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Density of points and lower bound for the height. (Densité de points et minoration de hauteur.) (French) Zbl 1065.11049
Let $$K\supset k$$ be two number fields, let $$A$$ be an abelian variety defined over $$k$$ and $$L$$ a very ample line bundle over $$A$$. Let $$V$$ be an algebraic subvariety defined and irreducible over $$K$$, such that there does not exist a finite union of translated of proper abelian subvarieties of $$A$$ by torsion points containing $$V$$. The author proves that for all $$\varepsilon>0$$, the set of points $$x\in V(\overline{K})$$ which are of infinite order in $$A$$ modulo every proper abelian subvariety of $$A$$, and of canonical height: $\widehat{h}_L(x)\leq \frac{\widehat{h}_L(V)}{\deg_L(V)}+\varepsilon$ is Zariski-dense in $$V$$. This result can be used to prove two corollaries in the case $$A$$ is of complex multiplication type; we suppose here that $$K=k$$. The first corollary concerns the points of small canonical height lying over a variety $$V$$ as above. Given such a $$V$$, there exists a constant $$c>0$$, depending on the couple $$(A,L)$$ only, such that the set of points $$x\in V(\overline{K})$$ of canonical height $\widehat{h}_L(x)\leq\frac{c}{\delta} \left(\frac{ \log\log(3\delta)}{\log (3\delta)}\right)^{\kappa}$ is not Zariski-dense in $$V$$. Here $$\delta$$ is the obstruction index of $$V$$ and $$\kappa>0$$ is a constant, explicit and rather big, depending on $$\dim A$$ only. The second corollary is a minoration of the essential minimum of a subvariety $$V$$ of $$A$$, defined over $$k$$, which is not equal to a finite union of translated of proper abelian subvarieties of $$A$$ by torsion points: $\mu^{\text{ess}}_L(V)\geq c\deg_L(V)^{\frac{1} {s-\dim(V)}}\cdot (\log(3 \deg_L(V)))^{-\kappa(s)},$ where $$s$$ is the dimension of the smallest algebraic subgroup of $$A$$ containing $$V$$ (the constant $$c$$ is the same as above). The proof of these results heavily depends on techniques and results introduced and proved by F. Amoroso and S. David [Ramanujan J. 5, No.3, 237–246 (2001; Zbl 0996.11046)], S. David and M. Hindry [J. Reine Angew. Math. 529, 1–74 (2000; Zbl 0993.11034)] and S. Zhang [J. Am. Math. Soc. 8, No. 1, 187–221 (1995; Zbl 0861.14018)]. More precisely, the proof of the main result of the revieved article follows the work of Amoroso-David, while the first corollary depends on the “state of the art” of the abelian Lehmer problem, which is very refined only if $$A$$ is of C.M. type (work of David-Hindry), and the second corollary uses in addition a theorem of Zhang which says that: $\frac{\widehat{h}_L(V)}{\deg_L(V)}\geq \mu^{\text{ess}}_L(V)\geq \frac{\widehat{h}_L(V)}{\deg_L(V)(\dim V+1)}.$

##### MSC:
 11G50 Heights 11J95 Results involving abelian varieties 11G10 Abelian varieties of dimension $$> 1$$ 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14K15 Arithmetic ground fields for abelian varieties
##### Keywords:
Abelian varieties; normalised height; Lehmer problem
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##### References:
 [1] Amoroso, F.; David, S., Densité des points à coordonnées multiplicativement indépendantes, Ramanujan J., 5, 237-246, (2001) · Zbl 0996.11046 [2] D. Bertrand, Minimal heights and polarizations on abelian varieties, preprint of the MSRI, Berkeley, CA, June, 1987. [3] Bertrand, D., Minimal heights and polarizations on group varieties, Duke math. J., 80, 1, 223-250, (1995) · Zbl 0847.11036 [4] Bertrand, D.; Philippon, P., Sous-groupes algébriques de groupes algébriques commutatifs, Illinois J. math., 32, 263-280, (1988) · Zbl 0618.14020 [5] Bost, J.-B.; Gillet, H.; Soulé, C., Heights of projective varieties and positive Green forms, J. amer. math. soc., 7, 4, 903-1022, (1994) · Zbl 0973.14013 [6] David, S.; Hindry, M., Minoration de la hauteur de Néron-Tate sur LES variétés abéliennes de type C. M, J. reine angew. math., 529, 1-74, (2000) · Zbl 0993.11034 [7] S. David, P. Philippon, Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes, Number Theory (Tiruchirapalli, 1996), Contemporary Mathematics, Vol. 210, American Mathematical Society, Providence, RI, 1998, pp. 333-364. [8] Masser, D., Small values of the quadratic part of the Néron – tate height on an abelian variety, Compositio math., 53, 2, 153-170, (1984) · Zbl 0551.14015 [9] Masser, D.; Wüstholz, G., Periods and minimal abelian subvarieties, Ann. of math. (2), 137, 2, 407-458, (1993) · Zbl 0796.11023 [10] Philippon, P., Sur des hauteurs alternatives I, Math. ann., 289, 2, 255-283, (1991) · Zbl 0726.14017 [11] Philippon, P., Sur des hauteurs alternatives II, Ann. inst. Fourier (Grenoble), 44, 4, 1043-1065, (1994) · Zbl 0878.11024 [12] Philippon, P., Sur des hauteurs alternatives III, J. math. pures appl. (9), 74, 4, 345-365, (1995) · Zbl 0878.11025 [13] G. Rémond, Intersection de sous-groupes et de sous-variétés I, Preprint, Octobre 2003. [14] C. Soulé, Géométrie d’Arakelov et théorie des nombres transcendants, Journées Arithmétiques, 1989 (Luminy, 1989); Astérisque No. 198-200 (1991) pp. 355-371 (1992). [15] Zhang, S., Positive line bundles on arithmetic varieties, J. amer. math. soc., 8, 1, 187-221, (1995) · Zbl 0861.14018 [16] Zhang, S., Small points and adelic metrics, J. algebraic geom., 4, 2, 281-300, (1995) · Zbl 0861.14019
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