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 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.

Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields

“…As in the classical ramification theory of local fields, G has upper and lower ramification subgroups [14,17]. When the base field is of mixed characteristic, Tian proved that the upper and the lower ramification subgroups correspond to each other via usual Cartier duality if G is killed by p [28], and Fargues gave a much simpler proof of this theorem [14].…”

Canonical subgroups via Breuil–Kisin modules

“…Such a generalization was first obtained by Abbes and Mokrane via a calculation of p-adic vanishing cycles of abelian schemes ( [1]). Namely, for an abelian scheme A over O K with relative dimension g, they found a subgroup of Their work was followed by many improvements and generalizations with various methods, such as [2], [7], [10], [11], [17], [19], [21] and especially [8] and [22].…”

Canonical subgroups over Hilbert modular varieties

“…Since then, many authors have studied the canonical subgroup in various settings and with different approaches. We mention the works [11], [16], [29], [35], as well as yet unpublished results by K. Buzzard, E. Nevens and J. Rabinoff. Broadly speaking, the traditional approach to the canonical subgroup problem proceeds through a careful examination of subgroup schemes of either abelian varieties, or p-divisible groups, and, again broadly speaking, much of the complications arise from the fact that formal groups in several variables are hard to describe and one lacks a theory of Newton polygons for power-series in several variables.…”