Deforming endomorphisms of supersingular Barsotti-Tate groups

TL;DR

This paper studies the deformation space of supersingular Barsotti-Tate groups with additional endomorphisms, describing its structure, components, and intersections, extending Gross-Keating theory for applications in Shimura surface intersection theory.

Contribution

It provides a complete description of the irreducible components, multiplicities, and intersection numbers of the deformation locus with extra endomorphisms, extending existing theories.

Findings

Description of irreducible components of the deformation locus

Calculation of multiplicities of components

Intersection numbers with codimension one subschemes

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.