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A lower bound for the canonical height on abelian varieties over abelian extensions. (English) Zbl 1060.11041
Let $$A$$ be an Abelian variety defined over a number field $$K$$. Let $$L$$ be an ample symmetric line bundle on $$A$$ and $$\widehat h: A(\overline K)\to \mathbb{R}$$ be the associated canonical height function. The main result of the paper is the following theorem: There is a positive constant $$C= C(A, K,L)$$ such that $$\widehat h(P)\geq C$$ for all non-torsion points $$P\in A(K^{ab})$$, where $$K^{ab}$$ denotes the Abelian extension of $$K$$. This theorem was proven by the first author for elliptic curves with complex multiplication in [Int. Math. Res. Not. 2003, No. 29, 1571–1589 (2003; Zbl 1114.11058)] and for elliptic curves without complex multiplication by the second author in [J. Number Theory 104, No. 2, 353–372 (2004; Zbl 1053.11052)]. It is a weak version of the so-called generalized Lehmer conjecture, which gives a more precise lower bound and of which several special cases are known.

##### MSC:
 11G50 Heights 11G10 Abelian varieties of dimension $$> 1$$ 14K15 Arithmetic ground fields for abelian varieties
##### Keywords:
Abelian variety over number fields; height function
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