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

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 $p$.


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 Br(A) := H 2 ét (A, 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 F p one can no longer expect Br(A) to be finite (see [SZ08,§ 1]).On the other hand, if Br(A ks ) k is the transcendental Brauer group of A, namely the image of Br(A) → Br(A ks ) where k s is a separable closure of k, the group Br(A ks ) k [ 1 p ] is finite by [CS21,Theorem 16.2.3].In [SZ08, Question 1], Skorobogatov and Zarhin asked whether the p-primary torsion of Br(A ks ) 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 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 Br(A ks ) 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 (Ak, W O Ak ) is a finite W ( k)-module, then Br(A ks ) k is finite.
The condition on H 2 (Ak, W O Ak ) 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 , where Hom(B, B ∨ ) denotes the group of homomorphisms B → B ∨ as abelian varieties over k.
Note that if we replace T p (Br(Ak)) with the -adic Tate module T (Br(Ak)), where is a prime different from p, then T (Br(A ks )) = T (Br(Ak)) has no non-trivial Galois-fixed points.
These 'exceptional classes' in T p (Br(Ak)) Γ 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 Y → X a smooth proper morphism.For every geometric point η of X there is an As it is well-known, the theorem is false when K = Fp (see [MP12,Remark 1.12]).What we prove is that, in the known counterexamples, the elements in T p (Br(Ak)) Γ 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 F p with generic point η = Spec(k) and let f : A → X be an abelian scheme over X with constant Newton polygon.2For every closed point x = Spec(κ) of X we have Boundedness of the p-primary torsion of the Brauer group of an abelian variety 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 -adic analogue.Note also that T p (Br(A x)) Γκ = 0 by Corollary 5.3 since κ is a perfect field.
To prove Theorem 1.1 we use a flat variant of the Tate conjecture.For every n, let H 2 fppf (Ak, μ p n) k be the image of the extension of scalars morphism H 2 fppf (A, μ p n) → H 2 fppf (Ak, μ p n). Theorem 1.5 (Theorem 5.1).After possibly replacing k with a finite separable extension, the cycle class map 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 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 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].

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 k.In this section, we also prove in Proposition 3.8 the exactness of some fundamental sequences for the groups H 2 fppf (Xk, μ p n) k .In § 4, we construct a morphism which relates ) 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.

Notation
If k is a field, we write k for a fixed algebraic closure of k and k s (respectively, k i ) for the separable (respectively, purely inseparable) closure of k in k.We denote by Γ k the absolute Galois group of k.If x is a k-point of a scheme, we denote by x the induced k-point.For an abelian group M , we write T p (M ) for the p-adic Tate module of M , which is the projective limit lim and we write M ∧ for the p-adic completion of M .If M is endowed with a Γ k -action, we denote by M Γ k the subgroup of fixed points.For a scheme X and an fppf sheaf F, we denote by H • (X, F) the fppf cohomology groups and when X = Spec(k) we simply write H • (k, F).If f : X → Y is a morphism of schemes, we denote by R • f * F the fppf higher direct image functors over (Sch/Y ) fppf .Finally, if X is a scheme over F p , we write X perf for the where F is the absolute Frobenius of X.

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 F be a sheaf over (Sch/k) fppf such that q * F X = F and suppose that X has a k-rational point.The group In addition, the natural morphism H 2 (X, F X ) → H 0 (k, R 2 q * F X ) sits in an exact sequence where K is an extension of Proof.We consider the Leray spectral sequence ∞ is an extension of E 1,1 2 by E 2,0 2 , as we wanted.The obstruction for the map . This concludes the proof.Definition 3.2.We say that a presheaf F on (Sch qcqs /k) fppf is finitary if for every inverse system {T ( ) } ∈L of quasi-compact quasi-separated k-schemes with affine transition maps, the natural morphism is an isomorphism.
Lemma 3.3.Let G be a commutative finite-type group scheme over k.If X is quasicompact quasi-separated, then R i q * G X is finitary for i ≥ 0. In addition, the natural morphism Proof.Let 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 H i (q, G X ) is finitary for i ≥ 0. In other words, we want to prove that for every inverse system {T ( ) } ∈L of quasi-compact quasi-separated k-schemes, the natural morphism is an isomorphism, where X for every ∈ L {∞}, where HC(X ( ) ) is the category of fppf hypercoverings of X ( ) .Since each X ( ) is quasi-compact quasi-separated, by [Sta23, Tag 021P] we can replace the category HC(X ( ) ) in the colimit with the subcategory HC(X ( ) ) qcqs , consisting of those hypercoverings such that U ( ) ) qcqs and n ≥ 0 there exists an ∈ L and U where tr n (−) denotes the nth truncation of simplicial schemes and T (∞) := lim ∈L T ( ) .This implies that

