Apr 30, 2018 - below, is based in part on a result of Gordon [Go]. Thomas Geisser noted in [4] that the formula provided in Theorem 4.3 in [LLR1] need...

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CORRIGENDUM TO ´ NERON MODELS, LIE ALGEBRAS, AND REDUCTION OF CURVES OF GENUS ONE [LLR1] AND THE BRAUER GROUP OF A SURFACE [LLR2] QING LIU, DINO LORENZINI, AND MICHEL RAYNAUD†

1. Introduction Let k be a finite field of characteristic p. Let V /k be a smooth projective geometrically connected curve with function field K. Let X/k be a proper smooth and geometrically connected surface endowed with a proper flat map f : X → V such that the generic fiber XK /K is smooth and geometrically connected of genus g ≥ 1. Let AK /K denote the Jacobian of XK /K. The proof of Theorem 4.3 in [LLR1], which we state in corrected form below, is based in part on a result of Gordon [Go]. Thomas Geisser noted in [4] that the formula provided in Theorem 4.3 in [LLR1] needs to be corrected, due to the fact that Lemma 4.2 in [Go] is missing a hypothesis. He provides a corrected formula in [4], Theorem 1.1, and his method applies also to the number field case (up to a power of 2 if not totally imaginary). Several of the intermediate results in [Go] are only valid under the assumption that Pic0 (XK ) = AK (K). We revisit the paper [Go] in this corrigendum to remove this hypothesis in all arguments. In doing so, we also avoid using Lemma 4.3 in [Go], whose proof is incorrect, and whose statement might be wrong in general. 2. Corrected Statements We start by recalling the notation needed to state our main theorem. Let X(AK ) denote the Shafarevich-Tate group of the abelian variety AK /K. Let Br(X) denote the Brauer group of X. It is well-known that if either X(AK ) or Br(X) is finite, then so is the other (see [12], section 3, or [7], section 4). Our coauthor, mentor, and friend Michel Raynaud fell ill soon after we started writing this corrigendum. We are profoundly sad by his passing on March 10, 2018. All mistakes in this corrigendum are ours only (Qing Liu and Dino Lorenzini). 1

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The index δ := δ(XK ) of a curve over a field K is the least positive degree of a divisor on XK . The period δ ′ := δ ′ (XK ) of XK is the order of the cokernel of the degree map PicXK /K (K) → Z. When v ∈ V is a closed point, we denote by Kv the completion of K at v, and let δv := δ(XKv ), and δv′ := δ ′ (XKv ). Recall that we have an exact sequence 0 −→ Pic0 (XK ) −→ AK (K) −→ Br(K). Since the Brauer group Br(K) is a torsion group, and since AK (K) is a finitely generated abelian group, the quotient AK (K)/ Pic0 (XK ) is finite, and Pic0 (XK ) and AK (K) have the same rank. Let a := |AK (K)/ Pic0 (XK )|. We find in [LLR1],Q Proof of 4.6, based on the proofs of 2.3 and 2.5 in [5], that a divides ( δv′ )/ lcm(δv′ ). We are now ready to state the main result of this corrigendum. Corrected Theorem 4.3. Let X/k and f : X → V be as above. Assume that X(AK ) and Br(X) are finite. Then the equivalence of the Artin-Tate and Birch-Swinnerton-Dyer conjectures holds exactly when Y (2.1) |X(AK )| δv δv′ = a2 δ 2 |Br(X)|. v

The statement of Theorem 4.3 of [LLR1] unfortunately omits the factor a2 in the above formula. This omission leads to the following change in Corollary 4.7 of [LLR1]. The paragraph after Corollary 4.7 in [LLR1] can now be completely omitted. Corrected Corollary 4.7 Assume that X(AK ) and Br(X) are finite. Then the conjectures of Artin–Tate and Birch–Swinnerton-Dyer are equivalent if and only if δa = δ ′ bcǫ. Theorem 4.3 in [LLR1] is used in the proof of Corollary 3 of [LLR2]. The corrected version of Theorem 4.3 can be used in that proof to produce exactly the same result. We restate below Corollary 3 of [LLR2] with the correct formula relating the orders of X(AK ) and Br(X). Corrected Corollary 3. Let f : X → V be as above. Assume that for some prime ℓ, the ℓ-part group Br(X) or of the group X(AK ) is Q of the ′ finite. Then |X(AK )| v δv δv = a2 δ 2 |Br(X)|, and |Br(X)| is a square.

