The conjectures of Artin–Tate and Birch–Swinnerton-Dyer

. We provide two proofs that the conjecture of Artin–Tate for a ﬁbered surface is equivalent to the conjecture of Birch–Swinnerton-Dyer for the Jacobian of the generic ﬁbre. As a byproduct, we obtain a new proof of a theorem of Geisser relating the orders of the Brauer group and the Tate–Shafarevich group.


Introduction and statement of results
Let k = F q be a finite field of characteristic p and let S be a smooth projective (geometrically connected) curve over T = Spec k and let F = k(S) = F q (S) be the function field of S. Let X be a smooth proper surface over T with a flat proper morphism π : X → S with smooth geometrically connected generic fiber X 0 over Spec F. The Jacobian J of X 0 is an Abelian variety over F.
Our first main result is a proof of the following statement conjectured by Artin and Tate [Tat66, Conjecture (d)]: Theorem 1.1. The Artin-Tate conjecture for X is equivalent to the Birch-Swinnerton-Dyer conjecture for J.
Recall that these conjectures concern two (conjecturally finite) groups: the Tate Our second main result is a new proof of a beautiful result (2.18) of Geisser [Gei20, Theorem 1.1] that relates the conjectural finite orders of X(J/F) and Br(X); special cases of (2.18) are due to Milne-Gonzales-Aviles [Mil81,GA03].
We actually provide two proofs of Theorem 1.1; while our first proof uses Geisser's result (2.18), the second (and very short) proof in §4, completely due to the third-named author, does not.

Our approach
Our first proof depends on [Gor79] only for the elementary result (2.9). As in [Gor79,LLR04,LLR18], this proof also follows the strategy in [Tat66,§4]. We use the localization sequence to record a short proof 3 of the Tate-Shioda relation (Corollary 2.2). In turn, this gives a quick calculation (2.17) of the height pairing ∆ ar (NS(X)) on the Néron-Severi group of X. The same calculation in [Gor79,LLR18] requires a detailed analysis of various subgroups of NS(X). A beautiful introduction to these results is [Ulm14]; see [Lic83,Lic05,GS20] for Weil-étale analogues.
The second proof ( §4) of Theorem 1.1 uses only (2.5) and the Weil-étale formulations of the two conjectures. In this proof, we do not compare each term of the two special value formulas and entirely work in derived categories.

Notations
Throughout, k = F q is a finite field of characteristic p and T = Spec k; ifk is an algebraic closure of k, let T = Speck. The function field of S is F = k(S). Let X be a smooth proper surface over T with a flat proper morphism π : X → S with smooth geometrically connected generic fiber X 0 over Spec F. The Jacobian J of X 0 is an Abelian variety over F.

The Artin-Tate conjecture
Let k = F q and F = k(S). For any scheme V of finite type over T , the zeta function ζ(V , s) is defined as the product is over all closed points v of V and q v is the size of the finite residue field k(v) of v. If V is smooth proper (geometrically connected) of dimension d, then the zeta function ζ(V , s) factorizes as is the characteristic polynomial of Frobenius acting on the -adic étale cohomology H i (V × TT , Q ) for any prime not dividing q; by Grothendieck and Deligne, P j (V , t) is independent of . One has the factorization [Tat66, (4.1)] (the second equality uses Poincaré duality) Let ρ(X) be the rank of the finitely generated Néron-Severi group NS(X). The intersection D · E of divisors D and E provides a symmetric non-degenerate bilinear pairing on NS(X); the height pairing D, E ar [LLR18, Remark 3.11] on NS(X) is related to the intersection pairing as follows: Let A be the reduced identity component Pic red,0 X/k of the Picard scheme Pic X/k of X.
We write [G] for the order of a finite group G.

Discriminants
For more details on the basic notions recalled next, see [Yun15, §2.8] and [Blo87]. Let N be a finitely generated Abelian group N and let ψ : N × N → K be a symmetric bilinear form with values in any field K of characteristic zero. If ψ : N /tor × N /tor → K is non-degenerate, the discriminant ∆(N ) is defined as the determinant of the matrix ψ(b i , b j ) divided by (N : N ) 2 where N is the subgroup of finite index generated by a maximal linearly independent subset {b i } of N . Note that ∆(N ) is independent of the choice of the subset {b i } and the subgroup N and incorporates the order of the torsion subgroup of N . For us, K = Q or Q(log q).
Given a short exact sequence 0 → N → N → N → 0 which splits over Q as an orthogonal direct sum N Q N Q ⊕ N Q with respect to a definite pairing ψ on N , one has the following standard relation

