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Comparison of different Tate conjectures

In this paper, we study the relation between different versions of Tate conjectures for smooth projective varieties over arbitrary finitely generated fields of positive characteristic, generalizing a theorem of Tate for smooth projective varieties over finite fields.


Tate conjectures in different forms
In [Tat1], Tate formulated [Tat1,Conj. 1] and [Tat1,Conj. B+SD,Conj. 2,Conj. B+SD+2] in terms of L-functions. He asked about the relations between these conjectures and proved some equivalences between them for smooth projective varieties over finite fields(cf. [Tat2,Thm. 2.9]). In this paper, We will try to answer his questions on the relations between these conjectures for a smooth projective variety over a finitely generated field of positive characteristic.
Let X be a smooth projective geometrically connected variety over a finitely generated field K (i.e. finitely generated over a finite field or Q). One can then construct a projective and smooth morphism f : X −→ Y of schemes of finite type over Z, with Y irreducible and regular, whose generic fiber is X/K. For each closed point y ∈ Y, let k(y) denote the residue field of y and N(y) = q y denote the cardinality of k(y).
Set P y,i (T ) := det(1 − σ −1 y T |H i (Xȳ, Q ℓ )), where σ y ∈ Gal(k(y)/k(y)) is the arithmetic Frobenius element and ℓ = char(k(y)) be a prime. Set where Y • denotes the set of closed points. One can show that the product (1) converges absolutely for Re(s) > dim Y + i/2. If we replace Y by a nonempty open subscheme in (1), we divide Φ i (s) by a product which converges absolutely for Re(s) > dim Y + i/2 − 1, so the zeros and poles of Φ i in the strip depend only on X/K and not on our choice of X /Y (cf. Proposition 2.1). The zeta function of X is defined as where X y denotes the special fiber over y. By Grothendieck trace formula, we have Let ℓ = char(K) be a prime number. Let A i (X) be the group of classes of algebraic

Main Theorems
For any noetherian scheme X, the cohomological Brauer group Br(X) := H 2 (X, G m ) tor is defined to be the torsion part of the etale cohomology group H 2 (X, G m ).
Theorem 1.6. Let X be a smooth projective variety over a finitely generated field K of characteristic p > 0. Let m denote the transcendence degree of K over its prime field.
with equality if and only if E i (X, ℓ) and T i (X, ℓ) hold.
This was proved by Tate for the case that K is a finite field (cf. [Tat2,Thm. 2.9]). Our proof is based on the idea of his proof for finite fields and Deligne's theory of weights.
Corollary 1.7. The following statements are equivalent.
Remark 1.8. The equivalence between (1) and (3) is well-known (cf. Proposition 3.2) and the above result was proved by Tate in the case that K is a finite field. Theorem 1.9. Let A be an abelian variety over a finitely generated field K of characteristic p > 0. Let Y be a smooth and irreducible variety over a finite field with function field K. Set m = dim(Y). Define where Y 1 denotes the set of points of codimension 1 and K sh s denotes the quotient field of a strict local ring at s, then we have rank A(K) ≤ ord s=m Φ 1 (s), and the following statements are equivalent.
(i) X Y (A)(ℓ) is finite for some prime ℓ = p.
(iii) The BSD conjecture for A holds i.e. the rank of A(K) is equal to ord s=m Φ 1 (s).
Remark 1.10. This kind of Tate-Shafarevich group was studied by Keller in [Kel1]. The above theorem was proved for the case m = 1 by Schneider [Sch] and for the case that A can extend to an abelian scheme over a smooth projective variety by Keller [Kel2]. Our proof is based on a combination of their ideas. We will show that V ℓ X Y (A) is independent of the choice of Y (cf. Lemma 5.2), so we will write V ℓ X K (A) for V ℓ X Y (A).
Theorem 1.11. Let π : X −→ Y be a dominant morphism between smooth geometrically connected varieties over a finitely generated field k. Let K be the function field of Y. Assume that the generic fiber X of π is smooth projective geometrically connected over K. Set K ′ := Kk s . Let ℓ = char(k) be a prime, and set Then we have canonical exact sequences Here we write V ℓ X K ′ (Pic 0 X/K ) G k for V ℓ X Y k s (Pic 0 X/K ) G k which only depends on Pic 0 X/K (cf. Lemma 5.2). Moreover, if char(k) > 0, and B denotes the K/k-trace of Pic 0 X/K , then we have a canonical exact sequence In particular, if k is finite, we have V ℓ X K (Pic 0 X/K ) = V ℓ X K ′ (Pic 0 X/K ) G k . Remark 1.12. The relation between Tate-Shafarevich group and Brauer group was studied by Artin and Grothendieck (cf.[Gro3,§ 4] or [Ulm,Prop. 5.3]), Milne [Mil2], Gonzales-Aviles [Goa] and Geisser [Gei1] for fibrations of relative dimension 1 over curves, by Keller [Kel1,Thm. 4.27] for relative dimension 1 over high dimensional bases, by Geisser [Gei2] for high relative dimension over curves. For fibrations over arbitrary finitely generated fields, Ulmer [Ulm,§7.3] formulated some questions on relations between Tate conjeture and BSD conjecture in terms of L-functions. The key idea of our proof is using a Lefschetz hyperplane argument developed by Colliot-Thélène and Skorobogatov (cf. [CTS] or [Yua2] ).
Corollary 1.13. Let π : X −→ Y be as in the above theorem, and assume that k is a finite field. Then Conjecture 1.5 for X is equivalent to Conjecture 1.5 for Y + BSD for Pic 0 X/K + Conjecture 1.4 for X when i = 1. Remark 1.14. For the case that Y is a smooth projective curve and π is proper, this result was firstly proved by Geisser (cf.[Gei2,Thm. 1.1]). Corollary 1.15. Assuming that T 1 (X, ℓ) holds for all smooth projective varieties X over finite fields of characteristic p, then Conjecture 1.1 for i = 1, Conjecture 1.3, Conjecture 1.4 for i = 1 and Conjecture 1.5 hold for the case in characteristic p.
Remark 1.16. T 1 (X, ℓ) for smooth projective varieties over finite fields has been reduced to T 1 (X, ℓ) for smooth projective surfaces over finite fields with H 1 (X, O X ) = 0 by the work of de Jong [deJ]

Notation and Terminology
Fields By a finitely generated field, we mean a field which is finitely generated over a prime field. For any field k, denote by k s the separable closure. Denote by G k = Gal(k s /k) the absolute Galois group of k.

