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.


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