2.1.1.

Let K be a field, equipped with a complete p-adic valuation v with $v(p) = 1,$ and with perfect residue field k. We denote by $\mathcal O_K$ the ring of integers of K and by $\mathfrak m_K \subset \mathcal O_K$ the maximal ideal. Fix an algebraic closure $\bar K$ of K, and denote by C its completion. We write $\Gamma_K = \mathrm{Gal}(\bar K/K)$ for the absolute Galois group of K.

For G a p-divisible group over K, we denote by $T_p G$ its Tate module. Let ${\mathcal G}$ be a p-divisible group over $\mathcal O_K,$ with generic fiber G. We will sometimes write $T_p {\mathcal G}$ for $T_pG$ . Applying [Reference FarguesFar19, Theorem 0.1] to the base change of G to C gives a functorial injection

(2.1) \begin{equation}f_{{\mathcal G}}: \mathrm{Lie\,} {\mathcal G} \otimes _{\mathcal O_K} C(1) \rightarrow T_p G\otimes _{\mathbb Z_p} C,\end{equation}

where we abuse notation and write $\mathrm{Lie\,} {\mathcal G}$ for the Lie algebra of the formal group associated to ${\mathcal G}$ . In particular, (2.1) is $\Gamma_K$ -equivariant. We denote by ${\mathscr F}({\mathcal G})$ the triple $(T_pG, \mathrm{Lie\,} {\mathcal G}, f_{{\mathcal G}})$ .

By a result of Fargues [Reference FarguesFar19, Theorem 0.1] and of Scholze and Weinstein [Reference Scholze and WeinsteinSW13, Theorem B], ${\mathscr F}$ induces an equivalence between the category of p-divisible groups over $\mathcal O_K$ and the category of triples (T,W, f) consisting of a $\mathbb Z_p[\Gamma_K]$ -module T, which is finite free over $\mathbb Z_p,$ a K-vector space W, and a $\Gamma_K$ -equivariant injection $f: W\otimes _K C(1) \rightarrow T\otimes _{\mathbb Z_p}C$ .

2.1.2.

Let $\mathcal A$ be a semi-abelian scheme over $\mathcal O_K,$ such that $A = \mathcal A\otimes K$ is an abelian variety. We want to extend the above theory to $\mathcal A,$ and compare it to the Hodge–Tate decomposition for A, when it applies.

To explain this, let $\mathcal H$ be a quasi-finite flat separated group scheme over $\mathcal O_K$ . As $\mathcal O_K$ is Henselian, we may write $\mathcal H$ as a disjoint union of schemes $\mathcal H = \mathcal H^\circ \coprod \mathcal H_{\eta},$ where $\mathcal H^\circ$ is finite over $\mathcal O_K,$ and $\mathcal H_{\eta}$ has empty special fiber [Reference GrothendieckGro61, IV, 18.5.11(c)]. In particular, $\mathcal H_{\eta}$ is a finite K-scheme and $\mathcal H$ is affine. If we write $\mathcal H = {\operatorname{ Spec\,}} R,$ then ones sees from this decomposition that $\mathcal H^\circ = {\operatorname{ Spec\,}} \hat R,$ where $\hat R$ is the p-adic completion of R. As $\hat R$ is a finitely generated, torsion-free $\mathcal O_K$ -module, it is finite free over $\mathcal O_K$ (see [Reference KaplanskyKap52, Theorem 1]) [Reference ScholzeSch13, Proposition 2.8], so $\mathcal H^\circ$ is a finite flat $\mathcal O_K$ -group scheme. We will apply this discussion to $\mathcal H = \mathcal A[p^n]$ (cf. [Reference Grothendieck and DeligneGD72, IX, Lemma 2.2.1]).

1. (1) $\{\mathcal A[p^n]^\circ\}_{n \geq 1}$ forms a p-divisible group, $\mathcal A[p^\infty]^\circ,$ over $\mathcal O_K$ ;

2. (2) $\mathrm{Lie\,} \mathcal A[p^\infty]^\circ \simeq \mathrm{Lie\,} \mathcal A$ and, in particular, $\mathrm{dim\,} \mathcal A[p^\infty]^\circ = g = : \mathrm{dim\,} A$ ;

3. (3) if K is discretely valued, then the subspace

   \begin{align*} \mathrm{Lie\,} \mathcal A \otimes _{\mathcal O_K} C (1) \simeq \mathrm{Lie\,} \mathcal A[p^\infty]^\circ \otimes _{\mathcal O_K} C (1) \subset T_p \mathcal A[p^\infty]^\circ\otimes _{\mathbb Z_p} C \subset T_p A\otimes _{\mathbb Z_p} C\end{align*}
   given by applying (2.1) to $\mathcal A[p^\infty]^\circ$ coincides with the subspace $\mathrm{Lie\,} \mathcal A \otimes C(1)$ under the Hodge–Tate decomposition [Reference FontaineFon81, Theorem 2]
   \begin{align*} T_p A\otimes C \simeq \mathrm{Lie\,} \mathcal A\otimes C(1) \oplus H^1(A,\mathcal O_A)^{\vee}\otimes C.\end{align*}

Proof. To check part (1), we have to check that for $m,n \geq 1$ the sequence

\begin{align*} 0 \rightarrow \mathcal A[p^n]^\circ \rightarrow \mathcal A[p^{m+n}]^\circ \overset {p^n} \rightarrow \mathcal A[p^m]^\circ \rightarrow 0 \end{align*}

is exact. Using the fiber-by-fiber criterion for flatness, one sees that it suffices to check that the special fiber of this sequence is exact. This follows from the fact that $\mathcal A\otimes _{\mathcal O_K} k$ is a semi-abelian variety, so $\{\mathcal A\otimes _{\mathcal O_K}k[p^n]\}_{n \geq 1}$ is a p-divisible group.

Now, from the definitions, one sees that the formal group attached to $\mathrm{Lie\,} \mathcal A[p^\infty]^\circ$ is a subgroup of the formal group of $\mathcal A$ . Using an argument with the special fiber, as in the previous paragraph, one sees that it is equal to the formal group of $\mathcal A$ . In particular, this implies part (2).

Finally, for part (3), note that, by what we have seen above, $\mathrm{Lie\,} \mathcal A[p^\infty]^\circ \otimes _{\mathcal O_K} C (1)$ has dimension g, hence it must coincide with

\begin{align*} (T_p A[p^\infty] \otimes _{\mathbb Z_p}C(-1))^{\Gamma_K}\otimes _K C(1) \simeq \mathrm{Lie\,} \mathcal A\otimes _K C(1). \end{align*}

2.2.1.

We retain the notation introduced in § 2.1.

Let ${\mathcal G}$ be a quasi-finite flat group scheme over $\mathcal O_K,$ such that ${\mathcal G}(C)$ has p-power order. We define the slope $\mu({\mathcal G})$ of ${\mathcal G},$ following Fargues [Reference FarguesFar10]: the height of ${\mathcal G}$ is defined as $\mathrm{ht\,} {\mathcal G} = \mathrm{log}_p |{\mathcal G}(C)|$ . Let $s: {\operatorname{ Spec\,}} \mathcal O_K \rightarrow {\mathcal G}$ be the identity section. Then $\omega_{{\mathcal G}} : = s^* \Omega^1_{{\mathcal G}/\mathcal O_K}$ is a finitely presented $\mathcal O_K$ -module. Hence, by [Reference WarfieldWar70, Theorem 1] and [Reference ScholzeSch13, Proposition 2.10], it is of the form $\omega_{{\mathcal G}} = \bigoplus_{i=1}^n \mathcal O_K/a_i\mathcal O_K$ with $a_i \in \mathcal O_K$ . The degree ${\mathcal G}$ is defined as

\begin{align*}\deg {\mathcal G} = v(\mathrm{Fitt}(\omega_{{\mathcal G}})) = \sum_{i=1}^n v(a_i),\end{align*}

where $\mathrm{Fitt}(\omega_{{\mathcal G}})$ denotes the Fitting ideal of $\omega_{{\mathcal G}}$ . Finally, we set

\begin{align*} \mu({\mathcal G}) = \frac{\deg {\mathcal G}}{\mathrm{ht\,} {\mathcal G}} . \end{align*}

Using the notation of § 2.1.2, we have

\begin{align*}\mu({\mathcal G}) = \mu({\mathcal G}^\circ) \cdot \frac{\mathrm{ht\,} {\mathcal G}^\circ}{\mathrm{ht\,} {\mathcal G}},\end{align*}

as $s^* \Omega^1_{{\mathcal G}/\mathcal O_K} \simeq s^{\circ*} \Omega^1_{{\mathcal G}^\circ/\mathcal O_K},$ where $s^{\circ*}$ denotes the identity section of ${\mathcal G}^{\circ}$ .

These invariants satisfy the following properties [Reference FarguesFar10, §§ 3, 4]:

1. (1) $\deg({\mathcal G})$ is additive in short exact sequences of quasi-finite flat group schemes;

2. (2) if ${\mathcal G}$ is a finite flat group scheme with Cartier dual ${\mathcal G}^*,$ then

   \begin{align*} \mu({\mathcal G}) + \mu({\mathcal G}^*) = 1;\end{align*}

3. (3) $\mu({\mathcal G}) \in [0,1];$

4. (4) if $f:{\mathcal G} \rightarrow {\mathcal G}'$ is a map of quasi-finite, flat $\mathcal O_C$ -group schemes, with $f\otimes C$ an isomorphism, then $\mu({\mathcal G}) \leq \mu({\mathcal G}'),$ with equality if and only if f is an isomorphism;

5. (5) if $\mathcal H$ is a p-divisible group over $\mathcal O_C,$ having dimension d and height h, then for any $n \geq 1,$ $\mu(\mathcal H[p^n]) = d/h$ .

For properties (1), (3), and (4), using the decomposition in § 2.1.2, one easily reduces to the case of finite flat group schemes considered in [Reference FarguesFar10]. Note that property (1) implies that for a short exact sequence of quasi-finite flat group schemes

\begin{align*} 0 \rightarrow {\mathcal G}' \rightarrow {\mathcal G} \rightarrow {\mathcal G}'' \rightarrow 0,\end{align*}

we have

(2.2) \begin{equation} \min(\mu({\mathcal G}'), \mu({\mathcal G}'')) \leq \mu({\mathcal G}) \leq \max (\mu({\mathcal G}'), \mu({\mathcal G}'')).\end{equation}

We will use this property repeatedly. We need the following lemma, which complements property (4).

Lemma 2.2. Let $f: {\mathcal G} \rightarrow {\mathcal G}'$ be a map of finite flat group schemes over $\mathcal O_K,$ such that $f\otimes K$ is an isomorphism. Let $a \in \mathcal O_K$ be non-zero and such that

\begin{align*} a\cdot \mathcal O_{{\mathcal G}} \subset \mathcal O_{{\mathcal G}'} \subset \mathcal O_{{\mathcal G}}.\end{align*}

Then

\begin{align*}\mu({\mathcal G}') \leq \mu({\mathcal G}) + 3v(a).\end{align*}

Let $I \subset \mathcal O_{{\mathcal G}}\otimes _{\mathcal O_K}\mathcal O_{{\mathcal G}}$ and $I' \subset \mathcal O_{{\mathcal G}'}\otimes _{\mathcal O_K} \mathcal O_{{\mathcal G}'}$ denote the ideals of the diagonal, and $J \subset \mathcal O_{{\mathcal G}}$ and $J' \subset \mathcal O_{{\mathcal G}'}$ the ideals of the identity section. As multiplication by a maps $ \mathcal O_{{\mathcal G}}$ into $ \mathcal O_{{\mathcal G}'},$ multiplication by $a^2$ induces a map $I \rightarrow I'$ . Multiplying this last map by a, we obtain maps

\begin{align*} \omega_{{\mathcal G}'} \rightarrow \omega_{{\mathcal G}} = (I/I^2)/J(I/I^2) \overset{\cdot a^3}\longrightarrow (I'/I^{\prime 2})/J'(I'/I^{\prime 2}) = \omega_{{\mathcal G}'} \end{align*}

whose composite is multiplication by $a^3$ .

By [Reference FontaineFon77, Proposition 4.3, 4.5.(i)], $\mathrm{dim}_k J'/J^{\prime 2}\otimes k \leq \mathrm{ht\,} {\mathcal G},$ so a minimal set of generators for $\mathcal O_{{\mathcal G}'}$ as an $\mathcal O_K$ -algebra has size at most $\mathrm{ht\,} {\mathcal G},$ as we are assuming that $\mathcal O_{{\mathcal G}'}$ is local. It follows that $\Omega^1_{{\mathcal G}'/\mathcal O_K}$ is generated by $\mathrm{ht\,} {\mathcal G}$ elements as an $\mathcal O_{{\mathcal G}'}$ -module, so $\omega_{{\mathcal G}'}$ is generated by $\mathrm{ht\,} {\mathcal G}$ elements as an $\mathcal O_K$ -module.

Now, let $C = \mathrm{ker}\,(\omega_{{\mathcal G}'} \rightarrow \omega_{{\mathcal G}})$ be the kernel of the natural map, that is the first map above. Then $a^3\cdot C = 0,$ and the previous paragraph implies that C is a quotient of $(\mathcal O_K/a^3)^{\mathrm{ht\,} {\mathcal G}},$ so $v(\mathrm{Fitt}(C)) \leq 3 v(a) \mathrm{ht\,} {\mathcal G} $ . This implies

\begin{align*} v(\mathrm{Fitt}(\omega_{{\mathcal G}'})) \leq v(\mathrm{Fitt}(\omega_{{\mathcal G}})) + v(\mathrm{Fitt}(C)) \leq v(\mathrm{Fitt}(\omega_{{\mathcal G}})) + 3 v(a)\mathrm{ht\,} {\mathcal G} . \end{align*}

Dividing both sides by $\mathrm{ht\,} {\mathcal G}$ proves the lemma.

