THE BRAUER-MANIN OBSTRUCTION FOR SUBVARIETIES OF ABELIAN VARIETIES OVER FUNCTION FIELDS

BJORN POONEN AND JOSÉ FELIPE VOLOCH

Abstract. We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.

1. Introduction

The notation in this section remains in force throughout the paper, except in Section 3.3, and in Section 4 where we allow also the possibility that K is a number field.

Let k be a field. Let K be a finitely generated extension of k of transcendence degree 1. We assume that k is relatively algebraically closed in K, since the content of our theorems will be unaffected by this assumption. Let \overline{K} be an algebraic closure of K. We will use this notation consistently for an algebraic closure, and we will choose algebraic closures compatibly whenever possible: thus \overline{k} is the algebraic closure of k in \overline{K}. Let K^{s} be the separable closure of K in \overline{K}. Let Ω_{all} be the set of all nontrivial valuations of K that are trivial on k. Let Ω be a cofinite subset of Ω_{all}; if k is finite, we may weaken the cofiniteness hypothesis to assume only that Ω ⊆ Ω_{all} has Dirichlet density 1. For each v ∈ Ω, let K_{v} be the completion of K at v, and let F_{v} be the residue field. Equip K_{v} with the v-adic topology. Define the ring of adèles A as the restricted direct product ∏_{v∈Ω}(K_{v}, O_{v}) of the K_{v} with respect to their valuation subrings O_{v}. Then A is a topological ring, in which ∏_{v∈Ω} O_{v} is open and has the product topology.

If A is an abelian variety over K, then A(K) embeds diagonally into A(A) ≃ ∏_{v} A(K_{v}). Define the adelic topology on A(K) as the topology induced from A(A). For any fixed v define the v-adic topology on A(K) as the topology induced from A(K_{v}). Let \overline{A(K)} be the closure of A(K) in A(A).

For any extension of fields F′ ⊃ F and any F -variety X, let X_{F′} be the base extension of X to F′. Call a K-variety X constant if X ≅ Y_{K} for some k-variety Y, and call X isotrivial if X\overline{K} ≅ Y\overline{K} for some variety Y defined over \overline{k}.

From now on, X is a closed K-subscheme of A. Call X coset-free if X\overline{K} does not contain a translate of a positive-dimensional abelian subvariety of A\overline{K}.

Date: July 20, 2008.

2000 Mathematics Subject Classification. Primary 11G35; Secondary 14G25, 14K12.

Key words and phrases. Brauer-Manin obstruction, Mordell-Lang conjecture, abelian varieties, function fields.

When k is finite and Ω = Ω_{all}, the intersection X(A) ∩ \overline{A(K)} ⊂ A(A) is closely related to the Brauer-Manin obstruction to the Hasse principle for X/K: see Section 4. For curves over number fields, V. Scharaschkin and A. Skorobogatov independently raised the question of whether the Brauer-Manin obstruction is the only obstruction to the Hasse principle, and proved that this is so when the Jacobian has finite Mordell-Weil group and finite Shafarevich-Tate group. The connection with the adelic intersection is stated explicitly in [Sch99], and is based on global duality statements originating in the work of Cassels: see Remark 4.4. See also [Sko01,Fly04,Poo06,Sto06]: the last of these in particular contains many conjectures and theorems relating descent information, the method of Chabauty and Coleman, the Brauer-Manin obstruction, and Grothendieck’s section conjecture.

In this paper we answer (most cases of) a generalization of the function field analogue of a question raised for curves over number fields in [Sch99], concerning whether the Brauer-Manin condition cuts out exactly the rational points: see Theorem D. This question is still wide open in the number field case. Along the way, we prove results about adelic intersections along the lines of the “adelic Mordell-Lang conjecture” suggested in [Sto06, Question 3.12]. Again, these are open in the number field case. In particular, we prove the following theorems.

Theorem A. If char k = 0, then X(K) = X(A) ∩ \overline{A(K)}.

Theorem B. Suppose that char k = p > 0, that A\overline{K} has no nonzero isotrivial quotient, and that A(K^{s})[p^{∞}] is finite. Suppose that X is coset-free. Then X(K) = X(A) ∩ \overline{A(K)}.

Remark 1.1. The proposition in [Vol95] states that in the “general case” in which A is ordinary and the Kodaira-Spencer class of A/K has maximal rank, we have A(K^{s})[p^{∞}] = 0.

Conjecture C. For any closed K-subscheme X of any A, we have \overline{X(K)} = X(A) ∩ \overline{A(K)}, where \overline{X(K)} is the closure of X(K) in X(A).

Remark 1.2. If A\overline{K} has no nonzero isotrivial quotient and X is coset-free, then X(K) is finite [Hru96, Theorem 1.1] so \overline{X(K)} = X(K). So Conjecture C predicts in particular that the hypothesis on A(K^{s})[p^{∞}] in Theorem B is unnecessary.

Remark 1.3. Here is an example to show that the statement X(K) = X(A) ∩ \overline{A(K)} can fail for a constant curve in its Jacobian. Let X be a curve of genus ≥ 2 over a finite field k. Choose a divisor of degree 1 on X to embed X in its Jacobian A. Let F: A → A be the k-Frobenius map. Let K be the function field of X. Let P ∈ X(K) be the point corresponding to the identity map X → X. Let P_{v} ∈ X(F_{v}) be the reduction of P at v.

For each v, the Teichmüller map F_{v} → K_{v} identifies F_{v} with a subfield of K_{v}. Any Q ∈ A(K_{v}) can be written as Q = Q_{0} +Q_{1} with Q_{0} ∈ A(F_{v}) and Q_{1} in the kernel of the reduction map A(K_{v}) → A(F_{v}); then lim_{m→∞} F^{m}(Q_{1}) = 0, so lim_{n→∞} F^{n!}(Q) = lim_{n→∞} F^{n!}(Q_{0}) = Q_{0}. In particular, taking Q = P, we find that (F^{n!}(P))_{n≥1} converges in A(A) to the point (P_{v}) ∈ X(A) = ∏_{v} X(K_{v}), where we have identified P_{v} with its image under the Teichmüller map X(F_{v}) ↪ X(K_{v}). If (P_{v}) were in X(K), then in X(K_{v}) we would have P_{v} ∈ X(F_{v}) ∩ X(K) = X(k), which contradicts the definition of P_{v} if v is a place of degree greater than 1 over k. Thus (P_{v}) is in X(A) ∩ \overline{A(K)} but not in X(K).

In the final section of this paper, we restrict to the case of a global function field, and extend Theorem B to prove (under mild hypotheses) that for a subvariety of an abelian variety, the Brauer-Manin condition cuts out exactly the rational points: see Section 4 for the definitions of X(A)^{Br} and Ŝel. Our result is as follows.

Theorem D. Suppose that K is a global function field of characteristic p, that A\overline{K} has no nonzero isotrivial quotient, and that A(K^{s})[p^{∞}] is finite. Suppose that X is coset-free. Then X(K) = X(A)^{Br} = X(A) ∩ Ŝel.

