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NS(X) is a saturated Z-submodule of NS(X¯). We have 2 ¯ Gal(k) 2 ¯ NS(X) ⊗ Zℓ ⊂ H´et(X, Zℓ(1)) ⊂ H´et(X, Zℓ(1)).

Since NS(X) ⊗ Zℓ is a saturated Zℓ-submodule of NS(X¯) ⊗ Zℓ, it is also a saturated 2 ¯ Zℓ-submodule of H´et(X, Zℓ(1)). The intersection pairing NS(X¯) × NS(X¯) → Z, α,β 7→ α · β is non-degenerate, see [9, Prop. 3, p. 64] and also [11, Lemma V.3.27]. In particular, the discriminant dX of this pairing is a non-zero integer. On the other hand, there is the Poincar´eduality pairing 2 ¯ 2 ¯ eX,ℓ : H´et(X, Zℓ(1)) × H´et(X, Zℓ(1)) → Zℓ, which is perfect and Galois-invariant. The restriction of eX,ℓ to NS(X¯) coincides with the intersection pairing. We define the group of transcendental cycles T (X¯)ℓ ¯ 2 ¯ as the orthogonal complement to NS(X)⊗Zℓ in H´et(X, Zℓ(1)) with respect to eX,ℓ. ¯ 2 ¯ It is clear that T (X)ℓ is a saturated Galois-stable Zℓ-submodule of H´et(X, Zℓ(1)). It is also clear that if ℓ does not divide dX , then 2 ¯ ¯ ¯ H´et(X, Zℓ(1)) = (NS(X) ⊗ Zℓ) ⊕ T (X)ℓ. For any ℓ we have (NS(X¯) ⊗ Zℓ) ∩ T (X¯)ℓ =0, ¯ ¯ 2 ¯ and the direct sum (NS(X)⊗Zℓ)⊕T (X)ℓ is a subgroup of finite index in H´et(X, Zℓ(1)). ¯ 2 ¯ It is clear that T (X)ℓ/ℓ is a Gal(k)-submodule of H´et(X,µℓ). 2 ¯ 2 ¯ Let us consider the Qℓ-vector space H´et(X, Qℓ(1)) := H´et(X, Zℓ(1)) ⊗Zℓ Qℓ with its natural structure of a Gal(k)-module. We have a natural embedding of Galois ¯ 2 ¯ modules NS(X) ⊗ Qℓ ⊂ H´et(X, Qℓ(1)), from which we obtain 2 ¯ Gal(k) NS(X) ⊗ Qℓ ⊂ H´et(X, Qℓ(1)) .

K. Madapusi Pera [14, Thm. 1] proved that if k is finitely generated over Fp and p> 2, then 2 ¯ Gal(k) NS(X) ⊗ Qℓ = H´et(X, Qℓ(1)) . This is an important special case of the celebrated Tate conjecture on algebraic cycles [20]. Closely related results were previously obtained by D. Maulik [10] and F. Charles [2, Cor. 2]. Since NS(X) ⊗ Zℓ is a saturated Zℓ-submodule in 2 ¯ H´et(X, Zℓ(1)), the theorem of Madapusi Pera is equivalent to 2 ¯ Gal(k) NS(X) ⊗ Zℓ = H´et(X, Zℓ(1)) . Another restatement of the same result is ¯ Gal(k) T (X)ℓ =0. Note that this still holds if k is replaced by any finite separable field extension. Our results are based on the Tate conjecture for K3 surfaces and the existence of the Kuga–Satake variety in odd characteristic, as established by Madapusi Pera in [14, Thm. 4.17], as well as on the results of one of the authors [24].

Theorem 1.1. Let k be a field finitely generated over Fp, where p 6=2, and let X 2 ¯ be a K3 surface over k. For any prime ℓ 6= p the Galois module H´et(X, Qℓ(1)) is semisimple. This is proved in Theorem 4.1 (i). Our main results are the two following state- ments whose proofs can be found in the final section of this note. AFINITENESSTHEOREMFORTHEBRAUERGROUP 3

Theorem 1.2. Let k be a field finitely generated over Fp, where p 6=2, and let X be 2 ¯ a K3 surface over k. Then for all but finitely many ℓ the Galois module H´et(X,µℓ) 2 ¯ Gal(k) is semisimple and H´et(X,µℓ) = NS(X)/ℓ. Using the method of [19] we deduce from the above results the following finiteness statement.

