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Partial Hasse Invariants, Partial Degrees, and the Canonical Subgroup

Published online by Cambridge University Press:  20 November 2018

Stephane Bijakowski
Stephane Bijakowski*
Affiliation:
Imperial College, Department of Mathematics, 180 Queen's Gate, London SW7 2AZ UK email: [email protected]
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Abstract

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If the Hasse invariant of a $P$ -divisible group is small enough, then one can construct a canonical subgroup inside its $P$-torsion. We prove that, assuming the existence of a subgroup of adequate height in the $P$-torsion with high degree, the expected properties of the canonical subgroup can be easily proved, especially the relation between its degree and the Hasse invariant. When one considers a $P$-divisible group with an action of the ring of integers of a (possibly ramified) finite extension of ${{\mathbb{Q}}_{P}}$ , then much more can be said. We define partial Hasse invariants (which are natural in the unramified case, and generalize a construction of Reduzzi and Xiao in the general case), as well as partial degrees. After studying these functions, we compute the partial degrees of the canonical subgroup.

Keywords

11F8511F4611S15canonical subgroupHasse invariantp-divisible group
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 4 , 01 August 2018 , pp. 742 - 772
Copyright
Copyright © Canadian Mathematical Society 2018

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