Elliptic Surfaces and Arithmetic Equidistribution for $\Mathbb {R

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E → B is not isotrivial. Let hÊ denote the Néron-Tate canonical height on E(k), associated ˆ to the choice of divisor O on E; let hEt denote the corresponding canonical height on the smooth fibers Et(K) for (all but finitely many) t ∈ B(K). By non-isotriviality, a point ˆ P ∈ E(k) satisfies hE(P ) = 0 if and only if it is torsion on E. We denote the specializations of P by Pt in the fiber Et. Tate showed in [Ta] that the canonical height function ˆ (1.1) hP (t) := hEt (Pt) is a Weil height on the base curve B(K), up to a bounded error. More precisely, there exists ˆ a Q-divisor DP on B of degree equal to hE(P ) so that hP (t)= hDP (t)+ O(1), where hDP is a Weil height on B(K) associated to DP . In [DM], we showed that we can also understand the small values of the function (1.1), from the point of view of equidistribution. Assume that hÊ(P ) > 0 (so that the function hP is nontrivial) and that, as a section, P : B →E is defined over the number field K. Building on work of Silverman [Si2, Si4, Si5], we showed

that hP is the height induced by an ample line bundle on B (with divisor DP ) equipped with a continuous, adelic metric of non-negative curvature deﬁned over K, denoted by DP and satisfying

DP · DP =0 for the Arakelov-Zhang intersection number introduced in [Zh4]. In particular, we can then apply the equidistribution theorems of [CL1, Th, Yu] to deduce that the Gal(K/K)-orbits

of points tn ∈ B(K) with height hP (tn) → 0 are uniformly distributed on B(C) with respect to the curvature distribution ωP for DP at an archimedean place of K. A similar equidistri- an bution occurs at each place v of K to a measure ωP,v on the Berkovich analytiﬁcation Bv [DM, Corollary 1.2]. As a consequence of our main result in [DM], and combined with the results of Masser and Zannier [MZ1, MZ2], we have

(1.2) DP · DQ ≥ 0 for all P, Q ∈ E(k), and

DP · DQ =0 ⇐⇒ either P or Q is torsion, or

∃ α> 0 such that hP (t)= α hQ(t) for all t ∈ B(K) ⇐⇒ ∃ (n, m) ∈ Z2 \{(0, 0)} such that nP = mQ

In particular, by the non-degeneracy of the Néron-Tate bilinear form h·, ·iE on E(k), we have ˆ ˆ 2 (1.3) DP · DQ =0 ⇐⇒ hE(P )hE(Q)= hP, QiE for all P, Q ∈ E(k). ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 3 The main result of this article is the proof of a stronger version of (1.3): Theorem 1.1. Let E → B be a non-isotrivial elliptic surface defined over a number field K. Let E be the corresponding elliptic curve over the field k = K(B). There exists a constant c> 0 so that ˆ ˆ 2 −1 ˆ ˆ 2 c hE(P )hE(Q) −hP, QiE ≤ DP · DQ ≤ c hE(P )hE(Q) −hP, QiE for all P, Q ∈ E(k), where h·, ·iE is the Néron-Tate bilinear form on E(k).

Remark 1.2. The upper bound on DP · DQ in Theorem 1.1 is relatively straightforward. The diﬃculty lies in the lower bound; in Section 7, we observe that this is equivalent to

proving that DX · DY > 0 for all independent X,Y ∈ E(k) ⊗ R. 1.2. Motivation and context. Theorem 1.1 was inspired by the statements and proofs of the Bogomolov Conjecture [Ul, Zh3, SUZ], extending Raynaud’s theorem that settled the Manin-Mumford Conjecture [Ra].

Theorem 1.1 can be seen as a Bogomolov-type bound. The intersection number DP · DQ ˆ ˆ is related to the small values of the heights hEt (Pt)+ hEt (Qt) in the ﬁbers Et(K). Indeed, as a consequence of Zhang’s Inequality [Zh4, Theorem 1.10] applied to the sum DP + DQ, and the fact that hP (t) ≥ 0 at all points t ∈ B(K) for every P ∈ E(k)[DM, Proposition 4.3], we have

1 DP · DQ (1.4) ess.min. (hP + hQ) ≤ ≤ ess.min. (hP + hQ) 2 hÊ(P )+ hÊ(Q) for every pair of non-torsion P, Q ∈ E(k). The essential minimum is defined by ess.min.(f)=

supF infB\F f over all finite sets F in B(K). Bogomolov-type bounds have found many applications in problems of unlikely intersections. In Section 7, we explain that Theorem 1.1 is equivalent to the following: Theorem 1.3. Let E → B be a non-isotrivial elliptic surface defined over a number field, and let π : E m → B be its m-th fibered power with m ≥ 2. Let E m,{2} denote the union of flat subgroup schemes of E m of codimension at least 2, and consider the tubular neighborhood

m,{2} m ′ m,{2} ′ ′ T (E , ǫ)= P ∈E : ∃P ∈E (Q) with π(P )= π(P ) and hˆ m (P − P ) ≤ ǫ Eπ(P ) Then, for any irreduciblen curve C in E m, defined over a number field and dominatingoB, there exists ǫ> 0 such that C ∩ T (E m,{2}, ǫ) is contained in a finite union of flat subgroup schemes of positive codimension. See, e.g., [BC, Lemma 2.2] for definitions and a classification of flat subgroup schemes. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 4 Remark 1.4. The conclusion of Theorem 1.3 with ǫ = 0 is a result of Barroero and Capuano [BC, Theorem 2.1]: using techniques involving o-minimality and transcendence theory, similar to those of [MZ1, MZ2] (which treated the intersections of curves C with T (E m,{m}, 0), the torsion subgroups), they show that C ∩ T (E m,{2}, 0) is contained in a finite union of flat subgroup schemes of positive codimension. Thus Theorem 1.3 may be seen as a Bogomolov-type extension of [BC, Theorem 2.1], while providing a new proof of results in [BC, MZ1, MZ2]. Our main result in [DM] treated the intersections of C with the smaller tube T (E m,{m}, ǫ). In [Zh2], Zhang proposed the investigation of a function on the base curve B that detects drops in rank of the specializations of a subgroup of E(k): given a finitely-generated subgroup

Λ of E(k) of rank m ≥ 1, if the torsion-free part of Λ is generated by S1,...,Sm, let

(1.5) hΛ(t) := det(hSi,Sjit)i,j ≥ 0

on B(K), where defined, where h·, ·it is the Néron-Tate bilinear form on the specialization Λt in the fiber Et. We propose the following result as the analog of [Zh2, §4 Conjecture] for elliptic surfaces; Zhang’s conjecture was formulated for geometrically simple families of abelian varieties A→ B of relative dimension > 1, and it does not hold as stated for elliptic surfaces [Zh2, §4 Remark 3]. Theorem 1.5. Let E → B be a non-isotrivial elliptic surface defined over a number field K, and let E be the corresponding elliptic curve over the field k = K(B). Let Λ ⊂ E(k) be a subgroup of rank m ≥ 2, with torsion-free part generated by S1,...,Sm ∈ E(k). For each i = 1,...,m, let Λi ⊂ Λ be generated by {S1,...,Sm}\{Si}. There is a constant ǫ = ǫ(Λ) > 0 so that

{t ∈ B(K): hΛ1 (t)+ ··· + hΛm (t) ≤ ε} is ﬁnite. In Section 7, we observe that Theorem 1.5 is equivalent to Theorems 1.1 and 1.3.

Remark 1.6. Note that, for rank 1 groups Λ, the value hΛ(t) is the canonical height of the ˆ generating point St in Et. In general, recall that the Néron-Tate height hEt on a smooth fiber defines a positive definite quadratic form in Et(Q) ⊗ R; see e.g. [Si6, Ch. VIII, Prop. 9.6]. Thus, hΛ will vanish at t ∈ B(K) if and only if rank Λt < rank Λ. The sum hΛ1 (t)+ ··· + hΛm (t) will be zero if and only if the points S1,t,...,Sm,t satisfy (at least) two independent linear relations over Z in the fiber Et.

Remark 1.7. The independence of the points S1,...,Sm ⊂ Λ in Theorem 1.5 is necessary for the ﬁniteness statement to hold. Indeed, suppose that Sm is a linear combination of S1,...,Sm−1, and suppose that {tn} ⊂ B(K) is any inﬁnite non-repeating sequence for which ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 5 hSm (tn) → 0 (for example, we can take tn where Sm,tn is torsion; see e.g. [DM, Proposition

6.2]). As we will see in Proposition 5.3, we then have hΛ1 (tn)+ ··· + hΛm (tn) → 0. If the elliptic surface E → B is isotrivial, the conclusion of Theorem 1.3 with ǫ = 0 was established by Viada [Vi1] and Galateau [Ga]. Moreover, in this isotrivial setting, Viada proved the analogue of Theorem 1.3 (for positive effective ǫ) in [Vi2, Theorem 1.4], [Vi1, Theorem 1.2], providing in particular new proofs of instances of earlier results by Poonen [Po] and Zhang [Zh1] and extending the work in [RV]. It is worth pointing out that the aforementioned results invoked a different Bogomolov-type bound than the one in Theorem 1.1, established by Galateau [Ga]. Our techniques provide an alternative approach to their results, as long as no complex multiplication is involved. The isotrivial case for ǫ = 0 is completed for curves in arbitrary abelian varieties in [HP] using amongst others techniques n from o-minimality. In the setting of the multiplicative group Gm, Habegger established results of this flavor in arbitrary dimension [Ha1], generalizing a result of Bombieri-Masser- Zannier [BMZ]. Finally, we remark that analogues of Theorem 1.1, Theorem 1.3 and Theorem 1.5 can be formulated for products of non-isogenous elliptic surfaces, as we did in [DM, Theorem 1.4]. Our methods here would yield these results and, in particular, Barroero-Capuano’s [BC2, Theorem 1.1]. We omit them in this article to simplify our exposition.

1.3. Metrized R-divisors on curves. For each t ∈ B(K) with Et smooth, the canonical ˆ height hEt induces a positive definite quadratic form on Et(K) ⊗ R. The height functions hP on B(K), defined by (1.1) for P ∈ E(k), therefore make sense for elements of the finite-dimensional vector space E(k) ⊗ R. In Theorem 4.6, we prove that every nonzero

element X ∈ E(k) ⊗ R gives rise to a continuous, adelic, semi-positive metrization DX of an ample R-divisor on the base curve B, defined over a number field K, with height function ˆ hX (t)= hEt (Xt) for t ∈ B(K) when Et is smooth, satisfying DX · DX = 0. Consequently, we are able to employ results of Moriwaki [Mo2] in our proof of Theorem 1.1. Specifically, we use his arithmetic Hodge index theorem for adelically metrized R-divisors on

curves defined over a number field [Mo2, Corollary 7.1.2] to understand when DX · DY =0 for X,Y ∈ E(k) ⊗ R. Although not needed for Theorem 1.1, we provide another application of Moriwaki’s arithmetic Hodge index theorem [Mo2, Theorem 7.1.1], combined with the continuity of the arithmetic volume function [Mo2, Theorem 5.3.1] and a generalization of Zhang’s fundamental inequality [CLT, Lemme 5.1]. We prove that metrizations of R-divisors on curves with these nice properties (continuous, adelic, semipositive) obey an equidistribution law. This extends the equidistribution theorems of Chambert-Loir, Thuillier, and Yuan [CL1, Th, Yu] and follows a proof strategy originating in [SUZ]: ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 6 Theorem 1.8. Let B be a smooth projective curve defined over a number field K. Fix an ample R-divisor D on B, equipped with an adelic, semipositive and normalized metrization D over K. Let M be any admissible adelic metrization on an R-divisor M over K. For any

(non-repeating) sequence xn ∈ B(K) with hD(xn) → 0, we have D · M h (x ) → . M n deg D Corollary 1.9. Fix any ample R-divisor D on B, equipped with an adelic, semipositive and normalized metrization D over K. For each place v of K and for any (non-repeating) inﬁnite

sequence xn ∈ B(K) with hD(xn) → 0, the discrete probability measures 1 µn = δy | Gal(K/K) · xn| y∈Gal(XK/K)·xn an 1 converge weakly in Bv to the probability measure µD,v = deg D ωD,v. an Here, Bv denotes the Berkovich analytification of the curve B over the complete and algebraically closed field Cv. Returning to the setting of E → B and its sections, Corollary 1.9 applies to sequences in the base curve B where the specializations of points in E(k) satisfy non-trivial linear relations. For example, generalizing [DM, Corollary 1.2], we show: Theorem 1.10. Let E → B be a non-isotrivial elliptic surface defined over a number field K, and let E be the corresponding elliptic curve over the field k = K(B). Suppose that

P1,...,Pm is a collection of m ≥ 1 linearly independent points in E(k), also deﬁned over K as sections of E → B. Suppose that {tn} ⊂ B(K) is an inﬁnite non-repeating sequence where

(1.6) a1,nP1,tn + a2,nP2,tn + ··· + am,nPm,tn = Otn m−1 for ai,n ∈ Z, with [a1,n : ··· : am,n] → [x1 : ··· : xm] in RP as n →∞. Set

X = x1P1 + ··· + xnPn ∈ E(k) ⊗ R. Then

hX (tn) → 0 for the height function associated to the metrized R-divisor DX . Moreover, for each place v an of K, the Gal(K/K)-orbits of tn in B(K) are uniformly distributed on Bv with respect to the curvature distribution

2 ωX,v = xi − xixj ωPi,v + xixj ωPi+Pj ,v. i ! i

Remark 1.11. For nonzero X ∈ E(k) ⊗ R, the height hX will have only ﬁnitely many zeros unless a positive real multiple cX is represented by an element of E(k); see Proposition 5.5.

