A REMARK ON UNIFORM BOUNDEDNESS FOR BRAUER GROUPS

Transcription

1 A REMARK ON UNIFORM BOUNDEDNESS FOR BRAUER GROUPS ANNA ADORET AND FRANÇOIS HARLES Abstract The Tate conjecture for divisors on varieties over number fieds is equivaent to finiteness of -primary torsion in the Brauer group We show that this finiteness is actuay uniform in one-dimensiona famiies for varieties that satisfy the Tate conjecture for divisors eg abeian varieties and K3 surfaces 1 Introduction 11 Let k be a fied of characteristic p and et k be a separabe cosure of k Let X be a smooth proper variety over k For any prime number p, consider the cyce cass map c 1 : Pic(X k ) Q H 2 (X k, Q (1)) The absoute Gaois group Ga(k/k) of k acts naturay on the étae cohomoogy group H 2 (X k, Q (1)), and the image of the cyce cass map c 1 is contained in the group H 2 (X k, Q (1)) U, U where U runs through the open subgroups of Ga(k/k) We say that X satisfies the -adic Tate conjecture for divisors if the image of c 1 is equa to the group above As is we-known, the Tate conjecture for divisors is reated to finiteness resuts for the Brauer group Indeed, given X and as above, X satisfies the -adic Tate conjecture for divisors if and ony if for every subgroup U of finite index in Ga(k/k), the group is finite Br(X k ) U [ ] 12 In this paper, we investigate whether one might be abe to find uniform bounds on the Brauer groups above when the variety X varies in a famiy Foowing Váriy-Avarado s conjectures on K3 surfaces [V17, par 4], such a question might be regarded as an anaogue of resuts of Manin [Ma69], Fatings- Frey [Fr94], Mazur [Ma77] and Mere [Me96] on uniform bounds for torsion of eiptic curves Reated boundedness resuts for famiies of smooth proper varieties have been deveoped by adoret-tamagawa [T12], [T13], [T17], [T16] It shoud be noted that a the aforementioned resuts can ony treat the case of one-dimensiona bases Deaing with the higher-dimensiona case seems to be much harder, as rationa points are not we-understood on varieties of dimension at east 2 The main resut of this paper is as foows Theorem 121 Let k be a finitey generated fied of characteristic zero Let S be a curve over k, and et π : X S be a smooth, proper morphism Let be a prime number Then for every positive integer d, there exists a constant d satisfying the foowing property: for every fied extension k K k with degree [K : k] d and every K-point s such that X s satisfies the -adic Tate conjecture for divisors Br(X s,k ) Ga(k/K) [ ] d Exampes of projective varieties for which the Tate conjecture for divisors is known and so for which Theorem 121 appies incude abeian varieties and K3 surfaces [Fa83, T94] In the case of certain one-parameter famiies of abeian and K3 surfaces which are actuay parametrized by Shimura curves the theorem above was proved by Váriy-Avarado-Viray [VAV16, Theorem 15, oroary 16] Whie their proof reies on deep arithmetic properties of eiptic curves, Theorem 121 is 1

2 derived as a consequence of the resuts of adoret-tamagawa in [T13] We wi aso prove a version of Theorem 121 that does not invove the Tate conjecture see Theorem 351 Outine of the paper Section 2 is devoted to some genera statements and questions about Brauer groups and their behavior in famiies In Section 3 we prove Theorem 121 and its unconditiona variant Acknowedgements The work on this note started after istening to an inspiring tak of Váriy- Avarado on his joint work with Viray at the conference Arithmetic Agebraic Geometry at the ourant Institute in september 216 We thank Tony Váriy-Avarado for usefu discussions and sharing his conjectures with us We thank Emiiano Ambrosi for hepfu discussions This project has received funding from the ANR grant ANR-15-E4-2-1 and, for the second author, from the European Research ounci (ER) under the European Union s Horizon 22 research and innovation programme (grant agreement No ) Notation If A is an abeian group, and n is an integer, we write A[n] for the n-torsion subgroup of A If is a prime number, we write A[ ] := im A[ n ] for the -primary torsion subgroup of A, and T (A) := im A[ n ] for its -adic Tate modue Let