arXiv:2012.01337v1 [math.AG] 2 Dec 2020 Introduction 1 Contents Applications 8 group Tate-Shafarevich and conjecture BSD 7 groups Brauer and Fibrations finitel 6 over varieties abelian for group Tate-Shafarevich 5 Theorem of Proof groups Brauer and 4 divisors for conjecture Tate 3 2 . h iadnme fAeinvreisand varieties Abelian of number Picard The Corollary of 8.3 Proof Corollary of 8.2 Proof . . 8.1 ...... pairing Yoneda and paring N´eron-Tate height The Theorem of 7.3 Proof of order vanishing 7.2 the of interpretation . Cohomological . . 7.1 ...... Theorem . . of . . Proof . . . . 6.1 ...... L . . Terminology . and . . Notation . . . Theorems 1.3 Main forms different in conjectures 1.2 Tate 1.1 fntosadconvergences and -functions bla ait ...... variety abelian .. oeapiig...... paring Yoneda . . and . pairing . height . . the . between . Compatibility . . . pairing 7.3.3 Yoneda . . . 7.3.2 N´eron model 7.3.1 oprsno ieetTt conjectures Tate different of Comparison 1.6 1.11 1.9 1.15 1.13 ...... eebr3 2020 3, December ...... asuiQin Yanshuai 1 K ufcs...... surfaces 3 L eeae fields generated y ucinfran for function . . . . . 41 30 22 13 16 42 42 41 38 36 33 33 31 30 30 9 8 2 6 4 2 1 Introduction

1.1 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)= qy denote the cardinality of k(y). Set −1 i Py,i(T ) := det(1 − σy T |H (Xy¯, Qℓ)),

where σy ∈ Gal(k(y)/k(y)) is the arithmetic Frobenius element and ℓ =6 char(k(y)) be a prime. Set 1 (1) Φ (s) := , i Y P (q−s) y∈Y◦ y,i y 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 Φi(s) by a product which converges absolutely for Re(s) > dim Y + i/2 − 1, so the zeros and poles of Φi in the strip

dim Y + i/2 − 1 < Re(s) ≤ dim Y + i/2

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 1 ζ(X ,s) := . Y 1 − N(x)−s x∈X ◦ It follows that ζ(X ,s)= Y ζ(Xy,s), y∈Y◦

where Xy denotes the special fiber over y. By Grothendieck trace formula, we have

2dim X (−1)i (2) ζ(X ,s)= Y Φi(s) . i=0

2 Let ℓ =6 char(K) be a prime number. Let Ai(X) be the group of classes of algebraic cycles of codimension i on X, with coefficents in Q, for ℓ-adic homological equivalence. Let N i(X) ⊆ Ai(X) denote the group of classes of cycles that are numerically equivalent to zero. Let d denote the dimension of X.

Conjecture 1.1. (Conjecture T i(X,ℓ)) the cycle class map

i 2i GK A (X) ⊗Q Qℓ −→ H (XKs , Qℓ(i)) is surjective.

Conjecture 1.2. (Conjecture Ei(X,ℓ)) N i(X)=0, i.e. numerical equivalence is equal to ℓ-adic homological equivalence for algebraic cycles of codimension i on X.

0 Conjecture 1.3. (BSD) The rank of PicX/K (K) is equal to the order of the zero of Φ1(s) at s = dim(Y) (and of Φ2d−1(s) at s = dim X − 1 by duality).

Conjecture 1.4 (Conjcture 2). The dimension of Ai(X)/N i(X) is equal to the order of the poles of Φ2i(s) at s = dim Y + i(and of Φ2d−2i(s) at s = dim X − i by duality).

Let X be a regular scheme of finite type over Z whose 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

0 ∗ 1 ∗ z(X ) = rank H (X , OX ) − rank H (X , OX ) − e(X ).

One can show that z(X ) is a birational invariant (cf. Proposition 2.2). Let f : X −→Y be a morphism as discussed at the beginning, by Proposition 2.2, any two of the following statements imply the third:

(i) BSD for X/K.

(ii) The Conjecture 1.4, for i = 1, for X/K.

(iii) The equality z(X )= z(Y).

One can show that z(Y) = 0 if Y is of dimension 1, this leads to the following conjecture:

Conjecture 1.5 (BSD+2). If X is a regular scheme of finite type over Z, then the order 0 ∗ 1 ∗ of ζ(X ,s) at the point s = dim X − 1 is equal to rank H (X , OX ) − rank H (X , OX ).

For a morphsim f : X −→ Y as above with dim Y = 1, Tate conjectured that the above statement is equivalent to BSD for X/K and the Conjecture 1.4, for i = 1, for X/K. By our result Corollary 1.13, this is true when Y is a curve over a finite field.

3 1.2 Main Theorems For any noetherian scheme X, the cohomological Brauer group

2 Br(X) := H (X, Gm)tor

2 is defined to be the torsion part of the etale cohomology group H (X, Gm). 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. Then i i dimQ(A (X)/N (X)) ≤ −ords=m+iΦ2i(s) with equality if and only if Ei(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.

GK (1) Br(XKs ) (ℓ) is finite for some prime ℓ =6 p.

GK (2) Br(XKs ) (ℓ) is finite for all primes ℓ =6 p. (3) T 1(X,ℓ) holds for all primes ℓ =6 p.

(4) The rank of NS(X) is equal to −ords=m+1Φ2(s). 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 charac- teristic p> 0. Let Y be a smooth and irreducible variety over a finite field with function field K. Set m = dim(Y). Define

X 1 1 sh Y (A) := Ker(H (K, A) −→ Y H (Ks , A)), s∈Y1

1 sh where Y denotes the set of points of codimension 1 and Ks denotes the quotient field of a strict local ring at s, then we have

rank A(K) ≤ ords=mΦ1(s), and the following statements are equivalent.

(i) XY (A)(ℓ) is finite for some prime ℓ =6 p.

(ii) XY (A)(ℓ) is finite for all primes ℓ =6 p.

(iii) The BSD conjecture for A holds i.e. the rank of A(K) is equal to ords=mΦ1(s).

4 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 VℓXY (A) is independent of the choice of Y (cf. Lemma 5.2), so we will write VℓXK(A) for VℓXY (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′ := Kks. Let ℓ =6 char(k) be a prime, and set

Gk GK K := Ker(VℓBr(Xks ) −→ VℓBr(XKs ) ). Then we have canonical exact sequences

Gk X ′ 0 Gk 0 −→ VℓBr(Yks ) −→ K −→ Vℓ K (PicX/K ) −→ 0,

Gk GK 0 −→ K −→ VℓBr(Xks ) −→ VℓBr(XKs ) −→ 0.

X ′ 0 Gk X 0 Gk 0 Here we write Vℓ K (PicX/K ) for Vℓ Yks (PicX/K ) which only depends on PicX/K (cf. 0 Lemma 5.2). Moreover, if char(k) > 0, and B denotes the K/k-trace of PicX/K , then we have a canonical exact sequence

X X 0 X ′ 0 Gk 0 −→ Vℓ k(B) −→ Vℓ K (PicX/K ) −→ Vℓ K (PicX/K ) −→ 0.

X 0 X ′ 0 Gk In particular, if k is finite, we have Vℓ K(PicX/K )= Vℓ K (PicX/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-Th´el`ene 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 0 Conjecture 1.5 for Y + BSD for PicX/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 1 1 to T (X,ℓ) for smooth projective surfaces over finite fields with H (X, OX ) = 0 by the work of de Jong [deJ], Morrow[Mor, Thm. 4.3] and Yuan [Yua1, Thm. 1.6].

5 1.3 Notation and Terminology Fields By a finitely generated field, we mean a field which is finitely generated over a prime field. s s For any field k, denote by k the separable closure. Denote by Gk = Gal(k /k) the absolute Galois group of k.

Henselization Let R be a noetherian local ring, denote by Rh (resp. Rsh) the henselization (resp. strict henselization) of R at the maximal ideal. If R is a domain, denote by Kh (resp. Ksh) the fraction field of Rh (resp. Rsh).

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 S1 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. 0 For a smooth proper geometrically connected variety X over a field k, we use PicX/k to denote the underlying reduced closed subscheme of the identity component of the Picard

scheme PicX/k.

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

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

2 Br(X) := H (X, Gm)tor.

Abelian group For any abelian group M, integer m and prime ℓ, we set

n M[m]= {x ∈ M|mx =0}, Mtor = [ M[m], M(ℓ)= [ M[ℓ ], m≥1 n≥1 T M = Hom (Q /Z , M) = lim M[ℓn], V M = T (M) ⊗ Q . ℓ Z ℓ ℓ ← ℓ ℓ Zℓ ℓ n

6 Acknowledgements: I would like to thank my advisor Xinyi Yuan for important communications and for telling me his idea of Lefschetz hyperplane arguments.

7 2 L-functions and convergences

Proposition 2.1. Notations as in the introduction. Then Φi(s) defined by (1) converges absolutely for Re(s) > dim Y + i/2 and the zeros and poles of Φi in the strip dim Y + i/2 − 1 < Re(s) ≤ dim Y + i/2 does not depend on the choice of a smooth projective model X −→Y.

Proof. By Deligne’s theorem, Py,i(T ) are of fixed degree with reciprocal roots of absolute i/2 value qy , 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 Φi(s) by −s Y Pz,i(qz ), z∈Z◦ which converges absolutely for Re(s) > dim Y + i/2 − 1 since dim Z ≤ dim Y − 1. It follows that the zeros and poles of Φi in the strip dim Y + i/2 − 1 < Re(s) ≤ dim Y + i/2 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

0 ∗ 1 ∗ z(X ) = rank H (X , OX ) − rank H (X , OX ) − e(X ). 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. (ii) The Conjecture 1.4, for i =1, for X/K. (iii) The equality z(X )= z(Y). Proof. If one removes from X a closed irreducible subscheme Z, then we have

ζ(X − Z,s)= ζ(X ,s)/ζ(Z,s).

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

0 ∗ 0 ∗ 0 −→ H (X , OX ) −→ H (U, OU ) −→ Z −→ Pic(X ) −→ Pic(U) −→ 0.

8 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 0 ∗ 0 ∗ and Y is normal, we have f∗OX = OY . Thus H (X , OX ) = H (Y, OY ). 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

p q p+q H (Y, R f∗Gm) ⇒ H (X , Gm),

and f∗Gm = Gm, we get an injection

1 1 H (Y, Gm) −→ H (X , Gm). Thus, we get an exact sequence

0 −→ Pic(Y) −→ Pic(X ) −→ Pic(X) −→ 0.