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3. Proof of the Corrected Theorem 4.3 We follow closely the paper [Go] of Gordon, and indicate below every change that needs to be made to the statements in [Go] to obtain a complete proof of Formula (2.1). 3.1. It may be of interest to first quickly indicate why the change in the formula occurs as a ‘square’. This fact is essential for the proof of Corollary 3 in [LLR2] to remain correct. The conjectures of Birch– Swinnerton-Dyer and of Artin–Tate require the explicit computation on one hand of the determinant of the height pairing on the lattice AK (K)/AK (K)tors , and on the other hand of the determinant of the intersection pairing on the free part NS(X)/NS(X)tors of the N´eron– Severi group NS(X). For this, it suffices to construct explicit bases for sublattices of finite index in these lattices (see, e.g., 3.7, 3.10), and the following well-known lemma then introduces ‘squares’ in the formuli. Lemma 3.2. Let Λ be a free abelian group of finite rank n, and let Λ′ ⊆ Λ be a sublattice of finite index [Λ : Λ′ ]. Let B : Λ × Λ → R be a bilinear form. Consider a basis λ1 , . . . , λn for Λ, and a basis λ′1 , . . . , λ′n for Λ′ . Let d := det((B(λi , λj ))1≤i,j≤n), and similarly, let d′ := det((B(λ′i , λ′j ))1≤i,j≤n). Then d′ = [Λ : Λ′ ]2 d. 3.3. We introduce below a finite group E. This group is claimed in [Go], Lemma 4.3, to be always trivial, but the proof provided in [Go] is unfortunately incorrect (in the last paragraph, the computation of π ∗ C is wrong). This group will appear in two quotients of the filtration of NS(X) introduced in 3.8. The final index discussed in 3.9 however does not depend on |E|. We follow below the notation in [Go] on page 177. Let k denote an algebraic closure of k, and for any k-scheme S, set as usual S := S ×k k. The natural map X → X defines an injection Div(X) → Div(X) which is compatible with the intersection pairings ( , )X and ( , )X . We identify Div(X) with its image in Div(X). Similarly, we use the maps f : X → V and f : X → V to identify Div(V ) and Div(V ) with their images in Div(X) and Div(X), respectively. Let us now define some natural subgroups of Div(X). First, Divvert (X) is the subgroup generated by the irreducible curves C on X for which f (C) is a single point. We denote by Div0 (X) the subgroup generated by the irreducible curves C on X which are algebraically equivalent to zero. Finally, let Div0 (V ) denote the image in Div(X) of the subgroup of divisors on V algebraically equivalent

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0 of the form to P zero. The subgroup Div (V ) is the set of all divisors P v av Xv , where Xv is the fiber over v ∈ V and v av = 0. The 0 intersection of Div(X) with the subgroup Div (V ), resp. with Div0 (X) or Divvert (X), is denoted by Div0 (V ), resp. by Div0 (X), or Divvert (X). It is clear that Div0 (V ) is contained in Div0 (X) ∩ Divvert (X). We let

E :=

Div0 (X) ∩ Divvert (X) . Div0 (V )

