight
argument (cf. [23, §§1.2.3])

$[\Gamma_{\operatorname{tr}}]_{2}\coloneqq\pi^{2}_{\operatorname{tr},Y}\circ[\Gamma_{\operatorname{tr}}]\circ\pi^{2}_{\operatorname{tr},X}\colon\mathfrak{h}^{2}_{\operatorname{tr}}(X)\xrightarrow{\sim}\mathfrak{h}^{2}_{\operatorname{tr}}(Y).$

Since twisted derived equivalent algebraic surfaces have same Picard number
(over $k$), one can choose a invertible correspondence

$[\Gamma_{\operatorname{alg}}]\colon\mathfrak{h}^{2}_{\operatorname{alg}}(X)\xrightarrow{\sim}\mathfrak{h}^{2}_{\operatorname{alg}}(Y),$

whose inverse is given by its transpose (see [23, §3.1] for more details).
This gives an isomorphism

$[\Gamma]\coloneqq[\Gamma_{\operatorname{tr}}]_{2}+[\Gamma_{\operatorname{alg}}]\colon\mathfrak{h}^{2}(X)\xrightarrow{\sim}\mathfrak{h}^{2}(Y).$

Any cohomologlical realization of such isomorphism clearly preserves the
Poincaré pairing by the construction. Therefore, by taking the corresponding
cohomological realization, we obtain

######  Proposition 3.1.1.

Assume $\mathrm{char}(k)=p\neq 2$. Let $\ell$ be a prime not equal to $p$. If
$X$ and $Y$ are twisted derived equivalent over $k$, then $[\Gamma]$ will
induce a $\mathrm{Gal}(\bar{k}/k)$-equivariant isometry

$\varphi_{\ell}\colon\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Q}_{\ell})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(Y_{\bar{k}},\mathbb{Q}_{\ell}).$
(3.1.1)

Suppose $k$ is perfect, it will induce an isometry between $F$-isocrystals

$\varphi_{K}\colon\mathrm{H}^{2}_{\operatorname{crys}}(X/K)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(Y/K).$
(3.1.2)

######  Remark 3.1.2.

The weight-argument in [13, §§1.2.3] actually provides an isomorphism

$\mathfrak{h}(X)\xrightarrow{\sim}\mathfrak{h}(Y),$

which preserves the even-degree parts

$\mathfrak{h}^{even}(-)\coloneqq\bigoplus^{2}_{k=0}\mathfrak{h}^{2k}(-)\cong\bigoplus^{2}_{k=0}\bigwedge^{2k}\mathfrak{h}^{1}(-).$

The cohomological realizations in Proposition 3.1.1 are not integral in
general. We can introduce the prime-to-$\ell$ derived isogeny via the integral
cohomological realizations, which will be used in the rest of the paper.

######  Definition 3.1.3.

Let $\ell$ be a prime and $\operatorname{char}(k)=p$. When $\ell\neq p$, a
derived isogeny $\operatorname{D}^{b}(X)\sim\operatorname{D}^{b}(Y)$ given by

${\operatorname{D}^{b}(X,\alpha)}$${\operatorname{D}^{b}(X_{1},\beta_{1})}$${\operatorname{D}^{b}(X_{1},\alpha_{2})}$${\operatorname{D}^{b}(X_{2},\beta_{2})}$${\vdots}$${\operatorname{D}^{b}(X_{n},\alpha_{n+1})}$${\operatorname{D}^{b}(Y,\beta_{n})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

is called prime-to-$\ell$ if each cohomological realization in the zig-zag
sequence

$\varphi_{\ell}^{i}\colon\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X_{i-1,\bar{k}},\mathbb{Q}_{\ell})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X_{i,\bar{k}},\mathbb{Q}_{\ell})$

is integral, i.e.
$\varphi_{\ell}\left(\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Z}_{\ell})\right)=\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(Y_{\bar{k}},\mathbb{Z}_{\ell})$.
In the case $\ell=p$, it is called prime-to-$p$ if each
$\varphi_{p}^{i}\colon\mathrm{H}^{2}_{\operatorname{crys}}(X_{i-1}/K)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(X_{i}/K)$
is integral.

###  3.2. Mukai lattices and $\mathbf{B}$-fields

At the beginning, we shall remark that we are able to transfer many
cohomological statements for twisted K3 surfaces to the case of twisted
abelian surfaces via the the Kummer construction thanks to Proposition 2.1.1
and 2.1.4. For this reason, we will omit many technical details which are
well-known in the case of K3 surfaces in the following discussions.

If $X$ is a complex abelian surface, the _Mukai lattice_ is defined as

$\widetilde{\mathrm{H}}(X,\mathbb{Z})\coloneqq\mathrm{H}^{0}(X,\mathbb{Z}(-1))\oplus\mathrm{H}^{2}(X,\mathbb{Z})\oplus\mathrm{H}^{4}(X,\mathbb{Z}(1))$

with the Mukai pairing

$\langle(r,c,\chi),(r^{\prime},c^{\prime},\chi^{\prime})\rangle\coloneqq
cc^{\prime}-r\chi^{\prime}-r^{\prime}\chi,$ (3.2.1)

and a pure $\mathbb{Z}$-Hodge structure of weight $2$.

For general algebraically closed field $k$ and an abelian surface $X$ over
$k$, we also have the following notion of Mukai lattices [41, §2].

  * •

Let $\widetilde{N}(X)$ be the _extended Néron–Severi lattice_ defined as

$\widetilde{N}(X)\coloneqq\mathbb{Z}\oplus{\rm NS}(X)\oplus\mathbb{Z},$

with Mukai pairing

$\langle(r_{1},c_{1},\chi_{1}),(r_{2},c_{2},\chi_{2})\rangle=c_{1}c_{2}-r_{1}\chi_{2}-r_{2}\chi_{1}.$

The Chow realization $\operatorname{CH}^{*}_{\mathbb{Q}}(-)$ of

$\mathfrak{h}^{0}(X)\oplus\mathfrak{h}^{2}_{\operatorname{alg}}(X)\oplus\mathfrak{h}^{4}(X)$

can be identified with $\widetilde{N}(X)_{\mathbb{Q}}$.

  * •

if $\operatorname{char}(k)\geq 0$, then the $\ell$-adic Mukai lattice is
defined on the even degrees of integral $\ell$-adic cohomology of $X$ for
$\ell$ coprime to $\operatorname{char}(k)$

$\mathrm{H}^{0}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(-1))\oplus\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell})\oplus\mathrm{H}^{4}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1)),$

with Mukai pairing defined in a similar formula as (3.2.1) denoted by
$\widetilde{\mathrm{H}}(X,\mathbb{Z}_{\ell})$; or

  * •

if $\operatorname{char}(k)=p>0$, then the $p$-adic Mukai lattice
$\widetilde{\mathrm{H}}(X,W)$ is defined on the even degrees of crystalline
cohomology of $X$ with coefficients in $W(k)$

$\mathrm{H}^{0}_{\operatorname{crys}}(X/W(k))(-1)\oplus\mathrm{H}^{2}_{\operatorname{crys}}(X/W(k))\oplus\mathrm{H}^{4}_{\operatorname{crys}}(X/W(k))(1),$

where the twist $(i)$ is given by changing the Frobenius $F\mapsto p^{-i}F$,
and the Mukai pairing is given similarly in the formula (3.2.1).

### Hodge $\mathbf{B}$-field

For any $B\in\mathrm{H}^{2}(X,\mathbb{Q})$, we define the twisted Mukai
lattice as

$\widetilde{\mathrm{H}}(X,\mathbb{Z};B)\coloneqq\exp(B)\cdot\widetilde{\mathrm{H}}(X,\mathbb{Z})\subset\widetilde{\mathrm{H}}(X,\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Q},$

which is naturally a lattice in $\widetilde{\mathrm{H}}(X,\mathbb{Z})$, and is
equipped with a induced pure Hodge structure of weight 2 from
$\widetilde{\mathrm{H}}(X,\mathbb{Q})$ (cf. [33, Definition 2.3]) i.e.,

$\widetilde{\mathrm{H}}^{0,2}(X;B)=\exp(B)\widetilde{\mathrm{H}}^{0,2}(X).$

The (extended) twisted Néron–Severi lattice is defined to be ${\rm
NS}(X;B)\coloneqq\widetilde{\mathrm{H}}^{1,1}(X,\mathbb{Z};B)$.

For such $B$, we can associate a Brauer class $\alpha_{B}=\exp(B^{0,2})$ via
the exponential sequence

$\mathrm{H}^{2}(X,\mathbb{Z})\to\mathrm{H}^{2}(X,\mathcal{O}_{X})\xrightarrow{\exp}\mathrm{H}^{2}(X,\mathcal{O}^{*}_{X})=\operatorname{Br}(X).$

Conversely, given $\alpha\in\operatorname{Br}(X)$, one can find a lift $B$ of
$\alpha$ in $\mathrm{H}^{2}(X,\mathcal{O}_{X})$ because $\operatorname{Br}(X)$
is torsion and $\mathrm{H}^{3}(X,\mathbb{Z})$ is torsion-free. The exponential
sequence implies $nB\in\mathrm{H}^{2}(X,\mathbb{Z})$ for the integer $n$ such
that $\alpha^{n}=1$, and so we have $B\in\mathrm{H}^{2}(X,\mathbb{Q})$. Any
such $B$ is called a $\mathbf{B}$-field lift of $\alpha$. It is clear that a
different choice of such lift $B^{\prime}$ satisfies
$B-B^{\prime}\in\mathrm{H}^{2}(X,\mathbb{Z})$ by the exponential sequence, and
thus there is a Hodge isometry

$\exp(B-B^{\prime})\colon\widetilde{\mathrm{H}}(X,\mathbb{Z};B^{\prime})\xrightarrow{\sim}\widetilde{\mathrm{H}}(X,\mathbb{Z};B).$

This implies that for any Brauer class $\alpha\in\operatorname{Br}(X)$, the
twisted Mukai lattice $\widetilde{\mathrm{H}}(X,\mathbb{Z};B)$ and the twisted
Néron–Severi lattice $\widetilde{N}(X;B)$ is independent of the choice of
$\mathbf{B}$-field lift $B$ up to isometry. Thus for any
$\mathbb{G}_{m}$-gerbe $\mathscr{X}\to X$ over a complex abelian surface, we
also denote $\widetilde{N}(\mathscr{X})$ for the twisted Néron–Severi lattice.

As shown in [33], for any twisted derived equivalence
$\Phi^{\mathcal{P}}\colon\operatorname{D}^{b}(X,\alpha)\xrightarrow{\sim}\operatorname{D}^{b}(Y,\beta)$,
we can associated it with a Hodge isometry

$\varphi=\varphi_{B,B^{\prime}}\colon\widetilde{\mathrm{H}}(X,\mathbb{Z};B)\xrightarrow{\sim}\widetilde{\mathrm{H}}(Y,\mathbb{Z};B^{\prime})$
(3.2.2)

for suitable $\mathbf{B}$-field lifts $B,B^{\prime}$ of $\alpha$ and $\beta$
respectively.

###  $\ell$-adic and crystalline $\mathbf{B}$-field

For the sake of completeness, we will briefly recall the following generalized
notions of B-fields in both $\ell$-adic cohomology (cf. [40, §3.2]) and
crystalline cohomology (cf. [6, §3]) as an analogue to that in Betti
cohomology. We refer [9, §2] for the full consideration of both $\ell$-adic
and $p$-adic case, which is for K3 surfaces, but also works for abelian
surfaces. The readers who are only interested on our main results may skip
this part as we only use these generalized $\mathbf{B}$-fields in the next
subsection and in the supersingular twisted derived Torelli theorem in
§§6.6.1.

For a prime $\ell\neq p$ and $n\in\mathbb{N}$, the Kummer sequence of étale
sheaves

$1\to\mu_{\ell^{n}}\to\mathbb{G}_{m}\xrightarrow{(\cdot)^{n}}\mathbb{G}_{m}\to
1,$ (3.2.3)

induces a long exact sequence

$\cdots{\rm Pic}(X)\xrightarrow{\cdot l^{n}}{\rm
Pic}{X}\to\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mu_{\ell^{n}})\rightarrow\operatorname{Br}(X)[\ell^{n}]\to
0.$

Taking the inverse limit $\varprojlim_{n}$, we get a map

$\pi_{\ell}\colon\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1))=\varprojlim_{n}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mu_{\ell^{n}})\to\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mu_{\ell^{n}})\twoheadrightarrow\operatorname{Br}(X)[\ell^{n}].$

######  Lemma 3.2.1.

The map $\pi_{\ell}$ is surjective.

###### Proof.

We have a short exact sequence (cf. [45, Chap.V, Lemma 1.11])

$0\to\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1))/\ell^{n}\to\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mu_{\ell^{n}})\to\mathrm{H}^{3}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1))[\ell^{n}]\to
0.$

As $\mathrm{H}^{3}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1))$ is
torsion-free for any abelian surface $X$, we have an isomorphism

$\mathrm{H}_{\operatorname{{\acute{e}t}}}^{2}(X,\mathbb{Z}_{\ell}(1))/\ell^{n}\cong\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mu_{\ell^{n}}).$

Therefore, the reduction morphism
$\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1))\to\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mu_{\ell^{n}})$
can be identified with

$\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1))\twoheadrightarrow\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell}(1))/\ell^{n},$

which is surjective. The assertion then follows from it. ∎

For any $\alpha\in\operatorname{Br}(X)[\ell^{n}]$ such that $\ell\neq p$, let
$B_{\ell}(\alpha)\coloneqq\pi^{-1}_{\ell}(\alpha)$, which is non-empty by
Lemma 3.2.1.

For Brauer class $\alpha\in\operatorname{Br}(X)[p^{n}]$, we need the following
commutative diagram via the de Rham-Witt theory (cf. [35, I.3.2, II.5.1,
Théorème 5.14])

${0}$${\mathrm{H}^{2}(X,\mathbb{Z}_{p}(1))}$${\mathrm{H}_{\operatorname{crys}}^{2}(X/W)}$${\mathrm{H}_{\operatorname{crys}}^{2}(X/W)}$${\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{p^{n}})}$${\mathrm{H}^{2}_{\operatorname{crys}}(X/W_{n})}$$\scriptstyle{p_{n}\coloneqq(\otimes
W_{n})}$$\scriptstyle{p-F}$$\scriptstyle{d\log}$ (3.2.4)

where
$\mathrm{H}^{2}(X,\mathbb{Z}_{p}(1))\coloneqq\varprojlim_{n}\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{p^{n}})$.
The $d\log$ map is known to be injective by flat duality (cf. [50, Proposition
1.2]). Since the crystalline cohomology groups of an abelian surface are
torsion-free, the mod $p^{n}$ reduction map $p_{n}$ is surjective. Consider
the canonical surjective map

$\pi_{p}\colon\mathrm{H}^{2}_{\operatorname{fl}}(X,\mu_{p^{n}})\twoheadrightarrow\operatorname{Br}(X)[p^{n}],$

induced by the Kummer sequence. We set

$B_{p}(\alpha)\coloneqq\left\\{b\in\mathrm{H}^{2}_{\operatorname{crys}}(X/W)|p_{n}(b)=d\log(t)\text{
such that }\pi_{p}(t)=\alpha\right\\}.$

Following [9, Definition 2.16, 2.17], we can introduce the (mixed) $B$-fields
for twisted abelian surfaces.

######  Definition 3.2.2.

Let $\ell$ be a prime and let $\alpha\in\operatorname{Br}(X)[\ell^{n}]$ be a
Brauer class of $X$ of order $\ell^{n}$.

  * •

If $\ell\neq p$, an $\ell$-adic B-field lift of $\alpha$ on $X$ is an element
$B=\frac{b}{\ell^{n}}\in\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Q}_{\ell})$
for some
$b\in\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mathbb{Z}_{\ell})$ such
that $b\in B_{\ell}(\alpha)$.

  * •

If $\ell=p$, a crystalline B-field lift of $\alpha$ is an element
$B=\frac{b}{p^{n}}\in\mathrm{H}^{2}_{\operatorname{crys}}(X/W)[\frac{1}{p}]$
with $b\in\mathrm{H}^{2}_{\operatorname{crys}}(X/W)$ such that $b\in
B_{p}(\alpha)$ .

More generally, for any $\alpha\in\operatorname{Br}(X)$, a mixed
$\mathbf{B}$-field lift of $\alpha$ is a set $B=\\{B_{\ell}\\}\cup\\{B_{p}\\}$
consisting of a choice of an $\ell$-adic $\mathbf{B}$-field lift $B_{\ell}$ of
$\alpha$ for each $\ell\neq p$ and a crystalline $\mathbf{B}$-field lift
$B_{p}$ of $\alpha$.