Henselization
Let R be a noetherian local ring, denote by R h (resp. R sh ) the henselization (resp. strict henselization) of R at the maximal ideal. If R is a domain, denote by K h (resp. K sh ) the fraction field of R h (resp. R sh ).

Schemes
All schemes are assumed to be separated over their bases. For a noetherian scheme S, denote by S • the set of closed points and by S 1 the set of points of codimension 1. By a variety over a field k, we mean a scheme which is separated and of finite type over k. For a smooth proper geometrically connected variety X over a field k, we use Pic 0 X/k to denote the underlying reduced closed subscheme of the identity component of the Picard scheme Pic X/k .

Cohomology
The default sheaves and cohomology over schemes are with respect to the smallétale site. So H i is usually the abbreviation of H i et unless otherwise instructed. We use H i fppf to denote flat cohomology for sheaves on fppf site.

Brauer groups
For any noetherian scheme X, denote the cohomological Brauer group

Abelian group
For any abelian group M, integer m and prime ℓ, we set 2 L-functions and convergences Proposition 2.1. Notations as in the introduction. Then Φ i (s) defined by (1) converges absolutely for Re(s) > dim Y + i/2 and the zeros and poles of Φ i in the strip does not depend on the choice of a smooth projective model X −→ Y.
Proof. By Deligne's theorem, P y,i (T ) are of fixed degree with reciprocal roots of absolute value q i/2 y , therefore, by [Ser,Thm. 1] the product (1) converges absolutely for Re(s) > dim Y + i/2. If we replace Y by a nonempty open subscheme in (1), and let Z denote the complement, we divide Φ i (s) by It follows that the zeros and poles of Φ i in the strip depend only on X/K and not on our choice of X /Y.
Proposition 2.2. Let X be a regular scheme of finite type over Z. Assume that its zeta function ζ(X , s) can be meromorphically continued to the point s = dim X − 1. Let e(X ) be the order of ζ(X , s) at that point, and put Then z(X ) is a birational invariant. Let f : X −→ Y be a morphism as discussed at the beginning, then any two of the following statements imply the third: (i) BSD for X/K.
Since ζ(Z, s) converges abosultely for Re(s) > dim Z and has a simple pole at s = dim Z (cf. [Ser,Thm. 1,2,3]), the order of ζ(Z, s) at dim X − 1 is 0 if the codimension of Z is at least 2 and is 1 if the codimension of Z is 1. If Z is a divisor, set U = X − Z, there is an exact sequence It follows that e(X ) = e(U) − 1, so z(X ) = z(U).
For the second claim, since f is proper and the generic fiber is geometrically irreducible and Y is normal, we have . By [Har,Chap. III,Cor. 11.3], all fibers of f are geometrically connected. Let D be a vertical prime divisor on X . Since f is smooth proper and all fibers are connected, one can show that D = f −1 (f (D)) as a Weil divisor. Thus, any vertical divisor on X is the pullback of some divisor on Y. By the Leray spectral sequence Thus, we get an exact sequence , comparing the order of both sides of the equation (2) at s = dim X − 1, it suffices to show that Φ i (s) has no pole or zero at s = dim X − 1 for i = 2d − 1, 2d − 2, 2d. Since all reciprocal roots of P y,i (T ) have absolute values q i 2 y , the product (1) converges absolutely for Re(s) > dim Y + i/2, the claim follows.