On the other hand, there is always an inﬁnite sequence {tn} for which (1.6) is satisﬁed; see Proposition 5.1.

1.4. Measures at an archimedean place. In [DM], the curvature measures ωP for the metrized divisor DP associated to P ∈ E(k), at an archimedean place, are computed as the pullback by P of a certain (1, 1)-form on E(C), via a dynamical construction. In [CDMZ],

it is shown that ωP = db1 ∧ db2 in the Betti coordinates (b1, b2) of P . We explain in Section 8 that elements X ∈ E(k) ⊗ R are also represented by holomorphic curves in the surface E, and the Betti coordinates of X are real linear combinations of the Betti coordinates of points Pi ∈ E(k). We use this to prove that the measure ωX , at a single archimedean place of the number ﬁeld K, is enough to uniquely determine the pair of points X and −X: Theorem 1.12. Fix X and Y in E(k) ⊗ R and an archimedean place of the number ﬁeld

K. Let ωX and ωY denote the curvature distributions on B(C) at this place for the adelically metrized R-divisors DX and DY . Then

ωX = ωY ⇐⇒ X = ±Y. We are grateful to Lars Kühne for helping us with the proof of Theorem 1.12: we use the holomorphic-antiholomorphic trick of André, Corvaja, and Zannier [ACZ, §5] and a transcendence result of Bertrand [Be, Théorème 5]. A special case of Theorem 1.12 was proved by a different method in [DWY, Proposition 1.9].

1.5. Example. Let Et be the Legendre elliptic curve defined by y2 = x(x − 1)(x − t) for t ∈ Q\{0, 1}. By filling in the family over t =0, 1, ∞, we obtain an elliptic surface E → B with B = P1 defined over Q. Here k = Q(t). It is easy to see that rank E(k) = 0. However, 1 by choosing any collection of m distinct points x1, x2,...,xm ∈ P (Q) \{0, 1, ∞}, we can construct an elliptic surface E ′ → B′ with rank E′(k′) ≥ m where k′ = Q(B′). Indeed, we

let Pxi be a point with constant x-coordinate equal to xi. As the points xi are distinct, the structure of the field extensions ki/k, determined by each Pxi , implies that the points must ′ be independent. We pass to a branched cover B → B so that each Pxi defines a section over B′ and set k′ = Q(B′). These examples were first considered in [MZ1] and the associated ′ C P1 measures ωPxi on B ( ) (or rather, their projections to B = ) were computed in [DWY]. 1.6. Outline of the article. In Section 3, we introduce metrizations on R-divisors on curves defined over a number field and define their intersection numbers. We then prove the equidistribution results, Theorem 1.8 and Corollary 1.9. In Section 4, we prove that each ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 8 nonzero element X ∈ E(k) ⊗ R induces a continuous, adelic, semipositive metrization DX on an ample R-divisor on the base curve B. In Section 5, we study the sequences of small

points for the height function hX on B(Q) associated to DX . In Section 6 we lay out the basic properties of the intersection number (X,Y ) 7→ DX · DY as a biquadratic form on the vector space E(k) ⊗ R. In Section 7, we analyze the signiﬁcance of DX · DY = 0 for nonzero X,Y ∈ E(k) ⊗ R, and we explain how to relate Theorems 1.1, 1.3, and 1.5. Section 8 contains a proof of Theorem 1.12, and we complete the proofs of Theorems 1.1, 1.3, and 1.5 in Section 9.

1.7. Acknowledgements. We spoke with many people about this work, and we are grateful to all of them for helpful discussions. Special thanks go to Fabrizio Barroero, Daniel Bertrand, Laura Capuano, Gabriel Dill, Philipp Habegger, Lars K¨uhne, Harry Schmidt, Xinyi Yuan, and Umberto Zannier.

2. Notation

Throughout this article, K denotes a number ﬁeld. We let MK denote its set of places, with absolute values |·|v satisfying the product formula:

[Kv:Qv] (2.1) |x|v =1 v∈YMK for all nonzero x in K. Here Kv denotes the completion of K with respect to |·|v. We set [K : Q ] (2.2) r := v v . v [K : Q]

For each place v ∈ MK , we let Cv denote the completion of an algebraic closure of Kv. We let B denote a smooth projective curve defined over a number field K. For each an v ∈ MK , we let Bv denote the Berkovich analytification of B over the field Cv. We let DivZ(B) denote the group of divisors on B. Throughout, k denotes the function field K(B). Its places are in one-to-one correspondence

with the elements t ∈ B(K), with absolute values given by |f|t = exp(− ordt(f)) for each nonzero f ∈ K(B).

3. R-divisors and equidistribution In this section, we introduce metrizations on R-divisors on curves, following Moriwaki [Mo2]. We show that an equidistribution theorem holds for sequences of small height on a curve B deﬁned over a number ﬁeld, for adelic semipositive metrizations D associated to an ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 9 R-divisor, extending the equidistribution results for curves of Chambert-Loir, Thuillier, and Yuan [CL1, Th, Yu]. The proofs of Theorem 1.8 and Corollary 1.9 follow a known strategy for equidistribution. We mimic the presentation of Chambert-Loir and Thuillier in [CLT]; they themselves appeal to results of Yuan [Yu] and Zhang [Zh4], building on the ideas that originally appeared in [SUZ]. The key ingredient for passing from Q-divisors to R-divisors is the continuity of the arithmetic volume function on the space of metrized of R-divisors, as proved by Moriwaki [Mo2]. We provide the details for completeness.

3.1. Metrizations of R-divisors on curves. Let B be a smooth projective curve defined over a number field K. Let D = i aiDi be an ample R-divisor on B, with ai ∈ R and D ∈ Div (B) with support in B(K), invariant under the action of Gal(K/K). By rewriting i Z P the sum if necessary, we may assume that each Di is associated to an ample line bundle Li an that extends over the Berkovich analytification Bv for each place v of K. A continuous, adelic metrization for D is a collection of continuous functions

an gv : Bv \ supp D → R

for v ∈ MK, such that

(1) For each v, the locally-defined function ψv := gv + i ai log |fi|v extends continuously to the support of D, where f is a local defining equation for D defined over K; i P i (2) there exists a model (B, D) of (B,D) over the ring of integers OK so that gv is the associated model function for all but finitely many v, or equivalently, the function

ψv ≡ 0 at all but finitely many places v for the associated {fi} near each element of supp D. See [Mo2, §0.2] and [CL2, §1.3.2] for the definition of model functions. an The metrization is semipositive if each gv is subharmonic on Bv \ supp D. An R-divisor an D on B and collection of continuous functions gv : Bv \ supp D → R, for v ∈ MK, is said to be admissible if D = D1 − D2 and gv = gv,1 − gv,2 for two adelic, semipositive metrizations on R-divisors Di =(Di, {gi,v}). We write D = D1 −D2 for the data (D, {gv}). An associated height function is given by hD = hD1 − hD2 . We denote this data by D = (D, {gv}). Moriwaki calls D an adelic arithmetic R-divisor of (C0 ∩ PSH)-type on B [Mo2]. It extends Zhang’s notion of an adelic, semipositive metric on a line bundle to R-divisors [Zh4]. Indeed, for D an ample divisor on B associated to a line bundle L, equipped with an adelic metric {k·kv}v, and s a meromorphic section of L with (s)= D, we put gv = − log kskv at each place v of the number field K. an We let ωD,v denote the curvature distribution on Bv associated to the metrization D; by definition, this is a positive measure of total mass deg D, equal to the Laplacian of gv away from supp D. See, for example, [BR] for more information about the distribution-valued ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 10 Laplacian on Berkovich curves. The associated probability measure is denoted by 1 µ := ω . D,v deg D D,v There is an associated height function on B(K) defined by

rv ′ (3.1) hD(x) := gv(x ), v∈M | Gal(K/K) · x| ′ XK x ∈Gal(XK/K)·x

for x 6∈ supp D. Recall that rv was deﬁned in (2.2). For any rational function φ on B deﬁned over K, and for any real a ∈ R, note that

rv ′ hD(x)= (gv − a log |φ|v)(x ) | Gal(K/K) · x| ′ vX∈MK x ∈Gal(XK/K)·x away from (supp D)∪(supp(φ)), from the product formula (2.1). This allows deﬁnition (3.1) to extend to the points x ∈ supp D, by choosing any φ so that x ∈ supp(φ) and a so that ′ gv − a log |φ|v extends continuously at x for every v. For an R-divisor D = i bi [xi] with support in B(K), we will write P ′ hD(D ) := bi hD(xi). i X

3.2. Intersection. For divisors D1,D2 ∈ DivZ(B) associated to line bundles L1 and L2, respectively, equipped with continuous, adelic metrics D1 and D2, the arithmetic intersection number is deﬁned in [Zh4] (see also [CL2]) as

(3.2) D1 · D2 := hD1 ((s2)) + rv (− log ks2kD2,v) dωD1,v an v∈M Bv XK Z

= hD1 ((s2)) + hD2 ((s1)) + rv (− log ks2kD2,v)(dωD1,v − δ(s1)) an v∈M Bv XK Z

= hD1 ((s2)) + hD2 ((s1)) + rv (− log ks2kD2,v) ∆(− log ks1kD1,v) an Bv vX∈MK Z = D2 · D1, where si is a meromorphic section of Li deﬁned over K, for i =1, 2, with divisors (s1) and (s2) of disjoint support. For the continuous, adelic metrizations of R-divisors, we extend by R-linearity, so that

(3.3) D1 · D2 = hD1 (D2)+ rv gD2,v dωD1,v = D2 · D1. an v∈M Bv XK Z ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 11

Remark 3.1. The intersection number (3.3) coincides with deg(D1 · D2) of [Mo2]. Indeed, [Mo2, Theorem 4.1.3] states that each D can be uniformly approximated by metrizations associated to models, and it is known that the intersection numbersd coincide for these model metrics [Mo1, Proposition 2.1.1].

We say D is normalized if its self-intersection number satisﬁes D · D =0.

Note that any continuous, adelic metrization can be normalized by adding a constant to gv at some place.

