1. Introduction
In this article we want to study problems related to the finiteness of the 
$p$-primary torsion of the Brauer group of abelian varieties in positive characteristic 
$p$. If 
$k$ is a finite field and 
$A$ is an abelian variety over 
$k$, it is well-known that the Brauer group of 
$A$, defined as 
$\mathrm {Br}(A):=H^2_\mathrm {\acute {e}t}(A,\mathbb {G}_m)$, is a finite group [Reference TateTat94, Proposition 4.3]. The main input for this result is the Tate conjecture for divisors, proved by Tate in [Reference TateTat66]. If 
$k$ is replaced by a finitely generated field extension of 
$\mathbb {F}_p$ one can no longer expect 
$\mathrm {Br}(A)$ to be finite (see [Reference Skorobogatov and ZarhinSZ08, § 1]). On the other hand, if 
$\mathrm {Br}(A_{{k_s}}\!)^{k}$ is the transcendental Brauer group of 
$A$, namely the image of 
$\mathrm {Br}(A)\to \mathrm {Br}(A_{{k_s}}\!)$ where 
${k_s}$ is a separable closure of 
$k$, the group 
$\mathrm {Br}(A_{{k_s}}\!)^{k}[\frac {1}{p}]$ is finite by [Reference Colliot-Thélène and SkorobogatovCS21, Theorem 16.2.3]. In [Reference Skorobogatov and ZarhinSZ08, Question 1], Skorobogatov and Zarhin asked whether the 
$p$-primary torsion of 
$\mathrm {Br}(A_{{k_s}}\!)^{k}$ is also finite. This question has a negative answer already for abelian surfaces, as we show in Proposition 5.4. Nonetheless, we prove the following alternative finiteness result. Write 
${\bar {k}}$ for an algebraic closure of 
$k_s$.
Theorem 1.1 (Theorem 5.2)
Let 
$A$ be an abelian variety over a finitely generated field 
$k$ of characteristic 
$p>0$. The transcendental Brauer group 
$\mathrm {Br}(A_{{k_s}}\!)^{k}$ is a direct sum of a finite group and a finite exponent 
$p$-group. In addition, if the Witt vector cohomology group 
$H^2(A_{{\bar {k}}},W\mathcal {O}_{A_{\bar {k}}}\!)$ is a finite 
$W({\bar {k}})$-module, then 
$\mathrm {Br}(A_{{k_s}}\!)^{k}$ is finite.
The condition on 
$H^2(A_{{\bar {k}}},W\mathcal {O}_{A_{\bar {k}}}\!)$ is necessary to remove the ‘supersingular pathologies’ as the one of our counterexample and it is satisfied, for example, when the 
$p$-rank of 
$A$ is 
$g$ or 
$g-1$, where 
$g$ is the dimension of 
$A$ (see [Reference IllusieIll83, Corollary 6.3.16]). Note that if the formal Brauer group of 
$A_{\bar {k}}$, denoted by 
$\hat {\mathrm {Br}}(A_{\bar {k}})$, is a formal Lie group, then by [Reference Artin and MazurAM77, Corollary II.4.4] the cohomology group 
$H^2(A_{{\bar {k}}},W\mathcal {O}_{A_{\bar {k}}}\!)$ is a finite 
$W({\bar {k}})$-module if and only if 
$\hat {\mathrm {Br}}(A_{\bar {k}})$ has finite height. Note also that the formal Brauer group of abelian surfaces is always a formal Lie group by [Reference Artin and MazurAM77, Corollary II.2.12]. As a consequence of Theorem 1.1, we deduce that the subgroup of Galois-fixed points of 
$\mathrm {Br}(A_{{k_s}}\!)$, denoted by 
$\mathrm {Br}(A_{{k_s}}\!)^{\Gamma _k}$, has finite exponent (Corollary 5.3). This is a variant of [Reference Skorobogatov and ZarhinSZ08, Question 2] for abelian varieties.
In this article, we also study the Galois-fixed points of 
$\mathrm {Br}(A_{{\bar {k}}})$. Ulmer in [Reference UlmerUlm14, § 7.3.1] conjectured that 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))^{\Gamma _k}=0$ where 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))$ is the 
$p$-adic Tate module of 
$\mathrm {Br}(A_{{\bar {k}}})$. Even in this case, we provide a counterexample to this conjecture. We use the following result.
Proposition 1.2 (Proposition 6.6)
Let 
$B$ be an abelian variety over a finitely generated field 
$k$ of characteristic 
$p>0$. Write 
$A$ for 
$B\times _k B$ and 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))$ for the 
$p$-adic Tate module of 
$\mathrm {Br}(A_{\bar {k}})$. There is a natural exact sequence
\[ 0\to \operatorname{Hom}(B,B^\vee)_{\mathbb{Z}_{p}}\to \operatorname{Hom}(B_{\bar{k}}[p^\infty],B_{\bar{k}}^\vee[p^\infty])^{\Gamma_k}\to \mathrm{T}_p(\mathrm{Br}(A_{\bar{k}}))^{\Gamma_k}, \]
where 
$\operatorname {Hom}(B,B^\vee )$ denotes the group of homomorphisms 
$B\to B^\vee$ as abelian varieties over 
$k$.
The proposition implies, for example, that when 
$\operatorname {End}(B)=\mathbb {Z}$ the 
$\Gamma _k$-module 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))$ admits non-zero Galois-fixed points (Corollary 6.7). In this case, 
$\mathrm {Br}(A_{{\bar {k}}})^{\Gamma _k}$ has infinite exponent since
\[ \mathrm{T}_p(\mathrm{Br}(A_{\bar{k}})^{\Gamma_k})=\mathrm{T}_p(\mathrm{Br}(A_{\bar{k}}))^{\Gamma_k}. \]
Note that if we replace 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))$ with the 
$\ell$-adic Tate module 
$\mathrm {T_\ell }(\mathrm {Br}(A_{\bar {k}}))$, where 
$\ell$ is a prime different from 
$p$, then 
$\mathrm {T_\ell }(\mathrm {Br}(A_{{k_s}}\!))=\mathrm {T_\ell }(\mathrm {Br}(A_{\bar {k}}))$ has no non-trivial Galois-fixed points.
These ‘exceptional classes’ in 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))^{\Gamma _k}$ are naturally related to specialisation morphisms of Néron–Severi groups. We recall the following theorem, which was proved in [Reference AndréAnd96, Theorem 5.2] in characteristic 
$0$ (see also [Reference Maulik and PoonenMP12]) and in [Reference AmbrosiAmb23] and [Reference ChristensenChr18] in positive characteristic.
Theorem 1.3 (André, Ambrosi, Christensen)
Let 
$K$ be an algebraically closed field which is not an algebraic extension of a finite field, 
$X$ a finite-type 
$K$-scheme, and 
$\mathcal {Y}\to X$ a smooth proper morphism. For every geometric point 
$\bar {\eta }$ of 
$X$ there is an 
$x\in X(K)$ such that 
$\operatorname {rk}_\mathbb {Z}(\mathrm {NS}(\mathcal {Y}_{\bar {\eta }}))=\operatorname {rk}_\mathbb {Z}(\mathrm {NS}(\mathcal {Y}_x))$.Footnote 1
As it is well-known, the theorem is false when 
$K=\bar {\mathbb {F}}_p$ (see [Reference Maulik and PoonenMP12, Remark 1.12]). What we prove is that, in the known counterexamples, the elements in 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))^{\Gamma _k}$ explain the failure of Theorem 1.3. More precisely, we prove the following result.
Theorem 1.4 (Theorem 6.2)
Let 
$X$ be a connected normal scheme of finite type over 
$\mathbb {F}_p$ with generic point 
$\eta =\operatorname {Spec}(k)$ and let 
$f:\mathcal {A}\to X$ be an abelian scheme over 
$X$ with constant Newton polygon.Footnote 2 For every closed point 
$x=\operatorname {Spec}(\kappa )$ of 
$X$ we have
\[ \operatorname{rk}_\mathbb{Z}(\mathrm{NS}(\mathcal{A}_{\bar{x}})^{\Gamma_\kappa})-\operatorname{rk}_\mathbb{Z}(\mathrm{NS}(\mathcal{A}_{\bar{\eta}})^{\Gamma_k})\geq \operatorname{rk}_{\mathbb{Z}_{p}} ( \mathrm{T}_p(\mathrm{Br}(\mathcal{A}_{\bar{\eta}}))^{\Gamma_k}). \]
Note that in the inequality the left term is ‘motivic’, whereas the right term comes from some 
$p$-adic object which, as far as we know, has no 
$\ell$-adic analogue. Note also that 
$\mathrm {T}_p(\mathrm {Br}(\mathcal {A}_{\bar {x}}))^{\Gamma _\kappa }=0$ by Corollary 5.3 since 
$\kappa$ is a perfect field.
To prove Theorem 1.1 we use a flat variant of the Tate conjecture. For every 
$n$, let 
$H^2_\mathrm {fppf}(A_{\bar {k}},{\mu _{p^n}}\!)^k$ be the image of the extension of scalars morphism 
$H^2_\mathrm {fppf}(A,{\mu _{p^n}}\!)\to H^2_\mathrm {fppf}(A_{\bar {k}},{\mu _{p^n}}\!)$.
Theorem 1.5 (Theorem 5.1)
After possibly replacing 
$k$ with a finite separable extension, the cycle class map
\[ c_1:\mathrm{NS}(A)_{{\mathbb{Z}_{p}}}\to\varprojlim_n H^2_\mathrm{fppf}(A_{\bar{k}},{\mu_{p^n}}\!)^k \]
becomes an isomorphism.Footnote 3
We obtain this result by using the crystalline Tate conjecture for abelian varieties, proved by de Jong in [Reference de JongdeJ98, Theorem 2.6]. The main issue that we have to overcome is the lack of a good comparison between crystalline and fppf cohomology of 
${\mathbb {Z}_{p}}(1)$ over imperfect fields. To avoid this problem, we exploit the fact that we are working with abelian varieties. In this special case, the comparison is constructed using the 
$p$-divisible group of 
$A$ (and its dual).
The technical issue that we have to solve using the groups 
$H^2_{\mathrm {fppf}}(A_{\bar {k}},{\mu _{p^n}}\!)^k$ is that it is not clear a priori whether 
$H^2_{\mathrm {fppf}}(A,{\mathbb {Z}_{p}}(1))\to \varprojlim _n H^2_{\mathrm {fppf}}(A_{\bar {k}},{\mu _{p^n}}\!)^k$ is surjective. This is done (after inverting 
$p$) in Proposition 3.9, where we reduce to the case when 
$A$ is the Jacobian of a curve. This idea was inspired by the proof of [Reference Colliot-Thélène and SkorobogatovCS13, Theorem 2.1].
