3.1.1 Hilbert–Blumenthal abelian varieties

Let $N\geq 4$ be an integer with $p\nmid N$ and let $\mathfrak {c}$ be a nonzero fractional ideal of F. Let S be a scheme over $\mathcal {O}_{K}$ . A Hilbert–Blumenthal abelian variety over S, which we abbreviate as ‘HBAV’, is a quadruple $(A,\iota ,\lambda ,\psi )$ such that the following are true:

• • A is an abelian scheme over S of relative dimension g.

• • $\iota : \mathcal {O}_{F}\to \mathrm {End}_{S}(A)$ is a ring homomorphism.

• • $\lambda $ is a $\mathfrak {c}$ -polarization. Namely, $\lambda :A\otimes _{\mathcal {O}_{F}} \mathfrak {c}\simeq A^{\vee }$ is an isomorphism of abelian schemes to the dual abelian scheme $A^{\vee }$ compatible with $\mathcal {O}_{F}$ -action such that the map

  $$ \begin{align*} \mathrm{Hom}_{\mathcal{O}_{F}}\left(A,A^{\vee}\right)\simeq \mathrm{Hom}_{\mathcal{O}_{F}}\left(A,A\otimes_{\mathcal{O}_{F}} \mathfrak{c}\right),\quad f\mapsto \lambda^{-1}\circ f, \end{align*} $$
  induces an isomorphism of $\mathcal {O}_{F}$ -modules with notion of positivity $\left (\mathcal {P}_{A},\mathcal {P}_{A}^{+}\right )\simeq \left (\mathfrak {c},\mathfrak {c}^{+}\right )$ . Here $\mathcal {P}_{A}$ denotes the $\mathcal {O}_{F}$ -module of symmetric $\mathcal {O}_{F}$ -homomorphisms from A to $A^{\vee }$ , $\mathcal {P}_{A}^{+}$ is the subset of $\mathcal {O}_{F}$ -linear polarizations, and any element $\gamma \in \mathfrak {c}$ is identified with the element $(x\mapsto x\otimes \gamma )$ of $\mathrm {Hom}_{\mathcal {O}_{F}}\left (A,A\otimes _{\mathcal {O}_{F}} \mathfrak {c}\right )$ .

• • $\psi : \mathcal {D}_{F}^{-1}\otimes \mu _{N}\to A$ is an $\mathcal {O}_{F}$ -linear closed immersion of group schemes, which we call a $\Gamma _{00}(N)$ -structure.

Note that for such data, the $\mathcal {O}_{F}\otimes \mathcal {O}_{S}$ -module $\mathrm {Lie}(A)$ is locally free of rank $1$ [Reference Deligne and Pappas20, Corollaire 2.9]. While the notion of an HBAV depends on $\mathfrak {c}$ , we will omit the reference to $\mathfrak {c}$ when there is no risk of confusion.

Let $L/K$ be an extension of complete valuation fields and let $\mathfrak {p}$ be a prime ideal of $\mathcal {O}_{F}$ dividing p. Let $\mathcal {G}$ be a finite flat group scheme over $ \mathcal {O}_{L}$ with an $\mathcal {O}_{F_{\mathfrak {p}}}$ -action. We have decompositions

$$ \begin{align*} \omega_{\mathcal{G}}=\bigoplus_{\beta\in \mathbb{B}_{\mathfrak{p}}} \omega_{\mathcal{G},\beta},\qquad \mathrm{Lie}\left(\mathcal{G}\times \mathscr{S}_{L,n}\right)=\bigoplus_{\beta\in \mathbb{B}_{\mathfrak{p}}} \mathrm{Lie}\left(\mathcal{G}\times \mathscr{S}_{L,n}\right)_{\beta} \end{align*} $$

according with the decomposition $\mathcal {O}_{F_{\mathfrak {p}}}\otimes W\simeq \prod _{\beta \in \mathbb {B}_{\mathfrak {p}}} W$ . Write $\omega _{\mathcal {G},\beta }\simeq \bigoplus _{i} \mathcal {O}_{L}/(a_{i})$ with some $a_{i}\in \mathcal {O}_{L}$ , and define the $\beta $ -degree of $\mathcal {G}$ by $\deg _{\beta }(\mathcal {G})=\sum _{i} v_{p}(a_{i})$ . Similarly, for any finite flat group scheme $\mathcal {H}$ over $\mathcal {O}_{L}$ with an $\mathcal {O}_{F}$ -action, we have decompositions

$$ \begin{align*} \mathcal{H}=\bigoplus_{\mathfrak{p}\mid p} \mathcal{H}_{\mathfrak{p}},\qquad \omega_{\mathcal{H}}=\bigoplus_{\beta\in \mathbb{B}_{F}} \omega_{\mathcal{H},\beta} \end{align*} $$

such that $\mathcal {H}_{\mathfrak {p}}$ is a finite flat closed subgroup scheme of $\mathcal {H}$ over $\mathcal {O}_{L}$ and $\omega _{\mathcal {H},\beta }=\omega _{\mathcal {H}_{\mathfrak {p}},\beta }$ for any $\beta \in \mathbb {B}_{\mathfrak {p}}$ . We put $\deg _{\beta }(\mathcal {H})=\deg _{\beta }\left (\mathcal {H}_{\mathfrak {p}}\right )$ for any $\beta \in \mathbb {B}_{\mathfrak {p}}$ .

