2.1 Notation

Fix a prime p and a totally real field F of degree $d = [F:\mathbb {Q}]$ . We let ${\mathcal {O}}_F$ denote the ring of integers of F, ${\mathfrak {d}}$ its different, $S_p$ the set of primes over p. For each ${\mathfrak {p}} \in S_p$ , we write $F_{\mathfrak {p}}$ for the completion of F at ${\mathfrak {p}}$ , and we let $F_{{\mathfrak {p}},0}$ denote its maximal unramified subextension, $\mathbb {F}_{\mathfrak {p}}$ the residue field ${\mathcal {O}}_F/{\mathfrak {p}}$ , $f_{\mathfrak {p}}$ the residue degree $[F_{{\mathfrak {p}},0}:{\mathbb {Q}}_p] = [\mathbb {F}_{\mathfrak {p}}:\mathbb {F}_p]$ , and $e_{\mathfrak {p}}$ the ramification index $[F_{\mathfrak {p}}:F_{{\mathfrak {p}},0}]$ . We also fix a choice of totally positive $\varpi _{\mathfrak {p}} \in F$ so that $v_{\mathfrak {p}}(\varpi _{\mathfrak {p}}) = 1$ and $v_{{\mathfrak {p}}'}(\varpi _{\mathfrak {p}}) = 0$ for all other ${\mathfrak {p}}' \in S_p$ . In particular, $\varpi _{\mathfrak {p}}$ is a uniformizer in $F_{\mathfrak {p}}$ , and for each ${\mathfrak {p}} \in S_p$ , and let $E_{\mathfrak {p}}(u) \in W(\mathbb {F}_{\mathfrak {p}})[u]$ denote its minimal polynomial over $F_{{\mathfrak {p}},0}$ , whose ring of integers we identify with $W(\mathbb {F}_{\mathfrak {p}})$ .

We adopt much of the notation and conventions of [Reference DiamondDia23] for notions and constructions associated to embeddings of F. In particular, for each ${\mathfrak {p}} \in S_p$ , we let $\Sigma _{{\mathfrak {p}}}$ denote the set of embeddings $F_{{\mathfrak {p}}} \to \overline {{\mathbb {Q}}}_p$ , and we identify $\Sigma := \coprod _{{\mathfrak {p}} \in S_p} \Sigma _{{\mathfrak {p}}}$ with the set of embeddings $F \hookrightarrow \overline {{\mathbb {Q}}}_p$ via the canonical bijection. We also let $\Sigma _{{\mathfrak {p}},0}$ denote the set of embeddings $F_{{\mathfrak {p}},0} \to \overline {{\mathbb {Q}}}_p$ , which we identify with the set of embeddings $W(\mathbb {F}_{\mathfrak {p}}) \to W(\overline {{\mathbb {F}}}_p)$ , or equivalently $\mathbb {F}_{\mathfrak {p}} \to \overline {{\mathbb {F}}}_p$ , and we let $\Sigma _0 = \coprod _{{\mathfrak {p}} \in S_p} \Sigma _{{\mathfrak {p}},0}$ .

For each ${\mathfrak {p}} \in S_p$ , we fix a choice of embedding $\tau _{{\mathfrak {p}},0} \in \Sigma _{{\mathfrak {p}},0}$ , and for $i \in {\mathbb {Z}}/f_{\mathfrak {p}}{\mathbb {Z}}$ , we let $\tau _{{\mathfrak {p}},i} = \phi ^i\circ \tau _{{\mathfrak {p}},0}$ where $\phi $ is the Frobenius automorphism of $\overline {{\mathbb {F}}}_p$ . We also fix an ordering $\theta _{{\mathfrak {p}},i,1},\theta _{{\mathfrak {p}},i,2},\ldots ,\theta _{{\mathfrak {p}},i,e_{\mathfrak {p}}}$ of the embeddings $\theta \in \Sigma _{\mathfrak {p}}$ restricting to $\tau _{{\mathfrak {p}},i}$ , so that

$$ \begin{align*}\Sigma = \{\,\theta_{{\mathfrak{p}},i,j}\,|\,{\mathfrak{p}} \in S_p, i \in \mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z}, 1 \le j \le e_{\mathfrak{p}}\,\}.\end{align*} $$

Finally, we let $\sigma $ denote the permutation of $\Sigma $ defined by $\sigma (\theta _{{\mathfrak {p}},i,j}) = \theta _{{\mathfrak {p}},i,j+1}$ if $j \neq e_{\mathfrak {p}}$ and $\sigma (\theta _{{\mathfrak {p}},i,e_{\mathfrak {p}}}) = \theta _{{\mathfrak {p}},i+1,1}$ .