So z(X )= z(Y) ⇔ e(X ) − e(Y)= −rank Pic(X). Since Φ2d(s)= ζ(Y,s − d), comparing the order of both sides of the equation (2) at s = dim X − 1, it suffices to show that Φi(s) has no pole or zero at s = dim X − 1 for i =26 d − 1, 2d − 2, 2d. Since all reciprocal i 2 roots of Py,i(T ) have absolute values qy , the product (1) converges absolutely for Re(s) > dim Y + i/2, the claim follows.

3 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 =6 p be a prime number. Then the cycle class map

2 Gk Pic(X) ⊗Z Qℓ −→ H (Xks , Qℓ(1)) is surjective. This conjecture is called Tate conjecture for divisors on X. By the proposition below, Gk it is equivalent to the finiteness of Br(Xks ) (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 Gk-module

2 0 −→ NS(Xks ) ⊗Z Qℓ −→ H (Xks , Qℓ(1)) −→ VℓBr(Xks ) −→ 0

is split. Taking Gk-invariant, there is a canonical exact sequence

2 Gk Gk 0 −→ NS(X) ⊗Z Qℓ −→ H (Xks , Qℓ(1)) −→ VℓBr(Xks ) −→ 0.

9 Proof. See [Yua2, §2.2].

Gk In the following, we will show that VℓBr(Xks ) is a birational invariant for smooth varieties over finitely generated fields.

Proposition 3.3. ([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 =6 p be prime. Then the natural map

Gk Gk VℓBr(Xks ) −→ VℓBr(Uks ) is an isomorphism.

Proof. Extending k to a finite extension will not affect the results. By purity for Brauer groups [Cesˇ , Thm. 1.1], removing a closed subscheme of codimension ≥ 2 will not change the Brauer group. Therefore, we might assume that X − U = F Di, where Di is a smooth geometrically connected subvariety over k of codimension 1. There is an exact sequence

2 2 3 s Q s Q s Q H (Xk , ℓ(1)) −→ H (Uk , ℓ(1)) −→ M HDi (Xk , ℓ(1)). i

By the theorem of cohomological purity (cf. [Fuj]),

3 1 s Q ∼ s Q HDi (Xk , ℓ(1)) = H ((Di)k , ℓ)

is of weight ≥ 1. Consider the commutative diagram

2 / / H (Xks , Qℓ(1)) VℓBr(Xks ) 0

  2 / / H (Uks , Qℓ(1)) VℓBr(Uks ) 0

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 Gk 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 Gk varieties over finite fields and the finiteness of Br(Xk¯) (ℓ). Lemma 3.4. Let X be a smooth geometrically connected variety over a finite field k of characteristic p > 0. Let ℓ =6 p be a prime. Then, there is a split exact sequence of Gk-representations

2 0 −→ Pic(Xk¯) ⊗Z Qℓ −→ H (Xk¯, Qℓ(1)) −→ VℓBr(Xk¯) −→ 0.

10 Proof. Firstly, if the claim holds for Gk1 -actions where k1/k is a finite Galois extension, then it also holds for Gk-actions. Let V be a sub-representation of some Gk-representation

W over Qℓ. Let p : W −→ V be a Gk1 -equivariant projection. Then

1 σ X p |Gk/Gk1 | σ∈Gk/Gk1

is a Gk-equivariant projection. 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

/ / 2 0 Pic(Xk¯) ⊗Z Qℓ H (Xk¯, Qℓ(1))

a b   / / 2 0 Pic(Uk¯) ⊗Z Qℓ H (Uk¯, Qℓ(1))

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

0 −→ Pic(Uk¯) ⊗Z Qℓ −→ Im(b) is also split. By extending k, we might assume that the Gk-action on Pic(Uk¯) ⊗Z Qℓ is trivial. Then Pic(Uk¯) ⊗ Qℓ is contained in the generalized 1-eigenspace of the Frobenuis 2 action on H (Uk¯, Qℓ(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¯) ⊗ Qℓ is a direct summand of 2 the generaized 1-eigenspace, and therefore, it is also a direct summand of H (Uk¯, Qℓ(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 2 2 f∗ : H (Uk¯, Qℓ(1)) −→ H (Xk¯, Qℓ(1)) ∗ ∗ 2 2 such that f∗f = deg(f). It follows that f : H (Xk¯, Qℓ(1)) −→ H (Uk¯, Qℓ(1)) is injective. ∗ Through f , Pic(Xk¯) ⊗ Qℓ is a direct summand of Pic(Uk¯) ⊗ Qℓ, and therefore, is also a 2 direct summand of H (Uk¯, Qℓ(1)). It follows that Pic(Xk¯) ⊗ Qℓ is a direct summand of 2 H (Xk¯, Qℓ(1)). Theorem 3.5. Let X be a smooth and irreducible variety over a finite field of character- istic p. Let ℓ =6 p be a prime. Then the Conjecture 1.5 for X is equivalent to the finiteness Gk of Br(Xks ) (ℓ).

Proof. Assuming that dim X = n and k = Fq, let F denote the geometric Frobenius element in Gk. By Grothendieck’s formula, we have

2n −s i (−1)i+1 ζ(X ,s)= Y det(1 − q F |Hc(Xk¯, Qℓ)) . i=0

11 i 2n By Deligne’s theorem, Hc(Xk¯, Qℓ) is mixed of weight ≤ i and Hc (Xk¯, Qℓ) is pure of weight −s 2n−2 −s 2n−1 2n. Therefore, only det(1 − q F |Hc (Xk¯, Qℓ)) and det(1 − q F |Hc (Xk¯, Qℓ)) have contributions to the order of the pole of ζ(X ,s) at s = n − 1. By Poincar´eduality,

2n−1 ∼ 1 ∨ Hc (Xk¯, Qℓ) = H (Xk¯, Qℓ) (−n) and 2n−2 ∼ 2 ∨ Hc (Xk¯, Qℓ) = H (Xk¯, Qℓ) (−n). n−1 1 ∨ Next, we will show that the dimension of the generalized q -eigenspace of H (Xk¯, Qℓ) (−n) 0 ∗ is equal to rank H (X , OX ). By Kummer exact sequence, we get an exact sequence

0 1 1 0 −→ H (Xk¯, Gm) ⊗Z Qℓ −→ H (Xk¯, Qℓ(1)) −→ VℓH (Xk¯, Gm).

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∗Gm → Gm whose composition with the canonical map Gm → f∗Gm is equal to the multiplication by deg(f). Since p ′ p H (X , Gm)= H (X , f∗Gm), it follows that the composition

p p ′ p p H (X , Gm) −→ H (X , Gm)= H (X , f∗Gm) −→ H (X , Gm)

is equal to the multiplication by deg(f). It follows that the natural map

1 1 ′ VℓH (Xk¯, Gm) −→ VℓH (Xk¯, Gm)

is injective. Assuming that X ′ ⊂ X¯ where X¯ is irreducible and smooth projective over k. Since X¯ is regular, the natural map

¯ ′ Pic(Xk¯) −→ Pic(Xk¯) ¯ is surjective and has a finitely generated kernel. Since Pic(Xk¯) is an extension of a finitely 0 ¯ generated abelian group by a ℓ-divisible torsion group PicX¯/k(k), it follows that the natural map ¯ ′ VℓPic(Xk¯) −→ VℓPic(Xk¯) ¯ ∼ 1 ¯ 1 is an isomorphism. Since VℓPic(Xk¯) = H (Xk¯, Qℓ(1)), so VℓH (Xk¯, Gm) is pure of weight 1 −1. It follows that the generalized 1-eigenspace of H (Xk¯, Qℓ(1)) can be identified with 0 Gk 0 H (Xk¯, Gm) ⊗Z Qℓ = H (X , Gm) ⊗Z Qℓ. This implies that the dimension of the gener- n−1 1 ∨ 0 alized q -eigenspace of H (Xk¯, Qℓ) (−n) is equal to rank H (X , Gm). 2 Thus, it suffices to show that the dimension of the generalized 1-eigenspace of H (Xk¯, Qℓ(1)) Gk is equal to rank Pic(X ) if and only if VℓBr(Xk¯) vanishes. By Lemma 3.4, there is an exact sequence

2 Gk Gk 0 −→ Pic(X ) ⊗Z Qℓ −→ H (Xk¯, Qℓ(1)) −→ VℓBr(Xk¯) −→ 0.

12 2 Assuming that the dimension of the generalized 1-eigenspace of H (Xk¯, Qℓ(1)) is equal to rank Pic(X ), then 2 Gk Pic(X ) ⊗Z Qℓ −→ H (Xk¯, Qℓ(1))

Gk Gk is an isomorphism. It follows that VℓBr(Xk¯) = 0. Assuming that VℓBr(Xk¯) = 0, then

2 Gk Pic(X ) ⊗Z Qℓ −→ H (Xk¯, Qℓ(1)) is an isomorphism. Since the Gk-action on Pic(Xk¯) ⊗Z Qℓ is semisimple. By Lemma 3.4, 2 Gk 2 H (Xk¯, Qℓ(1)) is a direct summand of H (Xk¯, Qℓ(1)) as a Gk-representation. It follows 2 that the generalized 1-eigenspace of H (Xk¯, Qℓ(1)) is same as the 1-eigenspace which is equal to Pic(X ) ⊗Z Qℓ. 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 : X ′ −→ X such that X ′ admits a smooth projective com- ′ 1 ′ ′ Gk pactification X . Assuming T (X ,ℓ), we have VℓBr(X ks ) = 0. By Proposition 3.3, ′ Gk ′ Gk Gk ′ Gk VℓBr(Xks ) = VℓBr(X ks ) = 0. Since VℓBr(Xks ) −→ VℓBr(Xks ) is injective, so Gk VℓBr(Xks ) = 0. 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 Qℓ-linear continuous rep- resentation of Gk. Let W ⊆ V be a subrepresentation. Assuming that there exists a Gk-equivariant paring ′ V × V −→ Qℓ ′ where V is a Qℓ-linear representaion of a finite quotient of Gk such that the restriction of the pairing to W × V ′ is left non-degenerate, then W is a direct summand of V as a Gk-representation. Proof. Since the pairing W × V ′ is left non-degenerate, it induces a surjection

V ′ −→ W ∗,

∗ ′ where W is the dual representation of W . Since the Gk-action on V factors through a finite quotient, so V ′ is semisimple. Thus, there exists a subrepresentation W ′ of V ′ such that the paring W × W ′ is perfect. Define (W ′)⊥ := {v ∈ V |(v, W ′)=0}. It follows that V = W ⊕ (W ′)⊥. This proves the claim.

