2.1 Group schemes of constant type

Here we collect some basic results about group schemes to be used later.

$p$-Lie algebras and group schemes. Here we recall the infinitesimal Lie theory of [Reference Demazure and GrothendieckSGA3, VII$_A$]. For simplicity, we only restrict to the commutative case here. Let $S$ be a noetherian scheme over $\mathbb {F}_p$. Recall that a commutative $p$-Lie algebra over $S$ is a coherent sheaf $\mathfrak {g}$ on $S$, endowed with an additive morphism

\[ \mathfrak{g} \longrightarrow \mathfrak{g},\quad \delta\longmapsto \delta^{[p]} \]

which is $p$-linear in the sense that

\[ (a\delta)^{[p]}=a^{p}\delta^{[p]}, \quad a\in {\mathcal{O}}_S,\ \delta\in \mathfrak{g}. \]

The additive morphism is called the $p$-th power map on $\mathfrak {g}$. We say that $\mathfrak {g}$ is locally free if it is locally free as an ${\mathcal {O}}_S$-module.

We can interpret the $p$-th power map on $\mathfrak {g}$ as an ${\mathcal {O}}_S$-linear map as follows. Recall the absolute Frobenius morphism $F_S: S\to S$. The pull-back $F_S^{*}\mathfrak {g}$ is still a vector bundle on $S$. The additive map

\[ F_S^{*}: \mathfrak{g}\longrightarrow F_S^{*}\mathfrak{g} \]

is $p$-linear in that $F_S^{*}(a\delta )=a^{p}F_S^{*}\delta .$ It follows that we have a well-defined ${\mathcal {O}}_S$-linear map given by

\[ F_S^{*}\mathfrak{g} \longrightarrow \mathfrak{g}, \quad F_S^{*}\delta \longmapsto \delta^{[p]}. \]

For a commutative group scheme ${\mathcal {G}}$ over $S$, the ${\mathcal {O}}_S$-module ${\mathrm {Lie}}({\mathcal {G}}/S)$ of invariant derivations on ${\mathcal {G}}$ is a natural commutative $p$-Lie algebra over $S$. By [Reference Demazure and GrothendieckSGA3, VII$_A$, Theorems 7.2 and 7.4, Remark 7.5], the functor ${\mathcal {G}}\mapsto {\mathrm {Lie}}({\mathcal {G}}/S)$ is an equivalence between the following two categories:

1. (1) the category of finite and flat commutative group schemes of height one over $S$;

2. (2) the category of locally free commutative $p$-Lie algebras over $S$.

Here a group scheme ${\mathcal {G}}$ over $S$ is of height one if the relative Frobenius morphism $F_{{\mathcal {G}}/S}:{\mathcal {G}}\to {\mathcal {G}}^{(p)}$ is zero. Furthermore, if ${\mathcal {G}}$ is in the first category, then $\bar {\Omega }_{{\mathcal {G}}/S}=e^{*}\Omega _{{\mathcal {G}}/S}$ and ${\mathrm {Lie}}({\mathcal {G}}/S)$ are locally free and canonically dual to each other. Here $e:S\to {\mathcal {G}}$ is the identity section. See [Reference Demazure and GrothendieckSGA3, VII$_A$, Proposition 5.5.3].

For some treatments in special cases, see [Reference MumfordMum74, § 15] for the case that $S$ is the spectrum of an algebraically closed field, and [Reference Conrad, Gabber and PrasadCGP10, § A.7] for the case that $S$ is affine.

Group schemes of constant type. The following results, except Lemma 2.1(1), also hold in characteristic zero. We restrict to positive characteristics for simplicity.

Let $S$ be a scheme over a field $k$ of characteristic $p>0$. A group scheme (respectively, scheme, coherent sheaf, $p$-Lie algebra, $p$-divisible group) ${\mathcal {G}}$ over $S$ is called of constant type over $S$ if it is isomorphic to the base change (respectively, base change, pull-back, pull-back, base change) $G_S$ by $S\to \operatorname {Spec} k$ of some group scheme (respectively, scheme, coherent sheaf, $p$-Lie algebra, $p$-divisible group) $G$ over $k$. Note that a finite flat group scheme of height one over $S$ is of constant type if and only if its $p$-Lie algebra is of constant type.

It is also reasonable to use the term ‘constant’ instead of ‘of constant type’ in the above definition. However, a ‘constant group scheme’ usually means a group scheme associated to an abelian group in the literature, so we choose the current terminology to avoid confusion.

Proof. We first prove part (1). Denote $\mathfrak {g}={\mathrm {Lie}}({\mathcal {G}}/S)$ and $\mathfrak {g}_0=\Gamma (S, \mathfrak {g})$. By assumption, the canonical morphism

\[ \mathfrak{g}_0 \otimes_k {\mathcal{O}}_S \longrightarrow \mathfrak{g} \]

is an isomorphism of ${\mathcal {O}}_S$-modules. It suffices to prove that the $p$-th power map of $\mathfrak {g}$ comes from a $p$-th power map of $\mathfrak {g}_0$. Note that $\mathfrak {g}_0$ has a canonical $p$-th power map coming from global sections of $\mathfrak {g}$, but we do not need this fact.

Note that the $p$-th power map of $\mathfrak {g}$ is equivalent to an ${\mathcal {O}}_S$-linear map $F_S^{*}\mathfrak {g} \to \mathfrak {g}$. It is an element of

\begin{align*} {\mathrm{Hom}}_{{\mathcal{O}}_S}(F_S^{*}\mathfrak{g},\mathfrak{g}) &=\Gamma(S, F_S^{*}(\mathfrak{g}^{{\vee}})\otimes_{{\mathcal{O}}_S}\mathfrak{g}) \\ &=\Gamma(S, (F_k^{*}(\mathfrak{g}_0^{{\vee}})\otimes_k\mathfrak{g}_0)\otimes_k{{\mathcal{O}}_S}) =F_k^{*}(\mathfrak{g}_0^{{\vee}})\otimes_k\mathfrak{g}_0 ={\mathrm{Hom}}_{k}(F_k^{*}\mathfrak{g}_0,\mathfrak{g}_0). \end{align*}

In other words, it is the base change of a $p$-th power map of $\mathfrak {g}_0$. This proves part (1).

The proof of part (2) is similar. In fact, denote ${\mathcal {F}}=\pi _*{\mathcal {O}}_{\mathcal {G}}$ and ${\mathcal {F}}_0=\Gamma (S, \pi _*{\mathcal {O}}_{\mathcal {G}})$. The canonical morphism

\[ {\mathcal{F}}_0 \otimes_k {\mathcal{O}}_S \longrightarrow {\mathcal{F}} \]

is an isomorphism of ${\mathcal {O}}_S$-modules. Note that the structure of ${\mathcal {G}}$ as a group scheme over $S$ is equivalent to a structure of ${\mathcal {F}}$ as a Hopf ${\mathcal {O}}_S$-algebra. For these, the extra data on ${\mathcal {F}}$ consist of an identity map ${\mathcal {O}}_S\to {\mathcal {F}}$, a multiplication map ${\mathcal {F}}\otimes _{{\mathcal {O}}_S}{\mathcal {F}} \to {\mathcal {F}}$, a co-identity map ${\mathcal {F}}\to {\mathcal {O}}_S$, a co-multiplication map ${\mathcal {F}} \to {\mathcal {F}}\otimes _{{\mathcal {O}}_S}{\mathcal {F}}$, and an inverse map ${\mathcal {F}} \to {\mathcal {F}}$. There are many compatibility conditions on these maps. All these maps are ${\mathcal {O}}_S$-linear. We claim that all these maps are coming from similar maps on ${\mathcal {F}}_0$. For example, the co-multiplication map is an element of

\begin{align*} {\mathrm{Hom}}_{{\mathcal{O}}_S}({\mathcal{F}}, {\mathcal{F}}\otimes_{{\mathcal{O}}_S}{\mathcal{F}}) &=\Gamma(S, {\mathcal{F}}^{{\vee}}\otimes_{{\mathcal{O}}_S}{\mathcal{F}}\otimes_{{\mathcal{O}}_S}{\mathcal{F}}) \\ &=\Gamma(S, ({\mathcal{F}}_0^{{\vee}}\otimes_k{\mathcal{F}}_0\otimes_k{\mathcal{F}}_0)\otimes_k{{\mathcal{O}}_S})\\ &={\mathcal{F}}_0^{{\vee}}\otimes_k{\mathcal{F}}_0\otimes_k{\mathcal{F}}_0 ={\mathrm{Hom}}_{k}({\mathcal{F}}_0, {\mathcal{F}}_0\otimes_k{\mathcal{F}}_0). \end{align*}