1.6 Outline of the article
In § 3 we prove some general results on the cohomology of fppf sheaves. In particular, we prove Corollary 3.4, which is a first result on the relation between the Brauer group of a scheme over 
${k_s}$ and 
${\bar {k}}$. In this section, we also prove in Proposition 3.8 the exactness of some fundamental sequences for the groups 
$H^2_\mathrm {fppf}(X_{\bar {k}},{\mu _{p^n}}\!)^k$. In § 4, we construct a morphism which relates 
$H^2_\mathrm {fppf}(A,{\mathbb {Z}_{p}}(1))$ with 
$\operatorname {Hom}(A[p^\infty ],A^\vee [p^\infty ])$ and we prove basic properties of this morphism as Propositions 4.5 and 4.6. In § 5, we prove the flat variant of the Tate conjecture (Theorem 5.1) and the finiteness result for the transcendental Brauer group (Theorem 5.2). Finally, in § 6, we look at the relation of our results with the theory of specialisation of Néron–Severi groups. In particular, we prove Theorem 6.2.
2. Notation
If 
$k$ is a field, we write 
${\bar {k}}$ for a fixed algebraic closure of 
$k$ and 
${k_s}$ (respectively, 
${k_i}$) for the separable (respectively, purely inseparable) closure of 
$k$ in 
${\bar {k}}$. We denote by 
$\Gamma _k$ the absolute Galois group of 
$k$. If 
$x$ is a 
$k$-point of a scheme, we denote by 
$\bar {x}$ the induced 
${\bar {k}}$-point. For an abelian group 
$M$, we write 
$\mathrm {T}_p(M)$ for the 
$p$-adic Tate module of 
$M$, which is the projective limit 
$\varprojlim _{n}M[p^n]$, we write 
$\mathrm {V}_p(M)$ for 
$\mathrm {T}_p(M)[\frac {1}{p}]$, and we write 
$M^\wedge$ for the 
$p$-adic completion of 
$M$. If 
$M$ is endowed with a 
$\Gamma _k$-action, we denote by 
$M^{\Gamma _k}$ the subgroup of fixed points. For a scheme 
$X$ and an fppf sheaf 
$\mathcal {F}$, we denote by 
$H^{\bullet }(X,\mathcal {F})$ the fppf cohomology groups and when 
$X=\operatorname {Spec}(k)$ we simply write 
$H^{\bullet }(k,\mathcal {F})$. If 
$f:X\to Y$ is a morphism of schemes, we denote by 
$R^{\bullet }f_*\mathcal {F}$ the fppf higher direct image functors over 
$(\mathbf {Sch}/Y)_\mathrm {fppf}$. Finally, if 
$X$ is a scheme over 
$\mathbb {F}_p$, we write 
$X^{\mathrm {perf}}$ for the projective limit 
$\varprojlim (\cdots \xrightarrow {F}X\xrightarrow {F}X\xrightarrow {F}X)$ where 
$F$ is the absolute Frobenius of 
$X$.
3. Preliminary results
In this section we start by proving some results that we will use later on. We work over a field 
$k$ of arbitrary characteristic and we consider a scheme 
$X$ over 
$k$ with structural morphism 
$q$.
Lemma 3.1 Let 
$\mathcal {F}$ be a sheaf over 
$(\mathbf {Sch}/k)_\mathrm {fppf}$ such that 
$q_*\mathcal {F}_{X}=\mathcal {F}$ and suppose that 
$X$ has a 
$k$-rational point. The group 
$H^0(k,R^1q_*\mathcal {F}_{X}\!)$ is canonically isomorphic to 
$H^1(X,\mathcal {F}_X\!)/H^1(k,\mathcal {F})$. In addition, the natural morphism 
$H^2(X,\mathcal {F}_X\!)\to H^0(k,R^2q_*\mathcal {F}_{X}\!)$ sits in an exact sequence
\[ 0\to K\to H^2(X,\mathcal{F}_X\!)\to H^0(k,R^2q_*\mathcal{F}_{X}\!)\to H^2(k,R^1q_*\mathcal{F}_{X}\!), \]
where 
$K$ is an extension of 
$H^1(k,R^1q_*\mathcal {F}_{X}\!)$ by 
$H^2(k,\mathcal {F})$.
Proof. We consider the Leray spectral sequence
\[ E^{i,j}_2=H^i(k,R^jq_{*}\mathcal{F}_X\!)\Rightarrow H^{i+j}(X,\mathcal{F}_X\!). \]
The morphisms 
$E^{i,0}_2=H^i(k,q_*\mathcal {F}_X\!)=H^i(k,\mathcal {F})\to H^i(X,\mathcal {F}_X\!)$ are injective since 
$X$ admits a 
$k$-rational point. We deduce that 
$E^{1,1}_2=E^{1,1}_\infty$ and 
$E^{2,0}_2=E^{2,0}_\infty$. This implies that the kernel of 
$H^2(X,\mathcal {F}_X\!)\to E^{0,2}_\infty$ is an extension of 
$E^{1,1}_2$ by 
$E^{2,0}_2$, as we wanted. The obstruction for the map 
$H^2(X,\mathcal {F}_X\!)\to E^{0,2}_2=H^0(k,R^2q_*\mathcal {F}_{X}\!)$ to be surjective lies in 
$E^{2,1}_2=H^2(k,R^1q_*\mathcal {F}_X\!)$. This concludes the proof.
Definition 3.2 We say that a presheaf 
$\mathcal {F}$ on 
$(\mathbf {Sch}^{\mathrm {qcqs}}/k)_{\mathrm {fppf}}$ is finitary if for every inverse system 
$\{T^{(\ell )}\}_{\ell \in L}$ of quasi-compact quasi-separated 
$k$-schemes with affine transition maps, the natural morphism
\[ \mathrm{colim}_{\ell\in L}\mathcal{F}(T^{(\ell)})\to \mathcal{F}\big(\lim_{\ell\in L} T^{(\ell)}\big) \]
is an isomorphism.
Lemma 3.3 Let 
$G$ be a commutative finite-type group scheme over 
$k$. If 
$X$ is quasi-compact quasi-separated, then 
$R^iq_*G_X$ is finitary for 
$i\geq 0$. In addition, the natural morphism 
$H^0(k,R^i q_*G_X\!)\to H^{i}(X_{\bar {k}},G_{X_{\bar {k}}}\!)$ is injective.
Proof. Let 
${\mathcal {H}}^i(q,G_{X}\!)$ be the higher presheaf pushforward of 
$G_{X}$ on 
$X$ with respect to 
$q$. We first want to prove that 
${\mathcal {H}}^i(q,G_{X}\!)$ is finitary for 
$i\geq 0$. In other words, we want to prove that for every inverse system 
$\{T^{(\ell )}\}_{\ell \in L}$ of quasi-compact quasi-separated 
$k$-schemes, the natural morphism
\[ \mathrm{colim}_{\ell\in L}H^i(X^{(\ell)},G_{X^{(\ell)}})\to H^i(X^{(\infty)},G_{X^{(\infty)}}\!) \]
is an isomorphism, where 
$X^{(\ell )}:=X\times _k T^{(\ell )}$ and 
$X^{(\infty )}:=\lim _{\ell \in L}X^{(\ell )}$. By [Sta23, Tag 01H0],
\[ H^i(X^{(\ell)},G_{X^{(\ell)}})=\mathrm{colim}_{U_\bullet^{(\ell)}\in \mathrm{HC}(X^{(\ell)})}\check{H}^i(U_\bullet^{(\ell)},G_{U_\bullet^{(\ell)}}) \]
for every 
$\ell \in L \coprod \{\infty \}$, where 
$\mathrm {HC}(X^{(\ell )})$ is the category of fppf hypercoverings of 
$X^{(\ell )}$. Since each 
$X^{(\ell )}$ is quasi-compact quasi-separated, by [Sta23, Tag 021P] we can replace the category 
$\mathrm {HC}(X^{(\ell )})$ in the colimit with the subcategory 
$\mathrm {HC}(X^{(\ell )})^{\mathrm {qcqs}}$, consisting of those hypercoverings such that 
$U^{(\ell )}_n$ is quasi-compact quasi-separated for every 
$n\geq 0$. By [Sta23, Lemma 01ZM], for 
$U^{(\infty )}_\bullet \in \mathrm {HC}(X^{(\infty )})^{\mathrm {qcqs}}$ and 
$n\geq 0$ there exists an 
$\ell \in L$ and 
$U_\bullet ^{(\ell,n)}\in \mathrm {HC}(X^{(\ell )})^{\mathrm {qcqs}}$ such that
\[ \mathrm{tr}_n(U_\bullet^{(\ell,n)}\times_{T^{(\ell)}} T^{(\infty)})\simeq \mathrm{tr}_n(U^{(\infty)}_\bullet), \]
where 
$\mathrm {tr}_n(-)$ denotes the 
$n$th truncation of simplicial schemes and 
$T^{(\infty )}:=\lim _{\ell \in L}T^{(\ell )}$. This implies that
\[
H^i(X^{(\infty)},G_{X^{(\infty)}}\!)=\mathrm{colim}_{\ell\in
L}\mathrm{colim}_{U_\bullet^{(\ell)}\in
\mathrm{HC}(X^{(\ell)})}\check{H}^i(U_\bullet^{(\ell)}\times_{T^{(\ell)}}
T^{(\infty)},G_{U_\bullet^{(\ell)}\times_{T^{(\ell)}}
T^{(\infty)}}). \]
We are reduced to proving that for every 
$\ell \in L$ and 
$U_\bullet ^{(\ell )}\in \mathrm {HC}(X^{(\ell )})^{\mathrm {qcqs}}$ we have that
\[ \check{H}^i(U_\bullet^{(\ell)}\times_{T^{(\ell)}}T^{(\infty)},G_{U_\bullet^{(\ell)}\times_{T^{(\ell)}}T^{(\infty)}})=\mathrm{colim}_{\ell\leq \ell'}\check{H}^i(U_\bullet^{(\ell)}\times_{T^{(\ell)}}T^{(\ell')},G_{U_\bullet^{(\ell)}\times_{T^{(\ell)}}T^{(\ell')}}). \]
Since 
$G$ is of finite type over 
$k$, this follows from [Sta23, Lemma 01ZM] and the exactness of filtered colimits.
Knowing that 
${\mathcal {H}}^i(q,G_{X}\!)$ is finitary, in order to prove that 
$R^iq_*G_X\!$ is finitary as well it is enough to prove that for every finitary presheaf 
$\mathcal {F}$ on 
$(\mathbf {Sch}^{\mathrm {qcqs}}/k)_{\mathrm {fppf}}$, the ‘partial’ sheafification 
$\mathcal {F}^+$ (defined as in [Sta23, § 00W1]) is finitary. Similarly to the previous paragraph, the proof of this fact follows from the observation that each finite quasi-compact quasi-separated fppf covering of 
$X^{(\infty )}$ descends to a covering of 
$X^{(\ell )}$ for some 
$\ell \in L$ and Čech cohomology commutes with filtered colimits (
$\check {H}^0$ is enough in this case).