13 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 =6 p be a prime. Then we have

i i 2i GK s dimQ(A (X)/N (X)) ≤ dimQℓ H (XK , Qℓ(i)) ≤ −ords=m+iΦ2i(s).

i i i i Moreover, dimQ(A (X)/N (X)) = −ords=m+iΦ2i(s) if and only if T (X,ℓ) and E (X,ℓ) hold.

d−i 2d−2i Proof. The cycle classes of A (X) generate a Qℓ-subspace in H (XKs , Qℓ(d − i)). d−i Let b1, ..., bs ∈ A (X) such that their cycle classes form a basis of this subspace. Let d−i B ⊆ A (X) be the subspace spanned by bi. Then the intersection pairing Ai(X)/N i(X) × B −→ Q

is left non-degenerate. Thus, the induced map Ai(X)/N i(X) −→ B∗ is injective. It i i i follows that A (X)/N (X) has a finite dimension. Let a1, .., at ∈ A (X) such that they i i ∗ ∗ ∗ form a basis in A (X)/N (X). Then there exist a1, ..., a1 ∈ B such that ai.aj = δij. So ∗ 2i GK cl(ai).cl(aj )= δij. Thus cl(a1), ..., cl(at) are Qℓ-linear independent in H (XKs , Qℓ(i)) . This proves the first inequality. i Let a ∈ N (X) with cl(a) =6 0, then cl(a1), ..., cl(at),cl(a) are Qℓ-linear independent. Let λi ∈ Qℓ such that

λ1cl(a1)+ ... + λtcl(at)+ λt+1cl(a)=0

∗ Intersecting with aj , we get λj = 0 for 1 ≤ j ≤ t. Since cl(a) =6 0, so λt+1 = 0. This implies that i i 2i GK s dimQ(A (X)/N (X)) < dimQℓ H (XK , Qℓ(i)) if N i(X) =6 0. 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 K′ s 2d−2i denote Kk . Let F denote R π∗Qℓ. F is lisse since π is smooth and proper. By Deligne’s theorem, F is pure of weight 2d − 2i. By Grothendieck’s formula, we have

2m F −s j F (−1)i+1 Φ2d−2i(s)= L(Y, ,s)= Y det(1 − q F |Hc (Yk¯, )) , j=0

j F where F denotes the geometric Frobenius element in Gk. Since Hc (Yk¯, ) is mixed of weight ≤ j+2d−2i, thus −ords=d+m−iΦ2d−2i(s) is equal to the dimension of the generalized d+m−i 2m F q -eigenspace of F on Hc (Yk¯, ). By Poincar´eduality, 2m F ∼ 0 F ∨ ∨ Hc (Yk¯, ) = H (Yk¯, ) (−m).

Therefore −ords=d+m−iΦ2d−2i(s) is equal to the dimension of the generalized 1-eigenspace 0 F ∨ F ∨ 2d−2i ∨ of H (Yk¯, )(i − d). corresponds to the GK -representation on H (XKs , Qℓ)

14 2i which is isomorphic to H (XKs , Qℓ)(d). It follows that −ords=d+m−iΦ2d−2i(s) is equal to 2i G ′ the dimension of the 1-eigenspace of H (XKs , Qℓ(i)) K . By Poincar´eduality, we have

Φ2d−2i(s) = Φ2i(s − d +2i).

2i G ′ Thus −ords=m+iΦ2i(s) is equal to the dimension of the generalized 1-eigenspace of H (XKs , Qℓ(i)) K 2i GK which contains H (XKs , Qℓ(i)) . This proves the second inequality. For “ ⇐′′ direction, assuming T i(X,ℓ) and Ei(X,ℓ), then we have

i 2i GK s A (X)Qℓ = H (XK , Qℓ(i)) .

Consider the Gk-equivariant pairing

2i G ′ d−i s K (3) H (XK , Qℓ(i)) × A (X)Qℓ −→ Qℓ.

Ei(X,ℓ) implies that i d−i A (X)Qℓ × A (X)Qℓ −→ Qℓ is left nondegenerate. Thus, the restriction of (3)

2i GK d−i s H (XK , Qℓ(i)) × A (X)Qℓ −→ Qℓ

2i GK is also left non-degenerate. By Lemma 4.1, H (XKs , Qℓ(i)) is a direct summand of 2i G ′ H (XKs , Qℓ(i)) K as a Gk-representation. This implies that the 1-eigenspace of F on 2i G ′ H (XKs , Qℓ(i)) K is a direct summand, and therefore, is equal to the generalized 1- eigenspace of F . It follows that

2i GK s dimQℓ H (XK , Qℓ(i)) = −ords=m+iΦ2i(s).

This completes the proof.

Corollary 4.3. The following statements are equivalent.

GK (1) Br(XKs ) (ℓ) is finite for some prime ℓ =6 p.

GK (2) Br(XKs ) (ℓ) is finite for all primes ℓ =6 p.

(3) T 1(X,ℓ) holds for all primes ℓ =6 p.

(4) The rank of NS(X) is equal to −ords=m+1Φ2(s).

Proof. By Proposition 3.2, (1) ⇔ T 1(X,ℓ). Since it is well-known that E1(X,ℓ) is true (cf., e.g., SGA 6 XIII, Theorem 4.6), by the theorem above, we have T 1(X,ℓ) ⇔ (4). Since (4) ⇒ (3) ⇒ (2) ⇒ (1), the claim follows.

15 5 Tate-Shafarevich group for abelian varieties over finitely generated fields

In this section, we will define Tate-Shafarevich group for abelian varieties over finitely gen- erated 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

X 1 1 sh S(A) := Ker(H (K, A) −→ Y H (Ks , A)), s∈S1 where S1 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. Write K′ for Kks. Let U ⊆ S be an open dense subscheme. Let ℓ =6 char(k) be a prime. Then the inclusion X Gk X Gk Sks (A) (ℓ) −→ Uks (A) (ℓ) has a finite cokernel. Furthermore, if A extends to an abelian scheme A −→ U, then we have a canonical isomorphism

1 X ∼ s A Uks (A)(ℓ) = H (Uk , )(ℓ). And we have X Gk 1 ′ Gk Vℓ Sks (A) = VℓH (K , A) ,

Gk which only depends on A/K. We will denote it by VℓXK′ (A) . If k is a finite field, we have 1 ∼ ′ Gk VℓXS(A)= VℓH (K, A) = VℓXK (A) .

We will write VℓXK(A) for VℓXS(A). Proof. It is easy to see that

G h X Gk 1 ′ Gk 1 sh Ks Sks (A) = Ker(H (K , A) −→ Y H (Ks , A) ). s∈S1

1 sh GKh By Lemma 5.4 below, H (Ks , A) s (ℓ) is finite. It follows that the inclusion

X Gk X Gk Sks (A) (ℓ) −→ Uks (A) (ℓ)

X Gk 1 ′ Gk has a finite cokernel and Vℓ Sks (A) = VℓH (K , A) . In the case that k is a finite field, by the same argument, the inclusion

XS(A)(ℓ) −→ XU (A)(ℓ)

16 1 has a finite cokernel and VℓXS(A)= VℓH (K, A). If A/K extends to an abelian scheme A −→ U, we will show that

1 A n 1 ′ n 1 sh n (4) H (Uks , [ℓ ]) = Ker(H (K , A[ℓ ]) −→ Y H (Ks , A[ℓ ])). 1 s∈Uks This will imply that the composition of

1 1 ∗ s A s A X H (Uk , )(ℓ) −→ H (Uk , j∗j ) −→ Uks (A)(ℓ)

′ n is surjective, where j : Spec K −→ Uks is the generic point. Since A [ℓ ] is a locally constant sheaf, we have n ∼ ∗ n A [ℓ ] = j∗j A [ℓ ]. Thus, we get an exact sequence

1 A n 1 ′ n 1 sh 0 −→ H (Uks , [ℓ ]) −→ H (K , A[ℓ ]) −→ Y H (Ks , A). s∈Uks Thus 1 A n 1 ′ n 1 sh n H (Uks , [ℓ ]) ⊆ Ker(H (K , A[ℓ ]) −→ Y H (Ks , A[ℓ ])). 1 s∈Uks Next, we will show that the inclusion is actually an equality. Since

H1(K′, A[ℓn]) = lim H1(V, A [ℓn]), −→ V ⊆Uks

1 ′ n let x ∈ H (K , A[ℓ ]), then there exists an open subscheme V ⊆ Uks such that x ∈ 1 n n H (V, A [ℓ ]). Let Y denote the reduced closed scheme Uks − V . Since A [ℓ ] is a locally constant sheaf, by the semi-purity [Fuj, §8], removing a closed subscheme of codimension 1 n ≥ 2 from Uks will not change H (Uks , A [ℓ ]). So by shrinking Y , we might assume that Y 2 A n ∼ 0 A n is regular and of codimension 1 in Uks , then HY (Uks , [ℓ ]) = H (Y, [ℓ ](−1)) by purity 1 sh n ∼ A n [Fuj, §0]. Let s be a generic point of Y , by purity, we have H (Ks , A[ℓ ]) = ( [ℓ ](−1))s¯. Thus, we get an exact sequence

1 A n 1 A n 1 sh n 0 −→ H (Uks , [ℓ ]) −→ H (V, [ℓ ]) −→ Y H (Ks , A[ℓ ]). s∈Y 0 Therefore, if x lies in

1 ′ n 1 sh n Ker(H (K , A[ℓ ]) −→ Y H (Ks , A[ℓ ])), 1 s∈Uks

1 n then it also lies in H (Uks , A [ℓ ]). This proves (4). By [Kel1, Thm. 3.3], we have ∼ ∗ A = j∗j A . Thus 1 A 1 ′ 1 sh H (Uks , ) = Ker(H (K , A) −→ Y H (Ks , A)). 1 s∈Uks

17 1 1 s A X s A It follows that H (Uk , )(ℓ) ⊆ Uks (A)(ℓ). Since the natural map H (Uk , (ℓ)) → 1 1 X s A s A X Uks (A)(ℓ) factors through H (Uk , )(ℓ) and H (Uk , (ℓ)) −→ Uks (A)(ℓ) is surjec- tive, thus 1 s A X H (Uk , )(ℓ)= Uks (A)(ℓ). By the same argument, one can show that

1 H (U, A )(ℓ)= XU (A)(ℓ).

∼ ′ Gk It remains to prove that VℓXK(A) = VℓXK (A) when k is a finite field. Firstly, we will show that the natural map

Gk VℓXK(A) −→ VℓXK′ (A)

is surjective. Since there exists an abelian variety B/K such that A × B is isogenous 0 to PicX/K for some smooth projective geometrically connected curve X/K, we might 0 assume that A = PicX/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

/ X 0 VℓBr(X ) Vℓ K(PicX/K )

  Gk / X ′ 0 Gk / VℓBr(Xks ) Vℓ K (PicX/K ) 0

Then the surjectivity of the second column will follow from the surjectivity of the first column. By the Leray sepectral sequence

p,q p q p+q E2 = H (Y, R π∗Gm) ⇒ H (X , Gm) we get an exact sequence

2 2 0 2 1 1 H (Y, Gm) −→ Ker(H (X , Gm) −→ H (Y, R π∗Gm)) −→ H (Y, R π∗Gm).