Tate conjecture for divisors and Brauer groups
Recall Conjecture 1.1 for i = 1, Conjecture 3.1. (Conjecture T 1 (X, ℓ)). Let X be a projective and smooth variety over a finitely generated field k of characteristic p ≥ 0, and ℓ = p be a prime number. Then the cycle class map This conjecture is called Tate conjecture for divisors on X. By the proposition below, it is equivalent to the finiteness of Br(X k s ) G k (ℓ).
Proposition 3.2. Let X be a smooth projective geometrically connected variety over a field k. Let ℓ be a prime different from char(k), then the exact sequence of G k -module In the following, we will show that V ℓ Br(X k s ) G k is a birational invariant for smooth varieties over finitely generated fields. Tat2,Thm. 5.2]) Let X be a smooth geometrically connected variety over a finitely generated field k of characteristic p ≥ 0 and U ⊆ X be an open dense subvariety. Let ℓ = p be prime. Then the natural map Proof. Extending k to a finite extension will not affect the results. By purity for Brauer groups [Čes,Thm. 1.1], removing a closed subscheme of codimension ≥ 2 will not change the Brauer group. Therefore, we might assume that X − U = D i , where D i is a smooth geometrically connected subvariety over k of codimension 1. There is an exact sequence By the theorem of cohomological purity (cf. [Fuj]), is of weight ≥ 1. Consider the commutative diagram Since the cokernel of the first column is of weight ≥ 1, the cokernel of the second column is also of weight ≥ 1. Since the second column is injective, taking G k invariants, the claim follows.
The following lemma generalizes Proposition 3.2 to noncompact varieties over finite fields. We will use this lemma to prove the equivalence between Conjecture 1.5 for smooth varieties over finite fields and the finiteness of Br(Xk) G k (ℓ).
Lemma 3.4. Let X be a smooth geometrically connected variety over a finite field k of characteristic p > 0. Let ℓ = p be a prime. Then, there is a split exact sequence of G k -representations Proof. Firstly, if the claim holds for G k 1 -actions where k 1 /k is a finite Galois extension, then it also holds for G k -actions. Let V be a sub-representation of some G k -representation W over Q ℓ . Let p : W −→ V be a G k 1 -equivariant projection. Then Secondly, we will show that the claim holds for open subvarieties of a smooth projective geometrically connected varietiy over k. Let U ⊆ X be an open subset and X is projective. There is a commutative diagram By the theorem of cohomological purity(cf. [Fuj]), Ker(b) = Ker(a) is generated by cycle classes of divisors contained in X − U. Since the first row is split, it follows that is also split. By extending k, we might assume that the G k -action on Pic(Uk) ⊗ Z Q ℓ is trivial. Then Pic(Uk) ⊗ Q ℓ is contained in the generalized 1-eigenspace of the Frobenuis action on H 2 (Uk, Q ℓ (1)). This eigenspace is a direct summand and is also contained in Im(b) by the proof of Proposition 3.3. It follows that Pic(Uk) ⊗ Q ℓ is a direct summand of the generaized 1-eigenspace, and therefore, it is also a direct summand of H 2 (Uk, Q ℓ (1)). Thirdly, by de Jong's theorem, there exists an alteration f : U −→ X such that U admits a smooth projective compactification. Since f is proper and generic finite, there is a pushfoward map Through f * , Pic(Xk) ⊗ Q ℓ is a direct summand of Pic(Uk) ⊗ Q ℓ , and therefore, is also a direct summand of H 2 (Uk, Q ℓ (1)). It follows that Pic(Xk) ⊗ Q ℓ is a direct summand of H 2 (Xk, Q ℓ (1)).
Theorem 3.5. Let X be a smooth and irreducible variety over a finite field of characteristic p. Let ℓ = p be a prime. Then the Conjecture 1.5 for X is equivalent to the finiteness of Br(X k s ) G k (ℓ).
Proof. Assuming that dim X = n and k = F q , let F denote the geometric Frobenius element in G k . By Grothendieck's formula, we have By Deligne's theorem, H i c (Xk, Q ℓ ) is mixed of weight ≤ i and H 2n c (Xk, Q ℓ ) is pure of weight 2n. Therefore, only det(1 − q −s F |H 2n−2 c (Xk, Q ℓ )) and det(1 − q −s F |H 2n−1 c (Xk, Q ℓ )) have contributions to the order of the pole of ζ(X , s) at s = n − 1. By Poincaré duality, Next, we will show that the dimension of the generalized q n−1 -eigenspace of H 1 (Xk, Q ℓ ) ∨ (−n) is equal to rank H 0 (X , O * X ). By Kummer exact sequence, we get an exact sequence There exists an alteration f : X ′ −→ X such that X ′ admits a smooth projective compactification over k. Since z(X ) is a birational invariant, we might shrink X such that f is finite flat. There is a canonical norm map f * G m → G m whose composition with the canonical map G m → f * G m is equal to the multiplication by deg(f ). Since is equal to the multiplication by deg(f ). It follows that the natural map is injective. Assuming that X ′ ⊂X whereX is irreducible and smooth projective over k.
SinceX is regular, the natural map is surjective and has a finitely generated kernel. Since Pic(Xk) is an extension of a finitely generated abelian group by a ℓ-divisible torsion group Pic 0X /k (k), it follows that the natural map This implies that the dimension of the generalized q n−1 -eigenspace of H 1 (Xk, Q ℓ ) ∨ (−n) is equal to rank H 0 (X , G m ).
Thus, it suffices to show that the dimension of the generalized 1-eigenspace of H 2 (Xk, Q ℓ (1)) is equal to rank Pic(X ) if and only if V ℓ Br(Xk) G k vanishes. By Lemma 3.4, there is an exact sequence Assuming that the dimension of the generalized 1-eigenspace of H 2 (Xk, Q ℓ (1)) is equal to rank Pic(X ), then (1)) as a G k -representation. It follows that the generalized 1-eigenspace of H 2 (Xk, Q ℓ (1)) is same as the 1-eigenspace which is equal to Pic(X ) ⊗ Z Q ℓ . This completes the proof.
Corollary 3.6. Assuming that T 1 (X , ℓ) holds for all smooth projective varieties over a finite field k, then Conjecture 1.5 holds for all smooth varieties over k.
Proof. Let X be a smooth variety over k. By shrinking X , we might assume that there is a finite flat morphism f : By the above theorem, Conjecture 1.5 holds for X . 4 Proof of Theorem 1.6 Lemma 4.1. Let k be a field and V be a finite-dimensional Q ℓ -linear continuous rep- This proves the claim.
Theorem 4.2. Let X be a smooth projective geometrically connected variety over a finitely generated field K of characteristic p > 0. Let π : X −→ Y be a smooth projective model of X/K as in section 1, where Y is a smooth geometrically connected variety over a finite field k. Set m = dim Y, d = dim X and let ℓ = p be a prime. Then we have

Then the intersection pairing
Let a ∈ N i (X) with cl(a) = 0, then cl(a 1 ), ..., cl(a t ), cl(a) are Q ℓ -linear independent. Let λ i ∈ Q ℓ such that Assuming the second inequlity, we proved the " ⇒ "direction of the second claim. Next we will prove the second inequality. Let q denote the cardinality of k and This proves the second inequality.
For " ⇐ ′′ direction, assuming T i (X, ℓ) and E i (X, ℓ), then we have This implies that the 1-eigenspace of F on H 2i (X K s , Q ℓ (i)) G K ′ is a direct summand, and therefore, is equal to the generalized 1eigenspace of F . It follows that This completes the proof.
Corollary 4.3. The following statements are equivalent.