For the second part, we note that for every presheaf 
$\mathcal {F}$ on 
$(\mathbf {Sch}/k)_{\mathrm {fppf}}$ with sheafification 
$\mathcal {F}^\sharp$, the natural morphism
\[ \mathcal{F}(\operatorname{Spec}({\bar{k}}))\to \mathcal{F}^\sharp(\operatorname{Spec}({\bar{k}})) \]
is an isomorphism because every fppf covering of 
$\operatorname {Spec}({\bar {k}})$ admits a section. This implies that
\[ H^0({\bar{k}},R^iq_*G_X\!)= H^{i}(X_{\bar{k}},G_{X_{\bar{k}}}). \]
Thanks to the previous part, we deduce that the composition
\[ H^0(k,R^i q_*G_X\!)\hookrightarrow
\mathrm{colim}_{{k'/k\ \mathrm{fin.}}}{H}^0(k',R^i
q_*G_X\!)\xrightarrow{\sim} H^0({\bar{k}},R^iq_*G_X\!)=
H^{i}(X_{\bar{k}},G_{X_{\bar{k}}}) \]
is injective, where the colimit runs over all finite field extensions of 
$k$. This ends the proof.
With the previous results we can prove [Reference GrothendieckGro68, Proposition 5.6], which was stated by Grothendieck without a complete proof.Footnote 4
Corollary 3.4 If 
$k$ is separably closed and 
$X$ is a proper 
$k$-scheme, then there is a natural exact sequence
\[ 0\to H^1(k,\mathrm{Pic}_{X/k})\to\mathrm{Br}(X)\to \mathrm{Br}(X_{{\bar{k}}}). \]
In particular, if 
$\mathrm {Pic}_{X/k}$ is smooth, then the natural morphism 
$\mathrm {Br}(X)\to \mathrm {Br}(X_{{\bar {k}}})$ is injective.
Proof. As in Lemma 3.1, we consider the Leray spectral sequence
\[
E^{i,j}_2=H^i(k,R^jq_{*}\mathbb{G}_{m,X}\!)\Rightarrow
H^{i+j}(X,\mathbb{G}_{m,X}\!). \]
Since 
$X$ is proper over 
$k$, by [Sta23, Tag 0BUG] we deduce that 
$A:=H^0(X,\mathcal {O}_X\!)$ is a finite 
$k$-algebra. This implies that 
$q_*\mathbb {G}_m$ is represented by a smooth group scheme over 
$k$. Thanks to [Reference GrothendieckGro68, Theorem 11.7], we deduce that 
$E^{i,j}_2=0$ for 
$i>0$ and 
$j=0$, so that 
$E^{1,1}_2=E^{1,1}_\infty$. The Leray spectral sequence produces then the exact sequence
\[ 0\to H^1(k,\mathrm{Pic}_{X/k})\to\mathrm{Br}(X)\to H^0(k,R^2q_*\mathbb{G}_{m,X}\!). \]
To get the first part of the statement it is then enough to apply Lemma 3.3. For the second part, we note that when 
$\mathrm {Pic}_{X/{k}}$ is smooth, thanks to [Reference GrothendieckGro68, Theorem 11.7], the group 
$H^1(k,\mathrm {Pic}_{X/{k}})$ vanishes.
Definition 3.5 For a scheme 
$X$ over 
$k$ and a prime 
$p$, we define 
$H^2(X,\mathbb {Z}_p(1))$ as the projective limit
\[ \varprojlim_{n} H^2(X,{\mu_{p^n}}\!). \]
Remark 3.6 Note that we are defining 
$H^2(X,\mathbb {Z}_p(1))$ without taking into account higher inverse limits. Nonetheless, if 
$k$ is algebraically closed of characteristic 
$p$ and 
$X$ is smooth and proper over 
$k$, then 
$R^1\varprojlim _n H^1(X,{\mu _{p^n}}\!)=R^1\varprojlim _n \mathrm {Pic}(X)[p^n]=0$ since 
$\mathrm {Pic}(X)[p^\infty ]$ is a direct sum of a 
$p$-divisible group and a finite group and 
$R^1\varprojlim _n H^2(X,{\mu _{p^n}}\!)=0$ by [Reference IllusieIll79, Chapter II, Proposition 5.9].
Construction 3.7 The Kummer exact sequences for 
$X$ and 
$X_{{\bar {k}}}$ (for the fppf topology) induce the following commutative diagram with exact rows.
We write
\[ C_n(X):=(\mathrm{Pic}(X_{\bar{k}})/p^n)^k\to H^2(X_{\bar{k}},{\mu_{p^n}}\!)^{k}\to (\mathrm{Br}(X_{{\bar{k}}})[p^n])^{k}\to 0\to \cdots \]
for the complex obtained by taking images of the vertical arrows. Note that a priori 
$(\mathrm {Br}(X_{{\bar {k}}})[p^n])^{k}$ is smaller than 
$\mathrm {Br}(X_{\bar {k}})^k[p^n]$, where 
$\mathrm {Br}(X_{\bar {k}})^k:=\mathrm {im}(\mathrm {Br}(X)\to \mathrm {Br}(X_{\bar {k}}))$.
Since both 
$R^1\varprojlim _{n}\mathrm {Pic}(X)/p^n$ and 
$R^1\varprojlim _{n}\mathrm {Pic}(X_{\bar {k}})/p^n$ vanish, we can also consider the following commutative diagram with exact rows:
obtained by taking the projective limit of the diagrams (3.7.1) for various 
$n$. We denote by
\[ \hat{C}(X):= (\mathrm{Pic}(X_{\bar{k}})^\wedge)^k\to H^2(X_{\bar{k}},\mathbb{Z}_p(1))^{k}\to \mathrm{T}_p(\mathrm{Br}(X_{{\bar{k}}}))^{k}\to 0\to \cdots \]
the complex obtained by taking images of the vertical arrows.
Proposition 3.8 If 
$\mathrm {char}(k)=p$ and 
$A$ is an abelian variety over 
$k$ such that the morphism 
$\mathrm {Pic}(A)\to \mathrm {NS}(A_{{\bar {k}}})$ is surjective, then the complexes 
$C_n(A)$ and 
$\hat {C}(A)$ are acyclic.
Proof. If 
$K_{1,n}$ is the kernel of 
$H^2_{}(A,{\mu _{p^n}}\!)\to H^2(A_{\bar {k}},{\mu _{p^n}}\!)$ and 
$K_{2}$ is the kernel of 
$\mathrm {Br}(A)\to \mathrm {Br}(A_{\bar {k}})$, in order to prove that 
$C_n(A)$ is acyclic we have to show that 
$K_{1,n}\to K_{2}[p^n]$ is surjective. Combining Lemmas 3.1 and 3.3, we deduce the following commutative diagram with exact rows.
The morphism of exact sequences factors through the complex
\[ \mathrm{Br}(k)[p^n]\to K_2[p^n]\to H^1_{}(k,\mathrm{Pic}_{A/k})[p^n]\to 0\to \cdots, \]
which is acyclic because 
$\mathrm {Br}(k)$ is 
$p$-divisible by [Reference Colliot-Thélène and SkorobogatovCS21, Theorem 1.3.7]. The image of
\[ H^1(k,\mathrm{Pic}_{A/k}[p^n])\to H^1_{}(k,\mathrm{Pic}_{A/k}^\circ) \]
is 
$H^1_{}(k,\mathrm {Pic}_{A/k}^\circ )[p^n]$, thus we are reduced to prove that
\[ H^1_{}(k,\mathrm{Pic}_{A/k}^\circ)[p^n]\to H^1_{}(k,\mathrm{Pic}_{A/k})[p^n] \]
is surjective. Since 
$\mathrm {Pic}(A)\to \mathrm {NS}(A_{\bar {k}})$ is surjective, we know that 
$\pi _0(\mathrm {Pic}_{A/k})$ is a constant finitely generated torsion-free group over 
$k$ such that 
$\mathrm {Pic}_{A/k}(k)\to \pi _0(\mathrm {Pic}_{A/k})(k)$ is surjective. Looking at the cohomology long exact sequence associated to
\[ 0\to \mathrm{Pic}_{A/k}^\circ\to \mathrm{Pic}_{A/k}\to \pi_0(\mathrm{Pic}_{A/k})\to 0, \]
we then deduce that 
$H^1_{}(k,\mathrm {Pic}_{A/k}^\circ )\xrightarrow {\sim } H^1_{}(k,\mathrm {Pic}_{A/k})$, which yields the desired result.
We now prove that 
$\hat {C}(A)$ is acyclic. The kernel of 
$H^2(A,{\mathbb {Z}_{p}}(1))\to H^2(A_{\bar {k}},{\mathbb {Z}_{p}}(1))$ is 
$\varprojlim _n K_{1,n}$ and the kernel of 
$\mathrm {T}_p(\mathrm {Br}(A))\to \mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))$ is 
$\mathrm {T}_p(K_2)$. Thus, again, we have to prove that 
$\varprojlim _n K_{1,n}\to \mathrm {T}_p(K_2)$ is surjective. Combining the previous discussion and the fact that 
$\mathrm {Br}(k)$ is 
$p$-divisible, we deduce that the two groups sit in the following diagram with exact rows.
For every 
$n>0$, the kernel of 
$H^1_{}(k,\mathrm {Pic}_{A/k}[p^n]) \to H^1_{}(k,\mathrm {Pic}_{A/k})[p^n]$ is 
$\mathrm {Pic}(A)/p^n$ and the groups 
$(\mathrm {Pic}(A)/p^n)_{n>0}$ form a Mittag–Leffler system. We deduce that the morphism
\[ \varprojlim_{n} H^1_{}(k,\mathrm{Pic}_{A/k}[p^n])\to \mathrm{T}_p(H^1_{}(k,\mathrm{Pic}_{A/k})) \]
is surjective. This implies that 
$\hat {C}(A)$ is acyclic, as we wanted.
The proof of the following proposition was inspired by [Reference Colliot-Thélène and SkorobogatovCS13, Theorem 2.1].
Proposition 3.9 If 
$\mathrm {char}(k)=p$ and 
$A$ is an abelian variety over 
$k$, we have 
$H^2(A_{\bar {k}},{\mathbb {Z}_{p}}(1))^{k}[\frac {1}{p}]=(\varprojlim _{n}H^2(A_{\bar {k}},{\mu _{p^n}}\!)^{k})[\frac {1}{p}]$ and 
$\mathrm {T}_p(\mathrm {Br}(A_{{\bar {k}}}))^{k}[\frac {1}{p}]=\mathrm {V}_p(\mathrm {Br}(A_{{\bar {k}}})^{k}).$
Proof. We first note that the four 
${\mathbb {Q}_{p}}$-vector spaces are invariant under isogenies of 
$A$ and finite separable extension of 
$k$. Indeed, for every isogeny 
$\varphi :B\to A$ there exists an isogeny 
$\psi : A\to B$ such that the composition 
$\varphi \circ \psi$ is the multiplication by some positive integer 
$n$. Since 
$n$ is invertible in 
${\mathbb {Q}_{p}}$, we deduce that 
$\varphi ^*$ is an isomorphism at the level of cohomology groups. Similarly, if 
$k'/k$ is a finite separable extension, then the pullback morphisms with respect to 
$A_{k'}\to A$ admit as inverse the morphisms 
$ ({1}/{[k':k]})\mathrm {Tr}_{A_{k'}/A}$.