To our knowledge, Theorem D is the first result giving a wide class of varieties of general type such that the Brauer-Manin condition cuts out exactly the rational points.

2. Characteristic 0

Throughout this section, we assume char k = 0. In this case, results follow rather easily.

Proposition 2.1. For any v, the v-adic topology on A(K) equals the discrete topology.

Proof. The question is isogeny-invariant, so we reduce to the case where A is simple. Let A(F_{v}) denote the group of F_{v}-points on the Néron model of A over O_{v}. Let A^{1}(K_{v}) be the kernel of the reduction map A(K_{v}) → A(F_{v}). The Lang-Néron theorem [LN59, Theorem 1] implies that either A is constant and A(K)/A(k) is finitely generated, or A is non-constant and A(K) itself is finitely generated. In either case, the subgroup A^{1}(K) := A(K) ∩ A^{1}(K_{v}) is finitely generated. By the theory of formal groups (cf. [Ser92, p. 118, Theorem 2]), A^{1}(K_{v}) has a descending filtration by open subgroups in which the quotients of consecutive terms are torsion-free (this is where we use char k = 0), so the induced filtration on the finitely generated group A^{1}(K) has only finitely many nonzero quotients. Thus A^{1}(K) is discrete. Since A^{1}(K_{v}) is open in A(K_{v}), the group A(K) is discrete in A(K_{v}). □

Remark 2.2. The literature contains results close to Proposition 2.1. It is mentioned in the third subsection of the introduction to [Man63] for elliptic curves with non-constant j-invariant, and it appears for abelian varieties with K/k-trace zero in [BV93].

Corollary 2.3. The adelic topology on A(K) equals the discrete topology.

Proof. The adelic topology is at least as strong as (i.e., has at least as many open sets as) the v-adic topology for any v. □

We can improve the result by imposing conditions in only the residue fields F_{v} instead of the completions K_{v}, that is, “flat” instead of “deep” information in the sense of [Fly04]. In fact, we have:

Proposition 2.4. There exist v, v′ ∈ Ω of good reduction for A such that A(K) → A(F_{v}) × A(F_{v′}) is injective.

Proof. Let B be the K/k-trace of A. Pick any v ∈ Ω of good reduction. The kernel H of A(K) → A(F_{v}) meets B(k) trivially. By Silverman’s specialization theorem [Lan83, Chapter 12, Theorem 2.3], there exists v′ ∈ Ω such that H injects under reduction modulo v′. □

Proof of Theorem A. By Corollary 2.3, X(A) ∩ \overline{A(K)} = X(A) ∩ A(K) = X(K). □

3. Characteristic p

Throughout this section, char k = p.

3.1. Field-theoretic lemmas.

Lemma 3.1. For any v, if α ∈ K_{v} is algebraic over K, then α is separable over K.

Proof. Replacing K by its relative separable closure in L := K(α), we may assume that L is purely inseparable over K. Then the valuation v on K admits a unique extension w to L, and the inclusion of completions K_{v} → L_{w} is an isomorphism. By [Ser79, I.§4, Proposition 10] (hypothesis (F) there holds for localizations of finitely generated algebras over a field), we have an “n = ∑ e_{i}f_{i}” result, which in our case says [L : K] = [L_{w} : K_{v}] = 1. So α ∈ K. □

If L is a finite extension of K, let A_{L} be the corresponding ring of adèles, defined as a restricted direct product over places of L. There is a natural inclusion A ↪ A_{L}.

Lemma 3.2. Let L be a finite extension of K. Then in A_{L} we have A ∩ L = K.

Proof. Fix v ∈ Ω. By [Bou98, VI.§8.5, Corollary 3] and the fact that hypothesis (F) in the previous proof holds, the natural map K_{v} ⊗_{K} L → ∏_{w|v} L_{w} is an isomorphism. Hence in ∏_{w|v} L_{w} we have K_{v} ∩ L = K. The result follows. □

3.2. Abelian varieties.

Lemma 3.3. For any n ∈ Z_{≥1}, the quotient A(K_{v})/nA(K_{v}) is Hausdorff.

Proof. Equivalently, we must show that nA(K_{v}) is closed in A(K_{v}). Suppose (P_{i}) is a sequence in nA(K_{v}) that converges to P ∈ A(K_{v}). Write P_{i} = nQ_{i} with Q_{i} ∈ A(K_{v}). Then n(Q_{i} − Q_{i+1}) → 0 as i → ∞.

Let O_{v} be the valuation ring of K_{v}, and let A over be the Néron model of A over O_{v}. Applying [Gre66, Corollary 1] to A[n] shows that for any sequence (R_{i}) in A(K_{v}) with nR_{i} → 0, the distance of R_{i} to the nearest point of A(K_{v})[n] tends to 0.

Thus by induction on i we may adjust each Q_{i} by a point in A(K_{v})[n] so that Q_{i}−Q_{i+1} → 0 as i → ∞. Since A(K_{v}) is complete, (Q_{i}) converges to some Q ∈ A(K_{v}), and nQ = P. Thus nA(K_{v}) is closed. □

Remark 3.4. In the case where k is finite, Lemma 3.3 is immediate since A(K_{v}) is compact and its image under multiplication-by-n is closed.

The following is a slight generalization of [Vol95, Lemma 2], with a more elementary proof.

Proposition 3.5. If A(K^{s})[p^{∞}] is finite, then for any v, the v-adic topology on A(K) is at least as strong as the topology induced by all subgroups of finite p-power index.

Proof. For convenience choose algebraic closures \overline{K}, \overline{K}_{v} of K, K_{v} such that K^{s} ⊆ \overline{K} ⊆ \overline{K}_{v}. As in the proof of Proposition 2.1, there is an open subgroup U of A(K_{v}) such that B := A(K) ∩ U is finitely generated. It suffices to show that for every e ∈ Z_{≥0}, there exists an open subgroup V of A(K_{v}) such that B ∩ V ⊆ p^{e}A(K).

Choose m such that p^{m}A(K^{s})[p^{∞}] = 0. Let M = e + m. Then B/p^{M}B is finite. By Lemma 3.3, A(K_{v})/p^{M}A(K_{v}) is Hausdorff, so the image of B/p^{M}B in A(K_{v})/p^{M}A(K_{v}) is discrete. Hence there is an open subgroup V of A(K_{v}) such that B ∩ V = ker(B → A(K_{v})/p^{M}A(K_{v})).

Suppose b ∈ B ∩ V. Then b = p^{M}c for some c ∈ A(K_{v}) ∩ A(\overline{K}). By Lemma 3.1, we obtain c ∈ A(K^{s}). If σ ∈ Gal(K^{s}/K), then ^{σ}c − c ∈ A(K^{s})[p^{M}] is killed by p^{m}. Thus p^{m}c ∈ A(K). So b = p^{e}p^{m}c ∈ p^{e}A(K). Hence B ∩ V ⊆ p^{e}A(K). □

Proposition 3.6. The adelic topology on A(K) is at least as strong as the topology induced by all subgroups of finite index.