0 2 ∼ GK By Lemma 6.1 below, H (Y, R π∗Gm)(ℓ) = Br(XKs ) (ℓ). Since X is a smooth projective curve over K, we have Br(XKs ) = 0. Thus, we get a natural map

1 1 VℓBr(X ) −→ VℓH (Y, R π∗Gm).

Let j : Spec K −→ Y be the generic point. By Lemma 6.1, we have

1 ∼ ∗ 1 R π∗Gm = j∗j R π∗Gm.

Thus, we have

1 1 ∼ 1 ∼ 1 0 ∼ X 0 H (Y, R π∗Gm)(ℓ) = H (Y, j∗PicX/K )(ℓ) = H (Y, j∗PicX/K )(ℓ) = Y (PicX/K)(ℓ).

18 Therefore, 1 1 ∼ X 0 VℓH (Y, R π∗Gm) = Vℓ Y (PicX/K ) By compositions, we get a natural map

X 0 VℓBr(X ) −→ Vℓ K(PicX/K ).

By Theorem 6.2, there is a surjective map compatible with the above map

Gk X ′ 0 Gk VℓBr(Xks ) −→ Vℓ K (PicX/K ) .

Gk It suffices to show that VℓBr(X ) −→ VℓBr(Xks ) is surjective. By Lemma 3.4, the natural map 2 Gk Gk H (Xks , Qℓ(1)) −→ VℓBr(Xks ) 2 is surjective. Since H (X , Qℓ(1)) −→ VℓBr(X ) is surjective by definition, it suffices to show that 2 2 Gk H (X , Qℓ(1)) −→ H (Xks , Qℓ(1)) is surjective. Consider the spectral sequence

p q p+q H (Gk,H (Xks ,µℓn )) ⇒ H (X ,µℓn )

q Since H (Gk, −) vanishes when q ≥ 2, we get a surjection

2 2 Gk H (X ,µℓn ) −→ H (Xks ,µℓn ) .

Taking limit, we get

2 2 Gk H (X , Qℓ(1)) −→ H (Xks , Qℓ(1)) −→ 0.

Next, we will show that the natural map

Gk VℓXK(A) −→ VℓXK′ (A) is injective. Consider the exact sequence

1 ′ 1 1 ′ 0 −→ H (Gk, A(K )) −→ H (K, A) −→ H (K , A).

′ s Let TrK/k(A) be the K/k trace of A. By the Lang-N´eron theorem, A(K )/TrK/k(A)(k ) 1 is finitely generated. Since H (Gk, TrK/k(A)) = 0 by Lang’s theorem, we have that 1 ′ H (Gk, A(K )) is finite. It follows that the natural map

1 1 ′ VℓH (K, A) −→ VℓH (K , A)

X 0 X ′ 0 Gk is injective. So Vℓ K(PicX/K ) −→ Vℓ K (PicX/K ) is also injective. This completes the proof.

19 Lemma 5.3. Let G be a smooth connected algebraic group over a finitely generated field K. Let ℓ =6 char(K) be a prime. Then the size of

n GK Hom(G[ℓ ], Qℓ/Zℓ)

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 0 −→ H −→ G −→ A −→ 0. ∼ s t Since H is commutative, by extending K, we can assume that H = Gm × Ga over K. Thus, we have 0 −→ H[ℓn] −→ G[ℓn] −→ A[ℓn] −→ 0. Taking dual, we get

0 −→ A[ℓn]∨ −→ G[ℓn]∨ −→ H[ℓn]∨ −→ 0

n s n Since H[ℓ ] = Gm[ℓ ], so it suffices to prove the claim for abelian varieties and Gm. 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. Gm ) over K, by shrinking S, we can assume that G extends to an abelian scheme GS (resp. Gm,S). Let s ∈ S be closed point with finite n n residue field k. Write F for the etale sheaf represented by GS[ℓ ] on S. Since GS[ℓ ] is finite etale over S, F is a locally constant sheaf of Z/ℓn-module on S. Let η be the ∼ n generic point of S. Since S is connected, we have Fη¯ = Fs¯. Since Fη¯ = G[ℓ ], we have n ∼ n G[ℓ ] = Gs[ℓ ] as Gk-modules. Through this isomorphism, there is an inclusion

n ∨ GK n ∨ Gk (G[ℓ ] ) ⊆ (Gs[ℓ ] ) .

Therefore, it suffices to prove the claim for the case that K is a finite field. In this case,

n ∨ GK n n GK |(G[ℓ ] ) | = |G[ℓ ]GK | = |G[ℓ ] |.

Since G[ℓn]GK ⊆ 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 Rsh denote a strict henselization of R and Ksh denote its quotient field. Let A be an abelian variety over K and ℓ be a prime different from char(k). Set G := Gal(ks/k) = Gal(Ksh/K). Then the size of

H1(Ksh, A[ℓn])G

is bounded by a constant independent of n. And H1(Ksh, A)G(ℓ) is a finite group.

20 s sh 1 n Proof. Let I denote Gal(K /K ) and I1 denote the wild inertia subgroup. Since H (I1, A[ℓ ]) = 0, by [Mil3, Chap I, Lem. 2.18], we have

1 sh n n H (K , A[ℓ ]) = (A[ℓ ](−1))I .

n n I Its dual Hom((A[ℓ ](−1))I , Qℓ/Zℓ) = Hom(A[ℓ ](−1), Qℓ/Zℓ) is canonically isomorphic to At[ℓn]I by Weil pairing, where At denotes the dual abelian variety of A. Let A t be a N´eron model of At/K and s : Spec k −→ S = Spec R be the closed point. Then

t n I A t n A [ℓ ] = s [ℓ ].

There is an exact sequence

A t 0 A t t 0 −→ ( s ) −→ s −→ Φ(A ) −→ 0,

where Φ(At) denotes the N´eron component group of At which is a finite etale scheme over k. Taking dual and then taking Gk-invariants, we get an exact sequence

t ∨ G A t n ∨ G A t 0 n ∨ G 0 −→ (Φ(A ) ) −→ (( s [ℓ ]) ) −→ ((( s ) [ℓ ]) )

Note that the size of (Φ(At)∨)G is bounded by a constant independent of n. Since 1 sh n G ∼ A t n ∨ G A t 0 n ∨ G H (K , A[ℓ ]) = (( s [ℓ ]) ) , it suffices to show that the size of ((( s ) [ℓ ]) ) 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

sh 1 sh 1 sh 0 −→ A(K ) ⊗Z Qℓ/Zℓ −→ H (K , A(ℓ)) −→ H (K , A)(ℓ) −→ 0.

Let A be a N´eron model of A over S and A 0 be the identity component of A . We have A(Ksh)= A (S) and an exact sequence

0 −→ A 0(S) −→ A (S) −→ Φ(A) −→ 0.

0 Since A (S) is ℓ-divisible and Φ(A) is finite, by tensoring Qℓ/Zℓ, we have

sh A(K ) ⊗Z Qℓ/Zℓ =0.

It follows H1(Ksh, A(ℓ)) ∼= H1(Ksh, A)(ℓ). Since H1(Ksh, A(ℓ))G = lim H1(Ksh, A[ℓn])G, −→ n by the first claim, the right side is finite. It follows that H1(Ksh, A)G(ℓ) is also finite.

21 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 1 ∗ 1 R π∗Gm −→ j∗j R π∗Gm is an isomorphism,

(b) the natural map 2 ∗ 2 R π∗Gm(ℓ) −→ j∗j R π∗Gm(ℓ) 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

1 ∗ 1 (R π∗Gm)s¯ = Pic(X ) and (j∗j R π∗Gm)s¯ = PicX/K (K).

Since Xs admits a section s −→ Xs and π is smooth, the section can be extended to a section U −→ X . Thus X(K) is not empty. So

PicX/K (K) = Pic(X).

Since X is regular, the natural map

Pic(X ) −→ Pic(X) 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

0 −→ Pic(Spec R) −→ Pic(X ) −→ Pic(X) −→ 0.

Since R is an UFD, we have Pic(Spec R) = 0. It follows that

Pic(X ) ∼= Pic(X).

This proves (a). Let I denote GK. For (b), the induced map on the stalk at s is

I Br(X )(ℓ) −→ Br(XKs ) (ℓ).

Since π is smooth and proper, we have

2 ∞ ∼ 2 ∞ 2 ∞ I H (X ,µℓ ) = H (XKs ,µℓ )= H (XKs ,µℓ ) .

22 Consider

/ / 2 / / 0 Pic(X ) ⊗ Qℓ/Zℓ H (X ,µℓ∞ ) Br(X )(ℓ) 0

   / / 2 / / 0 NS(XKs ) ⊗ Qℓ/Zℓ H (XKs ,µℓ∞ ) Br(XKs )(ℓ) 0

Since NS(XKs ) ⊗ Qℓ/Zℓ is I-invariant and Pic(X ) = Pic(X). It suffices to show that

I Pic(X) ⊗ Qℓ/Zℓ −→ (NS(XKs ) ⊗ Qℓ/Zℓ) is surjective. Write NS(XKs )free for NS(XKs )/NS(XKs )tor. The action of I on NS(XKs ) factors through a finite quotient I′. Consider the exact sequence

I′ I′ I 0 −→ (NS(XKs )free) ⊗ Zℓ −→ NS(XKs ) ⊗ Qℓ −→ (NS(XK¯ ) ⊗ Qℓ/Zℓ)

1 ′ −→ H (I , NS(XKs )free ⊗ Zℓ). 1 ′ ′ I H (I , NS(XKs )free ⊗ Zℓ) is killed by the order of I . Since (NS(XKs ) ⊗ Qℓ/Zℓ) = NS(XKs ) ⊗ Qℓ/Zℓ is divisible, so the image of the last map is zero. Since

I Pic(X) ⊗ Qℓ −→ NS(XKs ) ⊗ Qℓ is surjective (cf. [Yua2, §2.2]), the claim follows. By Snake Lemma, the natural map

I Br(X )(ℓ) −→ Br(XKs ) (ℓ) 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′ := Kks and let ℓ =6 p be a prime. Set

Gk GK K := Ker(VℓBr(Xks ) −→ VℓBr(XKs ) ), then we have canonical exact sequences

Gk X ′ 0 Gk 0 −→ VℓBr(Yks ) −→ K −→ Vℓ K (PicX/K ) −→ 0,

Gk GK 0 −→ K −→ VℓBr(Xks ) −→ VℓBr(XKs ) −→ 0. Proof. By shrinking Y, we might assume that π is projective and smooth. Firstly, we will show that the natural map

GK′ VℓBr(Xks ) −→ VℓBr(XKs )

23 is surjective. Consider the Leray spectral sequence

p,q p q p+q E2 = H (Yks , R π∗Qℓ(1)) ⇒ H (Xks , Qℓ(1)). By Deligne’s Lefschetz criteria(cf. [Del1] ), the above spectral sequence degenerates at E2. Thus, we get a canonical surjective map

2 0 2 H (Xks , Qℓ(1)) −→ H (Yks , R π∗Qℓ(1)).