For each a ∈ R and D = (D, {gv}), we write aD for the pair (aD, {agv}). Normalized metrized divisors D1 and D2 on B are isomorphic if D1 − D2 is principal, meaning that there are rational functions φ1,...,φm ∈ K(B) and real numbers a1,...,am, so that m

D1 − D2 = ai (φi), {− log |φi|v}v∈MK . i=1 X Note that by the product formula the height function hD depends only on the isomorphism class of D. We formulate Moriwaki’s arithmetic Hodge-index theorem as follows:

Theorem 3.2. [Mo2, Corollary 7.1.2] Suppose D1 and D2 are normalized continuous semipositive adelic metrizations on ample R-divisors with deg D1 = deg D2. Then D1 · D2 ≥ 0, and D1 · D2 =0 if and only if D1 and D2 are isomorphic.

Proof. Set D = D1 − D2, so that the underlying divisor D has degree 0, and

D · D = −2 D1 · D2 From [Mo2, Corollary 7.1.2], we have that D·D ≤ 0 with equality if and only if D is principal,

up to the addition of a constant to the metrization gv at some place. The normalization of D1 and D2 implies that this constant must be 0.

3.3. Essential minima. Following [Zh4], the essential minimum of the height hD is deﬁned as

(3.4) e1(D) := sup inf hD(x), F x∈B(K)\F over all ﬁnite subsets F of B(K), and we put

e2(D) := inf hD(x). x∈B(K) ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 12 Theorem 3.3. [Zh4, Theorem 1.10] For any adelic, semipositive metrization D of an ample R-divisor D, we have

D · D 1 e (D) ≥ ≥ e (D)+ e (D) 1 2 deg D 2 1 2 Proof. Zhang proved the result for ample line bundles equipped with adelic, semipositive metrics [Zh4, Theorem 1.10]. It holds also for metrizations of R-divisors because the height function associated to an R-divisor is a uniform limit of heights associated to Q-divisors, and the intersection number is a bilinear form on metrized divisors.

Using the upper bound on D · D in Theorem 3.3, we can extend Theorem 3.2 to:

Theorem 3.4. Suppose D1 and D2 are normalized semipositive adelic metrizations on ample R-divisors of the same degree, and suppose the essential minimum of at least one of the Di is 0. Then the following are equivalent:

(1) D1 · D2 =0 (2) D1 and D2 are isomorphic

(3) hD1 = hD2 on B(K)

(4) hD1 = hD2 at all but ﬁnitely many points of B(K) (5) there exists an inﬁnite non-repeating sequence tn in B(K) for which

lim hD (tn)+ hD (tn) =0. n→∞ 1 2 Proof. We have (1) ⇐⇒ (2) from Theorem 3.2. The deﬁnition of the height function, in view of the product formula, implies that (2) =⇒ (3), and we clearly have (3) =⇒ (4).

The essential minimum being 0 for D1 or for D2 gives (4) =⇒ (5). Finally, assume (5). Theorem 3.3 implies that e1(Di) ≥ 0, for i =1, 2, because Di is normalized. Therefore, we also have e1(D1 + D2) ≥ 0 for the essential minimum of the sum hD1 + hD2 . The existence of the sequence {tn} thus implies that e1(D1 + D2) = 0. As D1 · D2 ≥ 0 from Theorem 3.2 and Di · Di = 0 for i = 1, 2 by assumption, we apply Zhang’s inequality (Theorem 3.3) to D1 + D2 to obtain

2 D1 · D2 0= e1 D1 + D2 ≥ ≥ 0, deg D1 + deg D2 which allows us to deduce condition (1).

We will use the equivalences of Theorem 3.4 repeatedly in our proofs of Theorems 1.1, 1.3, and 1.5. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 13 3.4. Arithmetic volume. The rest of this section is devoted to the proof of the equidistribution results, Theorem 1.8 and Corollary 1.9. For any R-divisor D on B deﬁned over K, we set H0(X,D)= {φ ∈ K(B):(φ)+ D ≥ 0}.

For ample D ∈ DivZ(B), the volume of D is 1 vol(D) = lim dim H0(B, kD) = deg D k→∞ k from Riemann-Roch. For a Q-divisor D, the volume can be defined by taking the limit along sequences where kD ∈ Div(B) and again we have vol(D) = deg D when D is ample. The arithmetic volume of an adelically-metrized R-divisor D is defined as follows. We first fix a family of norms on H0(B,D) by

−gv(x) kφksup,v = sup |φ(x)|ve , an x∈Bv \supp D for each place v of K. Set µ((H0(B,D) ⊗ A )/H0(B,D)) χ(D)= − log K µ( v Uv) 0 where AK is the ring of adeles, µ is a Haar measureQ on H (B,D) ⊗ AK, and Uv is the unit 0 ball in H (B,D) ⊗ Cv in the induced norm. Then χ(kD) volχ(D) := lim sup 2 . k→∞ k /2

In [Mo2, Theorem 5.2.1], Moriwakic proves that volχ deﬁnes a continuous function on a space of continuous, adelic metrizations on R-divisors. As a consequence, he shows that for semipositive metrizations, we have volχ(D) = Dc· D [Mo2, Theorem 5.3.2]. Therefore, Zhang’s inequality (Theorem 3.3) implies that c (3.5) e1(D) ≥ volχ(D)/(2 deg D). for all continuous, semipositive, adelic metrizations of R-divisors on B. c

Remark 3.5. This volume function volχ is deﬁned diﬀerently than the one studied by Moriwaki in [Mo2], but they coincide. See, for example, Appendix C.2 of [BG] and the discussion on page 615, for the comparisonc of an adelic volume to a Euclidean volume. Proposition 3.6. For all admissible adelic metrizations on an ample R-divisor D on a curve, we have vol (D) e (D) ≥ χ 1 2 deg D c ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 14 Proof. From (3.5), the inequality holds for semipositive metrics. For admissible metrics,

we write D = D1 − D2 for semipositive Di and approximate each Di with semipositive adelic metrics on Q-divisors Di,n as n → ∞. Because D is ample, we can assume that D1,n − D2,n is ample for all n. In that setting, we apply [CLT, Lemme 5.1], recalling that vol(D1,n − D2,n) = deg(D1,n − D2,n). The result then follows by uniform convergence of the resulting height functions, so that e1 is continuous, and by continuity of the volume function volχ [Mo2, Theorem 5.2.1].

c 3.5. Proof of Equidistribution.

Proof of Theorem 1.8. Fix an ample R-divisor D, equipped with an adelic, semipositive and normalized metrization D. Let xn ∈ B(K) be any (non-repeating) sequence xn ∈ B(K) with hD(xn) → 0. Assume ﬁrst that M is an adelic, semi-positive metrization on an ample R-divisor M, not necessarily normalized. For each positive integer m, by Zhang’s inequality (Theorem 3.3) applied to (mD)+ M, we have

m2 D · D +2m D · M + M · M 2m D · M + M · M lim inf (mhD(xn)+ hM (xn)) ≥ = . n→∞ 2(m deg D + deg M) 2(m deg D + deg M)

As the sequence xn is small for D, this gives

2m D · M + M · M lim inf hM (xn) ≥ n→∞ 2(m deg D + deg M)

for all m. Letting m go to ∞, we obtain

D · M (3.6) lim inf hM (xn) ≥ . n→∞ deg D

For the reverse inequality, we choose m large enough so that deg(mD − M) > 0. We can therefore apply Proposition 3.6 to obtain

volχ mD − M lim inf(mhD(xn) − hM (xn)) ≥ n→∞ 2 deg( mD − M) c so that

volχ mD − M − lim sup hM (xn) ≥ n→∞ 2 deg( mD − M) c ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 15 On the other hand, again assuming that deg(mD − M) > 0, Moriwaki’s [Mo2, Theorem 7.1.1] implies that

volχ mD − M ≥ mD − M · mD − M 2 = m (D · D)− 2m D · M + M · M c = −2m D · M + M · M, with the last equality because D is normalized. (Compare [Yu, Theorem 2.2] and [CLT, Proposition 5.3].) Therefore, 2m D · M − M · M lim sup hM (xn) ≤ n→∞ 2 deg(mD − M) for all suﬃciently large m. As deg(mD − M) ≈ m deg D as m →∞, we obtain the desired upper bound of D · M (3.7) lim sup hM (xn) ≤ . n→∞ deg D Putting the two inequalities (3.6) and (3.7) together, we have D · M lim hM (xn)= . n→∞ deg D

Now suppose that M is admissible. By deﬁnition, we can write M = M 1 − M 2 for semipositive M i. By adding and subtracting any choice of ample divisor with semipositive metric, we can assume that each Mi is ample, and we apply the result above to each M i.

We have hM = hM 1 − hM 2 and D · M = D · M 1 − D · M 2. The theorem is a consequence of this linearity.

Proof of Corollary 1.9. Fix a place v ∈ MK , and let φ be a smooth real-valued function on an Bv . By density as in [Mo2, Theorem 3.3.3] it is enough to consider these functions. We denote by Oφ the trivial divisor on B equipped with the metrization given by gv = φ and gw = 0 for all w =6 v in MK . This metrization is clearly admissible. an Let µn denote the probability measure in Bv supported uniformly on the Galois conjugates of xn. Note that

hOφ (xn)= rv φdµn, an ZBv by the deﬁnition of the height function. From (3.3), we have

D · Ov(φ)= rv φdωD,v. an ZBv ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 16

Applying Theorem 1.8 to M = Ov(φ), we get that

1 [Kv : Qv] [Kv : Qv] lim hOv(φ)(xn)= φdωD,v = φdµD,v, n→∞ deg D [K : Q] an [K : Q] an ZBv ZBv an demonstrating weak convergence of µn to µD,v in Bv .

4. A metrized R-divisor for each element of E(k) ⊗ R Throughout this section, we let E → B be a non-isotrivial elliptic surface defined over a number field K, and let E be the corresponding elliptic curve over the field k = K(B). We denote the zero by O ∈ E(k). As E(k) is finitely generated, we enlarge K if needed so that all sections of E → B are defined over K. Recall that points P1,...,Pm are independent in E(k) if there is no nontrivial relation of the form

a1P1 + ··· + amPm = O

in E(k) with ai ∈ Z. In this section, we show that each nonzero element X ∈ E(k) ⊗ R naturally gives rise to

an adelic, semipositive continuous metrization DP associated to an ample R-divisor DX ; see Theorem 4.6. For P ∈ E(k), these metrizations on R-divisors coincide with the adelically metrized line bundles on B that we studied in [DM]. 4.1. Néron-Tate heights. Let F be a number field or a function field in characteristic 0. Let E/F be an elliptic curve with origin O, expressed in Weierstrass form as 2 3 2 E = {y + a1xy + a3y = x + a2x + a4x + a6} with discriminant ∆. Denote by

hÊ : E(F) → [0, ∞) a Néron-Tate canonical height function; it can be defined by 1 h(x(nP )) hÊ(P )= lim 2 n→∞ n2 where h is the naive Weil height on P1 and x : E → P1 is the degree 2 projection to the x-coordinate.

For each v ∈ MF , recall that Fv denotes the completion of F with respect to |·|v and Cv denote the completion of the algebraic closure of Fv. The canonical height has a decomposition into local heights, as 1 hÊ(P )= rv λÊ,v(Q) | Gal(F/F) · P | Q∈Gal(XF/F)·P vX∈MF ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 17 for all P ∈ E(F) \{O}, with rv defined by (2.2) in the number field case, and rv = 1 for ˆ function fields. The local heights λE,v are characterized by the three properties [Si3, Chapter 6, Theorem 1.1]:

(1) λˆE,v is continuous on E(Cv) \{O} and bounded on the complement of any v-adic neighborhood of O; ˆ 1 (2) the limit of λE,v(P ) − 2 log |x(P )|v exists as P → O in E(Cv); and (3) for all P =(x, y) ∈ E(Cv) with 2 P =6 O, 1 (4.1) λˆ (2 P )=4λˆ (P ) − log |2y + a x + a | + log |∆| . E,v E,v 1 3 v 4 v Property (3) may be replaced with the quasi-parallelogram law ˆ ˆ ˆ ˆ λE,v(P + Q)+ λE,v(P − Q)=2λE,v(P )+2λE,v(Q) (4.2) 1 − log |x(P ) − x(Q)| + log |∆| v 6 v ˆ under the assumption that none of P , Q, P + Q, nor P − Q is equal to O. Note that λE,v is independent of the choice of Weierstrass equation for E over F. It is useful to recall also the triplication formula; if 3P =6 O, then 2 (4.3) λˆ (3P )=9λˆ (P ) − log |(3x4 + b x3 +3b x2 +3b x + b )(P )| − log |∆| , E,v E,v 2 4 6 8 v 3 v where bi are the usual Weierstrass quantities; see e.g. [Si3, pg. 463].