This makes ${\mathcal {F}}_0$ a Hopf $k$-algebra, because the compatibility conditions hold by ${\mathcal {F}}_0=\Gamma (S,{\mathcal {F}})$. Finally, the Hopf algebra ${\mathcal {F}}$ is the base change of the Hopf algebra ${\mathcal {F}}_0$. Then the group scheme ${\mathcal {G}}$ is the base change of the group scheme corresponding to the Hopf algebra ${\mathcal {F}}_0$.

The proof of part (3) is similar by looking at ${\mathrm {Hom}}_{{\mathcal {O}}_S}((\pi _2)_*{\mathcal {O}}_{{\mathcal {G}}_2},(\pi _1)_*{\mathcal {O}}_{{\mathcal {G}}_1})$ with compatibility conditions.

The following result will be used for several times.

Lemma 2.2 Denote $S=\mathbb {P}_k^{1}$ for any field $k$ of characteristic $p>0$. Let

\[ 0 \longrightarrow {\mathcal{G}}_1\longrightarrow {\mathcal{G}}\longrightarrow {\mathcal{G}}_2\longrightarrow 0 \]

be an exact sequence of finite and flat commutative group schemes over $S$. If ${\mathcal {G}}_1$ and ${\mathcal {G}}_2$ are of constant type, then ${\mathcal {G}}$ is of constant type.

Proof. By Lemma 2.1(2), it suffices to prove that $\pi _*{\mathcal {O}}_{{\mathcal {G}}}$ is of constant type as a coherent ${\mathcal {O}}_S$-module. Here $\pi :{\mathcal {G}}\to S$ is the structure morphism. We can assume that $k$ is algebraically closed. In fact, the property that the canonical map

\[ \Gamma(S, \pi_*{\mathcal{O}}_{{\mathcal{G}}}) \otimes_k {\mathcal{O}}_S \longrightarrow \pi_*{\mathcal{O}}_{{\mathcal{G}}} \]

is an isomorphism can be descended from the algebraic closure of $k$ to $k$.

Once $k$ is algebraically closed, any finite group scheme over $k$ is a successive extension of group schemes in the following list:

\[ \mathbb{Z}/\ell\mathbb{Z}, \quad \mathbb{Z}/p\mathbb{Z},\quad \mu_p,\quad \alpha_p. \]

Here $\ell \neq p$ is any prime, and $\mathbb {Z}/\ell \mathbb {Z}$ is isomorphic to $\mu _\ell$.

For $i=1,2$, write ${\mathcal {G}}_i=G_i\times _k S$ for a finite group scheme $G_i$ over $k$. By induction, we can assume that $G_1$ is one of the four group schemes over $k$ in the list. View ${\mathcal {G}}$ as a $G_1$-torsor over ${\mathcal {G}}_2$. Then ${\mathcal {G}}$ corresponds to a cohomology class in the fppf cohomology group $H^{1}_\mathrm {fppf}({\mathcal {G}}_2,G_1)$. We first claim that the natural map

\[ H^{1}_\mathrm{fppf}(G_2, G_1) \longrightarrow H^{1}_\mathrm{fppf}({\mathcal{G}}_2,G_1) \]

is an isomorphism. If this holds, then ${\mathcal {G}}$ is a trivial torsor, and thus isomorphic to $G_1\times _k{\mathcal {G}}_2$ as a ${\mathcal {G}}_2$-scheme. In particular, it is a scheme of constant type over $S$.

It remains to prove the claim that the natural map

\[ H^{1}_\mathrm{fppf}(G_2, G_1) \longrightarrow H^{1}_\mathrm{fppf}({\mathcal{G}}_2,G_1) \]

is an isomorphism. Note the basic exact sequences

\begin{gather*} 0\longrightarrow \alpha_p \longrightarrow \mathbb{G}_a \stackrel{F}{\longrightarrow} \mathbb{G}_a \longrightarrow 0,\\ 0\longrightarrow \mathbb{Z}/p\mathbb{Z} \longrightarrow \mathbb{G}_a \stackrel{1-F}{\longrightarrow} \mathbb{G}_a \longrightarrow 0,\\ 0\longrightarrow \mu_\ell \longrightarrow \mathbb{G}_m \stackrel{[\ell]}{\longrightarrow} \mathbb{G}_m \longrightarrow 0. \end{gather*}

In the last one, $\ell =p$ is allowed. Then the claim is a consequence of explicit expressions on the relevant cohomology groups of $\mathbb {G}_a$ and $\mathbb {G}_m$ over ${\mathcal {G}}_2$ and $G_2$.

Now we compute the cohomology groups of $\mathbb {G}_a$ and $\mathbb {G}_m$ over ${\mathcal {G}}_2$ and $G_2$. Write $R=\Gamma (G_2,{\mathcal {O}}_{G_2})$. We first have

\[ H^{i}_\mathrm{fppf}({\mathcal{G}}_2, \mathbb{G}_a)=H^{i}_\mathrm{Zar}({\mathcal{G}}_2, {\mathcal{O}}_{{\mathcal{G}}_2}) =H^{i}_\mathrm{Zar}(S, R\times_k{\mathcal{O}}_S) =R\times_k H^{i}_\mathrm{Zar}(S, {\mathcal{O}}_S). \]

This gives

\[ H^{0}_\mathrm{fppf}({\mathcal{G}}_2, \mathbb{G}_a)=R, \quad H^{1}_\mathrm{fppf}({\mathcal{G}}_2, \mathbb{G}_a)=0,\quad H^{0}_\mathrm{fppf}({\mathcal{G}}_2, \mathbb{G}_m) =R^{{\times}}. \]

To compute $H^{1}_\mathrm {fppf}({\mathcal {G}}_2, \mathbb {G}_m)=H^{1}_\text {et}({\mathcal {G}}_2, \mathbb {G}_m)$, denote by $I=\ker (R\to R_{{\mathrm {red}}})$ the nilradical ideal of $R$, and by ${\mathcal {I}}=\ker ({\mathcal {O}}_{{\mathcal {G}}_2}\to {\mathcal {O}}_{({\mathcal {G}}_2)_{\mathrm {red}}})$ the nilradical ideal sheaf of ${\mathcal {G}}_2$. We have ${\mathcal {I}}= I\otimes _k {\mathcal {O}}_S$. There is an exact sequence of étale sheaves over ${\mathcal {G}}_2$ by

\[ 0\longrightarrow 1+{\mathcal{I}} \longrightarrow \mathbb{G}_{m,{\mathcal{G}}_2} \longrightarrow \mathbb{G}_{m,({\mathcal{G}}_2)_{\mathrm{red}}} \longrightarrow 0. \]