Boundedness of the p-primary torsion of the Brauer group of an abelian variety
We are reduced to proving that for every ∈ L and U ( ) • ∈ HC(X ( ) ) qcqs we have that Ȟi (U ( ) ).Since G is of finite type over k, this follows from [Sta23, Lemma 01ZM] and the exactness of filtered colimits.
Knowing that H i (q, G X ) is finitary, in order to prove that R i q * G X is finitary as well it is enough to prove that for every finitary presheaf F on (Sch qcqs /k) fppf , the 'partial' sheafification 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 (∞) descends to a covering of X ( ) for some ∈ L and Čech cohomology commutes with filtered colimits ( Ȟ0 is enough in this case).
For the second part, we note that for every presheaf F on (Sch/k) fppf with sheafification F , the natural morphism is an isomorphism because every fppf covering of Spec( k) admits a section.This implies that Thanks to the previous part, we deduce that the composition 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. 4orollary 3.4.If k is separably closed and X is a proper k-scheme, then there is a natural exact sequence In particular, if Pic X/k is smooth, then the natural morphism Br(X) → Br(Xk) is injective.
Proof.As in Lemma 3.1, we consider the Leray spectral sequence ).Since X is proper over k, by [Sta23, Tag 0BUG] we deduce that A := H 0 (X, O X ) is a finite k-algebra.This implies that q * 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 ∞ .The Leray spectral sequence produces then the exact sequence 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 Pic X/k is smooth, thanks to [Gro68,Theorem 11.7], the group H 1 (k, Pic X/k ) vanishes.Definition 3.5.For a scheme X over k and a prime p, we define H 2 (X, Z p (1)) as the projective limit lim Remark 3.6.Note that we are defining H 2 (X, 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 ] is a direct sum of a p-divisible group and a finite group and R 1 lim ← −n H 2 (X, μ p n) = 0 by [Ill79, Chapter II, Proposition 5.9].
Construction 3.7.The Kummer exact sequences for X and Xk (for the fppf topology) induce the following commutative diagram with exact rows. (3.7.1) We write for the complex obtained by taking images of the vertical arrows.Note that a priori (Br(Xk)[p n ]) k is smaller than Br(Xk) k [p n ], where Br(Xk) k := im(Br(X) → Br(Xk)).
Since both R 1 lim ← −n Pic(X)/p n and R 1 lim ← −n Pic(Xk)/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 Ĉ(X) : the complex obtained by taking images of the vertical arrows.
Proposition 3.8.If char(k) = p and A is an abelian variety over k such that the morphism Pic(A) → NS(Ak) is surjective, then the complexes C n (A) and Ĉ(A) are acyclic.
and K 2 is the kernel of Br(A) → Br(Ak), in order to prove that C n (A) is acyclic we have to show that K 1,n → 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 thus we are reduced to prove that Boundedness of the p-primary torsion of the Brauer group of an abelian variety is surjective.Since Pic(A) → NS(Ak) is surjective, we know that π 0 (Pic A/k ) is a constant finitely generated torsion-free group over k such that Pic A/k (k) → π 0 (Pic A/k )(k) is surjective.Looking at the cohomology long exact sequence associated to ), which yields the desired result.
We now prove that Ĉ(A) is acyclic.The kernel of H 2 (A, Z p (1)) → H 2 (Ak, Z p (1)) is lim ← −n K 1,n and the kernel of T p (Br(A)) → T p (Br(Ak)) is T p (K 2 ).Thus, again, we have to prove that lim Combining the previous discussion and the fact that 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 is Pic(A)/p n and the groups (Pic(A)/p n ) n>0 form a Mittag-Leffler system.We deduce that the morphism is surjective.This implies that Ĉ(A) is acyclic, as we wanted.
The proof of the following proposition was inspired by [CS13, Theorem 2.1].
Proposition 3.9.If char(k) = p and A is an abelian variety over k, we have Proof.We first note that the four Q p -vector spaces are invariant under isogenies of A and finite separable extension of k.Indeed, for every isogeny ϕ : B → A there exists an isogeny ψ : A → B such that the composition ϕ • ψ is the multiplication by some positive integer n.Since n is invertible in Q p , we deduce that ϕ * 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 → A admit as inverse the morphisms (1/[k : k])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 → A such that B := Jac(C) maps surjectively to A. By Poincaré's complete reducibility theorem, B is isogenous to a product A × k A with A an abelian variety over 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 Pic(B) → NS(Bk) is surjective, so that

