Distinguished Liftings and the André-oort Conjecture

Abstract

In this paper we study liftings of affine varieties from finite fields to number fields, such that the lifted varieties contain specified "canonical" lifts of points. If this canonical lifting of points corresponds to the Deuring-Serre-Tate lift of j-invariants of ordinary elliptic curves, then the resulting lifting problem is closely related to the André-Oort conjecture. We explore this connection, prove some results related to the André-Oort conjecture, and then apply these results together with other known special cases of the conjecture to our lifting problems.