Theorem 1.3. Let k be a field finitely generated over Fp, where p 6=2, and let X be a K3 surface over k. Then the group [Br(X)/Br0(X)](non − p) is finite. For a K3 surface X over a finitely generated field of characteristic zero, the finiteness of Br(X)/Br0(X) was proved in [19]. 2. Abelian varieties Let A be an abelian variety over a field k. We write End(A) for the ring of endomorphisms of A. For a positive integer n not divisible by char(k), we write An for the kernel of multiplication by n in A(k¯). Then An is a Gal(k)-submodule of A(k¯sep). Let ℓ 6= char(k) be a prime. Recall that the ℓ-adic Tate module Tℓ(A) is the projective limit of abelian groups Aℓi , where the transition maps Aℓi+1 → Aℓi are multiplications by ℓ. It is well known [16] that Tℓ(A) is a free Zℓ-module of rank 2dim(A) equipped with a natural continuous action

ρℓ,A : Gal(k) → AutZℓ (Tℓ(A)).

We write Vℓ(A) = Tℓ(A) ⊗Zℓ Qℓ, and view Tℓ(A) as a Zℓ-lattice in Vℓ(A). Then ρℓ,A gives rise to the ℓ-adic representation

ρℓ,A : Gal(k) → AutZℓ (Tℓ(A)) ⊂ AutQℓ (Vℓ(A)). i ∼ There are natural canonical isomorphisms of Galois modules Tℓ(A)/ℓ = Aℓi for all i ≥ 1. If we forget about the Galois action, then we can naturally identify the Zℓ-modules Tℓ(A) and Tℓ(A¯), and the Qℓ-vector spaces Vℓ(A) and Vℓ(A¯). There is a natural embedding of Z-algebras End(A) ֒→ End(A¯), whose image is a saturated subgroup [15, Sect. 4, p. 501]. We will identify End(A) with its image in End(A¯). The natural action of Gal(k) on End(A¯) is continuous when End(A¯) is given discrete topology, so the image of Gal(k) in Aut(End(A¯)) is finite. The ring End(A) coincides with the Galois-invariant subring End(A¯)Gal(k), see [18]. In other words, there is a finite Galois field extension k′/k such that Gal(k) → Aut(End(A¯)) factors through Gal(k) ։ Gal(k′/k) and ′ End(A) = End(A¯)Gal(k /k).

There are natural embeddings of Zℓ-algebras ¯ End(A) ⊗ Zℓ ⊂ End(A) ⊗ Zℓ ⊂ EndZℓ (Tℓ(A)), of Qℓ-algebras ¯ End(A) ⊗ Qℓ ⊂ End(A) ⊗ Qℓ ⊂ EndQℓ (Vℓ(A)) and of Fℓ-algebras ¯ End(A) ⊗ Fℓ ⊂ End(A) ⊗ Fℓ ⊂ EndFℓ (Aℓ).

The image of End(A)⊗Zℓ in EndZℓ (Tℓ(A)) is obviously contained in the centraliser

EndGal(k)(Tℓ(A)) of Gal(k) in EndZℓ (Tℓ(A)). Similar statements hold if Zℓ is re- placed by Qℓ or Fℓ. 4 ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN

For each prime ℓ 6= char(k) let us consider the Qℓ-bilinear trace form

ψℓ : EndQℓ (Vℓ(A)) × EndQℓ (Vℓ(A)) → Qℓ, u,v 7→ tr(uv). This form is Galois-invariant, symmetric and non-degenerate. The same formula defines a perfect (unimodular) Zℓ-bilinear form on EndZℓ (Tℓ(A)). The restriction of this form to End(A¯) takes values in Z and does not depend on the choice of ℓ, see [13, Sect. 19, Thm. 4]. Let us choose a polarisation on A and write u 7→ u′ for the corresponding Rosati involution on End(A¯) ⊗ Q, see [13, Sect. 20]. Then tr(u′u) is a positive rational number if u 6= 0, see [8, Ch. V, Sect. 3, Thm. 1] or [13, Sect. 21, Thm. 1]. This implies that the Gal(k)-invariant form

ψℓ : End(A¯) × End(A¯) → Z is non-degenerate (but not necessarily perfect) and does not depend on ℓ. Since the Rosati involution commutes with the action of Gal(k), and End(A) = End(A¯)Gal(k), the bilinear form