X be a smooth proper variety over a fied k We write Br(X) for the Brauer group of X, that is, the étae cohomoogy group H 2 (X, G m ), and NS(X) for the Néron-Severi group of X, that is, the image of P ic(x) in H 2 (X k, Q (1)), where k is a separabe cosure of k 2 A question on uniform finiteness for Brauer groups In this section, we reca the reationship between the Tate conjecture and finiteness of the Brauer group Whie the Tate conjecture is a quaitative statement, finiteness probems can be made quantitative, which eads us to Question The foowing proposition provides the basic reationship between the -adic Tate conjecture and finiteness for Brauer groups Let k be a fied of characteristic p Let X be a smooth proper variety over k hoose a prime different from p Let B := Br(X k ) be the Brauer group of X k, and set T := T (B), V := T Q, M := T Q /Z V /T For every integer n, we can identify M [ n ] with T/ n Proposition 211 The foowing assertions are equivaent (1) The -adic Tate conjecture hods for divisors on X; (2) B U [ ] is finite for every open subgroup U Ga(k/k); (3) M U is finite for every open subgroup U Ga(k/k); (4) T U = for every open subgroup U Ga(k/k) And, if this hods, for every open subgroup U Ga(k/k) one has (5) B U [ ] M U + H 3 (X k, Z (1))[ ] Proof We first prove the equivaence of (2), (3) and (4) We can restrict to the case where U is norma More precisey, we show that for any given norma subgroup U Ga(k/k) the foowing are equivaent (2) B U [ ] is finite ; (3) M U is finite; (4) T U =, and, that, if this hods, one has (5) B U [ ] M U + H 3 (X k, Z (1))[ ] 2

3 For every positive integer n, the cohomoogy of the Kummer short exact sequence 1 µ n G m G m 1 induces an exact sequence NS(X k ) Z/ n H 2 (X k, µ n) B[ n ] Letting n tend to, we get a commutative diagram with exact ines and coumns n NS(X k ) Z n H 2 (X k, Z (1)) K NS(X k ) Z H 2 (X k, Z (1)) T NS(X k ) Z/ n H 2 (X k, µ n) B[ n ] H 3 (X k, Z (1))[ n ] whence a short exact sequence T Z/ n B[ n ] H 3 (X k, Z (1))[ n ] The exactness on the eft foows from the snake, which ensures that K = n T Taking inductive imits and identifying T Z/ n M[ n ], we obtain a short exact sequence M B[ ] H 3 (X k, Z (1))[ ] and taking U-invariants, we obtain an exact sequence (2116) M U B U [ ] H 3 (X k, Z (1)) U [ ] This shows (2) (3) and that, in that case, (5) hods For the equivaence (3) (4), et U GL(T ) denote the image of U Ga(k/k) acting on T and consider the exact sequence T U V U M U δ H 1 (U, T ) Since U is a compact -adic Lie group, H 1 (U, T ) is a topoogicay finitey generated Z -modue [S64, Prop 9] In particuar, im(δ) is finite The concusion foows from the fact that either T U = V U = or V U/T U is infinite Now, assume X satisfies the -adic Tate conjecture for divisors Then, since NS(X k ) is a finitey generated abeian group ([N52, p 145, Thm 2], [SGA6, XIII, 51]), we can find an open subgroup U of Ga(k/k) such that the action of U on NS(X k ) is trivia, and the map NS(X k ) Q H 2 (X k, Q (1)) U is onto for every open subgroup U of U Furthermore, [T94, Prop (51)] shows that as a consequence of the Tate conjecture for X, the exact sequence of U-modues NS(X k ) Q H 2 (X k, Q (1)) V is spit, so that the surjectivity above impies V U = Since T is a torsion-free Z -modue, this impies T U = Since this vanishing hods for any sma enough open subgroup U of Ga(k/k), it hods for any open subgroup U of Ga(k/k) This shows (1) (4) For the converse impication, et U Ga(k/k) be an open subgroup such that H 2 (X k, Q (1)) U = H 2 (X k, Q (1)) U [Ga(k/k):U]< 3

