ON $p$-DIVISIBLE GROUPS WITH SATURATED NEWTON POLYGONS

Abstract

This paper concerns the classification of isogeny classes of $p$-divisible groups with saturated Newton polygons. Let $S$ be a normal Noetherian scheme in positive characteristic $p$ with a prime Weil divisor $D$. Let ${\mathcal{X}}$ be a $p$-divisible group over $S$ whose geometric fibers over $S\setminus D$ (resp. over $D$) have the same Newton polygon. Assume that the Newton polygon of ${\mathcal{X}}_{D}$ is saturated in that of ${\mathcal{X}}_{S\setminus D}$. Our main result (Corollary 1.1) says that ${\mathcal{X}}$ is isogenous to a $p$-divisible group over $S$ whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc. 17(2) (2004), 267–296; 6.9).

1 Introduction

Let $S$ be a scheme in positive characteristic $p$ . A $p$ -divisible group over $S$ is called NP-constant if all of its geometric fibers have the same Newton polygon. In [19], Zink proved that if $S$ is regular, then any NP-constant $p$ -divisible group over $S$ is isogenous to a $p$ -divisible group that has a slope filtration. The case that $S$ is finitely generated over a perfect field with $\dim (S)=1$ had already been shown by Katz [7, Corollary 2.6.3]. The result of Oort and Zink [13, Theorem 2.1] is quite general, where they showed that the same statement holds even when $S$ is a normal Noetherian scheme.

The aim of this paper is to weaken the NP-constancy condition. Since the condition on slope filtration makes sense only for NP-constant $p$ -divisible groups, we instead use the condition that all geometric fibers are minimal. The definition of minimality of [12, 1.1] is recalled in Definition 3.4. Note that any NP-constant $p$ -divisible group whose geometric fibers are all minimal has a slope filtration.

Let $S$ be a scheme in characteristic $p>0$ , and let $D$ be a closed subscheme on $S$ . An NP-saturated $p$ -divisible group over $(S,D)$ is a $p$ -divisible group ${\mathcal{X}}$ over $S$ such that ${\mathcal{X}}_{S\setminus D}$ and ${\mathcal{X}}_{D}$ are NP-constant and the Newton polygon of ${\mathcal{X}}_{D}$ is saturated in that of ${\mathcal{X}}_{S\setminus D}$ . Here, for two Newton polygons $\unicode[STIX]{x1D709},\unicode[STIX]{x1D701}$ where $\unicode[STIX]{x1D701}$ is less than $\unicode[STIX]{x1D709}$ , we say that $\unicode[STIX]{x1D701}$ is saturated in $\unicode[STIX]{x1D709}$ if there is no other Newton polygon between $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D709}$ . As a corollary of our main theorem (Theorem 4.2), we have the following.

Corollary 1.1. Assume that $S$ is Noetherian and normal and that $D$ is a prime Weil divisor. Then, any NP-saturated $p$ -divisible group over $(S,D)$ is isogenous to a $p$ -divisible group over $S$ whose geometric fibers are all minimal.

This means that in order to classify up to isogeny, NP-saturated $p$ -divisible groups over $(S,D)$ as in Corollary 1.1, it suffices to look into NP-saturated $p$ -divisible groups whose geometric fibers are all minimal. Such $p$ -divisible groups are very specific, and can be said to be concrete objects in the deformation theory at least for local $S$ , since the isomorphism class of every geometric fiber is determined.

This paper is organized as follows. In Section 2, we introduce the notion of quasi-saturated Newton polygons. The above corollary is regarded as a special case of a more general result on NP-quasi-saturated $p$ -divisible groups. In Section 3, we investigate the relation between the slope-divisibility and the minimality of $p$ -divisible groups. We introduce an isogeny $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}:X\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ in (18) and show some nice properties of the isogeny, which are used in the next section. The first part of Section 4 is the heart of this paper, where we prove the theorem in the case of $S=\operatorname{Spec}(R)$ with discrete valuation ring $R$ . In the second part, we extend it to general $(S,D)$ as in Corollary 1.1, using the ideas invented by [13]. In Section 5, as an application, we give a geometrical proof of the unpolarized analog of [2, Corollary 3.2] on the configuration of the minimal $p$ -kernel type, and show the unpolarized analog of Oort’s conjecture.

2 Quasi-saturated Newton polygons

A Newton polygon is a finite multiset of coprime pairs of nonnegative integers

(1) $$\begin{eqnarray}\{(m_{1},n_{1}),\ldots ,(m_{t},n_{t})\};\end{eqnarray}$$

that is to say, a function from the set of coprime pairs of nonnegative integers to the set of nonnegative integers with finite support. We define the addition of Newton polygons to be the addition of their functions, which are denoted by $+_{\operatorname{NP}}$ , so that we distinguish this from addition of two-dimensional vectors.

We regard Newton polygons as upward-convex line graphs defined in the following way. Let $\unicode[STIX]{x1D709}=\{(m_{1},n_{1}),\ldots ,(m_{t},n_{t})\}$ be a Newton polygon. Put $h=\sum _{i=1}^{t}(m_{i}+n_{i})$ and $d=\sum _{i=1}^{t}n_{i}$ . Set $\unicode[STIX]{x1D706}_{i}=n_{i}/h_{i}$ with $h_{i}:=m_{i}+n_{i}$ . We arrange the coprime pairs $(m_{i},n_{i})$ ( $i=1,\ldots ,t$ ) so that

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{1}\geqslant \unicode[STIX]{x1D706}_{2}\geqslant \cdots \geqslant \unicode[STIX]{x1D706}_{t}.\end{eqnarray}$$

To $\unicode[STIX]{x1D709}$ we associate the line graph obtained as the upper convex hull of the points $\sum _{i=1}^{j}(h_{i},n_{i})$ for $j=0,\ldots ,t$ . The line graph starts at $(0,0)$ and ends at $(h,d)$ . We call $(h_{i},n_{i})$ ( $i=1,\ldots ,t$ ) segments of $\unicode[STIX]{x1D709}$ .

Let $\unicode[STIX]{x1D709}$ be a Newton polygon. If a point $P$ is below or on $\unicode[STIX]{x1D709}$ , we write $P$ $\preccurlyeq \unicode[STIX]{x1D709}$ . For another Newton polygon $\unicode[STIX]{x1D701}$ whose end point is equal to that of $\unicode[STIX]{x1D709}$ , we say $\unicode[STIX]{x1D701}\preccurlyeq \unicode[STIX]{x1D709}$ if for every point $P$ on $\unicode[STIX]{x1D701}$ we have $P\preccurlyeq \unicode[STIX]{x1D709}$ . We say $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ if $\unicode[STIX]{x1D701}\preccurlyeq \unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D701}\neq \unicode[STIX]{x1D709}$ . Let $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D709}$ be Newton polygons with $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ . We say that $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is saturated if there is no Newton polygon $\unicode[STIX]{x1D702}$ such that $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D702}\prec \unicode[STIX]{x1D709}$ .

In the rest of this section, we introduce the notion of quasi-saturated pairs of Newton polygons, for which almost all arguments in this paper work, and give a numerical criterion for the saturatedness in the case that $\unicode[STIX]{x1D709}$ consists of two segments (see Lemma 2.2 below).

To a rational number $\unicode[STIX]{x1D706}=r/s$ with coprime nonnegative integers $r,s$ , we associate the two-dimensional vectors

(2) $$\begin{eqnarray}v_{\unicode[STIX]{x1D706}}=(s,r)\end{eqnarray}$$

and

(3) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})=\mathop{\sum }_{n_{i}/h_{i}>\unicode[STIX]{x1D706}}(h_{i},n_{i})\end{eqnarray}$$

for a Newton polygon $\unicode[STIX]{x1D709}$ of the form (1). We use the alternating form $\langle ~,~\rangle$ on two-dimensional vectors:

(4) $$\begin{eqnarray}\langle (a,b),(c,d)\rangle =ad-bc.\end{eqnarray}$$

If $\unicode[STIX]{x1D701}\preccurlyeq \unicode[STIX]{x1D709}$ , then we have $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})\rangle \geqslant 0$ for any $\unicode[STIX]{x1D706}$ . This is clear if we know the following graphical meaning of the value $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})\rangle$ : for $v_{\unicode[STIX]{x1D706}}=(s,r)$ , the line with slope $r/s$ that is tangent to $\unicode[STIX]{x1D709}$ is given by

$$\begin{eqnarray}\ell _{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709}):\quad y=\frac{r}{s}x+\frac{1}{s}\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})\rangle .\end{eqnarray}$$

See Figure 1. Note that $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})$ is the first point where $\ell _{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})$ is tangent to $\unicode[STIX]{x1D709}$ . If $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D701}$ , then $\ell _{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})$ is below or on $\ell _{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})$ , whence $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})\rangle \leqslant \langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})\rangle$ .

Figure 1. $\ell _{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})$ is the line with slope $\unicode[STIX]{x1D706}$ that is tangent to $\unicode[STIX]{x1D709}$ .

Definition 2.1. We say that $\unicode[STIX]{x1D701}\preccurlyeq \unicode[STIX]{x1D709}$ is quasi-saturated if for each slope $\unicode[STIX]{x1D706}$ of $\unicode[STIX]{x1D701}$ , we have $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})\rangle \leqslant 1$ .

Note that the condition of $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})\rangle \leqslant 1$ is equivalent to there being no lattice point properly between $\ell _{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})$ and $\ell _{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})$ .

Lemma 2.2. If $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is saturated, then $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is quasi-saturated. The converse holds if $\unicode[STIX]{x1D709}$ consists of two segments.

Proof. Let $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ be a saturated pair of Newton polygons. One can write

(5) $$\begin{eqnarray}\unicode[STIX]{x1D701}=\unicode[STIX]{x1D71A}+_{\operatorname{NP}}\unicode[STIX]{x1D701}^{\prime }\quad \text{and}\quad \unicode[STIX]{x1D709}=\unicode[STIX]{x1D71A}+_{\operatorname{ NP}}\unicode[STIX]{x1D709}^{\prime },\end{eqnarray}$$

so that $\unicode[STIX]{x1D701}^{\prime }\prec \unicode[STIX]{x1D709}^{\prime }$ is saturated and $\unicode[STIX]{x1D709}^{\prime }$ consists of only two segments. Write

(6) $$\begin{eqnarray}\unicode[STIX]{x1D701}^{\prime }=\{(m_{1},n_{1}),\ldots ,(m_{t},n_{t})\}\quad \text{and}\quad \unicode[STIX]{x1D709}^{\prime }=\{(a_{1},b_{1}),(a_{2},b_{2})\}.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D701}^{\prime }$ and $\unicode[STIX]{x1D71A}$ do not share any slope. For each slope $\unicode[STIX]{x1D706}$ of $\unicode[STIX]{x1D71A}$ , we have $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})\rangle =0$ .

