Serre–Tate theory for moduli spaces of PEL type

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We develop a Serre–Tate theory for moduli spaces of PEL type. This leads us to study Barsotti–Tate groups X equipped with an action ι of a Z p -algebra and possibly also a polarization λ. We define a notion of ordinariness for such triples X=(X,ι,λ) . If we work over k= k ̄ and fix suitable discrete invariants, such as the CM-type, we prove that there is a unique ordinary object, up to isomorphism. We introduce a new structure, called a cascade, that generalizes the notion of a biextension, and we show that the formal deformation space of an ordinary triple has a natural cascade structure. In particular, our theory gives rise to canonical liftings of ordinary objects. In the final section of the paper we apply our theory to the study of congruence relations.