Tate-Shafarevich group for abelian varieties over finitely generated fields
In this section, we will define Tate-Shafarevich group for abelian varieties over finitely generated fields of positive characteristics and study a geometric version of Tate-Shafarevich group. The Tate-Shafarevich group for abelian schemes over high dimensional bases was studied by Keller in [Kel1].
Definition 5.1. Let S be an integral regular noetherian scheme with function field K.
Let A be an abelian variety over K. Define where S 1 denotes the set points of codimension 1.
Lemma 5.2. Let S be a smooth geometrically connected variety over a finitely generated field k. Let K denote the function field of S and Let A be an abelian variety over K.
has a finite cokernel. Furthermore, if A extends to an abelian scheme A −→ U, then we have a canonical isomorphism which only depends on A/K. We will denote it by Proof. It is easy to see that has a finite cokernel and In the case that k is a finite field, by the same argument, the inclusion has a finite cokernel and V ℓ X S (A) = V ℓ H 1 (K, A).
If A/K extends to an abelian scheme A −→ U, we will show that ).
This will imply that the composition of Thus, we get an exact sequence Thus ).
Next, we will show that the inclusion is actually an equality. Since is a locally constant sheaf, by the semi-purity [Fuj,§8], removing a closed subscheme of codimension ≥ 2 from U k s will not change H 1 (U k s , A [ℓ n ]). So by shrinking Y , we might assume that Y is regular and of codimension 1 in Therefore, if x lies in then it also lies in H 1 (U k s , A [ℓ n ]). This proves (4). By [Kel1,Thm. 3.3], we have By the same argument, one can show that Firstly, we will show that the natural map Since there exists an abelian variety B/K such that A × B is isogenous to Pic 0 X/K for some smooth projective geometrically connected curve X/K, we might assume that A = Pic 0 X/K . One can spread out A/K to a smooth projective relative curve π : X −→ Y where Y is smooth and geometrically connected over k. In the following, we will show that there is commutative diagram with exact second row Then the surjectivity of the second column will follow from the surjectivity of the first column. By the Leray sepectral sequence . Since X is a smooth projective curve over K, we have Br(X K s ) = 0. Thus, we get a natural map Let j : Spec K −→ Y be the generic point. By Lemma 6.1, we have Therefore, By Theorem 6.2, there is a surjective map compatible with the above map Since H q (G k , −) vanishes when q ≥ 2, we get a surjection Taking limit, we get Next, we will show that the natural map Let Tr K/k (A) be the K/k trace of A. By the Lang-Néron theorem, A(K ′ )/Tr K/k (A)(k s ) is finitely generated. Since H 1 (G k , Tr K/k (A)) = 0 by Lang's theorem, we have that H 1 (G k , A(K ′ )) is finite. It follows that the natural map This completes the proof.
Lemma 5.3. Let G be a smooth connected algebraic group over a finitely generated field K. Let ℓ = char(K) be a prime. Then the size of is bounded by a constant independent of n.
Proof. Without loss of generality, we can replace K by its finite extension. By Chevally's Theorem, GK is an extension of an abelian variety by a linear algebraic group. By extending K, we can assume that this extension is defined over K. So we have an exact sequence Since H is commutative, by extending K, we can assume that H ∼ = G s m × G t a over K.
Taking dual, we get Since H[ℓ n ] = G s m [ℓ n ], so it suffices to prove the claim for abelian varieties and G m . Let S be an irreducible regular scheme of finite type over Spec Z[ℓ −1 ] with function field K. Let G be an abelian variety (resp. G m ) over K, by shrinking S, we can assume that G extends to an abelian scheme G S (resp. G m,S ). Let s ∈ S be closed point with finite residue field k. Write F for the etale sheaf represented by G S [ℓ n ] on S. Since G S [ℓ n ] is finite etale over S, F is a locally constant sheaf of Z/ℓ n -module on S. Let η be the generic point of S. Since S is connected, we have Fη ∼ = Fs. Since Fη = G[ℓ n ], we have G[ℓ n ] ∼ = G s [ℓ n ] as G k -modules. Through this isomorphism, there is an inclusion Therefore, it suffices to prove the claim for the case that K is a finite field. In this case, Since G[ℓ n ] G K ⊆ G(K), so the size is bounded by |G(K)|. This completes the proof.
Lemma 5.4. Let R be a henselian DVR with residue field k and quotient field K. Assume that k is a finitely generated field. Let R sh denote a strict henselization of R and K sh denote its quotient field. Let A be an abelian variety over K and ℓ be a prime different from char(k). Set G := Gal(k s /k) = Gal(K sh /K). Then the size of Proof. Let I denote Gal(K s /K sh ) and I 1 denote the wild inertia subgroup. Since H 1 (I 1 , A[ℓ n ]) = 0, by [Mil3,Chap I,Lem. 2.18], we have Its dual Hom((A[ℓ n ](−1)) I , Q ℓ /Z ℓ ) = Hom(A[ℓ n ](−1), Q ℓ /Z ℓ ) I is canonically isomorphic to A t [ℓ n ] I by Weil pairing, where A t denotes the dual abelian variety of A. Let A t be a Néron model of A t /K and s : Spec k −→ S = Spec R be the closed point. Then There is an exact sequence where Φ(A t ) denotes the Néron component group of A t which is a finite etale scheme over k. Taking dual and then taking G k -invariants, we get an exact sequence Note that the size of (Φ(A t ) ∨ ) G is bounded by a constant independent of n. Since it suffices to show that the size of (((A t s ) 0 [ℓ n ]) ∨ ) G is bounded independent of n. It follows directly from Lemma 5.3. This proves the first claim.
By Kummer exact sequence, we get an exact sequence Let A be a Néron model of A over S and A 0 be the identity component of A . We have A(K sh ) = A (S) and an exact sequence It follows H 1 (K sh , A(ℓ)) ∼ = H 1 (K sh , A)(ℓ). Since by the first claim, the right side is finite. It follows that H 1 (K sh , A) G (ℓ) is also finite.
6 Fibrations and Brauer groups Lemma 6.1. Let U be an irreducible regular scheme with function field K. Let π : X −→ U be a smooth proper morphism with a generic fiber geometrically connected over K. Let j : Spec K −→ U be the generic point of U. Then we have (a) the natural map is an isomorphism for any prime ℓ invertible on U.
Proof. It suffices to show that the induced maps on stalks are isomorphism. Thus, we might assume that U = Spec R where R is a strictly henselian regular local ring. Let X denote the generic fiber. Let s ∈ U be the closed point. Then we have (R 1 π * G m )s = Pic(X ) and (j * j * R 1 π * G m )s = Pic X/K (K).
Since X s admits a section s −→ X s and π is smooth, the section can be extended to a section U −→ X . Thus X(K) is not empty. So Pic X/K (K) = Pic(X).
Since X is regular, the natural map is surjective and has a kernel generated by vertical divisors. Since π is smooth and proper with a geometrically connected generic fiber, by the same arguments as in the proof of Proposition 2.2, there is an exact sequence Since R is an UFD, we have Pic(Spec R) = 0. It follows that Pic(X ) ∼ = Pic(X).
This proves (a). Let I denote G K . For (b), the induced map on the stalk at s is Br(X )(ℓ) −→ Br(X K s ) I (ℓ).
Since π is smooth and proper, we have Since NS(X K s ) ⊗ Q ℓ /Z ℓ is I-invariant and Pic(X ) = Pic(X). It suffices to show that The action of I on NS(X K s ) factors through a finite quotient I ′ . Consider the exact sequence , the claim follows. By Snake Lemma, the natural map is an isomorphism.
Theorem 6.2. Let π : X −→ Y be a dominant morphism between smooth geometrically connected varieties over a finitely generated field k of characteristic p ≥ 0. Let K be the function field of Y. Assume that the generic fiber X of π is smooth projective geometrically connected over K. Set K ′ := Kk s and let ℓ = p be a prime. Set then we have canonical exact sequences Proof. By shrinking Y, we might assume that π is projective and smooth. Firstly, we will show that the natural map (1)). By Deligne's Lefschetz criteria(cf. [Del1] ), the above spectral sequence degenerates at E 2 . Thus, we get a canonical surjective map