Let $\unicode[STIX]{x1D706}$ be a slope of $\unicode[STIX]{x1D701}^{\prime }$ . Let $j$ be the smallest index with $\unicode[STIX]{x1D706}=n_{j}/h_{j}$ with $h_{j}=m_{j}+n_{j}$ . Note that $v_{\unicode[STIX]{x1D706}}=(h_{j},n_{j})$ . Put $v=(a_{1}+b_{1},b_{1})$ and $u_{i}=(h_{i},n_{i})$ , which are considered as two-dimensional vectors. We have

(7) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D71A})+v\quad \text{and}\quad \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D71A})+\mathop{\sum }_{i<j}u_{i}.\end{eqnarray}$$

The condition $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})\rangle =1$ is equivalent to the condition that in the triangle with vertices $v,\sum _{i<j}u_{i}$ and $\sum _{i\leqslant j}u_{i}$ , there is no lattice point other than the vertices. (In this case, the same thing holds for the triangle with vertices $v,\sum _{i<l}u_{j}$ and $\sum _{i\leqslant l}u_{i}$ for all $l$ with $n_{l}/h_{l}=\unicode[STIX]{x1D706}$ .) Hence, the condition that $\langle v_{\unicode[STIX]{x1D706}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})\rangle =1$ for all slopes $\unicode[STIX]{x1D706}$ of $\unicode[STIX]{x1D701}^{\prime }$ is equivalent to there being no lattice point $P$ above $\unicode[STIX]{x1D701}^{\prime }$ with $P\preccurlyeq \unicode[STIX]{x1D709}^{\prime }$ except the breaking point of $\unicode[STIX]{x1D709}^{\prime }$ . This is equivalent to $\unicode[STIX]{x1D701}^{\prime }\prec \unicode[STIX]{x1D709}^{\prime }$ being saturated.◻

Example 2.3. Let $\unicode[STIX]{x1D709}=(0,1)+_{\operatorname{NP}}(1,3)+_{\operatorname{NP}}(3,1)+_{\operatorname{NP}}(1,0)$ , and let $\unicode[STIX]{x1D701}=(0,1)+_{\operatorname{NP}}(1,2)+_{\operatorname{NP}}(1,1)+_{\operatorname{NP}}(2,1)+_{\operatorname{NP}}(1,0)$ . Note that $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is saturated. In the proof of the lemma above, we use the notation $\unicode[STIX]{x1D709}^{\prime }=(1,3)+_{\operatorname{NP}}(3,1)$ and $\unicode[STIX]{x1D701}^{\prime }=(1,2)+_{\operatorname{NP}}(1,1)+_{\operatorname{NP}}(2,1)$ with $\unicode[STIX]{x1D70C}=(0,1)+_{\operatorname{NP}}(1,0)$ . See Figure 2 for the picture of $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ .

Figure 2. The picture of $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ .

In the second statement in Lemma 2.2, the condition that $\unicode[STIX]{x1D709}$ consists of two segments is necessary.

Example 2.4. Consider $\unicode[STIX]{x1D709}=(0,1)+_{\operatorname{NP}}(1,1)+_{\operatorname{NP}}(1,0)$ and $\unicode[STIX]{x1D701}=2(1,1)$ . Then, $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is not saturated, since $\unicode[STIX]{x1D701}\prec (0,1)+_{\operatorname{NP}}(2,1)\prec \unicode[STIX]{x1D709}$ . However, $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is quasi-saturated.

3 Slope-divisibility and minimality

A slope with exponent is a pair $(\unicode[STIX]{x1D706},e)$ of rational number $\unicode[STIX]{x1D706}$ with $0\leqslant \unicode[STIX]{x1D706}\leqslant 1$ and integer $e\neq 0$ . Let $\unicode[STIX]{x1D6EC}$ be the set of slopes with exponents

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}=\{(\unicode[STIX]{x1D706},e)\in \mathbb{Q}\times \mathbb{Z}\mid 0\leqslant \unicode[STIX]{x1D706}\leqslant 1,e\neq 0\}.\end{eqnarray}$$

For $\unicode[STIX]{x1D707}=(\unicode[STIX]{x1D706},e)\in \unicode[STIX]{x1D6EC}$ , we call $e$ the exponent of $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D706}$ the slope of $\unicode[STIX]{x1D707}$ , which will be denoted by $\overline{\unicode[STIX]{x1D707}}$ ,

(8) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D707}}:=\unicode[STIX]{x1D706}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D6EC}_{e}$ be the subset of $\unicode[STIX]{x1D6EC}$ consisting of elements with exponent $e$ . We identify $\unicode[STIX]{x1D6EC}_{1}$ with $\{\unicode[STIX]{x1D706}\in \mathbb{Q}\mid 0\leqslant \unicode[STIX]{x1D706}\leqslant 1\}$ , the set of usual slopes, by mapping $(\unicode[STIX]{x1D706},1)$ to $\unicode[STIX]{x1D706}$ . Let $\unicode[STIX]{x1D6EC}_{+}$ (resp. $\unicode[STIX]{x1D6EC}_{-}$ ) be the subset of $\unicode[STIX]{x1D6EC}$ consisting of elements with positive (resp. negative) exponents. We use the embedding of $\unicode[STIX]{x1D6EC}$ into $\mathbb{Z}^{2}$ sending $\unicode[STIX]{x1D707}=(r/s,e)$ with coprime integers $r,s\geqslant 0$ to

(9) $$\begin{eqnarray}v_{\unicode[STIX]{x1D707}}=e(s,r).\end{eqnarray}$$

Let $S$ be a scheme in characteristic $p>0$ . Let $\operatorname{Frob}_{S}:S\rightarrow S$ be the Frobenius morphism. Let $X$ be a $p$ -divisible group over $S$ . Set $X^{(p^{a})}=X\times _{\operatorname{Frob}_{S}^{a}}S$ . We denote by $\operatorname{Fr}:X\rightarrow X^{(p)}$ the relative Frobenius homomorphism and by $\operatorname{Ver}:X^{(p)}\rightarrow X$ the Verschiebung.

For $\unicode[STIX]{x1D707}=(\unicode[STIX]{x1D706},e)\in \unicode[STIX]{x1D6EC}$ , we write $\unicode[STIX]{x1D706}=r/s$ with coprime integers $r,s\geqslant 0$ , and consider the quasi-isogeny

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}=\left(p^{-r}\operatorname{Fr}^{s}\right)^{e}\end{eqnarray}$$

from $X$ to $X^{(p^{se})}$ if $e>0$ and from $X^{(p^{se})}$ to $X$ if $e<0$ . This is simply referred as “ $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ on $X$ ”.

Definition 3.1. Let $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}$ . We say that $X$ is slope divisible (resp. isoclinic and slope divisible) with respect to $\unicode[STIX]{x1D707}$ if the quasi-isogeny $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ on $X$ is an isogeny (resp. isomorphism), where $X=0$ is allowed.

Remark 3.2. If $X$ is slope divisible with respect to $\unicode[STIX]{x1D707}=(\unicode[STIX]{x1D706},e)$ , then its Serre dual is slope divisible with respect to $(1-\unicode[STIX]{x1D706},-e)$ , because the dual of $p^{-r}\operatorname{Fr}^{s}$ on $X$ is $p^{-r}\operatorname{Ver}^{s}=\left(p^{-(s-r)}\operatorname{Fr}^{s}\right)^{-1}$ on the Serre dual. In general, when we consider $\operatorname{Ver}$ -slopes, negative exponents appear naturally.

For $\unicode[STIX]{x1D707}=(\unicode[STIX]{x1D706},e)\in \unicode[STIX]{x1D6EC}$ , we set $\unicode[STIX]{x1D707}^{\ast }:=(\unicode[STIX]{x1D706},-e)$ . Note that $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}^{\ast }}=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}^{-1}$ .

Definition 3.3. Let $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}_{+}$ . Let $Y\subset X$ be a closed immersion of $p$ -divisible groups. We say that $Y$ in $X$ is slope bi-divisible with respect to $\unicode[STIX]{x1D707}$ if the quasi-isogeny $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ on $Y$ is an isogeny and also $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}^{\ast }}$ on $X/Y$ is an isogeny.

Let $\mathbb{D}$ be the covariant Dieudonné functor with $\mathbb{D}(\operatorname{Fr})=V$ and $\mathbb{D}(\operatorname{Ver})=F$ . Let $m$ and $n$ be coprime nonnegative integers. Let $H_{m,n}$ be the $p$ -divisible group over $\mathbb{F}_{p}$ whose Dieudonné module $N_{m,n}=\mathbb{D}(H_{m,n})$ is given by

(10) $$\begin{eqnarray}N_{m,n}=\bigoplus _{i=1}^{m+n}\mathbb{Z}_{p}\unicode[STIX]{x1D716}_{i}\end{eqnarray}$$

with $F\unicode[STIX]{x1D716}_{i}=\unicode[STIX]{x1D716}_{i+m}$ , $V\unicode[STIX]{x1D716}_{i}=\unicode[STIX]{x1D716}_{i+n}$ and $\unicode[STIX]{x1D716}_{i+m+n}=p\unicode[STIX]{x1D716}_{i}$ . Note that $H_{m,n}$ is a simple $p$ -divisible group with slope $n/(m+n)$ . Let $\unicode[STIX]{x1D71B}$ be the endomorphism of $H_{m,n}$ characterized by $\mathbb{D}(\unicode[STIX]{x1D71B})(\unicode[STIX]{x1D716}_{i})=\unicode[STIX]{x1D716}_{i+1}$ . It is straightforward to see that

(11) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D71B}^{\langle v_{\unicode[STIX]{x1D707}},(m+n,n)\rangle }.\end{eqnarray}$$

Let $K$ be a field of characteristic $p$ . A $p$ -divisible group over $K$ is called isoclinic and minimal if it is isomorphic over the algebraic closure $\overline{K}$ of $K$ to the product of some copies of $\left(H_{m,n}\right)_{\overline{K}}$ for a certain coprime pair $(m,n)$ of nonnegative integers. Clearly, an isoclinic minimal $p$ -divisible group with slope $\unicode[STIX]{x1D706}$ is slope divisible with respect to any $\unicode[STIX]{x1D707}$ with $\langle v_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D706}}\rangle \geqslant 0$ and is isoclinic and slope divisible with respect to $\unicode[STIX]{x1D706}$ .