4.2. Metrized divisors for elements of E(k). Fix non-torsion P ∈ E(k). Deﬁne ˆ DP := λE,ordγ (P )[γ]. γ∈XB(K) ˆ We remark that λE,ordγ (P ) ∈ Q [La, Chapter 11, Theorem 5.1], so DP is a Q-divisor on B. As P is deﬁned over K, the divisor is Gal(K/K)-invariant.

In [DM, Theorem 1.1] we established that DP can be equipped with an adelic, semipositive, continuous and normalized metrization

(4.4) DP := (DP , {λP,v}v∈MK ) ˆ an over the number ﬁeld K, where λP,v denotes the extension of t 7→ λEt,v(Pt) to Bv . It follows that the associated height functions satisfy ˆ hP (t) := hDP (t)= hEt (Pt)

for all t ∈ B(K) for which Et is smooth. Both minima e1(DP ) and e2(DP ) (deﬁned in §3.3) are equal to 0 [DM, Proposition 4.3]. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 18 For O ∈ E(k), we set

DO := (0, 0), the trivial divisor with all functions gv = 0. For torsion points T =6 O ∈ E(k), the metrized ˆ divisor DT can also be deﬁned by (4.4), with λT,v(t) := λEt,v(Tt) for all t ∈ B(K) with Et smooth ◮ and Tt =6 O ◭ . The following proposition is key for the passage from E(k) to E(k) ⊗ R.

Proposition 4.1. The metrized divisor DP is well deﬁned for P in E(k)/E(k)tors, up to isomorphism. Moreover, for each m ≥ 1 and any set of independent points P1,...,Pm ∈ E(k) and integers a1,...,am, the following metrized divisors are isomorphic: m 2 (4.5) Da1P1+···+amPm ≃ ai − ai aj DPi + aiajDPi+Pj i=1 ! 1≤i

Lemma 4.3. Fix P ∈ E(k). For each nonzero m ∈ Z with |m| ≥ 2 such that mP =6 O, there exist h ∈ K(B) and constant c ∈ Q so that ˆ 2 ˆ λEt,v(mPt)= m λEt,v(Pt)+ c log |h(t)|v

at every place v and for each t ∈ B(K) such that Et is smooth. If P is torsion of order m ≥ 2, we have ˆ λEt,v(Pt)= c log |h(t)|v, for some c ∈ Q and h ∈ K(B).

Proof. Upon replacing P by −P , it suffices to prove the statement for m ≥ 2. The duplication formula (4.1) provides the desired result for m = 2, assuming that 2P =6 O. Now fix any m ≥ 3 and P ∈ E(k), and assume that mP =6 O, (m − 1)P =6 O and (m − 2)P =6 O. Then the quasi-parallelogram law (4.2) implies ˆ ˆ ˆ ˆ (4.6) λEt,v(mPt) = 2 λEt,v((m − 1)Pt)+2 λEt,v(Pt) − λEt,v((m − 2)Pt) 1 − log |x((m − 1)P ) − x(P )| + log |∆ | , t t v 6 t v ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 19 for each t ∈ B(Cv) such that Et is smooth and mPt =6 Ot,(m−1)Pt =6 Ot and (m−2)Pt =6 Ot ˆ and therefore for all t ∈ B(Cv) by the continuity of the local height t 7→ λEt,v(Pt) ∈ R∪{±∞}. The desired relation, for all non-torsion points and for all m ≥ 3, then follows from (4.6) by an easy induction. Now suppose that 2P = O with P =6 O. Then we have 3P = P =6 O, and the triplication formula (4.3) implies that ˆ ˆ ˆ λEt,v(3Pt)=9λEt,v(Pt) − c log |h(t)|v = λEt,v(Pt), for a constant c ∈ Q and h ∈ K(B) and for all but finitely many t. The equation then ˆ holds for all t ∈ B(Cv) by the continuity of the local heights, and it implies that λEt,v(Pt)= c 8 log |h(t)|v. For a torsion point P of order 3, we have 2P = −P =6 O, so we may apply the duplication formula (4.1) to see that ˆ ˆ ˆ ˆ λEt,v(2Pt)=4λEt,v(Pt) − c log |h(t)|v = λEt,v(−Pt)= λEt,v(Pt), ˆ c for a constant c ∈ Q and h ∈ K(B). It follows that λEt,v(Pt)= 3 log |h(t)|v. Finally, suppose that P is torsion of order n ≥ 4, and note that (n − 1)P = −P =6 O, (n − 2)P =6 O and (n − 3)P =6 O. We infer from (4.6) with 3 ≤ m ≤ n − 1 inductively that ˆ 2ˆ ˆ ˆ λEt,v((n − 1)Pt)=(n − 1) λEt,v(Pt) − c log |h(t)|v = λEt,v(−Pt)= λEt,v(Pt) ˆ c for a rational function h ∈ K(B) and c ∈ Q, so that λEt,v(Pt)= n2−2n log |h(t)|v. Proof of Proposition 4.1. Lemma 4.3 implies that

DP ≃ DO for every torsion point P ∈ E(k). Furthermore, for any non-torsion point P , Lemma 4.3 also implies that 2 DaP ≃ a DP for all a ∈ Z, demonstrating (4.5) for m = 1. Therefore, if P is non-torsion and Q is torsion of order n ≥ 2, we have 1 1 D ≃ D = D ≃ D . P +Q n2 n(P +Q) n2 nP P

This proves that the metrized divisors depend only on the class in E(k)/E(k)tors, up to isomorphism.

Now ﬁx any m ≥ 2, and any collection of independent points P1,...,Pm ∈ E(k) and integers a1,...,am. Deﬁne a divisor on B by m ′ 2 D = ai − ai aj DPi + aiaj DPi+Pj , i=1 ! 1≤i

at all places v of K and for all but ﬁnitely many t ∈ B(Cv). Lemma 4.4. Let P, Q, R ∈ E(k) be independent points deﬁned over K. Then, there is a

rational function fP,Q,R ∈ K(B) such that ˆ ˆ ˆ ˆ λEt,v(Pt + Qt + Rt) = λEt,v(Pt + Rt)+ λEt,v(Pt + Qt)+ λEt,v(Qt + Rt) ˆ ˆ ˆ −λEt,v(Pt) − λEt,v(Qt) − λEt,v(Rt) − log |fP,Q,R(t)|v.

for all t ∈ B(K) such that Et is smooth and all v ∈ MK . Proof. The proof follows by applying the quasi-parallelogram law (4.2) for the pairs {P + R, Q}, {P, R−Q}, {P +Q, R} and {R, Q} and taking an alternating sum as in [Si6, Theorem 9.3]. Lemma 4.5. Fix independent P, Q ∈ E(k). For each (a, b) ∈ Z2 \{(0, 0)}, there is rational function ha,b ∈ K(B) such that ˆ 2 ˆ ˆ λEt,v(aPt + bQt) = (a − ab)λEt,v(Pt)+ abλEt,v(Pt + Qt)+ 2 ˆ (b − ab)λEt,v(Qt) − log |ha,b|v, for all t ∈ B(K) such that Et is smooth and all v ∈ MK . Proof. The proof follows from the quasi-parallelogram law by an easy induction. Lemma 4.3 provides the desired result if one of a or b is 0. Next we will show that for each n ∈ Z there is a rational function g ∈ K(B) so that ˆ 2 ˆ ˆ λEt,v(nPt + Qt)=(n − n)λEt,v(Pt)+ nλEt,v(Pt + Qt) (4.8) ˆ + (1 − n)λEt,v(Qt) − log |g|v. Replacing P by −P we may assume that n ≥ 1. For n = 1 the statement is clear. For n ≥ 1, the quasi-parallelogram law (4.2) implies that ˆ ˆ λEt,v((n + 1)Pt + Qt) = λEt,v(nPt +(P + Q)t) ˆ ˆ ˆ = 2λEt,v(nPt)+2λEt,v(Pt + Qt) − λEt,v((n − 1)Pt − Qt) 1 − log |x(nP ) − x(P + Q )| + log |∆| t t t v 6 v ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 21 and (4.8) follows inductively from Lemma 4.3. Using (4.8) we now have a rational function h ∈ K(B) so that ˆ 2 ˆ ˆ (4.9) λEt,v(aPt + bQt) = (a − a)λEt,v(Pt)+ aλEt,v(Pt + bQt)+ ˆ (1 − a)λEt,v(bQt) − log |h|v. The lemma then follows by another application of (4.3) and (4.8), exchanging the roles of P and Q. Finally, a simple induction using Lemmas 4.4 and 4.5 implies that for any m ≥ 2, and for any integers a1,...,am, the equality (4.7) holds, for some rational f. This completes the proof.

4.3. Metrized divisors for elements of E(k) ⊗ R. Fix nonzero X ∈ E(k) ⊗ R. Choose independent points P1,...,Pm ∈ E(k) that define a basis for E(k) ⊗ R, and write X = x1P1 + ··· + xmPm with xi ∈ R. With a slight abuse of notation, we identify the two isomorphic metrized divisors in Proposition 4.1 and define an adelically metrized R-divisor on B(K), over the number field K, by m 2 (4.10) DX := xi − xi xj DPi + xixj DPi+Pj i=1 1≤i

Theorem 4.6. Fix nonzero X ∈ E(k)⊗R. The metrized divisor DX of (4.10) is continuous, adelic, semipositive and normalized. The degree of the underlying R-divisor DX is hˆE(X) > 0. Its associated height function satisﬁes ˆ (4.12) hX (t)= hEt (Xt)

for all t ∈ B(K) with smooth ﬁber Et. Further, up to isomorphism, DX is independent of the choice of basis for E(k).

Proof. Fix x1,...,xm ∈ R and choose sequences of rational numbers an,i/an,0 → xi for i =1,...,m. From Proposition 4.1 we know that the functions 1 m a2 − a a λ (t)+ a a λ (t) a2 n,i n,i n,j Pj ,v n,i n,j Pi+Pj ,v n,0 i=1 1≤i

Dan,1P1+···+an,mPm · Dan,1P1+···+an,mPm =0, for all n ∈ N. In view of Proposition 4.1 we then have 2 1 m a2 − a a D + a a D =0, a4 n,i n,i n,j Pj n,i n,j Pi+Pj n,0 i=1 1≤i

section, we constructed metrized R-divisors DX and associated height functions hX for each element X ∈ E(k) ⊗ R. In this section, we look at the sets of “small” points for the height hX . We conclude the section with a proof of Theorem 1.10. 5.1. Small sequences exist. For an adelic, continuous, semipositive, and normalized metrized

R-divisor D, an inﬁnite sequence {tn} ⊂ B(K) is said to be small if

hD(tn) → 0 as n →∞.

Proposition 5.1. For every nonzero X ∈ E(k) ⊗ R, there exist small sequences for DX , so that the essential minimum is e1(DX )=0. More precisely, write X = x1P1 + ···+ xmPm for xi ∈ R and independent Pi ∈ E(k), and choose integers ai,n for i = 1,...,m and n ∈ N so m−1 that [a1,n : ··· : am,n] → [x1 : ··· : xm] as n → ∞ in the real projective space RP . Then there exists an inﬁnite non-repeating sequence of points tn ∈ B(K) at which

(a1,nP1 + ··· + am,nPm)tn is torsion in the ﬁber Etn (K). Moreover, for any such sequence {tn} ⊂ B(K) we have

hX (tn) → 0 as n →∞. To prove Proposition 5.1, we begin with a well-known statement that follows from Silver- man’s specialization theorem [Si1].

