Canonical subgroups via Breuil-Kisin modules

TL;DR

This paper establishes conditions under which the canonical subgroup of a truncated Barsotti-Tate group over a p-adic field is free of a certain rank, using Breuil-Kisin modules and ramification theory.

Contribution

It introduces new criteria involving the Hasse invariant for the existence and properties of canonical subgroups via Breuil-Kisin modules.

Findings

Upper ramification subgroup G^j+ is free of rank d under certain conditions.

Canonical subgroup properties are confirmed under the Hasse invariant threshold.

Provides a new approach to canonical subgroups using Breuil-Kisin modules.

Abstract

Let p>2 be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we show that an upper ramification subgroup G^j+ is free of rank d over Z/p^nZ if the Hasse invariant of G is less than 1/(2p^(n-1)). We also prove the usual properties as the canonical subgroup.