Moreover, $1+{\mathcal {I}}$ has a filtration

\[ 1+{\mathcal{I}} \supset 1+{\mathcal{I}}^{2} \supset 1+{\mathcal{I}}^{3} \supset \cdots, \]

whose $m$-th quotient admits an isomorphism given by

\[ (1+{\mathcal{I}}^{m})/(1+{\mathcal{I}}^{m+1}) \longrightarrow {\mathcal{I}}^{m}/{\mathcal{I}}^{m+1},\quad 1+t\longmapsto t. \]

Those quotients are coherent sheaves over ${\mathcal {G}}_2$. Then we have

\begin{align*} H^{i}_\text{et} ({\mathcal{G}}_2, {\mathcal{I}}^{m}/{\mathcal{I}}^{m+1}) &=H^{i}_\mathrm{Zar} ({\mathcal{G}}_2, {\mathcal{I}}^{m}/{\mathcal{I}}^{m+1})\\ &=H^{i}_\mathrm{Zar} ({\mathcal{G}}_2, (I^{m}/I^{m+1})\otimes_k{\mathcal{O}}_S) =(I^{m}/I^{m+1})\otimes_k H^{i}_\mathrm{Zar}(S, {\mathcal{O}}_S). \end{align*}

This vanishes for $i>0$. As a consequence, $H^{i}_\text {et} ({\mathcal {G}}_2, 1+{\mathcal {I}}) =0$ for $i>0$. Therefore,

\[ H^{1}({\mathcal{G}}_2, \mathbb{G}_m) =H^{1}({\mathcal{G}}_2, \mathbb{G}_{m,({\mathcal{G}}_2)_{\mathrm{red}}}) =\mathrm{Pic}(({\mathcal{G}}_2)_{\mathrm{red}}) \simeq \mathbb{Z}^{r}, \]

where $r$ is the number of connected components of ${\mathcal {G}}_2$. For the cohomology over $G_2$, similar computations give

\[ H^{0}_\mathrm{fppf}(G_2, \mathbb{G}_a)=R, \quad H^{1}_\mathrm{fppf}(G_2, \mathbb{G}_a)=0,\quad H^{0}_\mathrm{fppf}(G_2, \mathbb{G}_m)=R^{{\times}},\quad H^{1}_\mathrm{fppf}(G_2, \mathbb{G}_m)=0. \]

By these, it is easy to verify the claim.

2.2 The quotient process

The key to the proof of Theorem 1.1 is a quotient process. This quotient process is also introduced by Rössler [Reference RösslerRos20].

Roughly speaking, if the Hodge bundle $\bar {\Omega }_A$ of $A$ is not ample, then we take the maximal nef subbundle of ${\mathrm {Lie}}({\mathcal {A}}/S)=\bar {\Omega }_A^{\vee }$, ‘lift’ it to a local subgroup scheme of $A$, and take the quotient $A_1$ of $A$ by this subgroup scheme. If the Hodge bundle $\bar {\Omega }_{A_1}$ of $A_1$ is still not ample, then we perform the quotient process on $A_1$. Repeat this process. We obtain a sequence $A, A_1,A_2,\ldots$ of abelian varieties over $K$. The goal here is to introduce this quotient process. We start with some basic notions of vector bundles, Hodge bundles, and Lie algebras.

Vector bundles over a curve. Here we review some basic terminologies about stability and positivity of vector bundles over curves. We first introduce them for general curves, and then consider the case of the projective line. A basic reference is [Reference LazarsfeldLaz04, § 6.4].

Let $S$ be a projective and smooth curve over a field $k$. Let ${\mathcal {E}}$ be a vector bundle over $S$, i.e. a locally free sheaf of finite rank. The slope of ${\mathcal {E}}$ is defined as

\[ \mu({\mathcal{E}})=\frac{\deg({\mathcal{E}})}{{\mathrm{rank}}({\mathcal{E}})}=\frac{\deg(\det{\mathcal{E}})}{{\mathrm{rank}}({\mathcal{E}})}. \]

Let ${\mathcal {F}}$ be a coherent subsheaf of ${\mathcal {E}}$, which is automatically a vector bundle on $S$. We say that ${\mathcal {F}}$ is saturated in ${\mathcal {E}}$ if the quotient ${\mathcal {E}}/{\mathcal {F}}$ is torsion-free. Then ${\mathcal {E}}/{\mathcal {F}}$ is also a vector bundle on $S$. (In the literature, a saturated subsheaf is also called a subbundle.) Denote by $\eta$ the generic point of $S$. The functor ${\mathcal {F}}\mapsto {\mathcal {F}}_\eta$ is an equivalence of categories from the category of saturated subsheaves of ${\mathcal {E}}$ to the category of linear subspaces of ${\mathcal {E}}_\eta$. The inverse of the functor is $F\mapsto F\cap {\mathcal {E}}$, an intersection taken in ${\mathcal {E}}_\eta$.

We say that ${\mathcal {E}}$ is stable (respectively, semistable) if for any coherent subsheaf of ${\mathcal {F}}\subset {\mathcal {E}}$, one has $\mu ({\mathcal {F}})<\mu ({\mathcal {E}})$ (respectively, $\mu ({\mathcal {F}})\leq \mu ({\mathcal {E}})$).

The Harder–Narasimhan filtration of ${\mathcal {E}}$ is the unique filtration

\[ 0={\mathcal{E}}_0\subset {\mathcal{E}}_1\subset \cdots \subset {\mathcal{E}}_m={\mathcal{E}} \]

of saturated subsheaves of ${\mathcal {E}}$ such that each quotient ${\mathcal {E}}_i/{\mathcal {E}}_{i-1}$ is semistable and

\[ \mu({\mathcal{E}}_1/{\mathcal{E}}_0) > \mu({\mathcal{E}}_2/{\mathcal{E}}_1) > \cdots > \mu({\mathcal{E}}_m/{\mathcal{E}}_{m-1}). \]

The maximal slope and the minimal slope of ${\mathcal {E}}$ are defined as

\[ \mu_{{\max}}({\mathcal{E}})=\mu({\mathcal{E}}_1/{\mathcal{E}}_0), \quad \mu_{{\min}}({\mathcal{E}})=\mu({\mathcal{E}}_m/{\mathcal{E}}_{m-1}). \]

As in [Reference HartshorneHar66] and [Reference LazarsfeldLaz04, Chapter 6], the vector bundle ${\mathcal {E}}$ over $S$ is said to be ample (respectively, nef) if the tautological bundle ${\mathcal {O}}(1)$ over the projective space bundle $\mathbb {P}({\mathcal {E}})$ is an ample (respectively, nef) line bundle.

If $k$ has characteristic 0, then ${\mathcal {E}}$ is ample (respectively, $nef$) if and only if $\mu _{\rm min}({\mathcal {E}})>0$ (respectively, $\mu _{\rm min}({\mathcal {E}})\geq 0$). This is essentially Hartshorne [Reference HartshorneHar71, Theorem 2.4]; see also [Reference LazarsfeldLaz04, Theorem 6.4.15].

If $k$ has characteristic $p>0$, this property fails but can be remedied as follows. Define

\begin{gather*} \bar{\mu}_{{\max}}({\mathcal{E}})=\lim_{n\to \infty} p^{{-}n}\mu_{{\max}}((F^{n})^{*}{\mathcal{E}}),\\ \bar{\mu}_{{\min}}({\mathcal{E}})=\lim_{n\to \infty} p^{{-}n}\mu_{{\min}}((F^{n})^{*}{\mathcal{E}}). \end{gather*}

Here $F^{n}: S\to S^{(p^{n})}$ is the relative Frobenius morphism. Note that $(F^{n})^{*}$ may not preserve the Harder–Narasimhan filtration, but the sequences in both limits are eventually constant by Langer [Reference LangerLan04, Theorem 2.7, p. 259].