2.2.2.

Now let ${\mathcal G}$ be a p-divisible group over $\mathcal O_C,$ with generic fiber G, and let $H \subset G$ be a p-divisible subgroup. Denote by $\bar H[p^n] \subset {\mathcal G}$ the closure of $H[p^n]$ . We fix this notation for the remainder of this subsection. Our goal is to describe the limit $\lim_n \mu(\bar H[p^n])$ .

Proposition 2.3. The category of p-divisible groups $\mathcal J$ over $\mathcal O_C,$ equipped with a map $\mathcal J \rightarrow {\mathcal G}$ whose generic fiber factors through H, admits a final object $\mathcal H,$ whose generic fiber is H. We have

\begin{align*} {\mathscr F}(\mathcal H) = (T_pH, \mathrm{Lie\,}{\mathcal G}\otimes C(1) \cap T_pH\otimes C, f_{\mathcal H}),\end{align*}

where $f_{\mathcal H}$ is the restriction of $f_{{\mathcal G}}$ .

Proof. The triple

\begin{align*}(T_pH, \mathrm{Lie\,} {\mathcal G}\otimes C(1) \cap T_pH\otimes C, f_{{\mathcal G}}|_{\mathrm{Lie\,} {\mathcal G}\otimes C(1) \cap T_pH\otimes C})\end{align*}

defines a p-divisible group $\mathcal H$ over $\mathcal O_C,$ as ${\mathscr F}$ is an equivalence. If $h: \mathcal J \rightarrow {\mathcal G}$ is a map of p-divisible groups whose generic fiber $J \rightarrow G$ factors through H, then $T_p J \rightarrow T_p G$ factors through $T_p H,$ and one sees that the map of triples ${\mathscr F}(\mathcal J) \rightarrow {\mathscr F}({\mathcal G})$ factors through ${\mathscr F}(\mathcal H)$ . Hence, by full faithfulness of ${\mathscr F},$ h factors through $\mathcal H$ .

2.2.3.

We give another description of the p-divisible group $\mathcal H$ in the previous proposition. To do this, we need some results on $\mathcal O_C$ -modules.

We call an $\mathcal O_C$ -module M finite rank if $M \otimes C$ is finite-dimensional over C, in which case this dimension is called the rank of M. We call M bounded if it can be embedded in $\mathcal O_C^m$ for some integer m. For M a bounded $\mathcal O_C$ -module, write $M^+ = \mathrm{Hom}_{\mathcal O_C}(\mathfrak m_C, M)$ . Recall that a map of $\mathcal O_C$ -modules is called an almost isomorphism if its kernel and cokernel are killed by $\mathfrak m_C$ .

1. (1) $M^+$ is a finite free $\mathcal O_C$ -module;

2. (2) there is a natural map $M \rightarrow M^+,$ which is an isomorphism if and only if M is finitely generated over $\mathcal O_C$ ;

3. (3) for any bounded $\mathcal O_C$ -modules M,N, the natural map

   \begin{align*} M^+ \otimes _{\mathcal O_C} N^+ \rightarrow (M\otimes _{\mathcal O_C} N)^+\end{align*}
   is an isomorphism.

In general, suppose $M \subset \mathcal O_C^m$ . By intersecting M with finite free submodules of $\mathcal O_C^m$ of smaller rank, we may write M as successive extensions of bounded $\mathcal O_C$ -modules of rank 1. Since C is spherically complete, a result of Matlis [Reference MatlisMat59, Theorem 9] asserts that all such extensions split, so M is a direct sum of rank-1, bounded $\mathcal O_C$ -modules. Then part (1) follows from the rank-1 case proved above.

The map $M \rightarrow M^+$ is given by sending $m \in M$ to the restriction of the map $\mathcal O_C \rightarrow M$ sending 1 to m. If this map is an isomorphism, then M is finite free, and in particular, finitely generated, by part (1) of the lemma. If M is finitely generated, then by considering C-linear relations among the generators, one sees that M is finite free over $\mathcal O_C$ . Now any map $\mathfrak m_C \rightarrow M$ factors through $\mathfrak m_CM,$ as $\mathfrak m_C = \mathfrak m_C^2$ . As M is finite free, this implies $M \simeq M^+$ .

Finally, we show part (3). Since $M^+, N^+$ are finite free, the composite

\begin{align*} M^+ \otimes _{\mathcal O_C} N^+ \rightarrow (M\otimes _{\mathcal O_C} N)^+ \rightarrow (M^+\otimes _{\mathcal O_C} N^+)^+\end{align*}

is an isomorphism, and since both maps are injective, they are isomorphisms.

Lemma 2.5. With the notation of Proposition 2.3, § 2.2.2 for each $i \geq 1,$ there is a natural map

(2.3) \begin{equation}\widetilde{\mathcal {O}}(\bar H[p^i]) : = \underrightarrow {\mathrm{lim}}_j \mathcal O(\bar H[p^{i+j}]/\bar H[p^j]) \rightarrow \mathcal O(\mathcal H[p^i])\end{equation}

and this map is an almost isomorphism.

Proof. The transition maps in the limit correspond to the maps

\begin{align*}\bar H[p^{i+j}]/\bar H[p^j] \rightarrow \bar H[p^{i+j-1}]/\bar H[p^{j-1}]\end{align*}

induced by multiplication by p on ${\mathcal G}$ . The map $\mathcal H \rightarrow {\mathcal G}$ induces maps $\mathcal H[p^i] \rightarrow \bar H[p^i],$ and hence maps

\begin{align*} \mathcal H[p^i] \simeq \mathcal H[p^{i+j}]/\mathcal H[p^j] \rightarrow \bar H[p^{i+j}]/\bar H[p^j].\end{align*}

Passing to functions gives rise to the map in the proposition, and it remains to show this map is an almost isomorphism.

Since the functor ${\mathscr F}$ is compatible with base change in the field C, we may assume that C is spherically complete, and that the valuation $v: C\backslash\{0\} \rightarrow \mathbb R$ is surjective. Note that (2.3) is an isomorphism after inverting p, as the generic fiber of $\mathcal H$ is the p-divisible subgroup $H \subset G,$ and that $\mathcal O(\mathcal H[p^i])$ is a finite free $\mathcal O_C$ -module. Hence, by Lemma 2.4, (2.3) induces a map

(2.4) \begin{equation} \widetilde{\mathcal {O}}(\bar H[p^i])^+\rightarrow \mathcal O(\mathcal H[p^i])^+ \simeq \mathcal O(\mathcal H[p^i]) \end{equation}

and $\widetilde{\mathcal {O}}(\bar H[p^i])^+$ is a finite free $\mathcal O_C$ -module, which is almost isomorphic to $ \widetilde{\mathcal {O}}(\bar H[p^i])$ . Thus, it suffices to show that (2.4) is an isomorphism.

By Lemma 2.4(2), $\mathcal O(\bar H[p^i])$ is a finite free $\mathcal O_C$ -module, as it is finitely generated, and this implies that ${\mathcal G}$ induces the structure of a finite flat group scheme on $\bar H[p^i]$ . Hence, $\bar H[p^{i+j}]/\bar H[p^j]$ is also a finite flat group scheme, and $\mathcal O(\bar H[p^{i+j}]/\bar H[p^j])$ is a Hopf algebra. Thus, $ \widetilde{\mathcal {O}}(\bar H[p^i])$ is also a Hopf algebra, and the sequence of group schemes $\{{\operatorname{ Spec\,}} \widetilde{\mathcal {O}}(\bar H[p^i])\}_{i \geq 1}$ satisfies all the axioms of a p-divisible group, except possibly the condition that the elements of the sequence are finite flat.

Now using Lemma 2.4(1), one sees that $\{{\operatorname{ Spec\,}} \widetilde{\mathcal {O}}(\bar H[p^i])^+\}_{i \geq 1}$ is a sequence of finite flat group schemes, and we claim that it forms a p-divisible group. Indeed, this sequence inherits the other properties of a p-divisible group from the sequence $\{{\operatorname{ Spec\,}} \widetilde{\mathcal {O}}(\bar H[p^i])\}_{i \geq 1}$ : the maps $\widetilde{\mathcal {O}}(\bar H[p^i])^+ \rightarrow \widetilde{\mathcal {O}}(\bar H[p^{i-1}])^+$ are surjective, as they are almost-surjective maps between finite free $\mathcal O_C$ -modules. This implies the maps between the corresponding finite flat group schemes are closed immersions. The same argument shows that the maps

\begin{align*} {\operatorname{ Spec\,}} \widetilde{\mathcal {O}}(\bar H[p^{i+1}])^+/{\operatorname{ Spec\,}} \widetilde{\mathcal {O}}(\bar H[p])^+ \rightarrow {\operatorname{ Spec\,}} \widetilde{\mathcal {O}}(\bar H[p^i])^+ \end{align*}

induced by multiplication by p are isomorphisms. Thus, $\{{\operatorname{ Spec\,}} \widetilde{\mathcal {O}}(\bar H[p^i])^+\}_{i \geq 1}$ forms a p-divisible group, which we denote $\widetilde{\bar H}$ .

The maps in (2.4) induce a map of p-divisible groups $\iota: \mathcal H \rightarrow \widetilde{\bar H}$ . By construction, there is a map $\widetilde{\bar H} \rightarrow {\mathcal G}$ which is an isomorphism onto H on generic fibers. Hence, by the universal property for $\mathcal H,$ this map factors through $\mathcal H$ and induces an inverse to $\iota$ . It follows that (2.4) is an isomorphism.

Proposition 2.6. With the notation of § 2.2.2, the sequence $\{\mu(\bar H[p^i])\}_{i \geq 1}$ is monotone non-increasing. We have

\begin{align*} \lim_i \mu(\bar H[p^i]) = \frac{\mathrm{dim\,}\mathcal H}{\mathrm{ht\,} \mathcal H} = \frac{\mathrm{dim} (T_pH\otimes C \cap \mathrm{Lie\,} {\mathcal G}\otimes C(1))}{\mathrm{ht\,} H}.\end{align*}

Proof. The maps $\bar H[p^{i+1}]/\bar H[p^i] \rightarrow \bar H[p^i]/\bar H[p^{i-1}],$ induced by multiplication by p on H, are isomorphisms on generic fibers. Hence, by property (4) in § 2.2.1, the sequence $\{\mu(\bar H[p^{i+1}]/\bar H[p^i])\}_{i \geq 1}$ is monotone non-increasing. It then follows from property (1) in § 2.2.1 that $\{\mu(\bar H[p^i])\}_{i \geq 1}$ is monotone non-increasing, and we have

\begin{align*}\lim_i \mu(\bar H[p^i]) = \lim_i \mu(\bar H[p^{i+1}]/\bar H[p^i]).\end{align*}

The second equality in the proposition follows from Proposition 2.3. Moreover, by § 2.2.1, we have $ {\mathrm{dim\,} \mathcal H}/{\mathrm{ht\,} \mathcal H} = \mu(\mathcal H[p])$ . Thus, it remains to show that

\begin{align*} \lim_i \mu(\bar H[p^{i+1}]/\bar H[p^i]) = \mu(\mathcal H[p]). \end{align*}

By Lemma 2.5, for any $a \in \mathfrak m_C,$ we have $a\cdot \mathcal O(\mathcal H[p]) \subset \mathcal O(\bar H[p^{i+1}]/\bar H[p^i])$ for i sufficiently large. It follows from Lemma 2.2 that for any $\varepsilon \gt 0$ there exists i such that

\begin{align*} \mu(\bar H[p^{i+1}]/\bar H[p^i]) \leq \mu(\mathcal H[p]) +\varepsilon. \end{align*}

Thus, $ \lim_i \mu(\bar H[p^{i+1}]/\bar H[p^i]) \leq \mu(\mathcal H[p]),$ and the opposite inequality follows from property (4) in § 2.2.1, as $\mathcal H[p]$ maps to $\bar H[p^{i+1}]/\bar H[p^i]$ .

2.2.4.

We need a generalization of Proposition 2.6.

Proposition 2.7. Let $\mathcal A$ be a semi-abelian scheme over $\mathcal O_C,$ such that $A = \mathcal A\otimes C$ is an abelian variety. Let $H \subset A[p^\infty]$ be a p-divisible subgroup, and $\bar H[p^i]$ the closure of $H[p^i]$ in $\mathcal A$ . Then we have

\begin{align*} \lim_i \mu(\bar H[p^i]) = \frac{\mathrm{dim} (T_pH\otimes C \cap \mathrm{Lie\,} \mathcal A\otimes C(1))}{\mathrm{ht\,}H}.\end{align*}

Let $\bar H[p^i]$ denote the closure of $H[p^i]$ in $\mathcal A$ . Set $\bar H^\circ = \lim_i \bar H[p^i]^\circ$ as an fppf sheaf, and denote by $H^\circ$ its generic fiber. Let $H^{\circ\circ} \subset H^\circ$ be the p-divisible part of $H^\circ,$ so that $H^\circ/H^{\circ\circ}$ is finite, and write $\bar H^{\circ\circ}[p^i]$ for the closure of $H^{\circ\circ}[p^i]$ in $\mathcal A$ .