Suppose that $\mathcal {G}$ is a truncated Barsotti–Tate group of level n, height h, and dimension d over $\mathcal {O}_{L}$ . For the p-torsion part $\mathcal {G}[p]$ of $\mathcal {G}$ , the Lie algebra $\mathrm {Lie}\left (\mathcal {G}^{\vee }[p] \times \mathscr {S}_{L,1}\right )$ is a free $\mathcal {O}_{L,1}$ -module of rank $h-d$ . The Verschiebung of $\mathcal {G}^{\vee }[p]\times \mathscr {S}_{L,1}$ yields a map

$$ \begin{align*} \mathrm{Lie}\left(V_{\mathcal{G}^{\vee}[p]\times \mathscr{S}_{L,1}}\right): \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times \mathscr{S}_{L,1}\right)^{(p)} \to \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times \mathscr{S}_{L,1}\right). \end{align*} $$

Then the Hodge height $\mathrm {Hdg}(\mathcal {G})$ of $\mathcal {G}$ is by definition the truncated valuation for $v_{p}$ of the determinant of a representing matrix of this map. Moreover, if the ring $\mathcal {O}_{F_{\mathfrak {p}}}$ acts on $\mathcal {G}$ , then this map is also decomposed as the direct sum of maps

$$ \begin{align*} \mathrm{Lie}\left(V_{\mathcal{G}^{\vee}[p]\times\mathscr{S}_{L,1}}\right)_{\beta}: \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times\mathscr{S}_{L,1}\right)_{\sigma^{-1}\circ\beta}\to \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times\mathscr{S}_{L,1}\right)_{\beta}, \end{align*} $$

where $\sigma $ denotes the natural lift to W of the pth-power Frobenius map on k. If the $\mathcal {O}_{L,1}$ -module $\mathrm {Lie}\left (\mathcal {G}^{\vee }[p]\times \mathscr {S}_{L,1}\right )_{\beta }$ is free for any $\beta \in \mathbb {B}_{\mathfrak {p}}$ , we define the $\beta $ -Hodge height $\mathrm {Hdg}_{\beta }(\mathcal {G})$ of $\mathcal {G}$ as the truncated valuation of the determinant of the map $ \mathrm {Lie}\left (V_{\mathcal {G}^{\vee }[p]\times \mathscr {S}_{L,1}}\right )_{\beta }$ . These assumptions are satisfied when $\mathcal {G}$ is the $\mathfrak {p}^{n}$ -torsion part $A[\mathfrak {p}^{n}]=A[p^{n}]_{\mathfrak {p}}$ of an HBAV A over $\mathcal {O}_{L}$ or, more generally, when $\mathcal {G}$ is an $\mathcal {O}_{F_{\mathfrak {p}}}$ - $\mathrm {ADBT}_{n}$ [Reference Hattori29, §3]. For any $\beta \in \mathbb {B}_{\mathfrak {p}}$ , we put $ \mathrm {Hdg}_{\beta }(A)=\mathrm {Hdg}_{\beta }(A[\mathfrak {p}^{n}])$ .

3.1.2 Moduli spaces and toroidal compactifications

Let $M(\mu _{N},\mathfrak {c})$ be the Hilbert modular variety over $\mathcal {O}_{K}$ which parametrizes the isomorphism classes of HBAVs $(A,\iota , \lambda , \psi )$ such that $ \lambda $ is a $\mathfrak {c}$ -polarization and $\psi $ is a $\Gamma _{00}(N)$ -structure. The scheme $M(\mu _{N},\mathfrak {c})$ is smooth over $\mathcal {O}_{K}$ [Reference Goren26, Chapter 3, Theorem 6.9]. We denote by $A^{\mathrm {un}}$ the universal HBAV over $M(\mu _{N},\mathfrak {c})$ .

An unramified cusp for $M(\mu _{N},\mathfrak {c})$ is a triple $(\mathfrak {a},\mathfrak {b}, \phi _{N})$ of fractional ideals $\mathfrak {a},\mathfrak {b}$ of F satisfying $\mathfrak {a}\mathfrak {b}^{-1}=\mathfrak {c}$ and an isomorphism of $\mathcal {O}_{F}$ -modules

$$ \begin{align*} \phi_{N}: \mathfrak{a}^{-1}/N\mathfrak{a}^{-1}\simeq \mathcal{O}_{F}/N\mathcal{O}_{F}. \end{align*} $$

For each cusp, we have a Tate object $\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$ over a certain base scheme [Reference Rapoport36, §4], which is used to construct a toroidal compactification $\bar {M}(\mu _{N},\mathfrak {c})$ of $M(\mu _{N},\mathfrak {c})$ . We recall the definition for unramified cusps. Put $M=\mathfrak {a} \mathfrak {b}$ , $M_{\mathbb {R}}=M\otimes \mathbb {R}$ , and $M^{*}_{\mathbb {R}}=\mathrm {Hom}(M,\mathbb {R})$ . We identify $M\otimes \mathbb {Q}$ with F. Then any $\mathscr {C}\in \mathrm {Dec}(\mathfrak {a},\mathfrak {b})$ gives a rational polyhedral cone decomposition of

$$ \begin{align*} M^{*,+}_{\mathbb{R}}=\left\{f\in M^{*}_{\mathbb{R}}\mid f(M^{+})\subseteq \mathbb{R}_{>0}\right\}. \end{align*} $$

For each $\sigma \in \mathscr {C}$ , put

$$ \begin{align*} \sigma^{\vee} = \{ m\in M_{\mathbb{R}}\mid l(m)\geq 0 \text{ for any } l\in \sigma\}. \end{align*} $$

Then we have an affine torus embedding

$$ \begin{align*} S=\mathrm{Spec}(\mathcal{O}_{K}[q^{m}\mid m\in M])\to S_{\sigma}= \mathrm{Spec}\left(\mathcal{O}_{K}\left[q^{m}\mid m \in M\cap \sigma^{\vee}\right]\right). \end{align*} $$