We say that ${\mathcal {E}}$ is strongly stable (respectively, strongly semistable) if for any coherent subsheaf of ${\mathcal {F}}\subset {\mathcal {E}}$, one has $\bar {\mu }({\mathcal {F}})<\bar {\mu }({\mathcal {E}})$ (respectively, $\bar {\mu }({\mathcal {F}})\leq \bar {\mu }({\mathcal {E}})$).

Finally, by Barton [Reference BartonBar71, Theorem 2.1], the vector bundle ${\mathcal {E}}$ is ample (respectively, nef) if and only if $\bar {\mu }_{\rm min}({\mathcal {E}})>0$ (respectively, $\bar {\mu }_{\rm min}({\mathcal {E}})\geq 0$).

Vector bundles on $\mathbb {P}^{1}$. Now we consider the above terminologies over $\mathbb {P}^{1}$, which turns out to be very concrete. By the Birkhoff–Grothendieck theorem (cf. [Reference Huybrechts and LehnHL97, Theorem 1.3.1]), any nonzero vector bundle ${\mathcal {E}}$ on $S=\mathbb {P}^{1}$ (over any base field $k$) can be decomposed as

\[ {\mathcal{E}}\simeq {\mathcal{O}}(d_1) \oplus {\mathcal{O}}(d_2) \oplus \cdots \oplus {\mathcal{O}}(d_r), \]

with uniquely determined integers $d_1\geq d_2 \geq \cdots \geq d_r.$

The slope of ${\mathcal {E}}$ is defined as

\[ \mu({\mathcal{E}})=\frac{\deg({\mathcal{E}})}{{\mathrm{rank}}({\mathcal{E}})} =\frac 1r (d_1+ \cdots + d_r). \]

We also have the maximal slope and the minimal slope

\[ \mu_{{\max}}({\mathcal{E}})=d_1, \quad \mu_{{\min}}({\mathcal{E}})=d_r. \]

The bundle ${\mathcal {E}}$ is semistable if $\mu _{\rm max}({\mathcal {E}})=\mu _{\rm min}({\mathcal {E}})$.

We easily see that $\bar {\mu }_{\rm max}=\mu _{\rm max}$ and $\bar {\mu }_{\rm min}=\mu _{\rm min}$ if $k$ has a positive characteristic.

Under the decomposition, ${\mathcal {E}}$ is ample if and only if $d_r>0$; ${\mathcal {E}}$ is nef if and only if $d_r\geq 0$. As we are mainly concerned with vector bundles on $\mathbb {P}^{1}$, these can serve as our definitions of ampleness and nefness.

For any nonzero vector bundle ${\mathcal {E}}$ on $S=\mathbb {P}^{1}$, define the maximal nef subbundle (or just the nef part) of ${\mathcal {E}}$ to be

\[ {\mathcal{E}}_{\mathrm{nef}}={\mathrm{Im}}(\Gamma(S,{\mathcal{E}})\otimes_k {\mathcal{O}}_S \to {\mathcal{E}}). \]

In terms of the above decomposition, we simply have

\[ {\mathcal{E}}_{\mathrm{nef}}= \oplus_{d_i\geq 0}{\mathcal{O}}(d_i). \]

Note that ${\mathcal {E}}_{\mathrm {nef}}=0$ if and only $\mu _{\rm max}({\mathcal {E}})<0$.

Hodge bundles. Let ${\mathcal {G}}$ be a group scheme over a scheme $S$. The Hodge bundle of ${\mathcal {G}}$ over $S$ is the ${\mathcal {O}}_S$-module

\[ \bar{\Omega}_{{\mathcal{G}}}=\bar{\Omega}_{{\mathcal{G}}/{\mathcal{S}}} = e^{*}\Omega_{{\mathcal{G}}/{\mathcal{S}}}^{1}, \]

where $\Omega _{{\mathcal {G}}/{\mathcal {S}}}^{1}$ the relative differential sheaf, and $e:S\to {\mathcal {G}}$ denotes the identity section of ${\mathcal {G}}$.

Recall that if ${\mathcal {G}}$ is a finite and flat commutative group scheme of height one over $S$, then $\bar {\Omega }_{{\mathcal {G}}}$ and ${\mathrm {Lie}}({\mathcal {G}}/S)$ are locally free and canonically dual to each other; see [Reference Demazure and GrothendieckSGA3, VII$_A$, Proposition 5.5.3].

The definition particularly applies to Néron models of abelian varieties. Let $S$ be a connected Dedekind scheme, and $K$ be its function field. Let $A$ be an abelian variety over $K$. Then we write

\[ \bar{\Omega}_A=\bar{\Omega}_{\mathcal{A}}=\bar{\Omega}_{{\mathcal{A}}/S}. \]

Here ${\mathcal {A}}$ is the Néron model of $A$ over $S$.

For Hodge bundles of smooth integral models of abelian varieties, we have the following well-known interpretation as the sheaf of global differentials. We sketch the idea for lack of a complete reference.

Lemma 2.3 Let $S$ be an integral scheme. Let $\pi :{\mathcal {A}}\to S$ be a smooth connected group scheme whose generic fiber is an abelian variety. There are canonical isomorphisms

\[ \pi^{*}\bar{\Omega}_{{\mathcal{A}}/S}\longrightarrow \Omega_{{\mathcal{A}}/S}^{1}, \quad \bar{\Omega}_{{\mathcal{A}}/S}\longrightarrow \pi_*\Omega_{{\mathcal{A}}/S}^{1}. \]

Maximal nef subalgebra. The following result gives the notion of a maximal nef $p$-Lie subalgebra of a locally free commutative $p$-Lie algebra. It is a special case of [Reference RösslerRos20, Lemma 4.4], which works for general projective and smooth curves $S$ in the stability setting.

Quotient by subgroup scheme. Here we describe the quotient construction in the proof of Theorem 1.1.

Go back to the setting to Theorem 1.1. Namely, $k$ is a finite field of characteristic $p$, and $A$ is an abelian variety over $K= k(t)$. The $p$-Lie algebra ${\mathrm {Lie}}({\mathcal {A}}/S)$ of the Néron model ${\mathcal {A}}$ of $A$ is a vector bundle over $S$, canonically dual to the Hodge bundle $\bar {\Omega }_{{\mathcal {A}}/S}$. We also have a natural identity

\[ {\mathrm{Lie}}({\mathcal{A}}[F]/S)={\mathrm{Lie}}({\mathcal{A}}/S), \]

where ${\mathcal {A}}[F]=\ker (F:{\mathcal {A}}\to {\mathcal {A}}^{(p)})$ is the kernel of the relative Frobenius morphism.

For the sake of Theorem 1.1, assume that $\bar {\Omega }_{{\mathcal {A}}/S}$ is not ample, or equivalently $\mu _{\rm max}({\mathrm {Lie}}({\mathcal {A}}/S)) \geq 0$. Then the maximal nef subbundle ${\mathrm {Lie}}({\mathcal {A}}/S)_{\mathrm {nef}}$ of ${\mathrm {Lie}}({\mathcal {A}}/S)$ is a nonzero $p$-Lie subalgebra of ${\mathrm {Lie}}({\mathcal {A}}/S)$ by Lemma 2.4. By the correspondence between $p$-Lie algebras and group schemes, ${\mathrm {Lie}}({\mathcal {A}}/S)_{\mathrm {nef}}$ corresponds to a finite and flat group scheme ${\mathcal {A}}[F]_{\mathrm {nef}}$ of height one over $S$, which is a closed subgroup scheme of ${\mathcal {A}}[F]$ with $p$-Lie algebra isomorphic to ${\mathrm {Lie}}({\mathcal {A}}/S)_{\mathrm {nef}}$. Form the quotient

\[ {\mathcal{A}}_1'={\mathcal{A}}/({\mathcal{A}}[F]_{\mathrm{nef}}), \]

which is a smooth group scheme of finite type over $S$. We have a description of this quotient process in Theorem 2.8.