Using the definition of $\mu,$ as in § 2.2.1 and property (1) in 2.2.1, we obtain

(2.5) \begin{equation} \lim_i \mu(\bar H[p^i]) = \frac{\mathrm{ht\,} H^{\circ\circ}}{\mathrm{ht\,} H}{\lim_i \mu(\bar H[p^i]^{\circ})} = \frac{\mathrm{ht\,} H^{\circ\circ}}{\mathrm{ht\,} H}{\lim_i \mu(\bar H^{\circ\circ}[p^i]).}\end{equation}

We now apply Proposition 2.6 to $H^{\circ\circ} \subset \mathcal A[p^\infty]^\circ \otimes C,$ to obtain

(2.6) \begin{equation} \lim_i \mu(\bar H^{\circ\circ}[p^i]) = \frac{\mathrm{dim} (T_p H^{\circ\circ}\otimes C \cap \mathrm{Lie\,} \mathcal A \otimes C(1))}{\mathrm{ht\,} H^{\circ\circ}}.\end{equation}

From the definitions, one finds

\begin{align*}T_pH^{\circ\circ} = \mathrm{Hom}(\mathbb Q_p/\mathbb Z_p, H^{\circ\circ}(C)) \simeq \mathrm{Hom}(\mathbb Q_p/\mathbb Z_p, H^\circ(C)) = T_pH \cap T_p \mathcal A[p^\infty]^\circ \subset T_p A, \end{align*}

and on the other hand, from Lemma 2.1(3),

\begin{align*} \mathrm{Lie\,} \mathcal A\otimes C(1) \subset T_p\mathcal A[p^\infty]^\circ\otimes C,\end{align*}

which implies

(2.7) \begin{equation} T_p H^{\circ\circ}\otimes C \cap \mathrm{Lie\,} \mathcal A \otimes C(1) = T_p H \otimes C \cap \mathrm{Lie\,} \mathcal A \otimes C(1).\end{equation}

Combining (2.5), (2.6), and (2.7), we obtain the lemma.

2.3.1.

Let $\mathsf{K}$ be a number field with ring of integers $\mathcal O_\mathsf{K},$ $\bar{\mathsf{K}}$ an algebraic closure of $\mathsf{K},$ and $\Gamma_\mathsf{K} = \mathrm{Gal}(\bar{\mathsf{K}}/\mathsf{K})$ . Let $\mathcal H$ be a quasi-finite flat group scheme over $\mathcal O_\mathsf{K}$ . For p a prime, write $\mathcal H_p = \mathcal H[p^\infty] \subset \mathcal H$ for the p-power torsion subgroup of $\mathcal H,$ so that $\mathcal H \simeq \prod_p \mathcal H_p$ . When $\mathcal H = \mathcal H_p$ for some p, we set

\begin{align*} \nu(\mathcal H) = \frac 1 {[\mathsf{K}:\mathbb Q]} \sum_{w|p} [\mathsf{K}_{w}:\mathbb Q_p]\mu_w(\mathcal H),\end{align*}

where for $w|p$ a prime of $\mathsf{K}$ , we write $\mu_w(\mathcal H) = \mu(\mathcal H|_{\mathcal O_{\mathsf{K}_w}})$ . For general $\mathcal H$ we set

\begin{align*} \nu(\mathcal H) = \sum_p \frac{\mathrm{log}|\mathcal H_p(\bar{\mathsf{K}})|}{\mathrm{log}|\mathcal H(\bar{\mathsf{K}})|} \nu(\mathcal H_p). \end{align*}

Note that the sum of the coefficients ${\mathrm{log}|\mathcal H_p(\bar{\mathsf{K}})|}/{\mathrm{log}|\mathcal H(\bar{\mathsf{K}})|}$ is 1. We call $ \nu(\mathcal H)$ the global slope of $\mathcal H$ . Set $\omega_{\mathcal H/\mathcal O_\mathsf{K}} = s^*\Omega^1_{\mathcal H/\mathcal O_\mathsf{K}},$ and denote by $|\omega_{\mathcal H/\mathcal O_\mathsf{K}}\!|$ the order of $\omega_{\mathcal H/\mathcal O_\mathsf{K}}$ .

Lemma 2.8. We have

\begin{align*} \nu(\mathcal H) = \frac{\mathrm{log}|\omega_{\mathcal H/\mathcal O_{\mathsf{K}}}\!|}{[K:\mathbb Q]\mathrm{log}|\mathcal H(\bar{\mathsf{K}})|}.\end{align*}

Proof. Since $\omega_{\mathcal H/\mathcal O_{\mathsf{K}}} = \prod_p \omega_{\mathcal H_p/\mathcal O_{\mathsf{K}}},$ we have

\begin{align*} \frac{\mathrm{log}|\omega_{\mathcal H/\mathcal O_{\mathsf{K}}}\!|}{\mathrm{log}|\mathcal H(\bar{\mathsf{K}})|}= \sum_p \frac{\mathrm{log}|\mathcal H_p(\bar{\mathsf{K}})|}{\mathrm{log}|\mathcal H(\bar{\mathsf{K}})|} \frac{\mathrm{log}|\omega_{\mathcal H_p/\mathcal O_{\mathsf{K}}}\!|}{\mathrm{log}|\mathcal H_p(\bar{\mathsf{K}})|}, \end{align*}

so it suffices to consider the case when $\mathcal H = \mathcal H_p$ for some p, which we now assume.

For $w|p$ a prime of $\mathsf{K},$ as in § 2.2.1, write

\begin{align*} \omega_{\mathcal H/\mathcal O_{\mathsf{K}_w}} : = \omega_{\mathcal H/\mathcal O_{\mathsf{K}}}\otimes _{\mathcal O_{\mathsf{K}}}\mathcal O_{\mathsf{K}_{w}} = \bigoplus_{i=1}^n \mathcal O_{\mathsf{K}_w}/a_i\mathcal O_{\mathsf{K}_w}.\end{align*}

Then

\begin{align*} \mathrm{log}_p |\omega_{\mathcal H/\mathcal O_{\mathsf{K}_w}}\!| = [\mathsf{K}_w:\mathbb Q_p]\bigg(\sum_{i=1}^n v(a_i)\bigg) = [\mathsf{K}_w:\mathbb Q_p]\deg \mathcal H|_{\mathcal O_{\mathsf{K}_w}}.\end{align*}

Dividing both sides by $\mathrm{ht\,} \mathcal H = \mathrm{log}_p |\mathcal H(\bar{\mathsf{K}})|$ and summing over $w|p$ gives

\begin{align*} \frac{\mathrm{log}|\omega_{\mathcal H/\mathcal O_{\mathsf{K}}}\!|}{\mathrm{log}|\mathcal H(\bar{\mathsf{K}})|} = \frac{\mathrm{log}_p|\omega_{\mathcal H/\mathcal O_{\mathsf{K}}}\!|}{\mathrm{log}_p|\mathcal H(\bar{\mathsf{K}})|}= \sum_{w|p} [\mathsf{K}_{w}:\mathbb Q_p]\mu_w(\mathcal H) = [K:\mathbb Q]\nu(\mathcal H). \end{align*}

2.3.2.

The following lemma shows that the invariant $\nu$ satisfies the analogue of the same properties in § 2.2.1. We use these repeatedly in what follows. It will be convenient to define the global degree $\deg \mathcal H$ by $\deg \mathcal H = \nu(\mathcal H)\mathrm{log}|\mathcal H(\bar{\mathsf{K}})|$ .

1. (1) For a quasi-finite flat group scheme $\mathcal H/\mathcal O_{\mathsf{K}},$ we have $\nu(\mathcal H) \in [0,1]$ .

2. (2) If $\mathcal H \rightarrow \mathcal H'$ is a map of quasi-finite flat group schemes which is an isomorphism over $\mathsf{K},$ then

3. (3) We have $\nu(\mathcal H) \leq \nu(\mathcal H')$ .

4. (4) The function taking $\mathcal H$ to $\deg \mathcal H$ is additive in short exact sequences.

5. (5) If

   \begin{align*} 0 \rightarrow \mathcal H' \rightarrow \mathcal H \rightarrow \mathcal H'' \rightarrow 0 \end{align*}
   is a short exact sequence of quasi-finite flat group schemes over $\mathcal O_{\mathsf{K}}$ , then
   \begin{align*} \min(\nu(\mathcal H'), \nu(\mathcal H'')) \leq \nu(\mathcal H) \leq \max (\nu(\mathcal H'), \nu(\mathcal H'')).\end{align*}

2.3.3.

For an abelian variety A over $\mathsf{K},$ we denote by h(A) the Faltings’ height of A (see [Reference FaltingsFal83, p. 354]). Suppose that A extends to a semi-abelian scheme $\mathcal A$ over $\mathcal O_\mathsf{K}$ . The following lemma is due to Faltings.

Lemma 2.10. Let $\phi: \mathcal A \rightarrow \mathcal A'$ be an isogeny of semi-abelian $\mathcal O_\mathsf{K}$ -schemes, with kernel $\mathcal H$ , and write $A' = \mathcal A\otimes \mathsf{K}$ . Then

\begin{align*} h(A') - h(A) = \mathrm{log} \deg (\phi)(\tfrac{1}{2} - \nu(\mathcal H)).\end{align*}

Proof. By [Reference FaltingsFal83, Lemma 5], we have

\begin{align*} h(A') - h(A) = \frac 1 2 \mathrm{log} \deg (\phi) - \frac 1 {[\mathsf{K}:\mathbb Q]} \mathrm{log} (|\omega_{\mathcal H/\mathcal O_\mathsf{K}}\!|).\end{align*}

The lemma now follows from Lemma 2.8.

2.3.4.

Let $\mathcal A$ be as above, and $H \subset A_{\bar{\mathsf{K}}}[p^\infty]$ a finite subgroup, defined over a finite extension $\mathsf{K}' \subset \bar{\mathsf{K}}$ . For $w'|p$ a prime of $\mathsf{K}',$ we write $\mu_{w'}(H) = \mu_{w'}(\mathcal H)$ and $\nu(H) = \nu(\mathcal H),$ where $\mathcal H$ denotes the closure of H in $\mathcal A$ . Similarly, we sometimes write $\deg(H)$ for $\deg(\mathcal H)$ .

For $\sigma \in \Gamma_\mathsf{K},$ we denote by $\sigma(H) \subset A$ the subgroup $\sigma^* H \subset \sigma^*A \simeq A$ . Note that, with this notation, $\sigma(H)(\bar{\mathsf{K}}) = \sigma(H(\bar{\mathsf{K}})) \subset A(\bar{\mathsf{K}})$ . We denote by $d\sigma$ the Haar measure of volume 1 on the compact group $\Gamma_\mathsf{K}$ . For each place $w|p$ of $\mathsf{K},$ we extend w to a place of $\bar{\mathsf{K}},$ again denoted w.

Lemma 2.11. Let $H \subset A_{\bar{\mathsf{K}}}[p^\infty]$ be a closed, finite subgroup. Then

\begin{align*} \nu(H) = \sum_{w|p} \frac{[\mathsf{K}_w:\mathbb Q_p]}{[\mathsf{K}:\mathbb Q]}\int_{\Gamma_\mathsf{K}}\mu_w(\sigma(H))\,d\sigma,\end{align*}

where w runs over place of $\mathsf{K}$ dividing p.

Proof. Let $\mathsf{K}' \subset \bar{\mathsf{K}}$ be a finite Galois extension of $\mathsf{K},$ such that H is defined over $\mathsf{K}'$ . For $\sigma \in \mathrm{Gal}(\mathsf{K}'/\mathsf{K})$ , we have

\begin{align*}\mu_{\sigma(w)}(H) = \mu_{w}(\sigma^*H) =\mu_{w}(\sigma(H)).\end{align*}

Thus, $\nu(H)$ is given by

\begin{align}\frac 1 {[\mathsf{K}':\mathbb Q]} \sum_{w'|p} [\mathsf{K}'_{w'}:\mathbb Q_p]\mu_{w'}(H)&= \frac 1 {[\mathsf{K}:\mathbb Q]} \sum_{w|p} \frac{[\mathsf{K}_w:\mathbb Q_p]}{[\mathsf{K}':\mathsf{K}]}\sum_{\sigma \in \mathrm{Gal}(\mathsf{K}'/\mathsf{K})} \mu_w(\sigma(H)) \nonumber\\[5pt]&= \sum_{w|p} \frac{[\mathsf{K}_w:\mathbb Q_p]}{[\mathsf{K}:\mathbb Q]}\int_{\Gamma_\mathsf{K}}\mu_w(\sigma(H))d\sigma,\end{align}

where, as usual, w (respectively w’) runs over primes of $\mathsf{K}$ (respectively $\mathsf{K}'$ ) dividing p.

2.4.1.

Let C be the completion of an algebraic closure of $\mathbb Q_p$ . It will be convenient to fix an isomorphism of $\mathbb C \simeq C$ . We continue to use the notation of § 2.3.1. For any embedding $w: \bar{\mathsf{K}} \rightarrow C,$ we denote by $\tilde w: \bar{\mathsf{K}} \rightarrow \mathbb C$ the corresponding embedding. Fix an embedding $w_0: \bar{\mathsf{K}} \rightarrow C$ .

For any ring R and a finite free R-module M, we denote by $M^\otimes $ the direct sum $\bigoplus_{m,n} M^{\otimes m} \otimes M^{*\otimes n}$ .