Recall the definition [12, 1.1] of minimal $p$ -divisible groups. For a Newton polygon $\unicode[STIX]{x1D709}=\{(m_{1},n_{1}),\ldots ,(m_{t},n_{t})\}$ , we set

(12) $$\begin{eqnarray}H(\unicode[STIX]{x1D709}):=\bigoplus _{i=1}^{t}H_{m_{i},n_{i}}.\end{eqnarray}$$

Definition 3.4. A $p$ -divisible group over $K$ is called minimal if it is isomorphic over $\overline{K}$ to $H(\unicode[STIX]{x1D709})_{\overline{K}}$ for some Newton polygon $\unicode[STIX]{x1D709}$ .

Also recall the definition of completely slope divisible $p$ -divisible groups, which is slightly generalized from that in [13, 1.2] for later use. Let us introduce a partial variant at the same time.

Definition 3.5. Let $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }\in \unicode[STIX]{x1D6EC}_{+}$ , with $\overline{\unicode[STIX]{x1D707}_{1}}>\cdots >\overline{\unicode[STIX]{x1D707}_{\ell }}$ . A $p$ -divisible group $X$ is called partially completely slope divisible with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ if there exists a filtration by closed immersions of $p$ -divisible groups

(13) $$\begin{eqnarray}0\subset X_{0}\subset X_{1}\subset \cdots \subset X_{\ell -1}\subset X_{\ell }=X\end{eqnarray}$$

such that

1. (i) the $X_{j}$ $(j\leqslant i)$ are slope divisible with respect to $\unicode[STIX]{x1D707}_{i}$ for $1\leqslant i\leqslant \ell$ ;

2. (ii) $\operatorname{Gr}_{i}(X):=X_{i}/X_{i-1}$ is isoclinic and slope divisible with respect to $\unicode[STIX]{x1D707}_{i}$ for $1\leqslant i\leqslant \ell$ ;

3. (iii) all of the slopes of $X_{0}$ are greater than $\overline{\unicode[STIX]{x1D707}_{1}}$ .

When $X_{0}=0$ , we remove partially.

In the same way as in [19, Corollary 11], one can show the following.

Lemma 3.6. Assume that $K$ is a perfect field of characteristic $p$ . Let $X$ be a partially completely slope divisible $p$ -divisible group over $K$ with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ . Then, $X$ is isomorphic to $X_{0}\oplus \bigoplus _{i=1}^{\ell }\operatorname{Gr}_{i}(X)$ .

Let us define the bi-divisible variant.

Definition 3.7. Let $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }\in \unicode[STIX]{x1D6EC}_{+}$ , with $\overline{\unicode[STIX]{x1D707}_{1}}>\cdots >\overline{\unicode[STIX]{x1D707}_{\ell }}$ . A $p$ -divisible group $X$ is called partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ if there exists a filtration by closed immersions of $p$ -divisible groups

(14) $$\begin{eqnarray}0\subset X_{0}\subset X_{1}\subset \cdots \subset X_{\ell -1}\subset X_{\ell }=X\end{eqnarray}$$

such that

1. (i) the $X_{j}$ $(j\leqslant i)$ are slope divisible with respect to $\unicode[STIX]{x1D707}_{i}$ for $1\leqslant i\leqslant \ell$ ;

2. (ii) the $X/X_{j}$ $(j\geqslant i-1)$ are slope divisible with respect to $\unicode[STIX]{x1D707}_{i}^{\ast }$ for $1\leqslant i\leqslant \ell$ ;

3. (iii) all of the slopes of $X_{0}$ are greater than $\overline{\unicode[STIX]{x1D707}_{1}}$ .

When $X_{0}=0$ , we remove partially.

Lemma 3.8. Let $X$ be a $p$ -divisible group. Assume that $X$ is partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ . Then, we have the following.

1. (1) $X$ is partially completely slope divisible with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ .

2. (2) For $i=1,\ldots ,\ell$ , we have that $\operatorname{Gr}_{i}(X)$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{a}$ for $a\geqslant i$ and $\unicode[STIX]{x1D707}_{b}^{\ast }$ for $b\leqslant i$ .

Proof. (1) The quasi-isogeny $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{i}}:X_{i}/X_{i-1}\rightarrow X_{i}/X_{i-1}$ is an isogeny, because this is induced by the isogeny $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{i}}$ on $X_{i}$ . Consider the composition $X_{i}/X_{i-1}\rightarrow X_{i}/X_{i-1}\rightarrow X/X_{i-1}$ of $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{i}}$ and the restriction to $X_{i}/X_{i-1}$ of $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{i}^{\ast }}$ on $X/X_{i-1}$ . Since $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{i}^{\ast }}=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{i}}^{-1}$ , this composition is identical on $X_{i}/X_{i-1}$ . In particular, the kernel of $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{i}}:X_{i}/X_{i-1}\rightarrow X_{i}/X_{i-1}$ is zero. Hence, $X_{i}/X_{i-1}$ is isoclinic and slope divisible with respect to $\unicode[STIX]{x1D707}_{i}$ .

(2) It suffices to show this for each geometric fiber. Hence, we may assume that $X$ is a $p$ -divisible group over an algebraically closed field. By Lemma 3.6, $X$ is isomorphic to $X_{0}\oplus \bigoplus _{i=1}^{\ell }\operatorname{Gr}_{i}(X)$ . Since $X_{i}$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{a}$ for $a\geqslant i$ , its direct summand $\operatorname{Gr}_{i}(X)$ is also slope divisible with respect to $\unicode[STIX]{x1D707}_{a}$ for $a\geqslant i$ . Since $X/X_{i-1}$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{b}^{\ast }$ for $b\leqslant i$ , its direct summand $\operatorname{Gr}_{i}(X)$ is also slope divisible with respect to $\unicode[STIX]{x1D707}_{b}^{\ast }$ for $b\leqslant i$ .◻

Remark 3.9. Let $X$ be a minimal $p$ -divisible group over a field $K$ of characteristic $p$ . Then, $X$ is completely slope bi-divisible with respect to its slopes.

Example 3.10. Let $N_{3,2}=\bigoplus _{i=1}^{5}\mathbb{Z}_{p}\unicode[STIX]{x1D716}_{i}$ be as in (10). Let $M$ be the Dieudonné submodule of $N_{3,2}$ generated by $\unicode[STIX]{x1D716}_{1},p\unicode[STIX]{x1D716}_{2},\unicode[STIX]{x1D716}_{3},\unicode[STIX]{x1D716}_{4},\unicode[STIX]{x1D716}_{5}$ . Let $Y$ be a $p$ -divisible group over $\mathbb{F}_{p}$ whose Dieudonné module is isomorphic to $M$ . Let $X=H_{1,1}\oplus Y$ . Set $\unicode[STIX]{x1D707}_{1}=(1/2,1)$ and $\unicode[STIX]{x1D707}_{2}=(2/5,1)$ . Note that $X$ is completely slope divisible with respect to $\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}$ , whose slope filtration is $0\subset H_{1,1}\subset X$ . However, $X$ is not completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}$ , since $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{1}^{\ast }}=p^{-1}\operatorname{Ver}^{2}$ is not isogeny on $Y$ .

Lemma 3.11. Let ${\mathcal{X}}$ be an NP-constant $p$ -divisible group over $S$ . Then, the subset of points of $S$ over which the fiber of ${\mathcal{X}}$ is completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ is closed in $S$ .

Proof. Write $v_{\unicode[STIX]{x1D707}_{i}}=(s_{i},r_{i})$ . Let $s$ be the least common multiple of $s_{1},\ldots ,s_{\ell }$ . Let $\unicode[STIX]{x1D707}_{i}^{\prime }$ be the elements of $\unicode[STIX]{x1D6EC}_{+}$ such that $v_{\unicode[STIX]{x1D707}_{i}^{\prime }}=(s/s_{i})v_{\unicode[STIX]{x1D707}_{i}}$ . By [13, 2.3], the subset of points of $S$ over which the fiber of ${\mathcal{X}}$ is completely slope divisible with respect to $\unicode[STIX]{x1D707}_{1}^{\prime },\ldots ,\unicode[STIX]{x1D707}_{\ell }^{\prime }$ is closed in $S$ . Then, the lemma follows from the fact [15, Proposition 2.9] that for a quasi-isogeny $\unicode[STIX]{x1D70C}:X\rightarrow Y$ of $p$ -divisible groups over $S$ , the subset of points of $S$ over which $\unicode[STIX]{x1D70C}$ is an isogeny is closed in $S$ .◻

We see from Remark 3.9 that any minimal $p$ -divisible group is completely slope divisible. Let us study when a completely slope divisible $p$ -divisible group is minimal.

Proposition 3.12. Let $\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{1}$ . Let $X$ be a $p$ -divisible group over a field $K$ of characteristic $p$ that is isoclinic and slope divisible with respect to $\unicode[STIX]{x1D706}$ . The following are equivalent.

1. (1) $X$ is minimal.

2. (2) For any $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}$ with $\langle v_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D706}}\rangle >0$ , the quasi-isogeny $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ on $X$ is an isogeny.

3. (3) For a $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}$ with $\langle v_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D706}}\rangle =1$ , the quasi-isogeny $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ on $X$ is an isogeny.

Proof. It suffices to show the case that $K$ is algebraically closed. For $v_{\unicode[STIX]{x1D706}}=(m+n,n)$ , we write

(15) $$\begin{eqnarray}H_{\unicode[STIX]{x1D706}}:=(H_{m,n})_{K}.\end{eqnarray}$$

(1) $\Rightarrow$ (2): Let $X$ be an isoclinic and minimal $p$ -divisible group, say

$$\begin{eqnarray}X=H_{\unicode[STIX]{x1D706}}^{\oplus \unicode[STIX]{x1D708}}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}$ with $\langle v_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D706}}\rangle >0$ . As seen in (11), $\mathbb{D}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}})$ on $\mathbb{D}(H_{\unicode[STIX]{x1D706}})$ is the map sending $\unicode[STIX]{x1D716}_{i}$ to $\unicode[STIX]{x1D716}_{i+\langle v_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D706}}\rangle }$ . Thus, $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ on $H_{\unicode[STIX]{x1D706}}^{\oplus \unicode[STIX]{x1D708}}$ is an isogeny.

(2) $\Rightarrow$ (3) is obvious.