The affine schemes $\{S_{\sigma }\}_{\sigma \in \mathscr {C}}$ can be glued via $S_{\sigma }\cap S_{\tau }=S_{\sigma \cap \tau }$ to define a torus embedding $S \to S_{\mathscr {C}}$ . We denote by $S^{\infty }_{\sigma }$ and $S^{\infty }_{\mathscr {C}}=\bigcup _{\sigma \in \mathscr {C}} S^{\infty }_{\sigma }$ the complements of S in these embeddings with reduced structures. The formal completions along these closed subschemes are denoted by $\hat {S}_{\sigma }=\mathrm {Spf}\left (\hat {R}_{\sigma }\right )$ and $\hat {S}_{\mathscr {C}}$ . By assumption, we can construct the quotient $\hat {S}_{\mathscr {C}}/U_{N}$ by regluing $\left \{\hat {S}_{\sigma }\right \}_{\sigma \in \mathscr {C}}$ via the action $\epsilon : \hat {S}_{\sigma }\simeq \hat {S}_{\epsilon \sigma }$ for any $\epsilon \in U_{N}$ . The closed subscheme $S^{\infty }_{\sigma }$ is defined by a principal ideal $\hat {I}_{\sigma }$ of the ring $\hat {R}_{\sigma }$ satisfying $\sqrt {\hat {I}_{\sigma }}=\hat {I}_{\sigma }$ . The ring $\hat {R}_{\sigma }$ is a Noetherian normal excellent ring which is complete with respect to the $\hat {I}_{\sigma }$ -adic topology. Put $\bar {S}_{\sigma }=\mathrm {Spec}\left (\hat {R}_{\sigma }\right )$ , $\bar {S}_{\sigma }^{\infty }=V\left (\hat {I}_{\sigma }\right )$ , and $\bar {S}^{0}_{\sigma }=\bar {S}_{\sigma }\setminus \bar {S}_{\sigma }^{\infty }$ , where the latter is an affine scheme and we denote its affine ring by $\hat {R}_{\sigma }^{0}$ .

Note that the torus with character group $\mathfrak {a}$ is $(\mathfrak {a}\mathcal {D}_{F})^{-1}\otimes \mathbb {G}_{\mathrm {m}}$ . For any $\eta \in \mathfrak {a}$ , we denote by $\mathfrak {X}^{\eta }$ the element of $\mathcal {O}\left ((\mathfrak {a}\mathcal {D}_{F})^{-1}\otimes \mathbb {G}_{\mathrm {m}}\right )$ which the character $\eta $ defines. We have an $\mathcal {O}_{F}$ -linear homomorphism

$$ \begin{align*} q: \mathfrak{b} \to (\mathfrak{a}\mathcal{D}_{F})^{-1}\otimes \mathbb{G}_{\mathrm{m}}\left(\bar{S}_{\sigma}^{0}\right) \end{align*} $$

defined by $\xi \mapsto \left (\mathfrak {X}^{\eta }\mapsto q^{\xi \eta }\right )$ with $\xi \in \mathfrak {b}$ and $\eta \in \mathfrak {a}$ . By Mumford’s construction, we obtain the semi-abelian scheme $\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$ over $\bar {S}_{\sigma }$ such that its restriction to $\bar {S}_{\sigma }^{0}$ is an abelian scheme [Reference Rapoport36, §4]. It admits a natural $\mathcal {O}_{F}$ -action. Over $\bar {S}_{\sigma }^{0}$ , we have a natural exact sequence

which defines, for any unramified cusp $(\mathfrak {a},\mathfrak {b},\phi _{N})$ , a $\Gamma _{00}(N)$ -structure on $\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)\rvert _{\bar {S}_{\sigma }^{0}}$ using $\phi _{N}$ . Moreover, the natural isomorphism

$$ \begin{align*} \left((\mathfrak{a}\mathcal{D}_{F})^{-1}\otimes \mathbb{G}_{\mathrm{m}}\right)\otimes_{\mathcal{O}_{F}}\mathfrak{c} \to (\mathfrak{b}\mathcal{D}_{F})^{-1}\otimes \mathbb{G}_{\mathrm{m}} \end{align*} $$

induces a $\mathfrak {c}$ -polarization

$$ \begin{align*} \lambda_{\mathfrak{a},\mathfrak{b}}: \mathrm{Tate}_{\mathfrak{a},\mathfrak{b}}(q)\rvert_{\bar{S}_{\sigma}^{0}} \otimes_{\mathcal{O}_{F}} \mathfrak{c}\to \mathrm{Tate}_{\mathfrak{b},\mathfrak{a}}(q)\rvert_{\bar{S}_{\sigma}^{0}} \simeq \left(\mathrm{Tate}_{\mathfrak{a},\mathfrak{b}}(q)\rvert_{\bar{S}_{\sigma}^{0}}\right)^{\vee}. \end{align*} $$

By these data we consider the Tate object $\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)\rvert _{\bar {S}_{\sigma }^{0}}$ as an HBAV over $\bar {S}_{\sigma }^{0}$ , which yields a morphism $\bar {S} {}_{\sigma }^{0}\to M(\mu _{N},\mathfrak {c})$ . Then the toroidal compactification $\bar {M}^{\mathscr {D}(\mathfrak {c})}(\mu _{N},\mathfrak {c})$ of $M(\mu _{N},\mathfrak {c})$ over $\mathcal {O}_{K}$ with respect to $\mathscr {D}(\mathfrak {c})$ – which we also denote by $\bar {M}(\mu _{N},\mathfrak {c})$ if no confusion will occur – is constructed in such a way as to satisfy the following (see [Reference Rapoport36, Théorème 6.18] and [Reference Dimitrov23, Théorème 7.2 (i)]):

• • $\bar {M}(\mu _{N},\mathfrak {c})$ is projective and smooth over $\mathcal {O}_{K}$ .

• • $M(\mu _{N},\mathfrak {c})$ is an open subscheme of $\bar {M}(\mu _{N},\mathfrak {c})$ which is fiberwise dense and the complement D of $M(\mu _{N},\mathfrak {c})$ is a normal crossing divisor. In particular, $M(\mu _{N},\mathfrak {c})$ is quasi-compact.