Proof. As in the proof of Proposition 2.1, the Lang-Néron theorem implies that A(A) has an open subgroup whose intersection with A(K) is finitely generated. It suffices to study the topology induced on that finitely generated subgroup, so we may reduce to the case in which k is finitely generated over a finite field F_{q}. This case is proved in [Mil72], which adapts and extends [Ser64] and [Ser71]. (The paper [Mil72] uses not the adelic topology as we have defined it, but the topology coming from the closed points of a finite-type Z-scheme with function field K. Since the adelic topology is stronger, [Mil72] contains what we want.) □

Lemma 3.7. Suppose that A(K) is finitely generated. Then (\overline{A(K)})_{tors} = A(K)_{tors}.

Proof. Let

T := ker (A(K) → ∏_{v∈Ω} \frac{A(F_{v})}{A(F_{v})_{tors}} ) ,

where A(F_{v}) is the group of F_{v}-points on the Néron model of A. Since A(K) is finitely generated and the groups A(F_{v})/A(F_{v})_{tors} are torsion-free, there is a finite subset S ⊂ Ω such that T = A(K) ∩ U for the open subgroup

U := ker (A(A) → ∏_{v∈S} \frac{A(F_{v})}{A(F_{v})_{tors}} )

of A(A). The finitely generated group A(K)/T is contained in the torsion-free group ∏_{v∈S} \frac{A(F_{v})}{A(F_{v})_{tors}} , so A(K)/T is free, and we have A(K) ≅ T ⊕ F as topological groups, where F is a discrete free abelian group of finite rank.

We claim that the topology of T is that induced by the subgroups nT for n ≥ 1. For n ≥ 1, the subgroup nT is open in T by Proposition 3.6. If t ∈ T, then some positive multiple of t is in the kernel of A(K_{v}) → A(F_{v}), and then p-power multiples of this multiple tend to 0. Applying this to a finite set of generators of T, we see that any open neighborhood of 0 in T contains nT for some n ∈ Z_{>0}.

It follows that \overline{T} ≅ T ⊗ Ẑ. Now

(\overline{A(K)})_{tors} = (\overline{T} ⊕ F)_{tors} = \overline{T}_{tors} ≅ (T ⊗ Ẑ)_{tors} ≅ T_{tors}.
□

Remark 3.8. When k is finite, an easier proof of Lemma 3.7 is possible: combined with the fact that A(A) is profinite, Proposition 3.6 implies that \overline{A(K)} ≅ A(K) ⊗ Ẑ; the torsion subgroup of the latter equals A(K)_{tors}.

The following proposition is a function field analogue of [Sto06, Proposition 3.6]. Our proof must be somewhat different, however, since [Sto06] made use of strong “image of Galois” theorems whose function field analogues have recently been disproved [Zar06].

Proposition 3.9. Suppose that A(K^{s})[p^{∞}] is finite. Let Z be a finite K-subscheme of A. Then Z(A) ∩ \overline{A(K)} = Z(K).

Proof. In this first paragraph we show that replacing K by a finite extension L does not destroy the hypothesis that A(K^{s})[p^{∞}] is finite. This is obvious if L is separable over K, so assume that L is purely inseparable. Choose n ∈ Z_{≥0} with L^{p^{n}} ⊆ K. Then (L^{s})^{p^{n}} ⊆ K^{s}, so p^{n}A(L^{s})[p^{∞}] ⊆ A(K^{s})[p^{∞}]. Thus p^{n}A(L^{s})[p^{∞}] is finite. But multiplication-by-p^{n} has finite fibers, so A(L^{s})[p^{∞}] itself is finite.

Next we claim that if we prove the conclusion after base extension to a finite extension L, then the desired conclusion over K holds. Namely, suppose that we prove Z(A_{L}) ∩ \overline{A(L)} = Z(L). Then

Z(A) ∩ \overline{A(K)} ⊆ Z(A_{L}) ∩ \overline{A(L)} = Z(L)

so

Z(A) ∩ \overline{A(K)} ⊆ Z(A) ∩ Z(L) = Z(K),

where the last equality uses Lemma 3.2.

Thus we may replace K by a finite extension to assume that Z consists of a finite set of K-points of A. (The same idea was used in [Sto06].) A point P ∈ \overline{A(K)} is represented by a sequence (P_{n})_{n≥1} in A(K) such that for every v, the limit lim_{n→∞} P_{n} exists in A(K_{v}). If in addition P ∈ Z(A), then there is a point Q_{v} ∈ Z(K) whose image in Z(K_{v}) equals lim_{n→∞} P_{n} ∈ A(K_{v}). The P_{n} − Q_{v} are eventually contained in the kernel of A(K) → A(F_{v}), which is finitely generated, so there are finitely generated subfields k_{0} ⊆ k, K_{0} ⊆ K with K_{0}/k_{0} a function field such that all the P_{n} and the points of Z(K) are in A(K_{0}). By Proposition 3.5, the sequence (P_{n} − Q_{v})_{n≥1} is eventually divisible in A(K_{0}) by an arbitrarily high power of p. For any other v′ ∈ Ω, then the same is true of (P_{n} − Q_{v′})_{n≥1}. Then Q_{v′} − Q_{v} ∈ A(K_{0}) is divisible by every power of p. Since A(K_{0}) is finitely generated, Q_{v′} − Q_{v} is a torsion point in A(K_{0}). This holds for every v′ ∈ Ω, and A(K_{0})_{tors} is finite, so R := P − Q_{v} ∈ \overline{A(K_{0})} is a torsion point in \overline{A(K_{0})}. Lemma 3.7 applied to K_{0} yields R ∈ A(K_{0})_{tors}. Thus P = R + Q_{v} ∈ A(K). So P ∈ Z(A) ∩ A(K) = Z(K). □ Lemma 3.10. Fix v ∈ Ω. Let Γ_{v} be the closure of A(K) in A(K_{v}). Then for every e ∈ Z_{≥0}, the map A(K)/p^{e}A(K) → Γ_{v}/p^{e}Γ_{v} is surjective. Proof. Let O_{v} be the valuation ring of K_{v}, and let 𝔪_{v} be its maximal ideal. Let A over O_{v} be the Néron model. For r ∈ Z_{≥1}, let G_{r} be the kernel of A(K_{v}) = A(O_{v}) → A(O_{v}/𝔪_{v}^{r}). It follows from [Ser92, p. 118, Theorem 2] that G_{r}/G_{r+1} is isomorphic to (O_{v}/𝔪_{v})^{dim A}, which is killed by p, so each G_{r} is an abelian pro-p-group, and hence a topological Z_{p}-module. There are only finitely many points of order p in A(K_{v}), and ⋂_{r≥1} G_{r} = {0}, so some G_{r} contains no nontrivial p-torsion points, and hence is torsion-free. In particular, A(K_{v}) has an open subgroup A^{∘}(K_{v}) that is a torsion-free topological Z_{p}-module, and we may choose A^{∘}(K_{v}) so that A^{∘}(K) := A(K) ∩ A^{∘}(K_{v}) is finitely generated.