It follows that the canonical map
is surjective. Consider the following commutative diagram with exact rows Since the first column is surjective, the map is also surjective. Secondly, we will show the exactness of the first sequence in the statement of theorem. By the Leray spectral sequence we get a long exact sequence Note that without loss of generality, we can always replace k by its finite Galois extension. There exists a finite Galois extension L/K such that X(L) is not empty. Choose a Kmorphism Spec L −→ X and we might assume that L is the function field of smooth variety Z over k. By shrinking Y and Z, we can assume that Spec L −→ X extends to a morphism Z −→ X such that Z −→ Y is finite flat. Let π ′ denote Z −→ Y. Then we have a commutative diagram induced by the Leray spectral sequences for π and π ′ H 1 (Y k s , R 1 π * G m ) Since π ′ is finite, we have R i π ′ * G m = 0 for i > 0 and H 3 (Y k s , π ′ * G m ) = H 3 (Z k s , G m ). There is a canonical norm map π ′ * G m −→ G m which induces a norm map is equal to the multiplication by deg(π ′ ). It follows that the second column in the above diagram has a kernel killed by deg(π ′ ). Therefore, the first row has an image killed by deg(π ′ ) and the sequence 0 Thus, we get a split exact sequence of G k -representations In fact, the norm map N induces a projection So we get an isomorphism where the direct sum decomposition depends on the choice of π ′ . Next, we will show that there is a canonical isomorphism Let j : Spec K −→ Y be the generic point. By Lemma 6.1, the natural map There is an exact sequence Let X Y k s (Pic X/K ) denote the kernel of the third arrow. It suffices to show that By the canonical exact sequence we get a long exact sequence Let L/K be a finite Galois extension such that Pic(X L ) −→ NS(X K s ) is surjective. One can show that H 1 (K ′ , NS(X K s )) is killed by [L : K]#NS(X K s ) tor and the cokernel of Thus, the kernel and the cokernel of H 1 (K ′ , Pic 0 X/K ) −→ H 1 (K ′ , Pic X/K ) are killed by [L : K]#NS(X K s ) tor . By the same argument, the kernel and cokernel of are also killed by [L : K]#NS(X K s ) tor . Taking V ℓ , they become isomorphisms. Therefore, we have a canonical isomorphsim X/K ). By Lemma 6.1, the natural map is an isomorphism. It follows directly Thus, there is an exact sequence where the second arrow depends on the choice of π ′ . Taking G k -invariants, we proved the exactness of the first sequence in the theorem. It remains to show that the natural map is surjective. To prove this, we will use a pull-back trick (cf. [CTS] or [Yua2]). Let W be a smooth projective geometrically connected curve over K contained in X which is a complete intersection of hyperplane sections. By Shrinking Y, we might assume that W adimits a smooth projective model W −→ Y where W is the Zariski closure of W in X .
By shrinking Y, we might assume that the map Z −→ X chosen before factors through Z −→ W. Thus, we get a commutative diagram Since V ℓ Br(W K s ) G K = 0, the second arrow in the bottom vanishes. Thus, to show that a is surjective, it suffices to show that c is injective. This actually follows from the fact By our choice of W and the Lefschetz hyperplane section theorem, the pullback map is injective. This implies that the induced map Pic 0 X/K −→ Pic 0 W/K has a finite kernel. Therefore, there exists an abelian variety A/K and an isogeny Pic 0 . This proves the splitness. Thus, the natural map In the theorem above, we assume that the generic fiber is projective over K. In the following corollary, we will deduce the surjectivity of V ℓ Br(X k s ) G k −→ V ℓ Br(X K s ) G K from the above theorem for any fibration with a smooth generic fiber. Corollary 6.3. Let k be a finite field or a number field. Let π : X −→ Y be a dominant morphism between smooth geometrically connected varieties over k. Let K be the function field of Y and X be the generic fiber of π. Let ℓ = char(k) be a prime. Assuming that X is smooth over K, then the natural map Proof. Firstly, note that without loss of generality, we can always extend k to its finite extension.
Secondly, we might assume that X is geometrically irreducible over K. Let L be the algebraic closure of K in K(X). Then L/K is a finite separable extension and X is smooth and geometrically irreducible over Spec L. By speading out X −→ Spec L, we get a morphsim with a smooth geometrically connected generic fiber. Since Br(X ⊗ L L s ) G L = Br(X K s ) G K , it suffices to prove the claim for this morphism.
Thirdly, if X/K is birational equivalent to a smooth projective variety X ′ over K, then the claim will follow from Theorem 6.2 since V ℓ Br(X K s ) G K is a birational invariant.
Next, we will use de Jong's alteration theorem to reduce the question to this case. Let X ′K −→ XK be an alteration such that X ′K admits a smooth projective compactification. We might assume that X ′ and the alteration are defined over a finite normal extension L of K. By shrinking X, we can assume that the alteration X ′ −→ X L is finite flat. By spreading out X ′ −→ Spec L, we get a commutative diagram where Y ′ has function field L. By shrinking Y, we can assume that f is finite flat. Thinking f as a Y-morphsim and base change to Spec K s , we get X ′ K s −→ X K s which is also finite flat. It induces a norm map Br(X ′ K s ) −→ Br(X K s ) which is compatible with the norm map Br(X ′ k s ) −→ Br(X k ). Therefore we get a commutative diagram Since vertical maps are surjective, it suffices to show that the first row is surjective. Since the claim holds for X ′ → Y ′ , the natural map Fix an algebraic closureK of K and assume that L and L s are contained inK. Since Set K 1 = L ∩ K s , by our assumption that L/K is a normal extension, so K 1 /K is finite Galois. Write G = G K and H = G K 1 . We have One can show that L ⊗ K 1 ,σ K s is a separable closure of L and admits a H-action (acting on K s ). We denote it by L s σ . Then Taking H-invariant, we get Now G/H acts as permutations on the right side, so the G/H-invariant of right side can be identified with the factor with the index σ equal to the inclusion map K 1 ⊂ K s . The factor can be identified with Br(X ′ L s ) G L through the natural isomorphism L ⊗ K 1 K s ∼ = L s . It follows that there is a natural isomorphism This completes the proof.
Corollary 6.4. Let Y be a smooth geometrically connected variety over a finitely generated field k with function field K of characteristic p > 0. Let A be an abelian variety over K. Let B = Tr K/k (A) denote the K/k trace of A. Let ℓ = p be a prime. Then there is a canonical exact sequence Proof. Consider the exact sequence we will show that V ℓ H 1 (G k , A(K ′ )) is naturally isomorphic to V ℓ H 1 (k, B). This will imply the left exactness of the sequence in the claim. By the Lang-Néron theorem, A(K ′ )/B(k s ) is finitely generated, so H 1 (G k , A(K ′ )/B(k s )) is of finite exponent. By the exact sequence where the last equality follows from Lemma 5.2. It follows that To show the surjectivity of the last arrow, we might assume A = Pic 0 X/K where X is a smooth projective geometrically connected curve over K. Spreading out X/K, we get a smooth projective relative curve π : X −→ Y. Since Br(X K s ) = 0, by Theorem 6.2, we get a surjection Let k 0 be the algebraic closure of the prime field of k in k. Then there exists a smooth geometrically connected variety Z over k 0 with function field k. By shrinking Z, we can spread out π to a smooth projective morphsim π Z : X Z −→ Y Z . By shrinking Z further, we can assume that Y Z is smooth over Z. Using Theorem 6.2 for π Z , we get a surjection is surjective. Since X can be identified with the generic fiber of X Z −→ Z, then the claim follows from Corollary 6.3.