######  Remark 3.2.3.

Not all elements in $\mathrm{H}^{2}_{\operatorname{crys}}(X/W)[\frac{1}{p}]$
are crystalline B-fields since the map $d\log$ is not surjective. From the
first row in the diagram (3.2.4), we can see
$B\in\mathrm{H}^{2}_{\operatorname{crys}}(X/W)[\frac{1}{p}]$ is a B-field lift
of some Brauer class if and only if $F(B)=pB$.

For an $\ell$-adic or crystalline $B$-field $B=\frac{b}{m}$, let
$\exp(B)=1+B+\frac{B^{2}}{2}$. We define the twisted Mukai lattice as

$\widetilde{\mathrm{H}}(X,B)=\begin{cases}\exp(B)\widetilde{\mathrm{H}}(X,\mathbb{Z}_{\ell})&\text{if}~{}p\nmid
m\\\ ~{}\\\ \exp(B)\widetilde{\mathrm{H}}(X,W)&\text{if}~{}m=p^{n}\end{cases}$
(3.2.5)

under the Mukai pairing (3.2.1). Moreover, for crystalline $\mathbf{B}$-field
$B$, $\widetilde{\mathrm{H}}(X,B)$ is a $W$-lattice in
$\widetilde{\mathrm{H}}(X,K)$ stable under the Frobenius action. Sometimes, we
denote $\widetilde{\mathrm{H}}(\mathscr{X},\mathbb{Z}_{\ell})$ and
$\widetilde{\mathrm{H}}(\mathscr{X},W)$ for the twisted Mukai lattices if we
want to emphasis the coefficient other than the choice of the
$\mathbf{B}$-field lift.

Now let $\mathscr{X}\to X$ be a $\mu_{n}$-gerbe over $X$ whose associated
Brauer class is $\alpha$. The category $\mathtt{Coh}(\mathscr{X})$ of
_$\alpha$ -twisted coherent sheaves_ consists of $1$-fold
$\mathscr{X}$-twisted coherent sheaves in the sense of Lieblich (cf. [39]),
which is proven to be a Grothendieck category. Let
$\operatorname{D}^{b}(\mathscr{X})$ be the bounded derived category of
$\mathtt{Coh}(\mathscr{X})$. Consider the Grothendieck group
$\mathrm{K}_{0}(\mathscr{X})$ of $\mathtt{Coh}(\mathscr{X})$. There is a
_twisted Chern character_ map

$\operatorname{ch}_{B}\colon\mathrm{K}_{0}(\mathscr{X})\to\widetilde{\mathrm{H}}(X,B),$

see [40, §3.3] and [6, Appendix A3] for $\ell$-adic and crystalline cases
respectively. The twisted Chern character $\operatorname{ch}_{B}$ factors
through the rational extended Néron-Severi lattice
$\widetilde{N}(X)_{\mathbb{Q}}$:

${\mathrm{K}_{0}(\mathscr{X})}$${\widetilde{\mathrm{H}}(X,B)}$${\widetilde{N}(X)_{\mathbb{Q}},}$$\scriptstyle{\operatorname{ch}_{B}}$$\scriptstyle{\operatorname{ch}_{\mathscr{X}}}$$\scriptstyle{\exp(B)\operatorname{cl}_{\mathrm{H}}}$

where $\operatorname{cl}_{\mathrm{H}}$ is the cycle class map. The image of
$\mathrm{K}_{0}(\mathscr{X})$ in $\widetilde{N}(X)_{\mathbb{Q}}$ under
$\operatorname{ch}_{B}$ is denoted by $\widetilde{\mathrm{N}}(\mathscr{X})$.
For any $\mathscr{X}$-twisted sheaf $\mathcal{E}$ on $X$, the Mukai vector
$v_{B}(\mathcal{E})$ is defined to be

$\operatorname{ch}_{B}([\mathcal{E}])\sqrt{\operatorname{Td}(X)}\in\widetilde{\mathrm{H}}(X,B).$

Since the Todd class $\operatorname{Td}(X)$ is trivial when $X$ is an abelian
surface,
$v_{B}(\mathcal{E})=\operatorname{ch}_{B}([\mathcal{E}])\in\widetilde{\mathrm{H}}(X,B)$.

For any Fourier–Mukai transform
$\Phi^{\mathcal{P}}\colon\operatorname{D}^{b}(\mathscr{X})\to\operatorname{D}^{b}(\mathscr{Y})$,
[9, Thereom 3.6] shows that there is an isometry of Mukai lattices for
suitable (mixed) $\mathbf{B}$-field lifts $B$ and $B^{\prime}$

$\varphi_{B,B^{\prime}}\colon\widetilde{\mathrm{H}}(X,B)\to\widetilde{\mathrm{H}}(Y,B).$
(3.2.6)

###  3.3. A filtered Torelli Theorem

In [41, 42], Lieblich and Olsson introduce the notion of filtered derived
equivalence and show that filtered derived equivalent K3 surfaces are
isomorphic. In this part, we will give an analogue for (twisted) abelian
surfaces, whose proof is much more simple than the K3 surface case as the
bounded derived category of a (twisted) abelian surface is a generic K3
category in the sense of [32].

The rational numerical Chow ring
$\operatorname{CH}^{*}_{\operatorname{num}}(X)_{\mathbb{Q}}$ is equipped with
a codimension filtration

$\operatorname{Fil}^{i}\operatorname{CH}^{*}_{\operatorname{num}}(X)_{\mathbb{Q}}\coloneqq\bigoplus_{i\geq
k}\operatorname{CH}^{k}_{\operatorname{num}}(X)_{\mathbb{Q}}.$

As $X$ is a surface, we have a natural identification
$\widetilde{N}(X)_{\mathbb{Q}}\cong\operatorname{CH}^{*}_{\operatorname{num}}(X)_{\mathbb{Q}}$,
which gives a filtration of the rational extended Néron-Severi lattice. Let
$\Phi^{\mathcal{P}}$ be a Fourier-Mukai transform with respect to
$\mathcal{P}\in\operatorname{D}^{b}(X\times Y)$. The equivalence
$\Phi^{\mathcal{P}}$ is called _filtered_ if the induced numerical Chow
realization $\Phi^{P}_{\operatorname{CH}}$ preserves the codimension
filtration. It is not hard to see that $\Phi^{\mathcal{P}}$ is filtered if and
only it sends the Mukai vector $(0,0,1)$ to $(0,0,1)$. A filtered twisted
Fourier-Mukai transform is defined in a same way since the twisted Chern
character $\operatorname{ch}_{\mathscr{X}}$ maps onto
$\widetilde{N}(\mathscr{X})\subset\widetilde{N}(X)_{\mathbb{Q}}$.

At the cohomological level, the codimension filtration on
$\widetilde{\mathrm{H}}(X)[\frac{1}{\ell}]$ (the prime $\ell$ depends on the
choice of $\ell$-adic or crystalline twisted Mukai lattice) is given by
$F^{i}=\oplus_{r\geq i}\mathrm{H}^{2r}(X)[\frac{1}{\ell}]$. Let $B$ be a
B-field lift of $[\mathscr{X}]$. The filtration on
$\widetilde{\mathrm{H}}(X,B)$ is defined by

$F^{i}\widetilde{\mathrm{H}}(X,B)=\widetilde{\mathrm{H}}(X,B)\cap
F^{i}\widetilde{\mathrm{H}}(X)[\frac{1}{\ell}].$

A direct computation shows that the graded pieces of $F^{\bullet}$ are

$\displaystyle{\rm
Gr}^{0}_{F}\widetilde{\mathrm{H}}(X,B)=\left\\{(r,rB,\frac{rB^{2}}{2})\Big{|}r\in\mathrm{H}^{0}(X)\right\\},$
(3.3.1) $\displaystyle{\rm
Gr}^{1}_{F}\widetilde{\mathrm{H}}(X,B)=\left\\{(0,c,c\cdot
B)|c\in\mathrm{H}^{2}(X)\right\\}\cong\mathrm{H}^{2}(X),$ $\displaystyle{\rm
Gr}_{F}^{2}\widetilde{\mathrm{H}}(X,B)=\left\\{(0,0,s)|s\in\mathrm{H}^{4}(X)\right\\}\cong\mathrm{H}^{4}(X)(1).$

######  Lemma 3.3.1.

A twisted Fourier-Mukai transform
$\Phi^{\mathcal{P}}:\mathrm{D}^{b}(\mathscr{X})\to\operatorname{D}^{b}(\mathscr{Y})$
is filtered if and only if its cohomological realization is filtered for
certain B-field lifts.

###### Proof.

It is clear that being filtered implies being cohomologically filtered. This
is because the map
$\exp(B)\cdot\operatorname{cl}_{H}\colon\widetilde{N}(X,\mathbb{Q})\to\widetilde{\mathrm{H}}(X,B)$
preserves the filtrations for any B-field lift $B$ of $[\mathscr{X}]$.

For the converse, just notice that $\Phi^{\mathcal{P}}$ is filtered if and
only if the induced map $\Phi^{\mathcal{P}}_{\operatorname{CH}}$ takes the
vector $(0,0,1)$ to $(0,0,1)$. As $\Phi^{\mathcal{P}}$ is cohomologically
filtered for $B$, we can see the cohomological realization of
$\Phi^{\mathcal{P}}$ preserves the graded piece ${\rm Gr}_{F}^{2}$ in (3.3.1).
This implies that $\Phi^{\mathcal{P}}_{\operatorname{CH}}$ takes $(0,0,1)$ to
$(0,0,1)$. ∎

######  Proposition 3.3.2 (filtered Torelli theorem for twisted abelian
surfaces).

Suppose $k=\bar{k}$. Let $\mathscr{X}\to X$ and $\mathscr{Y}\to Y$ be
$\mu_{n}$-gerbes on abelian surfaces. Then following statements are equivalent

  1. (1)

There is an isomorphism between associated $\mathbb{G}_{m}$-gerbes
$\mathscr{X}_{\mathbb{G}_{m}}$ and $\mathscr{Y}_{\mathbb{G}_{m}}$.

  2. (2)

There is a filtered Fourier-Mukai transform $\Phi^{\mathcal{P}}$ from
$\mathscr{X}$ to $\mathscr{Y}$.

###### Proof.

For untwisted case, i.e. $\mathscr{X}=X$ and $\mathscr{Y}=Y$, this is exactly
[28, Proposition 3.1]. Here we extend it to the twisted case. As one direction
is obvious, it suffices to show that (2) can imply (1). Set

$\mathcal{P}_{x}\coloneqq\Phi^{\mathcal{P}}(k(x))=\mathcal{P}|_{\\{x\\}\times
Y},$

the image of the skyscraper sheaf $k(x)$ for a closed point $x\in X$. Since
$\mathtt{Coh}(\mathscr{Y})$ admits no spherical objects (cf. [32, §§3.2]),
$\operatorname{D}^{b}(\mathscr{Y})$ are generic K3-categories and the semi-
rigid objects in $\operatorname{D}^{b}(\mathscr{Y})$ are in
$\mathtt{Coh}(\mathscr{Y})$ up to shift of degree. We can see there is an
integer $m$ such that $H^{i}(\mathcal{P}_{x})=0$ for all $i\neq m$ and closed
points $x$ (cf. [32, Proposition 3.18]). Therefore, there is a
$\mathscr{X}^{(-1)}\times\mathscr{Y}$-twisted sheaf
$\mathcal{E}\in\mathtt{Coh}(\mathscr{X}^{(-1)}\times\mathscr{Y})$ such that
$\mathcal{P}\cong\mathcal{E}[m]$. Since
$\Phi^{\mathcal{P}}_{\mathscr{X}\to\mathscr{Y}}$ sends $(0,0,1)$ to $(0,0,1)$,
$\mathcal{E}_{x}$ is just a skyscraper sheaf on $\\{x\\}\times Y$. Then one
can proceed the arguments as in [14, Corollary 5.3] or [29, Corollary 5.22,
5.23] to show that there is an isomorphism $f\colon X\to Y$ such that
$f^{*}([\mathscr{Y}_{\mathbb{G}_{m}}])=[\mathscr{X}_{\mathbb{G}_{m}}]$. ∎

###  3.4. Twisted FM partners via moduli space of twisted sheaves

Keep the notations as before, we denote by $\mathscr{M}_{H}(\mathscr{X},v)$
(or $\mathscr{M}_{H}^{\alpha}(X,v)$) the moduli stack of $H$-semistable
$\mathscr{X}$-twisted sheaves with Mukai vector
$v\in\widetilde{\mathrm{N}}(\mathscr{X})$, where $H$ is a $v$-generic ample
divisor on $X$ and $\alpha=[\mathscr{X}]$ the associated Brauer class of $X$
(cf. [39] or [66]). To characterize the Fourier–Mukai partners of twisted
abelian surfaces via the moduli space of twisted sheaves, we first need the
following criterion on non-emptiness of moduli space of (twisted) sheaves on
an abelian surface $X$ in positive characteristic. In the rest of this
section, we will always assume that $k=\bar{k}$ and
$\operatorname{char}(k)\neq 2$.

######  Proposition 3.4.1 (Minamide–Yanagida–Yoshioka, Bragg–Lieblich).

Let $n$ be a positive integer. Assume that either $p\nmid n$ or $X$ is
supersingular and $n=p$. Let $\mathscr{X}\to X$ be a $\mu_{n}$-gerbe on $X$.
Let $v=(r,\ell,s)\in\widetilde{\mathrm{N}}(\mathscr{X})$ be a primitive Mukai
vector such that $v^{2}=0$. Fix a $v$-generic ample divisor $H$. If one of the
following holds (called _positive_):

  1. (1)

$r>0$.

  2. (2)

$r=0$ and $\ell$ is effective.

  3. (3)

$r=\ell=0$ and $s>0$.

then the coarse moduli space $M_{H}(\mathscr{X},v)\neq\emptyset$ and the
moduli stack $\mathscr{M}_{H}(\mathscr{X},v)$ is a $\mathbb{G}_{m}$-gerbe on
$M_{H}(\mathscr{X},v)$. Moreover, its coarse moduli space
$M_{H}(\mathscr{X},v)$ is an abelian surface.

###### Proof.

If $\mathscr{X}\to X$ is a $\mu_{n}$-gerbe such that $p\nmid n$, then the
statements are proven in [46, Proposition A.2.1] which is based on a statement
of lifting a Brauer classes on $X$ to characteristic 0 which requires the
condition $p\nmid n$ (see Lemma A.2.3 in loc.cit.).

If $X$ is supersingular and $\mathscr{X}\to X$ is a $\mu_{p}$-gerbe, then the
assertion will follow from a same argument in [7, Proposition 4.1.20], as we
will see in §6.6.2 that the twisor space of a supersingular abelian surface
can be constructed. ∎

######  Remark 3.4.2.

Actually, one can obtain the non-emptiness of $\mathscr{M}_{H}(\mathscr{X},v)$
for a $\mu_{n}$-gerbe $\mathscr{X}\to X$ over an abelian surface of finite
height with $p\mid n$ by following [46, Proposition A.2.1] together with the
lifting result 2.2.1.

######  Remark 3.4.3.

In the case $\mathscr{X}\to X$ is a essentially-trivial $\mu_{p}$-gerbe over a
supersingular abelian surface $X$, this can be proved by a standard lifting
argument (see also [22, Proposition 6.9]). When $\mathscr{X}\to X$ is non-
trivial, Bragg–Lieblich’s approach is to take the universal family of
$\mu_{p}$-gerbes

$f\colon\mathfrak{X}\to\mathbb{A}^{1}$

on the connected component
$\mathbb{A}^{1}\subset\operatorname{R}^{2}\pi_{*}\mu_{p}$ which contains
$\mathscr{X}$ (cf. Corollary 6.6.6). The fibers of $f$ contain $\mathscr{X}\to
X$ and the trivial $\mu_{p}$-gerbe over $X$. By taking the relative moduli
space of twisted sheaves (with suitable $v$-generic polarization) on
$\mathfrak{X}\to\mathbb{A}^{1}$, one can see the non-emptiness of
$M_{H}(\mathscr{X},v)$ from the case of essentially trivial gerbes.