The Birch-Swinnerton-Dyer conjecture
For more details on the basic notions recalled next, see [GS20]. Let J be the Jacobian of X 0 . Recall that the complete L-function [Ser70,Mil72], [GS20,§4] of J is defined as a product of local factors .
For any closed point v of S, the local factor L v (J, t) is the characteristic polynomial of Frobenius on where F v is the complete local field corresponding to v and I v is the inertia group at v. By [GS20, Proposition 4.1], L v (J, t) has coefficients in Z and is independent of , for any prime distinct from the characteristic of k. Let X(J/F) be the Tate Let J → S be the Néron model of J; for any closed point • Comparison of χ(S, Lie J ) and α(X) given in (2.5). This is known [LLR04,p. 483]. For the convenience of the reader, we recall it in §2.2. • (Proposition 2.4) ord s=1 P 2 (X, q −s ) = ρ(X) if and only if ord s=1 L(J, s) = r. • ( §3) P * 2 (X, 1) satisfies (1.4) if and only if L * (J, 1) satisfies (1.10). The first two parts are not difficult and we provide elementary proofs of the last two parts.
the morphism X → Spec k and the étale sheaf G m provides the first exact sequences [BLR90, Proposition 4, p. 204] below: Let P be the identity component of the Picard scheme Pic S/k of S. Let B be the cokernel of the natural injective map π * : P → A. So one has short exact sequences (using Lang's theorem [Tat66,p. 209] for the last sequence)

Comparison of χ(S, Lie J ) and α(X)
It is known [LLR04, p. 483] that We include their proof here for the convenience of the reader. A special case of this is due to Gordon [Gor79, Proposition 6.5]. The Leray spectral sequence for π and O X provides Recall that J is the Néron model of the Jacobian J of X 0 . As the kernel and cokernel of the natural map 5 φ : Thus,

The Tate-Shioda relation about the Néron-Severi group
The structure of NS(X) depends on the singular fibers of the morphism π : X → S.

Singular fibers.
R v = Z G v Z of the free Abelian group generated by the irreducible components of X v by the subgroup generated by the cycle associated with The following proposition provides a description of Proposition 2.1.
(i) The natural maps π * : Pic(S) → Pic(X) and π * : Proof. (i) From the Leray spectral sequence for π : X → S and the étale sheaf G m on X, we get the exact sequence . Now X 0 being geometrically connected and smooth over F implies [Mil81, Remark 1.7a] that π * G m is the sheaf G m on S. This provides the injectivity of the first map. The same argument with U in place of S provides the injectivity of the second.
(ii) The class group Cl(Y ) and the Picard group Pic(Y ) are isomorphic for regular schemes Y such as S and X. The localization sequences for X U ⊂ X and U ⊂ S can be combined as The induced exact sequence on the cokernels of the vertical maps is In particular, we get this sequence for Z and U . By assumption, X v is geometrically irreducible for any v Z; so R v = 0 for any v Z. So this means that, for any U = S − Z contained in U , the induced maps are isomorphisms. Taking the limit over Z gives us the exact sequence in the proposition.
(ii) This follows from the diagram Relating the order of vanishing at s = 1 of P 2 (X, q −s ) and L(J, s) , see §2.3.1 for notation. Using we can rewrite The precise relation between P 2 (X, q −s ) and L(J, s) is given by (2.11).
Proposition 2.3. One has ord s=1 Q 2 (s) = m and Proof. Observe that (2.10) is elementary: for any positive integer r, one has For each v ∈ Z, this shows that Therefore, we obtain that We now prove (2.11). Simplifying the identity On reordering, this becomes Let T J be the -adic Tate module of the Jacobian J of X. For any v ∈ S, the Kummer sequence on X and J provides a Gal(F as J is a self-dual Abelian variety: this provides the isomorphisms From [Del80, Théorème 3.6.1, pp.213-214] (the arithmetic case is in [Blo87, Lemma 1.2]), we obtain an isomorphism

Pairings on NS(X)
Our next task is to compute ∆(NS(X)).

Definition 2.5.
(i) Let Pic 0 (X 0 ) be the kernel of the degree map deg : Pic(X 0 ) → Z; the order δ of its cokernel is, by definition, the index of X 0 over F. (ii) Let α be the order of the cokernel of the natural map Pic 0 (X 0 ) → J(F). (iii) Let H (horizontal divisor on X) be the Zariski closure in X of a divisor d on X 0 , rational over F, of degree δ. (iv) The (vertical) divisor V on X is π −1 (s) for a divisor s of degree one on S. Such a divisor s exists as k is a finite field and so the index of the curve S over k is one. Writing s = a i v i as a sum of closed points v i on S gives V = a i π −1 (v i ). Note that V generates π * NS(S) ⊂ NS(X).