(3) $\Rightarrow$ (1): Write $v_{\unicode[STIX]{x1D706}}=(m+n,n)$ and $v_{\unicode[STIX]{x1D707}}=(a+b,b)$ . Since $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}}=p^{-n}\operatorname{Fr}^{m+n}$ and $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}=p^{-b}\operatorname{Fr}^{a+b}$ , we have

(16) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}}^{-b}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}^{n}=\operatorname{Fr},\quad \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D706}}^{-a}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}^{m}=\operatorname{Ver}.\end{eqnarray}$$

Let $G_{i}=X[\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}^{i}]$ be the kernel of the isogeny $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}^{i}$ on $X$ for $i=0,1,\ldots ,m+n$ . We have a filtration of $X[p]$ :

$$\begin{eqnarray}0=G_{0}\subset G_{1}\subset \cdots \subset G_{m+n}=X[p].\end{eqnarray}$$

By (16), we have $\operatorname{Fr}\,G_{i}=G_{i-n}$ and $\operatorname{Ver}\,G_{i}=G_{i-m}$ . Since $m$ and $n$ are coprime, $\{G_{i}/G_{i-1}\mid i=1,\ldots ,m+n\}$ consists of one $(\operatorname{Ver},\operatorname{Fr}^{-1})$ -cycle (cf. [8]), whence $G_{i}/G_{i-1}$ ( $i=1,\ldots ,m+n$ ) have the same rank, say $\unicode[STIX]{x1D708}$ . Thus, $X[p]$ is isomorphic to $(H_{m,n}^{\oplus \unicode[STIX]{x1D708}}[p])_{K}$ , and therefore $X$ is minimal by [12].◻

Let us give an alternative proof of a special case of [11, 2.2].

Corollary 3.13. Let ${\mathcal{X}}$ be an NP-constant $p$ -divisible group over $S$ . Then, the subset of points of $S$ over which the fiber of ${\mathcal{X}}$ is minimal is closed in $S$ .

Proof. Let $\unicode[STIX]{x1D706}_{1}>\cdots >\unicode[STIX]{x1D706}_{\ell }$ be the slopes of ${\mathcal{X}}$ . In a similar way to that in Lemma 3.11, the subset of points of $S$ over which the fiber of ${\mathcal{X}}$ is completely slope divisible with respect to $\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{\ell }$ is closed in $S$ . Hence, we may assume that ${\mathcal{X}}$ is completely slope divisible with respect to $\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{\ell }$ . Let $0={\mathcal{X}}_{0}\subset {\mathcal{X}}_{1}\subset \cdots \subset {\mathcal{X}}_{\ell }={\mathcal{X}}$ be the slope filtration. Let $s$ be a point of $S$ . Note that ${\mathcal{X}}_{s}$ is minimal if and only if $({\mathcal{X}}_{i}/{\mathcal{X}}_{i-1})_{s}$ is minimal for all $i=1,\ldots ,\ell$ (cf. Lemma 3.6). By Proposition 3.12, $({\mathcal{X}}_{i}/{\mathcal{X}}_{i-1})_{s}$ is minimal if and only if $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ on $({\mathcal{X}}_{i}/{\mathcal{X}}_{i-1})_{s}$ is an isogeny for some $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}$ with $\langle v_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D706}}\rangle =1$ . Hence, the corollary follows from [15, Proposition 2.9].◻

Let $K$ be a field of characteristic $p$ . Recall the definition of the small image of a homomorphism of $p$ -divisible groups over $K$ . This notion was introduced by Zink in [19, Section 3]. Let $g:G\rightarrow H$ be a homomorphism of $p$ -divisible groups over $K$ . It is shown in [19, Proposition 8] that $g$ has a unique factorization in the category of $p$ -divisible groups

$$\begin{eqnarray}G\rightarrow G^{\prime }\rightarrow H^{\prime }\rightarrow H,\end{eqnarray}$$

where $G^{\prime }\rightarrow H^{\prime }$ is an isogeny, $H^{\prime }\rightarrow H$ is a monomorphism of $p$ -divisible groups and $G\rightarrow G^{\prime }$ is a homomorphism satisfying that $G[p^{n}]\rightarrow G^{\prime }[p^{n}]$ is an epimorphism for each natural number $n$ . We call $G^{\prime }$ the small image of $g$ . In the proof of [19, Proposition 8], the small image $G^{\prime }$ is given by the quotient of $G$ by $A^{\prime }$ , where $A^{\prime }$ is the unique $p$ -divisible subgroup of $\operatorname{Ker}(g)$ such that $\operatorname{Ker}(g)/A^{\prime }$ is a finite group scheme. If $K$ is perfect, then $\mathbb{D}(G^{\prime })$ is the image of $\mathbb{D}(g)$ , and $\mathbb{D}(H^{\prime })$ is the smallest direct summand of $\mathbb{D}(H)$ containing $\mathbb{D}(G^{\prime })$ .

Let $X$ be a $p$ -divisible group over $K$ . Let $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}_{+}$ , and write $v_{\unicode[STIX]{x1D707}}=(s,r)$ . Let $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ be the small image of

(17)

where the second morphism is the addition of $X$ . Let $A=\operatorname{Ker}(f_{\unicode[STIX]{x1D707}})$ . Consider the homomorphism $g:X^{(p^{s})}\rightarrow A$ sending $y$ to $(\operatorname{Ver}^{s}y,-p^{s-r}y)$ . The kernel and the cokernel of $g$ are finite, since both are killed by $p^{s-r}$ . Hence, the image $Z$ of $g$ is the maximal $p$ -divisible subgroup of $A$ . By the construction of the small image explained above, we have $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)=X\times X^{(p^{s})}/Z$ . Composing $(\operatorname{id},0):X\rightarrow X\times X^{(p^{s})}$ and $X\times X^{(p^{s})}\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ , we have an isogeny

(18)

Since the kernel of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ is the intersection of $Z$ and $X\times \{0\}$ , we have the following.

Lemma 3.14. We have

$$\begin{eqnarray}\operatorname{Ker}(\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}})=\operatorname{Im}(\operatorname{Ver}^{s}:X^{(p^{s})}[p^{s-r}]\rightarrow X[p^{s-r}]).\end{eqnarray}$$

Here, the right-hand side is the image as the fppf sheaf, which is represented by a group scheme $X^{(p^{s})}[p^{s-r}]/\operatorname{Ker}(\operatorname{Ver}_{X^{(p^{s})}[p^{s-r}]}^{s})$ .

Remark 3.15. Assume that $K$ is a perfect field. Let $M$ be the Dieudonné module of $X$ . Then, the Dieudonné module of $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ is

$$\begin{eqnarray}\mathbb{D}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X))=p^{s-r}M+F^{s}M,\end{eqnarray}$$

which is isomorphic to $p^{-r}V^{s}M+M$ . The isogeny $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ in (18) corresponds to the isogeny $M\rightarrow p^{s-r}M+F^{s}M$ sending $m$ to $p^{s-r}m$ . The Dieudonné module of $\operatorname{Ker}(\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}})$ is

$$\begin{eqnarray}(p^{s-r}M+F^{s}M)/p^{s-r}M,\end{eqnarray}$$

which is isomorphic to $(M+p^{-(s-r)}F^{s}M)/M$ .

Lemma 3.16. Let $\unicode[STIX]{x1D707}=(\unicode[STIX]{x1D706},e)\in \unicode[STIX]{x1D6EC}_{+}$ . The following are equivalent.

1. (1) $\log _{p}\deg (\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}})=0$ .

2. (2) $X$ is slope divisible with respect to $\unicode[STIX]{x1D707}^{\ast }=(\unicode[STIX]{x1D706},-e)$ .

In this case, in particular, the slopes of $X$ are less than or equal to $\unicode[STIX]{x1D706}$ .

Proof. From the above remark, it is clear that (1) is equivalent to $p^{-(s-r)}\operatorname{Ver}^{s}$ is an isogeny on $X$ . Since $p^{-(s-r)}\operatorname{Ver}^{s}=(p^{-r}\operatorname{Fr}^{s})^{-1}$ , we have the lemma.◻

For $\unicode[STIX]{x1D707}=(\overline{\unicode[STIX]{x1D707}},e)\in \unicode[STIX]{x1D6EC}_{+}$ , we set $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})$ to be $\unicode[STIX]{x1D6FC}_{\overline{\unicode[STIX]{x1D707}}}(\unicode[STIX]{x1D709})$ .

Proposition 3.17. Let $\unicode[STIX]{x1D709}$ be the Newton polygon of $X$ . Assume that there is a short exact sequence

of $p$ -divisible groups over $K$ , which splits over $\overline{K}$ . Let $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}_{+}$ such that the slopes of $Y$ are greater than or equal to $\overline{\unicode[STIX]{x1D707}}$ and the slopes of $Z$ are less than or equal to $\overline{\unicode[STIX]{x1D707}}$ . Then, we have

$$\begin{eqnarray}\log _{p}\deg (\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}})\geqslant \langle v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})\rangle ,\end{eqnarray}$$

where the equality holds if and only if $Y$ in $X$ is slope bi-divisible with respect to $\unicode[STIX]{x1D707}$ .

Proof. We may assume that $K$ is an algebraically closed field and $X=Y\times Z$ . Let $h$ (resp. $d$ ) be the height (resp. the dimension) of $Y$ . Then, $\langle v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})\rangle =\langle v_{\unicode[STIX]{x1D707}},(h,d)\rangle$ . Let $M$ be the Dieudonné module of $Y$ . Set $A=F^{s}M$ and $B=p^{s-r}M$ . Then,

$$\begin{eqnarray}\log _{p}\deg (\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}})=\operatorname{length}(A+B)/B+\log _{p}\deg (\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}\text{ on }Z).\end{eqnarray}$$

Lemma 3.16 says that $\log _{p}\deg (\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}\text{ on }Z)=0$ if and only if $Z$ is slope divisible with respect to $\unicode[STIX]{x1D707}^{\ast }$ . Since $\operatorname{Coker}((A+B)/A\rightarrow M/A)=\operatorname{Coker}((A+B)/B\rightarrow M/B)$ , we have

$$\begin{eqnarray}\displaystyle \operatorname{length}(A+B)/B & = & \displaystyle \operatorname{length}M/B-\operatorname{length}M/A+\operatorname{length}(A+B)/A\nonumber\\ \displaystyle & = & \displaystyle (s-r)h-s(h-d)+\operatorname{length}(A+B)/A\nonumber\\ \displaystyle & = & \displaystyle \langle v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})\rangle +\operatorname{length}(A+B)/A.\nonumber\end{eqnarray}$$

Obviously, $(A+B)/A\simeq (M+p^{-r}V^{s}M)/M$ is zero if and only if $Y$ is slope divisible with respect to $\unicode[STIX]{x1D707}$ .◻

Lemma 3.18. Let $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}_{+}$ , and let $\unicode[STIX]{x1D707}^{\prime }\in \unicode[STIX]{x1D6EC}$ . If $X$ is slope divisible with respect to $\unicode[STIX]{x1D707}^{\prime }$ , then $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ is slope divisible with respect to $\unicode[STIX]{x1D707}^{\prime }$ . If $X$ is a minimal $p$ -divisible group, then so is $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ .

