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Elliptic Surfaces and Arithmetic Equidistribution for $\Mathbb {R ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES LAURA DE MARCO AND NIKI MYRTO MAVRAKI Abstract. Suppose E → B is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve B. Let k denote the function field Q(B) and E the associated elliptic curve over k. In this article, we construct adelically metrized R-divisors DX on the base curve B over a number field, for each X ∈ E(k) ⊗ R. We prove non-degeneracy of the Arakelov-Zhang intersection numbers DX · DY , as a biquadratic form on E(k) ⊗ R. As a consequence, we have the following Bogomolov-type statement for the N´eron- Tate height functions on the fibers Et(Q) of E over t ∈ B(Q): given points P1,...,Pm ∈ E(k) with m ≥ 2, there exist a non-repeating infinite sequence tn ∈ B(Q) and small- ′ ′ ′ height perturbations Pi,tn ∈ Etn (Q) of specializations Pi,tn so that the set {P1,tn ,...,Pm,tn } satisfies at least two independent linear relations for all n, if and only if the points P1,...,Pm are linearly dependent in E(k). This gives a new proof of results of Masser and Zannier [MZ1, MZ2] and of Barroero and Capuano [BC] and extends our earlier results [DM]. We also prove a general equidistribution theorem for adelically metrized R-divisors on curves (over a number field), extending the equidistribution theorems of Chambert-Loir, Thuillier, and Yuan [CL1, Th, Yu], and using results of Moriwaki [Mo2]. 1. Introduction Suppose E → B is an elliptic surface defined over a number field K. That is, E is a projective surface, B is a smooth projective curve, and there exists a section O : B → E, all defined over K, so that all but finitely many fibers Et, for t ∈ B(K), are smooth elliptic arXiv:2012.14529v1 [math.NT] 28 Dec 2020 curves with zero Ot. We say that the elliptic surface E → B is isotrivial if all of the smooth fibers Et are isomorphic over K. Let k denote the function field K(B); we also view the surface E as an elliptic curve E over the field k. In this article, we study the geometry and arithmetic of the set E(k) of rational points over the function field k when E → B is not isotrivial. To this end, we consider height functions associated to adelically metrized R-divisors on the base curve B over a number field K, and we prove an equidistribution theorem for these heights. We study the Arakelov- Zhang intersection number of metrized R-divisors and prove that it induces a non-degenerate biquadratic form on E(k) ⊗ R. We relate this theorem to existing results, and provide, for Date: January 1, 2021. 1 ELLIPTIC SURFACES AND ARITHMETIC EQUIDISTRIBUTION FOR R-DIVISORS ON CURVES 2 example, a new proof of results of Masser and Zannier and of Barroero and Capuano on linear relations between specializations of independent sections. 1.1. Heights and the Arakelov-Zhang intersection of points in E(k). Assume that E → B is not isotrivial. Let hˆE denote the N´eron-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 hˆE(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 N´eron-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 N´eron-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 hˆE(P )+ hˆE(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, simi- lar 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 sub- group 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 N´eron-Tate bilinear form on the specialization Λt in the fiber Et.
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