Choose a finite extension K of ${\mathbb {Q}}_p$ sufficiently large to contain the images of all $\theta \in \Sigma $ ; let ${\mathcal {O}}$ denote its ring of integers, $\varpi $ a uniformizer, and k its residue field.

For $\tau \in \Sigma _{{\mathfrak {p}},0}$ , we define $E_\tau (u) \in {\mathcal {O}}[u]$ to be the image of $E_{\mathfrak {p}}$ under the homomorphism induced by $\tau $ , so that

$$ \begin{align*}{\mathcal{O}}_F \otimes {\mathcal{O}} = \bigoplus_{{\mathfrak{p}} \in S_p} \bigoplus_{\tau \in \Sigma_{{\mathfrak{p}},0}} {\mathcal{O}}_{F,{\mathfrak{p}}} \otimes_{W(\mathbb{F}_{\mathfrak{p}}),\tau} {\mathcal{O}} \cong \bigoplus_{\tau \in \Sigma_0} {\mathcal{O}}[u]/(E_\tau).\end{align*} $$

For any ${\mathcal {O}}_F \otimes {\mathcal {O}}$ -module M, we obtain a corresponding decomposition $M = \bigoplus _{\tau \in \Sigma _{0}} M_\tau $ , and we also write $M_{{\mathfrak {p}},i}$ for $M_\tau $ if $\tau = \tau _{{\mathfrak {p}},i}$ . Similarly, if S is a scheme over ${\mathcal {O}}$ and ${\mathcal {M}}$ is a quasi-coherent sheaf of ${\mathcal {O}}_F \otimes {\mathcal {O}}_S$ -modules on S, then we write ${\mathcal {M}}_\tau = {\mathcal {M}}_{{\mathfrak {p}},i}$ for the corresponding summand of ${\mathcal {M}}$ .

For each $\theta = \theta _{{\mathfrak {p}},i,j} \in \Sigma $ , we factor $E_{\tau } = s_\theta t_\theta $ where $\tau = \tau _{{\mathfrak {p}},i}$ ,

(2.1) $$ \begin{align} \begin{array}{rcccc} s_{\theta} &=&s_{\tau,j}& =& (u-\theta_{{\mathfrak{p}},i,1}(\varpi_{\mathfrak{p}}))\cdots(u-\theta_{{\mathfrak{p}},i,j}(\varpi_{\mathfrak{p}}))\\ \mbox{and}\quad t_{\theta} &=& t_{\tau,j}& = & (u-\theta_{{\mathfrak{p}},i,j+1}(\varpi_{\mathfrak{p}}))\cdots(u-\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}(\varpi_{\mathfrak{p}}));\end{array}\end{align} $$

note that the ideals $(s_\theta )$ and $(t_\theta )$ in ${\mathcal {O}}[u]/(E_\tau )$ are each other’s annihilators, and that the corresponding ideals in ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W(\mathbb {F}_{\mathfrak {p}}),\tau } {\mathcal {O}}$ are independent of the choice of the uniformizer $\varpi _{\mathfrak {p}}$ .

2.2 The Pappas-Rapoport model

We now recall the definition, due to Pappas and Rapoport, for smooth integral models of Hilbert modular varieties of level prime to p,

We let ${\mathbb {A}}_{F,\mathbf {f}} = F \otimes \widehat {\mathbb {Z}}$ denote the finite adeles of F, and we let ${\mathbb {A}}_{F,\mathbf {f}}^{(p)} = {\mathbb {Q}} \otimes \widehat {{\mathcal {O}}}_F^{(p)} = F \otimes \widehat {\mathbb {Z}}^{(p)}$ , where $\widehat {{\mathcal {O}}}_F^{(p)}$ (resp. $\widehat {\mathbb {Z}}^{(p)}$ ) is the prime-to-p completion of ${\mathcal {O}}_F$ (resp. $\mathbb {Z}$ ), so $\widehat {\mathbb {Z}}^{(p)} = \prod _{\ell \neq p} \mathbb {Z}_\ell $ and $\widehat {{\mathcal {O}}}_F^{(p)} = {\mathcal {O}}_F \otimes \widehat {\mathbb {Z}}^{(p)} = \prod _{v\nmid p} {\mathcal {O}}_{F,v}$ .