Let A be an abelian variety over $\mathsf{K}$ . We consider the complex points $A(\mathbb C),$ via the embedding $\tilde w_0|_\mathsf{K}$ . The Betti cohomology $H^1(A(\mathbb C),\mathbb Q)$ carries a Hodge structure, and we denote by $G \subset \mathrm{GL}(H^1(A(\mathbb C),\mathbb Q))$ the Mumford–Tate group of A, defined as the subgroup fixing all Hodge cycles in $H^1(A(\mathbb C),\mathbb Q)^\otimes $ of bidegree (0,0). It is a connected, reductive algebraic $\mathbb Q$ -group, which depends on $w_0|_K$ ; groups corresponding to different embeddings may be inner forms [Reference SerreSer94, § 2].

Let $\{s_{\alpha}\}_\alpha \subset H^1(A(\mathbb C),\mathbb Q)^\otimes $ be a finite collection of Hodge cycles whose stabilizer is G. Using our fixed embedding $\tilde w_0$ , we have a canonical isomorphism

(2.9) \begin{equation} H^1_{{\mathrm{dR}}}(A_{\bar{\mathsf{K}}})\otimes _{\bar{\mathsf{K}},\tilde w_0} \mathbb C \simeq H^1(A(\mathbb C),\mathbb C),\end{equation}

where $H^1_{{\mathrm{dR}}}$ denotes the algebraic de Rham cohomology, and for each prime p canonical isomorphisms

(2.10) \begin{equation} H^1(A_{\bar{\mathsf{K}}}, \mathbb Q_p) \underset{\tilde w_0}\simeq H^1(A_{\mathbb C}, \mathbb Q_p) \simeq H^1(A(\mathbb C),\mathbb Q_p).\end{equation}

By a result of Deligne [Reference Deligne, Milne, Ogus and ShihDMO⁺82, I, Corollaries 2.7, 2.9, 2.11], for all $\alpha,$ there exists $s_{\alpha,{\mathrm{dR}}} \in H^1_{{\mathrm{dR}}}(A_{\bar{\mathsf{K}}})^\otimes $ and, for each p, $s_{\alpha,p} \in H^1(A_{\bar{\mathsf{K}}}, \mathbb Q_p)^\otimes $ mapping to $s_{\alpha}$ under the above isomorphisms. Moreover, for any embedding w, if we use $\tilde w$ to define isomorphisms analogous to (2.9) and (2.10), then $s_{\alpha,{\mathrm{dR}}}$ and $s_{\alpha,p}$ map to a Hodge cycle $s_{\alpha,w} \in H^1(A(\mathbb C),\mathbb Q)^\otimes $ .

Let $\mathsf{K}' \subset \bar{\mathsf{K}}$ be the smallest extension of $\mathsf{K}$ such that $s_{\alpha,{\mathrm{dR}}} \in H^1_{{\mathrm{dR}}}(A_{\mathsf{K}'})^\otimes $ for all $\alpha$ . It follows from the above that $s_{\alpha,w}$ depends only on $w|_{\mathsf{K}'},$ and hence that $\Gamma_{\mathsf{K}'}$ acts trivially on $s_{\alpha,p}$ for all p. Thus, the action of $\Gamma_\mathsf{K}$ on $H^1(A_{\bar{\mathsf{K}}}, \mathbb Q_p)$ induces a map

\begin{align*} \varrho_A: \Gamma_{\mathsf{K}'} \rightarrow G(\mathbb Q_p).\end{align*}

Recall that the Mumford–Tate conjecture for A asserts that the image of $\varrho_A$ is Zariski dense. In particular, we have seen that $G \otimes \mathbb Q_p$ does not depend on $w_0$ and that G has a canonical $\mathsf{K}'$ -form, the stabilizer of $\{s_{\alpha,{\mathrm{dR}}}\}_\alpha \subset H^1_{{\mathrm{dR}}}(A_{\mathsf{K}'})^\otimes $ .

We call $\mathsf{K}'$ the reflex field of A. Although this terminology is not standard, it seems appropriate. Note that for any Hodge cycle s on A, of bidegree 0, $\varrho_A(\Gamma_{\mathsf{K}'})$ fixes $s_p,$ for all p, and hence $s_{{\mathrm{dR}}}$ is defined over $\mathsf{K}'$ . Thus, the definition of $\mathsf{K}'$ does not depend on the choice of $\{s_{\alpha}\}_{\alpha}$ . For the rest of this subsection we assume that $\mathsf{K}$ contains the reflex field of A.

2.4.2.

Let $H^1_{{\mathrm{HT}}}(A) = \mathrm{gr}^\bullet H^1_{{\mathrm{dR}}}(A_\mathsf{K}),$ the associated grading of the de Rham cohomology of A. We also fix an isomorphism $\mathbb Z_p \simeq \mathbb Z_p(1)$ . For any w, the Hodge–Tate decomposition gives an isomorphism

(2.11) \begin{equation}H^1(A_{\bar{\mathsf{K}}},\mathbb Q_p)\otimes _{\mathbb Q_p} C \simeq H^1(A_{\bar{\mathsf{K}}_w},\mathbb Q_p)\otimes _{\mathbb Q_p} C \simeq H^1_{{\mathrm{HT}}}(A)\otimes _{\mathsf{K},w} C.\end{equation}

This induces a grading on the left-hand side, and we denote by $\boldsymbol{\mu}_w$ the inverse of the corresponding $\mathrm{GL}(H^1(A_{\bar{\mathsf{K}}}, \mathbb Q_p))$ -valued cocharacter. That is, $\boldsymbol{\mu}_w(z)$ acts as $z^{-i}$ in degree i. This is the Hodge–Tate cocharacter of the $\Gamma_{\mathsf{K}_w}$ -representation $H^1(A_{\bar{\mathsf{K}}_w},\mathbb Q_p),$ where we use the convention that the cyclotomic character has Hodge–Tate weight 1.

We denote by $\boldsymbol{w}: \mathbb G_m \rightarrow G$ the weight cocharacter. For $y \in \mathbb G_m,$ $\boldsymbol{w}(y)$ acts via the scalar y on $H^1(A(\mathbb C), \mathbb Q)$ . For two G-valued cocharacters $\mu_1,\mu_2,$ we write $\mu_1 \sim \mu_2$ if $\mu_1$ and $\mu_2$ are conjugate over some sufficiently large field, containing the fields of definition of $\mu_1$ and $\mu_2$ .

1. (1) The cocharacters $\boldsymbol{\mu}_w$ are G-valued.

2. (2) For any $\sigma \in \mathrm{Aut}(C),$ we have $\boldsymbol{\mu}_{\sigma\circ w} \sim \sigma^*(\boldsymbol{\mu}_w)$ .

3. (3) Let c on $\mathbb C \simeq C$ denote complex conjugation. Then $\boldsymbol{\mu}_{c\circ w}\ \sim \boldsymbol{w}^{-1}\cdot \boldsymbol{\mu}_w^{-1}$ .

4. (4) The images of the G(C)-conjugates of the cocharacters $\boldsymbol{\mu}_w,$ where w runs over all embeddings $\bar{\mathsf{K}} \rightarrow C,$ generate the group G.

Proof. Recall the p-adic period ring $B_{{\mathrm{dR}}}$ constructed from the field C. The isomorphism (2.11) arises from the p-adic de Rham comparison isomorphism

(2.12) \begin{equation}H^1(A_{\bar{\mathsf{K}}_w},\mathbb Q_p)\otimes _{\mathbb Q_p} B_{{\mathrm{dR}}} \simeq H^1_{{\mathrm{dR}}}(A)\otimes _{\mathsf{K},w} B_{{\mathrm{dR}}}\end{equation}

by passing to $\mathrm{gr}^0,$ and using the isomorphism $C \simeq C(1)$ . Here, the embedding $\mathsf{K} \rightarrow B_{{\mathrm{dR}}}$ is the unique embedding lifting w. A result of Blasius and Wintenberger [Reference BlasiusBla94] asserts that (2.12) takes $s_{\alpha,p}$ to $s_{\alpha,{\mathrm{dR}}}$ . Hence, one sees that (2.11) takes $s_{\alpha,p}$ to the image of $s_{\alpha,{\mathrm{dR}}},$ $s_{\alpha,{\mathrm{HT}}} \in \mathrm{gr}^0(H^1_{{\mathrm{HT}}}(A)^\otimes )$ . In particular, the image of $\boldsymbol{\mu}_w$ acts trivially on $s_{\alpha,p},$ so $\boldsymbol{\mu}_w$ is G-valued.

We now consider two isomorphisms

\begin{align*} H^1(A_{\bar{\mathsf{K}}},\mathbb Q_p)\otimes _{\mathbb Q_p} C \simeq H^1_{{\mathrm{HT}}}(A)\otimes _{\mathsf{K},\sigma\circ w} C. \end{align*}

The first is obtained by composing (2.11) with $1\otimes \sigma,$ and the second is the Hodge–Tate isomorphism (2.11) with $\sigma\circ w$ in place of w. By what we have just seen, each of these isomorphisms takes $s_{\alpha,p}$ to $s_{\alpha,{\mathrm{HT}}}$ . Hence, they differ by an element of G(C). The gradings induced on $H^1(A_{\bar{\mathsf{K}}},\mathbb Q_p)\otimes _{\mathbb Q_p} C$ by these isomorphisms correspond to the cocharacters $\sigma^*(\boldsymbol{\mu}_w)$ and $\boldsymbol{\mu}_{\sigma\circ w}$ , respectively, which proves part (2).

Next we show part (3). Consider the isomorphisms

(2.13) \begin{equation} H^1(A_{\bar{\mathsf{K}}},\mathbb Q_p)\otimes _{\mathbb Q_p} \mathbb C \simeq H^1(A(\mathbb C),\mathbb Q)\otimes _\mathbb Q \mathbb C \simeq H^1_{{\mathrm{HT}}}(A)\otimes _{\mathsf{K},\tilde w} \mathbb C, \end{equation}

where the first isomorphism uses the embedding $\tilde w,$ and the second is given by the Hodge decomposition. As we remarked above, the first isomorphism takes $s_{\alpha,p}$ to an element of $H^1(A(\mathbb C),\mathbb Q)^\otimes $ which maps to $s_{\alpha, {\mathrm{dR}}}$ under the (classical) de Rham isomorphism. Thus, (2.13) takes $s_{\alpha,p}$ to $s_{\alpha,{\mathrm{HT}}}$ . Denote by $\boldsymbol{\mu}'_w$ the inverse of the cocharacter corresponding to the grading on $H^1(A_{\bar{\mathsf{K}}},\mathbb Q_p)\otimes _{\mathbb Q_p} \mathbb C$ induced by (2.13). As above, $\boldsymbol{\mu}'_w \sim \boldsymbol{\mu}_w$ . Thus, using part (2), we have

\begin{align*} \boldsymbol{\mu}_{c\circ w} \sim c^*(\boldsymbol{\mu}_w) \sim c^*(\boldsymbol{\mu}'_w). \end{align*}

As $\boldsymbol{\mu}^{\prime-1}_w$ corresponds to the Hodge decomposition on $H^1(A(\mathbb C),\mathbb Q)\otimes _\mathbb Q \mathbb C,$ we have $c^*(\boldsymbol{\mu}'_w)\boldsymbol{\mu}'_w = \boldsymbol{w}^{-1},$ so that

\begin{align*} \boldsymbol{\mu}_{c\circ w} \sim c^*(\boldsymbol{\mu}'_w) = \boldsymbol{w}^{-1} \boldsymbol{\mu}^{\prime-1}_w \sim \boldsymbol{w}^{-1} \boldsymbol{\mu}_w^{-1},\end{align*}

which proves part (3).

Finally, we show part (4). As $\boldsymbol{\mu}'_w \sim \boldsymbol{\mu}_w,$ it suffices to prove part (4) with $\boldsymbol{\mu}'_w$ in place of $\boldsymbol{\mu}_w$ . As in part (2), for any $\sigma \in \mathrm{Aut}(\mathbb C),$ we have $\boldsymbol{\mu}'_{\sigma\circ w_0} \sim \sigma^*(\boldsymbol{\mu}'_{w_0}),$ so it suffices to show that the images of $\sigma^*(\boldsymbol{\mu}'_{w_0})$ generate G when $\sigma$ runs over $ \mathrm{Aut}(\mathbb C)$ . This is one possible definition of the Mumford–Tate group. To see that it agrees with our definition, let $H \subset G$ be the subgroup generated by the $\sigma^*(\boldsymbol{\mu}'_{w_0})$ . Then H is a $\mathbb Q$ -subgroup. The element $\boldsymbol{\mu}'_{w_0}(-1)\boldsymbol{w}(\sqrt{-1}) \in G(\mathbb C)$ is a Cartan involution, which is contained in $H(\mathbb C)$ . Thus, $H_{\mathbb R}/\boldsymbol{w}(\mathbb G_m)$ has a compact form, which implies that H is reductive. Any $t \in H^1(A(\mathbb C),\mathbb Q)^\otimes $ fixed by H is a Hodge cycle, as $\boldsymbol{\mu}_{w_0}$ and $c^*(\boldsymbol{\mu}_{w_0})$ act trivially on t, so H and G fix the same tensors, which implies that $H=G$ as H is reductive.

2.4.3.

Let V be a G-representation over C, $L \subset V$ a non-zero subspace, and $\boldsymbol{\mu}$ a cocharacter of G. We call V pure if the action of $\mathbb G_m$ on V via the cocharacter $\boldsymbol{w}: \mathbb G_m \rightarrow G$ has only one weight.