2 Since R π∗Qℓ(1) is lisse, we have

0 2 2 GK′ H (Yks , R π∗Qℓ(1)) = H (XKs , Qℓ(1)) . It follows that the canonical map

2 2 GK′ H (Xks , Qℓ(1)) −→ H (XKs , Qℓ(1)) is surjective. Consider the following commutative diagram with exact rows

2 / / H (Xks , Qℓ(1)) VℓBr(Xks ) 0

  2 GK′ / G ′ / H (XKs , Qℓ(1)) VℓBr(XKs ) K 0

Since the first column is surjective, the map

GK′ VℓBr(Xks ) −→ VℓBr(XKs ) is also surjective. Secondly, we will show the exactness of the first sequence in the statement of theorem. By the Leray spectral sequence

p q p+q H (Yks , R π∗Gm) ⇒ H (Xks , Gm), we get a long exact sequence

0 2 1 1 3 Br(Yks ) −→ Ker(Br(Xks ) −→ H (Yks , R π∗Gm)) −→ H (Yks , R π∗Gm) −→ H (Yks , Gm). 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 K- morphism 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′

1 1 / 3 H (Yks , R π∗Gm) H (Yks , Gm)

  1 1 ′ / 3 ′ H (Yks , R π∗Gm) H (Yks , π∗Gm)

24 ′ i ′ 3 ′ 3 Since π is finite, we have R π∗Gm = 0 for i > 0 and H (Yks , π∗Gm) = H (Zks , Gm). ′ There is a canonical norm map π∗Gm −→ Gm which induces a norm map

i i N : H (Zks , Gm) −→ H (Yks , Gm)

i i such that the composition of N with the pull back map H (Yks , Gm) −→ H (Zks , Gm) 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 −→ VℓBr(Yks ) −→ VℓBr(Xks ) is split as Gk-representations. Thus, we get a split exact sequence of Gk-representations

0 2 1 1 0 → VℓBr(Yks ) → Ker(VℓBr(Xks ) → VℓH (Yks , R π∗Gm)) → VℓH (Yks , R π∗Gm) → 0.

In fact, the norm map N induces a projection

0 2 Ker(VℓBr(Xks ) −→ VℓH (Yks , R π∗Gm)) −→ VℓBr(Yks ).

So we get an isomorphism

0 2 ∼ 1 1 Ker(VℓBr(Xks ) −→ VℓH (Yks , R π∗Gm)) = VℓBr(Yks ) ⊕ VℓH (Yks , R π∗Gm),

where the direct sum decomposition depends on the choice of π′. Next, we will show that there is a canonical isomorphism

1 1 0 s ∼ X VℓH (Yk , R π∗Gm) = Vℓ Yks (PicX/K ).

Let j : Spec K −→ Y be the generic point. By Lemma 6.1, the natural map

1 ∗ 1 R π∗Gm −→ j∗j R π∗Gm is an isomorphism. It follows that

1 1 ∼ 1 ∗ 1 H (Yks , R π∗Gm) = H (Yks , j∗j R π∗Gm).

There is an exact sequence

1 ∗ 1 1 ′ 1 sh 0 −→ H (Yks , j∗j R π∗Gm) −→ H (K , PicX/K ) −→ Y H (Ks , PicX/K ). s∈Yks

X Let Yks (PicX/K ) denote the kernel of the third arrow. It suffices to show that

X 0 ∼ X Vℓ Yks (PicX/K ) = Vℓ Yks (PicX/K ).

By the canonical exact sequence

0 0 −→ PicX/K −→ PicX/K −→ NS(XKs ) −→ 0,

25 we get a long exact sequence

0 ′ 1 ′ 0 1 ′ 1 ′ H (K , NS(XKs )) −→ H (K , PicX/K ) −→ H (K , PicX/K ) −→ H (K , NS(XKs )).

Let L/K be a finite Galois extension such that Pic(XL) −→ NS(XKs ) is surjective. One 1 ′ can show that H (K , NS(XKs )) is killed by [L : K]#NS(XKs )tor and the cokernel of 0 ′ 0 ′ H (K , PicX/K ) −→ H (K , NS(XKs )) is killed by [L : K]. Thus, the kernel and the cokernel of 1 ′ 0 1 ′ H (K , PicX/K ) −→ H (K , PicX/K ) are killed by [L : K]#NS(XKs )tor. By the same argument, the kernel and cokernel of

1 sh 0 1 sh Y H (Ks , PicX/K ) −→ Y H (Ks , PicX/K ) s∈Yks s∈Yks are also killed by [L : K]#NS(XKs )tor. Taking Vℓ, they become isomorphisms. Therefore, we have a canonical isomorphsim

X 0 ∼ X Vℓ Yks (PicX/K ) = Vℓ Yks (PicX/K ). Thus, we get a canonical isomorphism

1 1 0 s ∼ X VℓH (Yk , R π∗Gm) = Vℓ Yks (PicX/K ). By Lemma 6.1, the natural map

2 ∗ 2 R π∗Gm(ℓ) −→ j∗j R π∗Gm(ℓ)

is an isomorphism. It follows directly

0 2 ∼ GK′ H (Yks , R π∗Gm)(ℓ) = Br(XKs ) (ℓ).

Thus, there is an exact sequence

′ X ′ 0 GK 0 −→ VℓBr(Yks ) ⊕ Vℓ K (PicX/K ) −→ VℓBr(Xks ) −→ VℓBr(XKs ) −→ 0,

′ where the second arrow depends on the choice of π . Taking Gk-invariants, we proved the exactness of the first sequence in the theorem. It remains to show that the natural map

Gk GK VℓBr(Xks ) −→ VℓBr(XKs )

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 .

26 By shrinking Y, we might assume that the map Z −→ X chosen before factors through Z −→ W. Thus, we get a commutative diagram

′ / X ′ 0 / / GK / 0 VℓBr(Yks ) ⊕ Vℓ K (PicX/K ) VℓBr(Xks ) VℓBr(XKs ) 0

   ′ / X ′ 0 / / GK / 0 VℓBr(Yks ) ⊕ Vℓ K (PicW/K) VℓBr(Wks ) VℓBr(WKs ) 0

Taking Gk invariants, we get

a Gk / GK / 1 X ′ 0 VℓBr(Xks ) VℓBr(XKs ) H (Gk,VℓBr(Yks ) ⊕ Vℓ K (PicX/K ))

c    Gk / GK / 1 X ′ 0 VℓBr(Wks ) VℓBr(WKs ) H (Gk,VℓBr(Yks ) ⊕ Vℓ K (PicW/K))

GK Since VℓBr(WKs ) = 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

X ′ 0 X ′ 0 Vℓ K (PicX/K ) −→ Vℓ K (PicW/K) is split as Gk-representations. By our choice of W and the Lefschetz hyperplane section theorem, the pullback map

1 1 H (XKs , Qℓ) −→ H (WKs , Qℓ) 0 0 is injective. This implies that the induced map PicX/K −→ PicW/K has a finite kernel. 0 0 Therefore, there exists an abelian variety A/K and an isogeny PicX/K × A −→ PicW/K. It follows that X ′ 0 X ′ ∼ X ′ 0 Vℓ K (PicX/K ) ⊕ Vℓ K (A) = Vℓ K (PicW/K). This proves the splitness. Thus, the natural map

Gk GK VℓBr(Xks ) −→ VℓBr(XKs ) is surjective.

In the theorem above, we assume that the generic fiber is projective over K. In the Gk GK following corollary, we will deduce the surjectivity of VℓBr(Xks ) −→ VℓBr(XKs ) 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 =6 char(k) be a prime. Assuming that X is smooth over K, then the natural map

Gk GK VℓBr(Xks ) −→ VℓBr(XKs ) is surjective.

27 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 s GL a morphsim with a smooth geometrically connected generic fiber. Since Br(X ⊗L L ) = GK Br(XKs ) , it suffices to prove the claim for this morphism. Thirdly, if X/K is birational equivalent to a smooth projective variety X′ over K, GK then the claim will follow from Theorem 6.2 since VℓBr(XKs ) is a birational invariant. Next, we will use de Jong’s alteration theorem to reduce the question to this case. Let ′ ′ XK¯ −→ XK¯ be an alteration such that XK¯ 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 −→ XL is finite flat. By spreading out X′ −→ Spec L, we get a commutative diagram X ′ / Y′

f   X / Y where Y′ has function field L. By shrinking Y, we can assume that f is finite flat. s ′ Thinking f as a Y-morphsim and base change to Spec K , we get XKs −→ XKs which is ′ also finite flat. It induces a norm map Br(XKs ) −→ Br(XKs ) which is compatible with ′ the norm map Br(Xks ) −→ Br(Xk). Therefore we get a commutative diagram

′ Gk / ′ GK VℓBr(Xks ) VℓBr(XKs )

  Gk / GK VℓBr(Xks ) VℓBr(XKs ) 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

′ Gk ′ GL VℓBr(Xks ) −→ VℓBr(XLs ) is surjective. Thus, it suffices to show that there is a natural isomorphism

′ GK ∼ ′ s GL Br(XKs ) = Br(X ⊗L L ) . Fix an algebraic closure K¯ of K and assume that L and Ls are contained in K¯ . Since ′ ′ s s s XKs = X ⊗L L ⊗K K , the natural map L ⊗K K −→ L induces a map ′ s ′ s Br(X ⊗L L ⊗K K ) −→ Br(X ⊗L L ). s Set K1 = L ∩ K , by our assumption that L/K is a normal extension, so K1/K is finite

Galois. Write G = GK and H = GK1 . We have

s s s L ⊗K K = L ⊗K1 K1 ⊗K K = Y L ⊗K1,σ K s σ:K1֒→K

28 s One can show that L ⊗K1,σ K is a separable closure of L and admits a H-action (acting s s on K ). We denote it by Lσ. Then

′ s ′ Br(X ⊗ L ⊗ K ) ∼ Br(X s ). L K = Y Lσ s σ:K1֒→K Taking H-invariant, we get

′ s H ′ H Br(X ⊗ L ⊗ K ) ∼ Br(X s ) . L K = Y Lσ s σ:K1֒→K

Now G/H acts as permutations on the right side, so the G/H-invariant of right side can s be identified with the factor with the index σ equal to the inclusion map K1 ⊂ K . The ′ GL s ∼ s factor can be identified with Br(XLs ) through the natural isomorphism L⊗K1 K = L . It follows that there is a natural isomorphism

′ GK ∼ ′ s GL Br(XKs ) = Br(X ⊗L L ) . 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 = TrK/k(A) denote the K/k trace of A. Let ℓ =6 p be a prime. Then there is a canonical exact sequence

Gk 0 −→ VℓXk(B) −→ VℓXK(A) −→ VℓXK′ (A) −→ 0.