• • The formal completion $\bar {M}(\mu _{N},\mathfrak {c})\rvert ^{\wedge }_{D}$ of $\bar {M}(\mu _{N},\mathfrak {c})$ along the boundary divisor D contains

  $$ \begin{align*} \coprod \hat{S}_{\mathscr{C}(\mathfrak{a},\mathfrak{b})}/U_{N} \end{align*} $$
  as a formal open subscheme, where the disjoint union runs over the set of isomorphism classes of unramified cusps.

• • The universal HBAV $A^{\mathrm {un}}$ over $M(\mu _{N},\mathfrak {c})$ extends to a semi-abelian scheme $\bar {A}^{\mathrm {un}}$ with $\mathcal {O}_{F}$ -action over $\bar {M}(\mu _{N},\mathfrak {c})$ such that for any $\sigma \in \mathscr {C}(\mathfrak {a},\mathfrak {b})$ , the pullback of $\bar {A}^{\mathrm {un}}$ by the restriction to $\bar {S}_{\sigma }^{0}$ of the unique algebraization $\bar {S} {}_{\sigma }\to \bar {M}(\mu _{N},\mathfrak {c})$ of the map $\hat {S}_{\sigma }\to \bar {M}(\mu _{N},\mathfrak {c})\rvert _{D}^{\wedge }$ for any unramified cusp $(\mathfrak {a},\mathfrak {b},\phi _{N})$ is isomorphic to $ \mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)\rvert _{\bar {S}_{\sigma }^{0}}$ .

3.1.3 Strict neighborhoods of the ordinary locus and their integral models

Let $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$ be the p-adic formal completion of $\bar {M}(\mu _{N},\mathfrak {c})$ . Let $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$ be its Raynaud generic fiber. Let $ \mathcal {M}(\mu _{N},\mathfrak {c})$ be the analytification of the scheme $M(\mu _{N},\mathfrak {c})\otimes _{\mathcal {O}_{K}}K$ , which is a Zariski open subvariety of $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$ . The semi-abelian scheme $\bar {A}^{\mathrm {un}}$ defines semi-abelian objects $\bar {\mathfrak {A}}^{\mathrm {un}}$ over $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$ and $\bar {\mathcal {A}}^{\mathrm {un}}$ over $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$ by taking the p-adic completion and the Raynaud generic fiber. For the zero section e of $\bar {A}^{\mathrm {un}}$ , put $\omega _{\bar {A}^{\mathrm {un}}}=e^{*}\Omega ^{1}_{\bar {A}^{\mathrm {un}}/\bar {M}\left (\mu _{N},\mathfrak {c}\right )}$ . For any g-tuple $\kappa =\left (k_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$ in $\mathbb {Z}$ , we define

$$ \begin{align*} \omega_{\bar{A}^{\mathrm{un}},\beta}=\omega_{\bar{A}^{\mathrm{un}}}\otimes_{\mathcal{O}_{F}, \beta}\mathcal{O}_{K},\qquad \omega^{\kappa}_{\bar{A}^{\mathrm{un}}}=\bigotimes_{\beta\in \mathbb{B}_{F}} \omega_{\bar{A} {}^{\mathrm{un}},\beta}^{\otimes k_{\beta}}. \end{align*} $$

We also define $\omega _{\bar {\mathcal {A}}^{\mathrm {un}},\beta }$ and $\omega ^{\kappa }_{\bar {\mathcal {A}}^{\mathrm {un}}}$ similarly. For any $\beta \in \mathbb {B}_{F}$ , let $h_{\beta }$ be the $\beta $ -partial Hasse invariant, which is a section of the invertible sheaf $\omega _{\bar {A}^{\mathrm {un}},\sigma ^{-1} \circ \beta }^{p}\otimes \omega _{\bar {A}^{\mathrm {un}},\beta }^{-1}$ on $\bar {M}(\mu _{N},\mathfrak {c})\times \mathscr {S}_{1}$ [Reference Goren and Kassaei27, §2.5] (see also [Reference Andreatta and Goren2, §7] and [Reference Emerton, Reduzzi and Xiao24, §3.1]). For any extension $L/K$ of complete valuation fields, any HBAV A over $\mathcal {O}_{L}$ , and any $\beta \in \mathbb {B}_{F}$ , consider the element P of $\bar {\mathcal {M}} (\mu _{N},\mathfrak {c})(L)$ induced by A and a lift $\tilde {h}_{\beta }$ of $h_{\beta }$ as a section of $\omega _{\bar {A}^{\mathrm {un}},\sigma ^{-1}\circ \beta }^{p}\otimes \omega _{\bar {A}^{\mathrm {un}},\beta }^{-1}$ over an open neighborhood of P. Then we have the equality of truncated valuations

$$ \begin{align*} \mathrm{Hdg}_{\beta}(A)=v_{p}\left(\tilde{h}_{\beta}(P)\right). \end{align*} $$

If $P\in \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(L)$ corresponds to a semi-abelian scheme A over $\mathcal {O}_{L}$ which is not an abelian scheme, then we put $\mathrm {Hdg}_{\beta }(A)=v_{p}\left (\tilde {h}_{\beta }(P)\right )=0$ .