Let $\mathrm{Fil}^{\boldsymbol{\mu}}$ denote the decreasing filtration on V induced by $\boldsymbol{\mu}$ . If L is 1-dimensional, we denote by $\langle \boldsymbol{\mu}, L \rangle$ the least integer n such that $\mathrm{Fil}^{\boldsymbol{\mu}, n+1} L = 0$ . In general, we set $\langle \boldsymbol{\mu}, L \rangle = {\langle \boldsymbol{\mu}, \wedge^{\mathrm{dim\,} L} L \rangle}/{\mathrm{dim}_C L},$ where we think of $\wedge^{\mathrm{dim} L} L$ as a subspace of the G-representation $\wedge^{\mathrm{dim} L} V$ .

We set $\langle \boldsymbol{\mu}, L \rangle_G = \min_{g \in G(C)} \langle \boldsymbol{\mu}, g\cdot L \rangle$ . Note that we have $\langle \boldsymbol{\mu}, L \rangle_G = \langle \boldsymbol{\mu}, g\cdot L \rangle$ for g in a Zariski-dense subset of G.

Lemma 2.14. Let V be a pure G-representation, and $L \subset V$ and w as above. Then we have

(2.14) \begin{equation} \langle \boldsymbol{\mu}_w, L \rangle_G+ \langle \boldsymbol{\mu}_{c\circ w}, L \rangle_G\leq -\langle \boldsymbol{w}, L \rangle.\end{equation}

Moreover, if equality holds, then L is stable by the image of $\boldsymbol{\mu}_w$ .

Now let $\boldsymbol{\mu}'_w$ be as in the proof of Lemma 2.12. Write $V = \bigoplus_{p+q = -m} V^{p,q}$ , where $\boldsymbol{\mu}'_w(z)$ and $\boldsymbol{\mu}'_{c\circ w}(z)$ act on $V^{p,q}$ via $z^{p}$ and $z^{q}$ , respectively, and m is the weight of $\boldsymbol{w}$ on V. Let $p_0$ be the minimum integer such that $V^{p_0,q} \neq 0$ for some q, and define $q_0$ similarly. We have

\begin{align*} \langle \boldsymbol{\mu}'_w, L \rangle_G+ \langle c\circ\boldsymbol{\mu}'_w, L \rangle_G = p_0 + q_0 \leq -m = -\langle \boldsymbol{w}, L \rangle.\end{align*}

Here we are using that the G-translates of L span V. If equality holds in (2.14), then $p_0 + q_0 = -m$ . This implies that $V^{p,q} = 0$ for $(p,q) \neq (p_0,q_0)$ . That is, $\boldsymbol{\mu}_w$ acts as a scalar on V, and in particular leaves L stable.

Proposition 2.15. Let $H \subset A_{\bar{\mathsf{K}}}[p^\infty]$ be a p-divisible subgroup. Then the limit $\lim_n \nu(H[p^n])$ exists, and we have

(2.15) \begin{equation} \lim_n \nu(H[p^n]) \leq \tfrac{1}{2}. \end{equation}

If the Mumford–Tate conjecture holds for A, then equality holds in (2.15) only if H is defined over $\mathsf{K}$ .

Proof. Note that H is defined over $\mathsf{K}$ if and only if $T_pH \otimes \mathbb Q_p \subset T_pA\otimes \mathbb Q_p$ is stable by G. Thus, the proposition holds only if it holds after replacing $\mathsf{K}$ by a finite extension. We may assume that $\mathsf{K}$ is Galois over $\mathbb Q$ . Denote by s(p) the number of primes $w|p$ of $\mathsf{K}$ . For each embedding $w: \mathsf{K} \rightarrow C,$ we have

(2.16) \begin{align} \lim_n \int_{\Gamma_\mathsf{K}}\mu_w(\sigma(H[p^n])) \,d\sigma &= \int_{\Gamma_\mathsf{K}}\lim_n \mu_w(\sigma(H[p^n])) \,d\sigma \nonumber\\[5pt] & = \int_{\Gamma_\mathsf{K}} \frac{\mathrm{dim}(\sigma(T_p H) \otimes C \cap \mathrm{Lie\,} A \otimes C(1))}{\mathrm{ht\,} H} \, d\sigma, \end{align}

where, in the first equality, we are using that $\mu_w$ takes values in [0,1] and in the second equality, we have applied Proposition 2.7 to $\mathsf{K}_w \hookrightarrow C$ . Since we are assuming the Mumford–Tate conjecture holds for A, the image of $\Gamma_\mathsf{K}$ in $G(\mathbb Q_p)$ is open. Hence, for $\sigma$ contained in a subset of $\Gamma_\mathsf{K}$ of measure 1, we have

\begin{align*} \frac{\mathrm{dim}(\sigma(T_p H) \otimes C \cap \mathrm{Lie\,} A \otimes C(1))}{\mathrm{ht\,} H} = \langle \boldsymbol{\mu}_w,T_p H \otimes C \rangle_G, \end{align*}

where we view $T_pH \otimes C$ as a subspace of the G-representation $T_pA\otimes C$ .

Thus, using Lemma 2.14, we have

(2.17) \begin{align}\lim_n \nu(H[p^n]) &= \frac 1 {s(p)} \sum_{w|p} \langle \boldsymbol{\mu}_w,T_p H \otimes C \rangle_G \nonumber\\[5pt]&= \frac 1 {2s(p)} \bigg(\sum_{w|p} \ \langle \boldsymbol{\mu}_w,T_p H \otimes C \rangle_G+ \langle \boldsymbol{\mu}_{c\circ w},T_p H \otimes C \rangle_G\bigg) \nonumber\\[5pt]&\leq -\frac 1 {2s(p)} \bigg(\sum_{w|p} \langle \boldsymbol{w}, T_p H\otimes C \rangle\bigg) = \frac 1 2,\end{align}

with equality only if $T_p H\otimes C$ is stable by $\boldsymbol{\mu}_w$ for all $w|p$ . Now, if A satisfies the Mumford–Tate conjecture then, by Corollary 2.13, if $T_p H\otimes C$ is stable by all $\boldsymbol{\mu}_w,$ then $T_p H \subset T_p A$ is stable by G, and hence by $\Gamma_\mathsf{K}$ .

2.5.1.

We keep the notation introduced above. In particular, we continue to assume that $\mathsf{K}$ contains the reflex field of A, so that the G-valued representation $\varrho_A$ exists. The aim of this subsection is to show the Northcott property for the set of abelian varieties over $\bar{\mathsf{K}}$ which are related to $A_{\bar{\mathsf{K}}}$ via a p-isogeny. We do this by reducing to the case considered in Corollary 2.16, where the kernels of these p-isogenies are the levels of a p-divisible group.

2.5.2.

Let M be a torsion, finite $\mathbb Z_p$ -module. We call M uniform if $M \simeq (\mathbb Z/p^n\mathbb Z)^r$ for some n and r, in which case we say that M is uniform of rank r and level n. Fix r, and denote by $X_r(\mathbb Z/p^n\mathbb Z)$ the set of submodules $M \subset A[p^\infty](\bar{\mathsf{K}})$ which are uniform of rank r and level n. Then the set $X_r(\mathbb Q_p/\mathbb Z_p) = \underleftarrow{\mathrm{lim}} X_r(\mathbb Z/p^n\mathbb Z)$ is profinite, and equipped with its profinite topology. Here, the transition maps are given by sending $M \in X_r(\mathbb Z/p^n\mathbb Z)$ to $M[p^{n-1}]$ .

A point $H \in X_r(\mathbb Q_p/\mathbb Z_p)$ corresponds to a p-divisible subgroup $H \subset A[p^\infty](\bar{\mathsf{K}})$ . We write $\nu(H)$ for $\lim_n \nu(H[p^n])$ , which is well-defined by Equation (2.16).

Let $\{H_n\}_{n \geq 1}$ be an infinite sequence of distinct uniform subgroups of $A[p^\infty](\bar{\mathsf{K}})$ of fixed rank r, and $\tilde H_n \in X_r(\mathbb Q_p/\mathbb Z_p)$ a lift of $H_n$ . We say $\{H_n\}_{n \geq 1}$ converges to $H \in X_r(\mathbb Q_p/\mathbb Z_p)$ if $\{\tilde H_n\}_{n \geq 1}$ converges to H. Since the $H_n$ are distinct, their levels go to $\infty,$ so this condition is independent of the choice of $\tilde H_n$ .

Lemma 2.17. Let $\{H_n\}_{n \geq 1}$ be a sequence of distinct subgroups of $A[p^\infty](\bar K),$ which are uniform of fixed rank r, such that $\{H_n\}_{n \geq 1}$ converges to $H \in X_r(\mathbb Q_p/\mathbb Z_p)$ . Then

\begin{align*} \lim \sup_n \nu(H_n) \leq \nu(H).\end{align*}

Proof. Choose lifts $\tilde H_n \in X_r(\mathbb Q_p/\mathbb Z_p)$ of $H_n,$ and let $m \geq 1$ . Since $\lim \tilde H_n = H$ in $X_r(\mathbb Q_p/\mathbb Z_p),$ for n large enough $\tilde H_n[p^m] = H_n[p^m] = H[p^m]$ . Denote by $\bar H_n$ the closure of $H_n$ in $\mathcal A$ . Suppose $H_n$ has level $mk+t,$ with k a non-negative integer, and $0 \leq t \leq m-1$ . Then, for $j \leq k-1,$ multiplication by $p^{mj}$ on $\mathcal A$ induces a map

\begin{align*} \bar H_n[p^{m(j+1)}]/\bar H_n[p^{mj}] \rightarrow \bar H[p^m]\end{align*}

which is an isomorphism on generic fibers. Hence, using Lemma 2.9(2), we have

\begin{align*}\nu(\bar H_n[p^{m(j+1)}]/\bar H_n[p^{mj}]) \leq \nu(H[p^m]),\end{align*}

and so

\begin{align*}\deg(\bar H_n[p^{m(j+1)}]/\bar H_n[p^{mj}]) \leq \deg(H[p^m]).\end{align*}

Using the additivity of the global degree in short exact sequences Lemma 2.9(3), we obtain

\begin{align}\deg \bar H_n &= \deg \bar H_n /\bar H_n[p^{mk}] + \sum_{j=0}^{k-1} \deg(\bar H_n[p^{m(j+1)}]/\bar H_n[p^{mj}]) \nonumber\\&\leq k\deg H[p^m] + \deg \bar H_n /\bar H_n[p^{mk}].\end{align}

Hence,

\begin{align}\nu(H_n) &\leq \frac{k\deg H[p^m] + \deg \bar H_n /\bar H_n[p^{mk}]}{\mathrm{ht\,} H_n} \nonumber\\&\leq \frac{k\deg H[p^m] + (m-1)r}{k\mathrm{ht\,} H[p^m]} = \nu(H[p^m]) + \frac{(m-1)r}{k\mathrm{ht\,} H[p^m]}.\end{align}

Passing to the limit with n (so that $k \rightarrow \infty$ ), and then with m, gives the inequality of the lemma.

2.5.3.

Let M be a finitely generated, torsion $\mathbb Z_p$ -module. For each i consider the map $\epsilon_i:M[p^i] \overset {p^{i-1}} \rightarrow M[p]$ . We define a decreasing filtration $M^j$ on M by setting $M^j = M[p^{i_j}]$ , where $i_j$ is the greatest integer i for which the image of $\epsilon_i$ has $\mathbb F_p$ -dimension $\geq j$ .

Proof. Let $M = \bigoplus_s (\mathbb Z/p^{a_s}\mathbb Z)^{r_s},$ where $a_1 \lt a_2 \lt \cdots \lt a_s$ are integers, and $r_s \gt 0$ . Note that $i_j = a_{s_j}$ for some $s_j$ by maximality of $i_j$ . We have

\begin{align*}M^j = M[p^{i_j}] \simeq \bigoplus_{s\lt s_j} (\mathbb Z/p^{a_s}\mathbb Z)^{r_s} \bigoplus_{s \geq s_j} (\mathbb Z/p^{a_{s_j}}\mathbb Z)^{r_s}.\end{align*}

Thus

\begin{align*} \mathrm{gr}^j M = \bigoplus_{s \geq s_j} (\mathbb Z/p^{a_{s_j}- a_{s_{j+1}}}\mathbb Z)^{r_s},\end{align*}

so this module is uniform. Moreover, $ \mathrm{gr}^j M$ is non-zero if and only if $s_j \neq s_{j+1},$ or equivalently $i_j \neq i_{j+1},$ in which case its rank is

\begin{align*} \sum_{a_s \geq a_{s_j}} r_s = \sum_{a_s \geq i_j } r_s= \mathrm{rk\,} \epsilon_{i_j} = j. \end{align*}

Proposition 2.19. Suppose that A satisfies the Mumford–Tate conjecture, and let $\{H_\alpha\}_{\alpha}$ be an infinite collection of distinct, finite subgroups of $A[p^\infty](\bar{\mathsf{K}}),$ such that no $H_{\alpha}$ contains a non-trivial $\Gamma_\mathsf{K}$ -stable subgroup. Then there exists a sequence $\{H_n\}_{n \geq 1} \subset \{H_{\alpha}\}$ such that

\begin{align*} \lim_n \sup \nu(H_n) \lt \tfrac{1}{2}.\end{align*}

Proof. We consider the filtration $H^j_{\alpha}$ on $H_{\alpha}$ introduced in § 2.5.3. As usual, we denote by $\bar H^j_{\alpha}$ the closure of $H^j_{\alpha}$ in $\mathcal A$ . Multiplication by $p^{i_{j+1}}$ induces a map $\bar H^j_{\alpha}/\bar H^{j+1}_{\alpha} \rightarrow \bar H_{\alpha},$ which is an injection on generic fibers. We denote the image of the generic fiber of this map by $\underline {\mathrm{gr}}^j H_{\alpha} \subset H_{\alpha}$ . Then

(2.20) \begin{equation}\nu(\bar H^j_{\alpha}/\bar H^{j+1}_{\alpha}) \leq \nu (\underline {\mathrm{gr}}^j H_{\alpha}).\end{equation}

Now, by compactness of $X_j(\mathbb Q_p/\mathbb Z_p)$ and Lemma 2.18, we may choose a sequence $\{H_n\}_{n \geq 1} \subset \{H_{\alpha}\}$ such that, for each $j \geq 1,$ either $\underline {\mathrm{gr}}^j H_n$ is trivial for all n, or the sequence $\{\underline {\mathrm{gr}}^j H_n\}_n$ converges to a p-divisible group $H(j) \in X_j(\mathbb Q_p/\mathbb Z_p)$ . By Lemma 2.17 and Proposition 2.15, in the latter case we have

\begin{align*}\lim_n \sup \nu(\bar H^j_n/\bar H^{j+1}_n) \leq \lim_n\sup \nu(\underline {\mathrm{gr}}^j H_n) \leq \nu(H(j)) \leq \tfrac{1}{2},\end{align*}

with equality only if H(j) is defined over K. Hence,

(2.21) \begin{equation}\lim_n \sup \nu(H_n) \leq \tfrac{1}{2},\end{equation}

with equality only if each H(j) is defined over $\mathsf{K}$ .

