## Intersection of subgroups and subvarieties. II. (Intersection de sous-groupes et de sous-variétés. II.)(French)Zbl 1170.11014

Let $$X$$ denote an irreducible subvariety of an abelian variety $$A$$ defined over $$\overline{\mathbb{Q}}$$. In a previous paper [Math. Ann. 333, No. 3, 525–548 (2005; Zbl 1088.11047)] the author studied the intersection of $$X(\overline{\mathbb{Q}})$$ with the union of all codimension $$r$$ subgroups of $$A$$. The present paper studies the intersection of $$X(\overline{\mathbb{Q}})$$ with the union of all the subgroups obtained as a sum of a given finite rank subgroup $$\Gamma$$ and an algebraic subgroup of $$A$$ of given dimension $$d$$. The main result is that if a suitable exceptional subset is removed from $$X$$, then the intersection is a set of bounded height. In terms of boundedness of the height, an optimal statement is proved in the case of a curve $$X$$ and $$d=2$$. The proofs rely on a suitable uniform generalization of the method of Vojta and on computations of intersection numbers of real cycles on $$A$$.

### MSC:

 11G10 Abelian varieties of dimension $$> 1$$ 11G50 Heights 14K12 Subvarieties of abelian varieties 14G40 Arithmetic varieties and schemes; Arakelov theory; heights

### Keywords:

abelian varieties; heights; Pontryagin product

Zbl 1088.11047
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