Let U be an open compact subgroup of $\operatorname {GL}_2(\widehat {{\mathcal {O}}}_F) \subset \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}})$ of the form $U_pU^p$ , where $U_p = \operatorname {GL}_2({\mathcal {O}}_{F,p})$ . Consider the functor which associates, to a locally Noetherian ${\mathcal {O}}$ -scheme S, the set of isomorphism classes of data $(A,\iota ,\lambda ,\eta ,{\mathcal {F}}^\bullet )$ , where

• • $s:A \to S$ is an abelian scheme of relative dimension d;

• • $\iota : {\mathcal {O}}_F\to {\operatorname {End}\,}_S(A)$ is a homomorphism;

• • $\lambda $ is an ${\mathcal {O}}_F$ -linear quasi-polarization of A such that for each connected component $S_i$ of S, $\lambda $ induces an isomorphism ${\mathfrak {c}}_i{\mathfrak {d}} \otimes _{{\mathcal {O}}_F} A_{S_i} \to A_{S_i}^\vee $ for some fractional ideal ${\mathfrak {c}}_i$ of F prime to p;

• • $\eta $ is a level $U^p$ structure on A; that is, a $\pi _1(S_i,\overline {s}_i)$ -invariant $U^p$ -orbit of $\widehat {{\mathcal {O}}}_F^{(p)}$ -linear isomorphisms

  $$ \begin{align*}\eta_i :(\widehat{{\mathcal{O}}}_F^{(p)})^2 \to {\mathfrak{d}} \otimes_{{\mathcal{O}}_F} T^{(p)}(A_{\overline{s}_i})\end{align*} $$
  for a choice of geometric point $\overline {s}_i$ on each connected component $S_i$ of S, where $T^{(p)}$ denotes the product over $\ell \neq p$ of the $\ell $ -adic Tate modules, and $g \in U^p$ acts on $\eta _i$ by pre-composing with right multiplication by $g^{-1}$ ;

• • ${\mathcal {F}}^\bullet $ is a collection of Pappas–Rapoport filtrations; that is, for each $\tau = \tau _{{\mathfrak {p}},i} \in \Sigma _0$ , an increasing filtration of ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W(\mathbb {F}_{\mathfrak {p}}),\tau } {\mathcal {O}}_S$ -modules

  $$ \begin{align*}0 = {\mathcal{F}}_\tau^{(0)} \subset {\mathcal{F}}_\tau^{(1)} \subset \cdots \subset {\mathcal{F}}_\tau^{(e_{\mathfrak{p}} - 1)} \subset {\mathcal{F}}_\tau^{(e_{\mathfrak{p}})} = (s_*\Omega_{A/S}^1)_\tau\end{align*} $$
  such that for $j=1,\ldots ,e_{{\mathfrak {p}}}$ , the quotient $ {\mathcal {F}}_\tau ^{(j)}/{\mathcal {F}}_\tau ^{(j-1)}$ is a line bundle on S on which ${\mathcal {O}}_F$ acts via $\theta _{{\mathfrak {p}},i,j}$ .

If $U^p$ is sufficiently small, then the functor is representable by an infinite disjoint union $\widetilde {Y}_U$ of smooth, quasi-projective schemes of relative dimension d over ${\mathcal {O}}$ . Furthermore, we have an action of $\nu \in {\mathcal {O}}_{F,(p),+}^\times $ on $\widetilde {Y}_U$ defined by composing the quasi-polarization with $\iota (\nu )$ , and the action factors through a free action of ${\mathcal {O}}_{F,(p),+}^\times /(U\cap {\mathcal {O}}_F^\times )^2$ by which the quotient is representable by a smooth quasi-projective scheme of relative dimension d over ${\mathcal {O}}$ , which we denote by $Y_U$ . We also have a natural right action of $g \in \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on the inverse system of schemes $\widetilde {Y}_U$ defined by pre-composing the level structure $\eta $ with right-multiplication by $g^{-1}$ , and the action descends to one on the inverse system of schemes $Y_U$ . Furthermore, we have a compatible system of isomorphisms of the $Y_U(\mathbb {C})$ (for any choice of ${\mathcal {O}} \to \mathbb {C}$ ) with the Hilbert modular varieties

$$ \begin{align*} \operatorname{GL}_2(F)_+\backslash (\mathfrak{H}^\Sigma \times \operatorname{GL}_2({\mathbb{A}}_{F,\mathbf{f}})/U) \end{align*} $$

under which the action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ corresponds to right multiplication.

