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

Bijakowski, Stephane

Can. J. Math. 70, No. 4, 742-772 (2018).

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

Keywords:

canonical subgroup; Hasse invariant; \(p\)-divisible group