The group Γ_{v}^{∘} := Γ_{v} ∩ A^{∘}(K_{v}) is the closure of A^{∘}(K), so there is an isomorphism of topological groups Γ_{v}^{∘} ≅ Z_{p}^{⊕m} for some m ∈ Z_{≥0}. In particular, for any e ∈ Z_{≥0}, the group p^{e}Γ_{v}^{∘} is open in Γ_{v}^{∘}, which is open in Γ_{v}. So the larger group p^{e}Γ_{v} also is open in Γ_{v}. But the image of A(K) in the discrete group Γ_{v}/p^{e}Γ_{v} is dense, so the map A(K)/p^{e}A(K) → Γ_{v}/p^{e}Γ_{v} is surjective. □

3.3. A uniform Mordell-Lang conjecture. We thank Zoé Chatzidakis, Françoise Delon, and Thomas Scanlon for many of the ideas used in this section. See [Del98] for the definitions of separable, p-basis, p-free, p-components, etc. By iterated p-components we mean p-components of p-components of … of p-components (all with respect to a given p-basis).

The goal of this section is to deduce a uniform version (Proposition 3.16) of the function field Mordell-Lang conjecture from a version in [Hru96]. Under some hypotheses, the uniform version asserts the finiteness of the intersection of a subvariety X of an abelian variety A with any coset of a subgroup p^{e}A(F) of A(F), where F is allowed to range over p-basis-preserving extensions of an initial ground field K.

Remark 3.11. The p-basis condition on F, or something like it, is necessary for the truth of Proposition 3.16: with no condition, F might be algebraically closed, and then p^{e}A(F) = A(F), so the desired finiteness would fail assuming dim X > 0. The p-basis condition is used in the proof of Proposition 3.16 to imply separability of F over K, which guarantees that a non-isotriviality hypothesis on A over K is preserved by base extension to F: see Lemma 3.13 and its proof.

Lemma 3.12. Let B be a p-basis for a field K of characteristic p. Let L be an extension of K such that B is also a p-basis for L. Suppose that c is an element of L that is not algebraic over K. Then there exists a separably closed extension F of L such that B is a p-basis of F and the Aut(F/K)-orbit of c is infinite.

Proof. Fix a transcendence basis T for L/K. Let Ω be an algebraically closed extension of K such that the transcendence basis of Ω/K is identified with the set Z×T. Identify L with a subfield of Ω in such a way that each t ∈ T maps to the transcendence basis element for Ω/K labelled by (0, t) ∈ Z × T. The map of sets Z × T → Z × T mapping (i, t) to (i + 1, t) extends to an automorphism σ ∈ Aut(Ω/K). For i ∈ Z, let L_{i} = σ^{i}(L). Let L_{∞} be the compositum of the L_{i} in Ω. Then σ(L_{∞}) = L_{∞}. Let F be the separable closure of L_{∞} in Ω. So σ(F) = F. The σ-orbit of c is infinite, since L_{i} ∩ L_{j} is algebraic over K whenever i ≠ j.

The p-basis hypothesis implies that L is separable over K. Applying σ^{i} shows that L_{i} is separable over K. Moreover, the L_{i} are algebraically disjoint over K, so their compositum L_{∞} is separable over K, by the last corollary in [Bou03, V.§16.7] and Proposition 3(b) in [Bou03, V.§15.2]. Thus B is p-free in L_{∞}.

The p-basis hypothesis implies also that L = L^{p}(B). Thus L_{i} = L_{i}^{p}(B) for all i ∈ Z, and L_{∞} = L_{∞}^{p}(B).

Combining the previous two paragraphs shows that B is a p-basis for L_{∞}, and hence also for F. □

Lemma 3.13. Let K ⊆ F be a separable extension such that the field K^{p^{∞}} := ⋂_{n≥1} K^{p^{n}} is algebraically closed. Let A be an abelian variety over K such that no nonzero quotient of A\overline{K} is the base extension of an abelian variety over K^{p^{∞}}. Then no nonzero quotient of A\overline{F} is the base extension of an abelian variety over F^{p^{∞}}.

Proof. Suppose not. Thus there exists a nonzero abelian variety B over F^{p^{∞}} and a surjective homomorphism ϕ: A\overline{F} → B\overline{F}. Choose a finitely generated extension F_{0} of K^{p^{∞}} over which B is defined. Whenever A and B are abelian varieties over a field L, any homomorphism A → B is definable over L(A[ℓ^{∞}], B[ℓ^{∞}]) for any prime ℓ ≠ char L; thus, in our situation, ϕ is definable over a finite separable extension F_{1} of KF_{0}. Since K^{p^{∞}} is algebraically closed, we may choose a place F_{0} −−→ K^{p^{∞}} extending the identity on K^{p^{∞}}, such that B has good reduction at this place. By [Del98, Fact 1.4], K and F^{p^{∞}} are linearly disjoint over K^{p^{∞}}, so the place F_{0} −−→ K^{p^{∞}} extends to a place KF_{0} −−→ K that is the identity on K, and then to a place F_{1} −−→ K_{1} for some finite extension K_{1} of K. Reduction of ϕ: A_{F_{1}} → B_{F_{1}} yields a homomorphism A_{K_{1}} → B_{K_{1}}, and since the place restricted to F_{0} has values in K^{p^{∞}}, the abelian variety B_{K_{1}} is the base extension of an abelian variety over K^{p^{∞}}. This contradicts the hypothesis on A. □

Lemma 3.14 (a version of the Mordell-Lang conjecture). Let F be a separably closed field of characteristic p. Suppose that A is an abelian variety over F such that no nonzero quotient of A\overline{F} is the base extension of an abelian variety over F^{p^{∞}}. Suppose that X is a coset-free closed F -subvariety of A. Then there exists e ∈ Z_{≥1} such that for every a ∈ A(F), the intersection X(F) ∩ (a + p^{e}A(F)) is finite.

Proof. This is a special case of [Hru96, Lemma 6.2]. □

Using Lemma 3.14 and the compactness theorem in model theory, we can deduce a version of the Mordell-Lang conjecture that is more uniform as we vary the ground field, Proposition 3.16.

But first we introduce some notation. Suppose that K is a field of characteristic p such that [K : K^{p}] < ∞. Let F be any field extension with the same p-basis. Suppose that A is an abelian variety over K. Let R_{n} be the restriction of scalars of A from K to K^{p^{n}}, so R_{n} is an abelian variety over K^{p^{n}}. Let A_{n} be the base extension of R_{n} by the isomorphism K^{p^{n}} → K that takes an element to its p^{n}-th root, so A_{n}(F) = R_{n}(F^{p^{n}}) = A(F). For n ∈ Z_{≥0}, we have a natural morphism A_{n+1} → A_{n} such that A_{n+1}(F) → A_{n}(F) is compatible with the identifications A_{n+1}(F) = A(F) and A_{n}(F) = A(F).