Denote by $A_1$ and $A[F]_{\mathrm {nef}}$ the generic fibers of ${\mathcal {A}}_1$ and ${\mathcal {A}}[F]_{\mathrm {nef}}$. It follows that

\[ A_1=A/(A[F]_{\mathrm{nef}}) \]

is an abelian variety over $K$. In general, ${\mathcal {A}}_1'$ may fail to be the Néron model of $A_1$, or even fail to be an open subgroup scheme of the Néron model. Therefore, take ${\mathcal {A}}_1$ to be the Néron model of $A_1$ over $S$.

We see that there is a natural exact sequence

\[ 0\longrightarrow {\mathrm{Lie}}({\mathcal{A}}/S)_{\mathrm{nef}} \longrightarrow {\mathrm{Lie}}({\mathcal{A}}/S) \longrightarrow {\mathrm{Lie}}({\mathcal{A}}_1'/S). \]

It follows that the nef part of ${\mathrm {Lie}}({\mathcal {A}}/S)$ becomes zero in ${\mathrm {Lie}}({\mathcal {A}}_1'/S)$ and ${\mathrm {Lie}}({\mathcal {A}}_1/S)$. However, ${\mathrm {Lie}}({\mathcal {A}}_1/S)$ may obtain some new nef part. Thus, the quotient process does not solve Theorem 1.1 immediately. Our idea is that if ${\mathrm {Lie}}({\mathcal {A}}_1/S)_{\mathrm {nef}}\neq 0$, then we can further form the quotient

\[ {\mathcal{A}}_2'={\mathcal{A}}_1/({\mathcal{A}}_1[F]_{\mathrm{nef}}) \]

and let ${\mathcal {A}}_2$ be the Néron model of the generic fiber of ${\mathcal {A}}_2'$. Repeat the process, we obtain a sequence

\[ {\mathcal{A}}={\mathcal{A}}_0, \ {\mathcal{A}}_1', \ {\mathcal{A}}_1,\ {\mathcal{A}}_2',\ {\mathcal{A}}_2, \ {\mathcal{A}}_3',\ {\mathcal{A}}_3,\ \ldots \]

of smooth group schemes of finite type over $S$, whose generic fibers are abelian varieties isogenous to $A$.

To get more information from the sequence, the key is to consider the height of the above sequence. We see that the height sequence is decreasing. As each term is a non-negative integer, the height sequence is eventually constant, and thus ${\mathrm {Lie}}({\mathcal {A}}_n/S)_{\mathrm {nef}}$ is eventually a direct sum of the trivial bundle ${\mathcal {O}}_S$.

An intermediate result that the quotient process will give us is as follows. The proof is given in the next subsection.

The proposition is philosophically very similar to [Reference RösslerRos20, Proposition 2.6]. These two results are proved independently, but their proofs use similar ideas. For example, the ‘maximal nef subalgebra’ appears in [Reference RösslerRos20, Lemma 4.8], the ‘quotient process’ is used in the proof of [Reference RösslerRos20, Proposition 2.6] in pp. 1145–1146 of the paper, and the ‘control by height’, to be introduced in the following by us, is also used in the proof.

2.3 Control by heights

In this subsection, we prove Proposition 2.5. The main tool is the height of a group scheme over a projective curve.

Heights of smooth group schemes

Let $S$ be a projective and smooth curve over a field $k$, and let $K$ be the function field of $S$. Let ${\mathcal {G}}$ be a smooth group scheme of finite type over $S$. The height of ${\mathcal {G}}$ is defined to be

\[ h({\mathcal{G}})=\deg(\bar{\Omega}_{{\mathcal{G}}/S}) =\deg(\det\bar{\Omega}_{{\mathcal{G}}/S}). \]

Here the Hodge bundle $\bar {\Omega }_{{\mathcal {G}}/S}$ is the pull-back of the relative differential sheaf $\Omega _{{\mathcal {G}}/S}^{1}$ to the identity section of ${\mathcal {G}}$ as before.

Let $A$ be an abelian variety over $K$, and let ${\mathcal {A}}$ be the Néron model of $A$ over $S$. The height of $A$ is defined to be

\[ h(A)=\deg(\bar{\Omega}_{A})=\deg(\bar{\Omega}_{{\mathcal{A}}/S}) =\deg(\det\bar{\Omega}_{{\mathcal{A}}/S}). \]

If $k$ is finite, this definition was originally used by Parshin and Zarhin to prove the Tate conjecture of abelian varieties over global function fields. A number field analogue, introduced by Faltings [Reference FaltingsFal83] and called the Faltings height, was a key ingredient in his proof of the Mordell conjecture.

Theorem 2.6 Let $S$ be the projective and smooth curve over a field $k$. Let ${\mathcal {G}}$ be a smooth group scheme of finite type over $S$ whose generic fiber $A$ is an abelian variety. Then $h({\mathcal {G}})\geq h(A)\geq 0$. Moreover, the following hold:

1. (1) $h({\mathcal {G}})= h(A)$ if and only if ${\mathcal {G}}$ is an open subgroup of the Néron model of $A$ over $S$;

2. (2) $h({\mathcal {G}})=0$ if and only if ${\mathcal {G}}$ is isotrivial over $S$, i.e. for some finite étale morphism $S'\to S$, the base change ${\mathcal {G}}\times _SS'$ is constant over $S'$.

Proof. We first treat the inequality $h({\mathcal {G}})\geq h(A)$. Denote by ${\mathcal {A}}$ the Néron model of $A$ over $S$. By the Néron mapping property, there is a homomorphism $\tau :{\mathcal {G}}\to {\mathcal {A}}$ which is the identity map on the generic fiber. It induces morphisms $\bar {\Omega }_{{\mathcal {A}}/S}\to \bar {\Omega }_{{\mathcal {G}}/S}$ and $\det (\bar {\Omega }_{{\mathcal {A}}/S})\to \det (\bar {\Omega }_{{\mathcal {G}}/S})$ of locally free ${\mathcal {O}}_S$-modules. The morphisms are isomorphisms at the generic point of $S$, and thus are injective over $S$. Taking degrees, we have $h({\mathcal {A}})\leq h({\mathcal {G}})$.

If $h({\mathcal {A}})= h({\mathcal {G}})$, then $\det (\bar {\Omega }_{{\mathcal {A}}/S})\to \det (\bar {\Omega }_{{\mathcal {G}}/S})$ is an isomorphism, and thus $\bar {\Omega }_{{\mathcal {A}}/S}\to \bar {\Omega }_{{\mathcal {G}}/S}$ is also an isomorphism. By Lemma 2.3, the natural map $\tau ^{*}\Omega _{{\mathcal {A}}/S}\to \Omega _{{\mathcal {G}}/S}$ is also an isomorphism. Consequently, $\tau :{\mathcal {G}}\to {\mathcal {A}}$ is étale. Then it is an open immersion because it is an isomorphism between the generic fibers.

Part (2) is essentially due to Moret-Bailly. We first check $h(A)\geq 0$. If ${\mathcal {A}}$ is semi-abelian, then $h(A)\geq 0$ by [Reference Moret-BaillyMor85, IX, Theorem 2.1] or [Reference Faltings and ChaiFC90, § V.2, Proposition 2.2]. In general, by the semistable reduction theorem, there is a finite extension $K'$ of $K=k(S)$ such that $A_{K'}$ has everywhere semi-abelian reduction over the normalization $S'$ of $S$ in $K'$. It follows that $h({\mathcal {A}}_{S'})\geq h(A_{K'})\geq 0$, and thus $h({\mathcal {A}})\geq 0$.