If H(j) is defined over $\mathsf{K},$ then $H(j)[p^m] \subset A[p^m](\bar{\mathsf{K}})$ is stable by $\Gamma_\mathsf{K}$ for every m. Since $\{\underline {\mathrm{gr}}^j H_n\}_n$ converges to H(j), for n sufficiently large, $H(j)[p^m] \subset \underline {\mathrm{gr}}^j H_n,$ which contradicts the assumption that $H_{\alpha}$ contains no non-trivial $\Gamma_\mathsf{K}$ -stable subgroups. Therefore, (2.21) is strict.

2.5.4.

Let ${\mathscr I}_p(A)$ denote the p-isogeny class of A over $\bar{\mathsf{K}}$ . We denote by ${\mathscr I}_p(A)'$ the collection of subgroups $H \subset A[p^\infty](\bar{\mathsf{K}})$ such that H contains no non-trivial $\Gamma_\mathsf{K}$ -stable subgroup. Thus, ${\mathscr I}_p(A)' \subset {\mathscr I}_p(A)$ via $H \mapsto A/H$ .

Theorem 2.20. Let A be an abelian variety defined over a number field $\mathsf{K},$ and suppose that the Mumford–Tate conjecture holds for A. Then, for any $c \gt 0,$ the set

\begin{align*} \{ A' \in {\mathscr I}_p(A): h(A') \lt c \}\end{align*}

contains a finite number of isomorphism classes of abelian varieties.

Let $\{H_{\alpha}\}_{\alpha}$ be a collection of subgroups of $A[p^\infty](\bar{\mathsf{K}})$ . We have to show that the set $\{H_{\alpha}: h(A/H_{\alpha}) \lt c\}_{\alpha}$ contains only finitely many isomorphism classes of abelian varieties. Let $H_{\alpha}^0 \subset H_{\alpha}$ denote the maximal subgroup of $H_{\alpha}$ which is stable under $\Gamma_\mathsf{K}$ . As the sum of two $\Gamma_\mathsf{K}$ -stable subgroups is $\Gamma_\mathsf{K}$ -stable, $H_{\alpha}^0$ is well-defined. Moreover, if $H^1 \subset H_{\alpha}/H^0_{\alpha}$ is $\Gamma_\mathsf{K}$ -stable, then so is its preimage in $H_{\alpha}$ . Thus, $H_{\alpha}/H^0_{\alpha}$ contains no non-trivial $\Gamma_\mathsf{K}$ -stable subgroups.

By Faltings’ theorem, the set $\{A/H^0_{\alpha}\}_{\alpha}$ contains only finitely many isomorphism classes. Fix such an isomorphism class $\kappa,$ let $\alpha(\kappa)$ denote the set of $\alpha$ with $A/H^0_{\alpha}$ in $\kappa,$ and choose any $\alpha_0 \in \alpha(\kappa)$ . For $\alpha \in \alpha(\kappa),$ fix an isomorphism $A/H^0_{\alpha} \simeq A/H^0_{\alpha_0}$ . Then, for $\alpha \in \alpha(\kappa),$ we may view $H_{\alpha}/H^0_{\alpha}$ as a subgroup of $A/H^0_{\alpha_0},$ and it suffices to prove the finiteness statement with $A/H^0_{\alpha_0}$ in place of A, and $\{H_{\alpha}/H^0_{\alpha}: \alpha \in \alpha(\kappa)\}$ in place of $\{H_{\alpha}\}_{\alpha}$ . In particular, we may assume that no $H_{\alpha}$ has a non-trivial $\Gamma_\mathsf{K}$ -stable subspace.

We may assume that the $H_{\alpha}$ are distinct. If $\{H_{\alpha}: h(A/H_{\alpha}) \lt c\}_{\alpha}$ is infinite, then by Proposition 2.19 there is a sequence

\begin{align*}\{H_n\}_{n \geq 1} \subset \{H_{\alpha}: h(A/H_{\alpha}) \lt c\}_{\alpha}\end{align*}

such that $\lim_n \sup \nu(H_n) \lt {1}/{2}$ . By Faltings’ isogeny lemma, 2.3.4, this implies the set of heights $\{h(A/H_n)\}_{n \geq 1}$ is unbounded, contradicting our assumption. Hence, $\{H_{\alpha}: h(A/H_{\alpha}) \lt c\}_{\alpha}$ is finite.

2.5.5.

In fact, the final paragraph of the proof of the theorem gives a more precise statement, which we now record.

Corollary 2.21. Suppose $\mathsf{K}$ contains the reflex field of A. If the Mumford–Tate conjecture holds for A, then for any $c \gt 0,$ the set

\begin{align*} \{H \in {\mathscr I}_p(A)': h(A/H) \lt c\}\end{align*}

is finite.

3.1.1.

We now return to the notation of § 2.1.1, but we assume that K is a finite extension of $K_0 = W(k)[1/p],$ where k is a perfect field of characteristic p. We also fix a uniformizer $\pi \in \mathcal O_K,$ and denote by $E = E(u)$ the Eisenstein polynomial of $\pi$ . Fix a compatible sequence of p-power roots of $\pi$ in $\bar K,$ and set $K_{\infty} = K(\pi^{1/p^n})_{n \geq 1}$ .

Let $R = \underleftarrow{\mathrm{lim}}_{x \rightarrow x^p} \mathcal O_C/p\mathcal O_C$ be the integral subring of the tilt $C^\flat,$ and write $A_{\inf} = W(R)$ . Write $\xi = p - [\underline p],$ where $\underline p = (p, p^{1/p}, \dots) \in R$ . Set $\mathfrak S = W[\![ u ]\!]$ equipped with the Frobenius lift given by $u \mapsto u^p$ . We consider $\mathfrak S$ as a subring of $A_{\inf}$ via the map $u \mapsto [\underline \pi]$ , where $\underline \pi = (\pi, \pi^{1/p}, \dots) \in R$ . Note that this inclusion is compatible with Frobenius actions.

3.1.2.

Let A be a $\mathbb Z_p$ -algebra, equipped with a lift of Frobenius $\varphi,$ and a non-zero divisor $a \in A$ . We denote by $\mathrm{Mod}^{\varphi,c}_{/A}$ (respectively $\mathrm{Mod}^{\varphi}_{/A},$ respectively $\mathrm{Mod}^{\varphi,t}_{/A}$ ) the category of a-torsion free A-modules M, which are finite (respectively finite free, respectively finite, p-power torsion) equipped with an isomorphism $\varphi_M:\varphi^*(M)[1/a] \overset{\sim}{\longrightarrow} M[1/a]$ . We again denote by $\varphi_M: M \rightarrow M[1/a]$ the corresponding $\varphi$ -semi-linear map on M. We sometimes write $\varphi$ for $\varphi_M$ . We call M effective if $\varphi_M(M) \subset M$ .

As M has no a-torsion, $\varphi_M$ identifies $\varphi^*(M)$ with a submodule of $M[1/a]$ . For a non-negative integer h, we denote by $\mathrm{Mod}^{\varphi,\leq h}_{/A}$ (respectively $\mathrm{Mod}^{\varphi,t,\leq h}_{/A}$ ) the subcategory of $\mathrm{Mod}^{\varphi}_{/A}$ (respectively $\mathrm{Mod}^{\varphi,t}_{/A}$ ), consisting of effective objects M, such that $M/\varphi^*(M)$ is killed by $a^h$ .

We think of $A_{\inf}$ and R (respectively $\mathfrak S$ ) as equipped with the divisor $(\xi)$ (respectively (E)). We regard any quotient or localization of $A_{\inf}$ (respectively $\mathfrak S$ ) as equipped with the same element $\xi$ (respectively E).

Since $E \in \xi A_{\inf},$ we have functors

\begin{align*} \mathrm{Mod}^{\varphi}_{/\mathfrak S} \rightarrow \mathrm{Mod}^{\varphi}_{/A_{\inf}}\quad \text{ and }\quad\mathrm{Mod}^{\varphi}_{/k[\![ u ]\!] } \rightarrow \mathrm{Mod}^{\varphi}_{/R};\quad M \mapsto M\otimes _{\mathfrak{S}}A_{\inf}. \end{align*}

3.1.3.

Let $\mathrm{Rep}^{\mathrm{fin}}_{\mathbb Z_p}$ denote the category of finite $\mathbb Z_p$ -modules. For M in $\mathrm{Mod}^{\varphi,c}_{/A_{\inf}[1/\xi]},$ we write

\begin{align*} \textbf V(M) = (M\otimes _{A_{\inf}} W(C^\flat))^{\varphi = 1},\end{align*}

which is a finitely generated $\mathbb Z_p$ -module. As $W(C^{\flat})$ is a p-adically complete discrete valuation ring, with algebraically closed residue field (it is equal to the p-adic completion of $A_{\inf}[1/\xi]$ ), the natural map

\begin{align*} \textbf V(M)\otimes _{\mathbb Z_p}W(C^\flat) \rightarrow M\otimes _{A_{\inf}} W(C^\flat)\end{align*}

is an isomorphism. Thus, $\textbf V$ gives rise to an exact tensor functor.

\begin{align*} \mathrm{Mod}^{\varphi,c}_{/A_{\inf}[1/\xi]} \rightarrow \mathrm{Rep}^{\mathrm{fin}}_{\mathbb Z_p}; \quad M \mapsto \textbf V(M), \end{align*}

which induces an equivalence on p-power torsion objects. For M in $\mathrm{Mod}^{\varphi,c}_{/A_{\inf}},$ we will write $\textbf V(M)$ for $\textbf V(M[1/\xi])$ .

If M is in $\mathrm{Mod}^{\varphi,c}_{/\mathfrak S[1/E]},$ we likewise write

\begin{align*} \textbf V(M) = (M\otimes _{\mathfrak S} W(C^\flat))^{\varphi = 1},\end{align*}

when $\textbf V(M)$ inherits the structure of a $\Gamma_{K_{\infty}} : = \mathrm{Gal}(\bar K/K_{\infty})$ -module from the $\Gamma_{K_{\infty}}$ -action on $W(C^\flat)$ . If $\mathrm{Rep}_{\mathbb Z_p}^{\mathrm{fin}}(\Gamma_{K_{\infty}})$ denotes the category of $\mathbb Z_p-\Gamma_{K_{\infty}}$ -modules which are finitely generated over $\mathbb Z_p,$ then $\textbf V$ induces an exact tensor functor

\begin{align*} \mathrm{Mod}^{\varphi,c}_{/\mathfrak S[1/\xi]} \rightarrow \mathrm{Rep}_{\mathbb Z_p}^{\mathrm{fin}} (\Gamma_{K_{\infty}}); \quad M \mapsto \textbf V(M), \end{align*}

which is an equivalence on p-power torsion objects [Reference FontaineFon90, A.1.2.7]. As above, for M in $\mathrm{Mod}^{\varphi,c}_{/\mathfrak S},$ we write $\textbf V(M) = \textbf V(M[1/E])$ .

3.1.4.

We recall the definition of slopes of objects of $\mathrm{Mod}^{\varphi,t}_{/A_{\inf}}$ which was introduced by Cornut and Peche Irissarry [Reference Cornut and Peche IrissarryCPI19]; see also [Reference Wang-Erickson and LevinWEL20]. We ultimately make use of this only for modules in $\mathrm{Mod}^{\varphi}_{/R},$ so the reader may restrict themselves to this case, in which case the definitions are slightly simpler.

As in § 2.2.1, for a finitely presented torsion R-module $\mathfrak N,$ we have $\mathfrak N \simeq \bigoplus_{i=1}^n R/a_i R$ with $a_i \in R$ . We define the normalized length of $\mathfrak N$ as $\ell(\mathfrak N) = \sum_i v_R(a_i)$ , where $v_R$ is the valuation on R, normalized so that $v_R(\underline p) = 1$ . The normalized length is additive in short exact sequences. Now suppose that $\mathfrak N$ is a finitely presented $A_{\inf}$ -module, which is killed by some power of $(p,\xi)$ . Then $\mathfrak N$ has a filtration whose associated grading is a finitely presented torsion R-module, and we extend the definition of $\ell$ to such modules by additivity [Reference Cornut and Peche IrissarryCPI19, 2.2.13].