2.3 Automorphic bundles

Let $\underline {A} = (A,\iota ,\lambda ,\eta ,{\mathcal {F}}^\bullet )$ denote the universal object over $\widetilde {Y}_U$ . Recall that ${\mathcal {H}}^1_{\operatorname {dR}}(A/\widetilde {Y}_U)$ is a (Zariski-)locally free sheaf of rank two ${\mathcal {O}}_F \otimes {\mathcal {O}}_{\widetilde {Y}_U}$ -modules, and hence decomposes as $\oplus _{\tau \in \Sigma } {\mathcal {H}}^1_{\operatorname {dR}}(A/\widetilde {Y}_U)_\tau $ , where each ${\mathcal {H}}^1_{\operatorname {dR}}(A/\widetilde {Y}_U)_\tau $ is locally free of rank two over ${\mathcal {O}}_{\widetilde {Y}_U}[u]/(E_\tau )$ .

For $\tau = \tau _{{\mathfrak {p}},i}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ , we let ${\mathcal {L}}_\theta $ denote the line bundle ${\mathcal {F}}_\tau ^{(j)}/{\mathcal {F}}_\tau ^{(j-1)}$ on $\widetilde {Y}_U$ . We let ${\mathcal {G}}_\tau ^{(j)}$ denote the pre-image of ${\mathcal {F}}_\tau ^{(j-1)}$ in ${\mathcal {H}}^1_{\operatorname {dR}}(A/\widetilde {Y}_U)_\tau $ under $u - \theta (\varpi _{\mathfrak {p}})$ , so that ${\mathcal {P}}_\theta := {\mathcal {G}}_\tau ^{(j)}/{\mathcal {F}}_\tau ^{(j-1)}$ is a rank two vector bundle on $\widetilde {Y}_U$ , containing ${\mathcal {L}}_\theta $ as a sub-bundle. We let ${\mathcal {M}}_\theta = {\mathcal {P}}_\theta /{\mathcal {L}}_\theta = {\mathcal {G}}_\tau ^{(j)}/{\mathcal {F}}_\tau ^{(j)}$ and ${\mathcal {N}}_\theta = {\mathcal {L}}_\theta \otimes _{{\mathcal {O}}_{\widetilde {Y}_U}} {\mathcal {M}}_\theta = \wedge ^2_{{\mathcal {O}}_{\widetilde {Y}_U} }{\mathcal {P}}_\theta $ . For any $\mathbf {k}$ , $\mathbf {m} \in \mathbb {Z}^\Sigma $ , we let $\widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {m}}$ denote the line bundle

$$ \begin{align*}\bigotimes_{\theta\in \Sigma}\left( {\mathcal{L}}^{\otimes k_\theta}_\theta \otimes_{{\mathcal{O}}_{\widetilde{Y}_U}} {\mathcal{N}}_\theta^{\otimes m_\theta} \right)\end{align*} $$

on $\widetilde {Y}_U$ . We also let $\widetilde {\omega } = \widetilde {{\mathcal {A}}}_{\mathbf {{1}},\mathbf {0}}$ and $\widetilde {\delta } = \widetilde {{\mathcal {A}}}_{\mathbf {{0},\mathbf {1}}}$ . Note that $\widetilde {\omega }$ may be identified with $\wedge ^d_{{\mathcal {O}}_{\widetilde {Y}_U}} (s_*\Omega ^1_{A/\widetilde {Y}_U})$ ; we remark that, less obviously, $\widetilde {\delta }$ may be identified with $\wedge ^{2d}_{{\mathcal {O}}_{\widetilde {Y}_U}} ({\mathcal {H}}^1_{{\operatorname {dR}}}(A/\widetilde {Y}_U))$ (and more naturally with $\mathrm {Disc}_{F/\mathbb {Q}} \otimes \wedge ^{2d}_{{\mathcal {O}}_{\widetilde {Y}_U}} ({\mathcal {H}}^1_{{\operatorname {dR}}}(A/\widetilde {Y}_U))$ ).