ψℓ : End(A) × End(A) → Z is also non-degenerate, and therefore its discriminant dA is a non-zero integer that does not depend on ℓ. The following assertion was proved by one of the authors [23, 24] in char(k) > 2, by Faltings [5, 6] in char(k) = 0 and by Mori [12] in char(k) = 2. (See also [27, 29].) Theorem 2.1. Let k be a field finitely generated over its prime subfield and let A be an abelian variety over k. For a prime ℓ 6= char(k) we have

(i) End(A) ⊗ Zℓ = EndGal(k)(Tℓ(A)) and End(A) ⊗ Qℓ = EndGal(k)(Vℓ(A));

(ii) the Galois module Vℓ(A) is semisimple. Remark 2.2. When k is finite, the assertion (i) of Theorem 2.1 was proved by Tate [21], who conjectured that it holds for an arbitrary finitely generated field. He also proved that the two formulae in (i) are equivalent for any given k,A,ℓ. The semisimplicity of Vℓ(A) for finite k was established earlier by Weil [13]. The following assertion was proved in [25, 26, 29]. Theorem 2.3. Let k be a field finitely generated over its prime subfield, and let A be an abelian variety over k. Then for all but finitely many primes ℓ 6= char(k) the Galois module Aℓ is semisimple and End(A)/ℓ = EndGal(k)(Aℓ). Proposition 2.4. Let k be a field finitely generated over its prime subfield, and let A be an abelian variety over k. Let P be an infinite set of primes that does not contain char(k). Suppose that for each ℓ ∈ P we are given a Gal(k)-submodule

Uℓ ⊂ EndZℓ (Tℓ(A)) such that Uℓ is a saturated Zℓ-submodule of EndZℓ (Tℓ(A)) and Gal(k) Gal(k) Uℓ =0. Then for all but finitely many ℓ ∈ P we have (Uℓ/ℓ) =0.

Proof. By Theorem 2.1(ii) the Galois module Vℓ(A) is semisimple, and by a theorem of Chevalley [3, p. 88] this implies that EndQℓ (Vℓ(A)) is also semisimple. The Galois submodules Uℓ ⊗ Qℓ and EndGal(k)(Vℓ(A)) are orthogonal in EndQℓ (Vℓ(A)) with Gal(k) respect to the Galois-invariant bilinear form ψℓ, because otherwise (Uℓ ⊗Qℓ) 6= Gal(k) 0, which contradicts the assumption Uℓ = 0. It follows that End(A) ⊗ Zℓ and Uℓ are orthogonal, too. AFINITENESSTHEOREMFORTHEBRAUERGROUP 5

Now assume that ℓ ∈ P does not divide dA. Then the Zℓ-bilinear symmetric Galois-invariant form

ψℓ : End(A) ⊗ Zℓ × End(A) ⊗ Zℓ → Zℓ is perfect. Thus the Galois module EndZℓ (Tℓ(A)) splits into a direct sum

EndZℓ (Tℓ(A)) = (End(A) ⊗ Zℓ) ⊕ Sℓ, where Sℓ is the orthogonal complement to End(A)⊗Zℓ. It is clear that Sℓ is a Galois- stable saturated Zℓ-submodule of EndZℓ (Tℓ(A)). We have natural isomorphisms of Galois modules

EndZℓ (Tℓ(A))/ℓ = EndZℓ/ℓ(Tℓ(A)/ℓ) = EndFℓ (Aℓ). Thus reducing modulo ℓ, we obtain a Galois-invariant decomposition

EndFℓ (Aℓ) = End(A)/ℓ ⊕ Sℓ/ℓ. This gives rise to

Gal(k) EndGal(k)(Aℓ) = End(A)/ℓ ⊕ (Sℓ/ℓ) .

Gal(k) Theorem 2.3 implies that for all but finitely many ℓ ∈ P we have (Sℓ/ℓ) = 0. Since Uℓ is orthogonal to End(A)⊗Zℓ, we have Uℓ ⊂ Sℓ. Moreover, Uℓ is a saturated Zℓ-submodule of Sℓ, and this implies that the natural map Uℓ/ℓ → Sℓ/ℓ is injective. Gal(k) We conclude that (Uℓ/ℓ) = 0 for all but finitely many ℓ ∈ P . 