Proof. It suffices to show this over an algebraically closed field. Let $M$ be the Dieudonné module of $X$ . Write $v_{\unicode[STIX]{x1D707}}=(s,r)$ and $v_{\unicode[STIX]{x1D707}}^{\prime }=(s^{\prime },r^{\prime })$ . Obviously, if $p^{-r^{\prime }}V^{s^{\prime }}M\subset M$ , then $p^{-r^{\prime }}V^{s^{\prime }}N\subset N$ for $N=p^{s-r}M+F^{s}M$ . The second assertion follows from Proposition 3.12.◻

We collect some basic properties of the operators $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(-)$ for $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}_{+}$ .

Lemma 3.19. Let $X$ be a $p$ -divisible group over $K$ .

1. (1) We have $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(X))=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X))$ for $\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime }\in \unicode[STIX]{x1D6EC}_{+}$ .

2. (2) Let $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}_{+}$ . Let

   
   be an exact sequence of $p$ -divisible groups over $K$ that splits over $\overline{K}$ . Then, $Y\rightarrow X$ induces a monomorphism $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y)\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ , and we have a canonical isomorphism
   $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Z)\simeq \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)/\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y).\end{eqnarray}$$

Proof. (1) Consider the natural isogenies $X\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(X))$ and $X\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X))$ . (The former is the composition of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}^{\prime }}$ on $X$ and $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ on $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(X)$ , and the latter is obtained by exchanging the roles of $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D707}^{\prime }$ .) We claim that those kernels are the same. It suffices to see this over $\overline{K}$ . Let $M=\mathbb{D}(X_{\overline{K}})$ , and set $U_{\unicode[STIX]{x1D707}}=p^{-r}V^{s}$ for $v_{\unicode[STIX]{x1D707}}=(s,r)$ . The claim over $\overline{K}$ follows from the equality

$$\begin{eqnarray}\mathbb{D}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(X_{\overline{K}})))=M+U_{\unicode[STIX]{x1D707}^{\prime }}M+U_{\unicode[STIX]{x1D707}}M+U_{\unicode[STIX]{x1D707}}U_{\unicode[STIX]{x1D707}^{\prime }}M=\mathbb{D}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X_{\overline{K}}))).\end{eqnarray}$$

(2) Since the kernel of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}:Y\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y)$ is contained in the kernel of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}:X\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ , we have a homomorphism $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y)\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)$ . It suffices to show that this is a monomorphism over $\overline{K}$ . We may assume $X_{\overline{K}}=Y_{\overline{K}}\times Z_{\overline{K}}$ . Then,

(19) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X_{\overline{K}})=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y_{\overline{K}})\times \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Z_{\overline{K}}).\end{eqnarray}$$

Hence, obviously, $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y_{\overline{K}})\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X_{\overline{K}})$ is a monomorphism.

Note that $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ on $X$ and that on $Y$ induce an isogeny $\unicode[STIX]{x1D717}:Z\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)/\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y)$ . It is sufficient to show that the kernel of $\unicode[STIX]{x1D717}$ is the same as the kernel of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}:Z\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Z)$ . This follows from the fact that over $\overline{K}$ there is a canonical isomorphism $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Z_{\overline{K}})\simeq \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X_{\overline{K}})/\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Y_{\overline{K}})$ , which is obtained from (19).◻

From now on, for $\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime }\in \unicode[STIX]{x1D6EC}_{+}$ , we write $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(X)$ for $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}^{\prime }}(X))$ and $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{2}(X)$ for $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X))$ , and so on.

In [19, Lemma 9] and the argument following it, Zink explicitly constructed an isogeny from a given $p$ -divisible group $X$ over $K$ to a $p$ -divisible group that is slope divisible with respect to the smallest slope of $X$ . In the next lemma, we generalize this a little for later use.

Lemma 3.20. Let $X$ be a $p$ -divisible group over $K$ of height $h$ . Let $\unicode[STIX]{x1D707}$ be an element of $\unicode[STIX]{x1D6EC}_{+}$ whose slope is less than or equal to the smallest slope of $X$ . Then, $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(X)$ is slope divisible with respect to $\unicode[STIX]{x1D707}$ . In particular, if $X$ is isoclinic of slope $\unicode[STIX]{x1D706}$ , then $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}}^{h-1}(X)$ is minimal for $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6EC}_{+}$ with $\langle v_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D706}}\rangle =1$ .

Proof. If suffices to show this over an algebraically closed field. Let $M$ be the Dieudonné module of $X$ . Write $v_{\unicode[STIX]{x1D707}}=(s,r)$ , and set $U_{\unicode[STIX]{x1D707}}=p^{-r}V^{s}$ . The Dieudonné module $\mathbb{D}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(X))$ of $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(X)$ is isomorphic to

$$\begin{eqnarray}M+U_{\unicode[STIX]{x1D707}}M+\cdots +U_{\unicode[STIX]{x1D707}}^{h-1}M.\end{eqnarray}$$

We have $U_{\unicode[STIX]{x1D707}}\mathbb{D}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(X))\subset \mathbb{D}(\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(X))$ , since the proof of [19, Lemma 9] works without change. Thus, we obtain the first assertion. The second one follows from Lemma 3.18 and Proposition 3.12.◻

The following bi-divisible variant of Lemma 3.20 plays an important role in the proof of our main results.

Lemma 3.21. Let $X,Y,Z$ and $\unicode[STIX]{x1D707}$ be as in Proposition 3.17. Let $h$ be the height of $X$ . Then, $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(Y)$ in $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(X)$ is slope bi-divisible with respect to $\unicode[STIX]{x1D707}$ .

Proof. It is sufficient to show this over an algebraically closed field. We may assume $X=Y\times Z$ . Applying Lemma 3.20 to $Y$ , we have that $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(Y)$ is slope divisible with respect to $\unicode[STIX]{x1D707}$ . It remains to show that $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(Z)$ is slope divisible with respect to $\unicode[STIX]{x1D707}^{\ast }$ . Let $N$ be the Dieudonné module of $Z$ . The Dieudonné module of $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(Z)$ is $p^{-r}V^{s}N+N$ , which is isomorphic to $N+U_{\unicode[STIX]{x1D707}^{\ast }}N$ , where $U_{\unicode[STIX]{x1D707}^{\ast }}=(p^{-r}V^{s})^{-1}=p^{-(s-r)}F^{s}$ . Hence, the Dieudonné module $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{h-1}(Z)$ is isomorphic to

$$\begin{eqnarray}M+U_{\unicode[STIX]{x1D707}^{\ast }}M+\cdots +U_{\unicode[STIX]{x1D707}^{\ast }}^{h-1}M.\end{eqnarray}$$

One can show that this is slope divisible with respect to $\unicode[STIX]{x1D707}^{\ast }$ , in the same way as in Lemma 3.20, considering $\operatorname{Ver}$ -slope instead of slope ( $=\operatorname{Fr}$ -slope).◻

The next proposition is used in induction steps when we construct an isogeny from a given $p$ -divisible group over $K$ to a completely slope bi-divisible $p$ -divisible group.

Proposition 3.22. Let $X$ be a partially completely slope bi-divisible $p$ -divisible group over $K$ with respect to $\unicode[STIX]{x1D707}_{2},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ with filtration

$$\begin{eqnarray}0\subset X_{1}\subset X_{2}\subset \cdots \subset X_{\ell -1}\subset X_{\ell }=X.\end{eqnarray}$$

Let $\unicode[STIX]{x1D707}_{1}$ be an element of $\unicode[STIX]{x1D6EC}_{+}$ whose slope is greater than $\overline{\unicode[STIX]{x1D707}_{2}}$ and is less than or equal to the smallest slope of $X_{1}$ . Let $e$ be a nonnegative integer such that $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}X_{1}$ in $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}X$ is slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1}$ . (Lemma 3.21 says that $e=h-1$ satisfies this condition.) Then, $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}X$ is partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ .

Proof. Set $Y:=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}X$ and $Y_{i}:=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}X_{i}$ for $i=1,\ldots ,\ell$ . As obtained in [19, (11) on p. 89], there is an exact sequence of $p$ -divisible groups

(20)

where  and $Y_{1}^{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{1}}-\text{nul}}$ are characterized by the property that $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{1}}$ induces an isomorphism on  and is nilpotent on $Y_{1}^{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{1}}-\text{nul}}[p^{n}]$ for all $n$ . Put $Y_{0}:=Y_{1}^{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}_{1}}-\text{nul}}$ . We claim that $Y$ is partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ with filtration

$$\begin{eqnarray}0\subset Y_{0}\subset Y_{1}\subset \cdots \subset Y_{\ell }=Y.\end{eqnarray}$$

We need to check that this filtration $Y_{\bullet }$ satisfies the conditions (i), (ii), (iii) in Definition 3.7.

By the definition of $Y_{0}$ , all of the slopes of $Y_{0}$ are greater than $\overline{\unicode[STIX]{x1D707}_{1}}$ , whence $Y_{\bullet }$ satisfies (iii).

As $X_{j}$ ( $j\leqslant i$ ) are slope divisible with respect to $\unicode[STIX]{x1D707}_{i}$ for $2\leqslant i\leqslant \ell$ , so are $Y_{j}$ ( $j\leqslant i$ ), by Lemma 3.18. By the assumption, $Y_{1}$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{1}$ . As $X_{1}$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{i}$ ( $i\geqslant 2$ ), so is $Y_{1}=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}X_{1}$ , by Lemma 3.18. Since $(Y_{0})_{\overline{K}}$ is a direct summand of $(Y_{1})_{\overline{K}}$ , we have that $Y_{0}$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{i}$ ( $i\geqslant 1$ ). Hence, $Y_{\bullet }$ satisfies (i).

