Partial Hasse invariants, partial degrees, and the canonical subgroup.(English)Zbl 1446.11105

Summary: 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 D. A. Reduzzi and L. Xiao [Ann. Sci. Éc. Norm. Supér. (4) 50, No. 3, 579–607 (2017; Zbl 1430.11061)] in the general case), as well as partial degrees. After studying these functions, we compute the partial degrees of the canonical subgroup.

MSC:

 11F85 $$p$$-adic theory, local fields 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11S15 Ramification and extension theory 14L05 Formal groups, $$p$$-divisible groups

Zbl 1430.11061
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