Recall if char(k) does not divide a positive integer n, then we have a canonical isomorphism of Galois modules 1 ¯ H´et(A, Z/n) = HomZ/n(An, Z/n). For a prime ℓ 6= char(k) we deduce canonical isomorphisms of Galois modules 1 ¯ 1 ¯ H´et(A, Zℓ) = HomZℓ (Tℓ(A), Zℓ), H´et(A, Qℓ) = HomQℓ (Vℓ(A), Qℓ), 1 ¯ ∼ EndZℓ (H´et(A, Zℓ)) = EndZℓ (Tℓ(A)), where the last identification is an anti-isomorphism of rings. This allows us to restate Proposition 2.4 in the following form. Corollary 2.5. Let k be a field finitely generated over its prime subfield, and let A be an abelian variety over k. Let P be an infinite set of primes that does not contain char(k). Suppose that for each ℓ ∈ P we are given a Gal(k)-submodule 1 ¯ Uℓ ⊂ EndZℓ (H´et(A, Zℓ)) 1 ¯ Gal(k) such that Uℓ is a saturated Zℓ-submodule of EndZℓ (H´et(A, Zℓ)) and Uℓ = 0. Gal(k) Then for all but finitely many ℓ ∈ P we have (Uℓ/ℓ) =0. Theorem 2.6. Let k be a field finitely generated over its prime subfield, and let A be an abelian variety over k. Then 1 ¯ (i) the Gal(k)-module EndQℓ (H´et(A, Qℓ)) is semisimple for any prime ℓ 6= char(k); 1 ¯ (ii) the Gal(k)-module EndFℓ (H´et(A, Z/ℓ)) is semisimple for all but finitely many primes ℓ 6= char(k). 6 ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN

Proof. (i) We have a Qℓ-algebra anti-isomorphism ∼ 1 ¯ EndQℓ (Vℓ(A)) = EndQℓ (H´et(A, Qℓ)), which is also an isomorphism of Galois modules. By Theorem 2.1 the Galois module Vℓ(A) is semisimple, and then Chevalley’s theorem [3, p. 88] implies the semisim- plicity of EndQℓ (Vℓ(A)). (ii) We have an Fℓ-algebra anti-isomorphism 1 ¯ ∼ EndFℓ (H´et(A, Z/ℓ)) = EndFℓ (Aℓ), which is also an isomorphism of Galois modules. By Theorem 2.3 the Galois module Aℓ is semisimple for all but finitely many ℓ. By a theorem of Serre [17, Cor. 1], if ℓ is greater than 2 dimFℓ (Aℓ) − 2 = 4 dim(A) − 2, then the semisimplicity of Aℓ implies the semisimplicity of the Galois module EndFℓ (Aℓ). This finishes the proof of (ii). 

3. Clifford Algebras and Kuga–Satake construction Let Λ be a principal ideal domain and let E be a free Λ-module of finite rank with a non-degenerate quadratic form q : E → Λ. Let C(E, q) be the Clifford algebra of (E, q). Let ̺ : E → C(E, q) be the canonical Λ-linear homomorphism satisfying ̺(x)2 = q(x) ∈ Λ ⊂ C(E, q). The map ̺ is injective by [1, Sect. 19.3, Cor. 2 to Thm. 1], so that ̺(E) =∼ E. It follows from [1, Sect. 19.3, Thm. 1] that C(E, q) is a free Λ-module of finite rank, and the Λ-submodule ̺(E) is a direct summand of the Λ-module C(E, q); in particular, it is a saturated submodule. Let

multL : C(E, q) → EndΛ(C(E, q)) be the homomorphism of Λ-algebras that sends a ∈ C(E, q) to the endomorphism x 7→ ax. Since 1 ∈ C(E, q), the homomorphism multL is injective. In particular, the Λ-algebras C(E, q) and multL(C(E, q)) are isomorphic. We claim that multL(̺(E)) is a saturated Λ-submodule of EndΛ(C(E, q)). Tak- ing into account that ̺(E) is saturated in C(E, q), it is enough to show that multL(C(E, q)) is saturated in EndΛ(C(E, q)). Indeed, if a ∈ C(E, q) is such that multL(a) is a divisible element in EndΛ(C(E, q)), then multL(a)(x) is divisible in C(E, q) for all x ∈ C(E, q). In particular, if we put x = 1, then we see that a = multL(a)(1) is divisible in C(E, q). This proves our claim.

