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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
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