Now, we are going to define the twisted Poincaré bundle for a gerbe on a given
abelian surface. Let $\mathscr{X}\to X$ be a $\mu_{n}$-gerbe on $X$ such that
$p\nmid n$. As an element in
$\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X,\mu_{n})$, we can (uniquely)
associate $\mathscr{X}$ with a symmetric morphism

$\varphi_{n}\colon X[n]\to\widehat{X}[n]$

by the Weil pairing (cf. [59, Lemma 16.22]). Dually, we have
$\varphi_{n}^{t}\colon\widehat{X}[n]\to X[n]$, which corresponds to a
$\mu_{n}$-gerbe on $\widehat{X}$, denoted by $\widehat{\mathscr{X}}$. We can
take a separable isogeny $f\colon Y\to X$ such that
$\widehat{f}[n]\circ\varphi_{n}\circ f[n]=0$. This implies
$f^{*}\mathscr{X}=0\in\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(Y,\mu_{n})$.
Then there is also a separable isogeny $f^{t}\colon\widehat{Y}\to\widehat{X}$
given by the Cartier dual $\ker(f)^{D}\subset\widehat{Y}$, which satisfies
$f^{t*}\widehat{\mathscr{X}}=0$. Let $\mathcal{P}_{0}$ be the Poincaré bundle
on $Y\times\widehat{Y}$. Consider

$f\times f^{t}\colon Y\times\widehat{Y}=V\to X\times\widehat{X}$

as a finite étale covering which trivializes the $\mu_{n}$-gerbe
$\mathscr{X}\times\widehat{\mathscr{X}}$, we will get a
$\mathscr{X}\times\widehat{\mathscr{X}}$-twisted sheaf
$\mathcal{P}_{\mathscr{X}}$ on $X\times\widehat{X}$ by the étale descent. We
have the following commutative diagram

${V}$${Y}$${X\times\widehat{X}}$${\widehat{Y}}$${X}$${\widehat{X}}$$\scriptstyle{p_{Y}}$$\scriptstyle{q_{Y}}$$\scriptstyle{f\times
f^{t}}$$\scriptstyle{f}$$\scriptstyle{p_{X}}$$\scriptstyle{q_{X}}$$\scriptstyle{f^{t}}$

######  Proposition 3.4.4.

The Fourier–Mukai functor
$\Phi^{\mathcal{P}_{\mathscr{X}}}\colon\operatorname{D}^{b}(\mathscr{X}^{(-1)})\to\operatorname{D}^{b}(\widehat{\mathscr{X}})$
is a derived equivalence.

###### Proof.

This statement can be checked étale locally on $\widehat{X}$. Then this
follows from the Bridgeland’s criterion (Theorem 2.3 and Theorem 3.3 in [11])
as in [29, Proposition 9.19], since $\mathcal{P}_{X}$ is étale locally the
Poincaré bundle: For any skyscraper sheaf $k(x)$ on $X$, which is naturally a
$\mathscr{X}^{(-1)}$-twisted sheaf, we have the following yoga

$\displaystyle\Phi^{\mathcal{P}_{X}}(k(x))|_{\widehat{Y}}$
$\displaystyle=(f^{t})^{*}q_{X*}(\mathcal{P}_{\mathscr{X}}\otimes
p^{*}_{X}k(x))$ $\displaystyle=q_{Y*}(f\times
f^{t})^{*}(\mathcal{P}_{\mathscr{X}}\otimes p_{X}^{*}k(x))$
$\displaystyle\cong q_{Y*}(\mathcal{P}_{0}\otimes p_{Y}^{*}f^{*}k(x))$
$\displaystyle\cong\bigoplus_{y\in f^{-1}(x)}q_{Y*}(\mathcal{P}_{0}\otimes
p_{Y}^{*}(k(y)))=\bigoplus_{y\in f^{-1}(x)}\mathcal{P}_{0,y}.$

where $\mathcal{P}_{0,y}$ is the line bundle on $\\{x\\}\times\widehat{Y}$
corresponding to $y\in Y\cong{\rm Pic}^{0}(\widehat{Y})$. ∎

The following is an extension of [28, Theorem 1.2].

######  Theorem 3.4.5.

With the same assumptions as in Proposition 3.4.1. Let $\mathscr{X}\to X$ be
$\mu_{n}$-gerbe on an abelian surface $X$ such that $p\nmid n$. Then the
associated $\mathbb{G}_{m}$-gerbe of any Fourier-Mukai partner of
$\mathscr{X}$ is isomorphic to a $\mathbb{G}_{m}$-gerbe on the moduli space of
$\mathscr{Y}$-twisted sheaves $M_{H}(\mathscr{Y},v)$ with $\mathscr{Y}$ being
$\mathscr{X}$ or $\widehat{\mathscr{X}}$.

###### Proof.

Let $\mathscr{M}$ be a Fourier-Mukai partner of $\mathscr{X}$. Let
$\Phi^{\mathcal{P}}_{\mathscr{M}\to\mathscr{X}}$ be the Fourier-Mukai
transform. Let $v$ be the image of $(0,0,1)$ under
$\Phi^{\mathcal{P}}_{\mathscr{M}\to\mathscr{X}}$. We can assume $v$ satisfying
one of the conditions in Proposition 3.4.1 by changing $\mathscr{X}$ to
$\widehat{\mathscr{X}}$ if necessary. It is proved that the moduli stack
$\mathscr{M}_{H}(\mathscr{X},v)$ is a $\mathbb{G}_{m}$-gerbe on
$M_{H}(\mathscr{X},v)$ in Proposition 3.4.1. Then there is a Fourier-Mukai
transform

$\Phi^{\mathcal{P}}\colon\operatorname{D}^{b}(\mathscr{M}_{H}(\mathscr{X},v)^{(-1)})\to\operatorname{D}^{b}(\mathscr{X}^{(1)})$
(3.4.1)

induced by the tautological sheaf $\mathcal{P}$ on
$\mathscr{M}_{H}(\mathscr{X},v)\times\mathscr{X}$, whose cohomological
realization maps the Mukai vector $(0,0,1)$ to $v$. Combining it with the
derived equivalence

$\Phi\colon\operatorname{D}^{b}(\mathscr{X})\to\operatorname{D}^{b}(\mathscr{M}),$

we will obtain a filtered derived equivalence from
$\mathscr{M}_{H}(\mathscr{X},v)^{(-1)}$ to $\mathscr{M}^{(1)}$. This induces
an isomorphism from $\mathscr{M}_{H}(\mathscr{X},v)^{(-1)}$ to
$\mathscr{M}^{(1)}_{\mathbb{G}_{m}}$ by Theorem 3.3.2. ∎

##  4\. Shioda’s Torelli theorem for abelian surfaces

In [58], Shioda noticed that there is a way to extract the information of the
$1^{\text{st}}$-cohomology of a complex abelian surface from its
$2^{\text{nd}}$-cohomology, called Shioda’s trick. This established a global
Torelli theorem for complex abelian surfaces via the
$2^{\text{nd}}$-cohomology, which is also a key step in Pjateckii-
Šapiro–Šafarevič’s proof of the Torelli theorem for K3 surfaces (cf. [53,
Lemma 4, Theorem 1]).

The aim of this section is to generalize Shioda’s method to all fields and
establish an isogeny theorem for abelian surfaces via the
$2^{\text{nd}}$-cohomology. We will deal with Shioda’s trick for Betti
cohomology, étale cohomology and crystalline cohomology separately.

###  4.1. Recap of Shioda’s trick for Hodge isometry

We first recall Shioda’s construction. Suppose $X$ is a complex abelian
surface. Its singular cohomology ring $\mathrm{H}^{\bullet}(X,\mathbb{Z})$ is
canonically isomorphic to the exterior algebra
$\wedge^{\bullet}\mathrm{H}^{1}(X,\mathbb{Z})$. Let $V$ be a free
$\mathbb{Z}$-module of rank $4$. We denote by $\Lambda$ the lattice
$(\wedge^{2}V,q)$ where $q:\wedge^{2}V\times\wedge^{2}V\to\mathbb{Z}$ is the
wedge product. After choosing a $\mathbb{Z}$-basis $\\{v_{i}\\}_{1\leq i\leq
4}$ for $\mathrm{H}^{1}(X,\mathbb{Z})$, we have an isometry of
$\mathbb{Z}$-lattice $\Lambda\xrightarrow{\sim}\mathrm{H}^{2}(X,\mathbb{Z})$.
The set of vectors

$\\{v_{ij}\coloneqq v_{i}\wedge v_{j}\\}_{0\leq i<j\leq 4}$

clearly forms a basis of $\mathrm{H}^{2}(X,\mathbb{Z})$, which will be called
an _admissible basis_ of $A$ for its second singular cohomology. For another
complex abelian surface $Y$, a Hodge isometry

$\psi\colon\mathrm{H}^{2}(Y,\mathbb{Z})\xrightarrow{\sim}\mathrm{H}^{2}(X,\mathbb{Z})$

will be called _admissible_ if $\det(\psi)=1$, with respect to some admissible
bases on $X$ and $Y$. It is clear that the admissibility of a morphism is
independent of the choice of admissible bases.

In terms of admissible bases, we can view $\psi$ as an element in
$\operatorname{SO}(\Lambda)$. On the other hand, we have the following exact
sequence of groups

$1\to\\{\pm
1\\}\to\operatorname{SL}_{4}(\mathbb{Z})\xrightarrow{\wedge^{2}}\operatorname{SO}(\Lambda)$
(4.1.1)

Shioda observed that the image of $\operatorname{SL}_{4}(\mathbb{Z})$ in
$\operatorname{SO}(\Lambda)$ is a subgroup of index two and does not contain
$-\operatorname{id}_{\Lambda}$. From this, he proved the following (cf. [58,
Theorem 1])

######  Theorem 4.1.1 (Shioda).

For any admissible integral Hodge isometry $\psi$, there is an isomorphism of
integral Hodge structures

$\varphi\colon\mathrm{H}^{1}(Y,\mathbb{Z})\xrightarrow{\sim}\mathrm{H}^{1}(X,\mathbb{Z})$

such that $\wedge^{2}(\varphi)=\psi$ or $-\psi$.

This is what we call “Shioda’s trick”. As we can assume a Hodge isometry being
admissible after possibly taking the dual abelian variety for one of them, we
can obtain the Torelli theorem for complex abelian surfaces by using the
weight two Hodge structures, i.e., $X$ is isomorphic to $Y$ or its dual
$\widehat{Y}$ if and only if there is an integral Hodge isometry
$\mathrm{H}^{2}(X,\mathbb{Z})\cong\mathrm{H}^{2}(Y,\mathbb{Z})$ (cf. [58,
Theorem 1]).

###  4.2. Admissible basis

In order to extend Shioda’s work to arbitrary fields, we need to define
admissibility for various cohomology theories (e.g. étale cohomology and
crystalline cohomology).

Let $k$ be a perfect field with $\operatorname{char}(k)=0$ or $p\geq 2$.
Suppose $X$ is an abelian surface over $k$ and $\ell\nmid p$ is a prime. For
simplicity of notations, we will denote $\mathrm{H}^{\bullet}(-)_{R}$ for one
of the following cohomology theories:

  1. (1)

if $k\hookrightarrow\mathbb{C}$ and $R=\mathbb{Z}$ or any number field $E$,
then $\mathrm{H}^{\bullet}(X)_{R}=\mathrm{H}^{\bullet}(X(\mathbb{C}),R)$ the
singular cohomology.

  2. (2)

if $R=\mathbb{Z}_{\ell}$ or $\mathbb{Q}_{\ell}$, then
$\mathrm{H}^{\bullet}(X)_{R}=\mathrm{H}^{\bullet}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},R)$,
the $\ell$-adic étale cohomology.

  3. (3)

if $\operatorname{char}(k)=p>0$ and $R=W$ or $K$, then
$\mathrm{H}^{\bullet}(X)_{R}=\mathrm{H}^{\bullet}_{\operatorname{crys}}(X_{k^{\operatorname{perf}}}/W)$
or
$\mathrm{H}^{\bullet}_{\operatorname{crys}}(X_{k^{\operatorname{perf}}}/W)\otimes
K$, the crystalline cohomology.

There is an isomorphism between the cohomology ring
$\mathrm{H}^{\bullet}(X)_{R}$ and the exterior algebra
$\wedge^{\bullet}\mathrm{H}^{1}(X)_{R}$. We denote by
$\operatorname{tr}_{X}\colon\mathrm{H}^{4}(X)_{R}\xrightarrow{\sim}R$ the
corresponding trace map. Then the Poincaré pairing $\langle-,-\rangle$ on
$\mathrm{H}^{2}(X)_{R}$ can be realized as

$\langle\alpha,\beta\rangle=\operatorname{tr}_{X}(\alpha\wedge\beta).$

Analogous to §4.1, a $R$-basis $\\{v_{i}\\}$ of $\mathrm{H}^{1}(X)_{R}$ will
be called a $d$-admissible basis if it satisfies

$\operatorname{tr}_{X}(v_{1}\wedge v_{2}\wedge v_{3}\wedge v_{4})=d$

for some $d\in R^{\ast}$. When $d=1$, it will be called an _admissible basis_.
For any $d$-admissible (resp. admissible) basis $\\{v_{i}\\}$, the associated
$R$-basis $\\{v_{ij}\coloneqq v_{i}\wedge v_{j}\\}_{i<j}$ of
$\mathrm{H}^{2}(X)_{R}$ will also be called $d$-admissible (resp. admissible).

######  Example 4.2.1.

Let $\\{v_{1},v_{2},v_{3},v_{4}\\}$ be a $R$-linear basis of
$\mathrm{H}^{1}(X)_{R}$. Suppose

$\operatorname{tr}_{X}(v_{1}\wedge v_{2}\wedge v_{3}\wedge v_{4})=t\in R^{*}.$

For any $d\in R^{*}$, there is a natural $d$-admissible $R$-linear basis
$\\{\frac{d}{t}v_{1},v_{2},v_{3},v_{4}\\}$

######  Definition 4.2.2.

Let $X$ and $Y$ be abelian surfaces over $k$.

  * •

a $R$-linear isomorphism
$\psi\colon\mathrm{H}^{1}(X)_{R}\to\mathrm{H}^{1}(Y)_{R}$ is $d$-admissible if
it takes an admissible basis to a $d$-admissible basis.

  * •

a $R$-linear isomorphism
$\varphi\colon\mathrm{H}^{2}(X)_{R}\to\mathrm{H}^{2}(Y)_{R}$ is $d$-admissible
if

$\operatorname{tr}_{Y}\circ\wedge^{2}(\varphi)=d\operatorname{tr}_{X}$

for some $d\in R^{*}$, or equivalently, it sends an admissible basis to a
$d$-admissible basis. When $d=1$, it will also be called admissible.

The set of $d$-admissible isomorphisms will be denoted by
$\operatorname{Iso}^{\operatorname{ad},(d)}(\mathrm{H}^{1}(X)_{R},\mathrm{H}^{1}(Y)_{R})$
and
$\operatorname{Iso}^{\operatorname{ad},(d)}(\mathrm{H}^{2}(X)_{R},\mathrm{H}^{2}(Y)_{R})$
respectively.

For any isomorphism
$\varphi\colon\mathrm{H}^{2}(X)_{R}\xrightarrow{\sim}\mathrm{H}^{2}(Y)_{R}$,
let $\det(\varphi)$ be the determinant of the matrix with respect to some
admissible bases. It is not hard to see $\det(\varphi)$ is independent of the
choice of admissible bases, and $\varphi$ is admissible if and only if
$\det(\varphi)=1$.

######  Example 4.2.3.

For the dual abelian surface $\widehat{X}$, the dual basis $\\{v_{i}^{*}\\}$
with respect to the Poincaré pairing naturally forms an admissible basis,
under the identification
$\mathrm{H}^{1}(X)_{R}^{\vee}\cong\mathrm{H}^{1}(\widehat{X})_{R}$. Let

$\psi_{\mathcal{P}}\colon\mathrm{H}^{2}(X)_{R}\to\mathrm{H}^{2}(\widehat{X})_{R}$

be the isomorphism induced by the Poincaré bundle $\mathcal{P}$ on
$X\times\widehat{X}$. A direct computation (see e.g. [29, Lemma 9.3]) shows
that $\psi_{\mathcal{P}}$ is nothing but

$-\operatorname{D}\colon\mathrm{H}^{2}(X)_{R}\xrightarrow{\sim}\mathrm{H}^{2}(X)_{R}^{\vee}\cong\mathrm{H}^{2}(\widehat{X})_{R},$

where $\operatorname{D}$ is the Poincaré duality. For an admissible basis
$\\{v_{i}\\}$ of $X$, its $R$-linear dual $\\{v^{*}_{i}\\}$ with respect to
Poincaré pairing forms an admissible basis of $\widehat{X}$. By our
construction, we can see

$\operatorname{D}(v_{12},v_{13},v_{14},v_{23},v_{24},v_{34})=(v_{34}^{*},-v_{24}^{*},v_{23}^{*},v_{14}^{*},-v_{13}^{*},v_{12}^{*}),$

which implies that $\operatorname{D}$ is of determinant $-1$ under these
admissible bases. Thus the determinant of $\psi_{\mathcal{P}}$ is not
admissible.