If $h({\mathcal {G}})=0$, the above arguments already imply that ${\mathcal {G}}$ is an open subgroup scheme of the Néron model of $A$, and $A$ has everywhere semi-abelian reduction. Now the result follows from [Reference Moret-BaillyMor85, XI, Theorem 4.5] or [Reference Faltings and ChaiFC90, § V.2, Proposition 2.2]. This finishes the proof.

Remark 2.7 If $k$ is finite, then there is a Northcott property for the height of abelian varieties, as an analogue of [Reference Faltings and ChaiFC90, Chapter V, Proposition 4.6] over global function fields. This is the crucial property which makes the height a powerful tool in diophantine geometry, but we do not use this property here.

Height under purely inseparable isogenies

Here we prove a formula on the change of height under purely inseparable isogenies of smooth group schemes. We start with the following result about a general quotient process.

Theorem 2.8 Let ${\mathcal {G}}$ be a smooth group scheme of finite type over a Dedekind scheme $S$, and let ${\mathcal {H}}$ be a closed subgroup scheme of ${\mathcal {G}}$ which is flat over $S$. Then the fppf quotient ${\mathcal {G}}'={\mathcal {G}}/{\mathcal {H}}$ is a smooth group scheme of finite type over $S$, and the quotient morphism ${\mathcal {G}}\to {\mathcal {G}}'$ is faithfully flat.

Proof. The essential part follows from [Reference AnantharamanAna73, Theorem 4.C], which implies that the quotient ${\mathcal {G}}'$ is a group scheme over $S$. It is easy to check that ${\mathcal {G}}'$ is flat over $S$. In fact, because ${\mathcal {G}}$ is flat over $S$, the sheaf ${\mathcal {O}}_G$ contains no ${\mathcal {O}}_S$-torsion. As ${\mathcal {G}}\to {\mathcal {G}}'$ is an epimorphism, ${\mathcal {O}}_{G'}$ injects into ${\mathcal {O}}_G$, and thus contains no ${\mathcal {O}}_S$-torsion either. Then ${\mathcal {G}}'$ is flat over $S$. To check that ${\mathcal {G}}'$ is smooth over $S$, it suffices to check that for any geometric point $s$ of $S$, the fiber ${\mathcal {G}}'_s$ is smooth. As ${\mathcal {G}}_s$ is reduced and ${\mathcal {G}}_s\to {\mathcal {G}}'_s$ is an epimorphism, ${\mathcal {G}}'_s$ is reduced. Then ${\mathcal {G}}'_s$ is smooth because it is a reduced group scheme over an algebraically closed field. This checks that ${\mathcal {G}}'$ is smooth over $S$. Moreover, ${\mathcal {G}}_s\to {\mathcal {G}}'_s$ is flat as ${\mathcal {G}}_s$ is an ${\mathcal {H}}_s$-torsor over ${\mathcal {G}}_s'$. It follows that ${\mathcal {G}}\to {\mathcal {G}}'$ is flat by [Reference Dieudonné and GrothendieckEGA, IV-3, Theorem 11.3.10]. This finishes the proof.

Now we introduce a theorem to track the change of heights of abelian varieties under the quotient process. The result is similar to [Reference RösslerRos20, Lemma 4.12].

Theorem 2.9 Let $S$ be a Dedekind scheme over $\mathbb {F}_p$ for a prime $p$. Let ${\mathcal {A}}$ be a smooth group scheme of finite type over $S$. Let ${\mathcal {G}}$ be a closed subgroup scheme of ${\mathcal {A}}[F]$ which is flat over $S$, and denote by ${\mathcal {B}}={\mathcal {A}}/{\mathcal {G}}$ the quotient group scheme over $S$. Then the following hold.

1. (1) There is a canonical exact sequence

   \[ 0 \longrightarrow {\mathrm{Lie}}({\mathcal{G}}/S)\longrightarrow {\mathrm{Lie}}({\mathcal{A}}/S)\longrightarrow {\mathrm{Lie}}({\mathcal{B}}/S) \longrightarrow F_S^{*}{\mathrm{Lie}}({\mathcal{G}}/S)\longrightarrow 0 \]
   of coherent sheaves over $S$. Here $F_S:S\to S$ is the absolute Frobenius morphism.

2. (2) If $S$ is a projective and smooth curve over a field $k$ of characteristic $p>0$, then

   \begin{align*} h({\mathcal{B}})=h({\mathcal{A}})-(p-1)\deg({\mathrm{Lie}}({\mathcal{G}}/S)).\\ \end{align*}

Proof. Part (2) is a direct consequence of part (1) by $\bar {\Omega }_{{\mathcal {A}}/S}={\mathrm {Lie}}({\mathcal {A}}/S)^{\vee }$. The major problem is to prove part (1). Consider the following commutative diagram.

Both rows are exact. The relative Frobenius $F:{\mathcal {G}}\to {\mathcal {G}}^{(p)}$ is zero as ${\mathcal {G}}$ has height one. By the snake lemma, we have an exact sequence

\[ 0\longrightarrow {\mathcal{G}} \longrightarrow {\mathcal{A}}[F] \longrightarrow {\mathcal{B}}[F] \longrightarrow {\mathcal{G}}^{(p)} \longrightarrow 0. \]

The Lie algebra of this sequence is exactly the sequence of the theorem.

It suffices to check the general fact that the Lie functor from the category of finite and flat commutative group schemes of height one over $S$ to the category of $p$-Lie algebras is exact. In fact, if

\[ 0\longrightarrow{\mathcal{H}}_1 \longrightarrow {\mathcal{H}} \longrightarrow {\mathcal{H}}_2 \longrightarrow 0 \]

is an exact sequence in the first category, then we first get a complex

\[ 0 \longrightarrow {\mathrm{Lie}}({\mathcal{H}}_1/S)\longrightarrow {\mathrm{Lie}}({\mathcal{H}}/S) {\longrightarrow} {\mathrm{Lie}}({\mathcal{H}}_2/S)\longrightarrow 0 \]

of locally free sheaves over $S$. By the canonical duality between the Lie algebra and the Hodge bundle, the Lie functor commutes with base change. For any point $s\in S$, consider the fiber of the complex of Lie algebras above $s$. It is exact by counting dimensions, because $\dim {\mathrm {Lie}}({\mathcal {H}}_s/s)$ equals the order of ${\mathcal {H}}_s$ and the order is additive under short exact sequences. This shows that the complex is fiber-wise exact. Then the complex is exact. This finishes the proof.

Stationary height

Now we prove Proposition 2.5. Let ${\mathcal {A}}={\mathcal {A}}_0$ be as in the proposition. By the quotient process, we obtain a sequence

\[ {\mathcal{A}}_0,\ {\mathcal{A}}_1',\ {\mathcal{A}}_1,\ {\mathcal{A}}_2',\ {\mathcal{A}}_2,\ \ldots \]

of abelian varieties over $K$. Here for any $n\geq 0$,

\[ {\mathcal{A}}_{n+1}'={\mathcal{A}}_n/{\mathcal{A}}_n[F]_{\mathrm{nef}} \]

and ${\mathcal {A}}_{n+1}$ is the Néron model of the generic fiber of ${\mathcal {A}}_{n+1}'$. We know that ${\mathcal {A}}_n'$ is a smooth group scheme of finite type over $S$ by Theorem 2.8.