and by the assumption Pic
. By [Gro68, Rmq.2.5.b], the group Br(Ck) vanishes, thus H 2 (Ck, Z p (1)) = Z p and the morphism ) k is surjective as well.This implies that T p (Br(Bk)) k → lim ← −n (Br(Bk)[p n ]) k is surjective.It remains to prove that for every n we have (Br(Bk)[p n ]) k = Br(Bk) k [p n ].Consider the natural morphism K 3 → Br(C) where K 3 is the kernel of Br(B) → Br(Bk).Thanks to Lemmas 3.1 and 3.3 and using the fact that Br(Ck) = 0, this morphism sits in the following commutative diagram with exact rows.
Since Pic(B) → NS(Bk) is surjective and C is a curve, we have that We deduce that K 3 → Br(C) is an isomorphism, thus Br(B) → Br(C) This implies that Br(B)[p n ] → Br(Bk) k [p n ] is surjective and this yields the desired result.

Constructing a morphism
Let A be an abelian variety over a field k.For a line bundle L of A we write ϕ L : A → A ∨ for the morphism which sends x → t * x L ⊗ 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.

Consider the morphism h
, where π 1 and π 2 are the two projections of A × k A. This morphism has the property that the first Chern class c Boundedness of the p-primary torsion of the Brauer group of an abelian variety Lemma 4.2.The image of h 1 lies in Proof.The spectral sequence (4.1.1)gives the exact sequence is the 0-morphism.By [BO21, Corollary 1.4], there exists a commutative linear algebraic group5 G over k which represents R 2 q * μ p n on the big fppf site (Sch/k) fppf .Since R 2 π 2 * μ p n is the restriction of R 2 q * μ p n from (Sch/k) fppf to (Sch/A) fppf , this implies that H 0 (A, R 2 π 2 * μ p n) can be computed as Mor Sch/k (A, G).Thanks to the fact that G is affine, every morphism A → G contracts A to a point.We deduce that Mor is given by the pullback via i 1 : A = A × k 0 A → A × k A followed by the extension of scalars to k.
By construction, we have that i 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 : Hom 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 (σ) := σ * P n for every σ ∈ 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] → G.By Cartier duality, this is the same as defining a morphism g n (P ) ∨ : To prove that g n • f n = id it is enough to note that for every σ ∈ Hom(A[n], G), every scheme T over k, and every τ ∈ Hom(G T , G m,T ) we have that

Thanks to Lemma 4.2, we can define h1 : H
as the composition of h 1 and the natural morphism In addition, by Lemma 4.3 applied to G = A ∨ [p n ], we get a morphism for the composition h 2 • h1 and we denote with the same letter the induced morphism Proposition 4.5.The square (4.5.1) is commutative.
Proof.We have to show that for every line bundle L ∈ Pic(A) we have Consider the Leray spectral sequence 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.
) sends c 1 (P) to [P ].For this purpose, we introduce the Leray spectral sequence The morphism δ : G m [1] → μ 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 P to id . This yields the desired result.
Proof.Suppose char(k) = p and write W for the ring of Witt vectors of 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 On the other hand, by [Ill79,Theorem II.5.14], there is a canonical isomorphism H 2 (Ak, Z p (1)) = H 2 crys (Ak/W ) F =p .This concludes the case when char(k) = p.If p is invertible in k one can replace crystalline cohomology with p-adic étale cohomology.

