Vol. 6, No. 2, 2012

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Other MSP Journals
An upper bound on the Abbes–Saito filtration for finite flat group schemes and applications

Yichao Tian

Vol. 6 (2012), No. 2, 231–242
Abstract

Let ${\mathsc{O}}_{K}$ be a complete discrete valuation ring of residue characteristic $p>0$, and $G$ be a finite flat group scheme over ${\mathsc{O}}_{K}$ of order a power of $p$. We prove in this paper that the Abbes–Saito filtration of $G$ is bounded by a linear function of the degree of $G$. Assume ${\mathsc{O}}_{K}$ has generic characteristic $0$ and the residue field of ${\mathsc{O}}_{K}$ is perfect. Fargues constructed the higher level canonical subgroups for a “near from being ordinary” Barsotti–Tate group $\mathsc{G}$ over ${\mathsc{O}}_{K}$. As an application of our bound, we prove that the canonical subgroup of $\mathsc{G}$ of level $n\ge 2$ constructed by Fargues appears in the Abbes–Saito filtration of the ${p}^{n}$-torsion subgroup of $\mathsc{G}$.

Keywords
finite flat group schemes, ramification filtration, canonical subgroups
Mathematical Subject Classification 2000
Primary: 14L15
Secondary: 14G22, 11S15