4 Since T U = impies V U =, we obtain an isomorphism hence, a fortiori, NS(X k ) Q H 2 (X k, Q (1)) U (NS(X k ) Q ) U H 2 (X k, Q (1)) U 22 With Proposition 211 in mind, we offer the foowing question on uniform boundedness for Brauer groups, which we view as an enhancement of the -adic Tate conjecture for divisors Theorem 121 settes the case where S is one-dimensiona for those points that satisfy the Tate conjecture Question 221 Let k be a finitey generated fied of characteristic zero, and et be a prime Let S be a quasi-projective variety over k Let π : X S be a smooth projective morphism Then for any positive integer d, there exists a constant d satisfying the foowing property: for any fied extension k K k of degree [K : k] d, and any K-point s S(K), we have Br(X s,k ) Ga(k/k) [ ] d For d = 1, Question 221 is cosey reated to (the -adic Tate conjecture for divisors and) the Bombieri- Lang conjecture (see eg [B16, onj 18]) More precisey, assuming that the X s, s S(k) satisfies the -adic Tate conjecture for divisors, Question 221 woud foow from the foowing Gaois-theoretic conjecture For every integer 1 et S (k) be the set of a s S(k) such that the image of Ga(k/k) acting on H 2 (X s, Q ) is open of index at most in the image of π 1 (S) and et S,ex (k) be the compement of S (k) in S(k) Then we have the foowing: onjecture 222 There exists an integer 1 such that S,ex (k) is not Zariski-dense in S When S is a curve, onjecture 222 is a specia case of the main resut of [T12] We expain the reation between the genera case of onjecture 222 and the Bombieri-Lang conjecture One can aways construct a stricty increasing sequence n, n of integers and a projective system S n+1 S n S n 1 of étae covers of S with the property that the image of S n (k) in S is S n,ex (k) (see [T12, 312]) So proving that S,ex (k) is not Zariski-dense in S for arge enough amounts to proving that S n (k) is not Zariski-dense in S n for n According to the Bombieri-Lang conjecture, this woud foow from the fact that S n is of genera type for n Whie the Bombieri-Lang conjecture is sti competey open, proving that S n is of genera type for n seems to be a more tractabe probem The projective system (S n ) can indeed be regarded as a generaization of towers of Shimura varieties In this setting, Brunebarbe recenty obtained an asymptotic hyperboicity resut (impying the genera type property) see [B16] and the references therein, as we as [AVA16] As a concrete exampe, we now expain how a positive answer to the question above in the case of famiies of K3 surfaces for which the Tate conjecture is known yieds a positive answer to the -primary version of onjecture 46 of Váriy-Avarado in [V17] Proposition 223 Assume that Question 221 has a positive answer for famiies of K3 surfaces Let d be a positive integer, and et Λ be a attice Then there exists a positive constant d such that for any degree d number fied k with agebraic cosure k, and any K3 surface X over k with NS(X k ) Λ, the foowing inequaity hods: Br(X k ) Ga(k/k) [ ] d Using theorem 121, the proposition above is a consequence of the foowing resut Proposition 224 There exists a positive integer N with the foowing property: et Λ be a attice Then there exists a smooth projective famiy of K3 surfaces π : X S over a quasi-projective base S over Q such that given any fied k of characteristic zero with agebraic cosure k, and any K3 surface X over k with NS(X k ) Λ, there exists an extension K/k of degree N, and a K-point s of S with X s X K We start with a emma If L is a ine bunde on a K3 surface X, write c 1 (L) for its cass in NS(X) Pic(X) Lemma 225 There exists a positive integer N with the foowing property: et Λ be a attice There exists a positive integer δ Λ, depending ony on Λ, such that for any fied k with agebraic cosure k, and any K3 surface X over k with NS(X k ) Λ, there exists an extension K/k of degree N and an ampe ine bunde L on X K with c 1 (L) 2 = δ Λ 4

5 Proof Given any integer n, there exist ony finitey many finite groups that can occur as subgroups of GL n (Z) Since the continuous action of any profinite group on a discrete finitey generated abeian group factors through a finite quotient, and since the Néron-Severi group of any K3 surface is free of rank at most 22, this shows that there exist finitey many finite groups that can occur as the image of Ga(k/k) NS(X k ) for a K3 surface X over a fied k with agebraic cosure k In particuar, there exists an integer N, independent of X and k, with the foowing property: for any k, k and X as above, there exists a Gaois extension k K k of degree at most N such that the action of Ga(k/k) on NS(X k ) factors through Ga(K/k) Let X be a K3 surface over a fied k with agebraic cosure