By Lemma 3.19(2), the $p$ -divisible group $Y/Y_{j}$ is isomorphic to $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}(X/X_{j})$ for $j\geqslant 1$ . By Lemma 3.18, $Y/Y_{j}\simeq \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{1}}^{e}(X/X_{j})$ ( $j\geqslant 1$ ) is slope divisible with respect to $\unicode[STIX]{x1D707}_{i}^{\ast }$ for $i\leqslant j+1$ . Over the algebraic closure $\overline{K}$ of $K$ , we have $(Y/Y_{0})_{\overline{K}}\simeq (Y_{1}/Y_{0})_{\overline{K}}\oplus (Y/Y_{1})_{\overline{K}}$ . Since $(Y_{1}/Y_{0})_{\overline{K}}$ and $(Y/Y_{1})_{\overline{K}}$ are both slope divisible with respect to $\unicode[STIX]{x1D707}_{1}^{\ast }$ , we have that $(Y/Y_{0})_{\overline{K}}$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{1}^{\ast }$ and therefore so is $Y/Y_{0}$ . Thus, $Y_{\bullet }$ satisfies (ii).◻

Now we get the main result over a field of characteristic $p$ .

Corollary 3.23. Let $X$ be a $p$ -divisible group over $K$ of height $h$ . Let $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\cdots >\unicode[STIX]{x1D706}_{\ell }$ be the set of positive slopes of $X$ . Then, we have the following.

1. (1) $(\prod _{i=1}^{\ell }\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}_{i}}^{h-1})(X)$ is completely slope bi-divisible with respect to $\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{\ell },0$ .

2. (2) We choose $\unicode[STIX]{x1D707}_{i}\in \unicode[STIX]{x1D6EC}_{+}$ such that $\langle v_{\unicode[STIX]{x1D707}_{i}},v_{\unicode[STIX]{x1D706}_{i}}\rangle =1$ . Then, $(\prod _{i=1}^{\ell }\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{i}}^{h-1}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}_{i}}^{h-1})(X)$ is minimal.

Here, recall that $\unicode[STIX]{x1D706}_{i}$ is regarded as the element $(\unicode[STIX]{x1D706}_{i},1)$ of $\unicode[STIX]{x1D6EC}_{1}$ for each $i=1,\ldots ,\ell$ .

Proof. It suffices to show these over the algebraic closure of $K$ . Therefore, we assume that $K$ is algebraically closed. Then, as it suffices to show them for the formal part of $X$ , we may assume that $X$ is a formal $p$ -divisible group (i.e., every slope of $X$ is positive).

(1) By Proposition 3.22, inductively one can check that $(\prod _{i=j}^{\ell }\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}_{i}}^{h-1})(X)$ is partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D706}_{j},\ldots ,\unicode[STIX]{x1D706}_{\ell }$ .

(2) Set $Y:=(\prod _{i=1}^{\ell }\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D706}_{i}}^{h-1})(X)$ . By (1), $Y$ is completely slope bi-divisible with respect to $\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{\ell }$ . Set $Z:=(\prod _{i=1}^{\ell }\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{i}}^{h-1})(Y)$ , which is also completely slope bi-divisible with respect to $\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{\ell }$ . Since $\operatorname{Gr}_{j}(Y)$ is isoclinic and slope divisible with respect to $\unicode[STIX]{x1D706}_{j}$ , so is $\operatorname{Gr}_{j}(Z)=(\prod _{i=1}^{\ell }\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{i}}^{h-1})\operatorname{Gr}_{j}(Y)$ , by Lemma 3.18. Moreover, $\operatorname{Gr}_{j}(Z)=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{j}}^{h-1}(\prod _{i\neq j}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}_{i}}^{h-1}\operatorname{Gr}_{j}(Y))$ is slope divisible with respect to $\unicode[STIX]{x1D707}_{j}$ , by Lemma 3.20. Hence, $\operatorname{Gr}_{j}(Z)$ is minimal, by Proposition 3.12, and therefore so is $Z$ .◻

4 Proof

We start by proving our main result (Proposition 4.1) over a discrete valuation ring. Based on this result, we show the main theorem (Theorem 4.2).

The result over a discrete valuation ring is stated in terms of Raynaud’s flat extension. Let $R$ be a discrete valuation ring of characteristic $p$ . Let $K$ be the quotient ring of $R$ . Let ${\mathcal{X}}$ be a $p$ -divisible group over $R$ . Write $X={\mathcal{X}}_{K}$ . Let $G$ be a finite subgroup scheme of $X$ . This defines an isogeny $\unicode[STIX]{x1D70C}:X\rightarrow Y$ of $p$ -divisible groups with $G=\ker (\unicode[STIX]{x1D70C})$ . Let $N$ be a sufficiently large integer such that $G\subset X[p^{N}]$ . Let ${\mathcal{G}}$ be the schematic closure in ${\mathcal{X}}[p^{N}]$ of $G$ . Note that ${\mathcal{G}}$ is a flat subgroup scheme of ${\mathcal{X}}[p^{N}]$ (see [16, pp. 259–260]). By taking the quotient by ${\mathcal{G}}$ , we have an isogeny $\tilde{\unicode[STIX]{x1D70C}}:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ . This construction of the isogeny $\tilde{\unicode[STIX]{x1D70C}}:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ from given data $({\mathcal{X}},\unicode[STIX]{x1D70C}:X\rightarrow Y)$ is called the flat extension.

An NP-quasi-saturated $p$ -divisible group over $(S,D)$ is defined by replacing saturated by quasi-saturated in the definition of an NP-saturated $p$ -divisible group over $(S,D)$ . An NP-quasi-saturated $p$ -divisible group over $R$ is that over $(S,D)$ with $S=\operatorname{Spec}(R)$ and $D=\operatorname{Spec}(k)$ , where $k$ is the residue field of $R$ .

Proposition 4.1. Let ${\mathcal{X}}$ be an NP-quasi-saturated $p$ -divisible group over $R$ . Set $X={\mathcal{X}}_{K}$ . Let $\unicode[STIX]{x1D709}$ (resp. $\unicode[STIX]{x1D701}$ ) be the Newton polygon of $X$ (resp. ${\mathcal{X}}_{k}$ ). Let $\{\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }\}$ be a subset of $\unicode[STIX]{x1D6EC}_{+}$ containing all slopes of $\unicode[STIX]{x1D701}$ such that $\langle v_{\unicode[STIX]{x1D707}_{i}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}_{i}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}_{i}}(\unicode[STIX]{x1D701})\rangle \leqslant 1$ . Suppose that $\overline{\unicode[STIX]{x1D707}_{1}}>\cdots >\overline{\unicode[STIX]{x1D707}_{\ell }}$ . Then, there exists an isogeny $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over $K$ whose flat extension ${\mathcal{X}}\rightarrow {\mathcal{Y}}$ satisfies that ${\mathcal{Y}}_{K}$ is minimal and ${\mathcal{Y}}_{k}$ is completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ . Moreover, the isogeny $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can be taken as a composition of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}_{i}}$ s for $1\leqslant i\leqslant \ell$ . (See (18) for the definition of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ .)

Proof. We first reduce to the case where $X$ is minimal. If the theorem is true for minimal $X^{\prime }$ , choose an isogeny $X\rightarrow X^{\prime }$ with $X^{\prime }$ minimal (Corollary 3.23(2)), and let $X^{\prime }\rightarrow Y$ be an isogeny obtained from the theorem for $X^{\prime }$ ; then, the composition $\unicode[STIX]{x1D70C}:X\rightarrow X^{\prime }\rightarrow Y$ satisfies the properties of the theorem.

Therefore, we assume that $X$ is minimal. It suffices to show that if ${\mathcal{X}}_{\overline{k}}$ is partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{i+1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ , then there exists an isogeny $X\rightarrow Y$ such that $Y$ is minimal and ${\mathcal{Y}}_{\overline{k}}$ is partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{i},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ .

Set $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{i}$ , and write $v_{\unicode[STIX]{x1D707}}=(s,r)$ . Let $\text{}\underline{{\mathcal{G}}}$ be the fppf sheaf obtained as the sheafification of the functor sending an $R$ -algebra $A$ to

$$\begin{eqnarray}\operatorname{Im}(\operatorname{Ver}^{s}:{\mathcal{X}}^{(p^{s})}[p^{s-r}](A)\rightarrow {\mathcal{X}}[p^{s-r}](A)).\end{eqnarray}$$

For an $R$ -algebra $S$ , let $\text{}\underline{{\mathcal{G}}}_{S}$ be the functor obtained by restricting $\text{}\underline{{\mathcal{G}}}$ to $S$ -algebras. Note that $\text{}\underline{{\mathcal{G}}}_{k}$ (resp. $\text{}\underline{{\mathcal{G}}}_{K}$ ) is represented by a finite group scheme ${\mathcal{G}}_{k}$ (resp. ${\mathcal{G}}_{K}$ ). We see from Lemma 3.14 that ${\mathcal{G}}_{k}$ (resp. ${\mathcal{G}}_{K}$ ) is the kernel of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}:{\mathcal{X}}_{k}\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}}_{k})$ (resp. $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}:{\mathcal{X}}_{K}\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}}_{K})$ ). Set

$$\begin{eqnarray}{\mathcal{H}}:=\operatorname{Ker}(\operatorname{Ver}^{s}:{\mathcal{X}}^{(p^{s})}[p^{s-r}]\rightarrow {\mathcal{X}}[p^{s-r}]).\end{eqnarray}$$

By the upper-semicontinuity for the structure sheaf of ${\mathcal{H}}$ , we have

(21) $$\begin{eqnarray}\operatorname{rk}{\mathcal{G}}_{k}\leqslant \operatorname{rk}{\mathcal{G}}_{K}.\end{eqnarray}$$

We claim that $\operatorname{rk}{\mathcal{G}}_{k}=\operatorname{rk}{\mathcal{G}}_{K}$ if $({\mathcal{X}}_{k})_{i}$ in ${\mathcal{X}}_{k}$ is not slope bi-divisible with respect to $\unicode[STIX]{x1D707}$ . By Proposition 3.17 for ${\mathcal{X}}_{K}$ , we have

(22) $$\begin{eqnarray}\log _{p}\operatorname{rk}{\mathcal{G}}_{K}=\langle v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})\rangle .\end{eqnarray}$$

Also, by Proposition 3.17 again, we get

(23) $$\begin{eqnarray}\log _{p}\operatorname{rk}{\mathcal{G}}_{k}\geqslant \langle v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D701})\rangle ,\end{eqnarray}$$

where the equality holds if and only if $({\mathcal{X}}_{k})_{i}$ in ${\mathcal{X}}_{k}$ is slope bi-divisible with respect to $\unicode[STIX]{x1D707}$ . By our assumption, the difference of the right-hand sides of (22) and (23) is at most one:

(24) $$\begin{eqnarray}\langle v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D701})\rangle \leqslant 1.\end{eqnarray}$$

Clearly, (21)–(24) imply the claim.