P For v ∈ V , write Xv = a pva Xva with Xva /k(v) irreducible of multiplicity pva , and set dv := gcdv (pva ). The integer dv is called the multiplicity of the fiber Xv , and when dv > 1, Xv is called a multiple fiber. Clearly d1v Xv ∈ Div(X). If W ∈ Div0 (X) ∩ Divvert (X), then W is numerically equivalent to zero, and so (W · Xva )X = 0 for all Xva . It follows from the fact that d1v Xv generates the kernel of the intersection matrix assoP ciated with the fiber Xv that W = v mv ( d1v Xv ) for some integers mv . Since (W · Ω)X = 0 for any horizontal divisor Ω on X, we find that P 0 (m v /dv ) degk v = 0. Hence for any W ∈ Div (X) ∩ Divvert (X), we v have W ∈ Div0 (V ) if and only if mv ∈ dv Z for all v. This implies that E is isomorphic to a subgroup ofQ ⊕v Z/dv Z. Let ∆ := lcmv (dv ). Then E is killed by ∆ and |E| divides dv .

Let now Dℓ (X) denote the subgroup of divisors in Div(X) that are linearly equivalent to zero. Set Dℓ (X) := Dℓ (X) ∩ Div(X). Let Pic0X/k and Pic0V /k denote the Picard schemes of X/k and V /k, respectively. (Pic0V /k is nothing but the Jacobian of V /k.) The scheme Pic0X/k might not be reduced, and we denote by Pic0X/k,red the (reduced) abelian variety associated with Pic0X/k . We have Pic0X/k,red (k) = Div0 (X)/Dℓ (X) and Pic0V /k (k) = Div0 (V )/Dℓ (V ) because Br(k) is trivial. Lemma 3.4. Keep the above notation. Then a) We have (Div0 (X) ∩ Divvert (X)) ∩ (Div0 (V ) + Dℓ (X)) = Div0 (V ). b) We have a natural injection E −→ Pic0X/k,red (k)/ Pic0V /k (k)

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given explicitly as Div0 (X) ∩ (Divvert (X) + Dℓ (X)) Div0 (X) ∩ Divvert (X) = −→ Div0 (V ) Div0 (V ) + Dℓ (X) Div0 (X) . −→ Div0 (V ) + Dℓ (X) Proof. The proof of b) follows immediately from a). To prove Part a), it suffices to prove that Divvert (X) ∩ (Div0 (V ) + Dℓ (X)) = Div0 (V ). If D ∈ Divvert (X)∩(Div0 (V )+Dℓ (X)), thenP D ∈ Divvert (X)∩Div0 (X). As noted in 3.3, P we can then write D = v rv Xv for some rational numbers rv with v rv deg(v) = 0. On the other hand, by hypothesis, D = div(f ) + D0 for some f ∈ k(X)∗ and D0 ∈ Div0 (V ). Since k is finite, some multiple of D0 is linearly equivalent to zero. Thus, for some positive integer m, mD = div(f m h) for some h ∈ k(V )∗ . Since P mD = v mrv Xv ∈ Div0 (V ), we find that some positive multiple n of mD is of the form div(h′ ) for some h′ ∈ k(V )∗ . Hence, f mn ∈ k(V )∗ . Since we assume that the generic fiber of X → V is geometrically integral, we find that f ∈ k(V )∗ . Thus D ∈ Div0 (V ). We stray here a little bit from the notation used by [Go], and we define B/k to be the quotient abelian variety B := Pic0X/k,red / Pic0V /k . Since k is finite, we have B(k) := Pic0X/k,red (k)/ Pic0V /k (k). For use in the proof of 3.8 (iv), let us note that (3.1)

B(k) Div0 (X) = . E (Div0 (X) ∩ Divvert (X)) + Dℓ (X)

Remark 3.5. In [Go], just before Proposition 4.4 on page 180, B/k is defined to be the K/k-trace of AK /K. Then Proposition 4.4 asserts that the K/k-trace of AK /K is an abelian variety which is purely inseparably isogenous to the quotient abelian variety Pic0X/k,red / Pic0V /k . The proof of Proposition 4.4 in [Go] uses the fact that a = 1. We refer the reader to [3] for the definition and existence of the K/k-trace of AK /K. When k is algebraically closed, we find in [11], Theorem 2, a theorem of Raynaud which asserts that the K/k-trace of AK /K is k-isomorphic to Pic0X/k,red / Pic0V /k when f : X → V does not have any multiple fibers (i.e., dv = 1 for all v). The notion of K/k-trace is not needed in this corrigendum, and we do not use Proposition 4.4 in [Go].