######  Example 4.2.4.

Let $f\colon X\rightarrow Y$ be an isogeny of degree $d$ for some
$d\in\mathbb{Z}_{\geq 0}$ between two abelian surfaces. If $d$ is coprime to
$\ell$, then it will induce an isomorphism

$f^{\ast}\colon\mathrm{H}^{2}(Y)_{\mathbb{Z}_{\ell}}\xrightarrow{\sim}\mathrm{H}^{2}(X)_{\mathbb{Z}_{\ell}},$

which is $d$-admissible. If $d=n^{4}$, then $\frac{1}{n}f^{\ast}$ will be an
admissible $\mathbb{Z}_{\ell}$-integral isometry with respect to the Poincaré
pairing.

If $\ell\neq 2$, then $d$ or $-d$ is a square in $\mathbb{Z}_{\ell}$. Thus
there is some $\xi\in\mathbb{Z}_{\ell}^{*}$ such that $\pm d=\xi^{4}$.
Therefore, we can always find an admissible $\mathbb{Z}_{\ell}$-integral
isomorphism
$\frac{1}{\xi}f^{*}\colon\mathrm{H}^{1}(Y)_{\mathbb{Z}_{\ell}}\to\mathrm{H}^{1}(X)_{\mathbb{Z}_{\ell}}$
by possibly changing $Y$ to $\widehat{Y}$.

######  Example 4.2.5.

Suppose $X$ is an abelian surface over a perfect field $k$ with
$\operatorname{char}(k)=p>0$. Then $F$-crystal $\mathrm{H}^{1}(X)_{W}$
together with the trace map

$\operatorname{tr}_{X}\colon\mathrm{H}^{4}(X)_{W}\xrightarrow{\sim}W$

form an abelian crystal, in the sense of [50, §6]. We can see
$\mathrm{H}^{1}(X)_{W}\cong\mathrm{H}^{1}(Y)_{W}$ as abelian crystals if and
only if there is an admissible isomorphism
$\mathrm{H}^{1}(X)_{W}\xrightarrow{\sim}\mathrm{H}^{1}(Y)_{W}$.

###  4.3. More on admissible basis of $F$-crystals

In contrast to $\ell$-adic étale cohomology, the semilinear structure on
crystalline cohomology from its Frobenius is more tricky to work with.
Therefore, it seems necessary for us to spend more words on the interaction of
Frobenius with admissible bases.

We have the following Frobenius pull-back diagram:

${X}$${X^{(1)}}$${X}$${\operatorname{Spec}(k)}$${\operatorname{Spec}(k)}$$\scriptstyle{F_{X}^{(1)}}$$\scriptstyle{F_{X}}$$\scriptstyle{\sigma}$

Via the natural identification
$\mathrm{H}^{1}_{\operatorname{crys}}(X^{(1)}/W)\cong\mathrm{H}^{1}_{\operatorname{crys}}(X/W)\otimes_{\sigma}W$,
the $\sigma$-linearization of Frobenius action on
$\mathrm{H}^{1}_{\operatorname{crys}}(X/W)$ can be viewed as the injective
$W$-linear map

$F^{(1)}\coloneqq\left(F_{X}^{(1)}\right)^{*}\colon\mathrm{H}^{1}_{\operatorname{crys}}(X^{(1)}/W)\hookrightarrow\mathrm{H}^{1}_{\operatorname{crys}}(X/W).$

There is a decomposition
$\mathrm{H}^{1}_{\operatorname{crys}}(X/W)=H_{0}(X)\oplus H_{1}(X)$ such that

$F^{(1)}\left(\mathrm{H}^{1}_{\operatorname{crys}}(X^{(1)}/W)\right)\cong
H_{0}(X)\oplus pH_{1}(X),$ (4.3.1)

and ${\rm rank}_{W}H_{i}=2$ for $i=0,1$, which is related to the Hodge
decomposition of the de Rham cohomology of $X/k$ by Mazur’s theorem; see [4,
§8, Theorem 8.26].

The Frobenius map can be expressed in terms of admissible basis. We can choose
an admissible basis $\\{v_{i}\\}$ of
$\mathrm{H}^{1}_{\operatorname{crys}}(X/W)$ such that

$v_{1},v_{2}\in H_{0}(X)\quad\text{ and }\quad v_{3},v_{4}\in H_{1}(X).$

Then $\\{p^{\alpha_{i}}v_{i}\\}\coloneqq\\{v_{1},v_{2},pv_{3},pv_{4}\\}$ forms
an admissible basis of $\mathrm{H}^{1}_{\operatorname{crys}}(X^{(1)}/W)$ under
the identification (4.3.1), since
$\operatorname{tr}_{p}\circ\wedge^{4}F^{(1)}=p^{2}\sigma_{W}\circ\operatorname{tr}_{p}$.
In term of these basis, the Frobenius map can be written as

$F^{(1)}(p^{\alpha_{i}}v_{i})=\sum_{j}c_{ij}p^{\alpha_{j}}v_{j},$

where $C_{X}=(c_{ij})$ forms an invertible $4\times 4$-matrix with
coefficients in $W$.

Suppose $Y$ is another abelian surface over $k$ and
$\rho\colon\mathrm{H}^{1}_{\operatorname{crys}}(X/W)\to\mathrm{H}^{1}_{\operatorname{crys}}(Y/W)$
is an admissible map. Denote $\rho^{(1)}$ for the induced map
$\rho\otimes_{\sigma}W\colon\mathrm{H}^{1}_{\operatorname{crys}}(X^{(1)}/W)\to\mathrm{H}^{1}_{\operatorname{crys}}(Y^{(1)}/W)$.
The following lemma is clear.

######  Lemma 4.3.1.

The map $\rho$ is a morphism between $F$-crystals if and only if
$C_{Y}^{-1}\cdot\rho^{(1)}\cdot C_{X}=\rho$, where “$\cdot$” denotes by the
action of matrix with respect to the chosen admissible bases.

###  4.4. Generalized Shioda’s trick

Let us review some basic properties of the special orthogonal group scheme
over an integral domain. Let $\Lambda$ be an even $\mathbb{Z}$-lattice of rank
$2n$. Then we can associate it with a vector bundle $\underline{\Lambda}$ on
$\operatorname{Spec}(\mathbb{Z})$ with constant rank $2n$ equipped with a
quadratic form $q$ over $\operatorname{Spec}(\mathbb{Z})$ obtained from
$\Lambda$. Then the functor

$A\mapsto\left\\{g\in{\rm GL}(\Lambda_{A})\big{|}q_{A}(g\cdot
x)=q_{A}(x)\text{ for all }x\in\Lambda_{A}\right\\}$

represents a $\mathbb{Z}$-subscheme of ${\rm GL}(\Lambda)$, denoted by
$\operatorname{O}(\Lambda)$. There is a homomorphism between
$\mathbb{Z}$-group schemes

$D_{\Lambda}\colon\operatorname{O}(\Lambda)\to\underline{\mathbb{Z}/2\mathbb{Z}},$

which is called the Dickson morphism. It is surjective as $\Lambda$ is even,
and its formation commutes with any base change. The _special orthogonal group
scheme_ over $\mathbb{Z}$ with respect to $\Lambda$ is defined to be the
kernel of $D_{\Lambda}$, which is denoted by $\operatorname{SO}(\Lambda)$.
Moreover, we have

$\operatorname{SO}(\Lambda)_{\mathbb{Z}[\frac{1}{2}]}\cong\ker\left(\det\colon\operatorname{O}(\Lambda)\to\mathbb{G}_{m}\right)_{\mathbb{Z}[\frac{1}{2}]}.$

It is well-known that
$\operatorname{SO}(\Lambda)\to\operatorname{Spec}(\mathbb{Z})$ is smooth of
relative dimension $\frac{n(n-1)}{2}$ and with connected fibers; see [16,
Theorem C.2.11] for instance. Moreover, it is well-known that the special
orthogonal group scheme admits a universal covering (i.e., a simply connected
central isogeny)

$\operatorname{Spin}(\Lambda)\to\operatorname{SO}(\Lambda).$

See Appendix C.4 in loc.cit. for the construction. For any $\ell$, the special
orthogonal group scheme

$\operatorname{SO}(\Lambda_{\mathbb{Z}_{\ell}})\cong\operatorname{SO}(\Lambda)_{\mathbb{Z}_{\ell}}$

is smooth over $\mathbb{Z}_{\ell}$ with connected fibers, which implies its
generic fiber $\operatorname{SO}(\Lambda_{\mathbb{Q}_{\ell}})$ is connected.
Thus $\operatorname{SO}(\Lambda_{\mathbb{Z}_{\ell}})$ is clearly connected as
a group scheme over $\mathbb{Z}_{\ell}$ as
$\operatorname{SO}(\Lambda_{\mathbb{Q}_{\ell}})\subset\operatorname{SO}(\Lambda_{\mathbb{Z}_{\ell}})$
is dense.

Let $V$ be free $\mathbb{Z}$-module of rank $4$ and $\Lambda=\wedge^{2}V$ with
the natural pairing. Let $R$ be a ring of coefficients as listed in §§4.2.
Then we have

######  Lemma 4.4.1.

There is an exact sequence of smooth $R$-group schemes

$1\to\mu_{2,R}\to\operatorname{SL}(V)_{R}\xrightarrow{\wedge^{2}(-)_{R}}\operatorname{SO}(\Lambda)_{R}\to
1.$

(as fppf-sheaves if $\frac{1}{2}\notin R$.) Moreover, there is an exact
sequence

$1\to\\{\pm\operatorname{id}_{4}\\}\to\operatorname{SL}(V)(R)\xrightarrow{\wedge^{2}(-)_{R}}\operatorname{SO}(\Lambda)(R)\xrightarrow{\operatorname{SN}}R^{*}/(R^{*})^{2},$
(4.4.1)

where $\operatorname{SN}$ is the map of spinor norm (see [3, §3.3] for the
definition).

###### Proof.

For the first statement, it suffices to assume
$R=\operatorname{Spec}(\bar{k})$ for an algebraically closed field $\bar{k}$,
where it is clear from a computation.

Note that we have an exact sequence on rational points (cf. [25, Proposition
3.2.2])

$1\to\mu_{2}(R)\to\operatorname{SL}(V)(R)\to\operatorname{SO}(\Lambda)(R)\to\mathrm{H}^{1}(\operatorname{Spec}(R),\mu_{2}).$

From the Kummer sequence for $\mu_{2}$, we can see

$\mathrm{H}^{1}(\operatorname{Spec}(R),\mu_{2})\cong\mathrm{H}^{1}_{\operatorname{{\acute{e}t}}}(\operatorname{Spec}(R),\mu_{2})\cong
R^{*}/(R^{*})^{2}$

as ${\rm Pic}(R)[2]=0$.

For the last statement, it is sufficient to see that there is an isomorphism
of $R$-group schemes
$\operatorname{SL}(V)_{R}\xrightarrow{\sim}\operatorname{Spin}(\Lambda)_{R}$
such that the following diagram commutes

${\operatorname{Spin}(\Lambda)(R)}$${\operatorname{SL}(V)(R)}$${\operatorname{SO}(\Lambda)(R)}$${R^{*}/(R^{*})^{2}}$${R^{*}/(R^{*})^{2}}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{SN}}$$\scriptstyle{\sim}$

The group scheme $\operatorname{SL}(V)$ is simply-connected (as its geometric
fibers are semisimple algebraic group of type $A_{3}$). Thus the central
isogeny $\operatorname{SL}(V)_{R}\to\operatorname{SO}(\Lambda)_{R}$ forms the
universal covering of $\operatorname{SO}(\Lambda)_{R}$, which induces an
isomorphism
$\operatorname{SL}(V)_{R}\xrightarrow{\sim}\operatorname{Spin}(\Lambda)_{R}$
by using the Existence and Isomorphism Theorems over a general ring (see
e.g.,[16, Exercise 6.5.2]).

∎

######  Remark 4.4.2.

When $R=\mathbb{Z}_{\ell}$, we have

$\mathbb{Z}_{\ell}^{*}/(\mathbb{Z}_{\ell}^{*})^{2}\cong\begin{cases}\\{\pm
1\\}&\text{ if }\ell\neq 2,\\\ \\{\pm 1\\}\times\\{\pm 5\\}&\text{ if
}\ell=2.\end{cases}$

Thus the image of $\operatorname{SL}(V)(\mathbb{Z}_{\ell})$ is a finite index
subgroup in $\operatorname{SO}(\Lambda)(\mathbb{Z}_{\ell})$.

######  Remark 4.4.3.

When $R=W(k)$, we have

$W(k)^{*}/(W(k)^{*})^{2}\cong\begin{cases}\\{\pm 1\\}&\text{ if
$k=\mathbb{F}_{p^{s}}$ for $p>2,s\geq 1$}\\\ \\{\pm 1\\}\times\\{\pm
5\\}&\text{ if $k=\mathbb{F}_{p^{s}}$ for $p=2,s\geq 1$},\\\ \\{1\\}&\text{ if
$k=\bar{k}$ or $k^{s}\subset k,\operatorname{char}(k)>2$.}\end{cases}$

as $W(k)$ is Henselian. Thus the wedge map
$\operatorname{SL}(V)(W)\to\operatorname{SO}(\Lambda)(W)$ is surjective when
$k=\bar{k}$.

Let $V_{R}=\mathrm{H}^{1}(X)_{R}$. We can see the set

$\operatorname{Iso}^{\operatorname{ad},(d)}(\mathrm{H}^{1}(X)_{R},\mathrm{H}^{1}(Y)_{R})$

is a naturally (right) $\operatorname{SL}(V_{R})$-torsor if it is non-empty.
The wedge product provides a natural map

$\wedge^{2}\colon\operatorname{Iso}^{\operatorname{ad},(d)}\left(\mathrm{H}^{1}(X)_{R},\mathrm{H}^{1}(Y)_{R}\right)\to\operatorname{Iso}^{\operatorname{ad},(d)}\left(\mathrm{H}^{2}(X)_{R},\mathrm{H}^{2}(Y)_{R}\right).$

Let $\\{v_{i}\\}$ be an admissible basis of $\mathrm{H}^{1}(X)_{R}$ and let
$\\{v_{i}^{\prime}\\}$ be a $d$-admissible basis of $\mathrm{H}^{1}(Y)_{R}$
respectively. There is an $d$-admissible isomorphism
$\psi_{0}\in\operatorname{Iso}^{\operatorname{ad},(d)}(\mathrm{H}^{1}(X)_{R},\mathrm{H}^{1}(Y)_{R})$
such that $\psi_{0}(v_{i})=v_{i}^{\prime}$. For a $d$-admissible isometry
$\varphi\colon\mathrm{H}^{2}(X,R)\to\mathrm{H}^{2}(Y,R)$, we can see

$\varphi=\wedge^{2}(\psi_{0}^{-1})\circ g,~{}\text{for some
$g\in\operatorname{SO}(\Lambda_{R})$}.$

In this way, any $d$-admissible isomorphism $\varphi$ can be identified with
(unique) element $g\in\operatorname{SO}(\Lambda)(R)$ when the admissible bases
are fixed. This allows us to deal with $d$-admissible isomorphisms group-
theoretically. In particular, we have the following notion of spinor norm.

######  Definition 4.4.4.

The _spinor norm_ of the $d$-admissible isomorphism $\varphi$ is defined to
the image of $g$ under
$\operatorname{SN}\colon\operatorname{SO}(\Lambda)(R)\to R^{*}/(R^{*})^{2}$,
denoted by $\operatorname{SN}(\varphi)$.

######  Lemma 4.4.5.

The spinor norm $\operatorname{SN}(\varphi)$ is independent of the choice of
admissible bases.

