A Generalization of Conjectures of Bogomolov and Lang over Finitely Generated Fields
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چکیده
In this paper we prove a generalization of conjectures of Bogomolov and Lang in terms of an arithmetic Néron-Tate height pairing over a finitely generated field. §0. Introduction Let K be a finitely generated field over Q with d = tr. degQ(K), and let B be a big polarization of K . Let A be an abelian variety over K , and let L be a symmetric ample line bundle on A. In [4] we define the height pairing 〈 , 〉L : A ( K )× A(K) −→ R assigned to B and L with these properties: 〈x, x〉L ≥ 0 for all x ∈ A(K) and the equality holds if and only if x ∈ A(K)tor. For x1, . . . , xl ∈ A(K), we denote det(〈xi, xj 〉L) by δ L (x1, . . . , xl). The purpose of this paper is to prove the following theorem, which gives an answer to B. Poonen’s question in [5]. theorem 0.1 Let be a subgroup of finite rank in A(K) (i.e., is a subgroup of A(K) with dimQ( ⊗ Q) < ∞), and let X be a subvariety of AK . Fix a basis {γ1, . . . , γn} of ⊗ Q. If the set {x ∈ X(K) | δ L (γ1, . . . , γn, x) ≤ } is Zariski dense in X for every positive number , then X is a translation of an abelian subvariety of AK by an element of div, where div = {x ∈ A(K) | nx ∈ for some positive integer s}. In the case where d = 0, Poonen [5] and S.-W. Zhang [8] proved the equivalent result independently. Our argument for the proof of Theorem 0.1 essentially follows Poonen’s ideas. A new point is that we remove the measure-theoretic argument from his original one, so that we can apply it to our case. We note that Theorem 0.1 substantially includes Lang’s conjecture in the absolute form. DUKE MATHEMATICAL JOURNAL Vol. 107, No. 1, c © 2001 Received 14 September 1999. Revision received 8 May 2000. 2000 Mathematics Subject Classification. Primary 11G35, 14G25, 14G40; Secondary 11G10, 14K15.
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تاریخ انتشار 1999