k, such that NS(X k ) Λ By [LMS14, Lemma 232], there exists a constant Λ, depending ony on Λ, such that there exists an ampe ine bunde L on X k with c 1 (L) 2 = Λ Write α = c 1 (L) Then, N being as above, we can find a Gaois extension K of k of degree N such that α is Ga(k/K)-invariant onsider the exact sequence coming from the Hochschid-Serre spectra sequence and Hibert 9 NS(X K ) NS(X k ) Ga(k/K) Br(K) Br(X) The zero-cyce c 2 (X) on X has degree 24, and induces a morphism Br(X) Br(K) such that the composition Br(K) Br(X) Br(K) is mutipication by 24 This shows that the cokerne of the map NS(X) NS(X k ) Ga(k/K) is kied by 24 As a consequence, we can find a ine bunde L on X K with c 1 (L) = 24α The puback of L to X k is L 24, which is ampe Since ampeness is a geometric property, L itsef is ampe We have c 1 (L) 2 = 24 2 Λ =: δ Λ, which ony depends on Λ Proof of Proposition 224 Appy the previous emma to find an integer N such that for X as in the proposition, there exists an extension K of k of degree N, and an ampe ine bunde L on X K with c 1 (L) 2 = δ, where δ ony depends on Λ By Koár-Matsusaka s refinement of Matsusaka s big theorem [KM83], and since the canonica bunde of a K3 surface is trivia by definition, we can find integers i and r, depending ony on Λ, such that L i is very ampe, and induces an embedding of X into a projective space of dimension at most r as a subvariety of degree at most δi 2 The existence of the Hibert scheme (or the theory of how forms) aows us to concude 3 Proof of Theorem 121 and its unconditiona variant Let k be a finitey generated fied of characteristic 31 Let S be a smooth, separated and geometricay connected variety over k, with generic point η If s is a point of S, we wi aways write s for a geometric point ying above s Let π : X S be a smooth, proper morphism For every prime, et ρ : π 1 (S) GL(H 2 (X η, Q (1))) denote the corresponding representation 1 Let K be a fied, and et s be a K-point of S We write π(s) for the absoute Gaois group of K considered with its natura map to π 1 (S), and we write aso ρ for the representation of π 1 (s) obtained by composing the above with π 1 (s) π 1 (S) Let S gen denote the set of a points s of S with vaue in a fied such that ρ (π 1 (s)) is an open subgroup of ρ (π 1 (S)) Let S ex be the compement of S gen In other words, S gen is the set of points s such that the Zariski cosures of ρ (π 1 (s)) and ρ (π 1 (S)) have the same neutra component Note that the generic point η aways beongs to S gen 1 The choice of base points for étae fundamenta groups wi pay no part in the seque; we ignore such choices systematicay in the foowing and do not mention base points in the notation 5

6 32 We start with the foowing proposition, which wi aow us to compare the Gaois-invariant part of Brauer groups of points in S gen Proposition 321 For every s S gen, the natura speciaization maps NS(X η ) NS(X s ) and Br(X η ) Br(X s ) are isomorphisms In particuar, if X s satisfies the -adic Tate conjecture for divisors then X s satisfies the -adic Tate conjecture for divisors for a s S gen Proof For every prime, consider the commutative speciaization diagram, with exact rows (3211) NS(X η ) Z H 2 (X η, Z (1)) T (Br(X η )) NS(X s ) Z H 2 (X s, Z (1)) T (Br(X s )) By the smooth base change theorem, the midde vertica map is an isomorphism, so that the eftmost vertica map is injective, and the rightmost vertica map is surjective Since this hods for every, the speciaization map NS(X η ) NS(X s ) is injective Let U be an open subgroup of ρ (π 1 (s)) acting triviay on NS(X s ), and such that NS(X s ) Z injects into H 2 (X s, Z (1)) U Since s S gen, U is open in ρ (π 1 (S)) hence (3212) U ρ (π 1 (S k )) is open in ρ (π 1 (S k )) hoose an embedding k We may assume s S() Let [L] NS(X s ), where L is a ine bunde on X s From (3212) and comparison between the anaytic and étae sites, the image c 1 (L) of [L] in H 2 (X s, Q(1)) is invariant under a finite index subgroup of π 1 (S ) corresponding to a finite étae cover S of S Fix a smooth compactification X S of X S The theorem of the fixed part and the degeneration of the Leray spectra sequence