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

Rémond, Gaël

J. Inst. Math. Jussieu 6, No. 2, 317-348 (2007).

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 \(A\).

Reviewer: Herbert Lange (Erlangen)

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

Citations:

Zbl 1088.11047