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Let Div0 (X) denote the subgroup of Div(X) generated by the irreducible curves which intersect each complete vertical fiber Xv with total intersection multiplicity zero. We let Div0 (X) := Div0 (X) ∩ Div(X). Let Ω ∈ Div(X) be a horizontal divisor of degree δ, where δ is the index of XK over K. In the following modified version of Lemma 4.2 in [Go], the group AK (K) has now been replaced by Pic0 (XK ). Lemma 3.6. (see Lemma 4.2 in [Go]) There are natural isomorphisms of groups Div(X) Div0 (X) −→ −→ Pic0 (XK ). (Divvert (X) ⊕ ZΩ) + Dℓ (X) Divvert (X) + Dℓ (X) Proof. Same as in [Go], replacing when necessary AK (K) by Pic0 (XK ). 3.7. Let NS(X) := Div(X)/ Div0 (X). Let us now introduce further notation needed to define below the completely explicit subgroup N0 of NS(X). (a) Let r be the rank of AK (K), and let {α1 , . . . , αr } be a basis of the lattice Pic0 (XK )/ Pic0 (XK )tors . Choose divisors A1 , . . . , Ar in Div(X) such that for each i, the class in Pic0 (XK ) of the restriction of Ai to the generic fiber XK is αi . For the later purpose of computing the global height pairing hαi , αj i as in 3.11, we assume also that we have chosen the divisors A1 , . . . , Ar , such that the supports of the restrictions of Ai and Aj to the generic fiber XK are pairwise disjoint when i 6= j. P (b) SinceP XK /K has index δ, choose a divisor i si xi in Div(XK ) such that i sP i deg K (xi ) = δ. Let xi denote the closure of xi in X, and set Ω := i si xi in Div(X). (c) Since V /k is geometrically integral, its index δ(V P P/k) is equal to 1. Choose a divisor j tj vj in Div(V ) such that j tj degk (vj ) = 1. P Let F := j tj Xvj in Div(X). This definition agrees with [Go], 4.6, when XK has a k-rational point and the complete fiber in 4.6 is chosen to be above a k-rational point. Ph(v) (d) For each v ∈ V , write the fiber Xv as Xv = a=1 pva Xva , where the components Xva are irreducible. For each closed point v ∈ V such that Xv is reducible, consider the set {Xva , a > 1, v ∈ V } of irreducible divisors in Div(X). We let N0 denote the subgroup of NS(X) generated by NS(X)tors and the classes of {A1 , . . . , Ar }, Ω, F , and {Xva , a > 1, v ∈ V }. We will compute the index of N0 in NS(X) in Proposition 3.9.

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Denote by S1 the set of closed points v ∈ V such that Xv is reducible. Let S2 denote the set of closed points v ∈ V such that Xv is irreducible but not reduced. Set Σ := S1 ⊔ S2 . Let S3 denote the set of v ∈ V such that Xv is integral but not geometrically integral. The set Σ is finite, and thus we have ⊕v (⊕a ZXva ) ⊕a ZXva Divvert (X) . = = ⊕v∈Σ (3.2) Q := Div(V ) ⊕v ZXv ZXv Define NS(X)vert to be the image in NS(X) of the subgroup Divvert (X) of Div(X). Let [Ω] denote the class of Ω in NS(X). It is clear that NS(X)vert ∩ Z[Ω] = (0), and we write N := NS(X)vert ⊕ Z[Ω]. We may now state a modified version of Proposition 4.5 in [Go], where the group E occurs in two different factors. Proposition 3.8. (see Proposition 4.5 in [Go]) The group NS(X) has a filtration by subgroups 0 ⊆ f ∗ NS(V ) ⊆ NS(X)vert ⊆ N ⊆ NS(X) with respective quotients Z, Q/E, Z, and Pic0 (XK )/(B(k)/E). Proof. (i) The map f ∗ : NS(V ) → NS(X) is injective, and since NS(V ) is free of rank 1, so is f ∗ NS(V ). (ii) Let us first note that the natural map E=