show that H 2 (X s, Q(1)) π 1(S ) is a sub-hodge structure of H 2 (X s, Q(1)) onto which the Hodge structure H 2 (X S, Q(1)) surjects Semisimpicity of the category of poarized Hodge structures ensures that c 1 (L) can be ifted to a Hodge cass in H 2 (X S, Q(1)) By the Lefschetz (1, 1) theorem, this impies that some mutipe of [L] can be ifted to an eement of NS(X η ) As a consequence, the speciaization map NS(X η ) NS(X s ) has torsion cokerne Since for any prime number, both cyce cass maps to -adic cohomoogy have torsion-free cokerne as the Tate modues are torsion-free the speciaization map is surjective, so it is an isomorphism Finay, for every prime and positive integer n, consider the speciaization diagram NS(X η ) Z/ n Z H 2 (X η, µ n) Br(X η )[ s n ] NS(X s ) Z/ n Z H 2 (X s, µ n) Br(X s )[ n ] Both the eftmost and the midde vertica maps are isomorphisms, so the rightmost one is an isomorphism as we Since the Brauer groups are torsion, this proves that the speciaization map on Brauer groups is an isomorphism This concudes the proof of the first part of the assertion in Lemma 321 The second part of the assertion is a forma consequence of the first 33 For every open norma subgroup U ρ (π 1 (S)), et S U S denote the set of a points s with vaue in a fied such that ρ (π 1 (s)) U The set S U is we-defined since ρ (π 1 (s)) is we-defined up to conjugation by an eement of ρ (π 1 (S)) and U is norma in ρ (π 1 (S)) By definition, S U S gen and S gen is the union of the S U as U runs through a norma open subgroups of ρ (π 1 (S)) Proposition 331 Let U ρ (π 1 (S)) be a norma open subgroup (1) If there exists s S gen such that the variety X s satisfies the -adic Tate conjecture for divisors, then for every s S U, there exists a constant U 1 such that Br(X s ) π 1(s) [ ] U 6

7 (2) Assume that the Zariski-cosure G of ρ (π 1 (S)) is connected Then there exists a constant U 1 such that for every s S U [Br(X s ) π 1(s) [ ] : Br(X η ) π 1(S) [ ]] U Proof Write B = Br(X η ), T = T (B), V = T Q and M = V /T By the second part of Proposition 321 and Proposition 211, B U [ ] is finite By the first part of Proposition 321, the speciaization map B Br(X s ) is an isomorphism, which is obviousy Gaoisequivariant In particuar, since ρ (π 1 (s)) contains U, we have a natura injection This shows (1) Br(X s ) π 1(s) [ ] B U [ ] Let s S U From the exact sequence (2116), we find M π 1(s) /M π 1(S) B π1(s) [ ]/B π1(s) [ ] Q, where Q is a subquotient of H 3 (X η, Z (1))[ ] and M π 1(s) In particuar, (2) foows from Lemma 332 beow /M π 1(S) M U/M π 1(S) by definition of S U Let T be a free Z -modue of rank r and et Π be a cosed subgroup of GL(T ) Write again V = T Q and M = T Q /Z V/T For any positive integer n, we can identify M[ n ] with T Z/ n Z Let G denote the Zariski cosure of Π in GL(V ) Lemma 332 Assume that G is connected 2 Then, for every norma open subgroup U Π, the index of M Π in M U is finite Proof Since G is connected, we have V U = V Π and T U = T V U = T V Π = T Π The commutative diagram with exact ines V Π /T Π M Π δ Π H 1 (Π, T ) res V U /T U M U δ U H 1 (U, T ) yieds an isomorphism M U /M Π im(δ U )/im(δ Π res) In particuar, the group M U /M Π is a quotient of im(δ U ) and the concusion foows from the fact im(δ U ) is finite as a torsion submodue of H 1 (U, T ) 34 For every positive integer d, define Write S gen, d and Remark 312(1)] and Sex, d S d = [K:k] d S(K) accordingy The foowing key resut is a reformuation of [T13, Theorem 11 Fact 341 Assume that S is a curve Then for every positive integer d, the image of S, d ex in S is finite and there exists an open subgroup U d of ρ (π 1 (S)) such that S gen, d S U d Proof of Theorem 121 Without oss of generaity, we may repace k by a finite fied extension and S by its reduced subscheme Aso, from Proposition 211, we may repace S by an open subscheme Working separatey on each connected component, we may assume that S is smooth, separated and geometricay connected variety over k Let U d be an open subgroup of ρ (π 1 (S)) as in Fact 341 Let s S d such that X s satisfies the -adic Tate conjecture for divisors If s S gen, then Fact 341 and (1) in Proposition 331 yied Br(X s ) π1(s) [ ] Ud, for some constant Ud depending ony on U d 2 This assumption is necessary, as can be shown by considering the r = 1 case with Π acting through a finite quotient and U being trivia 7