Lemma 5.2. Fix any set of independent points P1,...,Pm in E(k), and let h be any Weil height function on B associated to a divisor of degree 1. The set of all t for which there exist

integers a1,...,am, not all zero, such that

a1P1,t + ··· + amPm,t = Ot

in Et has bounded h-height. Proof. For each non-torsion point Q ∈ E(k) we have [Si1] ˆ h(Qt) ˆ lim = hE(Q) > 0. h(t)→∞ h(t)

Because det(hPi,PjiE)i,j > 0, it follows that the set

R(P1,...,Pm)= {t ∈ B(K) : det(hPi,t,Pj,tit)=0} ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 24 also has bounded height. This set R(P1,...,Pm) clearly contains the set of t at which the points become linearly dependent.

Proof of Proposition 5.1. Write X = x1P1 + ··· + xmPm for independent P1,...,Pm ∈ E(k) and x1,...,xm ∈ R. Fix a sequence of positive integers Mn → ∞ as n → ∞. For i = 1,...,m, choose any sequence of integers ai,n so that ai,n/Mn → xi as n →∞, and set

Qn = a1,nP1 + ... + am,nPm ∈ E(k), so that 1 Q → X in E(k) ⊗ R. Mn n Consider the set

Tor(Qn)= {t ∈ B(K): Qn,t is torsion in Et}

For each n, the set Tor(Qn) is inﬁnite; in fact, it is dense in B(C)[DM, Proposition 6.2] [Za, §III.2 and Notes to Chapter III]. Moreover, from Lemma 5.2, this set has bounded height in the base curve B with respect to any chosen Weil height h, and the height is bounded independent of n. Therefore, from [Si1, Theorem A], we can ﬁnd H > 0 so that

hPi (t) ≤ H and hPi+Pj (t) ≤ H

for all t ∈ n Tor(Qn) and for all i, j. From the formula for the height h given in (4.11) and the formula for the height of Q S X n appearing in Proposition 4.1, we have the following. For any given ε> 0, there exists N > 0 so that m 2 1 2 ai,n ai,n aj,n hX (t) − 2 hQn (t) = xi − xi xj − 2 + hPi (t) Mn Mn Mn Mn ! i=1 j6=1 j6=i X X X

ai,naj,n + xixj − 2 hPi+Pj (t) <ε Mn ! 1≤i N and for all t where h (t) ≤ H and h (t) ≤ H for all i, j. In particular, the Pi Pi+Pj estimate holds for all t ∈ n≥1 Tor(Qn). For each n and for every t ∈ Tor(Q ), we have h (t) = 0. Choosing any sequence of S n Qn distinct points tn ∈ B(K) so that Qn,tn is torsion in Etn , we may conclude that

hX (tn) → 0 as n →∞.

5.2. Characterization of small sequences. Here, we observe that small sequences for real points X ∈ E(k) ⊗ R always arise from a construction similar to that of Proposition 5.1, where relations between the generators are “almost” satisﬁed. We will use this next proposition in the proof of Theorem 7.1. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 25

Proposition 5.3. Let M be a torsion-free subgroup of E(k) of rank m, generated by S1,...,Sm. Set hM (t) = det(hSi,t,Sj,tit), for the Néron-Tate bilinear form h·, ·it on the fiber Et(K). For a non-repeating infinite sequence tn ∈ B(K), the following are equivalent

(1) lim infn→∞ hM (tn) = 0; (2) there is a non-zero X ∈ M ⊗ R such that lim infn→∞ hX (tn)=0.

(3) there are sequences of points si,n ∈ Etn (K), for i =1,...,m, satisfying ˆ lim inf max hEt (si,n) =0 n→∞ i n and so that the points

S1,tn − s1,n,..., Sm,tn − sm,n

satisfy a linear relation over Z in Etn (K).

This proposition relies heavily on Silverman’s specialization results [Si1, Theorem A and Theorem B]. We point out that [Si1, Theorem B] holds for real points X ∈ E(k) ⊗ R by the bilinearity of the N´eron-Tate pairing. We begin with a lemma.

Lemma 5.4. Assume we are in the setting of Proposition 5.3. Assume further that there

are sequences of points si,n ∈ Etn (K), for i =1,...,m, satisfying ˆ sup max hEtn (si,n) < ∞ n i for which the points

S1,tn − s1,n,..., Sm,tn − sm,n satisfy a linear relation over Z in Etn (K). Then the sequence {tn} will have bounded height in B(K) with respect to any Weil height on B.

Proof. Fix any Weil height h on B(K) of degree 1. Consider the m × m matrix

An := (hSi,tn − si,n, Sj,tn − sj,nitn )i,j

where h·, ·itn is the N´eron-Tate inner product on the ﬁber Etn (K). Our assumption implies that

det An =0 for all n. Assume that h(tn) → ∞. Then by Silverman’s specialization theorem [Si1, Theorem B] we have

hSi,tn ,Sj,tn itn →hSi,SjiE, h(tn) ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 26 as n →∞ for all i, j =1,...,m. On the other hand, the bounded height of the perturbations si,n and the Cauchy-Schwarz inequality for h·, ·itn implies that ˆ ˆ hs ,s i hEtn (si,n)hEtn (sj,n) i,n j,n tn ≤ → 0. h(t ) q h(t ) n n

Using Silverman’s specialization [Si1 , Theorem A] we also have

ˆ ˆ hS ,s i hEtn (Si,tn )hEtn (sj,n) i,tn j,n tn ≤ → 0. h(t ) q h(t ) n n

Combining these estimates, we obtain

det An 0= m −→ det(hSi,SjiE)i,j =06 , (h(tn)) which is a contradiction.

Proof of Proposition 5.3. Assume condition (2). Let X ∈ M ⊗ R be nonzero and {tn} a sequence for which lim inf hX (tn)=0. Write X = x1S1 + ··· + xℓSℓ, for xi ∈ R not all equal n→∞ to 0. After reordering the points Si we may assume that x1 =6 0. Notice that ˆ hEtn (Xtn ) hXtn ,S2,tn i ··· hXtn ,Sℓ,tn i hS ,X i hˆ (S ) ··· hS ,S i  2,tn tn Etn 2,tn 2,tn ℓ,tn  2 det . . . = x1 hM (tn), . . .    hS ,X i hS ,S i ··· hˆ (S )   ℓ,tn tn ℓ,tn 2,tn Etn ℓ,tn    which easily follows by subtracting from the first column the sum of xi times the j-th column over all j =2,...,ℓ and then subtracting from the first row the sum of xi times the i-th row over all i =2,...,ℓ. Expanding the determinant along the first column we get ℓ 2 (5.1) x1 hM (tn)= hX (tn) f1,n + hSj,tn ,Xtn ifj,n, j=2 X

where for all n ∈ N the fj,n are polynomial functions of the quantities hSj,tn ,Xtn i and

hSj,tn ,Sk,tn i for j, k = 2,...,ℓ. Passing to a subsequence of {tn} we have lim hX (tkn ) = 0. n→∞

In particular, since X is non-trivial, [Si1, Theorem B] yields that {h(tkn )}n∈N is a bounded sequence. Using then [Si1, Theorem A], the functoriality of heights and the Cauchy-Schwarz inequality we get

(5.2) max{|f1,kn |,..., |fℓ,kn |} ≤ L, for some L> 0. Moreover, for all j =2,...,ℓ and all n ∈ N we have 2 ˆ ˆ (5.3) |hSj,t ,Xt i| ≤ hEt (Sj,t )hEt (Xt ) ≤ L hX (tkn ) → 0. kn kn kn kn kn kn ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 27 Our assumption on X together with (5.1), (5.2) and (5.3) yield

lim inf hM (tn)=0, n→∞ proving that condition (1) holds.

Now assume (1). Let At =(hSi,t,Sj,ti)i,j, so that hM (t) = det At, and consider the family of quadratic forms ˆ qt(~z) := hEt (z1S1,t + ··· zmSm,t) m 2 ˆ ⊤ = zk hEt (Sk,t)+2 zizjhSi,t,Sj,ti = ~zAt ~z , i

lim inf det Atn =0, n→∞

so, if λn is the smallest eigenvalue of Atn , then

lim inf λn =0. n→∞

Let ~vn =(v1,n,...,vm,n) =6 0 be an eigenvector of Atn corresponding to λn. Then ⊤ 2 ~vn Atn ~vn = λn||~vn|| , so that

~vn (5.4) lim inf qtn = lim inf λn =0. n→∞ ||~v || n→∞ n Passing to a subsequence of the {tn}, we have limn→∞ hM (tn) = 0, and passing to a further subsequence, we may set ~v ~x := lim n ∈ Rm \{~0}. n→∞ ||~vn||

By [Si1, Theorem B], the height of {tn} is bounded with respect to any choice of Weil height on B (because det(hSi,SjiE) =6 0). In view of [Si1, Theorem A], we get that the sequences

{hSi,tn ,Sj,tn i}n for i, j =1,...,ℓ are bounded. Thus (5.4) yields

lim qtn (~x)=0. n→∞

In other words, for X = x1S1 + ··· + xmSm we have limn→∞ hX (tn) = 0, providing condition (2). Assuming (2), we now prove (3). Reordering the points and rescaling X if necessary, we

may assume that x1 = 1. Passing to a subsequence, we have ˆ (5.5) hEtn (S1,tn + x2S2,tn + ··· + xmSm,tn ) → 0, ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 28 as n →∞. Let a2,n,...,am,n be inﬁnite sequences of integers satisfying ai,n/n → xi for each ˆ i =2,...,m. As hE(X) =6 0, we have by Silverman specialization [Si1, Theorem B] that the sequence {tn} has bounded height in B. Invoking [Si1, Theorem A] we get that all sequences ˆ {hSi,tn ,Sj,tn iEtn }n∈N are bounded. Using the fact that each hEtn (·) deﬁnes a quadratic form

on Etn (K), line (5.5) yields 1 (5.6) hˆ S + (a S + ··· + a S ) → 0. Etn 1,tn n 2,n 2,tn ℓ,n ℓ,tn

Since K is algebraically closed we may ﬁnd sn ∈ Etn (K) so that

(5.7) nsn = a2,nS2,tn + ··· + aℓ,nSℓ,tn .

Letting s1,n := S1,tn + sn and si,n := Otn for all i =2,...,m, equation (5.6) yields ˆ hEtn (si,n) → 0

for each i =1,...,m. Moreover by (5.7) we have that the set {S1,tn − s1,n,S2,tn ,...,Sℓ,tn } is linearly dependent in Etn for every n. Last, we assume condition (3) and prove (2). Pass to a subsequence so that

ˆ lim max hEt (si,n) =0. n→∞ i n Choose sequences of integers ai,n for i =1,...,m, not all 0, so that

a1,n(S1,tn − s1,n)+ ··· + am,n(Sm,tn − sm,n)= Otn for all n. Now, letting Mn = maxi ai,n, we can pass to a further subsequence so that

ai,n → xi ∈ R Mn as n →∞ for each i, with at least one xi nonzero. This implies that (5.8) 1 1 hˆ (a S + ··· + a S ) = hˆ (a s + ··· + a s ) → 0 Etn M 1,n 1,tn m,n m,tn Etn M 1,n 1,n m,n m,n n n as n →∞. Finally set

X = x1S1 + ... + xmSm.

From Lemma 5.4, we know that the sequence {tn} has bounded height and by [Si1, Theorem ˆ A] we get that the sequences {hEtn (Si,tn )}n are bounded. Therefore, from the deﬁnition of hX in (4.11), line (5.8) implies that hX (tn) → 0. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 29

5.3. Proof of Theorem 1.10. From Theorem 4.6, we know that DX is a continuous, adelic, semipositive, and normalized metrization on an ample R-divisor. Thus, Corollary 1.9 applies

to sequences with small height for hX . From Proposition 5.1, we have hX (tn) → 0 along any sequence tn for which i ri,nPi,t = Ot with ri,n ∈ Q satisfying ri,n → xi. The formula for ω at each place follows from the deﬁnition of D in (4.10). This completes the proof. X,v P X 5.4. Height 0. As we shall see, it follows from Theorem 1.1 that although small sequences exist as in Proposition 5.1, we don’t always have sequences with height 0: Proposition 5.5. Fix nonzero X ∈ E(k) ⊗ R. There exist inﬁnitely many t ∈ B(K) for

which hX (t)=0 if and only if there exists a real c > 0 so that cX is represented by an element of E(k). Proof. Suppose ﬁrst that cX is represented by an element P ∈ E(k) for some real c > 0. 1 Then hX (t)= c2 hP (t) at all t, so that hX (t) = 0 whenever Pt is torsion in Et. This holds at inﬁnitely many points t ∈ B(K). (See, e.g., [DM, Proposition 6.2].)

