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Mathematical Research Letters
Volume 25 (2018)
Number 4
Duality, refined partial Hasse invariants and the canonical filtration
Pages: 1109 – 1142
DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n4.a3
Author
Abstract
Let $G$ be a $p$-divisible group over the ring of integers of $\mathbb{C}_p$, and assume that it is endowed with an action of the ring of integers of a finite unramified extension $F$ of $\mathbb{Q}_p$. Let us fix the type $\mu$ of this action on the sheaf of differentials $\omega_G$. V. Hernandez, following a construction of Goldring and Nicole, defined partial Hasse invariants for $G$. The product of these invariants is the $\mu$-ordinary Hasse invariant, and it is non-zero if and only if the $p$-divisible group is $\mu$-ordinary (i.e. the Newton polygon is minimal given the type of the action).
We show that if the valuation of the $\mu$-ordinary Hasse invariant is small enough, then each of these partial Hasse invariants is a product of other sections, the refined partial Hasse invariants. We also give a condition for the construction of these invariants over an arbitrary scheme of characteristic $p$. We then give a simple, natural and elegant proof of the compatibility with duality for the classical Hasse invariant, and show how to adapt it to the case of the refined partial Hasse invariants. Finally, we show how these invariants allow us to compute the partial degrees of the canonical filtration (if it exists).
Keywords
$p$-divisible group, Hasse invariant, duality, $\mu$-ordinary locus, canonical filtration
2010 Mathematics Subject Classification
Primary 14L05. Secondary 11F55.
Received 14 April 2016
Accepted 11 April 2017
Published 16 November 2018