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 T p (Br(Ak) k ) = 0.Moreover, after possibly replacing k with a finite separable extension, the cycle class map becomes an isomorphism.
Proof.To prove the statement we may assume that Pic(A) → NS(Ak) is surjective by extending k.The Z p -module Hom . By Proposition 4.6, we know that h is injective and im( h) is contained in Hom sym (A[p ∞ ], A ∨ [p ∞ ]).In addition, by Boundedness of the p-primary torsion of the Brauer group of an abelian variety and using the fact that lim 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 ( k) is finite.In particular, the abstract group D is in Ab p .To bound U , we note that by [Mil86,Proposition 3.1] the dimension of the chain of algebraic groups G 1 ⊆ G 2 ⊆ • • • is eventually constant.Therefore, there exists N > 0 such that for every n ≥ N , the morphism (U n ) red → (U n+1 ) red is an isomorphism.This shows that U is a finite exponent p-group.
If We can finally prove that Br(Ak) k [p ∞ ] has finite exponent.Suppose by contradiction that this is not the case.Since Br(Ak) k [p ∞ ] ∈ Ab p , we deduce that it contains a copy of Q p /Z p .On the other hand, by Theorem 5.1, the group T p (Br(Ak) k ) vanishes, which leads to a contradiction.
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 × k E.
Proposition 5.4.After possibly extending k to a finite separable extension, the transcendental Brauer group Br(A ks ) k becomes infinite.
Proof.Even in this case we use that, thanks to Corollary 3.4, the transcendental Brauer group is the same as Br(Ak) k .Moreover, after extending the scalars we may assume that the morphism Pic(A) → NS(Ak) is surjective.Combining Proposition 3.8 and the fact that NS(Ak)/p is finite we deduce that it is enough to show that H 2 (Ak, μ p ) k is infinite.We look at the Leray spectral sequence with respect to the second projection π 2 : A = E × k E → E (both over k and over k).In the second page, we have that the boundary morphism H 1 (E, R 1 π 2 * μ p ) → H 3 (E, π 2 * μ p ) vanishes because H 3 (E, π 2 * μ p ) → H 3 (A, μ p ) admits a retraction induced by the zero section of π 2 .Since H 0 (Ek, R 2 π 2 * μ p ) = H 2 (Ek, μ p ) = Z/p, it is then enough to show that the image of Tate conjecture over finite fields (or Corollary 5.3), we get NS(A x) Γκ Qp = H 2 (Ak, Q p (1)) Γκ .Since H 2 (Ak, Q p (1)) Γ k is a subspace of H 2 (Ak, Q p (1)) Γκ we deduce that rk Z (NS(A x) Γκ ) = rk Zp (H 2 (Ak, Z p (1)) Γκ ) ≥ rk Zp (H 2 (Ak, Z p (1)) Γ k ).
We want to conclude this section with other examples of abelian varieties such that T p (Br(Ak)) Γ k = 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 = p.Proposition 6.6.Let A be an abelian variety which splits as a product B × k B with B an abelian variety over k.There is a natural exact sequence This concludes the proof.
Proof.By the assumption, Hom(B, B ∨ ) Zp is a Z p -module of rank 1.Therefore, by Proposition 6.6, it is enough to prove that the rank of Hom(Bk[p ∞ ], B ∨ k [p ∞ ]) Γ k is greater than 1.Since End(B) = Z, the abelian variety B is not supersingular, so that the p-divisible group B[p ∞ ] admits at least two slopes.By the Dieudonné-Manin classification, this implies that B k i [p ∞ ] is isogenous to a direct sum G 1 ⊕ G 2 of non-zero p-divisible groups over k i .Since End(G 1 )[ is infinite.By Lemma 4.3, we have that End(E[p])(respectively, End(Ek[p])) admits a natural embedding inH 1 (E, E[p]) (respectively, H 1 (Ek, Ek[p])).Since k = End(α p ) ⊆ End(E[p]) ⊆ End(Ek[p])by the assumption that E is supersingular, we deduce the desired result.
.2.1], we have thatH * crys (Ak ×k Ak/W ) = H * crys (Ak/W ) ⊗ H * crys (Ak/W ) so that m : A × k A → A inducesBoundedness of the p-primary torsion of the Brauer group of an abelian variety H 2 (Ak, W O Ak ) is a finite W ( k)-module, then the formal group Φ 2 fl (Ak, G m ) does not contain any copy of Ĝa .Indeed, by [BO21, Corollary 12.5], the group H 2 (Ak, W O Ak ) is the Cartier module of Φ 2 fl (Ak, G m ) and, by the assumption, it cannot contain k[[V ]], the Cartier module of Ĝa .Therefore, in this case, we have that each group U n ( k) is trivial, so that H[p] = D[p] is finite.