For the converse, write X = x1P1 + ··· + xmPm for independent P1,...,Pm ∈ E(k) and xi ∈ R, and assume that hX (t) = 0 for inﬁnitely many t. We can rewrite X as

X = α1Q1 + ··· + αsQs

for α1,...,αs ∈ R a basis for the span of {x1,...,xm} over Q and Q1,...,Qs ∈ E(k). For s = 1, we see that are we back in the setting where a multiple of X is represented by an element of E(k), so we may assume s > 1. But, for each t where hX (t) = 0, we must have that Xt = 0 in Et(Q)⊗R. By the choices of the αi, this means that each of the specializations Qi,t must be 0 in Et(Q) ⊗ R. (Compare [Mo2, Lemma 1.1.1].) In other words, the points Q1,...,Qs are simultaneously torsion at inﬁnitely many t. From Theorem 1.1, combined with (1.4), this implies that each pair Qi and Qj is linearly related. (Alternatively, here one could use the main results of [MZ1, MZ2].) Thus we infer that X = c Q for some c ∈ R and Q ∈ E(k).

6. The intersection number as a biquadratic form on E(k) ⊗ R Let E → B be a non-isotrivial elliptic surface defined over a number field K, and let E be the corresponding elliptic curve over the field k = K(B). Recall that we can enlarge the number field K, if necessary, to ensure that each section P : B → E is defined over K.

For each P ∈ E(k), a metrized divisor DP is defined by (4.4). We extended this definition to elements X ∈ E(k) ⊗ R with the definition (4.10). In this section, we study the basic properties of the Arakelov-Zhang intersection number

(X,Y ) 7→ DX · DY ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 30 as a biquadratic form on the ﬁnite-dimensional vector space E(k) ⊗ R. For X,Y ∈ E(k) ⊗ R, consider the metrized divisor

1 (6.1) D := D − D − D , hX,Y i 2 X+Y X Y of degree equal to the N´eron-Tate inner product of X and Y , 1 hX,Y i = hˆ (X + Y ) − hˆ (X) − hˆ (Y ) . E 2 E E E Our goal in this section is to prove:

Proposition 6.1. Fix X,Y,Z ∈ E(k) ⊗ R. The following hold:

(1) DX · DY = DY · DX ≥ 0, (2) DX · DY +Z + DX · DY −Z =2 DX · DY +2 DX · DZ, (3) DX · DX+Y = DX · DY , and 2 (4) for all x ∈ R, we have DX · DxY = x DX · DY .

For each X ∈ E(k) ⊗ R the map Y 7→ DX · DY deﬁnes a positive semideﬁnite quadratic form on E(k) ⊗ R, induced by the bilinear form (Y,Z) 7→ DX · DhY,Zi.

We begin with a lemma.

Lemma 6.2. We have DX · DY ≥ 0 for all X,Y ∈ E(k) ⊗ R.

Proof. From Theorem 4.6, both DX and DY are normalized, semipositive, continous adelic metrized divisors on B over K, so the lemma follows immediately from Theorem 3.2. Or we

can see it as a consequence of Theorem 1.8, because the height functions satisfy hX , hY ≥ 0 at all points of B(K).

The following lemma is a version of the Cauchy-Schwarz inequality.

Lemma 6.3. For each X ∈ E(k) ⊗ R, the intersection

(Y,Z) 7→ DX · DhY,Zi

is bilinear in Y,Z ∈ E(k) ⊗ R. Moreover,

2 (DX · DhY,Zi) ≤ (DX · DY )(DX · DZ ) for all X,Y,Z ∈ E(k) ⊗ R. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 31

Proof. Fix X,Y,Z,W ∈ E(k) ⊗ R, and a, b ∈ R. It is easy to see that DX · DhY,Zi = DX · DhZ,Y i. Then deﬁnitions (6.1) and (4.10) give

1 D · D = D · D − D · D − D · D X hY,aZ+bW i 2 X Y +aZ+bW X Y X aZ+bW 1 = D · (1 − a − b)D +(a2 − a − ab)D +(b2 − b − ab)D 2 X Y Z W

+ aDY+Z + bDY +W + abDZ+W − DX · DY 2 2 − DX · (a − ab)DZ +(b − ab)DW + abDZ+W 1 = aD · D − D − D + bD · D − D − D 2 X Y +Z Y Z X Y +W Y W

= a D X · DhY,Z i + b DX · DhY,W i, demonstrating bilinearity.

By Lemma 6.2 we have f(x)= DX · DY +xZ ≥ 0 for all x ∈ R. From deﬁnition (4.10), we have

2 DY +xZ = (1 − x)DY + xDY +Z +(x − x)DZ .

Deﬁnition (6.1) then yields

2 f(x)= DX · DY +2xDX · DhY,Zi + x DX · DZ ≥ 0

for all x ∈ R. Thus the quadratic f(x) has non-positive discriminant. The inequality follows.

We are now ready to prove the proposition.

Proof of Proposition 6.1. Fix X,Y,Z ∈ E(k) ⊗ R. The symmetry in (1) follows immediately from the symmetry of the intersection number, shown explicitly in (3.2) and extending to (3.3) by linearity. The non-negativity is the content of Lemma 6.2.

For (2), note that by Proposition 4.1 we have DY +Z + DY −Z ≃ 2DY +2DZ , for any X,Y,Z ∈ E(k). The same follows for X,Y,Z ∈ E(k) ⊗ R by deﬁnition (4.10). The intersection number is invariant under isomorphism for metrized divisors, and so

DX · (DY +Z + DY −Z )= DX · (2DY +2DZ).

For (3), we use Lemma 6.3 to compute that

2 DX · DhX,Y i ≤ DX · DX DX · DY =0 ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 32 because DX is normalized. Therefore,

0 = DX · DhX,Y i 1 = D · D − D · D − D · D 2 X X+Y X X X Y 1 = D · D − D · D , 2 X X+Y X Y so that

DX · DX+Y = DX · DY .

For (4), ﬁx x ∈ R. By the deﬁnition of DxY in (4.10), and the bilinearity of the intersection number, we have 2 DX · DxY = x DX · DY .

Properties (1)−(4) show that Y 7→ DX ·DY deﬁnes a positive semideﬁnite quadratic form as claimed. Finally, we note that 1 D · D = D · D − D · D − D · D X hY,Y i 2 X Y +Y X Y X Y 1 = D · D − 2 D · D = D · D , 2 X 2Y X Y X Y completing the proof of the proposition.

7. Equivalent formulations of Theorem 1.1 Recall that E → B denotes a non-isotrivial elliptic surface defined over a number field K, and let E be the corresponding elliptic curve over the field k = K(B). We extend K so that all sections of E → B are defined over K. In this section we prove: Theorem 7.1. Let E → B be a non-isotrivial elliptic surface defined over a number field K, and let E be the corresponding elliptic curve over the field k = K(B). Let Λ be any subgroup of E(k). The following are equivalent: (1) the conclusion of Theorem 1.1 holds for all P, Q ∈ Λ; (2) the conclusion of Theorem 1.3 holds for all sections C of E ℓ defined by the graph

t 7→ (Q1,t,...,Qℓ,t) for points Q1,...,Qℓ ∈ Λ, for all ℓ ≥ 2; (3) the conclusion of Theorem 1.5 holds for this Λ;

(4) the biquadratic form (X,Y ) 7→ DX · DY on Λ ⊗ R is non-degenerate, meaning that DX · DY =0 if and only if X and Y are linearly dependent over R; (5) for any pair X,Y ∈ Λ ⊗ R, if heights hX (t)= hY (t) for all t ∈ B(K), then X = ±Y ;

(6) for any pair X,Y ∈ Λ ⊗ R, if the Néron-Tate inner product satisfies hXt,YtiEt = 0 for all t ∈ B(K) with Et smooth, then either X or Y is 0; ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 33 (7) for any pair X,Y ∈ Λ ⊗ R, if there exists an infinite (non-repeating) sequence of

points tn ∈ B(K) for which

lim hX (tn)+ hY (tn)=0, n→∞ then X and Y are linearly dependent over R.

Throughout this section, we rely on the work carried out in Sections 3 – 6. Speciﬁcally, for each X ∈ E(k) ⊗ R, we can express X as a ﬁnite R-linear combination of elements

P1,...,Pm ∈ E(k). We appeal to Theorem 4.6 to know that DX is a well-defined, semipositive, normalized, continuous adelic metrization on B, defined over the number field K.

Further, (X,Y ) 7→ DX · DY is a well-deﬁned semipositive biquadratic form on E(k) ⊗ R by Proposition 6.1.

7.1. Intersection number 0. We begin by elaborating on the consequences of the existence of a pair X,Y ∈ E(k) ⊗ R for which DX · DY = 0. Recall that h·, ·it denotes the N´eron-Tate bilinear form on the ﬁber Et(K) ⊗ R, and h·, ·iE denotes the corresponding form on E(k) ⊗ R.

Proposition 7.2. Fix nonzero X,Y ∈ E(k) ⊗ R, and assume that DX · DY =0. Then for all t ∈ B(K) for which the ﬁber Et is smooth, we have

hÊ(X) hX,Y iE hX (t) = hY (t) and hXt,Ytit = hY (t). hÊ(Y ) hÊ(Y )

′ ′ Moreover, we have DX′ · DY ′ =0 for all X ,Y ∈ SpanR({X,Y }).

Proof. Assume that DX · DY = 0. From Theorem 4.6, each of DX and DY is a continuous, normalized, semipositive adelic metrization on an R-divisor on B. The degree of DX (respectively DY ) is hˆE(X) (respectively, hˆE(Y )). The relation between the heights hX and hY follows immediately from Theorem 3.4.

Using now part (3) of Proposition 6.1 our assumption that DX · DY = 0 implies that DX+Y · DY = DX+Y · DX = 0, and so

2 2 DxX+yY · DaX+bY = (x − xy)DX + xyDX+Y +(y − xy)DY 2 2 · (a − ab)DX + abDX+Y +(b − ab)DY = 0 for all x, y, a, b, ∈ R. In particular, we have ˆ ˆ hE(X + Y ) ˆ hEt (Xt + Yt)= hEt (Yt). hˆE(Y ) ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 34 so that ˆ ˆ ˆ hEt (Xt + Yt)= hEt (Xt)+ hXt, Qtit + hEt (Yt) implies hX,Y iE hXt,Ytit = hY (t) hˆE(Y )

for all t ∈ B(K) for which Et is smooth. The following proposition extends the observations of Proposition 5.3 to two independent relations.

Proposition 7.3. Let Λ be a subgroup of E(k) generated by independent, non-torsion ele-

ments P1,...,Pm, with m ≥ 2. The following are equivalent:

(1) there exist an inﬁnite, non-repeating sequence tn ∈ B(K) and points pi,n ∈ Etn (K) ˆ for i =1,...m, for which hEtn (pi,n) → 0 as n →∞, and the points

P1,tn − p1,n,...,Pm,tn − pm,n

satisfy two independent linear relations on Etn ; (2) there exist independent X,Y ∈ Λ ⊗ R for which

DX · DY =0.

