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Small values of the quadratic part of the Néron-Tate height on an Abelian variety. (English) Zbl 0551.14015
Let A be an abelian variety of dimension g, defined over the field \({\bar {\mathbb{Q}}}\) of algebraic numbers and embedded in projective space \({\mathbb{P}}_ N\). It is proved in this paper that there exist constants \(\kappa\),\(\lambda\) depending only on g, and a constant C depending only on A and the embedding, with the following properties. Let A(\({\bar {\mathbb{Q}}})\) denote the group of points on A defined over \({\bar {\mathbb{Q}}}\). For P in A(\({\bar {\mathbb{Q}}})\) write q(P) for the quadratic part of the associated (absolute) Néron-Tate height, and let \(D=D(P)\) be the degree of the smallest field of definition of P. Then if \(q(P)<C^{-1}D^{- \kappa}\) the point P is necessarily a torsion point (and so in fact \(q(P)=0)\); and furthermore its order is at most \(CD^{\lambda}\). This generalizes an earlier result of M. Anderson and the author [Math. Z. 174, 23-34 (1980; Zbl 0421.14006)] for elliptic curves. The proof involves methods from the theory of transcendental numbers.

MSC:
14K15 Arithmetic ground fields for abelian varieties
11J81 Transcendence (general theory)
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References:
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