Suppose moreover that X is a K-variety. Let Y_{n} be the set of (not necessarily closed) K-subvarieties Y ⊆ A_{n} × X such that the projection from Y to A_{n} has finite fibers. For such Y, use the identification A_{n}(F) = A(F) to view Y (F) as a subset of A(F) × X(F). Taking inverse images under A_{n+1} × X → A_{n} × X defines a map of sets Y_{n} → Y_{n+1}. Let Y = \underrightarrow{lim} Y_{n}. For Y ∈ Y, the set Y (F) ⊆ A(F) is independent of which Y_{n} we consider Y as coming from. Each Y_{n} is closed under taking finite unions of elements, so the same is true of Y.

Remark 3.15. Alternatively, one can think of Y as a “variety” defined by not only polynomial equations and inequations in the coordinates on A and X, but by polynomial equations and inequations in the iterated p-components of coordinates on A and usual coordinates on X.

Proposition 3.16 (more uniform version of the Mordell-Lang conjecture). Let k be an algebraically closed field of characteristic p. Let K be a finitely generated extension of k. Fix a (finite) p-basis B of K. Suppose that A is an abelian variety over K such that no nonzero quotient of A\overline{K} is the base extension of an abelian variety over k. Suppose that X is a coset-free closed K-subscheme of A. Define Y as in the preceding paragraph. Then there exists e ∈ Z_{≥1} and Y ∈ Y such that for every field extension F ⊇ K having B as p-basis, if a ∈ A(F) and c ∈ X(F) ∩ (a + p^{e}A(F)), then (a, c) ∈ Y (F).

Proof. Consider the language of fields augmented by a constant symbol α_{κ} for each element κ ∈ K and by additional finite tuples of constant symbols a and c (to represent coordinates of points on A and X, respectively). We construct a theory T in this language. Start with the field axioms, and the arithmetic sentences involving the α_{κ} that hold in K. Include a sentence saying that B is a p-basis for the universe field F. Include sentences saying that a, c ∈ A(F) (for some fixed representation of A as a definable set). For each e ∈ Z_{≥1}, include a sentence saying that

c ∈ X(F) ∩ (a + p^{e}A(F)).

For each Y ∈ Y, include a sentence saying that (a, c) ∉ Y (F); such a sentence may be constructed since iterated p-components admit a first-order definition.

Suppose that the conclusion of Proposition 3.16 fails. Then we claim that T is consistent. To prove this, we show that every finite subset T_{0} of T has a model. Given T_{0}, let e_{max} be the maximum e that occurs in the sentences, and let Y_{max} ∈ Y be the union of the Y ’s that occur. The negation of Proposition 3.16 implies that there exists a field extension F ⊇ K having B as a p-basis, together with a ∈ A(F) and c ∈ X(F) ∩ (a + p^{e_{max}}A(F)), such that (a, c) ∉ Y_{max}(F). This (F, a, c) is a model for T_{0}.

By the compactness theorem, there is a model (F, a, c) for T . So F is a field extension of K having B as a p-basis, and

c ∈ X(F) ∩ (a + p^{e}A(F)).

for all e ∈ Z_{≥1} and for every Y ∈ Y we have (a, c) ∉ Y (F). Let { \atop K(a)} be the smallest extension of K(a) in F that is closed under taking p-components with respect to B, so { \atop K(a)} too has B as p-basis. Then K(c) is not algebraic over { \atop K(a)}, because an algebraic relation could be used to define a Y ∈ Y containing (a, c). By Lemma 3.12 applied to F/{ \atop K(a)}, it is possible to enlarge F to assume that F is separably closed and the Aut(F/{ \atop K(a)})-orbit of c is infinite while B is still a p-basis for F. This infinite orbit is contained in the set X(F) ∩ (a + p^{e}A(F)), so the latter is infinite too, for every e ∈ Z_{≥1}. By Lemma 3.13 applied to K ⊆ F, the abelian variety A\overline{F} has no nonzero quotient defined over F^{p^{∞}}. Also, the fact that X is coset-free implies that X_{F} is coset-free (if X\overline{K} acquires a coset over some extension of \overline{K}, it acquires one over some finitely generated extension of \overline{K}, and then a standard specialization argument shows that it contains a coset already over \overline{K}). Thus X_{F} ⊆ A_{F} is a counterexample to Lemma 3.14. □

Remark 3.17. The proof of Lemma 3.14 involves more model theory than we have used elsewhere in this section. On the other hand, probably the proof of [AV92, Theorem A(3)] could be adapted to give a purely algebraic proof of Lemma 3.14 in the case where A is ordinary, and this would suffice to prove Proposition 3.16 in the case where A is ordinary.

3.4. Subvarieties of abelian varieties. We now return to our situation in which K is the function field of a curve over k.

Lemma 3.18. Suppose that A\overline{K} has no nonzero isotrivial quotient, and that X is coset-free. Then there exists a finite K-subscheme Z of X such that X(A) ∩ \overline{A(K)} ⊆ Z(A).

Proof. By replacing K with its compositum with \overline{k} in \overline{K}, we may reduce to the case in which k is algebraically closed.

We have [K : K^{p}] = p. Choose α ∈ K − K^{p}. Then {α} is a p-basis for K. For any v, the field K_{v} is generated by K and K_{v}^{p}, so [K_{v} : K_{v}^{p}] ≤ p. Moreover, Lemma 3.1 implies α ∉ K_{v}^{p}, so {α} is a p-basis for K_{v}.

Let e and Y be as in Proposition 3.16. We think of Y as a subvariety of A_{n}×X for some n. For each a ∈ A(K), let Y_{a} be the fiber of Y → A_{n} above the point of A_{n}(K) corresponding to a. View Y_{a} as a finite subscheme of X.

Since A\overline{K} has no nonzero isotrivial quotient, A(K) is finitely generated. Choose a (finite) set of representatives A ⊆ A(K) for A(K)/p^{e}A(K). Let Z = ⋃_{a∈A} Y_{a}, so Z is a finite subscheme of X. The conclusion of Proposition 3.16 applied to F = K_{v} says that

X(K_{v}) ∩ (a + p^{e}A(K_{v})) ⊆ Y_{a}(K_{v})

for each a ∈ A. Let Γ_{v} be the closure of A(K) in A(K_{v}). Then

X(K_{v}) ∩ (a + p^{e}Γ_{v}) ⊆ Y_{a}(K_{v}).

Taking the union over a ∈ A yields

X(K_{v}) ∩ Γ_{v} ⊆ Z(K_{v}),

since by Lemma 3.10, the a ∈ A represent all classes in Γ_{v}/p^{e}Γ_{v}. This holds for all v, so X(A) ∩ \overline{A(K)} ⊆ Z(A). □

Remark 3.19. In the number field case, one cannot expect finiteness of X(K_{v})∩Γ_{v} in general. Consider the case in which K_{v} = Q_{p}. If the “Chabauty condition” rank A(K) < dim A fails, it may happen that Γ_{v} is an open subgroup of A(K_{v}), in which case X(K_{v}) ∩ Γ_{v} will be infinite if nonempty.

3.5. End of proof of Theorem B. We now prove Theorem B. Let Z be as in Lemma 3.18. By Lemma 3.18 and then Proposition 3.9,

X(A) ∩ \overline{A(K)} ⊆ Z(A) ∩ \overline{A(K)} = Z(K) ⊆ X(K).