Proof. Assume first that DX · DY = 0. Write X = x1P1 + ... + xmPm and Y = y1P1 + ··· + m ymPm for linearly independent coefficient vectors ~x, ~y ∈ R . From Proposition 7.2, we can replace X and Y by linear combinations of X and Y (and relabel the points Pi if needed) and so assume that x1 =1= ym and xm = y1 = 0. From Theorem 4.6, we know that DX and DY are normalized, semipositive, continuous adelic metrizations. By Proposition 5.1, we know that e1(DX ) = e1(DY ) = 0. Theorem 3.4 then implies that there is an infinite non-repeating sequence {tn} ⊂ B(K) so that

(7.1) hX (tn)+ hY (tn) → 0 as n → ∞. We now apply Proposition 5.3 to each of hX and hY to show that small

perturbations of the specializations Pi,tn must satisfy two independent relations in the ﬁbers

Etn (K). More precisely, we choose integers ai,n, bi,n for each n ≥ 1 and each i =2,...,m−1, so that a b i,n → x and i,n → y n i n i as n →∞. As in the proof of Proposition 5.3 (2) =⇒ (3), we choose pn ∈ Etn (K) so that

npn = a2,nP2 + ··· + am−1,nPm−1. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 35

Set p1,n = P1,tn + pn ∈ Etn (K). Then 1 hˆ (p )= hˆ P + (a P + ··· + a P ) → 0, Etn 1,n Etn 1,tn n 2,n 2 m−1,n m−1

and {P1,tn − p1,tn ,P2,tn ,...,Pm−1,tn } satisfy a linear relation. On the other hand, we can

repeat the same argument with Y and ﬁnd point qn ∈ Etn (K) so that

n qn = b2,nP2 + ··· + bm−1,nPm−1

and set pm,n = Pm,tn + qn. Then 1 hˆ (p )= hˆ (b P + ··· + b P )+ P → 0, Etn m,n Etn n 2,n 2 m−1,n m−1 m,tn

and {P2,tn ,...,Pm−1,tn ,Pm,tn − pm,tn } satisfy a linear relation. It follows that the points

{P1,tn − p1,tn ,P2,tn ,...,Pm−1,tn ,Pm,tn − pm,tn }

satisfy two independent linear relations in Etn (K) for all n. For the converse direction, we assume there are an inﬁnite, non-repeating sequence tn ∈ ˆ B(K) and points pi,n ∈ Etn (K) for i = 1,...m, for which hEtn (pi,n) → 0 as n → ∞, and such that

{P1,tn − p1,n,...,Pm,tn − pm,n}

satisfy two independent linear relations on Etn . From Lemma 5.4, we know that the sequence {tn} must have bounded height. Choose integers ai,n, bi,n for n ≥ 1 and i =1,...,m so that the independent relations are expressed as

a1,n(P1,tn − p1,n)+ ··· + am,n(Pm,tn − pm,n)= Otn and

b1,n(P1,tn − p1,n)+ ··· + bm,n(Pm,tn − pm,n)= Otn Relabeling the points if necessary, we can rewrite the expressions as

(P1,tn − p1,n)+ r2,n(P2,tn − p2,n)+ ··· + rm,n(Pm,tn − pm,n)= Otn and ′ ′ r1,n(P1,tn − p1,n)+ ··· + rm−1,n(Pm−1,tn − pm−1,n)+(Pm,tn − pm,n)= Otn ′ ′ for bounded sequences of rational numbers r2,n,...,rm,n and r1,n,...,rm−1,n. Passing to a subsequence we may assume that

′ ri,n → xi ∈ R and ri,n → yi ∈ R ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 36 for each i. Then, recalling that {tn} has bounded height and that the perturbations pi,n have ˆ heights tending to 0 and using [Si1, Theorem A] to infer that {hEtn (Pi,tn )}n are bounded for each i, we conclude that

hX (tn) → 0 and hY (tn) → 0

along this subsequence, for X = P1 +x2P2 +...+xmPm and Y = y1P1 +···+ym−1Pm−1 +Pm. From Theorem 3.4, we have that DX · DY = 0. 7.2. Proof of Theorem 7.1. Throughout this proof, we ﬁx a ﬁnitely-generated subgroup

Λ ⊂ E(k). Assume it is of rank m ≥ 1 with torsion-free part generated by P1,...,Pm ∈ E(k).

ˆ (1) ⇐⇒ (4) Recall that the Néron-Tate height hE on Λ extends to a positive definite quadratic form on Λ ⊗ R. It follows (by Cauchy-Schwarz) that the Néron-Tate regulator ˆ ˆ 2 RE(X,Y ) := hE(X)hE(Y ) −hX,Y iE ≥ 0

extends to a biquadratic form on Λ ⊗ R satisfying RE(X,Y ) = 0 if and only if X and Y are linearly dependent over R. As

F (X,Y ) := DX · DY is also biquadratic on Λ⊗R and satisﬁes F (X,Y ) ≥ 0 (see Proposition 6.1), the upper bound

on DX · DY in Theorem 1.1 follows. Condition (1) is then equivalent to the statement that F (X,Y ) = 0 if and only if X and Y are linearly dependent over R. m m In details, if we assume (1), and if the pair X = i=1 xiPi and Y = i=1 yiPi with Pi ∈ Λ satisfy D ·D = 0, then we can approximate by rational combinations P = 1 a P → X X Y P Pn n i,n i and Q = 1 b P → Y with integers a , b , and compute that n n i,n i i,n i,n P 1 c D · D P= D · D ≥ R a P , b P = c R (P , Q ). Pn Qn n4 P ai,nPi P bi,nPi n4 E i,n i i,n i E n n Letting n →∞ shows that RE(X,Y ) = 0, implyingX that X,Y Xare linearly dependent over R. Now assume (4), so that F (·, ·) is nondegenerate on the ﬁnite-dimensional V =Λ⊗R. Using 1/2 the inner product h·, ·iE on V and associated norm k·k = hˆE(·) , we have (by continuity and compactness) uniform positive upper and lower bounds on DX · DY over all pairs X,Y ∈ V satisfying hX,Y iE = 0 and kXk = kY k = 1. On the other hand, RE(X,Y ) = 1 for all such pairs, and so there is a positive constant c = c(V ) so that −1 (7.2) c RE(X,Y ) ≤ DX · DY ≤ c RE(X,Y ) all pairs X,Y ∈ V satisfying hX,Y iE = 0 and kXk = kY k = 1. By scaling the points, this extends to orthogonal pairs of any norm. For an arbitrary pair X,Y ∈ V , we write ′ ′ Y = Y + xX with hY ,XiE = 0 and x ∈ R, and observe that DX · DY = DX · DY ′ from ′ ′ Proposition 6.1. We also have RE(X,Y + xX)= RE(X,Y ) and so (7.2) holds for all pairs X and Y in V . ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 37 (4) ⇐⇒ (7) Fix any pair X,Y ∈ Λ ⊗ R, and express X and Y as R-linear combinations of elements P1,...,Pm ∈ Λ. Theorem 4.6 shows that DX and DY are normalized, continuous, semipositive adelic metrizations on R-divisors, and Proposition 5.1 shows that each has

essential minimum equal to 0. Theorem 3.4 then implies that DX · DY = 0 if and only if the heights hX and hY have a common small sequence in B(K).

(3) ⇐⇒ (7) Assume that (7) holds. We aim to prove the conclusion of Theorem 1.5

for this Λ. Suppose that there is an inﬁnite, non-repeating sequence tn ∈ B(K) with

hΛi (tn) → 0 for all i = 1,...,m. From Lemma 5.3, we may choose point X1 ∈ Λ1 so that

lim infn→∞ hX1 (tn) = 0. Pass to a subsequence so that limn→∞ hX1 (tn) = 0. For each i =

2,...,m, we successively apply Lemma 5.3 to ﬁnd Xi ∈ Λi for which lim infn→∞ hXi (tn)=0 and then pass to a further subsequence so that limn→∞ hXi (tn) = 0. In this way, we have an inﬁnite, non-repeating sequence of points tn ∈ B(K) so that limn→∞ hXi (tn) = 0 for all m i. However, as i=1 Λi = {0}, at least two of the Xi must be independent. This contradicts (7). T Assume now that (3) holds. Fix a pair of independent nonzero points X,Y ∈ Λ ⊗ R, and

suppose that there is an inﬁnite nonrepeating sequence tn ∈ B(K) so that

hX (tn)+ hY (tn) → 0.

Then hX (tn) → 0 and hY (tn) → 0. We write

X = a1P1 + ··· + amPm

Y = b1P1 + ··· + bmPm,

with ai, bj ∈ R and independent Pi ∈ Λ. We want to show that

(7.3) lim inf hΛ (tn)=0 n→∞ i for all i = 1,...,m, contradicting (3). Fix i ∈ {1,...,m}. If bi = 0, then Y ∈ Λi ⊗ R and (7.3) follows from Lemma 5.3. If on the other hand b =6 0, then X − ai Y ∈ Λ ⊗ R and by i bi i the parallelogram law we also have

a hX− i Y (tn) → 0. bi As before, equation (7.3) follows by Lemma 5.3.

(7) =⇒ (6) Assume that (7) holds. Fix X,Y ∈ Λ ⊗ R, and suppose that hXt,Ytit = 0

for all t ∈ B(K) for which the ﬁber Et is smooth. By Proposition 5.1 there is an inﬁnite ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 38 sequence tn ∈ B(K) with hX−Y (tn) → 0. Since hXtn ,Ytn itn = 0 we have

hX (tn)+ hY (tn) = hX (tn) − 2hXtn ,Ytn itn + hY (tn)

= hX−Y (tn) → 0. Thus by (7) we get that either X or Y is 0 or there are non-zero a, b ∈ R such that aX = bY .

In the latter case, our assumption that hXt,Ytit = 0 for all t implies that both X and Y are 0. The assertion follows.

(6) =⇒ (5) Assume that (6) holds. Fix X,Y ∈ Λ ⊗ R, and suppose that hX (t) = hY (t) for all t ∈ B(K). If X = 0 or Y = 0 then our assumption that hX (t) = hY (t) for all t implies that X = Y = 0 in Λ ⊗ R. Thus we may assume that both X and Y are non-zero.

Since hX (t) = hY (t) for all t, Silverman’s specialization theorem [Si1, Theorem B] implies that hˆE(X) = hˆE(Y ). From Theorem 3.4 we know that DX · DY = 0, and therefore, from Proposition 7.2, we have hX,Y i hX ,Y i = E hˆ (Y ), t t t ˆ Et t hE(Y ) or equivalently,

hX,Y iE Xt − Yt,Yt =0, hˆ (Y ) * E +t for all t. By our assumption (6) and since Y =6 0, we have hX,Y i X = E Y. hˆE(Y )

Recalling that hˆE(X)= hˆE(Y ), we get X = ±Y in Λ ⊗ R, as claimed.

(5) =⇒ (7) Suppose there exist nonzero X,Y ∈ Λ ⊗ R and an inﬁnite, non-repeating

sequence tn ∈ B(K) for which hX (tn)+ hY (tn) → 0. By Theorem 4.6 we know that both hX and hY are induced by normalized semipositive adelic metrizations on ample divisors DX and DY on B, of degrees hÊ(X) and hÊ(Y ), respectively. We may thus apply Theorem 3.4 to get hÊ(X) hX (t)= hY (t)= hxY (t) hÊ(Y ) ˆ for all t in B(K), where x = hE (X) . Our assumption (5) then yields that X = ±xY as hÊ (Y ) claimed. r

(2) ⇐⇒ (4) Fix any collection of points Q1,...,Qℓ in Λ, and let C be the irreducible ℓ curve in E deﬁned by a section (Q1,...,Qℓ) over B. To say that C is not contained in ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 39

flat subgroup scheme of positive codimension means that the points Q1,...,Qℓ are linearly ℓ independent. To say that the curve C in E defined by (Q1,...,Qℓ) intersects the tube T (E m,{2}, ǫ) infinitely often for every ǫ > 0 means that there is an infinite non-repeating sequence of points tn ∈ B(K) and small points qi,n ∈ Etn (K) for each n so that the points

{Q1,tn − q1,n,...,Qℓ,tn − qℓ,n} satisfy two linear relations in Etn . Therefore the equivalence of (2) and (4) is the statement of Proposition 7.3. This completes the proof of the theorem.

8. Equality of measures In this section we prove Theorem 1.12. We begin by describing the complex geometry of elements X of the real vector space E(k) ⊗ R.