Now let Λ = Zℓ and let Eℓ be a free Zℓ-module of finite rank with a non- degenerate quadratic form qℓ : Eℓ → Zℓ. Let Aut(C(Eℓ, qℓ)) be the group of automorphisms of the Zℓ-algebra C(Eℓ, qℓ). If there is a continuous ℓ-adic repre- sentation ρℓ : Gal(k) → AutZℓ (Eℓ) that preserves qℓ, then, by a universal property of Clifford algebras, there is a representation

ρℓ,Cliff : Gal(k) → Aut(C(Eℓ, qℓ)) such that for any σ ∈ Gal(k) and any x ∈ Eℓ we have ρℓ,Cliff (σ)(̺(x)) = ̺(ρℓ(σ)(x)). Using [1, Sect. 19.3, Thm. 1] one easily checks that ρℓ,Cliff is continuous. Proposition 3.1. Let k be a field finitely generated over its prime subfield. Let P be an infinite set of primes not containing char(k). Assume that for each ℓ ∈ P we AFINITENESSTHEOREMFORTHEBRAUERGROUP 7 are given a free Zℓ-module Eℓ with a non-degenerate quadratic form qℓ : Eℓ → Zℓ and a continuous ℓ-adic representation ρℓ : Gal(k) → AutZℓ (Eℓ) preserving qℓ. Suppose that there is an abelian variety A over a finite separable field extension ′ k of k, and for each prime ℓ ∈ P there is an isomorphism of Zℓ-modules ιℓ : 1 ¯ C(Eℓ, qℓ)−→ ˜ H´et(A, Zℓ) such that the composition of injective maps of Zℓ-modules ̺ ιℓ multL 1 ¯ 1 ¯ Eℓ −−→ C(Eℓ, qℓ) −−−→ H´et(A, Zℓ) −−−−→ EndZℓ (H´et(A, Zℓ)) is a homomorphism of Gal(k′)-modules.

Then the Gal(k)-module Eℓ ⊗Zℓ Qℓ is semisimple for each ℓ ∈ P , and the Gal(k)- module Eℓ/ℓ is semisimple for all but finitely many ℓ ∈ P . Proof. Replacing k′ by its normal closure over k, we can assume that k′ is a finite Galois extension of k, so that Gal(k′) ⊂ Gal(k) is an (open) normal subgroup of ′ index [k : k]. In the beginning of this section we have seen that multL(̺(Eℓ)) is a saturated Zℓ-submodule of EndZℓ (C(Eℓ, qℓ)). Since ιℓ is an isomorphism, 1 multL(ιℓ(̺(Eℓ))) is a saturated Zℓ-submodule of EndZℓ (H´et(A, Zℓ)) isomorphic to ′ ′ Eℓ as a Gal(k )-module. This immediately implies that the Gal(k )-module Eℓ/ℓ is isomorphic to a submodule of 1 1 EndZℓ (H´et(A, Zℓ))/ℓ = EndFℓ (H´et(A, Fℓ)). ′ The Qℓ-vector space Eℓ ⊗Zℓ Qℓ is a Gal(k)-module. Considered as a Gal(k )-module, it is isomorphic to a submodule of 1 1 EndZℓ (H´et(A, Zℓ)) ⊗Zℓ Qℓ = EndQℓ (H´et(A, Qℓ)). By applying Theorem 2.6 to the abelian variety A over k′ we obtain that the Gal(k′)- ′ module Eℓ ⊗Zℓ Qℓ is semisimple for each ℓ ∈ P , and that the Gal(k )-module Eℓ/ℓ is semisimple for all but finitely many ℓ ∈ P . The semisimplicity of Eℓ ⊗Zℓ Qℓ as a Gal(k)-module follows from its semisimplicity as a Gal(k′)-module by [17, Lemma 5 (b), p. 523]. By the same lemma, if we further exclude the finitely many primes ′ dividing [k : k], the semisimplicity of the Gal(k)-module Eℓ/ℓ follows from the ′ semisimplicity of the Gal(k )-module Eℓ/ℓ.  We finish this section with the following complement to Proposition 3.1. Proposition 3.2. In the assumptions of Proposition 3.1 suppose that for each ′ prime ℓ ∈ P we have a Gal(k )-submodule Uℓ of Eℓ such that Uℓ is saturated in Eℓ ′ ′ Gal(k ) Gal(k ) and Uℓ =0. Then (Uℓ/ℓ) =0 for all but finitely many ℓ ∈ P .