###### Proof.

For different choice of admissible bases, we can see the resulted
$\widetilde{g}=KgK^{-1}$ for some $K\in\operatorname{SO}(\Lambda_{R})$.
Therefore $\operatorname{SN}(\widetilde{g})=\operatorname{SN}(g)$. ∎

######  Remark 4.4.6.

When $R$ is a field, the spinor norm can be computed by the Cartan-Dieudonné
decomposition. That means, we can write any
$g\in\operatorname{SO}(\Lambda)(R)$ as a the composition of reflections:

$\varphi_{b_{n}}\circ\varphi_{b_{n-1}}\circ\cdots\circ\varphi_{b_{1}}$

for some non-isotropic vectors $b_{1},\cdots,b_{n}\in\Lambda_{R}$, and
$\operatorname{SN}(g)=\left[(b_{1})^{2}\cdots(b_{n-1})^{2}(b_{n})^{2}\right]$.

######  Lemma 4.4.7.

The $d$-admissible isomorphism $\varphi$ is a wedge of some $d$-admissible
isomorphism $\psi\colon\mathrm{H}^{1}(X,R)\to\mathrm{H}^{1}(Y,R)$ if and only
if $\operatorname{SN}(\varphi)=1$.

###### Proof.

The exact sequence (4.4.1) shows that if
$\operatorname{SN}(\varphi)=\operatorname{SN}(g)=1$, then there is some
$h\in\operatorname{SL}(V_{R})$ such that $\wedge^{2}(h)=g$. Thus we can take
$\psi=\psi_{0}\circ h$ when $\operatorname{SN}(\varphi)=1$, and see that

$\wedge^{2}(\psi)=\wedge^{2}(\psi_{0})\circ\wedge^{2}(h)=\varphi.$

The converse is clear. ∎

###  4.5. Shioda’s trick for Hodge isogenies

When $k=\mathbb{C}$ and $d$ is an integer, we say an isometry

$\varphi\colon\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})(d)$

a Hodge isogeny of degree d if it is also a morphism of Hodge structures. In
particular, if $d=1$, then it is the classical Hodge isometry we usually talk
about. Clearly, a $d$-admissible rational Hodge isomorphism is a Hodge isogeny
of degree $d$. In terms of spinor norms, we can generalize Shioda’s theorem
4.1.1 to admissible rational Hodge isogenies.

######  Proposition 4.5.1 (Shioda’s trick on admissible Hodge isogenies).

  1. (1)

A $d$-admissible Hodge isogeny of degree $d$

$\varphi\colon\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})(d)$

is a wedge of some rational Hodge isomorphism
$\psi\colon\mathrm{H}^{1}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{1}(Y,\mathbb{Q})$,
if its spinor norm is a square in $\mathbb{Q}^{*}$. In this case, the Hodge
isogeny is induced by a quasi-isogeny of degree $d^{2}$.

  2. (2)

When $d=1$, any admissible Hodge isometry
$\varphi:\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})$
is induced by an isogeny $f\colon Y\to X$ of degree $n^{2}$ for some integer
$n$ such that $\varphi=\frac{f^{\ast}}{n}$.

###### Proof.

Under the assumption of $(1)$, we can find a $d$-admissible isomorphism $\psi$
by applying the Lemma 4.4.7. It remains to prove that $\psi$ preserves the
Hodge structure, which is essentially the same as in [58, Theorem 1].

For $(2)$, we shall suppose the spinor norm
$\operatorname{SN}(\varphi)=n\mathbb{Q}^{*2}\in\mathbb{Q}^{*}/\mathbb{Q}^{*2}$.
Let $E=\mathbb{Q}(\sqrt{n})$. We can see the base-change
$\mathrm{H}^{2}(X,E)\xrightarrow{\sim}\mathrm{H}^{2}(Y,E)$ is a Hodge isometry
with coefficients in $E$ such that $\operatorname{SN}(\varphi)=1\in
E^{*}/(E^{*})^{2}$. Then by applying Lemma 4.4.7, we will obtain an admissible
(fixing admissible bases for $\mathrm{H}^{1}(X,\mathbb{Q})$ and
$\mathrm{H}^{1}(Y,\mathbb{Q})$) Hodge isomorphism
$\psi\colon\mathrm{H}^{1}(X,E)\xrightarrow{\sim}\mathrm{H}^{1}(Y,E)$. Let

$\sigma\colon a+b\sqrt{n}\rightsquigarrow a-b\sqrt{n}$

be the generator of $\operatorname{Gal}(E/\mathbb{Q})$. As we have fixed the
$\mathbb{Q}$-linear admissible bases, the wedge map

$\operatorname{SL}_{4}(E)\xrightarrow{\wedge^{2}}\operatorname{SO}(\Lambda)(E)$

is defined over $\mathbb{Q}$, and so is $\sigma$-equivariant. Let $g$ be the
element in $\operatorname{SL}_{4}(E)$ corresponding to $\psi$. As
$\wedge^{2}(g)\in\operatorname{SO}(\Lambda)\subset\operatorname{SO}(\Lambda_{E})$,
we can see

$(\wedge^{2}(\sigma(g))=\sigma(\wedge^{2}(g))=\wedge^{2}(g).$

which implies that $\sigma(g)g^{-1}=\pm\operatorname{id}_{4}$ since
$\ker(\wedge^{2})=\\{\pm\operatorname{id}_{4}\\}$. If $\sigma(g)=g$, then
$g\in\operatorname{SL}_{4}(\mathbb{Q})$ and the statement trivially holds. If
$\sigma(g)=-g$, then $g_{0}=\sqrt{n}g$ is lying in ${\rm GL}_{4}(\mathbb{Q})$.
Let

$\psi_{0}\colon\mathrm{H}^{1}(X,\mathbb{Q})\to\mathrm{H}^{1}(Y,\mathbb{Q})$

be the corresponding element of $g_{0}$ in
$\operatorname{Iso}^{\operatorname{ad},(n^{2})}\left(\mathrm{H}^{1}(X,\mathbb{Q}),\mathrm{H}^{1}(Y,\mathbb{Q})\right)$.
As $\wedge^{2}\psi_{0}=n\varphi$ is a Hodge isogeny, the part (1) then implies
that $\psi_{0}$ is a Hodge isomorphism as well. Thus $\psi_{0}$ lifts to a
quasi-isogeny $f_{0}\colon Y\to X$ and we have

$\varphi=\wedge^{2}(\psi)=\frac{f_{0}^{*}}{n}\colon\mathrm{H}^{2}(X,\mathbb{Q})\to\mathrm{H}^{2}(Y,\mathbb{Q}).$

∎

######  Remark 4.5.2.

If a Hodge isometry
$\psi\colon\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})$
is not admissible, i.e., its determinant is $-1$ with respect to some
admissible bases, then we can take its composition with the isometry
$\psi_{\mathcal{P}}$ induced by the Poincaré bundle as in Example 4.2.3. After
that, we can see $\psi_{\mathcal{P}}\circ\psi$ is admissible and is induced by
an isogeny $f\colon\widehat{Y}\to X$.

###  4.6. $\ell$-adic and $p$-adic Shioda’s trick

For the integral $\ell$-adic étale cohomology, we have the following statement
similar to Shioda’s trick for integral Betti cohomology.

######  Proposition 4.6.1 ($\ell$-adic Shioda’s trick).

Suppose $\ell\neq 2$. For any $d$-admissible

$\varphi_{\ell}\colon\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(Y_{\bar{k}},\mathbb{Z}_{\ell})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Z}_{\ell}),$

we can find a $d$-admissible $\mathbb{Z}_{\ell}$-isomorphism $\psi_{\ell}$
such that $\wedge^{2}(\psi_{\ell})=\varphi_{\ell}$ or $-\varphi_{\ell}$.
Moreover, if $\varphi_{\ell}$ is $G_{k}$-equivariant, then $\psi_{\ell}$ is
also $G_{k}$-equivariant after replacing $k$ by some finite extension.

###### Proof.

As $\mathbb{Z}_{\ell}^{*}/(\mathbb{Z}_{\ell}^{*})^{2}=\\{\pm 1\\}$ for any
$\ell\neq 2$, the spinor norm of $\varphi_{\ell}$ is equal to $\pm 1$. Thus
$\varphi_{\ell}$ or $-\varphi_{\ell}$ is of spinor norm one. Then the first
statement follows from Lemma 4.4.7.

Suppose $\varphi_{\ell}$ is $G_{k}$-equivariant. We may assume
$\wedge^{2}(\psi_{\ell})=\varphi_{\ell}$ for simplicity. For any $g\in G_{k}$,
we have

$\wedge^{2}(g^{-1}\psi_{\ell}g)=g^{-1}\wedge^{2}(\psi_{\ell})g=\varphi_{\ell}=\wedge^{2}(\psi_{\ell}).$

Therefore, $g^{-1}\psi_{\ell}g=\pm\psi_{\ell}$. By passing to a finite
extension $k^{\prime}/k$, we always have $g^{-1}\psi_{\ell}g=\psi_{\ell}$ for
all $g\in G_{k^{\prime}}$ which proves the assertion. ∎

For $F$-crystals attached to abelian surfaces, we can also play Shioda’s
trick.

######  Proposition 4.6.2 ($p$-adic Shioda’s trick).

Suppose $k$ is a finite field $\mathbb{F}_{p^{s}}$ with odd prime $p$. For any
$d$-admissible $W$-linear isomorphism

$\varphi_{W}\colon\mathrm{H}^{2}_{\operatorname{crys}}(Y/W)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(X/W),$

we can find a $d$-admissible $W$-linear isomorphism
$\rho:\mathrm{H}^{1}_{\operatorname{crys}}(Y/W)\xrightarrow{\sim}\mathrm{H}^{1}_{\operatorname{crys}}(X/W)$
such that $\wedge^{2}(\rho)=\varphi_{W}$ or $-\varphi_{W}$. Moreover, if
$\varphi_{W}$ is a morphism of $F$-crystals, then $\rho$ is an isomorphism of
$2^{nd}$-iterate of $F$-crystals.

###### Proof.

The first statement follows from a similar reason as in Proposition 4.6.1 as
$W^{*}/(W^{*})^{2}=\\{\pm 1\\}$ (see Remark 4.4.3).

For the second statement, we assume $\wedge^{2}(\rho)=\varphi_{W}$. If
$\varphi_{W}$ commutes with the Frobenius action, then we have

$\wedge^{2}(C_{X}^{-1}\cdot\rho^{(1)}\cdot C_{Y})=\varphi_{W}.$

as in §4.3. Thus $C_{X}^{-1}\cdot\rho^{(1)}\cdot C_{Y}=\pm\rho$, which implies

$\rho\circ F_{X}=\pm F_{Y}\circ\rho$

by Lemma 4.3.1. Therefore, $\rho$ commutes with the $2^{nd}$-iterate Frobenius
$F^{2}_{X}$ and $F_{Y}^{2}$. ∎

######  Remark 4.6.3.

If $k$ is an algebraically closed field or the separable closure in an
algebraic closure such that $\operatorname{char}(k)>2$, then Proposition 4.6.2
also holds. In addition, the first statement can be enforced to
$\Lambda^{2}(\rho)=\varphi_{W}$; see Remark 4.4.3.

Combined with Tate’s isogeny theorem, we have the following direct
consequences of Propositions 4.6.1 and 4.6.2. It includes a special case of
Tate’s conjecture.

######  Corollary 4.6.4.

Suppose $k$ is a finitely generated field over $\mathbb{F}_{p}$ with $p$ an
odd prime. Let $\ell\neq 2$ be a prime not equal to $p$.

  1. (1)

For any admissible isometry of $\operatorname{Gal}(k^{s}/k)$-modules

$\varphi_{\ell}\colon\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(Y_{k^{s}},\mathbb{Z}_{\ell})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X_{k^{s}},\mathbb{Z}_{\ell}),$

we can find a $\mathbb{Z}_{\ell}$-quasi-isogeny
$f_{\ell}\in\operatorname{Hom}_{k^{\prime}}(X_{k^{\prime}},Y_{k^{\prime}})\otimes\mathbb{Z}_{\ell}$
for some finite extension $k^{\prime}/k$, which induces $\varphi_{\ell}$ or
$-\varphi_{\ell}$. In particular, $\varphi_{\ell}$ is algebraic.

  2. (2)

For any admissible isometry of $F$-crystals over the Cohen ring $W$

$\varphi_{W}\colon\mathrm{H}^{2}_{\operatorname{crys}}(Y/W)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(X/W),$

we can find a $\mathbb{Z}_{p^{2}}$-quasi-isogeny
$f_{p}\in\operatorname{Hom}_{k^{\prime}}(X_{k^{\prime}},Y_{k^{\prime}})\otimes\mathbb{Z}_{p^{2}}$
which induces $\varphi_{W}$ or $-\varphi_{W}$ for some finite extension
$k^{\prime}/k$, where $\mathbb{Z}_{p^{2}}=W(\mathbb{F}_{p^{2}})$. In
particular, $\varphi_{W}$ is algebraic.

###### Proof.

For $(1)$, Proposition 4.6.1 implies there is an
$\operatorname{Gal}(k^{s}/k)$-equivariant isomorphism

$\psi_{\ell}\colon\mathrm{H}^{1}_{\operatorname{{\acute{e}t}}}(Y_{k^{s}},\mathbb{Z}_{\ell})\xrightarrow{\sim}\mathrm{H}^{1}_{\operatorname{{\acute{e}t}}}(X_{k^{s}},\mathbb{Z}_{\ell}),$

inducing $\varphi_{\ell}$ or $-\varphi_{\ell}$ after a finite extension of
$k$. Then $f_{\ell}$ exists by the canonical bijection (cf. [20, VI, §3
Theorem 1])

$\operatorname{Hom}_{k}(X,Y)\otimes\mathbb{Z}_{\ell}\xrightarrow{\sim}\operatorname{Hom}_{\operatorname{Gal}(k^{s}/k)}\left(\mathrm{H}^{1}_{\operatorname{{\acute{e}t}}}(Y_{k^{s}},\mathbb{Z}_{\ell}),\mathrm{H}^{1}_{\operatorname{{\acute{e}t}}}(X_{k^{s}},\mathbb{Z}_{\ell})\right).$

For $(2)$, let $\bar{k}$ be an algebraic closure of $k$, then Proposition
4.6.2 and Remark 4.6.3 imply that there is an isomorphism

$\rho\colon\mathrm{H}^{1}_{\operatorname{crys}}(Y_{\bar{k}}/W(\bar{k}))\xrightarrow{\sim}\mathrm{H}^{1}_{\operatorname{crys}}(X_{\bar{k}}/W(\bar{k}))$

such that $F_{X_{\bar{k}}}\circ\rho=\pm\rho\circ F_{Y_{\bar{k}}}$. In fact,
the $\bar{k}$ in this formula can be replaced a finite extension $k^{\prime}$
of $k$ by a similar argument as the proof of (2) of Proposition 4.5.1.

Replace $k$ by $k^{\prime}$. If $F_{X}\circ\rho=\rho\circ F_{Y}$ then one can
conclude by the canonical isomorphisms

$\operatorname{Hom}_{k}(X,Y)\otimes\mathbb{Z}_{p}\xrightarrow{\sim}\operatorname{Hom}_{k}\left(X[p^{\infty}],Y[p^{\infty}]\right)\xrightarrow{\sim}\operatorname{Hom}_{F}\left(\mathrm{H}^{1}(Y/W),\mathrm{H}^{1}(X/W)\right),$
(4.6.1)

where the bijectivity of the first arrow is given by $p$-adic Tate’s isogeny
theorem (cf. [18, Theorem 2.6]) and the second one is the faithfulness of
taking Dieudonné module over $W$ (cf. [17, Theorem ]).

It remains to consider the case $F_{X}\circ\rho=-\rho\circ F_{Y}$. After
taking a finite extension of $k$, we may assume that
$\mathbb{Z}_{p^{2}}\subset W(k)$. Now there is $\xi\in W(k)$ such that
$\xi^{p-1}+1=0$. We can see that

$F_{X}\circ(\xi\rho)=\xi^{p}F_{X}\circ\rho=(\xi\rho)\circ F_{Y}.$

Again, the bijection (4.6.1) implies that $\xi\rho$ is induced by a
$\mathbb{Z}_{p}$-quasi-isogeny
$f_{0}\in\operatorname{Hom}_{k}(X,Y)\otimes\mathbb{Z}_{p}$. Note that
$\xi\in\mathbb{Z}_{p^{2}}^{*}$. We can take

$f_{p}=\frac{f_{0}}{\xi}\in\operatorname{Hom}_{k}(X,Y)\otimes\mathbb{Z}_{p^{2}}.$

∎

######  Remark 4.6.5.

In [67], Zarhin introduces the notion of _almost isomorphism_. Two abelian
varieties over $k$ are called almost isomorphic if their Tate modules
$T_{\ell}$ are isomorphic as Galois modules (replaced by $p$-divisible groups
when $\ell=p$). Proposition 4.6.1 and 4.6.2 imply that it is possible to
characterize almost isomorphic abelian surfaces by their
$2^{\text{nd}}$-cohomology groups.

