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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
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