By Theorem 2.6, $h({\mathcal {A}}_n)\leq h({\mathcal {A}}_n')$. By Theorem 2.9(2),

\[ h({\mathcal{A}}_{n+1}')=h({\mathcal{A}}_n)-(p-1)\deg({\mathrm{Lie}}({\mathcal{A}}_n/S)_{\mathrm{nef}}) \leq h({\mathcal{A}}_n). \]

It follows that the sequence

\[ h({\mathcal{A}}_0),\ h({\mathcal{A}}_1'),\ h({\mathcal{A}}_1),\ h({\mathcal{A}}_2'),\ h({\mathcal{A}}_2),\ \ldots \]

is decreasing. Note that each term of the sequence is a non-negative integer by Theorem 2.6. Therefore, there is an integer $n_0\geq 0$ such that

\[ h({\mathcal{A}}_{n}')=h({\mathcal{A}}_{n})=h({\mathcal{A}}_{n_0}), \quad \forall\ n\geq n_0. \]

It follows that for any $n\geq n_0$, ${\mathcal {A}}_{n}'$ is an open subgroup scheme of ${\mathcal {A}}_{n}$ and

\[ \deg({\mathrm{Lie}}({\mathcal{A}}_n/S)_{\mathrm{nef}})=\deg({\mathrm{Lie}}({\mathcal{A}}_n'/S)_{\mathrm{nef}})=0. \]

As a consequence, for any $n\geq n_0$, ${\mathrm {Lie}}({\mathcal {A}}_n/S)_{\mathrm {nef}}$ is a direct sum of copies of the trivial bundle ${\mathcal {O}}_S$, and $\bar {\Omega }_{{\mathcal {A}}_n/S}={\mathrm {Lie}}({\mathcal {A}}_n/S)^{\vee }$ is nef. Moreover, $\bar {\Omega }_{{\mathcal {A}}_n/S}$ is ample if and only if ${\mathcal {A}}_n\to {\mathcal {A}}_{n+1}$ is an isomorphism.

For Proposition 2.5, if none of $\bar {\Omega }_{{\mathcal {A}}_n/S}$ is ample, take ${\mathcal {B}}={\mathcal {A}}_{n_0}$. The group scheme $\ker ({\mathcal {A}}_{n_0}\to {\mathcal {A}}_n)$ has a degree going to infinity. It is of constant type over $S$, as an easy consequence of Lemmas 2.1(1) and 2.2. This proves Proposition 2.5.

2.4 Lifting $p$-divisible groups

In this subsection, we prove Theorem 1.1. Note that we have already proved Proposition 2.5. To finish the proof, it suffices to prove the following result.

We refer to Conrad [Reference ConradCon06] for Chow's theory of $(K/k)$-traces. Before proving Proposition 2.10, let us see how Propositions 2.5 and 2.10 imply Theorem 1.1. Let $A$ be as in Theorem 1.1. Apply Proposition 2.5 to $A$, which gives an abelian variety $B$ over $K$ with a purely inseparable isogeny $A\to B$. If $B$ satisfies Proposition 2.5(1), the result already holds. If $B$ satisfies Proposition 2.5(2), apply Proposition 2.10 to the Néron model ${\mathcal {B}}$ of $B$. Then $B$ has a non-trivial $(K/k)$-trace, and thus $A$ also has a non-trivial $(K/k)$-trace $A_0$, which is an abelian variety over $k$ with a homomorphism $A_{0,K}\to A$. By [Reference ConradCon06, Theorem 6.4], the homomorphism $A_{0,K}\to A$ is an isogeny to its image $A'$. Note that $A$ is isogenous to $A_{0,K}\times _K (A/A')$. Apply the same process to the abelian variety $A/A'$ over $K$. Note that the dimension of $A/A'$ is strictly smaller than that of $A$. The process eventually terminates. This proves Theorem 1.1.

The $p$-divisible group

To prove Proposition 2.10, the first step is to change the direct system $\{{\mathcal {G}}_n\}_n$ to a nonzero $p$-divisible group. For the basics of $p$-divisible groups, we refer to Tate [Reference TateTat67].

Let $S$ be any scheme. A direct system $\{G_n\}_{n\geq 1}$ of flat group schemes over $S$ is called an increasing system if the transition homomorphisms are closed immersions. A subsystem of the increasing system $\{G_n\}_{n\geq 1}$ is an increasing system $\{H_n\}_{n\geq 1}$ over $S$ endowed with an injection $\varinjlim H_n \to \varinjlim G_n$ as fppf sheaves over $S$.

There is a description of subsystem in terms of group schemes without going to the limit sheaves. In fact, an injection $\varinjlim H_n \to \varinjlim G_n$ as fppf sheaves over $S$ is equivalent to a sequence $\{\phi _n:H_n\to G_{\tau (n)}\}_{n\geq 1}$ of injections, compatible with the transition maps $H_n\to H_{n+1}$ and $G_{\tau (n)}\to G_{\tau (n+1)}$ for each $n\geq 1$, where $\{\tau (n)\}_{n\geq 1}$ is an increasing sequence of positive integers. For each $n\geq 1$, to find $\tau (n)$, it suffices to note that the identity map $i_n:H_n\to H_n$ is an element of $H_n(H_n)\subset H_\infty (H_n) \subset G_\infty (H_n)$, and thus it is contained in some $G_{\tau (n)}(H_n)$. This gives a morphism $H_n\to G_{\tau (n)}$.

We have the following basic result.

Lemma 2.11 Let $k$ be any field of characteristic $p>0$. Let $\{G_n\}_{n\geq 1}$ be an increasing system of finite commutative group schemes of $p$-power order over $k$. Assume that the order of $G_n$ goes to infinity as $n\to \infty$, but the order of $G_n[p]$ is bounded as $n\to \infty$. Then $\{G_n\}_{n\geq 1}$ has a subsystem $\{H_n\}_{n\geq 1}$ which is a nonzero $p$-divisible group over $k$.

Proof. The idea can be easily illustrated in terms of abelian groups. Assume for the moment that $\{G_n\}_{n\geq 1}$ is an increasing system of abelian groups satisfying similar conditions. Let $G_\infty$ be the direct limit of $\{G_n\}_{n\geq 1}$. By definition, $G_\infty$ is an infinite torsion group whose element has $p$-power orders, but $G_\infty [p]$ is finite. A structure theorem asserts that $G_\infty \simeq (\mathbb {Q}_p/\mathbb {Z}_p)^{r}\oplus F$ for some positive integer $r$ and some finite group $F$. Then $H_\infty =(\mathbb {Q}_p/\mathbb {Z}_p)^{r}$ is the subgroup of $G_\infty$ that gives us a $p$-divisible group. This subgroup consists of exactly the infinitely divisible elements of $G_\infty$, and thus can be extracted as $H_\infty =\cap _{a\geq 1} p^{a}G_\infty$. Then $H_m=H_\infty [p^{m}]=\cap _{a\geq 1} (p^{a}G_\infty )[p^{m}]$ for any $m\geq 1$.

Go back to the group schemes $G_n$ in the lemma. By assumption, the order of $G_n[p]$ is bounded by some integer $p^{r}$. For any $m$, the order of $G_n[p^{m}]$ is bounded by $p^{mr}$, which can be checked by induction using the exact sequence

\[ 0\longrightarrow G_n[p]\longrightarrow G_n[p^{m+1}]\stackrel{[p]}{\longrightarrow} G_n[p^{m}]. \]

As the order of $G_n$ goes to infinity, the exact exponent of $G_n$, which is the smallest positive integer $N_n$ such that the multiplication $[N_n]:G_n\to G_n$ is the zero map, also goes to infinity.

Now we construct the $p$-divisible group. Denote $G_\infty =\varinjlim G_n$ as an fppf sheaf over $\operatorname {Spec}(k)$. Denote $H_\infty =\cap _{a\geq 1} p^{a}G_\infty$ as a subsheaf of $G_\infty$. Denote $H_m=H_\infty [p^{m}]$ as a subsheaf of $H_\infty$ for any $m\geq 1$. We claim that the system $\{H_m\}_{m\geq 1}$, where the transition maps are injections as subsheaves of $H_\infty$, is a nonzero $p$-divisible group over $k$.