##  5\. Derived isogeny in characteristic zero

In this section, we follow [23] and [31] to prove the twisted Torelli theorem
for abelian surfaces over algebraically closed fields of characteristic zero.

###  5.1. Over $\mathbb{C}$: Hodge isogeny versus derived isogeny

Let $X$ and $Y$ be complex abelian surfaces.

######  Definition 5.1.1.

A rational Hodge isometry
$\psi_{b}\colon\mathrm{H}^{2}(X,\mathbb{Q})\to\mathrm{H}^{2}(Y,\mathbb{Q})$ is
called _reflexive_ if it is induced by a reflection on $\Lambda$ along a
vector $b\in\Lambda$:

$\varphi_{b}\colon\Lambda_{\mathbb{Q}}\xrightarrow{\sim}\Lambda_{\mathbb{Q}}\quad
x\mapsto x-\frac{2(x,b)}{(b,b)}b.$

######  Lemma 5.1.2.

Any reflexive Hodge isometry $\psi_{b}$ induces a Hodge isometry on twisted
Mukai lattices

$\widetilde{\psi}_{b}\colon\widetilde{\mathrm{H}}(X,\mathbb{Z};B)\to\widetilde{\mathrm{H}}(Y,\mathbb{Z};B^{\prime}),$

where $B=\frac{2}{(b,b)}b\in\mathrm{H}^{2}(X,\mathbb{Q})$ (via some marking
$\Lambda\cong\mathrm{H}^{2}(X,\mathbb{Z})$) and $B^{\prime}=-\psi_{b}(B)$.

###### Proof.

The proof can be found in [31, §1.2]. ∎

In analogy to [31, Theorem 1.1], the following result characterizes the
reflexive Hodge isometries between abelian surfaces.

######  Theorem 5.1.3.

Let $X$ and $Y$ be two complex abelian surfaces. If there is a reflexive Hodge
isometry

$\psi_{b}\colon\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q}),$

for some $b\in\Lambda$, then there exist $\alpha\in\operatorname{Br}(X)$ and
$\beta\in\operatorname{Br}(Y)$ such that $\psi_{b}$ is induced by a derived
equivalence

$\operatorname{D}^{b}(X,\alpha)\simeq\operatorname{D}^{b}(Y,\beta).$

Equivalently, $X$ or $\widehat{X}$ is isomorphic to the coarse moduli space of
twisted coherent sheaves on $Y$, and $\psi_{b}$ is induced by the twisted
Fourier-Mukai transform associated to the universal twisted sheaf.

###### Proof.

According to Lemma 5.1.2, there is a Hodge isometry

$\widetilde{\psi}_{b}\colon\widetilde{\mathrm{H}}(X,\mathbb{Z};B)\xrightarrow{\sim}\widetilde{\mathrm{H}}(Y,\mathbb{Z};B^{\prime}).$

Let $v_{B^{\prime}}$ be the image of Mukai vector $(0,0,1)$ under
$\widetilde{\psi}_{b}$. From our construction, there is a Mukai vector

$v=\exp(-B^{\prime})\cdot
v_{B^{\prime}}\in\widetilde{\mathrm{H}}(Y,\mathbb{Z})$

satisfying $v_{B^{\prime}}=\exp(B^{\prime})\cdot v$. We can assume that $v$ is
positive (see Proposition 3.4.1) by some suitable autoequivalences of
$\operatorname{D}^{b}(Y)$ as in [34, §2]. Let $\beta$ be the Brauer class on
$Y$ with respect to $B^{\prime}$ and $\mathscr{Y}\to Y$ be the corresponding
$\mathbb{G}_{m}$-gerbe. For some $v_{B^{\prime}}$-generic polarization $H$,
the moduli stack $\mathscr{M}_{H}(\mathscr{Y},v_{B^{\prime}})$ of
$\beta$-twisted sheaves on $Y$ with Mukai vector $v_{B^{\prime}}$ forms a
$\mathbb{G}_{m}$-gerbe on its coarse moduli space
$M_{H}(\mathscr{Y},v_{B^{\prime}})$ such that

$[\mathscr{M}_{H}(\mathscr{Y},v_{B^{\prime}})]\in\operatorname{Br}(M_{H}(\mathscr{Y},v_{B}))[r]$

(cf. [39, Proposition 2.3.3.4, Corollary 2.3.3.7]).

The kernel $\mathscr{P}$ is the tautological twisted sheaf on
$\mathscr{Y}\times\mathscr{M}_{H}(\mathscr{Y},v_{B^{\prime}})$ which induces a
twisted Fourier-Mukai transform

$\Phi_{\mathscr{P}}\colon\operatorname{D}^{b}(Y,\beta)\to\operatorname{D}^{b}(\mathscr{M}_{H}(\mathscr{Y},v_{B^{\prime}}))\simeq\operatorname{D}^{b}(M_{H^{\prime}}(\mathscr{Y},v_{B^{\prime}}),\alpha),$

where
$\alpha=[\mathscr{M}_{H}(\mathscr{Y},v_{B^{\prime}})]\in\operatorname{Br}(M_{H^{\prime}}(\mathscr{Y},v_{B^{\prime}}))$
(cf. [66, Theorem 4.3]). It induces a Hodge isometry

$\widetilde{\mathrm{H}}(Y,\mathbb{Z};B^{\prime})\xrightarrow{\sim}\widetilde{\mathrm{H}}(M_{H}(\mathscr{Y},v_{B^{\prime}}),\mathbb{Z};B^{\prime\prime}),$

where $B^{\prime\prime}$ is a $\mathbf{B}$-field lift of $\alpha$. Its
composition with $\widetilde{\psi}_{b}$ is a Hodge isometry

$\widetilde{\mathrm{H}}(X,\mathbb{Z};B)\xrightarrow{\sim}\widetilde{\mathrm{H}}(M_{H}(\mathscr{Y},v_{B^{\prime}}),\mathbb{Z};B^{\prime\prime}),$
(5.1.1)

sending the Mukai vector $(0,0,1)$ to $(0,0,1)$ and preserving the Mukai
pairing. We can see $(1,0,0)$ is mapping to $(1,b,\frac{b^{2}}{2})$ for some
$b\in\mathrm{H}^{2}(Y,\mathbb{Z})$ via (5.1.1). Thus we can replace
$B^{\prime\prime}$ by $B^{\prime\prime}+b$, which will not change the
corresponding Brauer class, to obtain a Hodge isometry which takes $(1,0,0)$
to $(1,0,0)$ and $(0,0,1)$ to $(0,0,1)$ at the same time. This yields a Hodge
isometry

$\mathrm{H}^{2}(X,\mathbb{Z})\xrightarrow{\sim}\mathrm{H}^{2}(M_{H^{\prime}}(\mathscr{Y},v_{B^{\prime}}),\mathbb{Z}).$

Then we can apply Shioda’s Torelli Theorem of abelian surfaces [58] to
conclude that

$M_{H^{\prime}}(\mathscr{Y},v_{B^{\prime}})\cong X$ or $\widehat{X}$.

When $X\cong M_{H^{\prime}}(\mathscr{Y},v_{B^{\prime}})$, $\Phi_{\mathscr{P}}$
gives the derived equivalence as desired. When $\widehat{X}\cong
M_{H^{\prime}}(\mathscr{Y},v_{B^{\prime}})$, we can prove the assertion by
using the fact $X$ and $\widehat{X}$ are derived equivalent. ∎

Next, we are going to show that any rational Hodge isometry can be decomposed
into a chain of reflexive Hodge isometries. This is a special case of Cartan-
Dieudonné theorem which says that any element
$\varphi\in\operatorname{SO}(\Lambda_{\mathbb{Q}})$ can be decomposed as
products of reflections:

$\varphi=\varphi_{b_{1}}\circ\varphi_{b_{2}}\circ\cdots\circ\varphi_{b_{n}},$
(5.1.2)

such that $b_{i}\in\Lambda$, and $(b_{i})^{2}\neq 0$. Then from the
surjectivity of period map [58, Theorem II], for any rational Hodge isometry

$\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q}),$

we can find a sequence of abelian surfaces $\\{X_{i}\\}$ with
$\Lambda$-markings and Hodge isometries
$\psi_{b_{i}}\colon\mathrm{H}^{2}(X_{i-1},\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(X_{i},\mathbb{Q})$,
where $X_{0}=X$ and $X_{n}=Y$, such that $\psi_{b_{i}}$ induces
$\varphi_{b_{i}}$ on $\Lambda_{\mathbb{Q}}$. We can arrange them as (1.1.1):

${\mathrm{H}^{2}(X,\mathbb{Q})}$${\mathrm{H}^{2}(X_{1},\mathbb{Q})}$${\mathrm{H}^{2}(X_{1},\mathbb{Q})}$${\mathrm{H}^{2}(X_{2},\mathbb{Q})}$${\vdots}$${\mathrm{H}^{2}(X_{n-1},\mathbb{Q})}$${\mathrm{H}^{2}(Y,\mathbb{Q}).}$$\scriptstyle{\psi_{b_{1}}}$$\scriptstyle{\psi_{b_{2}}}$$\scriptstyle{\psi_{b_{n}}}$
(5.1.3)

Finally, this yields

######  Corollary 5.1.4.

If there is a rational Hodge isometry
$\varphi:\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})$,
then there is a derived isogeny from $X$ to $Y$, whose Hodge realization is
$\varphi$.

As a consequence, we get

######  Corollary 5.1.5.

There is a rational Hodge isometry
$\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})$
if and only if there is a derived isogeny from $\operatorname{Km}(X)$ to
$\operatorname{Km}(Y)$.

###### Proof.

Witt’s cancellation theorem implies that

$\mathrm{H}^{2}(X,\mathbb{Q})\simeq\mathrm{H}^{2}(Y,\mathbb{Q})\Leftrightarrow\mathrm{T}(X)_{\mathbb{Q}}\simeq\mathrm{T}(Y)_{\mathbb{Q}},$

as Hodge isometries, where $\mathrm{T}(-)$ denotes the transcendental part of
$\mathrm{H}^{2}(-)$. According to [31, Theorem 0.1], $\operatorname{Km}(X)$ is
derived isogenous to $\operatorname{Km}(Y)$ if and only if there is a Hodge
isometry
$\mathrm{T}(\operatorname{Km}(X))_{\mathbb{Q}}\simeq\mathrm{T}(\operatorname{Km}(Y))_{\mathbb{Q}}$.
Then the statement is clear from the fact there is a canonical integral Hodge
isometry $\mathrm{T}(X)(2)\simeq\mathrm{T}(\operatorname{Km}(X))$ (cf. [47,
Proposition 4.3(i)]). ∎

######  Remark 5.1.6.

A consequence of Corollary 5.1.4 is that any rational Hodge isometry between
abelian surfaces is algebraic, which is a special case of Hodge conjecture on
product of two abelian surface. Unlike the case of K3 surfaces, the Hodge
conjecture for product of abelian surfaces were known for a long time. See
[56, Theorem 3.15] for example.

Moreover, we may call a reflexive Hodge isometry

$\psi_{b}\colon\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})$

induced by a primitive vector $b\in\Lambda$ _prime-to- $\ell$_ if $\ell\nmid
n=\frac{(b)^{2}}{2}$. The following results imply that the Hodge realization
of prime-to-$\ell$ derived isogeny is a composition of finitely many prime-
to-$\ell$ reflexive Hodge isometries.

######  Lemma 5.1.7.

If the induced derived isogeny
$\operatorname{D}^{b}(X)\sim\operatorname{D}^{b}(Y)$ in Corollary 5.1.4 is
prime-to-$\ell$, then each reflexive Hodge isometry $\psi_{b}$ in (5.1.3) is
prime-to-$\ell$.

###### Proof.

Otherwise, we can take $\ell^{k}$ to be the $\ell$-factor of $n$. As
$\psi_{b}$ restricts to an isomorphism

$\mathrm{H}^{2}(X,\mathbb{Z})\otimes\mathbb{Z}_{(\ell)}\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Z})\otimes\mathbb{Z}_{(\ell)},$

we have $\ell^{k}\mid(x,b)$ for any $x\in\Lambda$. This means $\ell^{k}$
divides the divisibility of $b$, which is impossible as $\Lambda$ is
unimodular. ∎

######  Remark 5.1.8.

With notations in Theorem 5.1.3, if $\psi_{b}$ is prime-to-$\ell$ with
$n=\frac{(b)^{2}}{2}$, then the Fourier-Mukai equivalence
$\operatorname{D}^{b}(X,\alpha)\xrightarrow{\sim}\operatorname{D}^{b}(Y,\beta)$
in Theorem 5.1.3 satisfies

$\alpha^{n}=\exp(nB)=1\in\operatorname{Br}(X),$

which implies $\alpha\in\operatorname{Br}(X)[n]$. Similarly, $n$ divides the
order of $\beta=\exp(B^{\prime})\in\operatorname{Br}(Y)$.

###  5.2. Isogeny versus derived isogeny

Let us now describe derived isogenies via suitable isogenies.

Recall that the isogeny category of abelian varieties
$\mathtt{AV}_{\mathbb{Q},k}$ consists of all abelian varieties over a field
$k$ as objects, and the Hom-sets are

$\operatorname{Hom}_{\mathtt{AV}_{\mathbb{Q},k}}(X,Y)\coloneqq\operatorname{Hom}_{\mathtt{AV}_{k}}(X,Y)\otimes_{\mathbb{Z}}\mathbb{Q},$

where $\operatorname{Hom}_{\mathtt{AV}_{k}}(X,Y)$ is the abelian group of
homomorphisms from $X$ to $Y$ with the natural addition. We may also write
$\operatorname{Hom}^{0}(X,Y)$ for
$\operatorname{Hom}_{\mathtt{AV}_{\mathbb{Q},k}}(X,Y)$ if there are no
confusion on the field of definition $k$. An isomorphism $f$ from $X$ to $Y$
in the isogeny category $\mathtt{AV}_{\mathbb{Q},k}$ is called a quasi-isogeny
from $X$ to $Y$. For any quasi-isogeny $f$, we can find a minimal integer $n$
such that

$nf\colon X\to Y$

is an isogeny, i.e., a finite surjective morphism of abelian varieties. When
$k=\mathbb{C}$, with the uniformization of complex abelian varieties, we have
a canonical bijection

$\operatorname{Hom}_{\mathtt{AV}_{\mathbb{Q},\mathbb{C}}}(X,Y)\xrightarrow{\sim}\operatorname{Hom}_{\operatorname{Hdg}}\left(\mathrm{H}^{1}(Y,\mathbb{Q}),\mathrm{H}^{1}(X,\mathbb{Q})\right),$

where the right-hand side is the set of $\mathbb{Q}$-linear morphisms
preserving Hodge structures. Then the integer $n$ for $f$ is also the minimal
integer such that
$(nf)^{*}(\mathrm{H}^{1}(Y,\mathbb{Z}))\subseteq\mathrm{H}^{1}(X,\mathbb{Z})$.

It is well-known that the functor $\underline{\operatorname{Hom}}(X,Y)$ of
homomorphisms from $X$ to $Y$ (not just as scheme morphisms) is representable
by an étale group scheme over $k$ (see [62, (7.14)] for example). Therefore,
via Galois descent, we have

$\operatorname{Hom}_{\mathtt{AV}_{\bar{k}}}(X_{\bar{k}},Y_{\bar{k}})\xrightarrow{\sim}\operatorname{Hom}_{\mathtt{AV}_{\bar{K}}}(X_{\bar{K}},Y_{\bar{K}}),$
(5.2.1)

for any algebraically closed field $\bar{K}\supset k$. A similar statement
holds for derived isogenies.

######  Lemma 5.2.1.

Let $X$ and $Y$ are abelian surfaces defined over $k$ with
$\operatorname{char}(k)=0$. Let $\bar{K}\supseteq k$ be an algebraically
closed field containing $k$. Let $\bar{k}$ be the algebraically closure of $k$
in $\bar{K}$. Then if $X_{\bar{K}}$ and $Y_{\bar{K}}$ are twisted derived
equivalent, so is $X_{\bar{k}}$ and $Y_{\bar{k}}$.