Let $\underline {v}=\left (v_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$ be a g-tuple in $[0,1]\cap \mathbb {Q}$ . We denote by $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ and $\mathcal {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ the admissible open subsets of $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$ and $\mathcal {M}(\mu _{N},\mathfrak {c})$ defined by $v_{p}\left (\tilde {h}_{\beta }(P)\right )\leq v_{\beta }$ for any $\beta \in \mathbb {B}_{F}$ , respectively. Note that $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ is quasi-compact. We define its integral model $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ as follows: Write $v_{\beta }=a_{\beta }/b_{\beta }$ with nonnegative integers $a_{\beta }$ and $b_{\beta }\neq 0$ . Take a formal open covering $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})=\bigcup \mathfrak {U}_{i}$ such that every $h_{\beta }$ lifts to a section $\tilde {h}_{\beta }$ on each $\mathfrak {U}_{i}$ . Consider the formal scheme whose restriction to each $\mathfrak {U}_{i}$ is the admissible blowup of $\mathfrak {U}_{i}$ along the ideal $\left (p^{a_{\beta }},\tilde {h}_{\beta }^{b_{\beta }}\right )$ , and its locus where this ideal is generated by $\tilde {h}_{\beta }^{b_{\beta }}$ . Repeat this for any $\beta \in \mathbb {B}_{F}$ and define $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ as the normalization in $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ of the resulting formal scheme. We denote the special fibers of $\bar {M}(\mu _{N},\mathfrak {c})$ and $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c}) \left (\underline {v}\right )$ by $\bar {M}(\mu _{N},\mathfrak {c})_{k}$ and $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )_{k}$ , respectively. We also denote by $\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ the complement in $\bar {\mathfrak {M}} (\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ of the boundary divisor $D_{k}$ of the special fiber. Then the open immersion $\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ is of finite type and [Reference de Jong19, Proposition 7.2.4] implies that $\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )^{\mathrm {rig}}$ is quasi-compact.

Let v be an element of $[0,1]\cap \mathbb {Q}$ . When $v_{\beta }=v$ for any $\beta \in \mathbb {B}_{F}$ , we write $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ as $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)$ . Moreover, we denote by $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$ the quasi-compact admissible open subset defined similarly to $\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)$ with the usual Hasse invariant

$$ \begin{align*} h_{\mathrm{tot}}=\prod_{\beta\in \mathbb{B}_{F}}h_{\beta} \end{align*} $$

instead of $h_{\beta }$ s. We also define similar spaces for these two variants, such as $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v)$ and $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$ . Note that $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$ is just the formal open subscheme of $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$ over which all the $\beta $ -partial Hasse invariants are invertible.

The formal scheme $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$ is also a formal open subscheme of $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ containing $D_{k}$ . Let $\mathrm {sp}:\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})_{k}$ be the specialization map with respect to $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$ . Then [Reference de Jong19, Propositions 7.2.1 and 7.2.4] yields

(3.1) $$ \begin{align} \mathfrak{M}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)^{\mathrm{rig}}=\bar{\mathcal{M}}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)\setminus \mathrm{sp}^{-1}(D_{k}). \end{align} $$

Let R be a topological $\mathcal {O}_{K}$ -algebra which is idyllic with respect to the p-adic topology [Reference Abbes1, Définition 1.10.1]. By [Reference Abbes1, Corollaire 2.13.9], any morphism $\hat {f}:\mathrm {Spf}(R)\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$ has a unique algebraization $f:\mathrm {Spec}(R)\to \bar {M}(\mu _{N},\mathfrak {c})$ , and we have a semi-abelian scheme $G_{R}=f^{*}\bar {A}^{\mathrm {un}}$ over $\mathrm {Spec}(R)$ . Taking the reduction modulo p, we see that $\hat {f}$ factors through $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$ if and only if $G_{R}$ is ordinary.

Let $\mathbf {NAdm}$ be the category of admissible p-adic formal $\mathcal {O}_{K}$ -algebras R such that R is normal. Note that we have $R[1/p]^{\circ }=R$ by [Reference Bosch, Güntzer and Remmert10, remark after Proposition 6.3.4/1]. By [Reference Rapoport36, Lemme 3.1], we can see as in [Reference Andreatta, Iovita and Pilloni3, Proposition 5.2.1.1] that any morphism $\mathrm {Sp}(R[1/p])\to \mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )^{\mathrm {rig}}$ corresponds uniquely to an isomorphism class of an HBAV A over $\mathrm {Spec}(R)$ such that $\mathrm {Hdg}_{\beta }(A_{x})\leq v_{\beta }$ for $x\in \mathrm {Sp}(R[1/p])$ .

We give a proof of the following lemma for lack of a reference:

Proof. Take an admissible affinoid covering $\mathcal {X}=\bigcup _{i\in I} \mathcal {X}_{i}$ such that $\mathcal {X}_{i}$ is connected and $\mathcal {F}$ is trivial on $\mathcal {X}_{i}$ for any $i\in I$ . We have $\mathcal {X}_{i_{0}}\cap \mathcal {U}\neq \emptyset $ for some $i_{0}$ . Then [Reference Fresnel and van der Put25, Exercise 4.6.3] implies $f\rvert _{\mathcal {X}_{i_{0}}}=0$ .

Put $I_{0}=\left \{i\in I\mid f\rvert _{\mathcal {X}_{i}}=0\right \}$ , which is nonempty. [Reference Fresnel and van der Put25, Exercise 4.6.3] also implies that $\mathcal {X}_{i}\cap \mathcal {X}_{j}=\emptyset $ for any $i\in I_{0}$ and $j\in I_{1}:=I\setminus I_{0}$ . Then for the subsets

$$ \begin{align*} \mathcal{X}_{0}=\bigcup_{i\in I_{0}} \mathcal{X}_{i},\qquad \mathcal{X}_{1}=\bigcup_{i\in I_{1}} \mathcal{X}_{i} \end{align*} $$

and $s\in \{0,1\}$ , the intersection $\mathcal {X}_{s}\cap \mathcal {X}_{i}$ equals $\mathcal {X}_{i}$ if $i\in I_{s}$ and $\emptyset $ if $i\notin I_{s}$ . Hence $\mathcal {X}=\mathcal {X}_{0}\coprod \mathcal {X}_{1}$ is an admissible covering of $\mathcal {X}$ . Since $\mathcal {X}$ is connected, we obtain $\mathcal {X}=\mathcal {X}_{0}$ and $f=0$ .