Next, thanks to [Reference KatzKat99, Theorem 11], we note that there exists a proper smooth connected curve 
$C$ with a rational point and a morphism 
$C\to A$ such that 
$B:=\mathrm {Jac}(C)$ maps surjectively to 
$A$. By Poincaré's complete reducibility theorem, 
$B$ is isogenous to a product 
$A\times _k A'$ with 
$A'$ an abelian variety over 
$k$. Since 
$H^2(A_{\bar {k}},{\mathbb {Z}_{p}}(1))^{k}$ (respectively, 
$\mathrm {Br}(A_{{\bar {k}}})^k$) is a direct summand of 
$H^2(A_{\bar {k}}\times _{\bar {k}} A'_{\bar {k}},{\mathbb {Z}_{p}}(1))^{k}$ (respectively, 
$\mathrm {Br}(A_{\bar {k}}\times _{\bar {k}} A'_{{\bar {k}}})^k$) and the property we want to prove is invariant by isogenies, it is then enough to prove the result for 
$B$. In addition, since in the statement it is harmless to extend 
$k$ to a finite separable extension, we may assume that 
$\mathrm {Pic}(B)\to \mathrm {NS}(B_{{\bar {k}}})$ is surjective, so that 
$H^1(k,\mathrm {Pic}^0_{B/k})=H^1(k,\mathrm {Pic}_{B/k})$.
Let 
$K_{1,n}$ be the kernel of the morphism 
$H^2(B,{\mu _{p^n}}\!)\to H^2(B_{\bar {k}},{\mu _{p^n}}\!)$. By Lemmas 3.1 and 3.3, the group 
$K_{1,n}$ is an extension of 
$H^1(k,\mathrm {Pic}_{B/k}[p^n])$ by 
$\mathrm {Br}(k)[p^n]$ and by the assumption 
$\mathrm {Pic}_{C/k}[p^n]=\mathrm {Pic}_{B/k}[p^n]$. We deduce that 
$K_{1,n}=\ker (H^2(C,{\mu _{p^n}}\!)\to H^2(C_{\bar {k}},{\mu _{p^n}}\!))$. By [Reference GrothendieckGro68, Rmq. 2.5.b], the group 
$\mathrm {Br}(C_{\bar {k}})$ vanishes, thus 
$H^2(C_{{\bar {k}}},{\mathbb {Z}_{p}}(1))={\mathbb {Z}_{p}}$ and the morphism 
$H^2(C,{\mathbb {Z}_{p}}(1))\to H^2(C_{{\bar {k}}},{\mathbb {Z}_{p}}(1))$ is surjective because 
$C$ has a rational point. This implies that 
$R^1 \varprojlim _{n}K_{1,n}\to R^1 \varprojlim _{n}H^2(C,{\mu _{p^n}}\!)$ is injective. Since
\[ R^1 \varprojlim_{n}K_{1,n}\to R^1 \varprojlim_{n}H^2(C,{\mu_{p^n}}\!) \]
factors through 
$R^1 \varprojlim _{n}H^2(B,{\mu _{p^n}}\!),$ we deduce that 
$R^1 \varprojlim _{n}K_{1,n}\to R^1 \varprojlim _{n}H^2(B,{\mu _{p^n}}\!)$ is injective as well. Therefore, the morphism 
$H^2(B,{\mathbb {Z}_{p}}(1))\to \varprojlim _{n}H^2(B_{\bar {k}},{\mu _{p^n}}\!)^{k}$ is surjective. Thanks to Proposition 3.8, for every 
$n$ the morphism 
$H^2(B_{\bar {k}},{\mu _{p^n}}\!)^{k}\to (\mathrm {Br}(B_{\bar {k}})[p^n])^k$ is surjective with finite kernel, so that 
$\varprojlim _{n}H^2(B_{\bar {k}},{\mu _{p^n}}\!)^{k}\to \varprojlim _n(\mathrm {Br}(B_{\bar {k}})[p^n])^k$ is surjective as well. This implies that 
$\mathrm {T}_p(\mathrm {Br}(B_{{\bar {k}}}))^{k}\to \varprojlim _n(\mathrm {Br}(B_{\bar {k}})[p^n])^k$ is surjective.
It remains to prove that for every 
$n$ we have 
$(\mathrm {Br}(B_{\bar {k}})[p^n])^k=\mathrm {Br}(B_{\bar {k}})^k[p^n]$. Consider the natural morphism 
$K_3\to \mathrm {Br}(C)$ where 
$K_3$ is the kernel of 
$\mathrm {Br}(B)\to \mathrm {Br}(B_{{\bar {k}}})$. Thanks to Lemmas 3.1 and 3.3 and using the fact that 
$\mathrm {Br}(C_{\bar {k}})=0$, this morphism sits in the following commutative diagram with exact rows.
Since 
$\mathrm {Pic}(B)\to \mathrm {NS}(B_{\bar {k}})$ is surjective and 
$C$ is a curve, we have that
\[ H^1(k,\mathrm{Pic}_{B/k})=H^1(k,\mathrm{Pic}^0_{B/k})\simeq H^1(k,\mathrm{Pic}^0_{C/k})=H^1(k,\mathrm{Pic}_{C/k}). \]
We deduce that 
$K_3\to \mathrm {Br}(C)$ is an isomorphism, thus 
$\mathrm {Br}(B)\to \mathrm {Br}(C)\xrightarrow {\sim } K_3$ provides a splitting of the exact sequence
\[ 0\to K_3\to \mathrm{Br}(B)\to \mathrm{Br}(B_{\bar{k}})^k\to 0. \]
This implies that 
$\mathrm {Br}(B)[p^n]\to \mathrm {Br}(B_{{\bar {k}}})^k[p^n]$ is surjective and this yields the desired result.
4. Constructing a morphism
Let 
$A$ be an abelian variety over a field 
$k$. For a line bundle 
$\mathcal {L}$ of 
$A$ we write 
$\varphi _{\mathcal {L}}:A\to A^\vee$ for the morphism which sends 
$x\mapsto t_x^*\mathcal {L}\otimes \mathcal {L}^{-1}$, where 
$t_x$ is the translation by 
$x$. In this section we want to complete the following solid square.
If 
$k$ is an algebraically closed field of characteristic 
$0$ such a commutative diagram is constructed in [Reference Orr, Skorobogatov and ZarhinOSZ21, Lemma 2.6] using an analytic method. We propose instead an algebraic construction which works for any field.
4.1
Consider the morphism 
$h_1:H^2(A,{\mu _{p^n}}\!)\to H^2(A\times _k A,{\mu _{p^n}}\!)$ which sends a class 
$\alpha$ to 
$m^*(\alpha )-\pi _1^*(\alpha )-\pi _2^*(\alpha )$, where 
$\pi _1$ and 
$\pi _2$ are the two projections of 
$A\times _k A$. This morphism has the property that the first Chern class 
$c_1(\mathcal {L})\in H^2(A,{\mu _{p^n}}\!)$ of a line bundle 
$\mathcal {L}$ is sent to 
$c_1(\Lambda (\mathcal {L}))$, the first Chern class of the associated Mumford bundle 
$\Lambda (\mathcal {L}):=m^*\mathcal {L}\otimes \pi _1^*\mathcal {L}^{-1}\otimes \pi _2^*\mathcal {L}^{-1}$. The Leray spectral sequence
\begin{equation} E^{i,j}_2:=H^i(A,R^j\pi_{2*}{\mu_{p^n}}\!)\Rightarrow H^{i+j}(A\times_k A,{\mu_{p^n}}\!) \end{equation}
induces a filtration 
$0\subseteq F^2 H^{2}(A\times _k A,{\mu _{p^n}}\!) \subseteq F^1 H^{2}(A\times _k A,{\mu _{p^n}}\!)\subseteq H^{2}(A\times _k A,{\mu _{p^n}}\!)$.
Lemma 4.2 The image of 
$h_1$ lies in 
$F^1H^{2}(A\times _k A,{\mu _{p^n}}\!)$.
Proof. The spectral sequence (4.1.1) gives the exact sequence
\[ 0\to F^1H^{2}(A\times_k A,{\mu_{p^n}}\!)\to H^{2}(A\times_k A,{\mu_{p^n}}\!)\to E^{0,2}_\infty\to 0. \]
Therefore, it is enough to check that the composition
\[ H^2(A,{\mu_{p^n}}\!)\xrightarrow{h_1} H^2(A\times_k A,{\mu_{p^n}}\!)\to H^0(A,R^2\pi_{2*}{\mu_{p^n}}\!) \]
is the 
$0$-morphism. By [Reference Bragg and OlssonBO21, Corollary 1.4], there exists a commutative linear algebraic groupFootnote 5 
$G$ over 
$k$ which represents 
$R^2q_{*}{\mu _{p^n}}$ on the big fppf site 
$(\mathbf {Sch}/k)_{\mathrm {fppf}}$. Since 
$R^2\pi _{2*}{\mu _{p^n}}$ is the restriction of 
$R^2q_{*}{\mu _{p^n}}$ from 
$(\mathbf {Sch}/k)_{\mathrm {fppf}}$ to 
$(\mathbf {Sch}/A)_{\mathrm {fppf}}$, this implies that 
$H^0(A,R^2\pi _{2*}{\mu _{p^n}}\!)$ can be computed as 
$\mathrm {Mor}_{\mathbf {Sch}/k}(A,G)$. Thanks to the fact that 
$G$ is affine, every morphism 
$A\to G$ contracts 
$A$ to a point. We deduce that 
$\mathrm {Mor}_{\mathbf {Sch}/k}(A,G)=\mathrm {Mor}_{\mathbf {Sch}/k}(0_A,G)=H^0(k,R^2q_{*}{\mu _{p^n}}\!)$. By Lemma 3.3, the group 
$H^0(k,R^2q_{*}{\mu _{p^n}}\!)$ is naturally a subgroup of 
$H^2(A_{\bar {k}},{\mu _{p^n}}\!)$ and the induced morphism
\[ H^2(A\times_k A,{\mu_{p^n}}\!)\to H^0(k,R^2q_{*}{\mu_{p^n}}\!)\hookrightarrow H^2(A_{\bar{k}},{\mu_{p^n}}\!) \]
is given by the pullback via 
$i_1:A=A\times _k 0_A \hookrightarrow A\times _k A$ followed by the extension of scalars to 
${\bar {k}}$. By construction, we have that 
$i_1^*\circ h_1=i_1^*\circ m^*-i_1^*\circ \pi _1^*=0$. This concludes the proof.
Lemma 4.3 Let 
$G$ be a finite commutative group scheme killed by a positive integer 
$n$. There is a natural injective morphism 
$f_n:\operatorname {Hom}(A[n],G)\to H^1(A,G)$ which admits a retraction 
$g_n$.
