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# Partial Hasse invariants, partial degrees, and the canonical subgroup

Published:2017-05-12

• Stephane Bijakowski,
Imperial College, Department of Mathematics, 180 Queen's Gate, London SW7 2AZ UK
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## Abstract

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 (they 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: canonical subgroup, Hasse invariant, $p$-divisible group
 MSC Classifications: 11F85 - $p$-adic theory, local fields [See also 14G20, 22E50] 11F46 - Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11S15 - Ramification and extension theory

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