Div0 (X) ∩ Divvert (X) Divvert (X) −→ Q = 0 Div(V ) Div (V )

is injective because (3.3)

Div0 (X) ∩ Divvert (X) ∩ Div(V ) = Div0 (V ).

Recall that NS(X)vert =

Divvert (X) , Divvert (X) ∩ Div0 (X)

and consider the natural map f ∗ Div(V ) −→ NS(X)vert . This map has kernel f ∗ Div0 (V ), by (3.3). Hence, we have an exact sequence 0 → f ∗ NS(V ) −→ NS(X)vert −→ Q/E −→ 0. (iii) By construction N /NS(X)vert = Z[Ω] ≃ Z.

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(iv) As in Part (4) of the proof in [Go], we have an exact sequence 0 −→

Div0 (X) −→ Div0 (X) ∩ (Divvert (X)) + Dℓ (X)) Div0 (X) NS(X) −→ −→ −→ 0. Divvert (X) + Dℓ (X) N

The first term in this sequence is identified with B(k)/E in (3.1) since Dℓ (X) ⊆ Div0 (X). The middle term is identified with Pic0 (XK ) in 3.6. We thus have an isomorphism NS(X)/N −→

Pic0 (XK ) . B(k)/E

Proposition 3.9. (see Proposition 4.6 in [Go]) Let N0 ⊆ NS(X) be as in 3.7. Then the quotient NS(X)/N0 is finite with Q | Pic0 (XK )tors | v∈Σ pv1 |NS(X)/N0| = · . |B(k)| |NS(X)tors | Proof. Let N ′ be the subgroup of NS(X) generated by the classes of Ω, F, and Xva for a > 1 and h(v) > 1, so that N ′ ⊆ N0 . Recall that N := NS(X)vert ⊕Z[Ω], so that N ′ ⊆ N . We have two exact sequences 0

N /N ′

0 /

N0 /N ′ /

A′ := NS(X)/N ′ /

NS(X)/N0 /

0

P := NS(X)/N

0 Let us start by computing the order of N /N ′. Write N ′′ for the subgroup of N ′ generated by the classes of F , and Xva , a > 1 for all v with h(v) > 1. Then N ′′ ⊆ NS(X)vert and N ′ = N ′′ ⊕ Z[Ω]. It follows that NS(X)vert NS(X)vert /f ∗ NS(V ) N = = . N′ N ′′ (N ′′ + f ∗ NS(V ))/f ∗ NS(V )

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The numerator of the group on the right is identified with Q/E in 3.8. One checks that N ′′ ∩ f ∗ NS(V ) = Z[F ]. With the group Q identified as in (3.2), let Q′ denote the subgroup of Q generated by the classes of the components Xva with a > 1 for all v with h(v) > 1. Then the ′ denominator in the above Q expression is equal ′to Q and it is clear that ′ Q/Q is isomorphic to v∈Σ Z/pv1 Z. Since Q is torsion free and E is torsion, we find that

N /N ′ ≃ (Q/E)/Q′ ≃ Q/(Q′ + E), Q so that N /N ′ is finite, of order ( v∈Σ pv1 )/|E|. Recall now from 3.8 that P ≃ Pic0 (XK )/(B(k)/E). Since B(k)/E is finite, we find that |Ptors | = | Pic0 (XK )tors |/|B(k)/E|,

(3.4)

and we also have a canonical isomorphism Pic0 (XK )/ Pic0 (XK )tors −→ P/Ptors .