The opposite inclusion X(K) ⊆ X(A) ∩ \overline{A(K)} is trivial. This completes the proof.

Remark 3.20. Our proof of Theorem B required both the flat information (from residue fields) implicit in Proposition 3.6 and the deep information (from each K_{v}) in Proposition 3.5. But it seems possible that the flat information suffices, even for Conjecture C. Specifically, if Ω is a cofinite subset of the set of places of good reduction for X/K, then it is possible that \overline{X(K)} = X (∏_{v∈Ω} F_{v})∩\overline{A(K)} always holds, where the closures are now taken in A (∏_{v∈Ω} F_{v}).

4. The Brauer-Manin obstruction

In this section we assume that K is either a global function field (i.e., as before, but with k finite) or a number field. We assume also that Ω is the set of all nontrivial places of K, and define the ring of adèles A in the usual way.

The purpose of this section is to relate the adelic intersections we have been considering to the Brauer-Manin obstruction. This relationship was discovered by Scharaschkin [Sch99] in the number field case. We will give a proof similar in spirit to his, and verify that it works in the function field case.

For convenience, if X is any topological space, let X_{•} denote the set of connected components, and equip X_{•} with the quotient topology: this will be needed to avoid annoyances with the archimedean places in the number field case.

Let A be an abelian variety over K. By the adelic topology on A(K), we mean the topology induced by A(A)_{•}; in the function field case this agrees with our terminology in Section 1.

Proposition 4.1. The adelic topology on A(K) equals the topology induced by subgroups of finite index.

Proof. Proposition 3.6 gives one inclusion. The other follows from the fact that A(A)_{•} = ∏_{v∈Ω} A(K_{v})_{•} is a profinite group. □

4.1. Global duality for abelian varieties. All cohomology below is fppf cohomology. As usual, for any n ∈ Z_{≥1}, define

Sel^{n} := ker (H^{1}(K, A[n]) → ∏_{v∈Ω} H^{1}(K_{v}, A))

� := ker (H^{1}(K, A) → ∏_{v∈Ω} H^{1}(K_{v}, A)) .

Following the notation of [Sto06, §2], define

{ \atop A(K)} := \underleftarrow{lim} \frac{A(K)}{nA(K)}
Ŝel := \underleftarrow{lim} Sel^{n}
T � := \underleftarrow{lim} � [n],

where each inverse limit is over positive integers n ordered by divisibility. Taking inverse limits of the usual descent sequence yields

(1)  0 → { \atop A(K)} → Ŝel → T � → 0.

For any abelian profinite group G and any prime ℓ, let G^{(ℓ)} be the maximal pro-ℓ quotient of G, so G ≅ ∏_{ℓ} G^{(ℓ)} ≅ \underleftarrow{lim} G/nG, where ℓ ranges over primes, and n ranges over positive integers ordered by divisibility.

Lemma 4.2. For each prime ℓ, each of { \atop A(K)}^{(ℓ)}, Ŝel^{(ℓ)}, T � ^{(ℓ)} is a finitely generated Z_{ℓ}- module.

Proof. For { \atop A(K)}^{(ℓ)} it holds simply because A(K) is finitely generated as a Z-module. For T � ^{(ℓ)}, it follows from the finiteness of � [ℓ], which is proved in [Mil70]. Now the result for

Ŝel^{(ℓ)} follows from (1). □

By definition, Sel^{n} maps to

ker (∏_{v∈Ω} H^{1}(K_{v}, A[n]) → ∏_{v∈Ω} H^{1}(K_{v}, A)) = ∏_{v∈Ω} \frac{A(K_{v})}{nA(K_{v})},

so as in [Sto06, §2] we obtain a map

Ŝel → A(A)_{•}.

For each v we have a pairing

A(K_{v})_{•} × H^{1}(K_{v}, A^{∨}) → H^{2}(K_{v}, G_{m}) ≅ Q/Z,

and in fact Tate local duality holds: each of the first two groups is identified with the Pontryagin dual of the other [Mil86, Theorem III.7.8]. An element of H^{1}(K, A^{∨}) maps to 0 in H^{1}(K_{v}, A^{∨}) for all but finitely many v, so summing the Tate local duality pairing over v defines a pairing

A(A)_{•} × H^{1}(K, A^{∨}) → Q/Z,

or equivalently a homomorphism

A(A)_{•} ⟶^{Tate} H^{1}(K, A^{∨})^{D},

where the superscript ^{D} denotes Pontryagin dual.

The following global duality statement is a version of the “Cassels dual exact sequence” in which finiteness of � is not assumed:

Proposition 4.3 (Cassels dual exact sequence). The sequence

0 ⟶ Ŝel ⟶ A(A)_{•} ⟶^{Tate} H^{1}(K, A^{∨})^{D}

is exact.

Proof. This is part of the main theorem of [GAT06]. □

Remark 4.4. Proposition 4.3 has a long history. For an elliptic curve A over a number field, Proposition 4.3 was proved by Cassels [Cas62, Cas64] under the assumption that � (A) is finite, using a result from [Ser64] to obtain exactness on the left. This, except for the exactness on the left, was generalized by Tate to abelian varieties over number fields with finite Shafarevich-Tate group, at around the same time, though the first published proof appeared much later [Mil86, Theorem I.6.13]. The latter reference also proved the global function field analogue except for the p-part in characteristic p. The Ŝel version (not requiring finiteness of � (A)), except for exactness on the left and the p-part, is implicit in the right three terms of the exact sequence in the middle of page 104 of [Mil86] if there we make the substitution

T � ≅ Ŝel/{ \atop A(K)}

from (1). Exactness on the left for the prime-to-p part is implicit in [Ser64, Ser71] and is mentioned explicitly in [Mil86, Corollary 6.23]. The full statement was proved in [GAT06].

4.2. Review of the Brauer-Manin obstruction. Let X be a projective K-scheme. Let X_{s} := X ×_{K} K^{s}. Define Br X := H^{2}(X, G_{m}), where here we may use étale cohomology since it gives the same result as fppf cohomology for smooth quasi-projective commutative group schemes [Mil80, Theorem III.3.9]. For simplicity, let \frac{Br X}{Br K} denote the cokernel of Br K → Br X even if this map is not injective. The Hochschild-Serre spectral sequence in étale cohomology gives an exact sequence

Br K → ker (Br X → Br X_{s}) → H^{1}(K, Pic X_{s}) → H^{3}(K, G_{m}),

and the last term is 0 for any global field: see [Mil86, Corollary 4.21], for instance. From this we extract an injection

H^{1}(K, Pic X_{s}) ↪ \frac{Br X}{Br K};

if Br X_{s} = 0, then this is an isomorphism. Composing with the map induced by Pic^{0} X_{s} ↪ Pic X_{s}, we obtain

(2)  H^{1}(K, Pic^{0} X_{s}) → \frac{Br X}{Br K} .

