Some unlikely intersections beyond André-Oort

According to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over Q. Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

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