Let $\mathfrak M$ be an effective object in $\mathrm{Mod}^{\varphi,t}_{/A_{\inf}}$ . Then $\mathfrak M/\varphi^*(\mathfrak M)$ is a finitely presented $A_{\inf}$ -module, killed by some power of $(p,\xi),$ and we define

\begin{align*} \deg \mathfrak M = \ell(\mathfrak M/\varphi^*(\mathfrak M)).\end{align*}

Set $\mathrm{rk\,} \mathfrak M = \ell_{W(C\flat)}(\mathfrak M\otimes W(C^\flat)),$ and define the slope of $\mathfrak M$ by

\begin{align*} \mu(\mathfrak M) = \frac{ \deg \mathfrak M}{\mathrm{rk\,} \mathfrak M}.\end{align*}

The definition of $\mathrm{rk\,} \mathfrak M$ is motivated by the case of $\mathfrak M$ in $\mathrm{Mod}^{\varphi}_{/R},$ when this is just the R-rank of $\mathfrak M$ . If $\mathfrak M$ is in $\mathrm{Mod}^{\varphi,t,\leq h}_{/A_{\inf}}$ , then $\mu(\mathfrak M) \in [0,h],$ but the converse does not hold.

Let $A_{\inf}(1)$ denote $A_{\inf}$ equipped with the Frobenius given by $a \mapsto \xi\varphi(a)$ . We may extend the functions $\deg$ and $\mu$ to all objects in $\mathrm{Mod}^{\varphi,t}_{/A_{\inf}}$ as follows. For any $\mathfrak M$ in $\mathrm{Mod}^{\varphi,t}_{/A_{\inf}},$ $\mathfrak M\otimes A_{\inf}(1)^{\otimes m}$ is effective for m large enough, and we set $\mu(\mathfrak M) = \mu(\mathfrak M\otimes A_{\inf}(1)^{\otimes m}) - m$ and $\deg \mathfrak M = \mu(\mathfrak M)\mathrm{rk\,} \mathfrak M$ . That these invariants are well-defined follows from Lemma 3.1(3) below applied to effective modules.

For the convenience of the reader, we summarize some of the properties of these invariants.

For part (2) we may assume $\mathfrak M_1$ and $\mathfrak M_2$ are effective. Let $\mathfrak C = \mathfrak M_2/\mathfrak M_1$ . Using the snake lemma, we get an exact sequence

\begin{align*} 0 \rightarrow \mathrm{ker}\,(\varphi^* \mathfrak C \rightarrow \mathfrak C) \rightarrow \mathfrak M_1/\varphi^*(\mathfrak M_1) \rightarrow \mathfrak M_2/\varphi^*(\mathfrak M_2)\rightarrow \mathrm{coker} (\varphi^* \mathfrak C \rightarrow \mathfrak C)\rightarrow 0. \end{align*}

Hence,

\begin{align} \ell(\mathfrak M_2/\varphi^*(\mathfrak M_2)) - \ell(\mathfrak M_1/\varphi^*(\mathfrak M_1))&= \ell(\mathrm{coker} (\varphi^* \mathfrak C \rightarrow \mathfrak C)) - \ell(\mathrm{ker}\, (\varphi^* \mathfrak C \rightarrow \mathfrak C)) \nonumber\\&= \ell(\mathfrak C)- \ell(\varphi^* \mathfrak C) = (1-p)\ell(\mathfrak C) \leq 0\end{align}

and we get the required inequality, with equality if and only if $\mathfrak C = 0$ .

3.1.5.

Let $\mathfrak M$ be in $\mathrm{Mod}^{\varphi}_{/R},$ and write $\mathfrak M/\varphi^*(\mathfrak M) \simeq \bigoplus_{i=1}^n R/a_i R$ . We set

\begin{align*}\mu_{ max}(\mathfrak M) = \max_i v_R(a_i),\end{align*}

and denote by $\mathcal H(\mathfrak M)$ the multiset $\{v_R(a_1), \dots, v_R(a_n)\}$ .

For $\mathfrak M$ in $\mathrm{Mod}^{\varphi}_{/k [\![ u ]\!]}$ , we write $\mu(\mathfrak M) = \mu(\mathfrak M\otimes _{k[\![ u ]\!]} R),$ and similarly for the other invariants defined above.

3.2.1.

We recall a classification of finite flat group schemes over $\mathcal O_K$ and $\mathcal O_C$ developed in [Reference KisinKis06, Reference KimKim12, Reference LauLau18, Reference Anschütz and Le BrasALB23]. Our goal is to explain the compatibility between the constructions over $\mathcal O_K$ and $\mathcal O_C,$ and also to relate the slopes of finite flat group schemes to slopes of objects in $\mathrm{Mod}^{\varphi,t}_{/A_{\inf}}$ . To do this, the approach in [Reference Anschütz and Le BrasALB23] via prismatic cohomology will be useful.

For any p-adically complete ring T, we denote by ${p{\text{- div}}}/T$ the category of p-divisible groups over T. We denote by $p{\text{- Gr}}/T$ the category of finite flat group schemes of p-power order over T, and by $p{\text{- Gr}}^1/T$ the full subcategory of $p{\text{- Gr}}/T$ consisting of objects killed by p.

Proposition 3.2. We have the following commutative diagram of exact functors, where the vertical maps are the natural ones and the horizontal maps are anti-equivalences of exact categories.

Proposition 3.3. We have the following commutative diagram of exact functors, where the vertical maps are the natural ones and the horizontal maps are exact anti-equivalences.

Proof. For ${\mathcal G}$ in $p{\text{- Gr}}/\mathcal O_K$ (respectively, $p{\text{- Gr}}/\mathcal O_C$ ), write ${\mathcal G}$ as the kernel of an isogeny of p-divisible groups over $\mathcal O_K$ (respectively, $\mathcal O_C$ ):

\begin{align*} 0 \rightarrow {\mathcal G} \rightarrow {\mathcal G}_1 \overset f \rightarrow {\mathcal G}_2 \rightarrow 0.\end{align*}

This is possible by [Reference Berthelot, Breen and MessingBBM82, Theorem 3.1.1]. Applying the functors of Proposition 3.2 to f, we obtain a map $\mathfrak M(f): \mathfrak M_2 \rightarrow \mathfrak M_1$ in $\mathrm{Mod}^{\varphi, \leq 1}_{/\mathfrak S}$ (respectively $\mathrm{Mod}^{\varphi, \leq 1}_{/A_{\inf}}$ ). We set $\mathfrak M({\mathcal G}) = \mathrm{coker}(\mathfrak M(f))$ .

If ${\mathcal G}$ is in $p{\text{- Gr}}/\mathcal O_K,$ then the fact that ${\mathcal G} \mapsto \mathfrak M({\mathcal G})$ is a well-defined functor to $\mathrm{Mod}^{\varphi, t,\leq 1}_{/\mathfrak S},$ and that the top map is an exact anti-equivalence, is already proved in [Reference KisinKis06, Theorem 2.3.5] for $p\gt2$ . The same proof applies when $p=2,$ using Proposition 3.2 in this case.

When ${\mathcal G}$ is in $p{\text{- Gr}}/\mathcal O_C,$ the same result was proved in [Reference LauLau18, Theorem 10.12, Remark 10.7] for $p \geq 3,$ and [Reference Anschütz and Le BrasALB23, Theorem 5.4] in general.

We remark that a finitely generated, p-power torsion $\mathfrak S$ (respectively $A_{\inf}$ ) module $\mathfrak M$ is E (respectively $\xi$ ) torsion free if and only if it has projective dimension 1 (cf. [Reference Cornut and Peche IrissarryCPI19, 2.3.4]). Thus, our categories $\mathrm{Mod}^{\varphi, t,\leq 1}_{/\mathfrak S}$ and $\mathrm{Mod}^{\varphi, t, \leq 1}_{/A_{\inf}}$ do coincide with the categories defined in the references above.

Corollary 3.4. We have the following commutative diagram of exact functors, where the vertical maps are the natural ones and the horizontal maps are exact anti-equivalences.

Lemma 3.5 For ${\mathcal G}$ in $p{\text{- Gr}}/\mathcal O_C,$ there is a canonical isomorphism of finite abelian groups

\begin{align*} \textbf V(\mathfrak M({\mathcal G})) \simeq {\mathcal G}(C)^\vee : = \mathrm{Hom}({\mathcal G}(C), \mathbb Q_p/\mathbb Z_p) .\end{align*}

In particular, for ${\mathcal G}$ in $p{\text{- Gr}}/\mathcal O_K,$ there is an isomorphism of $\Gamma_{K_{\infty}}$ -modules

\begin{align*} \textbf V(\mathfrak M({\mathcal G})) \simeq {\mathcal G}(C)^\vee. \end{align*}

Proof. Let ${\mathcal G}$ be in $p{\text{- Gr}}/\mathcal O_C,$ killed by $p^n$ . We denote by $\mathfrak M({\mathcal G})^\vee$ the $\mathfrak S/p^n\mathfrak S$ -linear dual of $\mathfrak M({\mathcal G})$ . Then, using that $\textbf V$ is compatible with duals, we obtain canonical isomorphisms

\begin{align}{\mathcal G}(C) &\simeq \mathrm{Hom}_{p{\text{- Gr}}/\mathcal O_C}(\mathbb Z/p^n\mathbb Z, {\mathcal G}) \nonumber\\& \simeq \mathrm{Hom}_{\mathrm{Mod}^{\varphi,t}_{/A_{\inf}}}(\mathfrak M({\mathcal G}), \mathfrak M(\mathbb Z/p^n\mathbb Z))\simeq (\mathfrak M({\mathcal G})^\vee)^{\varphi=1}\hookrightarrow \textbf V(\mathfrak M({\mathcal G}))^{\vee} .\end{align}

The final inclusion is a bijection, as both sides have the same length. Hence, $\textbf V(\mathfrak M({\mathcal G})) \simeq {\mathcal G}(C)^\vee$ . This proves the first claim, and the second claim follows from the functoriality of the isomorphism.

Proposition 3.6. Let $A/\mathcal O_C$ be an abelian scheme, and ${\mathcal G}$ its p-divisible group. Then there is a canonical isomorphism

\begin{align*} H^0(A, \Omega^1_{A/\mathcal O_C}) \simeq \mathfrak M({\mathcal G})/\varphi^*(\mathfrak M({\mathcal G})).\end{align*}

Proof. We use the description of $\mathfrak M({\mathcal G})$ in terms of prismatic cohomology of A proved in [Reference Anschütz and Le BrasALB23, Theorem 4.62]. In particular, there is a canonical isomorphism [Reference Bhatt and ScholzeBS22, Theorem 1.8(3)]

\begin{align*}\varphi^*(\mathfrak M({\mathcal G}))\otimes _{A_{\inf}} \mathcal O_C \simeq H^1_{{\mathrm{dR}}}(A) \end{align*}

and $\varphi^*(\mathfrak M({\mathcal G}))$ is equipped with a filtration $\mathrm{Fil}^1\varphi^*(\mathfrak M({\mathcal G})) \supset \xi\mathfrak M({\mathcal G}),$ which identifies $\mathrm{Fil}^1 \varphi^*(\mathfrak M({\mathcal G}))/\xi\varphi^*(\mathfrak M({\mathcal G}))$ with $\mathrm{Fil}^1 H^1_{{\mathrm{dR}}}(A) = H^0(A, \Omega^1_{A/\mathcal O_C})$ . Now the linearized Frobenius identifies $\mathrm{Fil}^1 \varphi^*(\mathfrak M({\mathcal G}))$ with $\xi\mathfrak M({\mathcal G})$ . This follows from [Reference Li and LiuLL22, Corollary 7.11], which relates the Nygaard filtration on prismatic cohomology with the Breuil–Kisin filtration of [Reference Li and LiuLL22]. Thus, we have

\begin{align*} H^0(A, \Omega^1_A) \simeq \mathrm{Fil}^1\varphi^*(\mathfrak M({\mathcal G}))/\xi\varphi^*(\mathfrak M({\mathcal G})) \simeq \xi\mathfrak M({\mathcal G})/\xi \varphi^*(\mathfrak M({\mathcal G}))\simeq \mathfrak M({\mathcal G}) /\varphi^*(\mathfrak M({\mathcal G})).\end{align*}

Proposition 3.7. Let ${\mathcal G}$ be in $p{\text{- Gr}}/\mathcal O_C$ . Then there is a canonical isomorphism

\begin{align*} \omega_{{\mathcal G}/\mathcal O_C} \simeq \mathfrak M({\mathcal G})/\varphi^*(\mathfrak M({\mathcal G})).\end{align*}

Proof. For an abelian scheme $A/\mathcal O_C,$ with identity section s, we set

\begin{align*} \omega_{A/\mathcal O_C} = s^*\Omega^1_{A/\mathcal O_C} \simeq H^0(A, \Omega^1_{A/\mathcal O_C}). \end{align*}

By [Reference Berthelot, Breen and MessingBBM82, Theorem 3.1.1], ${\mathcal G}$ is the kernel of an isogeny of abelian schemes $f: A_1 \rightarrow A_2,$ and we write $f: {\mathcal G}_1 \rightarrow {\mathcal G}_2$ for the corresponding map of p-divisible groups. Then, using Proposition 3.6, we have

\begin{align}\omega_{{\mathcal G}/\mathcal O_C} &\simeq \mathrm{coker} (\omega_{A_2/\mathcal O_C} \rightarrow \omega_{A_1/\mathcal O_C}) \nonumber\\&\simeq \mathrm{coker}(\mathfrak M({\mathcal G}_2)/\varphi^*\mathfrak M({\mathcal G}_2) \rightarrow \mathfrak M({\mathcal G}_1)/\varphi^*\mathfrak M({\mathcal G}_1))\simeq \mathfrak M({\mathcal G})/\varphi^*\mathfrak M({\mathcal G}).\end{align}

That this isomorphism is independent of the choice of f can be shown in the same way one shows the definition of $\mathfrak M({\mathcal G})$ is independent of $f:$ given two resolutions, one reduces to the case where there is a map between them, when the result follows from the functoriality of the isomorphism in Proposition 3.7.