###### Proof.

As $X_{\bar{K}}$ is twisted derived equivalent to $Y_{\bar{K}}$, by Theorem
3.4.5, there exist finitely many abelian surfaces $X_{0},X_{1},\ldots,X_{n}$
defined over $\bar{K}$ with $X_{0}=X_{\bar{K}}$ and

$X_{i}~{}\text{or}~{}\widehat{X_{i}}=M_{H_{i}}(\mathscr{X}_{i-1},v_{i})\quad
Y_{\bar{K}}~{}\hbox{or}~{}\widehat{Y}_{\bar{K}}\cong
M_{H_{n}}(\mathscr{X}_{n},v_{n})$

for some $[\mathscr{X}_{i-1}]\in\operatorname{Br}(X_{i-1})$. Let us construct
abelian surfaces over $\bar{k}$ to connect $X_{\bar{k}}$ and $Y_{\bar{k}}$ as
follows:

Set $X_{0}^{\prime}=X_{\bar{k}}$, then we take
$X_{1}^{\prime}=M_{H^{\prime}_{1}}(\mathscr{X}_{0}^{\prime},v_{1}^{\prime})$
where $\mathscr{X}_{0}^{\prime},H^{\prime}_{1}$ and $v_{1}^{\prime}$ are the
descent of $\mathscr{X}_{0},H_{1}$ and $v$ via the isomorphisms
$\operatorname{Br}(X_{\bar{K}})\cong\operatorname{Br}(X_{\bar{k}})$, ${\rm
Pic}(X_{\bar{K}})\cong{\rm Pic}(X_{\bar{k}})$ and
$\widetilde{\mathrm{H}}(X_{\bar{K}})\cong\widetilde{\mathrm{H}}(X_{\bar{k}})$.
Then inductively, we can define $X_{i}^{\prime}$ as the moduli space of
twisted sheaves
$M_{H_{i}^{\prime}}(\mathscr{X}_{i-1}^{\prime},v_{i}^{\prime})$ (or its dual
respectively) over $\bar{k}$. Note that we have natural isomorphisms

$(M_{H_{i}^{\prime}}(\mathscr{X}_{i-1}^{\prime},v_{i}^{\prime}))_{\bar{K}}\cong
M_{H_{i}}(\mathscr{X}_{i-1},v_{i})$

over $\bar{K}$. In particular,
$(M_{H_{i}^{\prime}}(\mathscr{X}_{n}^{\prime},v_{i}^{\prime}))_{\bar{K}}\cong
Y_{\bar{K}}$. It follows that
$M_{H_{i}^{\prime}}(\mathscr{X}_{n}^{\prime},v_{i}^{\prime})\cong
Y_{\bar{k}}.$ ∎

More generally, we can replace $\mathbb{Q}$ in $\mathtt{AV}_{\mathbb{Q},k}$ by
any ring $R$. Then any isomorphism from $X$ to $Y$ in $\mathtt{AV}_{R,k}$ will
be called a _$R$ -(quasi)-isogeny_. In particular,

######  Definition 5.2.2.

An element
$f\in\operatorname{Hom}_{k}(X,Y)\otimes_{\mathbb{Z}}\mathbb{Z}_{(\ell)}$ which
has an inverse in
$\operatorname{Hom}_{k}(Y,X)\otimes_{\mathbb{Z}}\mathbb{Z}_{(\ell)}$ is called
a prime-to-$\ell$ quasi-isogeny, where $\mathbb{Z}_{(\ell)}$ is the
localization of $\mathbb{Z}$ at $(\ell)$.

For any abelian surface $X_{\mathbb{C}}$ over $\mathbb{C}$, the spreading out
argument shows that there is a finitely generated field $k\subset\mathbb{C}$
and an abelian surface $X$ over $k$ such that $X\times_{k}\mathbb{C}\cong
X_{\mathbb{C}}$. We have the following Artin comparison

$\mathrm{H}^{i}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Z}_{\ell})\cong\mathrm{H}^{i}(X_{\mathbb{C}},\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Z}_{\ell},$
(5.2.2)

for any $i\in\mathbb{Z}$ and $\ell$ a prime. Suppose $Y$ is another abelian
surface defined over $k$. Suppose $f\colon Y_{\mathbb{C}}\to X_{\mathbb{C}}$
is a prime-to-$\ell$ quasi-isogeny. By definition, it induces an isomorphism
of $\mathbb{Z}_{(\ell)}$-modules

$f^{*}\colon\mathrm{H}^{1}(X_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Z}_{(\ell)}\xrightarrow{\sim}\mathrm{H}^{1}(Y_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Z}_{(\ell)},$

such that there is a commutative diagram

${\mathrm{H}^{i}(X_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Z}_{(\ell)}}$${\mathrm{H}^{i}(Y_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Z}_{(\ell)}}$${\mathrm{H}^{i}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Z}_{\ell})}$${\mathrm{H}^{i}_{\operatorname{{\acute{e}t}}}(Y_{\bar{k}},\mathbb{Z}_{\ell})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

for any $i$, under the comparison (5.2.2). For the converse, we have the
following simple fact given by a faithfully flat descent of modules along
$\mathbb{Z}_{(\ell)}\hookrightarrow\mathbb{Z}_{\ell}$ and the $\ell$-adic
Shioda thick.

######  Lemma 5.2.3.

A (quasi)-isogeny $f\colon Y_{\mathbb{C}}\to X_{\mathbb{C}}$ is prime-
to-$\ell$ if and only it induces an isomorphism of integral $\ell$-adic
realizations

$f^{*}\colon\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(X_{\bar{k}},\mathbb{Z}_{\ell})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(Y_{\bar{k}},\mathbb{Z}_{\ell}).$

Inspired by Shioda’s trick for Hodge isogenies 4.5.1, we introduce the
following notions.

######  Definition 5.2.4.

Let $X$ and $Y$ be $g$-dimensional abelian varieties over $k$. We say $X$ and
$Y$ are (prime-to-$\ell$) _principally isogenous_ if there is a (prime-
to-$\ell$) isogeny $f$ from $X$ or $\widehat{X}$ to $Y$ of square degree,
i.e., $\deg(f)=d^{2}$ for some $d\in\mathbb{Z}$. In this case, we may call $f$
a _principal isogeny_.

Furthermore, we say $f$ is quasi-liftable if $f$ can be written as the
composition of finitely many isogenies that are liftable to characteristic
zero.

Now, we can state the main result in this section.

######  Theorem 5.2.5.

Suppose $\mathrm{char}(k)=0$. The following statements are equivalent:

  1. (1)

$X$ is (prime-to-$\ell$) principally isogenous to $Y$ over $\bar{k}$.

  2. (2)

$X$ and $Y$ are (prime-to-$\ell$) derived isogenous over $\bar{k}$.

###### Proof.

$(1)\Rightarrow(2)$: we can assume that $f:X\to Y$ is a principal isogeny
defined over a finitely generated field $k^{\prime}$. By embedding
$k^{\prime}$ into $\mathbb{C}$, two complex abelian surfaces $X_{\mathbb{C}}$
and $Y_{\mathbb{C}}$ are derived isogenous since there is a rational Hodge
isometry

$\frac{1}{n}f^{\ast}\otimes\mathbb{Q}:\mathrm{H}^{2}(Y_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Q}\cong\mathrm{H}^{2}(X_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Q}$

where $\deg(f)=n^{2}$. By Lemma 5.2.1, one can conclude $X_{\bar{k}}$ and
$Y_{\bar{k}}$ are derived isogenous, with the rational Hodge realization
$\frac{1}{n}f^{\ast}\otimes\mathbb{Q}$.

If $f$ is a prime-to-$\ell$ isogeny, the map $\frac{1}{n}f^{\ast}$ restricts
to an isomorphism

$\mathrm{H}^{2}(Y_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Z}_{(\ell)}\xrightarrow{\sim}\mathrm{H}^{2}(X_{\mathbb{C}},\mathbb{Z})\otimes\mathbb{Z}_{(\ell)}.$

Then one can take a prime-to-$\ell$ Cartan-Dieudonné decomposition (see Lemma
5.2.6 below), which decomposes $\frac{1}{n}f^{\ast}\otimes\mathbb{Q}$ into a
sequence of prime-to-$\ell$ reflexive Hodge isometries. The assertion follows
immediately.

To deduce $(2)\Rightarrow(1)$, we may assume $X$ and $Y$ are derived isogenous
over a finitely generated field $k^{\prime}$. Embed $k^{\prime}$ into
$\mathbb{C}$, $X_{\mathbb{C}}$ and $Y_{\mathbb{C}}$ are derived isogenous as
well. According to Proposition 4.5.1, they are principally isogenous over
$\mathbb{C}$. It follows that $X$ and $Y$ are principally isogenous over
$\bar{k}$ by Lemma (5.2.1).

If $\operatorname{D}^{b}(X)\sim\operatorname{D}^{b}(Y)$ is prime-to-$\ell$,
then each reflexive Hodge isometry $\psi_{b}$ in (5.1.3) is prime-to-$\ell$ by
Lemma 5.1.7. The principal isogeny which induces $\psi_{b}$ is prime-to-$\ell$
by Lemma 5.2.3. This proves the assertion.

∎

######  Lemma 5.2.6 (prime-to-$\ell$ Cartan-Dieudonné decomposition).

Let $\Lambda$ be an integral lattice over $\mathbb{Z}$ whose reduction mod
$\ell$ is still non-degenerated. Any orthogonal matrix
$A\in\operatorname{O}(\Lambda)(\mathbb{Z}_{(\ell)})\subset\operatorname{O}(\Lambda)(\mathbb{Q})$,
with $(\ell>2)$, can be decomposed into a sequence of prime-to-$\ell$
reflections.

###### Proof.

To prove the assertion, we will follow the proof of [57] to refine Cartan-
Dieudonné decomposition for any field. In general, if $\Lambda_{k}$ is
quadratic space over a field $k$ with the Gram matrix $G$. Let $I$ be the
identity matrix and let $R_{b}$ be the reflection with respect to
$b\in\Lambda_{k}$. The proof of Cartan-Dieudonné decomposition in [57] relies
on the following facts: for any element $A\in\operatorname{O}(\Lambda_{k})$,
we have

  1. i)

$A$ is a reflection if ${\rm rank}(A-I)=1$ (cf. [57, Lemma 2])

  2. ii)

if $S=G(A-I)$ is not skew symmetric, there exists $a\in\Lambda$ satisfying
$a^{t}Sa\neq 0$ and

$S+S^{t}\neq\frac{1}{a^{t}Sa}(Sb\cdot b^{t}S+S^{t}b\cdot b^{t}S^{t}),$

then ${\rm rank}(AR_{b}-I)={\rm rank}(A-I)-1$ and $G(AR_{b}-I)$ is not skew
symmetric with $b=(A-I)a$ satisfying $b^{2}=-2a^{t}Sa$. Such $a$ always
exists. (cf. [57, Lemma 4, Lemma 5]).

  3. iii)

if $S=G(A-I)$ is skew symmetric, then there exists $b\in\Lambda$ such that
$G(AR_{b}-I)$ is not skew symmetric (cf. [57, Theorem 2]).

Then one can decompose $A$ as a series of reflections by repeatedly using ii).
In our case, it suffices to show that if $A$ is coprime to $\ell$, i.e. $nA$
is integral for some $n$ coprime to $\ell$, then

  1. i’)

$A$ is a corpime to $\ell$ reflection if ${\rm rank}(A-I)=1$;

  2. ii’)

if $S=G(A-I)$ is not skew symmetric and there exists $a\in\Lambda$ satisfying
$p\nmid a^{t}Sa$ and

$S+S^{t}\neq\frac{1}{a^{t}Sa}(Sb\cdot b^{t}S+S^{t}b\cdot b^{t}S^{t}),$

then $AR_{b}$ is coprime to $\ell$ and $G(AR_{b}-I)$ is not skew symmetric
with $b$ constructed above;

  3. iii’)

if $S=G(A-I)$ is skew symmetric, then there exists $b\in\Lambda$ such that
$AR_{b}$ is coprime to $\ell$ and $G(AR_{b}-I)$ is not skew symmetric.

This means that we only need to find some prime-to-$\ell$ reflections
satisfying the conditions as above. By our assumption, the modulo $\ell$
reduction $\Lambda_{\mathbb{F}_{\ell}}$ of $\Lambda$ remains non-degenerate.
If $A$ is coprime to $\ell$, then we can consider the reduction $A\mod\ell$
and apply i)-iii) to
$A\mod\ell\in\operatorname{O}(\Lambda_{\mathbb{F}_{\ell}})$ to obtain
reflections over $\mathbb{F}_{\ell}$. We can lift the reflections to
$\operatorname{O}(\Lambda_{\mathbb{Q}})$, which are obviously coprime to
$\ell$. One can easily check such reflections satisfy ii’) and iii’). ∎

###  5.3. Proof of Theorem 1.2.1 and Corollary 1.2.2

Let us summarize all the results which conclude Theorem 1.2.1. By a similar
argument in Theorem 5.2.5, we can reduce them to the case $k=\mathbb{C}$.

#### Proof of $(i)\Leftrightarrow(ii)$

This is Theorem 5.2.5.

#### Proof of
$(i)\Leftrightarrow(vi)\Leftrightarrow(vii)\Leftrightarrow(viii)$

The equivalence $(i)\Leftrightarrow(vi)$ is Corollary 5.1.4. The equivalence
$(vi)\Leftrightarrow(vii)$ is from Witt cancellation theorem. For
$(vi)\Leftrightarrow(viii)$, note that a rational Hodge isometry
$\varphi\colon\mathrm{H}^{2}(X,\mathbb{Q})\xrightarrow{\sim}\mathrm{H}^{2}(Y,\mathbb{Q})$
induces a rational isometry ${\rm NS}(X)_{\mathbb{Q}}\xrightarrow{\sim}{\rm
NS}(Y)_{\mathbb{Q}}$. Then we have a Hodge isometry
$\mathrm{T}(X)_{\mathbb{Q}}\xrightarrow{\sim}\mathrm{T}(Y)_{\mathbb{Q}}$ by
Witt cancellation theorem. The converse is clear.

#### Proof of $(i)\Leftrightarrow(iii)$

This is Corollary 5.1.5.

#### Proof of $(ii)\Rightarrow(iv)\Rightarrow(v)$

This is from the computation in [23, Proposition 4.6]. Indeed, one may take
the correspondence

$\Gamma\coloneqq\bigoplus_{i}\Gamma_{2i}\colon\mathfrak{h}^{even}(X)\xrightarrow{\sim}\mathfrak{h}^{even}(Y),$

where

$\Gamma_{2i}\coloneqq\frac{1}{n^{i}}f^{*}\circ\pi_{X}^{2i}\colon\mathfrak{h}^{2i}(X)\to\mathfrak{h}^{2i}(Y),$

and $f\colon X\to Y$ is the given principal isogeny.

#### Proof of $(v)\Rightarrow(ii)$

Let
$\Gamma\colon\mathfrak{h}^{even}(X)\xrightarrow{\sim}\mathfrak{h}^{even}(Y)$
be the isomorphism of Frobenius algebra objects. The Betti realization of its
second component is a Hodge isometry by the Frobenius condition (cf. [23,
Theorem 3.3]). Thus $X$ and $Y$ are derived isogenous by Corollary 5.1.4, and
hence principally isogenous.

##  6\. Derived isogeny in positive characteristic

In this section, we will prove the twisted derived Torelli theorem for abelian
surfaces over odd characteristic fields. The principal strategy is to lift
everything to characteristic zero.

###  6.1. Prime-to-$p$ derived isogeny in mixed characteristic

Let us start with an important lemma for prime-to-$p$ derived isogenies, which
is the only place we require the characteristic $p>3$.

######  Lemma 6.1.1.

Let $K$ be a complete discretely valued field in characteristic zero, whose
residue field is perfect with characteristic $p>3$. Denote by
$\mathcal{O}_{K}$ the ring of integers. Let $\mathfrak{X}\to\mathcal{X}$ and
$\mathfrak{Y}\to\mathcal{Y}$ be twisted abelian surfaces over
$\mathcal{O}_{K}$ whose special fibers are $\mathscr{X}_{0}\to X_{0}$ and
$\mathscr{Y}\to Y_{0}$, and generic fibers are
$\mathfrak{X}_{K}\to\mathcal{X}_{K}$ and $\mathfrak{Y}_{K}\to\mathcal{Y}_{K}$.
Suppose
$f_{0}\colon\operatorname{D}^{b}(\mathscr{X}_{0})\xrightarrow{\sim}\operatorname{D}^{b}(\mathscr{Y}_{0})$
is a prime-to-$p$ derived equivalence and
$f\colon\operatorname{D}^{b}(\mathfrak{X})\xrightarrow{\sim}\operatorname{D}^{b}(\mathfrak{Y})$
is a lifting of $f_{0}$, then
$f_{K}\colon\operatorname{D}^{b}(\mathfrak{X}_{K})\xrightarrow{\sim}\operatorname{D}^{b}(\mathfrak{Y}_{K})$
is also prime-to-$p$.