Proof. Since Uℓ is saturated in Eℓ and multL(ιℓ(̺(Eℓ))) =∼ Eℓ is saturated in 1 EndZℓ (H´et(A, Zℓ)), we see that multL(ιℓ(̺(Uℓ))) is a saturated Zℓ-submodule of 1 ′ EndZℓ (H´et(A, Zℓ)). It is Gal(k )-stable and isomorphic to Uℓ as a Galois module. The proposition now follows from Corollary 2.5. 

4. K3 surfaces Theorem 4.1. Let k be a field of odd characteristic p that is finitely generated over Fp. Let X be a K3 surface over k. Then 2 ¯ ¯ (i) for any prime ℓ 6= p the Gal(k)-modules H´et(X, Qℓ(1)) and T (X)ℓ ⊗Zℓ Qℓ are semisimple; 2 ¯ ¯ (ii) for all but finitely many primes ℓ the Gal(k)-modules H´et(X,µℓ) and T (X)ℓ/ℓ are semisimple; Gal(k) (iii) for all but finitely many primes ℓ we have (T (X¯)ℓ/ℓ) =0. 8 ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN

Proof. Choose a polarisation ¯ Gal(k) 2 ¯ Gal(k) ξ ∈ Pic(X) ⊂ NS(X) ⊂ H´et(X, Zℓ(1)) . 2 By the adjunction formula the degree of ξ is even, so that (ξ ) = eX,ℓ(ξ, ξ)=2d, where d is a positive integer. 2 ¯ 2 ¯ Let PH´et(X, Zℓ(1)) be the orthogonal complement to ξ in H´et(X, Zℓ(1)) with 2 ¯ 21 respect to eX,ℓ. Clearly, PH´et(X, Zℓ(1)) ≃ Zℓ is a Galois-stable saturated Zℓ- 2 ¯ submodule of H´et(X, Zℓ(1)). We have 2 ¯ Zℓ ξ ∩ PH´et(X, Zℓ(1)) = 0 2 ¯ 2 ¯ and the direct sum Zℓ ξ ⊕ PH´et(X, Zℓ(1)) is a subgroup of index 2d in H´et(X, Zℓ(1)). Furthermore, if ℓ does not divide 2d, then 2 ¯ 2 ¯ H´et(X, Zℓ(1)) = Zℓ ξ ⊕ PH´et(X, Zℓ(1)) (2) and the restriction 2 ¯ 2 ¯ eX,ℓ : PH´et(X, Zℓ(1)) × PH´et(X, Zℓ(1)) → Zℓ is a perfect Galois-invariant pairing. By a theorem of K. Madapusi Pera [14, Thm. 4.17] the family of ℓ-adic repre- 2 ¯ sentations Eℓ := PH´et(X, Zℓ(1)) preserving the quadratic form qℓ(x) = eX,ℓ(x, x), where ℓ is a prime not equal to p, satisfies the assumption of Proposition 3.1. It follows that the Gal(k)-module 2 ¯ 2 ¯ PH´et(X, Qℓ(1)) = PH´et(X, Zℓ(1)) ⊗Zℓ Qℓ is semisimple for each ℓ 6= p. This implies that the Gal(k)-module 2 ¯ 2 ¯ H´et(X, Qℓ(1)) = Qℓ ξ ⊕ PH´et(X, Qℓ(1)) ¯ is semisimple, and hence so is its Gal(k)-submodule T (X)ℓ ⊗Zℓ Qℓ, so (i) is proved. Suppose that (ℓ, 2d)=1. Let ξ¯ be the image of ξ in 2 ¯ 2 ¯ H´et(X,µℓ(1)) = H´et(X, Zℓ(1))/ℓ. From (2) we obtain a direct sum decomposition of Gal(k)-modules 2 ¯ ¯ 2 ¯ H´et(X,µℓ(1)) = Fℓξ ⊕ PH´et(X, Zℓ(1))/ℓ, ¯ 2 ¯ where ξ is Gal(k)-invariant. By Proposition 3.1 the Gal(k)-module PH´et(X, Zℓ(1))/ℓ 2 ¯ is semisimple for all but finitely many primes ℓ 6= p. Hence H´et(X,µℓ(1)) is semisim- ¯ 2 ¯ ple too. Since T (X)ℓ is a Galois-stable saturated Zℓ-submodule of PH´et(X, Zℓ(1)), ¯ 2 ¯ the Gal(k)-module T (X)ℓ/ℓ is isomorphic to a submodule of PH´et(X, Zℓ(1))/ℓ. This proves (ii). Part (iii) follows from Proposition 3.2 applied to the family of Gal(k)-submodules ¯ 2 ¯ Uℓ := T (X)ℓ ⊂ Eℓ = PH´et(X, Zℓ), for all ℓ 6= p. Indeed, by the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic, proved by Madapusi Pera [14, Thm. 1] (see also [10] ′ ¯ Gal(k ) ′  and [2]), we have T (X)ℓ = 0 for any finite separable field extension k of k. Proof of Theorem 1.2. The first statement is already proved in Theorem 4.1 (ii). As recalled in the introduction, for all but finitely many primes ℓ we have a direct sum decomposition of Gal(k)-modules 2 ¯ ¯ H´et(X, Zℓ(1)) = (NS(X) ⊗ Zℓ) ⊕ T (X)ℓ. AFINITENESSTHEOREMFORTHEBRAUERGROUP 9

