Canonical Models of (Mixed) Shimura Varieties and Automorphic Vector Bundles

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A Combination of the Conjectures of Mordell-Lang and André-Oort (2005, background): “(b) The generalized Hecke operator associated to an inner automorphism int(p) : p′ → pp′p−1 for an element p ∈ P (Q)+ is called a (usual) Hecke operator on S and is denoted by Tp.”
Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points (2013, result): “[22] P. Deligne, J. Milne, A. Ogus, and K. Shih.”
Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic (2010, background): “4 are analogues to two of the main theorems of the theory of complex multiplication for abelian varieties ([28], Chapter I, Corollary 5.”
Integrals of Borcherds Forms (2001, unknown): “Zðm;j;K Þ ¼ Zðm;j;K 0Þ; ð1:49Þ so that the special cycles are defined on the full Shimura variety associated to ðH;DÞ, [41].”