Proof. Write 
$P_{n}$ for the 
$A[n]$-torsor over 
$A$ given by the multiplication by 
$n$. The morphism 
$f_n$ is then defined by 
$f_n(\sigma ):=\sigma _*P_{n}$ for every 
$\sigma \in \operatorname {Hom}(A[n],G)$. We want to define now 
$g_n$ which sends a 
$G$-torsor 
$P$ over 
$A$ to an homomorphism 
$g_n(P): A[n]\to G$. By Cartier duality, this is the same as defining a morphism 
$g_n(P)^\vee :G^\vee \to (A[n])^\vee =A^\vee [n]$. For a scheme 
$T$ over 
$k$ and a 
$T$-point of 
$G^\vee$ corresponding to a morphism 
$\tau :G_T\to \mathbb {G}_{m,T}$ we define 
$g_n(P)^\vee (\tau )$ as 
$\tau _{*}P_T\in H^1(A_T,\mathbb {G}_{m,T})[n]=A^\vee [n](T)$. To prove that 
$g_n\circ f_n=\operatorname {id}$ it is enough to note that for every 
$\sigma \in \operatorname {Hom}(A[n],G)$, every scheme 
$T$ over 
$k$, and every 
$\tau \in \operatorname {Hom}(G_T,\mathbb {G}_{m,T})$ we have that 
$g_n(f_n(\sigma ))^\vee (\tau )=\tau _*(\sigma _*P_n)_T=(\tau \circ \sigma _T)_*P_{n,T}$ is the line bundle over 
$A_T$ associated to 
$\tau \circ \sigma _T\in (A[n])^\vee (T)$ under the identification 
$(A[n])^\vee =A^\vee [n]$.
4.4
Thanks to Lemma 4.2, we can define
\[ \bar{h}_1: H^2(A,{\mu_{p^n}}\!)\to H^1(A,A^\vee[p^n]) \]
as the composition of 
$h_1$ and the natural morphism
\[ F^1H^2(A\times_k A,{\mu_{p^n}}\!)\to H^1(A,R^1\pi_{2*}{\mu_{p^n}}\!)=H^1(A,A^\vee[p^n]). \]
In addition, by Lemma 4.3 applied to 
$G=A^\vee [p^n]$, we get a morphism
\[ h_2:H^1(A, A^\vee[p^n])\to\operatorname{Hom}(A[p^n],A^\vee[p^n]). \]
We write
\[ h: H^2(A,{\mu_{p^n}}\!)\to \operatorname{Hom}(A[p^n],A^\vee[p^n]) \]
for the composition 
${h_2}\circ \bar {h}_1$ and we denote with the same letter the induced morphism 
$H^2(A,{\mathbb {Z}_{p}}(1))\to \operatorname {Hom}(A[p^\infty ],A^\vee [p^\infty ])$.
Proposition 4.5 The square
is commutative.
Proof. We have to show that for every line bundle 
$\mathcal {L}\in \mathrm {Pic}(A)$ we have
\[ h(c_1(\mathcal{L}))=\varphi_{\mathcal{L}}|_{A[p^n]}. \]
Consider the Leray spectral sequence
\begin{equation} E^{i,j}_2=H^i(A^\vee,R^j\pi_{2*}{\mu_{p^n}}\!)\Rightarrow H^{i+j}(A\times_k A^\vee,{\mu_{p^n}}\!). \end{equation}
The morphism 
$A\times _k A\xrightarrow {\operatorname {id}_A\times \varphi _{\mathcal {L}}} A\times _k A^\vee$ induces via pullback a morphism from (4.5.2) to (4.1.1). This produces the commutative diagram
where the composition of the lower horizontal arrows is 
$h$. If 
$\mathcal {P}\in \mathrm {Pic}({A\times _k A^\vee })$ is the Poincaré bundle of 
$A$, we have that 
$\Lambda (\mathcal {L})=(\operatorname {id}_A\times \varphi _{\mathcal {L}})^* \mathcal {P}$. This implies that 
$h_1(c_1(\mathcal {L}))=c_1(\Lambda (\mathcal {L}))=(\operatorname {id}_A\times \varphi _{\mathcal {L}})^* c_1(\mathcal {P})$. In addition, by direct inspection, we note that 
$h_2(\varphi _{\mathcal {L}}^*([P]))=\varphi _\mathcal {L}|_{A[p^n]}$, where 
$[P]\in H^1(A^\vee,A^\vee [p^n])$ is the class of the torsor 
$A^\vee \xrightarrow {\cdot p^n} A^\vee$. It remains to prove that the morphism 
$F^1H^2(A\times _k A^\vee,{\mu _{p^n}}\!)\to H^1(A^\vee,A^\vee [p^n])$ sends 
$c_1(\mathcal {P})$ to 
$[P]$. For this purpose, we introduce the Leray spectral sequence
\begin{equation} E^{i,j}_2=H^i(A^\vee,R^j\pi_{2*}\mathbb{G}_m[1])\Rightarrow H^{i+j}(A\times_k A^\vee,\mathbb{G}_m[1]). \end{equation}
The morphism 
$\delta :\mathbb {G}_m[1]\to {\mu _{p^n}}$ associated to the Kummer exact sequence induces a morphism from (4.5.3) to (4.5.2) which we denote with the same symbol. In turn, this produces the following commutative diagram.
The upper horizontal arrow sends the line bundle 
$\mathcal {P}$ to 
$\operatorname {id}_{A^\vee }\in H^0(A^\vee,A^\vee )$, while 
$\delta$ sends 
$\operatorname {id}_{A^\vee }$ to 
$[P]$. This yields the desired result.
Proposition 4.6 The morphism 
$h:H^2(A_{\bar {k}},{\mathbb {Z}_{p}}(1))\to \operatorname {Hom}_{}(A_{\bar {k}}[p^\infty ],A^\vee _{\bar {k}}[p^\infty ])$ is an injective morphism with image 
$\operatorname {Hom}^{\mathrm {sym}}_{}(A_{\bar {k}}[p^\infty ],A^\vee _{\bar {k}}[p^\infty ])$, the group of homomorphisms which are fixed by the involution 
$\tau \mapsto \tau ^\vee$.
Proof. Suppose 
$\mathrm {char}(k)=p$ and write 
$W$ for the ring of Witt vectors of 
${\bar {k}}$. The crystalline cohomology groups of an abelian variety are torsion free by [Reference Berthelot, Breen and MessingBBM82, Corollary 2.5.5]. Therefore, thanks to the Künneth formula, [Reference BerthelotBer74, Theorem V.4.2.1], we have that 
${H^*_{{\mathrm {crys}}}(A_{\bar {k}}\times _{\bar {k}} A_{\bar {k}}/W)=H^*_{{\mathrm {crys}}}(A_{\bar {k}}/W)\otimes H^*_{{\mathrm {crys}}}(A_{\bar {k}}/W)}$ so that 
$m:A\times _k A\to A$ induces a morphism
\[ m^*:H^*_{{\mathrm{crys}}}(A_{\bar{k}}/W)\to H^*_{{\mathrm{crys}}}(A_{\bar{k}}/W)\otimes H^*_{{\mathrm{crys}}}(A_{\bar{k}}/W). \]
In degree 
$2$ we get a morphism
\[ m^*:H^2_{{\mathrm{crys}}}(A_{\bar{k}}/W)\to H^2_{{\mathrm{crys}}}(A_{\bar{k}}/W)\oplus H^1_{{\mathrm{crys}}}(A_{\bar{k}}/W)^{\otimes 2} \oplus H^2_{{\mathrm{crys}}}(A_{\bar{k}}/W), \]
which, in turn, induces a morphism
\[ m^*-\pi_1^*-\pi_2^*: H^2_{{\mathrm{crys}}}(A_{\bar{k}}/W)\to H^1_{{\mathrm{crys}}}(A_{\bar{k}}/W)^{\otimes 2}. \]
Write 
$\sigma : \bigwedge ^2 H^1_{{\mathrm {crys}}}(A_{\bar {k}}/W)\xrightarrow {\sim } H^2_{{\mathrm {crys}}}(A_{\bar {k}}/W)$ for the natural isomorphism induced by the cup product, as in [Reference Berthelot, Breen and MessingBBM82, Corollary 2.5.5]. For every 
$v\in H^1_{{\mathrm {crys}}}(A_{\bar {k}}/W)$, the pullback 
$m^*(v)$ is equal to 
$\pi _1^*(v)+\pi _2^*(v)$. Therefore, the composition 
$(m^*-\pi _1^*-\pi _2^*)\circ \sigma : \bigwedge ^2 H^1_{{\mathrm {crys}}}(A_{\bar {k}}/W)\hookrightarrow H^1_{{\mathrm {crys}}}(A_{\bar {k}}/W)^{\otimes 2}$ is equal to the natural embedding 
$v\wedge w\mapsto v\otimes w- w\otimes v$. By [Reference Berthelot, Breen and MessingBBM82, Theorem 5.1.8], we have that the 
$F$-crystal 
$H^1_{{\mathrm {crys}}}(A_{\bar {k}}/W)$ over 
${\bar {k}}$ is canonically isomorphic to 
$H^1_{{\mathrm {crys}}}(A_{\bar {k}}^\vee /W)^\vee$ with 
$F$-structure defined as the dual of the 
$F$-structure of 
$H^1_{{\mathrm {crys}}}(A_{\bar {k}}^\vee /W)$ multiplied by 
$p$. Thus, we have that
\[ (H^1_{{\mathrm{crys}}}(A_{\bar{k}}/W)^{\otimes 2})^{F=p}=\operatorname{Hom}_{\mathbf{F\textbf{-}Crys}({\bar{k}})}(H^1_{{\mathrm{crys}}}(A_{\bar{k}}^\vee/W),H^1_{{\mathrm{crys}}}(A_{\bar{k}}/W)), \]
where 
$\mathbf {F\textbf {-}Crys}({\bar {k}})$ is the category of 
$F$-crystals over 
${\bar {k}}$. By [Reference IllusieIll79, Rmq. II.3.11.2], the 
$F$-crystals 
$H^1_{{\mathrm {crys}}}(A_{\bar {k}}/W)$ and 
$H^1_{{\mathrm {crys}}}(A_{\bar {k}}^\vee /W)$ are the contravariant crystalline Dieudonné modules of the 
$p$-divisible groups 
$A_{\bar {k}}[p^\infty ]$ and 
$A^\vee _{\bar {k}}[p^\infty ]$, thus we get
\[ (H^1_{{\mathrm{crys}}}(A_{\bar{k}}/W)^{\otimes 2})^{F=p}=\operatorname{Hom}(A_{\bar{k}}[p^\infty],A^\vee_{\bar{k}}[p^\infty]). \]
On the other hand, by [Reference IllusieIll79, Theorem II.5.14], there is a canonical isomorphism 
$H^2(A_{\bar {k}},{\mathbb {Z}_{p}}(1))=H^2_{{\mathrm {crys}}}(A_{\bar {k}}/W)^{F=p}$. This concludes the case when 
$\mathrm {char}(k)=p$. If 
$p$ is invertible in 
$k$ one can replace crystalline cohomology with 
$p$-adic étale cohomology.
5. Main results
We are now ready to prove our main result, which is a flat version of the Tate conjecture for divisors of abelian varieties.
Theorem 5.1 If 
$A$ is an abelian variety over a finitely generated field 
$k$ of characteristic 
$p>0$, then 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}})^k)=0$. Moreover, after possibly replacing 
$k$ with a finite separable extension, the cycle class map
\begin{align*} c_1:\mathrm{NS}(A)_{{\mathbb{Z}_{p}}}\to\varprojlim_n H^2(A_{\bar{k}},{\mu_{p^n}}\!)^k \end{align*}
becomes an isomorphism.