Proof of Theorem 1.11
It follows directly from Theorem 6.2 and Corollary 6. 7 BSD conjecture and Tate-Shafarevich group 7.1 Cohomological interpretation of the vanishing order of L function for an abelian variety Lemma 7.1. Let A be an abelian variety defined over a finitely generated field K of characteristic p > 0. Assume that A extends to an abelian scheme π : A −→ Y, where Y is a smooth geometrically connected variety over a finite field k with function field K. Set m = dim Y and let ℓ = p be a prime. Then and ord s=m Φ 1 (s) is equal to the dimension of the generalized 1-eigenspace of the Frobenius action on H 1 (Yk, V ℓ A ).
Proof. Let q denote the cardinality of k and F denote the geometric Frobenius element in G k . Set F = R 1 π * Q ℓ , F is lisse since π is smooth and proper. By definition, By Grothendieck's formula, By Deligne's theorem, H i c (Yk, F ) is of mixed weight ≤ i + 1 and H 2m c (Yk, F ) is of pure weight 2m+1 by Poincaré duality. Therefore only det(1−q −s F |H 2m−1 c (Yk, F )) contributes to the order of zero at s = m. By Poincaré duality, It follows that ord s=m Φ 1 (s) is equal to the dimension of the generalized 1-eigenspace of F on H 1 (Yk, F ∨ ). Thus The lisse sheaf F corresponds to the representation of G K on H 1 (AK, Q ℓ ), therefore F ∨ corresponds to H 1 (AK, Q ℓ ) ∨ which can be identified with V ℓ A . Since π is an abelian scheme, there is a Kummer exact sequence of etale sheaves on Y Since A [ℓ n ] is a finite etale group scheme over Y, it represents a locally constant sheaf on Y. Let V ℓ A be the Q ℓ -sheaf associated to the inverse system A [ℓ n ]. It corresponds to the G K representation on V ℓ A. Therefore V ℓ A can be identified with F ∨ . From the Kummer exact sequence, we get By [Kel1,Thm. 3.3], A satisfies the Neron property Taking G k -invariants for the above exact sequence, we get 7.2 Proof of Theorem 1.9 Theorem 7.2. Let A be an abelian variety over a finitely generated field K of characteristic p > 0. Let ℓ = p be a prime. Then the BSD conjecture (i.e. Conjecture 1.4) for A is equivalent to the finiteness of X K (A)(ℓ).
Proof. Assuming the BSD conjecture for A, by Lemma 7.1 and 7.3 below, we have Then the claim follows from Lemma 7.1.
Lemma 7.3. Let A be an abelian variety defined over a finitely generated field K of characteristic p > 0. Assume that A extends to an abelian scheme π : A −→ Y, where Y is a smooth geometrically connected variety over a finite field k with function field K. Let ℓ = p be a prime. Then there is a split exact sequence of G k -representations Note that without loss of generality, we might extend k or shrink Y. By shrinking Y, we can assume that Y is an open subvariety of a projective normal varietyȲ over k. Let V denote the smooth locus ofȲ/k. Then Y −V has codimension ≥ 2 inȲ. Let j denote the open immersion Y ֒→ V and F denote V ℓ A on Y. By [Fu,Prop. 10.1.18 (iii)], R q j * F are constructible Q ℓ -sheaves for all q ≥ 0. Set D = V − Y, by removing a closed subset Z of codimension ≥ 2 from V , we might assume that D is a smooth divisor and R 1 j * F and j * R 1 π * Q ℓ are lisse on D. There is a canonical exact sequence We will show that H 0 (Dk, R 1 j * F ) is of weight ≥ 1. This will imply that H 1 (Vk, j * F ) and H 1 (Yk, F ) have the same generalized 1-eigenspace of the Frobenius action. Then the question will be reduced to show the splitness of and this will follow from Lemma 7.5 and Corollary 7.10 below. Let η be a generic point of D and k(η) be the residue field. Then we have Set I = Gal(K s /K sh η ), by [Mil3,Chap I,Lem. 2.18], there is an isomorphism of G k(η)representations as G k(η) -representations. Let s ∈ {η} be a closed point. By assumption, j * R 1 π * Q ℓ is lisse on D. So G k(s) acts on (j * R 1 π * Q ℓ )η = H 1 (A K s , Q ℓ ) I . By [Del2,Cor. 1.8.9], the G k(s) - is of weight ≥ 1. It follows that (R 1 j * F )s is of weight ≥ 1. Since this holds for all closed points in D, it follows that H 0 (Dk, R 1 j * F ) is of weight ≥ 1. This completes the proof.

