## Intersections of subgroups and subvarieties. I. (Intersection de sous-groupes et de sous-variétés. I.)(English)Zbl 1088.11047

Raynaud proved that for an irreducible subvariety $$X$$ of an abelian variety $$A$$ defined over $$\overline{Q}$$, the field of algebraic numbers, the intersection of the set of $$\overline{Q}$$-rational points with the torsion points $$A_{\text{tors}}$$ of $$A$$ is contained in the union of finitely many translates of proper abelian subvarieties. One can view $$A_{\text{tors}}$$ as the union of the zero-dimensional subgroups of $$A$$.
In the present paper the author studies the intersection of $$X(\overline{Q})$$ with $$A^{[r]}$$, the union of all codimension $$r$$ subgroups of $$A$$. The exceptional set is replaced by the set $$Z_{X,0}^{(r)}$$, the set of points $$P$$ such that there is a subgroup $$G$$ with $$\dim_P(X\cap G) \geq \max(1,r-\text{codim }G)$$. The question is now whether the intersection $X(\overline{Q})\backslash Z_{X,0}^{(r)} \cap A^{[r]}$ is finite. The author proves a positive answer under several assumptions. This result generalizes work of Viada and the proof is similar in spirit to work of Bombieri, Masser and Zannier on algebraic tori.

### MSC:

 11G10 Abelian varieties of dimension $$> 1$$ 14G40 Arithmetic varieties and schemes; Arakelov theory; heights

### Keywords:

abelian variety; height
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### References:

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