Proof. To prove the statement we may assume that 
$\mathrm {Pic}(A)\to \mathrm {NS}(A_{{\bar {k}}})$ is surjective by extending 
$k$. The 
${\mathbb {Z}_{p}}$-module 
$\operatorname {Hom}(A[p^\infty ],A^\vee [p^\infty ])$ embeds into 
$\operatorname {Hom}(A_{\bar {k}}[p^\infty ],A^\vee _{\bar {k}}[p^\infty ])$, therefore the morphism
\[ h:H^2(A,{\mathbb{Z}_{p}}(1))\to \operatorname{Hom}(A[p^\infty],A^\vee[p^\infty]) \]
induces a morphism 
$\tilde {h}:H^2(A_{\bar {k}},\mathbb {Z}_p(1))^{k}\to \operatorname {Hom}(A[p^\infty ],A^\vee [p^\infty ])$. By Proposition 4.6, we know that 
$\tilde {h}$ is injective and 
$\mathrm {im}(\tilde {h})$ is contained in 
$\operatorname {Hom}^{\mathrm {sym}}(A[p^\infty ],A^\vee [p^\infty ])$. In addition, by Proposition 4.5, we have the following commutative square.
The lower arrow is an isomorphism by [Reference de JongdeJ98, Theorem 2.6], and since 
$\mathrm {NS}(A)=\operatorname {Hom}^{\mathrm {sym}}(A,A^\vee )$, we deduce that 
$\mathrm {im}(\tilde {h}\circ c_1)=\operatorname {Hom}^{\mathrm {sym}}(A[p^\infty ],A^\vee [p^\infty ])$. This implies that
\[ c_1:\mathrm{NS}(A)_{\mathbb{Z}_{p}}\to H^2(A_{\bar{k}},\mathbb{Z}_p(1))^{k} \]
is surjective, thus by Proposition 3.9 we get that
\[ c_1:\mathrm{NS}(A)_{\mathbb{Q}_{p}}\to \varprojlim_n H^2(A_{\bar{k}},{\mu_{p^n}}\!)^k[\tfrac{1}{p}] \]
is surjective. Combining this with Proposition 3.8, we deduce that
\[ c_1:\mathrm{NS}(A)_{\mathbb{Z}_{p}}\to \varprojlim_n H^2(A_{\bar{k}},{\mu_{p^n}}\!)^k \]
is surjective. For the result about the Brauer group, we just note that by the previous argument and Proposition 3.8, the 
${\mathbb {Z}_{p}}$-module 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))^k$ vanishes. Therefore, thanks to Proposition 3.9, we deduce that 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}})^k)=\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}})^k)[\frac {1}{p}]=0$.
Theorem 5.2 Let 
$A$ be an abelian variety over a finitely generated field 
$k$ of characteristic 
$p>0$. The transcendental Brauer group 
$\mathrm {Br}(A_{{k_s}}\!)^{k}$ is a direct sum of a finite group and a finite exponent 
$p$-group. In addition, if the Witt vector cohomology group 
$H^2(A_{{\bar {k}}},W\mathcal {O}_{A_{\bar {k}}}\!)$ is a finite 
$W({\bar {k}})$-module, then 
$\mathrm {Br}(A_{{k_s}}\!)^{k}$ is finite.
Proof. By [Reference Colliot-Thélène and SkorobogatovCS21, Theorem 16.2.3], the group 
$\mathrm {Br}(A_{{k_s}}\!)^{k}[\frac {1}{p}]$ is finite. Moreover, thanks to Corollary 3.4, the morphism 
$\mathrm {Br}(A_{{k_s}}\!)\to \mathrm {Br}(A_{{\bar {k}}})$ is injective, which implies that the transcendental Brauer group is the same as 
$\mathrm {Br}(A_{{\bar {k}}})^{k}$. Write 
$\mathbf {Ab}_p^\star \subseteq \mathbf {Ab}$ for the full subcategory of the category of abstract abelian groups with objects those (possibly infinite) 
$p$-groups isomorphic to 
$({\mathbb {Q}_{p}}/{\mathbb {Z}_{p}})^{\oplus a} \oplus M$ for some 
$a\geq 0$ and 
$M$ a finite exponent 
$p$-group. Equivalently, 
$\mathbf {Ab}_p^\star$ is the subcategory of those 
$p$-groups 
$M$ such that 
$M[p^{n+1}]/M[p^n]$ is finite for 
$n$ big enough. Note that this subcategory is closed under the operation of taking subobjects, quotients, and finite direct sums. We first want to prove that 
$H:=\varinjlim _n H^2(A_{\bar {k}},{\mu _{p^n}}\!)\in \mathbf {Ab}_p^\star$ and when 
$H^2(A_{{\bar {k}}},W\mathcal {O}_{A_{\bar {k}}}\!)$ is a finite 
$W({\bar {k}})$-module then, in addition, 
$H[p]$ is finite (so that 
$H[p^n]$ is also finite for every 
$n\geq 0$). Thanks to the Kummer exact sequence this implies the same result for 
$\mathrm {Br}(A_{{\bar {k}}})^{k}[p^\infty ]$.
Write 
$q: A_{\bar {k}} \to \operatorname {Spec}({\bar {k}})$ for the structural morphism. By [Reference Bragg and OlssonBO21, Corollary 1.4], for every 
$n$ there exists a commutative linear algebraic group 
$G_n$ representing 
$R^2q_{*}\mu _{p^n}$.Footnote 6 Write 
$U_n$ for the unipotent radical of 
$G_n$ and 
$D_n$ for the reductive quotient 
$G_n/U_n$. Since 
$G_n$ is commutative, there is a canonical Levi decomposition 
$G_n=U_n\times D_n$. In particular, we have that 
$H=U\times D$, where 
$U:=\varinjlim _nU_n({\bar {k}})$ and 
$D:=\varinjlim _nD_n({\bar {k}})$. For every 
$n>0$, the group scheme 
$D_n$ is finite, because it is a reductive group killed by 
$p^n$. In addition, by [Reference Bragg and OlssonBO21, Proposition 10.7], there is a canonical isomorphism of formal groups 
$\varinjlim _n \hat {G}_n=\varinjlim _n \hat {U}_n=\Phi ^2_{\mathrm {fl}}(A_{\bar {k}},\mathbb {G}_m)$, where 
$\hat {G}_n$ and 
$\hat {U}_n$ are the formal completions at the identity of 
$G_n$ and 
$U_n$ and 
$\Phi ^2_{\mathrm {fl}}(-,-)$ is as in [Reference Bragg and OlssonBO21, § 10.6].
Applying 
$Rq_*$ to the exact sequence
\begin{equation} 1\to {\mu_{p^n}} \to \mu_{p^{n+1}}\xrightarrow{\cdot p^n} \mu_p\to 1 \end{equation}
and using the fact that 
$\varinjlim _n R^1q_{*}\mu _{p^n}=\mathrm {Pic}_{A_{{\bar {k}}}/{\bar {k}}}[p^\infty ]$ is a 
$p$-divisible group, we get the exact sequence
\begin{align*} 1\to G_n\to G_{n+1}\xrightarrow{\cdot p^n} G_1. \end{align*}
As a first consequence, we deduce that for every 
$n>0$ the group scheme 
$D_n$ is the same as 
$D_{n+1}[p^n]$, thus 
$D[p]=D_1({\bar {k}})$ is finite. In particular, the abstract group 
$D$ is in 
$\mathbf {Ab}_p^\star$. To bound 
$U$, we note that by [Reference MilneMil86, Proposition 3.1] the dimension of the chain of algebraic groups 
$G_1\subseteq G_2\subseteq \cdots$ is eventually constant. Therefore, there exists 
$N>0$ such that for every 
$n\geq N$, the morphism 
$(U_n)_{\mathrm {red}}\to (U_{n+1})_{\mathrm {red}}$ is an isomorphism. This shows that 
$U$ is a finite exponent 
$p$-group.
If 
$H^2(A_{\bar {k}},W\mathcal {O}_{A_{\bar {k}}}\!)$ is a finite 
$W({\bar {k}})$-module, then the formal group 
$\Phi ^2_{\mathrm {fl}}(A_{\bar {k}},\mathbb {G}_m)$ does not contain any copy of 
$\hat {\mathbb {G}}_a$. Indeed, by [Reference Bragg and OlssonBO21, Corollary 12.5], the group 
$H^2(A_{\bar {k}},W\mathcal {O}_{A_{\bar {k}}}\!)$ is the Cartier module of 
$\Phi ^2_{\mathrm {fl}}(A_{\bar {k}},\mathbb {G}_m)$ and, by the assumption, it cannot contain 
${\bar {k}}[[V]]$, the Cartier module of 
$\hat {\mathbb {G}}_a$. Therefore, in this case, we have that each group 
$U_n({\bar {k}})$ is trivial, so that 
$H[p]=D[p]$ is finite.
We can finally prove that 
$\mathrm {Br}(A_{{\bar {k}}})^{k}[p^\infty ]$ has finite exponent. Suppose by contradiction that this is not the case. Since 
$\mathrm {Br}(A_{{\bar {k}}})^{k}[p^\infty ]\in \mathbf {Ab}_p^\star$, we deduce that it contains a copy of 
${\mathbb {Q}_{p}}/{\mathbb {Z}_{p}}$. On the other hand, by Theorem 5.1, the group 
$\mathrm {T}_p(\mathrm {Br}(A_{{\bar {k}}})^{k})$ vanishes, which leads to a contradiction.
Corollary 5.3 The group 
$\mathrm {Br}(A_{{k_s}}\!)^{\Gamma _k}$ has finite exponent.
Proof. This follows from Theorem 5.2 thanks to [Reference Colliot-Thélène and SkorobogatovCS21, Theorem 5.4.12].
We end this section with some examples of abelian varieties over finitely generated fields with infinite transcendental Brauer group. Let 
$E$ be a supersingular elliptic curve over an infinite finitely generated field 
$k$ and let 
$A$ be the product 
$E\times _k E$.
Proposition 5.4 After possibly extending 
$k$ to a finite separable extension, the transcendental Brauer group 
$\mathrm {Br}(A_{{k_s}}\!)^k$ becomes infinite.
Proof. Even in this case we use that, thanks to Corollary 3.4, the transcendental Brauer group is the same as 
$\mathrm {Br}(A_{{\bar {k}}})^{k}$. Moreover, after extending the scalars we may assume that the morphism 
$\mathrm {Pic}(A)\to \mathrm {NS}(A_{{\bar {k}}})$ is surjective. Combining Proposition 3.8 and the fact that 
$\mathrm {NS}(A_{{\bar {k}}})/p$ is finite we deduce that it is enough to show that 
$H^2(A_{{\bar {k}}},\mu _p)^k$ is infinite. We look at the Leray spectral sequence with respect to the second projection 
$\pi _2:A=E\times _k E\to E$ (both over 
$k$ and over 
${\bar {k}}$). In the second page, we have that the boundary morphism 
$H^1(E,R^1\pi _{2*} \mu _p) \to H^3 (E,\pi _{2*}\mu _p)$ vanishes because 
$H^3 (E,\pi _{2*}\mu _p)\to H^3(A,\mu _p)$ admits a retraction induced by the zero section of 
$\pi _2$. Since 
$H^0(E_{\bar {k}},R^2\pi _{2*}\mu _p)=H^2(E_{\bar {k}},\mu _p)=\mathbb {Z}/p$, it is then enough to show that the image of
\[ H^1(E,E[p])=H^1(E,R^1\pi_{2*}\mu_p)\to H^1(E_{\bar{k}},R^1\pi_{2*}\mu_p)=H^1(E_{\bar{k}},E_{\bar{k}}[p]) \]
is infinite. By Lemma 4.3, we have that 
$\operatorname {End}(E[p])$ (respectively, 
$\operatorname {End}(E_{{\bar {k}}}[p])$) admits a natural embedding in 
$H^1(E,E[p])$ (respectively, 
$H^1(E_{{\bar {k}}},E_{{\bar {k}}}[p])$). Since
\[ k=\operatorname{End}(\alpha_p)\subseteq \operatorname{End}(E[p])\subseteq \operatorname{End}(E_{{\bar{k}}}[p]) \]
by the assumption that 
$E$ is supersingular, we deduce the desired result.