8.1. Real points as holomorphic curves. Given a non-isotrivial elliptic surface E → B defined over the number field K, let S ⊂ B be the finitely-punctured Riemann surface over which all fibers Et(C) are smooth. Write ES for the open subset of E over S. Recall that each rational point P ∈ E(k) determines a holomorphic section of E → B defined by t 7→ Pt ∈ Et(C) for t ∈ S(C). The Betti coordinates of P ∈ E(k) are defined as follows. Passing to the universal cover π : S → S, there is a holomorphic period function τ : S → H e taking values in the upper half plane, so that the fibers of E satisfy e S Eπ(s)(C) ≃ C/(Z ⊕ Zτ(s))

for all s ∈ S. Passing to the universal cover of Eπ(s)(C) for each ﬁber, we obtain a holomorphic line bundle over S, trivialized by sending generator 1 of the lattice to 1 ∈ C. For each P ∈ E(k),e the corresponding section of E → B therefore lifts to a holomorphic function e ξP : S → C The Betti map of P is the real-analytic map β : S → R2 given by e P β (s)=(x(s),y(s)) such that ξ (s)= x(s)+ y(s) τ(s). P e P The coordinates x and y themselves depend on the choices of τ and ξP , but as proved in [CDMZ], we have

(8.1) ωP = dx ∧ dy,

independent of the choices, for the curvature distribution of DP at an archimedean place of K. ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 40 For points Q ∈ E(k) and integers n satisfying nQ = P ∈ E(k), the holomorphic functions 1 ξ = ξ , Q n P

for any choice of ξP (associated to a ﬁxed choice of τ), deﬁne holomorphic curves in ES that are not necessarily sections over S. Letting P range over all elements of E(k) and passing to limits as n →∞, we obtain the following: Proposition 8.1. Fix a choice of period function τ : S → H. Each X ∈ E(k) ⊗ R is represented by a holomorphic function ξX : S → C given by ξX = xX + yX τ for which e ω = dx ∧ dy X e X X

on S. More precisely, if X = i xiPi for xi ∈ R and Pi ∈ E(k), then the Betti map of X can be taken to be P βX = xiβPi i X for choices of Betti maps βPi . The map βX is uniquely determined up to translation by a constant in R2.

Recall here that the curvature distribution for DX as in 4.10 at an archimedean place is given by

2 ωX = xi − xixj ωPi + xixj ωPi+Pj , i ! i

Remark 8.2. The holomorphic functions ξX of Proposition 8.1 project to holomorphic curves in ES over S. For torsion points of E(k) representing the 0 of E(k)⊗R, the holomorphic curves given by Proposition 8.1 are precisely the leaves of the Betti foliation, because we allow for arbitrary translation in R2. By deﬁnition, the leaves of the Betti foliation have constant Betti coordinates; see, e.g., [ACZ], [CDMZ], and [UU] for more information. For

each nonzero X ∈ E(k) ⊗ R, there is a corresponding foliation of ES. When an element X is represented by P ∈ E(k), the foliation is simply the corresponding Betti foliation for the elliptic surface with P chosen as the zero section. Proof of Proposition 8.1. For each P ∈ E(k) we have Betti coordinates as defined above. For 1 multiples n P , there are natural choices of the functions ξ, also as described above, defined 1 up to translation by elements of n (Z ⊕ Zτ). Letting Pn be a sequence in E(k) such that 1 R Qn := n Pn converges to X in E(k) ⊗ , the families of holomorphic functions ξQn converge, locally uniformly in S. This defines a family of limit holomorphic functions ξX . Because of the choices for ξPn and ξQn , we see that ξX is only defined up to translation by real multiples e ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 41 of lattice elements. Fix a choice of ξX and consider the measure νX = dxX ∧ dyX . This measure is clearly independent of the choices. Furthermore, it is the weak limit of measures

ωQn on S, by the formula (8.1) for ωQn and local uniform convergence of ξQn to ξX . We

already know that ωQn → ωX for the curvature distributions (at a ﬁxed archimedean place), from the deﬁnitions given in §4.3. It follows that νX = ωX .

8.2. Proof of Theorem 1.12. Fix X1,X2 ∈ E(k)⊗R, and let D1 and D2 be the associated metrized R-divisors on B, deﬁned over the number ﬁeld K. Fix an archimedean place of

K, and let ω1 and ω2 be the curvature measures on B(C) at this place. We assume that ω1 = ω2. As in §8.1, we ﬁx a period function τ : S → H, and we choose holomorphic functions ξi = xi + yiτ, i = 1, 2, representing the points X1 and X2. From (8.1), we know that e

(8.2) dx1 ∧ dy1 = dx2 ∧ dy2 on S. We break the proof into two steps. In the ﬁrst, we exploit the holomorphic-antiholomorphic

tricke of [ACZ, §5], applied to a relation between holomorphic functions ξ1, ξ2, τ (and their derivatives) and the anti-holomorphic functions ξ¯1, ξ¯2, andτ ¯ (and their derivatives) coming from (8.2); the result is a relation on the holomorphic input alone. In the second step, we apply the transcendence result of [Be, Th´eor`eme 5] to this relation and deduce that the points X1 and X2 must be related in E(k) ⊗ R.

Step 1: Holomorphic-Antiholomorphic. We are grateful to Lars K¨uhne for teaching us this step. Note that

dξi = dxi + yi dτ + τ dyi so that

(dξi − yidτ) ∧ (dξ¯i − yidτ¯)=(¯τ − τ) dxi ∧ dyi. Writing ξ − ξ¯ y = i i i τ − τ¯ we obtain a relation from (8.2) of the form

((τ − τ¯)dξ1 − (ξ1 − ξ¯1)dτ) ∧ ((τ − τ¯)dξ¯1 − (ξ1 − ξ¯1)dτ¯) ¯ ¯ ¯ = ((τ − τ¯)dξ2 − (ξ2 − ξ2)dτ) ∧ ((τ − τ¯)dξ2 − (ξ2 − ξ2)dτ¯). ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 42 Working in coordinates in the simply connected S, this gives

′ ′ ′ ′ 2 ′ ′ ′ (8.3) ξ1ξ1 − ξ2ξ2 (τ − τ¯) − (ξ1 − ξ1)ξ1 − (ξ2 − ξe2)ξ2 (τ − τ¯)τ ′ ′ ′ 2 2 ′ ′ − (ξ1 − ξ1)ξ1 − (ξ2 − ξ2)ξ2 (τ − τ¯)τ + (ξ1 − ξ1) − (ξ2 − ξ2) τ τ = 0. Equation (8.3) can be expressed as N

fj(z)gj(z) ≡ 0 j=1 X ′ ′ ′ for holomorphic functions fj ∈ Z[ξ1,ξ2,ξ1,ξ2,τ,τ ] and antiholomorphic functions gj ∈ ′ ′ ′ Z[ξ1, ξ2, ξ1, ξ2, τ, τ ] in z ∈ S. For each j, deﬁne holomorphic function e gˆj(w) := gj(¯w). Then N

(8.4) F (z,w) := fj(z)ˆgj(w) j=1 X is holomorphic on S × S and vanishes identically on the real-analytic subvariety {w =z ¯}, where it coincides with (8.3). It follows that F must vanish identically on S × S; see [ACZ,

Lemma 5.2]. In particular,e e if we ﬁx any w0 ∈ S, we have F (z,w0) ≡ 0 on S, and we obtain ′ ′ ′ a polynomial relation in the holomorphic functions ξ1,ξ2,ξ1,ξ2,τ,τ that holdse one all of S. e e Step 2: Algebraic Independence. Suppose that P ,...,P ∈ E(k) deﬁne a basis for 1 m e E(k) ⊗ R, so that m

Xi = ai,jPj j=1 X for ai,j ∈ R, i =1, 2. From Proposition 8.1, we know that we can write

ξi = ai,jξPj j X

for choices of lifts ξPj of each point Pj. From Step 1, for each w0 ∈ S, the function (8.4)

satisﬁes F (·,w0) ≡ 0 on S, giving a polynomial relation on the holomorphic functions e ξ ,...,ξ ,ξ′ ,...,ξ′ ,τ,τ ′. e P1 Pm P1 Pm

But the functions ξPj come from algebraic points Pj ∈ E(k) and so satisfy the hypothesis

of Théorème 5 in [Be]. Consequently, if the relation is nontrivial, the functions ξPj and therefore the points Pj must themselves satisfy a linear relation. As they were assumed to be a basis for E(k)⊗R, we conclude that the relation must have been trivial. In other words, ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 43 ′ ′ for any choice of w0, the coefficients of F (z,w0) – as polynomials in ξP1 ,...,ξPm ,ξP1 ,...,ξPm – must vanish. Examining the relation (8.3), we can determine these coefficients explicitly. The “con- ′ stant” term, having no dependence on the ξPj or ξPj , gives ′ ′ C1(w0) τ + C2(w0) ττ =0 as a function of z ∈ S, with coefficients C1,C2 that are antiholomorphic functions of w0 on S. For fixed w0, if C1 or C2 is nonzero, this would imply that τ is constant, which is absurd because the elliptic surfacee E → B is nonisotrivial. This implies that C2(w0) = 0 for all w0. But,e again looking at the formula from (8.3), we have

′ ′ C2(w0)= ξ1(w0)ξ1(w0) − ξ2(w0)ξ2(w0)=0 for all w0. Taking complex conjugates, we have m ′ ′ ′ 0 ≡ ξ1ξ1 − ξ2ξ2 = (a1,ja1,ℓ − a2,ja2,ℓ)ξPj ξPℓ . j,ℓX=1 ′ In other words, we find another relation on the holomorphic functions ξPj and ξPℓ which must therefore be trivial [Be, Théorème 5]. We conclude that either

a1,j = a2,j for all j or that

a1,j = −a2,j for all j. In other words,

X1 = ±X2. This completes the proof of Theorem 1.12.

9. Proofs of the main theorems In this section, we prove our main theorems. ˆ 9.1. Proof of Theorem 1.1. Recall that the Néron-Tate height hE on E(k) extends to a positive definite quadratic form on E(k) ⊗ R. It follows (by Cauchy-Schwarz) that the Néron-Tate regulator ˆ ˆ 2 RE(X,Y ) := hE(X)hE(Y ) −hX,Y iE ≥ 0

extends to a biquadratic form on E(k) ⊗ R satisfying RE(X,Y ) = 0 if and only if X and Y are linearly dependent over R. As

F (X,Y ) := DX · DY ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 44 is also biquadratic on E(k) ⊗ R and satisﬁes F (X,Y ) ≥ 0 (see Proposition 6.1), the upper

bound on DX · DY in Theorem 1.1 follows. From Theorem 7.1, we know that Theorem 1.1 holds for E → B if and only if DX ·DY =06 for all pairs of linearly independent X,Y ∈ E(k) ⊗ R. So assume we have nonzero elements

X,Y ∈ E(k)⊗R satisfying DX ·DY = 0. By scaling X and Y , we may assume that hˆE(X)= hˆE(Y ) = 1. We proved in Theorem 4.6 that DX and DY are normalized, semipositive, continuous adelic metrizations on R-divisors on B, each on divisors of degree 1. Theorem

3.2 then implies that the DX and DY are isomorphic, so the curvature forms for DX and for an DY on Bv must coincide at all places v of the number ﬁeld K. Fixing a single archimedean place, we deduce from Theorem 1.12 that X = ±Y . This completes the proof.

9.2. Proof of Theorem 1.3. Suppose that C is an algebraic curve in E m that dominates the base curve B. Passing to a finite branched cover B′ → B, we may view C as a section C′ of the m-th fibered power of the pull-back elliptic surface E ′ → B′. As Theorem 1.1 holds for E ′ → B′, we apply Theorem 7.1 to conclude that the intersection of C′ with the tube T ((E ′)m,{2}, ǫ) is contained in a finite union of flat subgroup schemes of positive dimension, for all sufficiently small ǫ> 0. Projecting back to E m → B, we can make the same conclusion about the intersection of C with T (E m,{2}, ǫ). This completes the proof.

9.3. Proof of Theorem 1.5. The theorem follows immediately from Theorem 1.1 and Theorem 7.1.

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