# An upper bound on the Abbes-Saito filtration for finite flat group schemes and applications - Mathematics > Number Theory

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Abstract: Let $\cO K$ be a complete discrete valuation ring of residue characteristic$p>0$, and $G$ be a finite flat group scheme over $\cO K$ of order a power of$p$. We prove in this paper that the Abbes-Saito filtration of $G$ is boundedby a simple linear function of the degree of $G$. Assume $\cO K$ has genericcharacteristic 0 and the residue field of $\cO K$ is perfect. Farguesconstructed the higher level canonical subgroups for a Barsotti-Tate group$\cG$ over $\cO K$ which is -not too supersingular-. As an application of ourbound, we prove that the canonical subgroup of $\cG$ of level $n\geq 2$constructed by Fargues appears in the Abbes-Saito filtration of the$p^n$-torsion subgroup of $\cG$.

Author: ** Yichao Tian**

Source: https://arxiv.org/