Canonical subgroups of Barsotti-Tate groups

Abstract

Let $S$ be the spectrum of a complete discrete valuation ring with fraction field of characteristic $0$ and perfect residue field of characteristic $p\geq 3$. Let $G$ be a truncated Barsotti-Tate group of level $1$ over $S$. If “$G$ is not too supersingular”, a condition that will be explicitly expressed in terms of the valuation of a certain determinant, then we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fiber to a subgroup scheme of $G$, finite and flat over $S$. We call it the canonical subgroup of $G$.