Each A ∈ Br X induces evaluation maps ev_{A} fitting in a commutative diagram

\xymatrix{ & X(K) \ar[r] \ar[d]_{_{ev_{A}}} & X(A)_{•} \ar[d]_{_{ev_{A}}} \ar@{.>}[dr] \\ 0 \ar[r] & Br K \ar[r] & ⨁_{v∈Ω} Br(K_{v}) \ar[r] & Q/Z \ar[r] & 0.}

The diagonal arrows give rise to a pairing

X(A)_{•} × \frac{Br X}{Br K} → Q/Z

which induces a map

X(A)_{•} ⟶^{BM} (\frac{Br X}{Br K} )^{D} .

The kernel (inverse image of 0) is called the Brauer set, and is denoted X(A)_{•}^{Br}: it contains X(K) and hence also its closure \overline{X(K)} in X(A)_{•}. One says that the Brauer-Manin obstruction to the Hasse principle is the only one for X if the implication

X(A)_{•}^{Br} ≠ ∅ ⟹ X(K) ≠ ∅

holds. (This is equivalent to the usual notion without the •.)

4.3. The Brauer set for abelian varieties.

Theorem E. We have

{ \atop A(K)} = \overline{A(K)} ⊆ A(A)_{•}^{Br} ⊆ Ŝel.

in A(A).

Proof. Proposition 4.1 shows that A(K) → A(A)_{•} induces an isomorphism { \atop A(K)} → \overline{A(K)}, and the first inclusion is automatic from the previous subsection.

So it remains to prove the second inclusion. Taking the dual of (2) for A yields the vertical arrow in

\xymatrix{0 \ar[r] & Ŝel \ar[r] & A(A)_{•} \ar[r]^{Tate} \ar[dr]_{BM} & H^{1}(K, Pic^{0} A_{s})^{D} \\  &  &  & _{(\frac{Br A}{Br K} )}_{D} \ar[u]}

Applying [Man71, Proposition 8] to each K_{v} shows that the triangle commutes (the hypothesis there that the ground field be perfect is not used). The diagram now gives ker(BM) ⊆ ker(Tate), or equivalently A(A)_{•}^{Br} ⊆ Ŝel. □

Remark 4.5. If � is finite (or more generally, if its maximal divisible part � _{div} is 0), then T � = 0 and the injection { \atop A(K)} → Ŝel is an isomorphism; in this case, the conclusion of Theorem E simplifies to

{ \atop A(K)} = \overline{A(K)} = A(A)_{•}^{Br} = Ŝel.

Closely related results in the number field case can be found in [Wan96].

4.4. The Brauer set for subvarieties of abelian varieties.

Proposition 4.6. Let i: X → A be a morphism from a projective K-scheme to an abelian variety over K. So i induces a map X(A)_{•} → A(A)_{•}.

(a) We have i(X(A)_{•}^{Br}) ⊆ Ŝel.

(b) Suppose that • X is a smooth projective geometrically integral curve,

• A is the Jacobian of X,

• i is an Albanese map (that is, i sends a point P to the class of P − D, where D ∈ Div A_{s} is a fixed divisor of degree 1 whose class is Gal(K^{s}/K)-invariant), and

• � is finite (or at least � _{div} = 0). Then X(A)_{•}^{Br} = i^{−1}(\overline{A(K)}).

Proof. (a) This follows immediately from Theorem E, since i maps X(A)_{•}^{Br} to A(A)_{•}^{Br}. (b) We use the diagram

\xymatrix{ &  &  & (\frac{Br X}{Br K} )^{D} \ar[d] \ar[dddd] \\  &  &  & H^{1}(K, Pic X_{s})^{D} \ar[d] \\  &  & X(A)_{•} \ar[uur]^{_{BM_{X}}} \ar[d]_{i} & H^{1}(K, Pic^{0} X_{s})^{D} \ar[d] \\ 0 \ar[r] & \overline{A(K)} \ar[r] & A(A)_{•} \ar[r]^{Tate} \ar[dr]_{_{BM_{A}}} & H^{1}(K, Pic^{0} A_{s})^{D} \\  &  &  & _{(\frac{Br A}{Br K} )}_{D} \ar[u]}

in which the horizontal sequence is the exact sequence of Proposition 4.3 in which we have used the finiteness of � to replace Ŝel with \overline{A(K)}. The lower triangle commutes: as mentioned earlier, this follows from a result of Manin. The “pentagon” at the far right also commutes, since the homomorphism (2) is functorial in X. Thus the whole diagram commutes.

We next claim that the three downward vertical arrows at the right are isomorphisms. The first is an isomorphism because Br X_{s} = 0 [Gro68, Corollaire 5.8]. The second is an isomorphism because Pic X_{s} ≅ Pic^{0} X_{s} ⊕ Z (where the Z is generated by the class of D) and H^{1}(K, Z) = 0. The third is an isomorphism because an Albanese map i induces an isomorphism i^{∗}: Pic^{0} A_{s} → Pic^{0} X_{s}.

The commutativity of the upper left “hexagon” now implies

X(A)_{•}^{Br} := ker (BM_{X}) = ker(Tate ∘i) = i^{−1}(ker(Tate)) = i^{−1}(\overline{A(K)}).

□

Remark 4.7. Part (b) was originally proved in the number field case in [Sch99] using a slightly different proof. For yet another proof of this case, see [Sto06, Corollary 7.4].

Remark 4.8. If K is a global function field, the conclusion of (b) can be written as X(A)^{Br} = X(A) ∩ \overline{A(K)} in A(A).

5. Intersections with Ŝel

From now on, we assume that K is a global function field.

The proof of Theorem D will follow that of Theorem B, with Ŝel playing the role of \overline{A(K)}. We begin by proving Ŝel-versions of several of the lemmas and propositions of Section 3.2. The following is an analogue of Lemma 3.7.

Lemma 5.1. The maps A(K)_{tors} → { \atop A(K)}_{tors} → Ŝel_{tors} are isomorphisms.

Proof. Proposition 4.1 yields { \atop A(K)} ≅ \overline{A(K)}, so the first map is an isomorphism by Lemma 3.7. The second map is an isomorphism by (1), since T � is torsion-free by definition. □

The following is an analogue of Proposition 3.5.

Proposition 5.2. If A(K^{s})[p^{∞}] is finite, then for any v, the map Ŝel^{(p)} → A(K_{v})^{(p)} is injective.

Proof. Let K′_{v} ⊆ K^{s} be the Henselization of K at v. Define Sel^{n}′ and Ŝel′ in the same way as Sel^{n} and Ŝel, but using K′_{v} in place of its completion K_{v}. By [Mil86, I.3.10(a)(ii)], the natural maps Sel^{n}′ → Sel^{n} and Ŝel′ → Ŝel are isomorphisms. Similarly, by [Mil86, I.3.10(a)(i)] we may replace A(K_{v})^{(p)} by A(K′_{v})^{(p)} := \underleftarrow{lim} A(K′_{v})/p^{n}A(K′_{v}). Hence it suffices to prove

injectivity of (Ŝel′)^{(p)} → A(K′_{v})^{(p)}.

