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Lower bounds for normalized heights of subvarieties of abelian varieties. (Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes.) (French) Zbl 0899.11027

Murty, V. Kumar (ed.) et al., Number theory. Proceedings of the international conference on discrete mathematics and number theory, Tiruchirapalli, India, January 3–6, 1996 on the occasion of the 10th anniversary of the Ramanujan Mathematical Society. Providence, RI: American Mathematical Society. Contemp. Math. 210, 333-364 (1998).
Let \(A\) be an abelian variety over a number field \(K\), embedded as a projectively normal subvariety into a projective space. Assume that the equivalence class associated with this embedding is symmetric. The second author [P. Philippon, Math. Ann. 289, 255-283 (1991; Zbl 0726.14017)] has defined a normalized height \(\widehat{h}\) which extends the Néron-Tate height to higher dimensional subvarieties of \(A\). He conjectured that for a subvariety \(X\) of \(A\) which is defined over an algebraic number field, \(\widehat{h}(X)\) vanishes if and only if \(X\) is a translate of an abelian subvariety of \(A\) by a torsion point. This conjecture is equivalent to another conjecture of Bogomolov on the density of points of small height on \(X\). The relation between these two questions is described in the present paper.
After the work of Szpiro, Burnol, Bombieri and Zannier, and Ullmo, this conjecture has been proved by S. Zhang [Equidistribution of small points on abelian varieties, Ann. Math. (2) 147, 159-165 (1998)]. For other surveys of this question, see A. Abbes, Hauteurs et discrétude (d’après L. Szpiro, E. Ullmo et S. Zhang, Sémin. Bourbaki, 49ème année 1996-97, Exp. No. 825, Astérisque 245, 141-166 (1997) and S. Zhang, Small points and Arakelov theory, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 217-225 (1998).
The first main result of this paper is a lower bound for \(\widehat{h}(X)\) when \(A\) is an abelian variety with complex multiplication defined over a number field \(K\) and \(X\) is an irreducible algebraic subvariety of \(A\) which is not a translate of an abelian subvariety of \(A\) by a torsion point. This lower bound is explicit in terms of the degree of \(X\) and the degree \([K':K]\) of a number field \(K'\) over which \(X\) is defined.
The second result does not assume complex multiplication any more, but deals only with the case where \(X\) is not a translate of any abelian subvariety of \(A\). The lower bound now does not depend on the degree \([K':K]\), hence is a geometric refinement (as well as an independent proof) of the result of Ullmo and Zhang.
The authors also raise a problem relating these questions with conjectures by Lehmer and Lang.
For the entire collection see [Zbl 0878.00049].

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11J81 Transcendence (general theory)
14K15 Arithmetic ground fields for abelian varieties

Citations:

Zbl 0726.14017
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