(3.5)

Since the group N /N ′ is finite, we find that |A′tors | = |N /N ′| · |Ptors |

(3.6) and that

A′ /A′tors −→ P/Ptors

(3.7)

is an isomorphism. By construction, the classes of the restrictions of A1 , . . . , Ar to the generic fiber are a basis of Pic0 (XK )/ Pic0 (XK )tors . Using the isomorphisms (3.5) and (3.7), we find that the classes of A1 , . . . , Ar are a basis of A′ /A′tors . This implies that NS(X)/N0 is torsion and that 0 −→ (N0 /N ′ )tors −→ A′tors −→ NS(X)/N0 −→ 0 is exact. It is clear that N0 = (h[A1 ], . . . , [Ar ]i + NS(X)tors ) ⊕ N ′. It follows that

A′tors . NS(X)tors The desired formula for the index follows from (3.4) and (3.6). NS(X)/N0 =

3.10. Let N 0 be the image of N0 in the lattice NS(X)/NS(X)tors . The computation of the discriminant of the intersection pairing on the sublattice N 0 is done exactly as in Proposition 5.1 of [Go], and the formula obtained is the same. The only difference now is that the discriminant of the height pairing | det hαi , αj i | that appears in the formula is the discriminant for the height pairing on Pic0 (XK )/ Pic0 (XK )tors ,

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and not anymore on AK (K)/AK (K)tors . Let af denote the index of Pic0 (XK )/ Pic0 (XK )tors in AK (K)/AK (K)tors . As indicated in Lemma 3.2, the two discriminants differ by a factor a2f . Similarly, the discriminant of the intersection pairing on N 0 differs from the discriminant of the intersection pairing on the full lattice NS(X)/NS(X)tors by the square of the index Q | Pic0 (XK )tors | v∈Σ pv1 · |B(k)| |NS(X)tors | obtained in 3.9. This index is exactly the same as the one obtained [Go], except that in [Go], the term | Pic0 (XK )tors | is replaced by |AK (K)tors |. Let ators := |AK (K)tors / Pic0 (XK )tors |. We have a = af ators , and we find that the final discrepancy is a factor of a2 . Remark 3.11. We supply in this remark some references for an important result stated just before Proposition 5.1 of [Go], and needed in its proof. Let α, β in Pic0 (XK )/ Pic0 (XK )tors . The global height P pairing hα, βi can be computed as a sum of local contributions v hα, βiv (see, e.g., [6], (4.6)). Each local contribution can be expressed as a local intersection number hα, βiv = −(α, β)v log(|k(v)|) (see, e.g., [6], (3.7)), where the contribution (α, β)v is the value of N´eron’s pairing at v on α and β. Let A, B ∈ Div(X)⊗Q be two divisors whose restrictions to XK are in Div(XK ) and equal the classes α and β, respectively, and have disjoint supports. Assume in addition that (A · Xva )X = 0 for all v and all a. Then (α, β)v = (A · B)v , where (A · B)v denotes the contribution of the points in Xv in the intersection number (A · B)X (see, e.g., [2], 4.3, or [10], 2.2). One then obtains that hα, βi = −(A · B)X log(|k|). 3.12. We recall below the formula of Gordon found in the middle of page 196 in [Go]. This formula is claimed to hold exactly when the Birch–Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture. This claim is incorrect when a > 1. In [Go], page 169, the integer α appearing below is defined to be the index δ. (3.8)

|X(AK )|

Y

d2v ǫv = α2 |Br(X)|.

v

This formula in [Go] is misleading, as the term ǫv is only introduced in the statement of Proposition 5.5 of [Go] when v ∈ S1 , but the formula (6.2) in [Go], from which (3.8) above is derived, involves a product over a set S (defined on page 165 of [Go]) which contains S1 , but which might also contain S2 and S3 (notation introduced in 3.7). Let us therefore state below the correct formula (3.9) that can