Proof. Consider the exact sequence

1 ′ 1 1 ′ 0 −→ H (Gk, A(K )) −→ H (K, A) −→ H (K , A),

1 ′ 1 we will show that VℓH (Gk, A(K )) is naturally isomorphic to VℓH (k, B). This will imply the left exactness of the sequence in the claim. By the Lang-N´eron theorem, A(K′)/B(ks) 1 ′ s is finitely generated, so H (Gk, A(K )/B(k )) is of finite exponent. By the exact sequence

0 −→ B(ks) −→ A(K′) −→ A(K′)/B(ks) −→ 0,

we get 1 ′ 1 s VℓH (Gk, A(K )) = VℓH (Gk, B(k )) = VℓXk(B), where the last equality follows from Lemma 5.2. It follows that

Gk 0 −→ VℓXk(B) −→ VℓXK(A) −→ VℓXK′ (A) is exact. 0 To show the surjectivity of the last arrow, we might assume A = PicX/K where X is a smooth projective geometrically connected curve over K. Spreading out X/K, we get

29 a smooth projective relative curve π : X −→Y. Since Br(XKs ) = 0, by Theorem 6.2, we get a surjection Gk Gk Br(Xks ) −→ VℓXK′ (A) .

Let k0 be the algebraic closure of the prime field of k in k. Then there exists a smooth geometrically connected variety Z over k0 with function field k. By shrinking Z, we can spread out π to a smooth projective morphsim πZ : XZ −→ YZ. By shrinking Z further, we can assume that YZ is smooth over Z. Using Theorem 6.2 for πZ , we get a surjection

Gk0 X Gk0 VℓBr(XZ s ) −→ Vℓ Kks (A) . k0 0 It remains to show that the natural map

Gk0 Gk VℓBr(XZ s ) −→ Br(Xks ) k0

is surjective. Since X can be identified with the generic fiber of XZ −→ Z, then the claim follows from Corollary 6.3.

6.1 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 =6 p be a prime. Then

rank A(K) ≤ ords=mΦ1(s),

and ords=mΦ1(s) is equal to the dimension of the generalized 1-eigenspace of the Frobenius 1 A action on H (Yk¯,Vℓ ). Proof. Let q denote the cardinality of k and F denote the geometric Frobenius element 1 in Gk. Set F = R π∗Qℓ, F is lisse since π is smooth and proper. By definition, −s −1 F −1 Φ1(s)= Y det(1 − qy σy | y¯) . y∈Y◦ By Grothendieck’s formula,

2m −s i F (−1)i+1 Φ1(s)= Y det(1 − q F |Hc(Yk¯, )) i=0

30 i F 2m F By Deligne’s theorem, Hc(Yk¯, ) is of mixed weight ≤ i + 1 and Hc (Yk¯, ) is of pure −s 2m−1 F weight 2m+1 by Poincar´eduality. Therefore only det(1−q F |Hc (Yk¯, )) contributes to the order of zero at s = m. By Poincar´eduality,

2m−1 F ∼ 1 F ∨ ∨ Hc (Yk¯, ) = H (Yk¯, ) (−m).

It follows that ords=mΦ1(s) is equal to the dimension of the generalized 1-eigenspace of 1 F ∨ F on H (Yk¯, ). Thus

1 F ∨ Gk dim H (Yk¯, ) ≤ ords=mΦ1(s).

F 1 F ∨ The lisse sheaf corresponds to the representation of GK on H (AK¯ , Qℓ), therefore 1 ∨ corresponds to H (AK¯ , Qℓ) which can be identified with VℓA . Since π is an abelian scheme, there is a Kummer exact sequence of etale sheaves on Y

n 0 −→ A [ℓn] −→ A −→ℓ A −→ 0.

Since A [ℓn] is a finite etale group scheme over Y, it represents a locally constant sheaf n on Y. Let VℓA be the Qℓ-sheaf associated to the inverse system A [ℓ ]. It corresponds ∨ to the GK representation on VℓA. Therefore VℓA can be identified with F . From the Kummer exact sequence, we get

0 A 1 A 1 A 0 −→ H (Yk¯, ) ⊗Z Qℓ −→ H (Yk¯,Vℓ ) −→ VℓH (Yk¯, ) −→ 0.

By [Kel1, Thm. 3.3], A satisfies the Neron property

∼ ∗ A = j∗j A ,

where j = Spec K −→ Y is the generic point of Y. Write K′ for Kk¯, it follows

0 A ′ H (Yk¯, )= A(K ).

Taking Gk-invariants for the above exact sequence, we get

0 A Gk 1 A Gk 1 A Gk 0 −→ H (Yk¯, ) ⊗Z Qℓ −→ H (Yk¯,Vℓ ) −→ VℓH (Yk¯, ) .

0 Gk Since H (Yk¯, A) = A(K), it follows

1 A Gk rank A(K) ≤ dim H (Yk¯,Vℓ ) ≤ ords=mΦ1(s).

7.2 Proof of Theorem 1.9 Theorem 7.2. Let A be an abelian variety over a finitely generated field K of character- istic p > 0. Let ℓ =6 p be a prime. Then the BSD conjecture (i.e. Conjecture 1.4) for A is equivalent to the finiteness of XK(A)(ℓ).

31 Proof. Assuming the BSD conjecture for A, by Lemma 7.1 and 7.3 below, we have Gk Gk VℓXK′ (A) = 0. By Lemma 5.2, VℓXK(A) = VℓXK′ (A) , so VℓXK (A) = 0. Thus XK(A)(ℓ) is finite. Gk Assuming that XK (A)(ℓ) is finite, then VℓXK′ (A) = 0. By Lemma 7.3 below,

0 A ∼ 1 A Gk H (Y, ) ⊗Z Qℓ = H (Yk¯,Vℓ )

0 A 1 A and H (Y, ) ⊗Z Qℓ is a direct summand of H (Yk¯,Vℓ ) as Gk-representations. It 1 A Gk 1 A follows that H (Yk¯,Vℓ ) is direct summand of H (Yk¯,Vℓ ) as Gk-representations. 1 A Gk 1 A This implies that H (Yk¯,Vℓ ) is equal to the generalized 1-eigenspace of H (Yk¯,Vℓ ). 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 =6 p be a prime. Then there is a split exact sequence of Gk-representations

0 A 1 A 1 A 0 −→ H (Yk¯, ) ⊗Z Qℓ −→ H (Yk¯,Vℓ ) −→ VℓH (Yk¯, ) −→ 0.

′ ¯ 0 A ′ Proof. Write K for Kk. It suffices to show that H (Yk¯, )⊗ZQℓ = A(K )⊗ZQℓ is a direct 1 A summand of H (Yk¯,Vℓ ) as Gk-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 variety Y¯ over k. Let V denote the smooth locus of Y¯/k. Then Y¯ −V has codimension ≥ 2 in Y¯. Let j denote the open immersion Y ֒→ V and F denote q VℓA on Y. By [Fu, Prop. 10.1.18 (iii)], R j∗F are constructible Qℓ-sheaves for all q ≥ 0. Set D = V −Y, by removing a closed subset Z of codimension ≥ 2 from V , we might 1 1 assume that D is a smooth divisor and R j∗F and j∗R π∗Qℓ are lisse on D. There is a canonical exact sequence

1 F 1 F 0 1 F 0 −→ H (Vk¯, j∗ ) −→ H (Yk¯, ) −→ H (Dk¯, R j∗ ).

0 1 F 1 F We will show that H (Dk¯, R j∗ ) is of weight ≥ 1. This will imply that H (Vk¯, j∗ ) 1 F and H (Yk¯, ) have the same generalized 1-eigenspace of the Frobenius action. Then the question will be reduced to show the splitness of

0 A 1 A 0 −→ H (Yk¯, ) ⊗Z Qℓ −→ H (Vk¯, j∗Vℓ ),

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

1 F 1 sh (R j∗ )η¯ = H (Kη ,VℓA).

s sh Set I = Gal(K /Kη ), by [Mil3, Chap I, Lem. 2.18], there is an isomorphism of Gk(η)- representations 1 sh ∼ H (Kη ,VℓA) = (VℓA)I (−1).

32 1 I ∨ Since (VℓA)I =(H (AKs , Qℓ) ) , it follows

1 sh ∼ 1 I ∨ H (Kη ,VℓA) = (H (AKs , Qℓ) ) (−1)

1 as Gk(η)-representations. Let s ∈ {η} be a closed point. By assumption, j∗R π∗Qℓ is lisse 1 1 I on D. So Gk(s) acts on (j∗R π∗Qℓ)η¯ = H (AKs , Qℓ) . By [Del2, Cor. 1.8.9], the Gk(s)- 1 I 1 I ∨ action on H (AKs , Qℓ) is of weight ≤ 1. Thus, the Gk(s)-action on (H (AKs , Qℓ) ) (−1) 1 is of weight ≥ 1. It follows that (R j∗F )s¯ is of weight ≥ 1. Since this holds for all 0 1 F closed points in D, it follows that H (Dk¯, R j∗ ) is of weight ≥ 1. This completes the proof.

7.3 The N´eron-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

′ 1 (5) 0 −→ A(K ) ⊗Z Qℓ −→ H (Vks , j∗VℓA ).

In the case that dim S = 1, V is equal to S. j∗A is the etale sheaf represented by a N´eron 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 Gk-equivariant pairings

0 A 0 Q 1 A 0 G Q Q H (V, ) ⊗Z × ExtVfppf ( , m) ⊗Z

−δ ←−lim rn 1 0 1 0 n H (V¯,V A )× lim Ext n (A [ℓ ],µ n ) ⊗Z Q Q k ℓ ←−n (ℓ )−Vet ℓ ℓ ℓ ℓ

where A 0 is a smooth group scheme over V with connected fibers (A 0 is taken to be the identity component of a N´eron 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.

7.3.1 N´eron model Lemma 7.4. Let A be an abelian variety over a finitely generated field K of characteristic p > 0. Let S be a projective normal geometrically connected variety over a finite field k with function field K. Let S1 denote the set of points of codimenson 1. Then there exists a regular open subscheme V ⊆ S with codim(S − V ) ≥ 2 and a smooth commutative V -group scheme A of finite type satisfying the following assumptions:

33 (i) A is the generic fiber of A and the restriction of A to Spec OS,s is a N´eron model of A/K for all s ∈ S1.