6. Specialisation of Néron–Severi groups
6.1
We want to start this section with an explicative example. Let 
$\mathcal {E}\to X$ be a non-isotrivial family of ordinary elliptic curves, where 
$X$ is a connected normal scheme of finite type over 
$\mathbb {F}_p$. Let 
$\mathcal {A}$ be the fibred product 
$\mathcal {E}\times _X \mathcal {E}$. We denote by 
$E$ and 
$A$ the generic fibres over the generic point 
$\operatorname {Spec}(k)\hookrightarrow X$. The Kummer exact sequence induces the exact sequence
\[ 0\to\mathrm{NS}(A_{\bar{k}})_{\mathbb{Z}_p} \to H^2_{}(A_{\bar{k}},\mathbb{Z}_p(1))\to \mathrm{T}_p(\mathrm{Br}(A_{\bar{k}}))\to 0. \]
The group 
$\mathrm {NS}(A_{\bar {k}})_{\mathbb {Z}_p}$ is of rank 
$2+\operatorname {rk}_\mathbb {Z}(\operatorname {End}(E_{\bar {k}}))=3$, whereas, by Proposition 4.6, the rank of 
$H^2_{}(A_{\bar {k}},\mathbb {Z}_p(1))$ is 
$2+\operatorname {rk}_{\mathbb {Z}_{p}}(\operatorname {End}(E_{\bar {k}}[p^\infty ]))=4$. This shows that 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))$ is of rank 
$1$. The endomorphisms of 
$E_{{\bar {k}}}[p^\infty ]$ are all defined over 
${k_i}$, which implies that the action of 
$\Gamma _k$ on 
$H^2_{}(A_{\bar {k}},\mathbb {Z}_p(1))$ is trivial. In particular, the morphism 
$\mathrm {NS}(A)_{\mathbb {Z}_p} \to H^2_{}(A_{\bar {k}},\mathbb {Z}_p(1))^{\Gamma _k}$ is not surjective and the cokernel 
$\mathrm {T}_p(\mathrm {Br}(A_{\bar {k}}))^{\Gamma _k}$ is isomorphic to 
${\mathbb {Z}_{p}}$.
In this case, the Galois action on 
$H^2_{}(A_{\bar {k}},\mathbb {Z}_p(1))$ is not enough to detect what classes are 
${\mathbb {Z}_{p}}$-linear combinations of algebraic cycles. There is an additional obstruction to descend cohomology classes through the purely inseparable extension 
${k_i}/k$. This extra purely inseparable obstruction gives an explanation of the failure of surjectivity of specialisation morphisms of Néron–Severi groups. In the example, if 
$\operatorname {Spec}(\kappa )\hookrightarrow X$ is a closed point, we have that 
$\mathrm {NS}(\mathcal {A}_{\kappa })=\mathrm {NS}(\mathcal {A}_{\bar {\kappa }})$ is of rank 
$4$ because 
$\operatorname {End}(\mathcal {E}_\kappa )=\operatorname {End}(\mathcal {E}_{\bar \kappa })$ is of rank 
$2$ (there is an extra Frobenius endomorphism). Thus, the specialisation map 
$\mathrm {NS}(A)\hookrightarrow \mathrm {NS}(\mathcal {A}_\kappa )$ is never surjective even if the rank of 
$H^2_{}(A_{\bar {\kappa }},\mathbb {Z}_p(1))$ is 
$4$ as the generic geometric fibre and the Galois action is trivial in both cases. One can interpret this failure by saying that the extra obstruction on 
$H^2_{}(A_{\bar {k}},\mathbb {Z}_p(1))$ coming from the purely inseparable extension 
${k_i}/k$ is trivial on 
$H^2_{}(A_{\bar {\kappa }},\mathbb {Z}_p(1))$ since 
$\kappa$ is perfect. In general, we prove the following theorem.
Theorem 6.2 Let 
$X$ be a connected normal scheme of finite type over 
$\mathbb {F}_p$ with generic point 
$\eta =\operatorname {Spec}(k)$ and let 
$f:\mathcal {A}\to X$ be an abelian scheme over 
$X$ with constant Newton polygon. For every closed point 
$x=\operatorname {Spec}(\kappa )$ of 
$X$ we have
\[ \operatorname{rk}_\mathbb{Z}(\mathrm{NS}(\mathcal{A}_{\bar{x}})^{\Gamma_\kappa})-\operatorname{rk}_\mathbb{Z}(\mathrm{NS}(\mathcal{A}_{\bar{\eta}})^{\Gamma_k})\geq \operatorname{rk}_{\mathbb{Z}_{p}} ( \mathrm{T}_p(\mathrm{Br}(\mathcal{A}_{\bar{\eta}}))^{\Gamma_k}). \]
Remark 6.3 Note that after replacing 
$X$ with a finite étale cover the action of 
$\Gamma _k$ on 
$\mathrm {NS}(\mathcal {A}_{\bar {\eta }})$ is trivial. Thus, we also get an inequality before taking Galois-fixed points.
To prove Theorem 6.2 we first need the following result.
Proposition 6.4 Under the assumptions of Theorem 6.2, the functor 
$\mathcal {F}$ which sendsFootnote 7 
$T\in (X^{\mathrm {perf}})_{\mathrm {pro}\mathrm {\acute {e}t}}$ to 
$\operatorname {Hom}^{{\mathrm {sym}}}(\mathcal {A}_T[p^\infty ],\mathcal {A}^\vee _T[p^\infty ])[\frac {1}{p}]$ is a semi-simple finite-rank 
${\mathbb {Q}_{p}}$-local system such that for every 
$\bar {x}\in X(\bar {\mathbb {F}}_p)$ we have
\[ \mathcal{F}_{\bar{x}}=\operatorname{Hom}^{{\mathrm{sym}}}(\mathcal{A}_{\bar{x}}[p^\infty],\mathcal{A}^\vee_{\bar{x}}[p^\infty])[\tfrac{1}{p}]. \]
Proof. Let 
$\widetilde {\mathcal {F}}$ be the functor which sends 
$T\in (X^{\mathrm {perf}})_{\mathrm {pro}\mathrm {\acute {e}t}}$ to 
$\operatorname {Hom}(\mathcal {A}_T[p^\infty ],\mathcal {A}^\vee _T[p^\infty ])[\frac {1}{p}]$. We first note that to prove the result we can replace 
$\mathcal {F}$ with 
$\widetilde {\mathcal {F}}$ since 
$\mathcal {F}$ is the kernel of the 
${\mathbb {Q}_{p}}$-linear endomorphism 
$\alpha -\operatorname {id}_{\widetilde {\mathcal {F}}}:\widetilde {\mathcal {F}}\to \widetilde {\mathcal {F}}$ where 
$\alpha$ sends 
$\tau \in \operatorname {Hom}(\mathcal {A}_T[p^\infty ],\mathcal {A}^\vee _T[p^\infty ])[\frac {1}{p}]$ to 
$\tau ^\vee$. Write 
$\mathbf {F\textrm {-}Crys}(X)$ for the category of 
$F$-crystals over the absolute crystalline site of 
$X$ and let 
$\mathcal {M}_1,\mathcal {M}_2\in \mathbf {F\textrm {-}Crys}(X)$ be the contravariant crystalline Dieudonné modules of 
$\mathcal {A}[p^\infty ]$ and 
$\mathcal {A}^\vee [p^\infty ]$ over 
$X$ constructed in [Reference Berthelot, Breen and MessingBBM82, Déf. 3.3.6]. By [Reference Berthelot, Breen and MessingBBM82, Theorem 5.1.8], we have that 
$\mathcal {M}_1= \mathcal {M}_2^\vee (-1)$ where 
$\mathcal {M}_2^\vee (-1)$ is the 
$F$-crystal 
$\mathcal {H}om(\mathcal {M}_2,\mathcal {O}_{X,\mathrm {cris}})$ endowed with the dual of the 
$F$-structure of 
$\mathcal {M}_2$ multiplied by 
$p$. Thus 
$\mathcal {M}_1^{\otimes 2}$ is equal to 
$\mathcal {H}om(\mathcal {M}_2,\mathcal {M}_1)$ endowed with the natural 
$F$-structure multiplied by 
$p$. By [Reference LauLau13, Theorem D], for every perfect scheme 
$T\to X$ we have canonical isomorphisms
\[ \Gamma(T,\mathcal{M}_1^{\otimes 2})^{F=p}=\operatorname{Hom}_{\mathbf{F\textrm{-}Crys}(T)}(\mathcal{M}_{2,T},\mathcal{M}_{1,T})=\operatorname{Hom}(\mathcal{A}_T[p^\infty],\mathcal{A}^\vee_T[p^\infty]), \]
where 
$\mathcal {M}_{1,T}$ and 
$\mathcal {M}_{2,T}$ are the inverse images of 
$\mathcal {M}_1$ and 
$\mathcal {M}_2$ to 
$T$. These isomorphisms are equivariant with respect to the action of the abstract group 
$\operatorname {Aut}(T/X)$.