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be inferred from Gordon’s work and which should be substituted for (3.8). Let Av /OKv denote the N´eron model of AKv /Kv . Let Φv /k(v) denote the group of components of the special fiber of Av . When v ∈ S2 ⊔ S3 , the fiber Xv is irreducible, say Xv = dv Γv for some irreducible curve Γv /k(v). Let qv denote the degree over k(v) of the algebraic closure of k(v) in the function field of Γv /k(v). It follows from the fact that k(v) is a finite field that δv = dv qv . Note that if v ∈ / S1 ⊔ S2 ⊔ S3 , ′ then δv = δv = 1. Then Gordon’s arguments, along with the removal of the hypothesis that X → V be cohomologically flat in dimension 0 in [LLR1] and the corrections given in this corrigendum, give the following formula.

(3.9) |X(AK )|

Y

v∈S1

d2v ǫv

!

Y

d2v |Φv (k(v))|qv

v∈S2 ⊔S3

!

= a2 δ 2 |Br(X)|.

The formula can be turned into Formula (2.1) as we did in the proof of Theorem 4.3 in [LLR1], using Theorem 1.17 of [1]. For instance, when v ∈ S2 ⊔ S3 , this theorem shows that |Φv (k(v))| = δv′ /dv . Since it follows from the adjunction formula that dv qv divides g − 1 in this case, Theorem 7 in [9] shows that δv = δv′ . It follows that d2v |Φv (k(v))|qv = δv δv′ , as desired, and Formula (2.1) is established. References [1] S. Bosch and Q. Liu, Rational points on the group of components of a N´eron model, Manuscripta Math. 98 (1999), 275–293. [2] S. Bosch, and D. Lorenzini, Grothendieck’s pairing on component groups of Jacobians, Invent. Math. 148 (2002), no. 2, 353–396. [3] B. Conrad, Chow’s K/k-image and K/k-trace, and the Lang–N´eron theorem, Enseign. Math. (2) 52 (2006), no. 1-2, 37–108. [4] T. Geisser, Comparing the Brauer group to the Tate-Shafarevich group, to appear in J. Inst. Math. Jussieu, arXiv:1712.06249v2 [5] C. Gonzalez-Aviles, Brauer groups and Tate–Shafarevich groups, J. Math. Sci. Univ. Tokyo 10 (2003), 391-419. [Go] W. Gordon, Linking the conjectures of Artin–Tate and Birch–SwinnertonDyer, Comp. Math. 38 (1979), 163–199. [6] B. Gross, Local heights on curves, Arithmetic geometry (Storrs, Conn., 1984), 327–339, Springer, New York, 1986. [7] A. Grothendieck, Le groupe de Brauer III, Exemples et compl´ements, (French) Dix expos´es sur la cohomologie des sch´emas, 88–188, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968. [8] S. Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983.

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[9] S. Lichtenbaum, Duality theorems for curves over p-adic fields, Invent. Math. 7 (1969), 120–136. [LLR1] Q. Liu, D. Lorenzini, and M. Raynaud, N´eron models, Lie algebras, and reduction of curves of genus one, Invent. Math 157 (2004), 455-518. [LLR2] Q. Liu, D. Lorenzini, and M. Raynaud, The Brauer group of a surface, Invent. Math. 159 (2005), 673-676. [10] C. P´epin, N´eron’s pairing and relative algebraic equivalence, Algebra Number Theory 6 (2012), no. 7, 1315–1348. [11] T. Shioda, Mordell-Weil lattices for higher genus fibration over a curve, New trends in algebraic geometry (Warwick, 1996), 359–373, London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press, Cambridge, 1999. [12] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analogue, S´eminaire Bourbaki 1965/66, Expos´e 306, Benjamin, New York. Institut de Math´ ematiques de Bordeaux, CNRS UMR 5251, Univer´ de Bordeaux, 33405 Talence cedex, France site E-mail address: [email protected] Department of Mathematics, University of Georgia, Athens, GA 30602, USA E-mail address: [email protected] e Paris-Sud, 91405 Orsay ematiques, Universit´ Laboratoire de Math´ Cedex, FRANCE