If $\operatorname{rk}{\mathcal{G}}_{k}=\operatorname{rk}{\mathcal{G}}_{K}$ , then ${\mathcal{H}}$ is flat over $R$ , whence $\text{}\underline{{\mathcal{G}}}$ is represented by a finite flat group scheme ${\mathcal{G}}$ that is isomorphic to the quotient ${\mathcal{X}}^{(p^{s})}[p^{s-r}]/{\mathcal{H}}$ (cf. [1, Example V]). Putting $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}})={\mathcal{X}}/{\mathcal{G}}$ , we have the canonical isogeny ${\mathcal{X}}\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}})$ . Note that $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}})_{k}=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}}_{k})$ and $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}})_{K}=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}}_{K})$ .

This argument can be applied to $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}})$ if $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}}_{k})_{i}$ in $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}}_{k})$ is not slope bi-divisible with respect to $\unicode[STIX]{x1D707}$ . Repeating this argument, we have the sequence of isogenies

$$\begin{eqnarray}{\mathcal{X}}\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}({\mathcal{X}})\rightarrow \cdots \rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{e}({\mathcal{X}}),\end{eqnarray}$$

where $e$ is the smallest nonnegative integer such that $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{e}({\mathcal{X}}_{k})_{i}$ in $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{e}({\mathcal{X}}_{k})$ is slope bi-divisible with respect to $\unicode[STIX]{x1D707}$ . Here, we use Lemma 3.21 for the existence of $e$ . This sequence is obtained by the flat extension of

$$\begin{eqnarray}X\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}(X)\rightarrow \cdots \rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{e}(X),\end{eqnarray}$$

where all $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{i}(X)$ are minimal. Let $X\rightarrow Y$ be the isogeny $X\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{e}(X)$ . Then, its flat extension ${\mathcal{X}}\rightarrow {\mathcal{Y}}$ coincides with ${\mathcal{X}}\rightarrow \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D707}}^{e}({\mathcal{X}})$ . It follows from Proposition 3.22 that ${\mathcal{Y}}_{\overline{k}}$ is partially completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{i},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ .◻

We generalize Proposition 4.1 to the case of general $(S,D)$ , using the same technique as in [13].

Theorem 4.2. Let $S$ be an integral Noetherian scheme with prime Weil divisor $D$ . Assume that $S$ is regular at the generic point of $D$ . Let ${\mathcal{X}}$ be an NP-quasi-saturated $p$ -divisible group over $(S,D)$ . Let $\unicode[STIX]{x1D709}$ (resp. $\unicode[STIX]{x1D701}$ ) be the Newton polygon of ${\mathcal{X}}_{S\setminus D}$ (resp. ${\mathcal{X}}_{D}$ ). Let $\{\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }\}$ be a subset of $\unicode[STIX]{x1D6EC}_{+}$ containing all slopes of $\unicode[STIX]{x1D701}$ such that $\langle v_{\unicode[STIX]{x1D707}_{i}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}_{i}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}_{i}}(\unicode[STIX]{x1D701})\rangle \leqslant 1$ . Suppose that $\overline{\unicode[STIX]{x1D707}_{1}}>\cdots >\overline{\unicode[STIX]{x1D707}_{\ell }}$ . Then, there is a finite birational morphism $\unicode[STIX]{x1D70B}:T\rightarrow S$ such that ${\mathcal{X}}_{T}$ is isogenous to a $p$ -divisible group ${\mathcal{Y}}$ over $T$ such that all of the geometric fibers over $T\setminus \unicode[STIX]{x1D70B}^{-1}(D)$ are minimal and ${\mathcal{Y}}_{\unicode[STIX]{x1D70B}^{-1}D}$ is completely slope bi-divisible with respect to $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ .

Proof. Let $\unicode[STIX]{x1D702}$ be the generic point of $D$ . Let $R={\mathcal{O}}_{S,\unicode[STIX]{x1D702}}$ , and let $K=\operatorname{frac}(R)$ . Set $X={\mathcal{X}}_{K}$ . Let $\unicode[STIX]{x1D70C}:X\rightarrow Y$ be the isogeny over $K$ constructed in Proposition 4.1.

Let $G$ be the kernel of $\unicode[STIX]{x1D70C}$ , and let $\overline{G}$ be the scheme-theoretic image of $G\rightarrow {\mathcal{X}}[p^{N}]$ for sufficient large $N$ . Let $V$ be the largest open subvariety such that $\overline{G}$ is flat over $V$ . Note that $V$ contains the generic point $\unicode[STIX]{x1D702}$ of $D$ . We have the $p$ -divisible group ${\mathcal{Y}}^{\prime }:={\mathcal{X}}_{V}/\overline{G}_{V}$ over $V$ with isogeny

Let $d$ be the degree of $\unicode[STIX]{x1D70C}$ . We make use of the moduli space ${\mathcal{M}}$ of isogenies from ${\mathcal{X}}$ of degree $d$ . This is defined to be the scheme over $S$ representing the following functor $\text{}\underline{{\mathcal{M}}}$ from the category of $S$ -schemes to that of sets. For an $S$ -scheme $T$ , an element of $\text{}\underline{{\mathcal{M}}}(T)$ is the isomorphism class of an isogeny ${\mathcal{X}}_{T}\rightarrow Z$ of degree $d$ over $T$ , where $Z$ is a $p$ -divisible group over $T$ . It is known that $\text{}\underline{{\mathcal{M}}}$ is represented by a projective scheme ${\mathcal{M}}$ over $S$ (see [13, 2.3]).

Now, $\unicode[STIX]{x1D70C}^{\prime }$ defines a morphism $V\rightarrow {\mathcal{M}}$ commuting the diagram

Let $\tilde{S}$ be the scheme-theoretic image of $V$ in ${\mathcal{M}}$ . Then, we have a morphism $f:\tilde{S}\rightarrow S$ , which is proper, surjective and birational. The inclusion $\tilde{S}\subset {\mathcal{M}}$ defines an isogeny ${\mathcal{X}}_{\tilde{S}}\rightarrow {\mathcal{Y}}^{\prime \prime }$ over $\tilde{S}$ . Since ${\mathcal{Y}}^{\prime \prime }$ is minimal over the generic point of $\tilde{S}\setminus f^{-1}(D)$ , by Corollary 3.13, ${\mathcal{Y}}^{\prime \prime }$ is minimal over $\tilde{S}\setminus f^{-1}(D)$ . Moreover, ${\mathcal{Y}}_{f^{-1}(D)}^{\prime \prime }$ is completely slope bi-divisible over every generic point, and therefore ${\mathcal{Y}}_{f^{-1}(D)}^{\prime \prime }$ is completely slope bi-divisible, by Lemma 3.11.

Let

be the Stein factorization with $f_{\ast }{\mathcal{O}}_{\tilde{S}}={\mathcal{O}}_{T}$ . Let $x\in T$ , and let $\tilde{S}_{x}$ be the fiber over $x$ of $\tilde{S}\rightarrow T$ . By [13, Lemma 2.5], the image of $\tilde{S}_{\overline{x}}\rightarrow {\mathcal{M}}$ is finite. Since $\tilde{S}_{\overline{x}}$ is connected, the image is a single point of ${\mathcal{M}}$ . From [13, Lemma 2.6], we have a morphism $T\rightarrow {\mathcal{M}}$ . This defines a desired isogeny

over $T$ .◻

The next corollary is the result in the NP-saturated case, from which Corollary 1.1 follows immediately.

Corollary 4.3. Let $S,D$ be as in Theorem 4.2. Let ${\mathcal{X}}$ be an NP-saturated $p$ -divisible group over $(S,D)$ . Then, there is a finite birational morphism $T\rightarrow S$ such that ${\mathcal{X}}_{T}$ is isogenous to a $p$ -divisible group ${\mathcal{Y}}$ over $T$ whose geometric fibers are all minimal.

Proof. Let $\unicode[STIX]{x1D709}$ (resp. $\unicode[STIX]{x1D701}$ ) be the Newton polygon of ${\mathcal{X}}_{S\setminus D}$ (resp. ${\mathcal{X}}_{D}$ ). As in (5), we write

(25) $$\begin{eqnarray}\unicode[STIX]{x1D701}=\unicode[STIX]{x1D71A}+_{\operatorname{NP}}\unicode[STIX]{x1D701}^{\prime }\quad \text{and}\quad \unicode[STIX]{x1D709}=\unicode[STIX]{x1D71A}+_{\operatorname{ NP}}\unicode[STIX]{x1D709}^{\prime },\end{eqnarray}$$

so that $\unicode[STIX]{x1D701}^{\prime }\prec \unicode[STIX]{x1D709}^{\prime }$ is saturated and $\unicode[STIX]{x1D709}^{\prime }$ consists of only two segments. Let $a$ (resp. $b$ ) be the smallest (resp. largest) slope of $\unicode[STIX]{x1D709}^{\prime }$ .

In order to apply Theorem 4.2 to ${\mathcal{X}}$ , we need to choose $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ as in Theorem 4.2. We define them as the union of three kinds of subsets of $\unicode[STIX]{x1D6EC}_{1}$ , which are labeled as $A,B$ and $C$ . (For the definition of $\unicode[STIX]{x1D6EC}_{1}$ , see the sentence following (8) in §3. Recall that $\unicode[STIX]{x1D6EC}_{1}$ is canonically identified with the set of slopes $\unicode[STIX]{x1D706}\in \mathbb{Q}$ with $0\leqslant \unicode[STIX]{x1D706}\leqslant 1$ .) First, $A$ is the set of slopes of $\unicode[STIX]{x1D701}$ . Let $B$ be the set of $\unicode[STIX]{x1D708}\in \unicode[STIX]{x1D6EC}_{1}$ such that $v_{\unicode[STIX]{x1D708}}$ or $-v_{\unicode[STIX]{x1D708}}$ is equal to $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D701})$ for some slope $\unicode[STIX]{x1D706}$ of $\unicode[STIX]{x1D701}^{\prime }$ . For each positive slope $\unicode[STIX]{x1D706}$ of $\unicode[STIX]{x1D70C}$ , we choose a $\unicode[STIX]{x1D708}\in \unicode[STIX]{x1D6EC}_{1}$ satisfying the following two properties: (i) $\langle v_{\unicode[STIX]{x1D708}},v_{\unicode[STIX]{x1D706}}\rangle =1$ and (ii) $\overline{\unicode[STIX]{x1D708}}$ is sufficiently close to $\unicode[STIX]{x1D706}$ so that $\overline{\unicode[STIX]{x1D708}}$ is distinct from the slope of any element of $A\cup B$ . Let $C$ be the set of such $\unicode[STIX]{x1D708}$ s. Let $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ be the union of $A,B$ and $C$ , and arrange them so that $\overline{\unicode[STIX]{x1D707}_{1}}>\cdots >\overline{\unicode[STIX]{x1D707}_{\ell }}$ . Theorem 4.2 is applicable for these $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell }$ . Indeed,

(26) $$\begin{eqnarray}\langle v_{\unicode[STIX]{x1D707}_{i}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}_{i}}(\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D707}_{i}}(\unicode[STIX]{x1D701})\rangle \leqslant 1\end{eqnarray}$$

holds for $i=1,\ldots ,\ell$ . For $\unicode[STIX]{x1D707}_{i}\in A$ , this follows from the fact that $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is quasi-saturated (Lemma 2.2). For $\unicode[STIX]{x1D707}_{i}\in B$ , the slope of $\unicode[STIX]{x1D707}_{i}$ is outside $[a,b]$ ; hence, the left-hand side of (26) is equal to zero. Moreover, for $\unicode[STIX]{x1D707}_{i}\in C$ , the inequality (26) holds.