There is a natural action of ${\mathcal {O}}_{F,(p),+}^\times $ on the vector bundles ${\mathcal {F}}_\tau ^{(j)}$ and ${\mathcal {G}}_\tau ^{(j)}$ over the one on $\widetilde {Y}_U$ , inducing actions on ${\mathcal {L}}_\theta $ , ${\mathcal {P}}_\theta $ , ${\mathcal {M}}_\theta $ and ${\mathcal {N}}_\theta $ ; however, the restriction to $(U\cap {\mathcal {O}}_F^\times )^2$ is nontrivial, so it fails to define descent data on these sheaves to $Y_U$ (see [Reference DiamondDia23, §3.2]). More precisely, if $\nu = \mu ^2$ for some $\mu \in U \cap {\mathcal {O}}_F^\times $ , then $\nu $ acts on ${\mathcal {L}}_\theta $ , ${\mathcal {P}}_\theta $ and ${\mathcal {M}}_\theta $ (resp. ${\mathcal {N}}_\theta $ ) by $\theta (\mu )$ (resp. $\theta (\nu )$ ). Thus, if $\mathbf {k} + 2\mathbf {m}$ is parallel, in the sense that $k_\theta + 2 m_\theta $ is independent of $\theta $ , then we obtain descent data on $\widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {m}}$ (provided $\mathrm {Nm}_{F/{\mathbb {Q}}}(U\cap {\mathcal {O}}_F^\times ) = \{1\}$ ), and we let ${\mathcal {A}}_{\mathbf {k},\mathbf {m}}$ denote the resulting line bundle on $Y_U$ ; note in particular this applies to $\widetilde {\omega }$ and $\widetilde {\delta }$ , yielding the line bundles on $Y_U$ which we denote $\omega $ and $\delta $ . More generally, if R is an ${\mathcal {O}}$ -algebra in which the image of $\prod _\theta \theta (\mu )^{k_\theta + 2m_\theta }$ is $1$ for all $\mu \in U \cap {\mathcal {O}}_F^\times $ , then $\widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {m},R}:= \widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {m}} \otimes _{{\mathcal {O}}} R$ descends to a line bundle on $Y_{U,R}: = Y_U \times _{{\mathcal {O}}} R$ which we denote by ${\mathcal {A}}_{\mathbf {k},\mathbf {m},R}$ ; note in particular that if $p^N R = 0$ for some N, then this applies to all $\mathbf {k},\mathbf {m}$ whenever U is sufficiently small (depending on N).

We also have natural left action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on the various vector bundles over its right action on the inverse system $\widetilde {Y}_U$ . More precisely, suppose that $g \in \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ is such that $g^{-1}Ug \subset U'$ , where U and $U'$ are sufficiently small open compact subgroups of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}})$ containing $\operatorname {GL}_2({\mathcal {O}}_{F,p})$ . Let $\underline {A}' = (A',\iota ',\lambda ',\eta ',{\mathcal {F}}^{\prime \bullet })$ denote the universal object over $\widetilde {Y}_{U'}$ , and similarly let ${\mathcal {L}}_\theta ^{\prime }$ , etc., denote the associated vector bundles. As described in [Reference DiamondDia23, §3.2], the morphism $\widetilde {\rho }_g: \widetilde {Y}_U \to \widetilde {Y}_{U'}$ is associated with a (prime-to-p) quasi-isogeny $A \to \widetilde {\rho }_g^{*}A'$ inducing isomorphisms $\widetilde {\rho }_g^{*}{\mathcal {F}}_\tau ^{\prime (j)} \stackrel {\sim }{\to } {\mathcal {F}}_\tau ^{(j)}$ , which in turn give rise to isomorphisms $\widetilde {\rho }_g^{*}{\mathcal {L}}^{\prime }_\theta \stackrel {\sim }{\to } {\mathcal {L}}_\theta $ , etc., satisfying the usual compatibilities. Furthermore, if $\mathbf {k} + 2\mathbf {m}$ is parallel, then the resulting isomorphisms $\widetilde {\rho }_g^{*}\widetilde {{\mathcal {A}}}^{\prime }_{\mathbf {k},\mathbf {m}} \stackrel {\sim }{\to } \widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {m}}$ descend to isomorphisms $\rho _g^{*}{{\mathcal {A}}}^{\prime }_{\mathbf {k},\mathbf {m}} \stackrel {\sim }{\to } {{\mathcal {A}}}_{\mathbf {k},\mathbf {m}}$ , where $\rho _g: {Y}_U \to {Y}_{U'}$ is the morphism obtained by descent from $\widetilde {\rho }_g$ , and more generally, we obtain isomorphisms $\rho _g^{*}{{\mathcal {A}}}^{\prime }_{\mathbf {k},\mathbf {m},R} \stackrel {\sim }{\to } {{\mathcal {A}}}_{\mathbf {k},\mathbf {m},R}$ whenever the image in R of $\prod _\theta \theta (\mu )^{k_\theta + 2m_\theta }$ is $1$ for all $\mu \in U' \cap {\mathcal {O}}_F^\times $ .