2 ¯ For these primes ℓ the Galois module H´et(X,µℓ) is isomorphic to the direct sum of NS(X)/ℓ and T (X¯)ℓ/ℓ. By Theorem 4.1 (iii), for all but finitely many ℓ we have Gal(k) (T (X¯)ℓ/ℓ) = 0. This implies 2 ¯ Gal(k) Gal(k) H´et(X,µℓ) = (NS(X)/ℓ) . The cokernel of the natural map NS(X¯)Gal(k) → (NS(X)/ℓ)Gal(k) is a subgroup of the Galois cohomology group H1(k, NS(X¯)). This group is finite, since NS(X¯) is a free abelian group of finite rank. Removing the primes that divide the order of H1(k, NS(X¯)) we see that the natural map ¯ Gal(k) 2 ¯ Gal(k) NS(X) = NS(X) −→H´et(X,µℓ) is surjective for all but finitely many primes ℓ.  Proof of Theorem 1.3. The Hochschild–Serre spectral sequence p q p+q H (k, H (X,¯ Gm)) ⇒ H (X, Gm) gives rise to the well known filtration Br0(X) ⊂ Br1(X) ⊂ Br(X), where Br0(X) is the image of the natural map Br(k) → Br(X) and Br1(X) is the kernel of the natural map Br(X) → Br(X¯). The spectral sequence shows that Br1(X)/Br0(X) 1 is contained in the finite group H (k, NS(X¯)), and Br(X)/Br1(X) is contained in Br(X¯)Gal(k). Hence it is enough to prove that Br(X¯)Gal(k) is finite. For a prime ℓ 6= p the Kummer exact sequence gives rise to the following exact sequence of Gal(k)-modules [19, (5), p. 486]: ¯ Gal(k) 2 ¯ Gal(k) ¯ Gal(k) 0 → (NS(X)/ℓ) → H (X,µℓ) → Br(X)ℓ → 1 1 2 H (k, NS(X¯)/ℓ) → H (k, H (X,µ¯ ℓ)). In the proof of Theorem 1.2 we have seen that for all but finitely many primes ℓ the 2 Galois module NS(X¯)/ℓ is a direct summand of H (X,µ¯ ℓ), and the second arrow ¯ Gal(k) in the displayed exact sequence is an isomorphism. It follows that Br(X)ℓ =0 for all but finitely many ℓ. To finish the proof we note that for any prime ℓ 6= p the ℓ-primary subgroup Br(X¯){ℓ}Gal(k) is finite. This follows from [19, Prop. 2.5 (c)] in view of the semisim- 2 licity of H (X,¯ Qℓ(1)), proved in Theorem 4.1 (i), and the Tate conjecture 2 ¯ Gal(k) NS(X) ⊗ Qℓ = H´et(X, Qℓ(1)) proved by Madapusi Pera [14, Thm. 1] for any field k of odd characteristic p finitely generated over Fp. 

