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Intersection of curves and of subgroups and lower bound problems for the height in CM abelian varieties. (Intersection de courbes et de sous-groupes et problèmes de minoration de dernière hauteur dans les variétés abéliennes C.M.) (French. English summary) Zbl 1156.11025
The main object of the paper under review is a special case of the conjecture of Zilber and Pink, generalising the Manin-Mumford conjecture. It states that if \(X\) is an irreducible curve on a semiabelian variety \(G\), and if \(G^{[2]}\) denotes the union of all subgroup varieties of \(G\) of codimension \(\geq 2\), then the intersection of \(X \cap G^{[2]}\) is finite, unless \(X\) is contained in a proper algebraic subgroup of \(G\).
The author proves this to hold in the case where \(G\) is isogenous to a power of a simple abelian variety of CM type. This generalizes the similar result for powers of CM elliptic curves obtained by G. Remond and E. Viada [Int. Math. Res. Not. 2003, No. 35, 1915–1931 (2003; Zbl 1072.11038)].
The proof combines the methods of Remond and Viada used in [loc. cit.], which in turn rely on techniques developed by E. Bombieri, D. Masser and U. Zannier, with a lower bound for the Néron-Tate height of points on an abelian variety of CM type, in the spirit of Lehmer’s problem.
This lower bound is analogous to a lower bound for heights on tori by F. Amoroso and S. David [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 3, No. 2, 325–348 (2004; Zbl 1150.11021)].

MSC:
11G50 Heights
11G10 Abelian varieties of dimension \(> 1\)
11J95 Results involving abelian varieties
14K22 Complex multiplication and abelian varieties
11R20 Other abelian and metabelian extensions
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