###### Proof.

It suffices to prove that the $p$-adic realization of $f_{K}$ is integral.
This can be deduced from an argument from Cais–Liu’s crystalline cohomological
description for integral $p$-adic Hodge theory (cf. [13]).

Let us sketch the proof. As $f_{0}$ is prime-to-$p$, its cohomological
realization restricts to an isometry of $F$-crystals

$\varphi_{p}\colon\mathrm{H}^{2}_{\operatorname{crys}}(X_{0}/W)\simeq\mathrm{H}^{2}_{\operatorname{crys}}(Y_{0}/W)$

by our definition. The base-extension $\varphi_{p}\otimes K$ can be identified
with the cohomological realization of $f_{K}$ on the de Rham cohomology

$\varphi_{K}\colon\mathrm{H}^{2}_{\operatorname{dR}}(\mathcal{X}_{K}/K)\simeq\mathrm{H}^{2}_{\operatorname{dR}}(\mathcal{Y}_{K}/K)$

by Berthelot–Ogus comparison (cf. [24, Theorem B.3.1]). It also preserves the
Hodge filtration. Let $S$ be the divided power envelope of pair $(W\llbracket
u\rrbracket,\ker(W\llbracket u\rrbracket\to\mathcal{O}_{K}))$. Then the map

$\varphi_{p}\otimes_{W}S\colon\mathrm{H}^{2}_{\operatorname{crys}}(X_{0}/S)\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{crys}}(Y_{0}/S)$

is an isomorphism of strongly divisible $S$-lattices (cf. [13, §4]). If $p>3$,
then according to [13, Theorem 5.4], one can apply Breuil’s functor $T_{st}$
on it to see that $\varphi_{K}$ restricts to an $\mathbb{Z}_{p}$-integral
$\operatorname{Gal}(\bar{K}/K)$-equivariant isometry

$\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(\mathcal{X}_{\bar{K}},\mathbb{Z}_{p})\xrightarrow{\sim}\mathrm{H}^{2}_{\operatorname{{\acute{e}t}}}(\mathcal{Y}_{\bar{K}},\mathbb{Z}_{p}).$

∎

######  Remark 6.1.2.

The technical requirement for $p>3$ is needed in [13, Theorem 4.3 (3),(4)].
When $\mathcal{O}_{K}=W(k)$ is unramified, this condition can be released to
$p>2$ by using Fontaine’s result [21, Theorem 2 (iii)]. In general, when
$p=3$, a possible approach is to prove the Shioda’s trick as in §4 for
strongly divisible $S$-lattices (cf. [10, Definition 2.1.1]), which can reduce
the statement to crystalline Galois representations of Hodge–Tate weight one.

###  6.2. Serre–Tate theory and lifting of prime-to-$p$ quasi-isogeny

The Serre–Tate theorem says that the deformation theory of an abelian scheme
in characteristic $p$ is equivalent to the deformation theory of its
$p$-divisible group (cf. [44, Chapter V.§2, Theorem 2.3]). Let
$S_{0}\hookrightarrow S$ be an infinitesimal thickening of schemes such that
$p$ is locally nilpotent on $S$. Let $\mathcal{D}(S_{0},S)$ be the category of
pairs $(\mathcal{X}_{0},\mathcal{G})$, where $\mathcal{X}_{0}$ is an abelian
scheme over $S_{0}$ and $\mathcal{G}$ is a lifting of $p$-divisible group
$\mathcal{X}_{0}[p^{\infty}]$ to $S$. There is an equivalence of categories

$\displaystyle\\{\text{abelian schemes over
}S\\}\xrightarrow{\sim}\mathcal{D}(S_{0},S)$
$\displaystyle\mathcal{X}\mapsto(\mathcal{X}\times_{S}S_{0},\mathcal{X}[p^{\infty}]).$

Now we set $S_{0}=\operatorname{Spec}(k)$ and
$S=\operatorname{Spec}(V/(\pi^{n+1}))$ for a perfect field $k$, $V$ is a
totally ramified finite extension of $W(k)$ and an integer $n\geq 1$. Since
there is an equivalence between the category of $p$-divisible groups over $V$
and the category of inductive systems of $p$-divisible groups over
$V/(\pi^{n})$, we have an identification

$\mathcal{D}(k,V)=\varprojlim_{n}\mathcal{D}(k,V/(\pi^{n})).$

As a consequence, we get

######  Lemma 6.2.1.

There is an equivalence of categories

$\displaystyle\\{\text{formal abelian schemes over
}V\\}\xrightarrow{\sim}\mathcal{D}(k,V)$ $\displaystyle\quad
A\mapsto\left(A\times_{V}k,A[p^{\infty}]\right).$

One can lift separable isogenies between abelian surfaces, which is well-known
to experts.

######  Proposition 6.2.2.

Suppose $p>2$. Let $f\colon X\to Y$ be a separable isogeny. There are liftings
$\mathcal{X}\to\mathrm{Spec}(V)$ and $\mathcal{Y}\to\mathrm{Spec}(V)$ of $X$
and $Y$ respectively, such that the isogeny $f$ can be lifted to an isogeny
$f_{V}\colon\mathcal{X}\to\mathcal{Y}$. In particular, every prime-to-$p$
isogeny is liftable.

###### Proof.

Suppose we are given a lifting

$\widetilde{f}[p^{\infty}]\colon\mathcal{G}_{X}\to\mathcal{G}_{Y}$

of the isogeny of $p$-divisible groups $f[p^{\infty}]\colon X[p^{\infty}]\to
Y[p^{\infty}]$. Then we can apply Lemma 6.2.1 to get a formal lifting of $f$
to $\operatorname{Spec}(V)$:

$\widetilde{f}\colon\mathcal{X}\to\mathcal{Y},$

such that $\widetilde{f}$ is finite and $\mathcal{Y}$ admits an
algebraization. It suffices to show $\widetilde{f}$ also admits an
algebraization, which can be done by [26, Proposition (5.4.4)].

The required lifting of $p$-divisible groups can be constructed as follows.
Since $f[p^{\infty}]$ is separable, its kernel is a finite étale group scheme,
which is freely liftable. Therefore, we may assume that $f[p^{\infty}]$ is an
isomorphism. Let us fix a lifting of $\mathcal{X}$ to $V$. The $p$-divisible
group $\mathcal{G}_{X}\coloneqq\mathcal{X}[p^{\infty}]$ over $V$ forms a
lifting of $X[p^{\infty}]$ to $V$, and such lifting gives a filtered Dieudonné
module structure on $\mathbb{D}(Y[p^{\infty}])$ under the isomorphism
$f[p^{\infty}]$. Then the statement follows from the Grothendieck–Messing
theory (see the proof of [36, Proposition A.6] for example). ∎

###  6.3. Specialization of derived isogenies

Next, we shall show that prime-to-$p$ geometric derived isogenies are
preserved under reduction. The basic idea is to show that two smooth
projective family of abelian or K3 surfaces over a complete discrete valuation
ring whose geometric generic fibers are Fourier–Mukai partners will have
special fibers which are moduli space of stable twisted sheaves on each other.
For this, we only need to specialize the datum that form a moduli space.

######  Theorem 6.3.1.

Let $V$ be a discrete valuation ring with residue field $k$ and let $\eta$ be
its generic point. Assume that $\mathrm{char}(k)=p>2$. Let
$\mathcal{X}\to\mathrm{Spec}(V)$ and $\mathcal{Y}\to\mathrm{Spec}(V)$ be two
projective families of abelian surfaces or K3 surfaces over
$\mathrm{Spec}(V)$. If there is a derived equivalence

$\Psi^{\mathcal{P}}\colon\operatorname{D}^{b}(\mathcal{X}_{\bar{\eta}},\alpha)\xrightarrow{\sim}\operatorname{D}^{b}(\mathcal{Y}_{\bar{\eta}},\beta)$

between geometric generic fibers such that ${\rm ord}(\alpha)$ and ${\rm
ord}(\beta)$ are prime-to-$p$, then their special fibers are twisted derived
equivalent.

###### Proof.

We denote by $X_{0}$ and $Y_{0}$ the geometric special fibers of
$\mathcal{X}/V$ and $\mathcal{Y}/V$ respectively. From Theorem 3.4.5, it is
known that there is an isomorphism

$\mathcal{Y}_{\bar{\eta}}\cong
M^{\alpha}_{\mathcal{H}_{\eta}}(\mathcal{X}_{\bar{\eta}},v_{\bar{\eta}}),$

for some twisted Mukai vector
$v_{\bar{\eta}}\in\widetilde{\mathrm{N}}(\mathcal{X}_{{\bar{\eta}}},\alpha)$,
after replacing $(\mathcal{X},\alpha)$ by
$(\widehat{\mathcal{X}},\widehat{\alpha})$ if it is necessary. Up to taking a
finite extension of $V$, we may assume that the Brauer class $\alpha$ can be
defined over $\eta$.

We claim that one can extend $\alpha$ to a class in the Brauer group of the
total space $\operatorname{Br}(\mathcal{X})$ if $p\nmid{\rm ord}(\alpha)$. For
simplicity, we assume ${\rm ord}(\alpha)=\ell^{n}$ for some prime $\ell$. In
this case, the Gysin sequence and Gabber’s absolute purity gives an exact
sequence

$0\to\operatorname{Br}(\mathcal{X})\\{\ell\\}\to\operatorname{Br}(\mathcal{X}_{\eta})\\{\ell\\}\to\varinjlim_{n}\mathrm{H}^{1}_{\operatorname{{\acute{e}t}}}(X_{0},\mathbb{Z}/\ell^{n}).$
(6.3.1)

(cf. [15, Theorem 3.7.1 (ii)]). If $\mathcal{X}$ is a K3 surface, then we have
$\mathrm{H}^{1}_{\operatorname{{\acute{e}t}}}(X_{0},\mathbb{Z}/\ell^{n})=0$
for all $n$ as it is simply-connected, and thus one can find a lift
$\tilde{\alpha}\in\operatorname{Br}(\mathcal{X})$ of $\alpha$ by (6.3.1). When
$\mathcal{X}$ is an abelian surface over $\operatorname{Spec}(V)$, the Gysin
sequence can not directly give the existence of a extension of $\alpha$.
Again, one can use the trick of Kummer surfaces. Consider the relative Kummer
surface $\mathrm{Km}(\mathcal{X})\to\operatorname{Spec}(V)$, we have a
commutative diagram

${\operatorname{Br}(\operatorname{Km}(\mathcal{X}))\\{\ell\\}}$${\operatorname{Br}(\operatorname{Km}(\mathcal{X}_{\eta}))\\{\ell\\}}$${\operatorname{Br}(\mathcal{X})\\{\ell\\}}$${\operatorname{Br}(\mathcal{X}_{\eta})\\{\ell\\}}$$\scriptstyle{\sim}$$\scriptstyle{\Theta}$$\scriptstyle{\Theta_{\eta}}$

from Proposition 2.1.1. After passing to a finite extension, we can assume
$\alpha$ lies in the image of $\Theta_{\eta}$. As the top arrow is surjective
and $\Theta_{\eta}$ is an isomorphism, we may find an extension
$\widetilde{\alpha}\in\operatorname{Br}(\mathcal{X})\\{\ell\\}$ whose
restriction on $\mathcal{X}_{\eta}$ is $\alpha$.

As the family $\mathcal{X}\to\mathrm{Spec}(V)$ is smooth and proper, the
relative Picard scheme ${\rm Pic}_{\mathcal{X}/V}$ is proper. Now, under the
specialization of the Picard group

${\rm Pic}(\mathcal{X}_{\eta})\xleftarrow{\sim}{\rm Pic}(\mathcal{X})\to{\rm
Pic}(\mathcal{X}_{0}),$

we can pick extensions $v\in\widetilde{\mathrm{N}}(\mathcal{X})$ and
$\mathcal{H}\in{\rm Pic}(\mathcal{X})$ of $v_{\eta}$ and $\mathcal{H}_{\eta}$,
so that $v|_{\mathcal{X}_{\eta}}=v_{\eta}$ and
$\mathcal{H}|_{\mathcal{X}_{\eta}}=\mathcal{H}_{\eta}$. In general, the line
bundle $H_{0}=\mathcal{H}_{k}$ on the special fiber is not ample. However, we
may replace $\mathcal{H}$ by $\mathcal{H}\otimes\mathcal{M}^{\otimes n}$ for a
relative ample line bundle on $\mathcal{X}/V$ and $n\gg 0$, such that
$\mathcal{H}_{0}$ and $\mathcal{H}_{\eta}$ are both ample and $v$-generic
(i.e., not lie in a wall of the Mukai vector $v$), since the $v$-walls in the
ample cones of $\mathcal{X}_{\bar{\eta}}$ and $\mathcal{X}_{0}$ are known to
be (locally) finitely many hyperplanes (cf. [66, Proposition 3.10] for
$\operatorname{char}(k)=0$ or [7, Proposition 4.1.14] for
$\operatorname{char}(k)=p$). Then we let
$M^{\tilde{\alpha}}_{\mathcal{H}}(\mathcal{X},v)$ be the corresponding
relative moduli space of $\mathcal{H}$-stable twisted sheaves, which is smooth
over $V$. The generic fiber of
$M^{\widetilde{\alpha}}_{\mathcal{H}}(\mathcal{X},v)\to\operatorname{Spec}(V)$
is isomorphic to $\mathcal{Y}_{\eta}$ after a finite base extension since it
is geometrically birational to $\mathcal{Y}_{\bar{\eta}}$ by the wall-crossing
(see [46] for example).

Set $\alpha_{0}=\widetilde{\alpha}|_{X_{0}}\in\operatorname{Br}(X_{0})$. Note
that its special fiber is also isomorphic to
$M^{\alpha_{0}}_{H_{0}}(X_{0},v_{0})$ after some finite field extension, we
have the following commutative diagram after taking a finite ring extension of
$V$:

${M^{\widetilde{\alpha}}_{\mathcal{H}}(\mathcal{X},v)}$${M^{\alpha}_{\mathcal{H}_{\eta}}(\mathcal{X}_{\eta},v_{\eta})}$${\mathcal{Y}_{\eta}}$${\mathcal{Y}}$${\mathrm{Spec}(V)}$${\operatorname{Spec}(k(\eta))}$${\operatorname{Spec}(k(\eta))}$${\mathrm{Spec}(V)}$$\scriptstyle{\cong}$

According to Matsusaka–Mumford [43, Theorem 1], the isomorphism can be
extended to the special fiber. Thus $Y_{0}$ is isomorphic to
$M_{H_{0}}^{\alpha_{0}}(X_{0},v_{0})$. It follows that
$\mathrm{D}^{b}(X_{0},\alpha_{0})\simeq\mathrm{D}^{b}(Y_{0},\beta_{0})$.

∎

Using Remark 5.1.8, one can easily deduce that every prime-to-$p$ derived
isogeny can be specialized.

######  Remark 6.3.2.

In Theorem 6.3.1, the original derived equivalence $\Psi^{\mathcal{P}}$ is
shown to be extended to the whole family. We only replaced it by a
Fourier–Mukai transform given by the universal family of twisted sheaves which
is naturally be specialized.

######  Remark 6.3.3.

Our proof fails when $k$ is imperfect and the twisted derived equivalence is
not prime-to-$p$. This is because if the associated Brauer class $\alpha$ has
order $p^{n}$, the map

$\operatorname{Br}(\mathcal{X})[p^{n}]\to\operatorname{Br}(\mathcal{X}_{\eta})[p^{n}]$

may not be surjective (cf. [55, 6.8.2]).

###  6.4. Proof of Theorem 1.4.1

When $X$ or $Y$ is supersingular, the assertion will be will be proved in
Proposition 6.6.8. So we can assume that $X$ and $Y$ both have finite height.

#### Proof of $(i^{\prime})\Rightarrow(ii^{\prime})$

We can assume the prime-to-$p$ derived isogeny is defined over a finitely
generated subfield of $\bar{k}$. By the definition of prime-to-$p$ derived