If $R = \mathbb {C}$ and $\mathbf {k} + 2\mathbf {m}$ is parallel, then we recover the usual automorphic line bundles on $Y_U({\mathbb {C}})$ whose global sections are Hilbert modular forms of weight $(\mathbf {k},\mathbf {m})$ and level U, along with the usualFootnote ⁵ Hecke action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on their direct limit over U.

Finally, recall from [Reference DiamondDia23, §3.2] that the quasi-polarization and Poincaré duality induce a perfect alternating pairing on ${\mathcal {P}}_\theta $ (depending on the choice of $\varpi _{\mathfrak {p}}$ ), and hence a trivialization of ${\mathcal {N}}_\theta $ , but their products do not descend to trivializations of the bundles ${\mathcal {A}}_{\mathbf {{0}},\mathbf {m},R}$ over $Y_U$ (if $\mathbf {m} \neq \mathbf {0}$ ). These are, however, torsion bundles, which are furthermore non-canonically trivializable for sufficiently small U. The Hecke action on their global sections is given by [Reference DiamondDia23, Prop. 3.2.2].

2.4 Iwahori–level structure

For ${\mathfrak {p}} \in S_p$ , we let $I_0({\mathfrak {p}})$ denote the Iwahori subgroup

$$ \begin{align*}\left\{\,\left. \left(\begin{array}{cc} a&b \\ c& d \end{array}\right) \in \operatorname{GL}_2({\mathcal{O}}_{F,{\mathfrak{p}}}) \,\right|\, c \in {\mathfrak{p}} {\mathcal{O}}_{F,{\mathfrak{p}}} \,\right\}.\end{align*} $$

In this section, we recall the definition of suitable integral models of Hilbert modular varieties with such level structure at a set of primes over p.

Fix an ideal ${\mathfrak {P}}$ of ${\mathcal {O}}_F$ containing the radical of $p{\mathcal {O}}_F$ , so that ${\mathfrak {P}} = \prod _{{\mathfrak {p}} \in P} {\mathfrak {p}}$ for some subset P of $S_p$ . We are mainly interested in the cases $P = \{{\mathfrak {p}}\}$ and $P = S_p$ , but the additional generality introduces no difficulties, and it may be instructive (and amusing) to note how some of our results specialize to well-known ones in the case $P = \emptyset $ . Let U be an open compact subgroup of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}})$ as above, so that $U = U_p U^p$ where $U_p = \operatorname {GL}_2({\mathcal {O}}_{F,p})$ and $U^p$ is a sufficiently small open subgroup of $\operatorname {GL}_2(\widehat {{\mathcal {O}}}_F^{(p)})$ , and we let

$$ \begin{align*}U_0({\mathfrak{P}}) = \{ \, g \in U \,|\, g_{\mathfrak{p}} \in I_0({\mathfrak{p}}) \mbox{ for all } {\mathfrak{p}}|{\mathfrak{P}}\,\}.\end{align*} $$

d We let $\varpi _{\mathfrak {P}} = \prod \varpi _{\mathfrak {p}}$ , $f_{\mathfrak {P}} = \sum f_{\mathfrak {p}}$ , $d_{\mathfrak {P}} = \sum e_{\mathfrak {p}} f_{\mathfrak {p}}$ , $\Sigma _{\mathfrak {P}} = \coprod \Sigma _{{\mathfrak {p}}}$ and $\Sigma _{{\mathfrak {P}},0} = \coprod \Sigma _{{\mathfrak {p}},0}$ , with the product, sums and unions taken over the set of ${\mathfrak {p}}$ dividing ${\mathfrak {P}}$ .

For a locally Noetherian ${\mathcal {O}}$ -scheme S, we consider the functor which associates to S the set of isomorphism classes of triples $(\underline {A}_1,\underline {A}_2,\psi )$ , where $\underline {A}_i = (A_i,\iota _i,\lambda _i,\eta _i,{\mathcal {F}}_i^\bullet )$ define elements of $Y_U(S)$ for $i = 1,2$ and $\psi :A_1 \to A_2$ is an isogeny of degree $p^{f_{\mathfrak {P}}}$ such that

• • $\ker (\psi ) \subset A_1[{\mathfrak {P}}]$ ;