(ii) A admits an open subgroup scheme A 0 which equals to the identity component of 1 A when restricted to Spec OS,s for all s ∈ S and there exists a closed reduced subscheme Y of V such that A is abelian over V − Y and the quotient of AY by A 0 Y is a finite etale group scheme over Y . (iii) All fibers of A 0 −→ V are geometrically connected.

Proof. For the case dim(S) = 1, we can take V = S and take A to be a N´eron model of A/K over S and A 0 to be its identity component. In general, if dim(S) > 1, the N´eron model of A over S might not exist. The idea is that extending A to a N´eron model over 1 Spec OS,s for all s ∈ S , then spreads out. Since a N´eron model of an abelian variety over a DVR is of finite type, therefore a N´eron model of A over Spec OS,s can spread out to a smooth group scheme of finite type over an open neighborhood of s ∈ S1. There exists a regular open dense subscheme U of S such that A extends to an abelian scheme AU −→ U. By [Kel1, Thm. 3.3], AU −→ U is a Neron model of A over U. Since 1 1 S − U is finite, therefore we can glue Neron models of A over Spec OS,s for all s ∈ S − U with AU −→ U to get a smooth commutative group scheme A of finite type over some open subset V that contains S1 satisfying the condition (i). For each s ∈ S1 − U, by removing the closure of the complement of the identity component of As, we get an open subscheme A 0 of A , then shrink V , this subscheme becomes an open subgroup scheme. A A 0 Since s/ s is finite etale over k(s). Taking Y = V − U, by shrinking V , we will show A A 0 1 that the quotient Y / Y exists and is finite etale over Y . Let s ∈ Y ∩ S . There is an exact sequence A 0 A F 0 −→ s −→ s −→ s −→ 0, where Fs is a finte etale group scheme over k(s). By shrinking Y , Fs extends to a finte etale group scheme F over Y and the map As −→ Fs extends to a faithful flat morphism of finite type AY −→ F . By shrinking Y further, the kernel of AY −→ F can be A 0 indentified with Y . We get an exact sequence of fppf sheaves on Y A 0 A F 0 −→ Y −→ Y −→ −→ 0.

A 0 A 0 Therefore, −→ V satisfies condition (ii). Since s is geometrically connected for all 1 A 0 s ∈ S , by shrinking Y , we can assume that all fibers of Y −→ Y are geometrically connected. In the lemma below, we will show that the A 0 constructed as above satisfies

0 ∼ VℓA = j∗Vℓ(A |U ).

As a result, there is a canonical isomorphism

1 A ∼ 1 A 0 H (Vk¯, j∗Vℓ( |U )) = H (Vk¯,Vℓ ).

34 Lemma 7.5. Notations as above, let U ⊆ V be an open dense subset such that A is an 0 abelian scheme over U. Let j : U −→ V be the inclusion. Denote by TℓA (resp. TℓA ) 0 n n the inverse system of ℓ-toriosn sheaves (A [l ])n≥0 ( resp. (A [l ])n≥0 ). By shrinking V , there is a canonical isomorphism between A-R ℓ-adic sheaves (cf. [Fu, Chap. 10] )

0 ∼ TℓA = j∗(TℓA |U ).

0 Proof. By the lemma below, TℓA is an A-R ℓ-adic sheaf. There is an exact sequence

0 −→ A 0[ln] −→ A [ln] −→ F [ln] −→ 0.

0 Since F is finite etale, TℓF is A-R zero. Therefore TℓA is A-R isomorphic to TℓA . Since TℓA |U is a ℓ-adic sheaf, j∗(TℓA |U ) is also A-R ℓ-adic. Since A is a N´eron model 1 ∼ of A when restricted to Spec OS,s for all s ∈ S , we have A = j∗(A |U ) over Spec OS,s. It follows n ∼ n A [ℓ ]s¯ = (j∗(A [ℓ ]|U ))s¯. It follows that the kernel and cokernel of

TℓA −→ j∗(TℓA |U )

are supported on a closed subset of codimension ≥ 2. Removing this closed subset from V , we have 0 ∼ TℓA = j∗(TℓA |U ).

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 ℓ- n torsion sheaves TℓG = (G[ℓ ])n≥1 is A-R ℓ-adic. If all fibers of G/S are geometrically connected, then n G −→ℓ G is surjective as a morphism between etale sheaves on S.

Proof. Firstly, note that G[ℓn] is etale and of finite type over S. Thus the etale sheaf represented by G[ℓn] is constructible. Secondly, let f : S1 −→ S be a surjective between notherian schemes. If the statement F n F F ∗ holds for GS1 /S1, then it holds for G/S. Set n = G[ℓ ] and = ( 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 ¯ 0 S. GK¯ is smooth and of finite type over K. Let GK¯ be the identity component of GK¯ , then there is an exact sequence

0 0 −→ GK¯ −→ GK¯ −→ π0(G) −→ 0,

35 0 where π0(G) is a finite etale group scheme. By Chevalley’s theorem, GK¯ is an extension r s of an abelian variety A by Gm × Ga i.e.

r s 0 0 −→ Gm × Ga −→ GK¯ −→ A −→ 0.

The above exact sequences can descend to a finite extension L of K. Choose a model S1 for L. By shrinking S1 and S, we can assume that there exists a flat surjective morphism S1 −→ S of finite type whose function field extension corresponds to L/K and exact sequences 0 0 −→ GS1 −→ GS1 −→ π0(G) −→ 0 and Gr Gs 0 A 0 −→ m × a −→ GS1 −→ S1 −→ 0, A where π0(G) is finite etale over S1 and S1 is an abelian scheme. We get exact sequences of systems of ℓ-torsion sheaves

0 0 −→ TℓGS1 −→ TℓGS1 −→ Tℓπ0(G) −→ 0

and Gr Gs 0 A 0 −→ Tℓ( m × a) −→ TℓGS1 −→ Tℓ S1 −→ 0. r s r A Since Tℓ(Gm × Ga)= TℓGm and Tℓ S1 are ℓ-adic sheaves, by [Fu, Prop. 10.1.7(iii)], TℓGS1 is A-R ℓ-adic. It follows that TℓG 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.

7.3.2 Yoneda pairing

n Following notations in [Sch, §2], let Vfppf (resp. (ℓ ) − Vet) denote the category of fppf- sheaves (resp. etale sheaves of Z/ℓnZ-modules) on V .

Lemma 7.7. Notations as before, assuming that V is regular, then there is a commutative diagram

0 A 0 1 A 0 G / 1 G H (V, ) × ExtVfppf ( , m) H (V, m)

−δ rn    1 0 n 1 0 n / 2 A n A n n H (V, [ℓ ]) × Ext(ℓ )−Vet ( [ℓ ],µℓ ) H (V,µℓ )

n 0 n 0 ℓ 0 where δ is induced by the exact sequence 0 −→ A [ℓ ] −→ A −→ A −→ 0 and rn is defined by

0 n 0 n (0 −→ Gm −→ X −→ A −→ 0) 7→ (0 −→ µℓn −→ X [ℓ ] −→ A [ℓ ] −→ 0).

36 The natural map 1 0 n 1 0 n A G n A n (6) ExtVfppf ( [ℓ ], m) −→ Ext(ℓ )−Vet ( [ℓ ],µℓ ) defined by 0 n n 0 n (0 −→ Gm −→ X −→ A [ℓ ] −→ 0) 7→ (0 −→ µℓn −→ X [ℓ ] −→ A [ℓ ] −→ 0) is an isomorphism. And it induces a natural map 1 0 n+1 1 0 n n A n+1 n A n Ext(ℓ )−Vet ( [ℓ ],µℓ ) −→ Ext(ℓ )−Vet ( [ℓ ],µℓ ) compatible with pairings in the above diagram. As a result, by taking limit, there is a commutative diagram

0 A 0 1 A 0 G / 1 G H (V, ) × ExtVfppf ( , m) H (V, m) −δ ←−lim rn    1 0 1 0 n / 2 H (V, T A ) × lim Ext n (A [ℓ ],µ n ) H (V, Z (1)) ℓ ←−n (ℓ )−Vet ℓ ℓ 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 1 0 n 1 0 n n A n A n Ext(ℓ )−Vet ( [ℓ ],µℓ ) −→ ExtVfppf ( [ℓ ],µℓ ) which induces a map 1 0 n 1 0 n n A n A G Ext(ℓ )−Vet ( [ℓ ],µℓ ) −→ ExtVfppf ( [ℓ ], m). And it is easy to check the induced map is the inverse of (6) by definition. Let (0 −→ 0 n 1 0 n n Y A n A n Y n µℓ −→ −→ [ℓ ] −→ 0) ∈ Ext(ℓ )−Vet ( [ℓ ],µℓ ). may be regarded as a µℓ - torsor over A 0[ℓn] and so is representable by an etale scheme of finite type over V (cf. [Mil1, Chap. III, Thm. 4.3]). Therefore the exact sequence can be regarded as an exact 1 0 n A n sequence of fppf sheaves on V which gives an element in ExtVfppf ( [ℓ ],µℓ ). This defines a map 1 0 n 1 0 n n A n A n Ext(ℓ )−Vet ( [ℓ ],µℓ ) −→ ExtVfppf ( [ℓ ],µℓ ). By composing with the natural map 1 0 n 1 0 n A n A G ExtVfppf ( [ℓ ],µℓ ) −→ ExtVfppf ( [ℓ ], m), we get the desired map. Define 1 0 n+1 1 0 n A G n A n ExtVfppf ( [ℓ ], m) −→ Ext(ℓ )−Vet ( [ℓ ],µℓ ) as 0 n+1 n 0 n (0 −→ Gm −→ X −→ A [ℓ ] −→ 0) 7→ (0 −→ µℓn −→ X [ℓ ] −→ A [ℓ ] −→ 0). Then the composition 1 0 n+1 1 0 n+1 1 0 n n A n+1 A G n A n Ext(ℓ )−Vet ( [ℓ ],µℓ ) −→ ExtVfppf ( [ℓ ], m) −→ Ext(ℓ )−Vet ( [ℓ ],µℓ ) gives a map compatible with the first diagram.

37 7.3.3 Compatibility between the height pairing and Yoneda paring Theorem 7.8. Notations as before. Let S be a projective normal geometrically connected m variety over a finite field k. Fix a closed k-immersion ι : S ֒→ Pk . 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 m Pk . Composing it with the Yoneda pairing in the lemma above, we get a pairing 0 A 0 1 A 0 G Z (7) H (V, ) × ExtVfppf ( , m) −→ .