8 If s S ex then, by Fact 341, there are ony finitey many possibiities for the image s of s in S Since ρ (π 1 (s )) is an -adic Lie group, it has ony finitey many open subgroups of index at most d The intersection of a these open subgroups is an open subgroup U d (s ) contained in ρ (π 1 (s)) By construction Br(X s ) π1(s) [ ] Br(X s ) U d(s ) [ ] From Proposition 211 appied to X s, Br(X s ) U d(s ) [ ] is finite 35 Using Proposition 331, (2) instead of (1) in the proof above, one obtains the foowing unconditiona variant of Theorem 121 Theorem 351 Let k be a finitey generated fied of characteristic zero and et S be a quasi-projective curve over k Let π : X S be a smooth, proper morphism Then for every prime such that the Zariski-cosure of the image of π 1 (S) acting on H 2 (X η, Q (1)) is connected and for every positive integer d, there exists a constant d satisfying the foowing property: for every s S gen, d [Br(X s ) π 1(s) [ ] : Br(X η ) π 1(S) [ ]] d References [AVA16] D Abramovich and A Váriy-Avarado, Leve structures on abeian varieties, Kodaira dimensions, and Lang s conjecture, Preprint 216 [B16] Y Brunebarbe, A strong hyperboicity property of ocay symmetric varieties, Preprint 216 [T12] A adoret and A Tamagawa, A uniform open image theorem for -adic representations I, Duke Math J 161, p , 212 [T13] A adoret and A Tamagawa, A uniform open image theorem for -adic representations II, Duke Math J 162, p , 213 [T16] A adoret and A Tamagawa, Gonaity of abstract moduar curves in positive characteristic, ompositio Math 152, p , 216 [T17] A adoret and A Tamagawa, Genus of abstract moduar curves with eve- structures, to appear in Journa für die reine und angewandte Mathematik [Fa83] G Fatings, Endichkeitssätze für abesche Varietäten über Zahkörpern, Invent Math 73, 1983, p [Fr94] G Frey urves with infinitey many points of fixed degree, Israe J Math 85, 1994, p [KM83] J Koar and T Matsusaka, Riemann-Roch Type Inequaities, American Journa of Mathematics, 15, p , 1983 [LMS14] M Liebich, D Mauik and A Snowden, Finiteness of K3 surfaces and the Tate conjecture, Annaes scientifiques de ENS 47, fascicue 2, p , 214 [Ma69] Ju I Manin, The p-torsion of eiptic curves is uniformy bounded, Math USSR Izv 3, p , 1969 [Ma77] B Mazur, Moduar curves and the Eisenstein idea, Inst Hautes Etudes Sci Pub Math 47, p , 1977 [Me96] L Mere, Bornes pour a torsion des courbes eiptiques sur es corps de nombres, Invent Math 124, p , 1996 [N52] A Néron, Probèmes arithmétiques et géométriques rattachés à a notion de rang d une courbe agébrique dans un corps, Bu Soc Math France 8, p , 1952 [S64] J-P Serre, Sur es groupes de congruence des variétés abéiennes Izv Akad Nauk SSSR Ser Mat 28, p 3 2, 1964 [S89] J-P Serre, Lectures on the Morde-Wei theorem, Aspects of Mathematics E15, Friedr Vieweg & Sohn, 1989 [SGA6] Théorie des intersections et théorème de Riemann-Roch, LNM 225, Springer-Verag, Berin, 1971 (French) Séminaire de Géométrie Agébrique du Bois-Marie (SGA 6); Dirigé par P Bertheot, A Grothendieck et L Iusie Avec a coaboration de D Ferrand, J P Jouanoou, O Jussia, S Keiman, M Raynaud et J P Serre [SkZ8] A Skorobogatov and Ju G Zarhin, A finiteness theorem for the Brauer group of abeian varieties and K3 surfaces, Journa of Agebraic Geometry 17, p , 28 [T94] J Tate, onjectures on agebraic cyces in -adic cohomoogy in Motives (Seatte, WA, 1991), Proc Symp Pure Math 55 Part I, AMS, p 71-83, 1994 [V17] A Váriy-Avarado, Arithmetic of K3 surfaces in Geometry over noncosed fieds, (F Bogomoov, B Hassett, and Yu Tschinke eds), Simons Symposia 5, p , 217 [VAV16] A Váriy-Avarado and B Viray, Abeian n-division fieds of eiptic curves and Brauer groups of product of Kummer and abeian surfaces, Forum of Mathematics, Sigma, to appear annacadoret@imj-prgfr IMJ-PRG Sorbonne Universités, UPM University Paris 6, 755 PARIS, FRANE francoischares@mathu-psudfr Université Paris-Sud, 9145 ORSAY, FRANE 8