UC Berkeley

This dissertation is concerned with problems related to unlikely intersections and is divided into three parts. The first part consists of background about unlikely intersection problems, with particular emphasis on the Andre–Oort conjecture and existential closedness problems. These problems will be the central focus of the subsequent parts. In the second part, wegive an explicit formula for heights of special points on quaternionic Shimura varieties using Faltings heights of CM abelian varieties. Special points are associated to CM-fields E and partial CM-types ϕ ⊂ Hom(E,C). We show that this quaternionic height is compatible with the canonical height of a partial CM-type given by Pila, Shankar, and Tsimerman. By doing so, we give another proof showing that the height of partial CM-types is bounded in terms of the discriminant of E. This height bound is a crucial ingredient in proving the Andre–Oort conjecture for general Shimura varieties. The third part is about the intersection of algebraic varieties with the graph of transcendental functions. Let q : Ω → S be the uniformization map of a Shimura variety. We prove two results that give geometric conditions for when an algebraic variety V ⊂ Ω×S contains a Zariski dense subset of points of the form (x, q(x)).