Corollary 3.8. Let ${\mathcal G}$ be in $p{\text{- Gr}}/\mathcal O_C$ . Then we have

\begin{align*}\text{$\mu({\mathcal G}) = \mu(\mathfrak M({\mathcal G}))$ and $\deg({\mathcal G}) = \deg(\mathfrak M({\mathcal G}))$.} \end{align*}

In this case, using Proposition 3.7, we have

\begin{align*}\mathfrak M({\mathcal G})/\varphi^*(\mathfrak M({\mathcal G})) \simeq \omega_{{\mathcal G}/\mathcal O_C} \simeq \bigoplus_{i=1}^n \mathcal O_C/a_i\mathcal O_C \end{align*}

for some $a_i \in \mathcal O_C$ with $v_p(a_i) \leq 1$ . Let $\tilde a_i \in R$ be any element with $v_R(\tilde a_i) = v_p(a_i)$ . As $v_p(a_i) \leq 1,$ we may rewrite the composite of the above isomorphisms as

\begin{align*} \mathfrak M({\mathcal G})/\varphi^*(\mathfrak M({\mathcal G})) \simeq \bigoplus_{i=1}^n R/\tilde a_i R.\end{align*}

Thus,

\begin{align*} \deg({\mathcal G}) = \sum_{i=1}^n v_p(a_i) =\sum_{i=1}^n v_R(\tilde a_i) = \deg(\mathfrak M({\mathcal G})).\end{align*}

3.3.1.

From now on, unless stated otherwise, we assume that k is algebraically closed, and that $K = K_0$ is absolutely unramified. We take $\pi = p,$ so that $E(u) = u-p$ .

3.3.2.

Let $h: \mathfrak M \rightarrow \mathfrak N$ be a $\varphi$ -semi-linear map of finite free $k [\![ u ]\!]$ -modules of the same rank. We define a decreasing filtration on $\mathfrak M$ by

\begin{align*} \mathrm{Fil}^i \mathfrak M = \{ m \in \mathfrak M: h(m) \in u^i \mathfrak N \}.\end{align*}

We define a map

\begin{align*} \tilde h^i: \mathrm{Fil}^i \mathfrak M \rightarrow \mathfrak N: m \mapsto u^{-i}h(m) \end{align*}

which induces

\begin{align*} \tilde h^i: \mathrm{gr}^i \mathfrak M \rightarrow \mathfrak N/u\mathfrak N. \end{align*}

1. (1) The map

   \begin{align*} \bigoplus_{i=0}^{p-1} \tilde h^i: \mathrm{gr}^{\lt p} \mathfrak M := \bigoplus_{i=0}^{p-1} \mathrm{gr}^i \mathfrak M \rightarrow \mathfrak N/u\mathfrak N \end{align*}
   is an isomorphism.

2. (2) The map $ \bigoplus_{i=0}^{p-1} \tilde h^i$ is surjective.

3. (3) The map $ \bigoplus_{i=0}^{p-1} \tilde h^i$ is injective and $\mathrm{Fil}^p \mathfrak M = u\mathfrak M$ .

4. (4) There exist choices of $k [\![ u ]\!]$ -bases for $\mathfrak M$ and $\mathfrak N$ such that h is given by a diagonal matrix with entries $u^{i_1}, \dots, u^{i_r},$ with $i_j \in [0,p-1]$ .

Proof. Note that

\begin{align*} u \mathfrak M \subset \mathrm{Fil}^p \mathfrak M \subset \mathrm{Fil}^{p-1} \mathfrak M \subset \dots \subset \mathfrak M,\end{align*}

so $\mathrm{dim}_k \mathrm{gr}^{\lt p} \mathfrak M \leq \mathrm{dim}_k \mathfrak M/u\mathfrak M = \mathrm{dim}_k \mathfrak N/u\mathfrak N,$ with equality if and only if $\mathrm{Fil}^p \mathfrak M = u \mathfrak M$ . This implies that conditions (1), (2), and (3) are equivalent.

For (4) $\implies$ (1), we reduce to the case where h is given by multiplication by $u^t$ for $t \in [0,p-1],$ when the result is clear. Now, to check (1) $\implies$ (4), choose a basis $\bar e_1, \dots , \bar e_r$ for $\mathrm{gr}^{\lt p} \mathfrak M$ such that $\bar e_j \in \mathrm{gr}^{i_j} \mathfrak M$ for some $i_j,$ and lift $\bar e_j$ to $e_j \in \mathrm{Fil}^{i_j}\mathfrak M$ . Then $e_1, \dots, e_r$ is a $k[\![ u ]\!]$ -basis for $\mathfrak M$ . Let $e_j' = u^{-i_j}h(e_j)$ . Then conditions (1) implies that $e_1', \dots, e_r'$ is a $k[\![ u ]\!]$ -basis for $\mathfrak N,$ and with our choice of bases, h is given by the diagonal matrix with entries $ u^{i_1}, \dots, u^{i_r}. $

3.3.3.

We say that a module $\mathfrak M$ in $\mathrm{Mod}^{\varphi}_{/k[\![ u ]\!]}$ is of Fontaine–Laffaille (FL) type if it is effective, and $\varphi: \mathfrak M \rightarrow \mathfrak M$ satisfies the equivalent conditions of Lemma 3.9. We denote by $\mathrm{Mod}^{\varphi, { FL}}_{/k[\![ u ]\!]} \subset \mathrm{Mod}^{\varphi}_{/k[\![ u ]\!]}$ the full subcategory of modules of FL type. For h a non-negative integer, we denote by $\mathrm{Mod}^{\varphi, \leq h, { FL}}_{/k[\![ u ]\!]} \subset \mathrm{Mod}^{\varphi, { FL}}_{/k[\![ u ]\!]}$ the full subcategory consisting of objects in $\mathrm{Mod}^{\varphi, \leq h}$ . With the notation of § 3.1.5, we have the following.

Lemma 3.10. The subcategory $\mathrm{Mod}^{\varphi, { FL}}_{/k[\![ u ]\!]} \subset \mathrm{Mod}^{\varphi, \leq p-1}_{/k[\![ u ]\!]}$ is closed under quotients and saturated subobjects. Moreover, if

\begin{align*} 0 \rightarrow \mathfrak M_1 \rightarrow \mathfrak M \rightarrow \mathfrak M_2 \rightarrow 0 \end{align*}

is an extension in $\mathrm{Mod}^{\varphi, { FL}}_{/k[\![ u ]\!]},$ then $\mathcal H(\mathfrak M) = \mathcal H(\mathfrak M_1) \cup \mathcal H(\mathfrak M_2)$ as multisets.

Proof. Suppose that $\mathfrak M$ is in $\mathrm{Mod}^{\varphi, { FL}}_{/k[\![ u ]\!]},$ and let

\begin{align*} 0 \rightarrow \mathfrak M_1 \rightarrow \mathfrak M \rightarrow \mathfrak M_2 \rightarrow 0 \end{align*}

be a short exact sequence in $\mathrm{Mod}^{\varphi}_{/k[\![ u ]\!]}$ . Consider the following commutative diagram, where the bottom row is exact.

Then the right vertical map is surjective, hence an isomorphism, and $\mathfrak M_2$ is of FL type. As $\mathfrak M_1 \subset \mathfrak M$ is saturated, we have $\mathrm{Fil}^i\mathfrak M_1 = \mathrm{Fil}^i\mathfrak M\cap \mathfrak M_1$ . Thus $\mathrm{gr}^{\lt p} \mathfrak M_1 \rightarrow \mathrm{gr}^{\lt p} \mathfrak M$ is injective, and $\mathrm{Fil}^p\mathfrak M_1= u\mathfrak M \cap \mathfrak M_1 = u\mathfrak M_1$ . Hence, $\mathfrak M_1$ satisfies condition (3) of Lemma 3.9, and so is of FL type. This proves the first claim, and also implies that the top line in the diagram is a short exact sequence, which implies that $\mathcal H(\mathfrak M) = \mathcal H(\mathfrak M_1) \cup \mathcal H(\mathfrak M_2)$ .

3.3.4.

We remark that the category $\mathrm{Mod}^{\varphi, { FL}}_{/k[\![ u ]\!]}$ is not closed under extensions, as the matrix $\left(\begin{smallmatrix}u & 1 \\0 & u\end{smallmatrix}\right)$ gives rise to a module $\mathfrak M$ in $\mathrm{Mod}^{\varphi,\leq 2}_{/k[\![ u ]\!]}$ which is a self extension of a rank-1 object of degree 1, but $\mathcal H(\mathfrak M) = \{0,2\} \neq \{1\} \cup \{1\}$ .

Proof. Since $\mathfrak M_1, \mathfrak M_2$ are in $\mathrm{Mod}^{\varphi, { FL}}_{/k[\![ u ]\!]},$ the composite of the maps

\begin{align*} \mathrm{gr}^{\lt p} \mathfrak M_1\otimes _k \mathrm{gr}^{\lt p} \mathfrak M_2 \rightarrow \mathrm{gr}^{\lt p} (\mathfrak M_1\otimes \mathfrak M_2) \rightarrow \mathfrak M_1/u\mathfrak M_1\otimes _k \mathfrak M_2/u\mathfrak M_2 \end{align*}

is surjective. Hence, the second map is surjective.

3.4.1.

We keep the assumptions of the previous subsection. We denote by $\bar{\mathbb {F}}_p$ the algebraic closure of $\mathbb F_p$ in k, and by $\mathbb F_q \subset \bar{\mathbb {F}}_p$ the subfield of order $q = p^n$ .

Recall that the fundamental character of level n, $\omega_n: \Gamma_K \rightarrow \mathbb F_q^\times,$ is defined by $\omega_n(g) = g(p^{{1}/{q-1}})/{p^{{1}/{q-1}}}$ . It is a character of order $q-1,$ and we will again denote by $\omega_n$ the field $\mathbb F_q$ equipped with the $\Gamma_K$ -action given by $\omega_n$ . Any character $\omega: \Gamma_K \rightarrow \bar{\mathbb {F}}_p^\times$ can be written as a power of $\omega_n$ for some n.

3.4.2.

Fix $n \geq 1$ . Let $\boldsymbol{ j} = (j_1, \dots, j_n)$ be a tuple of integers. Write $\overline{\boldsymbol{ j}}= \sum_{i=1}^n p^{n-i}j_i$ and $\sigma(\boldsymbol{ j}) = (j_2, j_3, \dots, j_n, j_1)$ . The ring $\mathbb F_q\otimes k [\![ u ]\!]$ has a distinguished idempotent $e_1,$ corresponding to the given embedding $\mathbb F_q \subset k;$ it satisifies $ae_1 = e_1 a$ for $a \in \mathbb F_q$ . We set $e_i = \varphi^{i-1}(e_1)$ for $i= 1,2, \dots, n,$ where $\varphi$ is the Frobenius on k. Then $ae_i = e_i \varphi^{i-1}(a)$ . For $\boldsymbol{ j}$ as above, we denote by $\mathfrak M(\boldsymbol{ j})$ the module in $\mathrm{Mod}^{\varphi}_{/k [\![ u ]\!]}$ whose underlying $\mathfrak S$ -module is $\mathbb F_q\otimes k [\![ u ]\!],$ with the Frobenius given by $\varphi(e_i) = u^{j_i} e_{i+1}$ .

Lemma 3.13. For any $\boldsymbol{ j}$ there is an isomorphism of $\mathbb F_q[\Gamma_{K_{\infty}}]$ -modules

\begin{align*} \textbf V(\mathfrak M(\boldsymbol{ j})) \simeq \omega_n^{-\overline{\boldsymbol{ j}}}.\end{align*}

Proof. Let $\boldsymbol{ j}_1, \boldsymbol{ j}_2$ be n-tuples as above. As $\textbf V$ is a tensor functor, we have

\begin{align*} \textbf V(\mathfrak M(\boldsymbol{ j}_1 + \boldsymbol{ j}_2)) \simeq \textbf V(\mathfrak M(\boldsymbol{ j}_1)\otimes \mathfrak M(\boldsymbol{ j}_2)) \simeq\textbf V(\mathfrak M(\boldsymbol{ j}_1))\otimes \textbf V(\mathfrak M(\boldsymbol{ j}_2)). \end{align*}

Hence it suffices to consider the case where $j_{i_0} = 1$ for some $i_0,$ and $j_i = 0$ for $i \neq i_0$ .

Let $\sigma$ denote the Frobenius on $\mathbb F_q$ . One checks that $\sigma(e_i) = e_{i-1}$ (when these idempotents are ordered cyclicly), and hence that $\sigma^*(\mathfrak M(\boldsymbol{ j})) = \mathfrak M(\sigma(\boldsymbol{ j}))$ . As $\omega_n$ has order $q-1,$ we have