By [Reference KatzKat79, Theorem 2.5.1], the slope filtration of the 
$F$-crystal 
$\mathcal {M}_{1,X^{\textrm {perf}}}^{\otimes 2}$ (which exists since 
$\mathcal {A}\to X$ has constant Newton polygon) splits uniquely up to isogeny. We denote by 
$\mathcal {N}^{[1]}_{X^\textrm {perf}}$ the slope 
$1$ subobject of 
$\mathcal {M}_{1,X^{\textrm {perf}}}^{\otimes 2}$, defined up to isogeny. Note that for every 
$T\to X^\textrm {perf}$ we have that
\[ \Gamma(T,\mathcal{M}_{1,T}^{\otimes 2})^{F=p}=\Gamma(T,\mathcal{N}^{[1]}_T)^{F=p}=\Gamma(T,\mathcal{N}^{[1]}_{T}(1))^{F=1}. \]
By construction, the 
$F$-crystal 
$\mathcal {N}^{[1]}_{X^{\textrm {perf}}}(1)$ is unit-root. Therefore, by [Reference KatzKat73, Proposition 4.1.1], we deduce that 
$\widetilde {\mathcal {F}}$ is a 
${\mathbb {Q}_{p}}$-local system. In addition, by [Reference KatzKat73, Lemma 4.3.15], for every 
$S=\operatorname {Spec}(R)\to X$ with 
$R$ strictly henselian perfect ring we have
\[ \operatorname{Hom}(\mathcal{A}_{S}[p^\infty],\mathcal{A}^\vee_{S}[p^\infty])[\tfrac{1}{p}]=\operatorname{Hom}(\mathcal{A}_{s}[p^\infty],\mathcal{A}^\vee_{s}[p^\infty])[\tfrac{1}{p}], \]
where 
$s$ is the closed point of 
$S$. This implies that for every 
$\bar {x}\in X(\bar {\mathbb {F}}_p)$ we have
\[ \mathcal{F}_{\bar{x}}=\operatorname{Hom}^{{\mathrm{sym}}}(\mathcal{A}_{\bar{x}}[p^\infty],\mathcal{A}^\vee_{\bar{x}}[p^\infty])[\tfrac{1}{p}]. \]
For the semi-simplicity, since 
$X$ is normal, we can shrink 
$X$ and assume it smooth. Write 
$\mathcal {N}$ for the 
$F$-isocrystal 
$(R^1f_{{\mathrm {crys}}*}\mathcal {O}_{\mathcal {A},{\mathrm {crys}}})^{\otimes 2}$ and 
$\mathcal {N}^{[1]}$ for the quotient 
$\mathcal {N}^{\leq 1}/\mathcal {N}^{<1}$, where 
$\mathcal {N}^{\leq 1}$ (respectively, 
$\mathcal {N}^{<1}$) is the subobject of 
$\mathcal {N}$ of slopes 
$\leq 1$ (respectively, 
$<1$). Note that by [Reference Berthelot, Breen and MessingBBM82, Theorem 2.5.6(ii)], the pullback of 
$\mathcal {N}^{[1]}$ to 
$X^{\mathrm {perf}}$ is isomorphic as an 
$F$-isocrystal with 
$\mathcal {N}^{[1]}_{X^\textrm {perf}}$ over 
$X^{\textrm {perf}}$ defined above. Thanks to [Reference D'AddezioD'Ad23, Theorem 1.1.2], we have that 
$\mathcal {N}^{[1]}$ is semi-simple as an 
$F$-isocrystal.
By [Reference CrewCre87, Theorem 2.1], there is an equivalence between unit-root 
$F$-isocrystals over 
$X$ and finite-rank 
${\mathbb {Q}_{p}}$-local systems. By construction, Crew's and Katz's correspondences are compatible, in the sense that they agree after pulling back the objects through 
$X^\mathrm {perf}\to X$. Since the étale fundamental groups of 
$X$ and 
$X^{\mathrm {perf}}$ are canonically isomorphic, we deduce that 
$\widetilde {\mathcal {F}}$ is semi-simple as well. This yields the desired result.
6.5
Proof of Theorem 6.2 We look at the exact sequence
\[ 0\to\mathrm{NS}(A_{{\bar{k}}})_{{\mathbb{Q}_{p}}}\to H^2(A_{{\bar{k}}},{\mathbb{Q}_{p}}(1))\to \mathrm{T}_p(\mathrm{Br}(A_{{\bar{k}}}))[\tfrac{1}{p}]\to 0. \]
Thanks to Proposition 6.4, the operation of taking Galois-fixed points is exact. We get the exact sequence
\[ 0\to\mathrm{NS}(A_{{\bar{k}}})_{{\mathbb{Q}_{p}}}^{\Gamma_k}\to H^2(A_{{\bar{k}}},{\mathbb{Q}_{p}}(1))^{\Gamma_k}\to \mathrm{T}_p(\mathrm{Br}(A_{{\bar{k}}}))[\tfrac{1}{p}]^{\Gamma_k}\to 0. \]
Looking at the ranks we deduce the following equality
\begin{equation} \operatorname{rk}_{\mathbb{Z}_{p}}(H^2(A_{{\bar{k}}},{\mathbb{Z}_{p}}(1))^{\Gamma_k})=\operatorname{rk}_\mathbb{Z}(\mathrm{NS}(A_{{\bar{k}}})^{\Gamma_k})+\operatorname{rk}_{\mathbb{Z}_{p}}(\mathrm{T}_p(\mathrm{Br}(A_{{\bar{k}}}))^{\Gamma_k}). \end{equation}
By Proposition 6.4, the action of 
$\Gamma _k$ on 
$H^2(A_{{\bar {k}}},{\mathbb {Q}_{p}}(1))=\operatorname {Hom}^{\mathrm {sym}}(A_{\bar {k}}[p^\infty ],A^\vee _{\bar {k}}[p^\infty ])[\frac {1}{p}]$ factors through the étale fundamental group of 
$X$ associated to 
$\bar {\eta }$, denoted by 
$\pi _1^{\mathrm {\acute {e}t}}(X,\bar {\eta })$. In addition, if 
$\kappa$ is the residue field of 
$x$, the inclusion 
$x\hookrightarrow X$ induces then an action of 
$\Gamma _\kappa$ on 
$H^2(A_{{\bar {k}}},{\mathbb {Q}_{p}}(1))$ which corresponds, up to conjugation, to the action of 
$\Gamma _\kappa$ on 
$H^2(A_{\bar {\kappa }},{\mathbb {Q}_{p}}(1))$. Therefore, by the Tate conjecture over finite fields (or Corollary 5.3), we get 
$\mathrm {NS}(\mathcal {A}_{\bar {x}})^{\Gamma _\kappa }_{\mathbb {Q}_{p}}=H^2(A_{{\bar {k}}},{\mathbb {Q}_{p}}(1))^{\Gamma _\kappa }$. Since 
$H^2(A_{{\bar {k}}},{\mathbb {Q}_{p}}(1))^{\Gamma _k}$ is a subspace of 
$H^2(A_{{\bar {k}}},{\mathbb {Q}_{p}}(1))^{\Gamma _\kappa }$ we deduce that
\begin{equation} \operatorname{rk}_\mathbb{Z}(\mathrm{NS}(\mathcal{A}_{\bar{x}})^{\Gamma_\kappa})=\operatorname{rk}_{\mathbb{Z}_{p}}(H^2(A_{{\bar{k}}},{\mathbb{Z}_{p}}(1))^{\Gamma_\kappa})\geq \operatorname{rk}_{\mathbb{Z}_{p}}(H^2(A_{{\bar{k}}},{\mathbb{Z}_{p}}(1))^{\Gamma_k}). \end{equation} We want to conclude this section with other examples of abelian varieties such that 
$\mathrm {T}_p(\mathrm {Br}(A_{{\bar {k}}}))^{\Gamma _k}\neq 0$. These are variants of the abelian surface of § 6.1 and they all provide counterexamples to the conjecture in [Reference UlmerUlm14, § 7.3.1] when 
$\ell =p$.
Proposition 6.6 Let 
$A$ be an abelian variety which splits as a product 
$B\times _k B$ with 
$B$ an abelian variety over 
$k$. There is a natural exact sequence
\[ 0\to \operatorname{Hom}(B,B^\vee)_{\mathbb{Z}_{p}}\to \operatorname{Hom}(B_{\bar{k}}[p^\infty],B_{\bar{k}}^\vee[p^\infty])^{\Gamma_k}\to \mathrm{T}_p(\mathrm{Br}(A_{{\bar{k}}}))^{\Gamma_k}. \]
Proof. We consider the exact sequence
\[ 0\to \mathrm{NS}(A_{{\bar{k}}})^{\Gamma_k}_{\mathbb{Z}_{p}}\to H^2(A_{{\bar{k}}}, {\mathbb{Z}_{p}}(1))^{\Gamma_k}\to \mathrm{T}_p(\mathrm{Br}(A_{{\bar{k}}}))^{\Gamma_k}. \]
Arguing as in the proof of Proposition 4.6, the 
${\mathbb {Z}_{p}}$-module 
$\operatorname {Hom}(B_{\bar {k}}[p^\infty ],B_{\bar {k}}^\vee [p^\infty ])^{\Gamma _k}$ is naturally a direct summand of 
$H^2(A_{{\bar {k}}}, {\mathbb {Z}_{p}}(1))^{\Gamma _k}$. Its preimage in 
$\mathrm {NS}(A_{{\bar {k}}})_{\mathbb {Z}_{p}}^{\Gamma _k}$ corresponds to the 
${\mathbb {Z}_{p}}$-module
\[ \operatorname{Hom}(B_{\bar{k}},B^\vee_{\bar{k}})_{\mathbb{Z}_{p}}^{\Gamma_k}=\operatorname{Hom}(B,B^\vee)_{\mathbb{Z}_{p}}. \]
This concludes the proof.
Corollary 6.7 If 
$\operatorname {End}(B)=\mathbb {Z}$, then 
$\mathrm {T}_p(\mathrm {Br}(A_{{\bar {k}}}))^{\Gamma _k}\neq 0$.
Proof. By the assumption, 
$\operatorname {Hom}(B,B^\vee )_{\mathbb {Z}_{p}}$ is a 
${\mathbb {Z}_{p}}$-module of rank 
$1$. Therefore, by Proposition 6.6, it is enough to prove that the rank of 
$\operatorname {Hom}(B_{\bar {k}}[p^\infty ],B_{\bar {k}}^\vee [p^\infty ])^{\Gamma _k}$ is greater than 
$1$. Since 
$\operatorname {End}(B)=\mathbb {Z}$, the abelian variety 
$B$ is not supersingular, so that the 
$p$-divisible group 
$B[p^\infty ]$ admits at least two slopes. By the Dieudonné–Manin classification, this implies that 
$B_{{k_i}}[p^\infty ]$ is isogenous to a direct sum 
$\mathcal {G}_1\oplus \mathcal {G}_2$ of non-zero 
$p$-divisible groups over 
${k_i}$. Since 
$\operatorname {End}(\mathcal {G}_1)[\frac {1}{p}]\oplus \operatorname {End}(\mathcal {G}_2)[\frac {1}{p}]$ embeds into 
$\operatorname {End}(B_{{k_i}}[p^\infty ])[\frac {1}{p}]\simeq \operatorname {Hom}(B_{\bar {k}}[p^\infty ],B_{\bar {k}}^\vee [p^\infty ])[\frac {1}{p}]^{\Gamma _k}$, we deduce that 
$\operatorname {rk}_{\mathbb {Z}_{p}}(\operatorname {Hom}(B_{\bar {k}}[p^\infty ],B_{\bar {k}}^\vee [p^\infty ])^{\Gamma _k})>1$, as we wanted.
Acknowledgements
I thank Emiliano Ambrosi for the discussions we had during the writing of [Reference Ambrosi and D'AddezioAD22], which inspired this article, Matthew Morrow and Kay Rülling for many enlightening conversations about the cohomology of 
$\mathbb {Z}_p(1)$, and Ofer Gabber, Luc Illusie, Peter Scholze, and Takashi Suzuki for answering some questions on the fppf site. I also thank Jean-Louis Colliot-Thélène, Bruno Kahn, Alexei Skorobogatov, and Takashi Suzuki for very useful comments on a first draft of this article. Finally, I thank the anonymous referees for their careful reading of the article and for the corrections they suggested.
The author was funded by the Deutsche Forschungsgemeinschaft (EXC-2046/1, project ID 390685689 and DA-2534/1-1, project ID 461915680) and by the Max-Planck Institute for Mathematics.
Conflicts of Interest
None.
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