Remark. The definitions show that the intersections of the divisor classes H and V in NS(X) are given by (2.13)
Also, since π : X → S is a flat map between smooth schemes, the map π * : CH(S) → CH(X) on Chow groups is compatible with intersection of cycles. Since V = π * (s) and the intersection s · s = 0 in CH(S), one has V · V = 0.
Let NS(X) 0 = (π * NS(S)) ⊥ ; as V generates π * NS(S), we see that NS(X) 0 is the subgroup of divisor classes Y such that Y · X v = 0 for any fiber π −1 (v) = X v of π; let Pic(X) 0 be the inverse image of NS(X) 0 under the projection Pic(X) → NS(X)

Pic(X)
A(k) . Lemma 2.6. NS(X) 0 is the subgroup of NS(X) generated by divisor classes whose restriction to X 0 is trivial.
Proof. We need to show that NS(X) 0 is equal to K := Ker(NS(X) → NS(X 0 )). If D is a vertical divisor (π(D) ⊂ S is finite), then D is clearly in K; by [Liu02, §9.1, Proposition 1.21], D is in NS(X) 0 .
If D has no vertical components, then D · V = deg(D 0 ). To see this, clearly we may assume D is reduced and irreducible (integral) and so flat over S. So O D is locally free over O S of constant degree n since S is connected. But then deg(D 0 ) is equal to n as is the integer D · V .

Lemma 2.7. Let us denote
One has the exact sequences Proof. Lemma 2.6 shows that R ⊂ Pic(X) 0 π * Pic(S) . As A(k) is the kernel of the map Pic(X) → NS(X), it follows that A(k) ⊂ Pic(X) 0 . Thus, B(k) is a subgroup of Pic(X) 0 π * Pic(S) . The first exact sequence follows from Lemma 2.6; the second one follows from Corollary 2.2 (ii).

Lemma 2.8. One has the equality
Proof. The exact sequence (2.14) splits orthogonally over Q: for any divisor γ representing an element of Pic(X 0 ), consider its Zariski closureγ in X. Since the intersection pairing on R v is negative-definite [Liu02, §9.1, Theorem 1.23], the linear map R v → Z defined by β → β ·γ is represented by a unique element Thus, the elementγ This says that (2.15) ∆ ar Pic 0 (X 0 ) = ∆ NT Pic 0 (X 0 ) .

The map
provides an orthogonal splitting of (2.14) (over Q). So where e = [E] as the size of E. As this proves the lemma.
With Lemma 2.8 at hand we are almost ready to compute ∆ ar (NS(X)). As the intersection pairing on NS(X) is not definite (Hodge index theorem), we cannot apply (1.5). Instead, we use a variant of a lemma of Z. Yun [Yun15].

A lemma of Yun.
Given a non-degenerate symmetric bilinear pairing Λ × Λ → Z on a finitely generated Abelian group Λ, an isotropic subgroup Γ , a subgroup Γ containing Γ and with finite index in Γ ⊥ , Proof. Applying (1.6) to the maps of triangles shows that z(D) · z(C) −1 = z(D 0 ) · z(C ).
Combining the previous proposition with Lemma 2.8 provides the identity For v ∈ S, we put δ v and δ v for the (local) index and period of X × F v over the local field F v .
Theorem 2.11. [Gei20, Theorem 1.1] Assume that Br(X) is finite. The following equality holds: Remark 2.12. Note that for v ∈ U , one has

Second proof of Theorem 1.1
We will give another more direct proof of Theorem 1.1 using Weil-étale cohomology. We refer the reader to [Lic05,Gei04,GS20] for basics about Weil-étale cohomology over finite fields. Throughout this section, we assume that Br(X) (and hence X(J/F)) is finite. with finite cohomology. Set C Q l = R lim ← − −n (C ⊗ L Z/l n Z) ⊗ Z l Q l , whose cohomologies are finite-dimensional vector spaces over Q l (by the finiteness of H * (C ⊗ L Z/lZ)) equipped with an action of the geometric Frobenius ϕ of k. Define

Setup
Assume that Z(C, t) ∈ Q(t) and is independent of l. Define Q(C, D) ∈ Q × >0 × (1 − t) Z to be the leading term of the (1 − t)-adic expansion of the function (the sign is the one that makes the coefficient positive). It is the defect of a zeta value formula of the form

Special cases
We give two special cases of the above constructions. First, let π X : Xé t → Té t be the structure morphism. Let P 2 (X, 1)(1 − t) ρ(X) be the leading term of the (1 − t)-adic expansion of P 2 (X, t/q). C) is finitely generated, Z(C, q −s ) = ζ(X, s + 1) and In particular, the statement Q(C, D) = 1 is equivalent to Conjecture 1.2. (1)). The finiteness assumption on Br(X) implies the Tate conjecture for divisors on X and hence the finite generation of H * W (X, Z(1)) by [Gei04, Theorems 8.4 and 9.3]. The object C ⊗ L Z/lZ Rπ X, * Z/lZ(1) ∈ D b (Té t ) is constructible and hence its cohomologies are finite. We have H i (C Q l ) R i π X, * Q l (1), which is the vector space H í et (X × kk , Q l (1)) equipped with the natural Frobenius action. It follows that Z(C, q −s ) = ζ(X, s + 1).

A new proof of Geisser's formula
The above proposition, combined with the results of the previous sections, also gives a new proof of Theorem 2.11 as follows.
Taking a suitable alternating product of these four equalities, we obtain (2.18).