Proof. Take a surjection $\mathcal {O}_{L}\langle T_{1},\dotsc ,T_{n}\rangle \to \mathfrak {R}$ . Then [Reference Bosch, Lütkebohmert and Raynaud11, Proposition 1.1] shows $(\mathfrak {R}[1/p])^{\circ }=\mathfrak {R}$ . Moreover, by [Reference Bosch, Güntzer and Remmert10, remark after Proposition 6.3.4/1] $\mathfrak {R}$ is integrally closed in $\mathfrak {R}[1/p]$ . Thus we have bijections between the sets of connected components

$$ \begin{align*} \pi_{0}(\mathrm{Sp}(\mathfrak{R}[1/p]))\simeq \pi_{0}(\mathrm{Spec}(\mathfrak{R}))\simeq \pi_{0}(\mathrm{Spec}(\mathfrak{R}/m_{L} \mathfrak{R})), \end{align*} $$

from which the lemma follows.

Lemma 3.3. Let $l/k$ be a finite extension and put $W^{\prime }=W(l)$ . Let $K^{\prime }$ be the composite field of K and $\mathrm {Frac}(W^{\prime })$ in $\bar {\mathbb {Q}}_{p}$ . Let R be a Noetherian W-algebra and let m be a maximal ideal of R with residue field l. Consider the natural maps

$$ \begin{align*} R\otimes_{W} W^{\prime}\to l,\qquad R\otimes_{W} \mathcal{O}_{K}\to l,\qquad R\otimes_{W} \mathcal{O}_{K^{\prime}}\to l \end{align*} $$

induced by $R\to R/m=l$ . We denote the kernels of these maps by $m^{\prime }$ , n, and $n^{\prime }$ , respectively. Then the natural maps between complete local rings

$$ \begin{align*} R^{\wedge}_{m}\to (R\otimes_{W} W^{\prime})^{\wedge}_{m^{\prime}},\qquad R_{m}^{\wedge}\hat{\otimes}_{W}\mathcal{O}_{K}\to (R\otimes_{W}\mathcal{O}_{K})_{n}^{\wedge} \end{align*} $$

are bijective. In particular, we have isomorphisms

$$ \begin{align*} (R\otimes_{W} \mathcal{O}_{K})^{\wedge}_{n}\to \left(R\otimes_{W}\mathcal{O}_{K^{\prime}}\right)_{n^{\prime}}^{\wedge}\gets(R\otimes_{W} W^{\prime})_{m^{\prime}}^{\wedge}\hat{\otimes}_{W^{\prime}}\mathcal{O}_{K^{\prime}}. \end{align*} $$

Lemma 3.4. Let $\mathcal {X}_{v}$ be either of the rigid analytic varieties

$$ \begin{align*} \bar{\mathcal{M}}(\mu_{N},\mathfrak{c})(v),\qquad \bar{\mathcal{M}}(\mu_{N},\mathfrak{c})(v_{\mathrm{tot}}). \end{align*} $$

Then $\mathcal {X}_{v}\hat {\otimes }_{K}{\mathbb {C}_{p}}$ is connected for any $v\in [0,1)\cap \mathbb {Q}$ .

Proof. Since $\mathcal {X}_{v}$ is separated, it is enough to show that for any sufficiently large finite extension $K^{\prime }/K$ , the base extension $\mathcal {X}_{v}\hat {\otimes }_{K}{K^{\prime }}$ is connected [Reference Conrad17, Theorem 3.2.1]. Replacing K by $K^{\prime }$ , we may assume that $K^{\prime }=K$ and every connected component of $\mathcal {X}_{v}$ has a K-rational point.

By Ribet’s theorem (see [Reference Goren26, Chapter 3, Theorem 6.19]), the ordinary locus ${\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}$ is geometrically connected. Then $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}$ is also geometrically connected, since it is smooth over k and contains ${\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}$ as a dense open subset. Since $\mathcal {X}_{0}$ is the tube for the immersion $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$ , [Reference Berthelot9, Proposition 1.3.3] yields the lemma for $v=0$ .

Consider the case of $v>0$ . Suppose that $\mathcal {X}_{v}$ is not connected. Then we can take its connected component $\mathcal {U}$ which does not intersect $\mathcal {X}_{0}$ . Since $\mathcal {U}$ is quasi-compact, there exists a finite admissible affinoid covering $\mathcal {U}=\bigcup _{i=1}^{m} \mathcal {U}_{i}$ of $\mathcal {U}$ such that any $\beta $ -partial Hasse invariant can be lifted to a section over $\mathcal {U}_{i}$ . Using the maximal modulus principle on each $\mathcal {U}_{i}$ , we see that there exists a positive rational number $\delta $ satisfying

$$ \begin{align*} \max\left\{\mathrm{Hdg}_{\beta}(x)\mid \beta\in \mathbb{B}_{F}\right\}\geq \delta \end{align*} $$

for any $x\in \mathcal {U}$ . Then for any rational number $\varepsilon $ satisfying $0<\varepsilon <\delta $ , we have $\mathcal {X}_{\varepsilon }\cap \mathcal {U}=\emptyset $ .

