Stratifications of Newton polygon strata and Traverso's conjectures for p-divisible groups

The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field of characteristic p is the least positive integer m such that D[pm] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-...

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Bibliographic Details
Published in:Annals of mathematics 2013-11, Vol.178 (3), p.789-834
Main Authors: Lau, Eike, Nicole, Marc-Hubert, Vasiu, Adrian
Format: Article
Language:English
Subjects:
Algebra
Homomorphisms
Integers
Mathematical rings
Mathematical theorems
Mathematics
Morphisms
Polygons
Spatial points
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Summary:The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field of characteristic p is the least positive integer m such that D[pm] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[pm] to D.
ISSN:0003-486X
DOI:10.4007/annals.2013.178.3.1