0 A 0 1 A 0 G → (There is a natural inclusion H (V, ) ֒→ A(K) and a natural map ExtVfppf ( , m 1 ˜ ˜ ExtK(A, Gm)= A(K), where A is the dual abelian variety of A. Then, up to a normalizing constant, the above pairing is compatible with the N´eron-Tate height pairing defined in [Con, Cor. 9.17] A(K) × A˜(K) −→ R.

Remark 7.9. This is a generalization of results in [Kel2, § 3] and [Sch, § 3]. The proof is just an imitation of arguments in [Blo, § 2], [Sch, § 3] and [Kel2, § 3]. As a consequence, the pairing (7) is left non-degenerate after tensoring Q. Proof. Firstly, we define the adele ring associated to S (cf.[Kel2, §3.1]. Define for T ⊂ S1 finite the T -adele ring of S as the restricted product

A O K,T = Y Ks × Y S,s, s∈T s∈S1\T b where OS,s is the completion of the local ring OS,s and Ks is the quotient field of OS,s, and theb adele ring of S as b

A = lim A . K −→ K,T T ⊂S1

1 m For each s ∈ S , we define deg(s) as the degree of the close subvariety {s} in Pk . This − deg(s).vs(·) gives an absolute value on Ks i.e. |·|s = q , 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. Leta ˜ = (0 −→ Gm −→ X −→ 0 1 0 0 A −→ 0) ∈ Ext (A , Gm). One can think X as a Gm-torsor on A 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

0 −→ Gm(K) −→ X (K) −→ A(K) −→ 0 and 0 0 −→ Gm(AK) −→ X (AK ) −→ A (AK ) −→ 0.

38 There is a natural homomorphism, the logrithmic modulus map, A l : Gm( K) −→ log q · Z ⊆ R, (as) 7→ X log |as|s = − log q · X deg(s) · vs(as). s∈S1 s∈S1 1 X 1 By the product formula, l(Gm(K)) = 0. Define Gm as the kernel of l, and as

1 1 X = {a ∈ X (AK ): ∃n ∈ Z≥1, na ∈ X (AK)}, g the rational saturation of X 1 with g X 1 1 X O X A = Gm · Y ( S,s) ⊆ ( K). g s∈S1 b

1 Then there exists a unique extension la˜ : X (AK) → R of l vanishing on X (cf.[Kel2, Lem. 3.1.4] or [Blo, Lem. 1.8]). It induces by restriction to X (K) a homomorphism

la˜ : A(K) −→ R.

Next, we will show that for any a ∈ A 0(S) ⊆ A(K), we have

la˜(a)= − log q · deg(a ∨ a˜), 0 A 0 1 A 0 G 1 G where ∨ denotes the Yoneda pairing H (V, ) × ExtVfppf ( , m) → H (V, m). By definition, a ∨ a˜ is defined by the following commutative diagram

/ / / / a ∨ a˜ : 0 Gm Y Z 0

id a    / / / 0 / a˜ : 0 Gm X A 0

By composition, one gets an extension

la˜ la∨a˜ : Y (AK ) −→ X (AK ) −→ R of l : Gm(AK ) → R to Y (AK ), which induces because of l(Gm(K)) = 0 in the exact sequence a ∨ a˜ by restriction to Y (K) a homomorphism

a la˜ la∨a˜ : Z −→ A(K) −→ R, so one obviously has la˜(a)= la∨a˜(1). Since X O la˜( Y ( S,s))=0, s∈S1 b hence Y O ℓa∨a˜( Y ( S,s))=0. s∈S1 b

39 Set e = a ∨ a˜, e is represented by an exact sequence of fppf sheaves on V

0 −→ Gm −→ Y −→ Z −→ 0, Y O and le is an extension of l and vanishes on Qs∈S1 ( S,s). By [Kel2, Lem 3.3.2]( replacing X by V in his argument) or [Sch, Lem. 12], b

la˜(a)= le(1) = − log q · deg(e).

Thirdly, we will show that the pairing h(a, a˜) := la˜(a) can be written as a sum of the local N´eron pairings which therefore coincides with the canonical N´eron-Tate height pairing by [BG, p.307, Cor. 9.5.14]. The proof for our case is exactly same as the argument 0 in [Blo, §2] and [Kel2, §3.4]. Given an extension (0 −→ Gm −→ X −→ A −→ 0) ∈ 1 A 0 G O 1 O ExtVfppf ( , m), its restriction to Spec S,s for s ∈ S is still an extension. Since S,s is a Dedekind domain, by[MW, p.53, Lem. 5.1], the push-forward of the sheaf

E 1 A 0 G xt(Spec OS,s)fppf ( , m)

O ˜ 0 to the smooth site of Spec S,s is represented by the N´eron model of A := PicA/K over 0 Spec O . By [Sch, Lem. 9], H om b (A , G ) = 0, it follows that S,s (Spec OS,s)fppf m

˜ 0 O E 1 A 0 1 A 0 A(Ks)= H ( S,s, xt Ob ( , Gm)) = Ext Ob ( , Gm). b (Spec S,s)fppf (Spec S,s)fppf 0 Therefore, the restriction of (0 −→ Gm −→ X −→ A −→ 0) to Spec OS,s gives an b element in A˜(Ks). Then one can define the local N´eron pairings as [Blo, (2.6)] for each s ∈ S1. By [Blo, Thm. 2.7] or [Kel2, Thm. 3.4.2], h(·, ·) is equal to the sum of the local N´eron 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 0 A 0 ′ H (Vk¯, ) ⊗Z Qℓ = A(K ) ⊗Z Qℓ

and the injective map of Gk-representations 0 A 0 1 A 0 H (Vk¯, ) ⊗Z Qℓ −→ H (Vk¯,Vℓ ) is split . ′ ¯ Proof. By the Lang-N´eron theorem [Con, Thm.7.1], the quotient group A(K )/TrK/k(A)(k) ¯ is finitely generated where TrK/k(A) is a K/k-trace of A/K. Since TrK/k(A)(k) is a ℓ- divisible torsion group, we have that

′ ′ A(K )⊗ˆ Qℓ = A(K ) ⊗Z Qℓ

has a finite dimension over Qℓ. Thus, there exists a finite extension l/k such that A(Kl)⊗Z ′ ′ Qℓ = A(K ) ⊗Z Qℓ. Replacing k by l, we might assume A(K) ⊗ Qℓ = A(K ) ⊗ Qℓ. Since

40 A(K) is finitely generated, by shrinking V , we can assume A (V ) = A(K). Shrinking V 0 0 further, since A (V )/A (V )is finite, hence A (V ) ⊗Z Qℓ = A(K) ⊗Z Qℓ. Assume that A˜(K) is generated by finitely many elementsa ˜i. Since ˜ 1 A 0 G A(Ks) = Ext(Spec OS,s)fppf ( , m)

1 for any s ∈ S , sofor agivena ˜ ∈ A˜(K), it corresponds to an extensiona ˜s : (0 −→ Gm −→ 0 X −→ A −→ 0) over Spec OS,s whose restriction to Spec K is independent of s. One 0 can think X as a Gm-torsor on A and so it is representable by a smooth commutative group scheme of finite type over Spec OS,s (cf. [Mil1, Chap III, Thm. 4.3]). Therefore, by 0 shrinking V , one can gluea ˜s to get an extension (0 −→ Gm −→ X −→ A −→ 0) over V . Thus, by shrinking V , we can assume that the natural map

1 A 0 G 1 G ˜ ExtVfppf ( , m) −→ ExtK (A, m)= A(K) is surjective. There is a commutative diagram

0 A 0 1 A 0 G 1 G deg(·) Z H (V, )× ExtVfppf ( , m) H (V, m)

−δ ←−lim rn d−1 1 0 1 0 n 2 tr(·∩η ) H (V¯,V A )× lim Ext n (A [ℓ ],µ n ) ⊗Z Q H (V¯, Q (1)) Q k ℓ ←−n (ℓ )−Vet ℓ ℓ ℓ k ℓ ℓ

O 2d ∼ where η denotes the cycle class of S(1) and tr denotes the trace map H (Sk¯, Qℓ(d)) = Qℓ. The commutativity of the second square follows from the definition of degree. By Lemma 7.7, the first square commutes. By Theorem 7.8, the top pairing is compatible with the N´eron-Tate pairing which is non-degenerate after tensoring Q (cf. [Con, Cor. 9.17]). 1 A 0 G 1 G ˜ Assuming the surjectivity of ExtVfppf ( , m) −→ ExtK(A, m) = A(K), then the top pairing is left non-degenerate after tensoring Qℓ. By Lemma 4.1, the injective map of Gk-representations 0 A 0 δ 1 A 0 H (Vk¯, ) ⊗Z Qℓ −→ H (Vk¯,Vℓ ) is split.

8 Applications

8.1 Proof of Corollary 1.13 Let π : X −→ Y be a dominant morphism between smooth geometrically connected varieties over a finite field k. Let K be the function field of Y. Assuming that the generic fiber X of π is smooth projective geometrically irreducible over K, then Conjecture 1.5 for X is equivalent to

0 Conjecture 1.5 for Y + BSD for PicX/K + Conjecture 1.4 for X for i =1.

41 Gk Proof. By Theorem 3.5, Conjecture 1.5 for X is equivalent to the vanishing of VℓBr(Xks ) Gk X 0 GK which is equivalent to the vanishing of VℓBr(Yks ) , Vℓ K(PicX/K ) and VℓBr(XKs ) by Theorem 6.2. By Corollary 1.7, Theorem 1.9 and Theorem 3.5, this is equivalent to

0 Conjecture 1.5 for Y + BSD for PicX/K + Conjecture 1.4 for X for i =1

8.2 Proof of Corollary 1.15 Proof. Assume that T 1(X,ℓ) holds for all smooth projective varieties X over finite fields of characteristic p. Let X be a smooth projective variety over a finitely generated field K of characteristic p. We can spread out X/K to get a smooth projective morphism π : X −→ Y as in Theorem 6.2. By Theorem 3.5, Conjecture 1.5 holds for X . By Corollary 1.13, we have Conjecture 1.1 for i = 1 holds for X , Conjecture 1.3 holds for 0 PicX/K , Conjecture 1.4 for i = 1 holds for X by Theorem 1.6.

8.3 The Picard number of Abelian varieties and K3 surfaces Proposition 8.1. Let X be an abelian variety (resp. a K3 surface ) over a finitely generated field K of characteristic p> 0 (resp. p> 2). Let m be the transcendence degree of K over its prime field. Then

rank NS(X)= −ords=m+1Φ2(s).

Proof. T 1(X,ℓ) holds for abelian varieties over finitely generated fields of positive char- acteristic (cf.[Zar]) and for K3 surfaces over finitely generated fields of characteristic p> 2(cf. [Cha] or [MP]). So the claim follows from Corollary 1.7.

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