Partial Hasse invariants, partial degrees and the canonical subgroup

TL;DR

This paper explores the relationship between partial Hasse invariants, partial degrees, and the canonical subgroup in p-divisible groups, extending existing concepts to ramified cases and providing new properties and computations.

Contribution

It introduces the notion of partial Hasse invariants and partial degrees in ramified settings, generalizing previous constructions and establishing their properties and relations.

Findings

Established the equality between Hasse invariant and dual canonical subgroup degree.

Defined partial Hasse invariants and degrees in ramified cases.

Computed partial degrees of the canonical subgroup.

Abstract

If the Hasse invariant of a $p$-divisible group is small enough, then one can construct a canonical subgroup inside its $p$-torsion. We remark that, assuming the existence of a subgroup of adequate height in the $p$-torsion whose dual has small degree, the expected properties of the canonical subgroup can be easily proven. A fundamental relation is the equality between the Hasse invariant and the degree of the dual of the canonical subgroup. When one considers a $p$-divisible group $G$ with an action of the ring of integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$, then much more can be said. One can define partial Hasse invariants ; they are natural in the unramified case, and generalize a construction of Reduzzi and Xiao in the general case. One can also define partial degrees for finite flat subgroups of $G$. We prove some properties for these partial Hasse invariants and partial degrees, and compute the partial degrees of the canonical subgroup.