The Néron-Tate height paring and Yoneda pairing
Let S be a projective normal geometrically connected variety over a finite field k with function field K. Let V ⊆ S be a regular open subscheme with S − V of codimension ≥ 2. Let A be an abelian variety over K. Let U ⊆ V be an open dense subscheme such that A extends to an abelian scheme π : A −→ U. Let j : U ֒→ V be the inclusion and K ′ denote Kk. The aim of this section is to show the splitness of In the case that dim S = 1, V is equal to S. j * A is the etale sheaf represented by a Néron model of A. The claim was proved by Schneider [Sch]. In the case that A extends to an abelian scheme over a smooth projective variety S over k, we have j * A = A . This case was proved by Keller [Kel2]. The idea is to construct compatible G k -equivariant pairings where A 0 is a smooth group scheme over V with connected fibers (A 0 is taken to be the identity component of a Néron model of A for two special cases above) and the top pairing is left non-degenerate. The the splitness of (5) will follow from Lemma 4.1. In the following, we will explain the construction of A 0 and the two pairings. Proof. For the case dim(S) = 1, we can take V = S and take A to be a Néron model of A/K over S and A 0 to be its identity component.
where F s is a finte etale group scheme over k(s). By shrinking Y , F s extends to a finte etale group scheme F over Y and the map A s −→ F s extends to a faithful flat morphism of finite type A Y −→ F . By shrinking Y further, the kernel of A Y −→ F can be indentified with A 0 Y . We get an exact sequence of fppf sheaves on Y Therefore, A 0 −→ V satisfies condition (ii). Since A 0 s is geometrically connected for all s ∈ S 1 , by shrinking Y , we can assume that all fibers of A 0 Y −→ Y are geometrically connected.
In the lemma below, we will show that the A 0 constructed as above satisfies As a result, there is a canonical isomorphism Lemma 7.5. Notations as above, let U ⊆ V be an open dense subset such that A is an abelian scheme over U. Let j : U −→ V be the inclusion. Denote by T ℓ A 0 (resp. T ℓ A ) the inverse system of ℓ-toriosn sheaves (A 0 [l n ]) n≥0 ( resp. (A [l n ]) n≥0 ). By shrinking V , there is a canonical isomorphism between A-R ℓ-adic sheaves (cf. [Fu,Chap. 10] ) Proof. By the lemma below, T ℓ A 0 is an A-R ℓ-adic sheaf. There is an exact sequence It follows that the kernel and cokernel of are supported on a closed subset of codimension ≥ 2. Removing this closed subset from V , we have Lemma 7.6. Let S be a noetherian scheme and G be a smooth commutative group scheme over S of finite type. Let ℓ be a prime invertible on S. Then the inverse system of ℓtorsion sheaves T ℓ G = (G[ℓ n ]) n≥1 is A-R ℓ-adic. If all fibers of G/S are geometrically connected, then G Secondly, let f : S 1 −→ S be a surjective between notherian schemes. If the statement holds for G S 1 /S 1 , then it holds for G/S. Set F n = G[ℓ n ] and F = (F n ). Since f * is an exact functor, one can show that if f * F satisfies the criterion of [Fu,Prop. 10.1.1], so does F .
Thirdly, we might assume that S is integral. By noetherian induction, it suffices to prove the statment for an open sense subscheme of S. Let K denote the function field of S. GK is smooth and of finite type overK. Let G 0K be the identity component of GK, then there is an exact sequence where π 0 (G) is a finite etale group scheme. By Chevalley's theorem, G 0K is an extension of an abelian variety A by G r m × G s a i.e.
The above exact sequences can descend to a finite extension L of K. Choose a model S 1 for L. By shrinking S 1 and S, we can assume that there exists a flat surjective morphism S 1 −→ S of finite type whose function field extension corresponds to L/K and exact sequences where π 0 (G) is finite etale over S 1 and A S 1 is an abelian scheme. We get exact sequences of systems of ℓ-torsion sheaves T ℓ G r m and T ℓ A S 1 are ℓ-adic sheaves, by [Fu,Prop. 10.1.7(iii)], T ℓ G S 1 is A-R ℓ-adic. It follows that T ℓ G is A-R ℓ-adic on S. This proves the first claim.
For the second claim, we might assume S = Spec R where R is a strict local ring with residue field k. It suffices to show that G(S) is ℓ-divisible. Let τ : S −→ G be a section. ℓ −1 (τ ) is etale over S. ℓ −1 (τ )(k) is not empty since G(k) is ℓ divisible. By Hensel's Lemma, it follows that ℓ −1 (τ )(S) is not empty. Thus, G(S) is ℓ divisible.
Lemma 7.7. Notations as before, assuming that V is regular, then there is a commutative diagram where δ is induced by the exact sequence 0 −→ A 0 [ℓ n ] −→ A 0 ℓ n −→ A 0 −→ 0 and r n is defined by

The natural map
is an isomorphism. And it induces a natural map , µ ℓ n ) compatible with pairings in the above diagram. As a result, by taking limit, there is a commutative diagram Proof. The two pairings in the first diagram are defined as Yoneda product of extensions and the commutativity of the diagram can be checked explicitly. We will construct a natural map . And it is easy to check the induced map is the inverse of (6) by definition.