• • $\psi $ is ${\mathcal {O}}_F$ -linear (i.e., $\psi \circ \iota _1(\alpha ) = \iota _2(\alpha )\circ \psi $ for all $\alpha \in {\mathcal {O}}_F$ );

• • $\lambda _1\circ \iota _1(\varpi _{\mathfrak {P}}) = \psi ^\vee \circ \lambda _2 \circ \psi $ ;

• • $\psi \circ \eta _1 = \eta _2$ (as $U^p$ -orbits on each connected component of S);

• • $\psi ^{*}{\mathcal {F}}_2^\bullet \subset {\mathcal {F}}_1^\bullet $ (i.e., $\psi ^{*}{\mathcal {F}}_{\tau ,2}^{(j)} \subset {\mathcal {F}}_{\tau ,1}^{(j)}$ for all ${\mathfrak {p}} \in S_p$ , $\tau \in \Sigma _{0,{\mathfrak {p}}}$ and $j = 1,\ldots ,e_{{\mathfrak {p}}}$ ).

For such an isogeny $\psi $ , consider also the isogeny $\xi :A_2 \to {\mathfrak {P}}^{-1}\otimes _{{\mathcal {O}}_F} A_1$ such that $\xi \circ \psi : A_1 \to {\mathfrak {P}}^{-1} \otimes _{{\mathcal {O}}_F} A_1$ is the canonical isogeny with kernel $A_1[{\mathfrak {P}}]$ . The compatibility with the quasi-polarizations $\lambda _1$ and $\lambda _2$ then implies the commutativity of the resulting diagram of p-integral quasi-isogenies (over each connected component of the base S):

This in turn implies the commutativity of the diagram

for each $\tau \in \Sigma _0$ , where ${\mathcal {H}}_{\tau ,i}$ is the $\tau $ -component of the locally free ${\mathcal {O}}_F\otimes {\mathcal {O}}_S$ -module ${\mathcal {H}}^1_{\operatorname {dR}}(A_i/S)$ for $i=1,2$ , the superscript ${}^\vee $ denotes its ${\mathcal {O}}_S[u]/(E_\tau )$ -dual, and the $\mu _{\tau ,i}$ are the $\tau $ -components of the isomorphisms obtained from the polarizations and Poincaré duality (see [Reference DiamondDia23, (4)]). The condition that $\psi ^{*}{\mathcal {F}}_{\tau ,2}^{(j)} \subset {\mathcal {F}}_{\tau ,1}^{(j)}$ for all j therefore implies that $\xi _\tau ^{*}({\mathfrak {P}}\otimes _{{\mathcal {O}}_F}({\mathcal {F}}_{\tau ,1}^{(j)})^\perp ) \subset ({\mathcal {F}}_{\tau ,2}^{(j)})^\perp $ , so it follows from [Reference DiamondDia23, Lemma 3.1.1] that

(2.2) $$ \begin{align} \xi_\tau^{*}({\mathfrak{P}}\otimes_{{\mathcal{O}}_F}{\mathcal{F}}_{\tau,1}^{(j)}) \subset ({\mathcal{F}}_{\tau,2}^{(j)}).\end{align} $$

The functor defined above is representable by a scheme $\widetilde {Y}_{U_0({\mathfrak {P}})}$ , projective over $\widetilde {Y}_U$ relative to either of the forgetful morphisms.Footnote ⁶ In the next section, we prove that the schemes $\widetilde {Y}_{U_0({\mathfrak {P}})}$ are syntomic of relative dimension d over ${\mathcal {O}}$ (see also [Reference Emerton, Reduzzi and XiaoERX17a, Prop. 3.3]). Note that we again have a free action of ${\mathcal {O}}_{F,(p),+}^\times /(U\cap {\mathcal {O}}_F^\times )^2$ for which the quotient is representable by a quasi-projective scheme of relative dimension d over ${\mathcal {O}}$ , which we denote by $Y_{U_0({\mathfrak {P}})}$ , and a natural right action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on the inverse system of the $\widetilde {Y}_{U_0({\mathfrak {P}})}$ (over $U^p$ ) descending to one on the inverse system of $Y_{U_0({\mathfrak {P}})}$ . Furthermore, the schemes $Y_{U_0({\mathfrak {P}})}$ are independent of the choices of $\varpi _{\mathfrak {p}}$ , and we again have isomorphisms of their sets of complex points with Hilbert modular varieties

$$ \begin{align*}Y_{U_0({\mathfrak{P}})}(\mathbb{C}) \cong \operatorname{GL}_2(F)_+\backslash (\mathfrak{H}^\Sigma \times \operatorname{GL}_2({\mathbb{A}}_{F,\mathbf{f}})/U_0({\mathfrak{P}}))\end{align*} $$

under which the action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ corresponds to right multiplication.