First, every $H_m$ is representable by a finite group scheme over $k$. In fact, $H_m$ is the intersection of the decreasing sequence $\{(p^{a}G_\infty )[p^{m}]\}_{a\geq 1}$. As the order of $\{G_n[p^{m}]\}_n$ is bounded, the increasing sequence $\{(p^{a}G_n)[p^{m}]\}_n$ of finite group schemes is eventually stationary. This stationary term is exactly $(p^{a}G_\infty )[p^{m}]$. The sequence $\{(p^{a}G_\infty )[p^{m}]\}_{a\geq 1}$ of finite group schemes is decreasing, and thus eventually stationary. This stationary term is exactly $H_m$.

Second, $H_1\neq 0$. Otherwise, $(p^{a}G_\infty )[p]=0$ for some $a\geq 1$. Then $p^{a}G_\infty =0$ and, thus, $p^{a}G_n=0$ for all $n$. This contradicts the fact that the exponent of $G_n$ goes to infinity. Thus, $H_1\neq 0$.

By definition, the map $[p^{m}]:H_\infty \to H_\infty$ is surjective with kernel $H_m$. It follows that the morphism $[p^{m}]:H_{m+1}\to H_{m+1}$ has kernel $H_m$ and image $H_1$. This implies that $\{H_m\}_{m\geq 1}$ is a $p$-divisible group. This finishes the proof.

Remark 2.12 If $k$ is perfect, one can prove the lemma by Dieudonné modules. In fact, take the covariant Dieudonné module of the sequence $\{G_n\}_{n\geq 1}$, apply the above construction of abelian groups to the Dieudonné modules, and transfer the result back to obtain a $p$-divisible group by the equivalence between finite group schemes and Dieudonné modules.

Algebraicity

Now we prove Proposition 2.10. Recall that we have an increasing system $\{{\mathcal {G}}_n\}_n$ of finite and flat closed subgroup schemes of ${\mathcal {A}}$ of constant type. The transition maps are necessarily of constant type by Lemma 2.1(3). Thus, $\{{\mathcal {G}}_n\}_n$ is the base change of an increasing system $\{G_n\}_n$ of finite group schemes over $k$. By Lemma 2.11, the system $\{G_n\}_n$ has a subsystem $H_\infty =\{H_n\}_n$, which is a nonzero $p$-divisible group over $k$. Denote by ${\mathcal {H}}_\infty =\{{\mathcal {H}}_n\}_n$ the base change of $\{H_n\}_n$ to $S$, which is a $p$-divisible group over $S$, and also a subsystem of $\{{\mathcal {G}}_n\}_n$. Then ${\mathcal {H}}_\infty =\{{\mathcal {H}}_n\}_n$ is a subsystem of ${\mathcal {A}}[p^{\infty }]=\{{\mathcal {A}}[p^{n}]\}_n$. We are going to ‘lift’ ${\mathcal {H}}_\infty =\{{\mathcal {H}}_n\}_n$ to an abelian scheme over $S$ of constant type.

By [Reference ConradCon06, Theorem 6.6], the $(K/k)$-trace of $A_{K}$ is nonzero if and only if the $(Kk'/k')$-trace of $A_{K'}$ is nonzero for any extension $k'/k$. Therefore, in the proposition, we can replace $k$ by any finite extension. In particular, we can assume that there is a point $s\in S(k)$ such that ${\mathcal {A}}$ has good reduction at $s$. The fiber $C={\mathcal {A}}_s$ is an abelian variety over $k$, and the $p$-divisible group $C[p^{\infty }]$ has a $p$-divisible subgroup ${\mathcal {H}}_{\infty ,s}$, which is canonically isomorphic to $H_\infty$. We prove that ${\mathrm {Hom}}(C_K,A)\neq 0$ from the fact that they share the same $p$-divisible subgroup $H_{\infty ,K}$.

To proceed, we need two fundamental theorems on $p$-divisible groups of abelian varieties over finitely generated fields.

Theorem 2.13 Let $K$ be a finitely generated field over a finite field $\mathbb {F}_p$. Let $A$ and $B$ be abelian varieties over $K$. Then the canonical map

\[ {\mathrm{Hom}}(A,B)\otimes_\mathbb{Z}\mathbb{Z}_p \longrightarrow {\mathrm{Hom}}(A[p^{\infty}], B[p^{\infty}]) \]

is an isomorphism.

Theorem 2.14 Let $A$ be an abelian variety over a finite field $k$ of characteristic $p>0$. Then the $p$-divisible group $A[p^{\infty }]$ is semisimple, i.e. isogenous to a direct sum of simple $p$-divisible groups over $k$.

The more classical $\ell$-adic analogues of theorems are the Tate conjectures and the semisimplicity conjecture proved by Tate and Zarhin. For the current $p$-adic version, Theorem 2.13 for a finite field $K$ and Theorem 2.14 can be proved by an easy modification of the $\ell$-adic argument of Tate [Reference TateTat66]. For general $K$, Theorem 2.13 is proved by de Jong [Reference de JongJon98]. For convenience readers, we sketch a proof of Theorem 2.14 later.

Return to the proof of Proposition 2.10. By Theorem 2.14, the injection $H_{\infty }\to C[p^{\infty }]$ implies the existence of a surjection $C[p^{\infty }]\to H_{\infty }$. Take a base change to $K$ and compose with $H_{\infty ,K}\to A[p^{\infty }]$. We have a nonzero element of ${\mathrm {Hom}}(C_K[p^{\infty }], A[p^{\infty }])$. By Theorem 2.13, we have ${\mathrm {Hom}}(C_K,A)\neq 0$. This proves the proposition.

Now we sketch a proof of Theorem 2.14. We refer to [Reference MilneMil16, IV, Theorem 2.5] for a modern treatment of the $\ell$-adic version, which we modify to the current $p$-adic version.

Proof of Theorem 2.14 The key is still the fact that there are only finitely abelian varieties (up to isomorphism) of a fixed dimension over a fixed finite field. This essentially follows from Zarhin's trick. See [Reference MilneMil16, I, Corollary 3.13] for example.

Let $G$ be a $p$-divisible subgroup of $A[p^{\infty }]$, and we are going to prove that $G$ has a complement in $A[p^{\infty }]$ up to isogeny. Denote $A_n=A/G[p^{n}]$, and denote by $f_n:A\to A_n$ the quotient map. By the finiteness, there is an abelian variety $B$ over $k$ and an infinite set $\Sigma$ of positive integers such that $A_n$ is isomorphic to $B$ for any $n\in \Sigma$. By the isomorphism, we obtain an isogeny $f_n:A\to B$ with $\ker (f_n)=G[p^{n}]$ for any $n\in \Sigma$.

By compactness, replacing $\Sigma$ by an infinite subset if necessary, we can assume that $f_n$ converges to $f\in {\mathrm {Hom}}(A,B)\otimes _\mathbb {Z}\mathbb {Z}_p$ for $n\in \Sigma$. By definition, the kernel of $f[p^{\infty }]:A[p^{\infty }]\to B[p^{\infty }]$ is exactly $G$. This result corresponds to [Reference MilneMil16, IV, Lemma 2.4].

The rest of the proof is similar to [Reference MilneMil16, IV, Theorem 2.5(a)]. In fact, composed with an isogeny $B\to A$, the element $f$ gives an element $g\in {\mathrm {End}}(A)\otimes _\mathbb {Z} \mathbb {Z}_p$. The algebra $R={\mathrm {End}}(A)\otimes _\mathbb {Z}\mathbb {Q}_p$ is semisimple over $\mathbb {Q}_p$, so the left ideal $Rg$ is generated by an idempotent $e\in R$. Then $\ker (e[p^{\infty }])$ is isogenous to $G$. Now we have a decomposition

\[ A[p^{\infty}] = \ker(e[p^{\infty}]) \oplus \ker((1-e)[p^{\infty}]), \]

which is understood up to isogeny. This finishes the proof.

3.1 Preliminary results on torsors