Compatibility between the height pairing and Yoneda paring
Theorem 7.8. Notations as before. Let S be a projective normal geometrically connected variety over a finite field k. Fix a closed k-immersion ι : S ֒→ P m k . Let V be a regular open subscheme of S with codim(S − V ) ≥ 2 satisfying conditions in the lemma above. Define a degree map Pic(S) → Z by sending a prime Weil divisor to its degree as a subvariety in P m k . Composing it with the Yoneda pairing in the lemma above, we get a pairing There is a natural inclusion H 0 (V, A 0 ) ֒→ A(K) and a natural map Ext 1 V fppf (A 0 , G m ) → Ext 1 K (A, G m ) =Ã(K), whereÃ is the dual abelian variety of A. Then, up to a normalizing constant, the above pairing is compatible with the Néron-Tate height pairing defined in [Con,Cor. 9.17] A(K) ×Ã(K) −→ R.
Proof. Firstly, we define the adele ring associated to S (cf. [Kel2,§3.1]. Define for T ⊂ S 1 finite the T -adele ring of S as the restricted product where O S,s is the completion of the local ring O S,s and K s is the quotient field of O S,s , and the adele ring of S as For each s ∈ S 1 , we define deg(s) as the degree of the close subvariety {s} in P m k . This gives an absolute value on K s i.e. | · | s = q − deg(s).vs(·) , where q denotes the cardinality of k. Since a principle Weil divisor has degree zero, thus the absolute value | · | s satisfies product formula. We call the field K equipped with the set of absolute values | · | s a generalized global field(cf. [Con,Def. 8.1]).
Secondly, we describe the first pairing explicitly. Letã = (0 −→ G m −→ X −→ A 0 −→ 0) ∈ Ext 1 (A 0 , G m ). One can think X as a G m -torsor on A 0 and so it is representable by a smooth commutative S-group scheme of finite type (cf. [Mil1,Chap III,Thm. 4.3]). By Hilbert's theorem 90, there are exact sequences There is a natural homomorphism, the logrithmic modulus map, l : G m (A K ) −→ log q · Z ⊆ R, (a s ) → s∈S 1 log |a s | s = − log q · s∈S 1 deg(s) · v s (a s ).
By the product formula, l(G m (K)) = 0. Define G 1 m as the kernel of l, and X 1 as the rational saturation of X 1 with Then there exists a unique extension lã : X (A K ) → R of l vanishing on X 1 (cf. [Kel2,Lem. 3.1.4] or [Blo,Lem. 1.8]). It induces by restriction to X (K) a homomorphism lã : A(K) −→ R.
Next, we will show that for any a ∈ A 0 (S) ⊆ A(K), we have lã(a) = − log q · deg(a ∨ã), where ∨ denotes the Yoneda pairing H 0 (V, A 0 ) × Ext 1 V fppf (A 0 , G m ) → H 1 (V, G m ). By definition, a ∨ã is defined by the following commutative diagram Set e = a ∨ã, e is represented by an exact sequence of fppf sheaves on V and l e is an extension of l and vanishes on s∈S 1 Y ( O S,s ). By [Kel2,Lem 3.3.2]( replacing X by V in his argument) or [Sch,Lem. 12], lã(a) = l e (1) = − log q · deg(e).
Thirdly, we will show that the pairing h(a,ã) := lã(a) can be written as a sum of the local Néron pairings which therefore coincides with the canonical Néron-Tate height pairing by [BG,p.307,Cor. 9.5.14]. The proof for our case is exactly same as the argument in [Blo,§2] and [Kel2,§3.4]. Given an extension (0 −→ G m −→ X −→ A 0 −→ 0) ∈ Ext 1 V fppf (A 0 , G m ), its restriction to Spec O S,s for s ∈ S 1 is still an extension. Since O S,s is a Dedekind domain, by [MW,p.53,Lem. 5.1], the push-forward of the sheaf Therefore, the restriction of (0 −→ G m −→ X −→ A 0 −→ 0) to Spec O S,s gives an element inÃ(K s ). Then one can define the local Néron pairings as [Blo,(2.6)] for each s ∈ S 1 . By [Blo,Thm. 2.7] or [Kel2,Thm. 3.4.2], h(·, ·) is equal to the sum of the local Néron pairings. This completes the proof of the theorem.
Corollary 7.10. Notations as before. Set d = dim(V ) and let K ′ denote Kk. Then, one can shrink V (with Codim (S − V ) ≥ 2 ) such that H 0 (Vk, A 0 ) ⊗ Z Q ℓ = A(K ′ ) ⊗ Z Q ℓ and the injective map of G k -representations is split .
Proof. By the Lang-Néron theorem [Con,Thm.7.1], the quotient group A(K ′ )/Tr K/k (A)(k) is finitely generated where Tr K/k (A) is a K/k-trace of A/K. Since Tr K/k (A)(k) is a ℓdivisible torsion group, we have that A(K ′ )⊗Q ℓ = A(K ′ ) ⊗ Z Q ℓ has a finite dimension over Q ℓ . Thus, there exists a finite extension l/k such that A(Kl)⊗ Z Q ℓ = A(K ′ ) ⊗ Z Q ℓ . Replacing k by l, we might assume A(K) ⊗ Q ℓ = A(K ′ ) ⊗ Q ℓ . Since