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Lower bounds for the canonical height on elliptic curves over abelian extensions. (English) Zbl 1114.11058

Summary: Let \(K\) be a number field and let \(E/K\) be an elliptic curve. If \(E\) has complex multiplication, we show that there is a positive lower bound for the canonical height of nontorsion points on \(E\) defined over the maximal abelian extension \(K^{ab}\) of \(K\). This is analogous to results of F. Amoroso and R. Dvornicich [J. Number Theory 80, No. 2, 260–272 (2000; Zbl 0973.11092)] and M. Amoroso and U. Zannier [Ann. Sc. Norm. Sup. Pisa, Cl. Sci. (4) 29, No. 3, 711–727 (2000; Zbl 1016.11026)] for the multiplicative group. We also show that if \(E\) has nonintegral \(j\)-invariant (so that, in particular, \(E\) does not have complex multiplication), then there exists \(C>0\) such that there are only finitely many points \(P\in E(K^{ab})\) of canonical height less than \(C\). This strengthens a result of M. Hindry et J. Silverman [Sémin Théor. Nombres, Paris 1988-89, Prog. Math. 91, 103–116 (1990; Zbl 0741.14013)] .

MSC:

11G50 Heights
11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
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