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Families of $$p$$-divisible groups with constant Newton polygon. (English) Zbl 1022.14013
Let $$S$$ be a scheme of characteristic $$p>0.$$ Let $$X$$ be a $$p$$-divisible group over $$S$$. A filtration $$0=X_{0}\subset X_{1}\subset \cdots \subset X_{m}=X$$ with each $$X_{i}$$ a $$p$$-divisible group is called a slope filtration of $$X$$ provided there exists a strictly decreasing set of rational numbers $$\lambda _{1},\lambda _{2},\ldots ,\lambda _{m}$$ in $$[0,1]$$ such that for all $$i>1$$ we have $$X_{i}/X_{i-1}$$ is isoclinic of slope $$\lambda _{i}.$$ Such a filtration may not exist, although it does in the case where $$S$$ is a field [see T. Zink, Duke Math. J. 109, 79-95 (2001; Zbl 1061.14045)]. If $$X$$ satisfies further conditions involving powers of the Frobenius, then $$X$$ is completely slope divisible. The issue in this paper is to investigate when a certain class of $$p$$-divisible groups admits a slope filtration.
Here the $$p$$-divisible groups that are considered have constant Newton polygon. An example is given to show that even in this case a slope filtration may not exist. However, the main result here is the following (from the paper): “Let $$S$$ be an integral, normal Noetherian scheme. Let $$X$$ be a $$p$$-divisible group over $$S$$ of height $$h$$ with constant Newton polygon. Then there is a completely slope divisible $$p$$-divisible group $$Y$$ over $$S$$, and an isogeny: $$\varphi :X\rightarrow Y$$ over $$S$$ with $$\deg \varphi \leq N(h)$$”. A corollary to this result is that every $$p$$-divisible group with constant Newton polygon over a normal base has, up to isogeny, a slope filtration.
From this theorem, the authors are able to prove the following. Let $$R$$ be a Henselian local ring with residue field $$k$$, and let $$h\in \mathbb{N.}$$ Let $$S=\text{Spec}\left( R\right)$$, and let $$X$$ and $$Y$$ be isoclinic $$p$$-divisible groups over $$S$$ with ht$$\left( X\right)$$, ht$$\left( Y\right) <h.$$ Then there exists a $$c$$ such that, given any homomorphism $$\psi :X_{k}\rightarrow Y_{k},$$ the homomorphism $$p^{c}\psi$$ lifts to a homomorphism $$X\rightarrow Y.$$ From this it follows that the category of isoclinic $$p$$-divisible groups up to isogeny over $$R$$ is equivalent to the same category over $$k.$$
Finally, examples are given. The first example is the one referred to above – a $$p$$-divisible group with constant Newton polygon which does not have a slope filtration. The other example given in the case $$S$$ is not normal is of an $$X$$ which is not isogenous to a $$Y$$ admitting a slope filtration. These illustrate the necessity of the conditions on $$S$$ as well as the “up to isogeny” portion of the corollary.

##### MSC:
 14L05 Formal groups, $$p$$-divisible groups 14F30 $$p$$-adic cohomology, crystalline cohomology 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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