On the other hand, let us consider the specialization map

$$ \begin{align*} \mathrm{sp}:\bar{\mathcal{M}}(\mu_{N},\mathfrak{c})\to \bar{\mathfrak{M}}(\mu_{N},\mathfrak{c})_{k} \end{align*} $$

with respect to $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$ . Take any $P\in \mathcal {U}(K)$ and consider its specialization $\bar {P}=\mathrm {sp}(P)$ . Since $P\notin \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(0)$ , the point $\bar {P}$ corresponds to an HBAV. By [Reference Goren and Kassaei27, (2.5.1)] and Lemma 3.3, the complete local ring of $M(\mu _{N},\mathfrak {c})$ at $\bar {P}$ is isomorphic to the ring $\mathcal {O}_{K}\left [\left [t_{\beta }\mid \beta \in \mathbb {B}_{F}\right ]\right ]$ . Then [Reference de Jong19, Lemma 7.2.5] gives an identification

(3.2) $$ \begin{align} \mathrm{sp}^{-1}\left(\bar{P}\right)=\prod_{\beta\in\mathbb{B}_{F}} \mathcal{A}_{\beta}[0,1), \end{align} $$

where for any interval I we denote by $\mathcal {A}_{\beta } I$ the annulus over K with parameter $t_{\beta }$ defined by the condition $\left \lvert t_{\beta }\right \rvert \in I$ . By [Reference Goren and Kassaei27, §4.2], we may assume that the parameter $t_{\beta }$ satisfies

(3.3) $$ \begin{align} \mathrm{Hdg}_{\beta}(A)= \begin{cases} \min\{v_p(t_\beta(Q)),1\} & \left(\beta\in \tau\left(\bar{P}\right)\right) \\ 0 & \left(\beta\notin \tau\left(\bar{P}\right)\right) \end{cases} \end{align} $$

for any $Q\in \mathrm {sp}^{-1}\left (\bar {P}\right )$ and for any $\beta \in \mathbb {B}_{F}$ , where A is the HBAV corresponding to Q and $\tau \left (\bar {P}\right )$ is defined by [Reference Goren and Kassaei27, (2.3.3)].

Suppose $\mathcal {X}_{v}=\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)$ . For any positive rational number $\varepsilon $ , put

$$ \begin{align*} \mathrm{sp}^{-1}\left(\bar{P}\right)(\varepsilon)^{\prime}=\prod_{\beta\in\tau\left(\bar{P}\right)} \mathcal{A}_{\beta}\left[p^{-\varepsilon},1\right)\times \prod_{\beta\notin\tau\left(\bar{P}\right)} \mathcal{A}_{\beta}[0,1). \end{align*} $$

Since $\mathrm {sp}^{-1}\left (\bar {P}\right )(v)^{\prime }$ is a connected admissible open subset of $\mathcal {X}_{v}$ containing P, it is contained in $\mathcal {U}$ . However, for any $\varepsilon $ satisfying $\varepsilon <\min \{\delta ,v\}$ , we have

$$ \begin{align*} \emptyset \neq \mathrm{sp}^{-1}\left(\bar{P}\right)(\varepsilon)^{\prime}\subseteq \mathcal{X}_{\varepsilon}\cap \mathcal{U}, \end{align*} $$

which is a contradiction.

When $\mathcal {X}_{v}=\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$ , take any rational number $\varepsilon $ satisfying $0<\varepsilon <\min \{\delta ,v\}$ and $v_{p}\left (t_{\beta }(P)\right )\geq \varepsilon $ for any $\beta \in \mathbb {B}_{F}$ . Put $w=g^{-1}\varepsilon $ . Let $\mathcal {V}$ be the admissible affinoid open subset of $\mathrm {sp}^{-1}\left (\bar {P}\right )$ defined as

$$ \begin{align*} \mathcal{V}=\left\{\left(t_{\beta}\right)_{\beta\in \mathbb{B}_{F}}\in \prod_{\beta\in\mathbb{B}_{F}} \mathcal{A}_{\beta}\left[0,p^{-w}\right]\ \middle|\ \prod_{\beta\in\tau\left(\bar{P}\right)}\left\lvert t_{\beta}\right\rvert\geq p^{-v}\right\}. \end{align*} $$

Since $P\in \mathcal {V}\subseteq \mathcal {X}_{v}$ and $\mathcal {V}\cap \mathcal {X}_{\varepsilon }\neq \emptyset $ , by an argument as before we reduce ourselves to showing that $\mathcal {V}$ is connected.

For this, replacing K by its finite extension, we may assume that there exist $a,b\in \mathcal {O}_{K}$ satisfying $v_{p}(a)=v$ and $v_{p}(b)=w$ . Consider the ring

$$ \begin{align*} \mathfrak{R}=\mathcal{O}_{K}\left\langle T_{\beta}, U_{\beta}\mid \beta\in \mathbb{B}_{F}\right\rangle\langle T\rangle/\left(Tf-a,b U_{\beta}-T_{\beta}\mid \beta\in \mathbb{B}_{F}\right) \end{align*} $$

with $f=\prod _{\beta \in \tau \left (\bar {P}\right )}T_{\beta }$ . Then $\mathcal {V}=\mathrm {Sp}(\mathfrak {R}[1/p])$ and the special fiber

$$ \begin{align*} \mathrm{Spec}\left(k\left[T_{\beta}, U_{\beta}\mid \beta\in \mathbb{B}_{F}\right][T]/\left(Tf,T_{\beta}\mid \beta\in \mathbb{B}_{F}\right)\right)=\mathrm{Spec}\left(k\left[U_{\beta}\mid \beta\in \mathbb{B}_{F}\right][T]\right) \end{align*} $$

is reduced and connected. Then Lemma 3.2 concludes the proof.

3.1.4 Canonical subgroups over moduli spaces

Let n be a positive integer. Let $\underline {v}=\left (v_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$ be a g-tuple satisfying

