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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.

14L05 Formal groups, \(p\)-divisible groups
14F30 \(p\)-adic cohomology, crystalline cohomology
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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