Boundedness of the p-primary torsion of the Brauer group of an abelian variety

Abstract

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant $p^\infty$-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$-divisible. We explain how the existence of these $p$-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic $p$.

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 [Tat94, Proposition 4.3]. The main input for this result is the Tate conjecture for divisors, proved by Tate in [Tat66]. 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 [SZ08, § 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 [CS21, Theorem 16.2.3]. In [SZ08, 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 [Ill83, 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 [AM77, 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 [AM77, 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 [SZ08, Question 2] for abelian varieties.

In this article, we also study the Galois-fixed points of $\mathrm {Br}(A_{{\bar {k}}})$. Ulmer in [Ulm14, § 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 [And96, Theorem 5.2] in characteristic $0$ (see also [MP12]) and in [Amb23] and [Chr18] 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 $\operatorname {rk}_\mathbb {Z}(\mathrm {NS}(\mathcal {Y}_{\bar {\eta }}))=\operatorname {rk}_\mathbb {Z}(\mathrm {NS}(\mathcal {Y}_x))$.¹

As it is well-known, the theorem is false when $K=\bar {\mathbb {F}}_p$ (see [MP12, 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.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.³

We obtain this result by using the crystalline Tate conjecture for abelian varieties, proved by de Jong in [deJ98, 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 [CS13, 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 [Gro68, Proposition 5.6], which was stated by Grothendieck without a complete proof.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 [Gro68, 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 [Gro68, 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 [Ill79, 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.

(3.7.1)

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 [CS21, 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 [CS13, 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 [Kat99, 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 [Gro68, 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 [OSZ21, 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

(4.1.1)\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 [BO21, Corollary 1.4], there exists a commutative linear algebraic group⁵ $G$ 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

(4.5.1)

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

(4.5.2)\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

(4.5.3)\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 [BBM82, Corollary 2.5.5]. Therefore, thanks to the Künneth formula, [Ber74, 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 [BBM82, 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 [BBM82, 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 [Ill79, 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 [Ill79, 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 [deJ98, 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 [CS21, 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 [BO21, Corollary 1.4], for every $n$ there exists a commutative linear algebraic group $G_n$ representing $R^2q_{*}\mu _{p^n}$.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 [BO21, 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 [BO21, § 10.6].

Applying $Rq_*$ to the exact sequence

(5.2.1)\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 [Mil86, 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 [BO21, 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 [CS21, 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 sends⁷ $T\in (X^{\mathrm {perf}})_{\mathrm {pro}\mathrm {\acute {e}t}}$ 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 [BBM82, Déf. 3.3.6]. By [BBM82, 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 [Lau13, 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 [Kat79, 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 [Kat73, Proposition 4.1.1], we deduce that $\widetilde {\mathcal {F}}$ is a ${\mathbb {Q}_{p}}$-local system. In addition, by [Kat73, 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 [BBM82, 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 [D'Ad23, Theorem 1.1.2], we have that $\mathcal {N}^{[1]}$ is semi-simple as an $F$-isocrystal.

By [Cre87, 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

(6.5.1)\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

(6.5.2)\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}

Combining (6.5.1) and (6.5.2) we get the desired result.

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 [Ulm14, § 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.