References

[1] N. Bourbaki, Alg´ebre, Chapitre 9, Hermann, Paris, 1959. [2] F. Charles, The Tate conjecture for K3 surfaces over finite fields, Invent. Math. 194 (2013), 119–145. [3] C. Chevalley, Th´eorie de groupes de Lie, tome III. Paris, Hermann 1954. [4] P. Deligne, La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206–226. [5] G. Faltings, Endlichkeitss¨atze f¨ur abelsche Variet¨aten ¨uber Z¨ahlkorpern, Invent. Math. 73 (1983) 349–366; Erratum 75 (1984) 381. [6] G. Faltings, Complements to Mordell, Chapter VI in: G. Faltings, G. Wustholz et al., Rational points. Aspects of Mathematics, E6. F. Vieweg & Sohn, Braunschweig, 1984. [7] A. Grothendieck, Le groupe de Brauer. I, II, III. In: Dix expos´es sur la Cohomologie des Sch´emas. North Holland, 1968, 46–188. [8] S. Lang, Abelian varieties. Second edition Springer-Verlag, New York, 1983. 10 ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN

[9] T. Matsusaka, The criteria for algebraic equivalence and the torsion group. Amer. J. Math. 79 (1957), 53–66. [10] D. Maulik, Supersingular K3 surfaces for large primes. arXiv:1203.2889 [11] J. Milne, Etale´ cohomology. Princeton University Press, Princeton, 1980. [12] L. Moret-Bailly, Pinceaux de vari´et´es ab´eliennes, Ast´erisque, 129, 1985. [13] D. Mumford, Abelian varieties, Second edition, Oxford University Press, London, 1974. [14] K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic. Preprint, April 2013, available at http://www.math.harvard.edu/∼keerthi/papers/tate.pdf. arXiv: 1301.6326 [15] J.-P. Serre and J. Tate, Good reduction of abelian varieties. Ann. Math. 88 (1968), 492–517. [16] J.-P. Serre, Abelian ℓ-adic representations and elliptic curves. Second edition. Addison-Wesley, New York, 1989. [17] J.-P. Serre, Sur la semi-simplicit´ede produit tensoriel de repr´esentations de groupes. Invent. Math. 116 (1994), 513–530. [18] A. Silverberg, Fields of definition for homomorphisms of abelian varieties. J. Pure Applied Algebra 77 (1992), 253–262. [19] A.N. Skorobogatov and Yu.G. Zarhin, A finiteness theorem for Brauer groups of abelian varieties and K3 surfaces. J. Alg. Geometry 17 (2008), 481–502. [20] J. Tate, Algebraic cycles and poles of zeta functions. In: Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, 93–110. [21] J. Tate, Endomorphisms of Abelian varieties over finite fields, Invent. Math. 2 (1966), 134– 144. [22] J. Tate, On the conjectures of Birch and Swinnerton–Dyer and a geometric analog. In: Dix expos´es sur la Cohomologie des Sch´emas. North Holland, 1968, 189–214. [23] Yu.G. Zarhin, Endomorphisms of Abelian varieties over fields of finite characteristic, Izv. Akad. Nauk SSSR Ser. Matem. 39 (1975) 272–277; Math. USSR Izv. 9 (1975) 255–260. [24] Yu.G. Zarhin, Abelian varieties in characteristic P , Mat. Zametki 19 (1976), 393–400; Math- ematical Notes 19 (1976), 240–244. [25] Yu.G. Zarhin, Endomorphisms of Abelian varieties and points of finite order in characteristic P , Mat. Zametki 21 (1977), 737–744; Mathematical Notes 21 (1978), 415–419. [26] Yu.G. Zarhin, A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Invent. Math. 79 (1985), 309–321. [27] Yu.G. Zarhin and A.N. Parshin, Finiteness problems in Diophantine geometry. Amer. Math. Soc. Transl. (2) 143 (1989), 35–102. [28] Yu.G. Zarhin, Homomorphisms of abelian varieties over finite fields. In: Higher-dimensional geometry over finite fields, D. Kaledin, Yu. Tschinkel, eds., IOS Press, Amsterdam, 2008, 315–343. [29] Yu.G. Zarhin, Abelian varieties over fields of finite characteristic. Central Eur. J. Math. 12 (2014), 659–674.

Department of Mathematics, South Kensington Campus, Imperial College, London, SW7 2BZ England, United Kingdom

Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow, 127994 Russia E-mail address: [email protected]

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Department of Mathematics, The Weizmann Institute of Science, POB 26, Rehovot 7610001, Israel

Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, Russia E-mail address: [email protected]