Let ${\mathcal{Y}}$ be the $p$ -divisible group obtained by Theorem 4.2. Let $s$ be any geometric point of $\unicode[STIX]{x1D70B}^{-1}(D)$ . Let $\unicode[STIX]{x1D706}$ be any slope of $\unicode[STIX]{x1D701}$ . Let $Z_{\unicode[STIX]{x1D706}}$ be the nonzero $\operatorname{Gr}_{i}({\mathcal{Y}}_{s})$ of slope $\unicode[STIX]{x1D706}$ . Since ${\mathcal{Y}}_{s}$ is completely slope divisible, $Z_{\unicode[STIX]{x1D706}}$ is slope divisible with respect to $\unicode[STIX]{x1D706}$ . If $\unicode[STIX]{x1D706}$ is zero, then $Z_{\unicode[STIX]{x1D706}}$ is étale and therefore $Z_{\unicode[STIX]{x1D706}}\simeq H_{1,0}$ , whence this is minimal. If $\unicode[STIX]{x1D706}>0$ , then there exists $\unicode[STIX]{x1D708}\in \{\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{\ell },\unicode[STIX]{x1D707}_{1}^{\ast },\ldots ,\unicode[STIX]{x1D707}_{\ell }^{\ast }\}$ such that $\langle v_{\unicode[STIX]{x1D708}},v_{\unicode[STIX]{x1D706}}\rangle =1$ , and $Z_{\unicode[STIX]{x1D706}}$ is slope divisible with respect to $\unicode[STIX]{x1D708}$ . It follows from Proposition 3.12 that $Z_{\unicode[STIX]{x1D706}}$ is minimal. Thus, every $\operatorname{Gr}_{i}({\mathcal{Y}}_{s})$ is minimal, and therefore so is ${\mathcal{Y}}_{s}$ .◻

Example 4.4. For the case of Example 2.3, we illustrate the subsets $A$ , $B$ and $C$ of $\unicode[STIX]{x1D6EC}_{1}$ that appear in the proof of Corollary 4.3. The saturated pair of Newton polygons is $\unicode[STIX]{x1D709}=(0,1)+_{\operatorname{NP}}(1,3)+_{\operatorname{NP}}(3,1)+_{\operatorname{NP}}(1,0)$ and $\unicode[STIX]{x1D701}=(0,1)+_{\operatorname{NP}}(1,2)+_{\operatorname{NP}}(1,1)+_{\operatorname{NP}}(2,1)+_{\operatorname{NP}}(1,0)$ . We use the identification of $\unicode[STIX]{x1D6EC}_{1}$ with the set of slopes. First, as $A$ is the set of slopes of $\unicode[STIX]{x1D701}$ , we have $A=\{1,2/3,1/2,1/3,0\}$ . In Figure 3, the dotted arrows correspond to the elements of $B$ . Therefore, $B=\{1,3/4,0\}$ .

Figure 3. The picture of $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ .

Finally, $n/(n+1)$ for any sufficiently large $n$ can be an element of $C$ . If we choose $4/5$ , then $C=\{4/5\}$ . Thus, the union of $A$ , $B$ and $C$ is

$$\begin{eqnarray}\{1,4/5,3/4,2/3,1/2,1/3,0\}.\end{eqnarray}$$

5 Application: the configuration of minimal $p$ -kernel types

Recall from [2, Corollary 3.2] that the central streams [11, 3.10] in the moduli space of principally polarized abelian varieties are configurated as given by the partial ordering on symmetric Newton polygons. As an application of Corollary 1.1, we show its unpolarized analog (Corollary 5.1) with a geometrical proof, whereas a combinatorial method is used in [2].

Let $h$ be a natural number. Let $c,d$ be nonnegative integers, with $c+d=h$ . Let $W$ be the Weyl group of $\operatorname{GL}_{h}$ . Let $\unicode[STIX]{x1D6E5}=\{\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{h-1}\}$ be the set of simple roots, as usual. Let $s_{i}$ be the simple reflection associated to $\unicode[STIX]{x1D6FC}_{i}$ . Set $I=\unicode[STIX]{x1D6E5}\setminus \{\unicode[STIX]{x1D6FC}_{c}\}$ . Let $W_{I}$ be the subgroup of $W$ generated by $s_{i}$ with $\unicode[STIX]{x1D6FC}_{i}\in I$ . Let $^{I}W$ be the set of the minimal-length representatives of $W_{I}\backslash W$ .

Let $k$ be an algebraically closed field. Recall the classification theory of truncated Barsotti–Tate groups of level one ( $\operatorname{BT}_{1}$ s) over $k$ found by Kraft [8], rediscovered by Oort and reproved and formulated as follows by Moonen and Wedhorn [9]. It says that there exists a canonical bijection from $^{I}W$ to the set of isomorphism classes of $\operatorname{BT}_{1}$ s over $k$ of codimension $c$ and of dimension  $d$ .

We use $F$ -zips, which in this paper mean those with support contained in $\{0,1\}$ in the terminology of [9]. Let $S$ be a scheme in characteristic $p>0$ . An $F$ -zip over $S$ is a quintuple $(N,C,D,\unicode[STIX]{x1D711},\dot{\unicode[STIX]{x1D711}})$ consisting of a locally free ${\mathcal{O}}_{S}$ -module $N$ and ${\mathcal{O}}_{S}$ -submodules $C,D$ of $N$ which are locally direct summands of $N$ with ${\mathcal{O}}_{S}$ -linear isomorphisms $\unicode[STIX]{x1D711}:(N/C)^{(p)}\rightarrow D$ and $\dot{\unicode[STIX]{x1D711}}:C^{(p)}\rightarrow N/D$ . Let $G$ be a $\operatorname{BT}_{1}$ over $k$ . To $G$ , we associate an $F$ -zip $(\mathbb{D}(G),V\mathbb{D}(G),F\mathbb{D}(G),F,V^{-1})$ . This gives a canonical bijection from the set of $\operatorname{BT}_{1}$ s over $k$ and the set of $F$ -zips over $k$ .

Let $w_{\unicode[STIX]{x1D709}}\in ^{I}W$ denote the $p$ -kernel type of the minimal $p$ -divisible group $H(\unicode[STIX]{x1D709})_{k}$ of the Newton polygon $\unicode[STIX]{x1D709}$ . For $v,w\in ^{I}W$ , we say that $v\subset w$ if there exists an $F$ -zip over a discrete valuation ring of which the generic fiber (resp. the special fiber) is of type $w$ (resp. of type $v$ ). It follows from [14, Theorem 12.17] that $\subset$ is a partial ordering on $^{I}W$ , and this coincides with the partial ordering introduced and investigated by He [5].

Corollary 5.1. $w_{\unicode[STIX]{x1D701}}\subset w_{\unicode[STIX]{x1D709}}$ if and only if $\unicode[STIX]{x1D701}\preccurlyeq \unicode[STIX]{x1D709}$ .

Proof. For the if part, since $\subset$ is a partial ordering, it is sufficient to show the case that $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ is saturated. Applying Corollary 1.1 to a family with saturated $\unicode[STIX]{x1D701}\prec \unicode[STIX]{x1D709}$ and with $a$ -number ${\leqslant}1$ , constructed in [10, (3.2)], we have $w_{\unicode[STIX]{x1D701}}\subset w_{\unicode[STIX]{x1D709}}$ .

Suppose that $w_{\unicode[STIX]{x1D701}}\subset w_{\unicode[STIX]{x1D709}}$ . There exists an $F$ -zip ${\mathcal{N}}$ over a discrete valuation ring $R$ with algebraically closed residue field whose special fiber is of type $w_{\unicode[STIX]{x1D701}}$ and whose generic fiber is of type $w_{\unicode[STIX]{x1D709}}$ . Then, there exists a display ${\mathcal{M}}$ over $R$ such that ${\mathcal{M}}/I_{R}{\mathcal{M}}$ is isomorphic to ${\mathcal{N}}$ (see [4, Lemma 4.1]). By [12], the special fiber (resp. the generic fiber) of ${\mathcal{M}}$ is minimal of Newton polygon $\unicode[STIX]{x1D701}$ (resp. $\unicode[STIX]{x1D709}$ ). By Grothendieck–Katz [7, Theorem 2.3.1 on p. 143], we have $\unicode[STIX]{x1D701}\preccurlyeq \unicode[STIX]{x1D709}$ .◻

Combining this with [4, Theorem 1.1], one can get the unpolarized analog of Oort’s conjecture [11, 6.9]. The original conjecture was proved in [3, 17] (see also [18] for a generalization to some Shimura varieties).

Corollary 5.2. If there exists a $p$ -divisible group with Newton polygon $\unicode[STIX]{x1D709}$ and $p$ -kernel type $w$ , then we have $w_{\unicode[STIX]{x1D709}}\subset w$ .

Proof. Let $\unicode[STIX]{x1D709}(w)$ be the supremum of Newton polygons of $p$ -divisible groups with $p$ -kernel type $w$ . We have $\unicode[STIX]{x1D709}\preccurlyeq \unicode[STIX]{x1D709}(w)$ . From Corollary 5.1, it follows that $w_{\unicode[STIX]{x1D709}}\subset w_{\unicode[STIX]{x1D709}(w)}$ . Recall [4, Theorem 1.1], which says that $\unicode[STIX]{x1D709}(w)$ is the maximal one among Newton polygons $\unicode[STIX]{x1D702}$ with $w_{\unicode[STIX]{x1D702}}\subset w$ . In particular, we have $w_{\unicode[STIX]{x1D709}(w)}\subset w$ .◻