Deforming endomorphisms of supersingular Barsotti-Tate groups

Abstract

The formal deformation space of a supersingular Barsotti-Tate group over of dimension two equipped with an action of Z_{p^2} is known to be isomorphic to the formal spectrum of a power series ring in two variables. If one chooses an extra Z_{p^2}-linear endomorphism of the p-divisible group then the locus in the formal deformation space formed by those deformations for which the extra endomorphism lifts is a closed formal subscheme of codimension two. We give a complete description of the irreducible components of this formal subscheme, compute the multiplicities of these components, and compute the intersection numbers of the components with a distinguished closed formal subscheme of codimension one. These calculations, which extend the Gross-Keating theory of quasi-canonical lifts, are used in the companion article "Intersection theory on Shimura surfaces II" to compute global intersection numbers of special cycles on the integral model of a Shimura surface.