Choose m such that p^{m}A(K^{s})[p^{∞}] = 0. Suppose b ∈ ker ((Ŝel′)^{(p)} → A(K′_{v})^{(p)}). For each

M ∈ Z_{≥0}, let b_{M} be the image of b in Sel′^{p^{M}} ⊆ H^{1}(K, A[p^{M}]). Then the image of b_{M} under

Sel′^{p^{M}} → \frac{A(K′_{v})}{p^{M}A(K′_{v})} ⊆ H^{1}(K′_{v}, A[p^{M}]) → H^{1}(K^{s}, A[p^{M}])

is 0. The Hochschild-Serre spectral sequence (see [Mil80, III.2.21]) gives an exact sequence

0 → H^{1}(Gal(K^{s}/K), A(K^{s})[p^{M}]) → H^{1}(K, A[p^{M}]) → H^{1}(K^{s}, A[p^{M}])

so b_{M} comes from an element of H^{1}(Gal(K^{s}/K), A(K^{s})[p^{M}]), which is killed by p^{m}. Thus p^{m}b_{M} = 0 for all M, so p^{m}b = 0.

By Lemma 5.1, b comes from a point in A(K)[p^{∞}], which injects into A(K′_{v})^{(p)}. □

The next result is an analogue of Proposition 3.9.

Proposition 5.3. Suppose that A(K^{s})[p^{∞}] is finite. Let Z be a finite K-subscheme of A. Then Z(A) ∩ Ŝel = Z(K).

Proof. One inclusion is easy: Z(K) ⊆ Z(A) and Z(K) ⊆ A(K) ⊆ { \atop A(K)} ⊆ Ŝel. So we focus on the other inclusion.

As in the proof of Proposition 3.9, we may replace K by a finite extension to assume that Z consists of a finite set of K-points of A. Suppose P ∈ Z(A) ∩ Ŝel. The v-adic component of P in Z(K_{v}) equals the image of a point Q_{v} ∈ Z(K). Then P − Q_{v} maps to 0 in A(K_{v}),

and in particular in A(K_{v})^{(p)}, so by Proposition 5.2 the image of P − Q_{v} in Ŝel^{(p)} is 0. This

holds for every v, so if v′ is another place, then Q_{v′} − Q_{v} maps to 0 in Ŝel^{(p)}. The kernel of

A(K) → { \atop A(K)}^{(p)} ↪ Ŝel^{(p)} is the prime-to-p torsion of A(K), so Q_{v′} − Q_{v} ∈ A(K)_{tors}. This holds for all v′, so the point R := P − Q_{v} belongs to Ŝel_{tors}. By Lemma 5.1, R ∈ A(K)_{tors}. Thus P = R + Q_{v} ∈ A(K). So P ∈ Z(A) ∩ A(K) = Z(K). □

Each τ ∈ Sel^{p^{e}} may be represented by a “covering space”: an fppf torsor T under A equipped with a morphism ϕ_{T} : T → A that after base extension to K^{s} becomes isomorphic to the base extension of [p^{e}]: A → A. (Passing to K^{s} is enough to trivialize T since A is smooth.)

Lemma 5.4. We have Ŝel ⊆ ⋃_{τ∈Sel^{p^{e}}} ϕ_{T} (T (A)).

Proof. In the commutative diagram

\xymatrix{ & T (A) \ar[d]^{ϕ_{T}} \\ Ŝel \ar@{^{(}->}[r] \ar[d] & A(A) \ar[d] \\ _{Sel_{p^{e}}} \ar@{^{(}->}[r] & \frac{A(A)}{p^{e}A(A)}}

the set ϕ_{T} (T (A)) is a coset of p^{e}A(A) in A(A), and its image in \frac{A(A)}{p^{e}A(A)} equals the image of τ under the bottom horizontal map, by the definition of this bottom map. Thus any element of Ŝel mapping to τ in Sel^{p^{e}} belongs to the corresponding ϕ_{T} (T (A)). □

Proof of Theorem D. We have X(K) ⊆ X(A)^{Br} ⊆ X(A) ∩ Ŝel, by Proposition 4.6(a), so it will suffice to show that X(A) ∩ Ŝel consists of K-rational points.

Let e and Y be as in Proposition 3.16. For each τ ∈ Sel^{p^{e}}, choose ϕ_{T} : T → A as above, choose t ∈ T (K^{s}), let a = ϕ_{T} (t) ∈ A(K^{s}), and let Y_{a} be the fiber of Y → A_{n} above the point corresponding to a. View Y_{a} as a finite subscheme of X_{K^{s}}. It follows from [Mil70] that Sel^{p^{e}} is finite, so we can choose a finite K-subscheme Z of X such that Z_{K^{s}} contains all the Y_{a} as τ ranges over Sel^{p^{e}}.

For each v, choose a separable closure K_{v}^{s} of K_{v} containing K^{s}. By the proof of Lemma 3.18, any p-basis of K is also a p-basis for K_{v}, and hence for K_{v}^{s}, so the conclusion of Proposition 3.16 may be applied with F = K_{v}^{s} to yield

X(K_{v}^{s}) ∩ (a + p^{e}A(K_{v}^{s})) ⊆ Y_{a}(K_{v}^{s})

for each a. By definition of ϕ_{T}, we have ϕ_{T} (T (K_{v}^{s})) = a + p^{e}A(K_{v}^{s}). Thus

X(K_{v}) ∩ ϕ_{T} (T (K_{v})) ⊆ X(K_{v}^{s}) ∩ (a + p^{e}A(K_{v}^{s})) ⊆ Y_{a}(K_{v}^{s}) ⊆ Z(K_{v}^{s}).

Hence

X(K_{v}) ∩ ϕ_{T} (T (K_{v})) ⊆ X(K_{v}) ∩ Z(K_{v}^{s}) = Z(K_{v}).

This holds for all v, so X(A) ∩ ϕ_{T} (T (A)) ⊆ Z(A). Taking the union over τ ∈ Sel^{p^{e}}, and applying Lemma 5.4, we obtain X(A) ∩ Ŝel ⊆ Z(A). Thus X(A) ∩ Ŝel ⊆ Z(A) ∩ Ŝel, and the latter equals Z(K) by Proposition 5.3, so we are done. □

Acknowledgements

This work was initiated while the authors were at the program on “Rational and integral points on higher-dimensional varieties” at the Mathematical Sciences Research Institute in Spring 2006. We thank Elisabeth Bouscaren, Zoé Chatzidakis, Jean-Louis Colliot-Thélène, Françoise Delon, Cristian Gonzalez-Aviles, David Harari, Thomas Scanlon, and Michael Stoll for discussions, and the referee for some suggestions. B.P. was partially supported by NSF grant DMS-0301280.

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Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA Current address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA

02139-4307, USA

E-mail address: poonen@math.mit.edu
URL: http://math.mit.edu/~poonen

Department of Mathematics, University of Texas, Austin, TX 78712, USA

E-mail address: voloch@math.utexas.edu
URL: http://www.ma.utexas.edu/~voloch