$$ \begin{align*} v_{\beta}\in [0,(p-1)/p^{n})\cap \mathbb{Q} \end{align*} $$

for any $\beta \in \mathbb {B}_{F}$ . Note that the $1/(p^{n}(p-1))$ th lower ramification subgroups can be patched into a rigid analytic family [Reference Hattori28, Lemma 5.6]. Let R be an object of $\mathbf {NAdm}$ and put $\mathcal {U}=\mathrm {Sp}(R[1/p])$ . Let $\mathcal {U}\to \mathfrak {M}(\mu _{N}, \mathfrak {c})\left (\underline {v}\right )^{\mathrm {rig}}$ be any morphism of rigid analytic varieties over K. This defines an HBAV $\bar {A}^{\mathrm {un}}\rvert _{R}$ over $\mathrm {Spec}(R)$ . For any rig-point $x\in \mathrm {Spec}(R)$ , we have the canonical subgroup $\mathcal {C}_{n}\left (\left (\bar {A}^{\mathrm {un}}\rvert _{R}\right )_{x}\right )$ by [Reference Hattori29, Theorem 8.1]. [Reference Hattori29, Theorem 8.1(7)] implies that they can be patched into an admissible open $\mathcal {O}_{F}$ -submodule $C_{n}$ of $\bar {\mathcal {A}}^{\mathrm {un}}[p^{n}]\rvert _{\mathcal {U}}$ . By [Reference Andreatta, Iovita and Pilloni3, Proposition 4.1.3], it uniquely extends to a finite flat subgroup scheme $\mathcal {C}_{n}$ of $\bar {A}^{\mathrm {un}}\rvert _{R}$ over $ \mathrm {Spec}(R)$ . Since $\mathcal {C}_{n}$ agrees with the scheme-theoretic closure of $C_{n}$ in $\left (\bar {A}^{\mathrm {un}}\rvert _{R}\right )[p^{n}]$ , we see that $\mathcal {C}_{n}$ is stable under the $\mathcal {O}_{F}$ -action.

On the other hand, on a formal open neighborhood $\mathfrak {U}$ of a point of the boundary satisfying $\mathfrak {U}\subseteq \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$ , the unit component $\bar {\mathfrak {A}}^{\mathrm {un}}[p^{n}]^{0}\rvert _{\mathfrak {U}}$ is quasi-finite and flat over $\mathfrak {U}$ with constant degree on each fiber by [Reference Rapoport36, p. 297(ii)]. Thus it is finite and flat. Then by gluing along $\mathfrak {M}(\mu _{N},\mathfrak {c})(0)$ , we obtain a finite flat formal subgroup scheme $\mathcal {C}_{n}$ of $\bar {\mathfrak {A}}^{\mathrm {un}}$ over $\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ and its generic fiber $C_{n}$ , which we refer to as the canonical subgroup of level n.

Let R be a topological $\mathcal {O}_{K}$ -algebra which is quasi-idyllic with respect to the p-adic topology [Reference Abbes1, 1.10.1.1]. Since any finitely generated R-module is automatically p-adically complete [Reference Abbes1, Proposition 1.10.2], any finitely presented flat formal group scheme over $\mathrm {Spf}(R)$ can be identified with a finitely presented flat group scheme over $\mathrm {Spec}(R)$ . Thus we have a theory of Cartier duality for any finitely presented flat formal group scheme $\mathcal {G}$ over any quasi-idyllic p-adic formal scheme, and we can define the Hodge–Tate map

$$ \begin{align*} \mathrm{HT}_{\mathcal{G}}:\mathcal{G}(R)\to \omega_{\mathcal{G}^{\vee}},\quad x\mapsto x^{*}\left(\frac{dT}{T}\right). \end{align*} $$

From the construction, we see that the restriction of the Cartier dual $\mathcal {C}_{n}^{\vee }\rvert _{\bar {\mathfrak {M}}\left (\mu _{N},\mathfrak {c}\right )(0)}$ to the ordinary locus is finite and étale.

Note that for the function

$$ \begin{align*} \delta_{K}(v)=\min\left([v/(p-1),+\infty)\cap e^{-1}\mathbb{Z}\right), \end{align*} $$

the ideal $m_{K}^{\geqslant v/(p-1)}$ is generated by $\pi _{K}^{e\delta _{K}(v)}$ . Then we have the following variant of [Reference Andreatta, Iovita and Pilloni3, Propositions 4.2.1 and 4.2.2]:

Lemma 3.5. Let $\underline {v}=\left (v_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$ be a g-tuple of nonnegative rational numbers satisfying

$$ \begin{align*} v:=\max\left\{v_{\beta}\mid \beta\in \mathbb{B}_{F}\right\}< (p-1)/p^{n}. \end{align*} $$

Let R be an object of $\mathbf {NAdm}$ . For any morphism of admissible formal schemes $\hat {f}:\mathrm {Spf}(R)\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$ over $\mathcal {O}_{K}$ , consider the pullback $G=\bar {A}^{\mathrm {un}}\rvert _{R}$ by the unique algebraization $\mathrm {Spec}(R)\to \bar {M}(\mu _{N},\mathfrak {c})$ of $\hat {f}$ and $\mathcal {H}_{n}=\mathcal {C}_{n}\rvert _{\mathrm {Spf}(R)}$ , which is a subgroup scheme of the formal completion of G.

1. (1) For any rational number $i\in e^{-1}\mathbb {Z}_{\geq 0}$ satisfying $i\leq n-v(p^{n}-1)/(p-1)$ , the natural map $\omega _{G}\otimes _{\mathcal {O}_{K}} \mathcal {O}_{K,i}\to \omega _{\mathcal {H}_{n}} \otimes _{\mathcal {O}_{K}} \mathcal {O}_{K,i}$ is an isomorphism.

2. (2) Assume that we have an isomorphism of $\mathcal {O}_{F}$ -modules $\mathcal {H}_{n}^{\vee }(R)\simeq \mathcal {O}_{F} /p^{n}\mathcal {O}_{F}$ . If there exists a rational number $i\in e^{-1}\mathbb {Z}_{\geq 0}$ satisfying $\delta _{K}(v)<i\leq n-v(p^{n}-1)/(p-1)$ , then the cokernel of the linearization of the Hodge–Tate map

   $$ \begin{align*} \mathrm{HT}_{\mathcal{H}^{\vee}_{n}}\otimes 1: \mathcal{H}_{n}^{\vee}(R)\otimes R\to \omega_{\mathcal{H}_{n}} \end{align*} $$
   is killed by $m_{K}^{\geqslant v/(p-1)}$ .