Consider also the forgetful morphisms $\widetilde {\pi }_i: \widetilde {Y}_{U_0({\mathfrak {P}})} \to \widetilde {Y}_U$ sending $(\underline {A}_1,\underline {A}_2,\psi )$ to $\underline {A}_i$ for $i =1,2$ , and their product

$$ \begin{align*}\widetilde{h}: = (\widetilde{\pi}_1,\widetilde{\pi}_2): \widetilde{Y}_{U_0({\mathfrak{P}})} \to \widetilde{Y}_U \times_{{\mathcal{O}}} \widetilde{Y}_U,\end{align*} $$

descending to morphisms $\pi _i:Y_{U_0({\mathfrak {P}})} \to Y_U$ and $h: Y_{U_0({\mathfrak {P}})} \to Y_U \times _{{\mathcal {O}}} Y_U$ , which are again independent of the choices of $\varpi _{\mathfrak {p}}$ .

Proof. Since $\widetilde {h}$ is projective, it suffices to prove that it is injective on geometric points and their tangent spaces. Furthermore, the assertion for h follows from the one for $\widetilde {h}$ .

Suppose then that $(\underline {A}_1,\underline {A}_2,\psi )$ and $(\underline {A}_1^{\prime },\underline {A}_2^{\prime },\psi ')$ correspond to geometric points of $\widetilde {Y}_{U_0({\mathfrak {P}})}$ with the same image under $\widetilde {h}$ . Thus, there are isomorphisms $f_i:A_i \to A_i^{\prime }$ for $i=1,2$ which are compatible with all auxiliary data. Let $\xi :A_2^{\prime } \to A_1$ be the unique isogeny such that $\xi \circ \psi '\circ f_1 = p$ , and consider the ${\mathcal {O}}_F$ -linear endomorphism $\alpha := \xi \circ f_2 \circ \psi $ of $A_1$ (where we identify ${\mathcal {O}}_F$ with a subalgebra of ${\operatorname {End}\,}(A_1)$ via $\iota _1$ ). We wish to prove that $\alpha = p$ , as this implies that $f_2 \circ \psi = \psi '\circ f_1$ , giving the desired injectivity on geometric points.

First, note that the compatibility of $\psi '$ and $f_1$ with quasi-polarizations implies that $\xi ^\vee \circ \lambda _1 \circ \varpi _{\mathfrak {P}} \circ \xi = p^2\lambda _2^{\prime }$ . Combining this with the compatibility of $\psi $ and $f_2$ with quasi-polarizations, it follows that $\alpha ^\vee \circ \lambda _1 \circ \alpha = p^2\lambda _1$ . In particular, $F(\alpha )$ is stable under the $\lambda _1$ -Rosati involution of ${\operatorname {End}\,}^0(A_1)$ , which sends $\alpha $ to $p^2\alpha ^{-1}$ . By the classification of endomorphism algebras of abelian varieties, it follows that either $\alpha \in F$ , in which case $\alpha = \pm p$ , or $F(\alpha )$ is a quadratic CM-extension of F, in which case $\alpha $ is in its ring of integers and $\alpha \overline {\alpha } = p^2$ .

Next, note that the compatibility of $\psi $ , $\psi '$ , $f_1$ and $f_2$ with level structures implies that if $U \subset U({\mathfrak {n}})$ , then $\alpha \circ \overline {\eta }_1 = p\overline {\eta }_1$ , where $\overline {\eta }_1: ({\mathcal {O}}_F/{\mathfrak {n}})^2 \cong A_1[{\mathfrak {n}}]$ is the isomorphism induced by $\eta _1$ . It follows that $\alpha - p$ annihilates $A_1[{\mathfrak {n}}]$ , and therefore so does $\overline {\alpha } - p$ (writing $\overline {\alpha } = \alpha $ if $\alpha \in F$ ). Letting $\beta $ denote the element $\alpha + \overline {\alpha } - 2p \in {\mathcal {O}}_F$ , it follows that $\beta $ annihilates $A_1[{\mathfrak {n}}]$ , and hence $\beta \in {\mathfrak {n}}$ . Furthermore, since $\alpha $ is a root of

$$ \begin{align*}X^2 - (\beta + 2p)X + p^2\end{align*} $$

and either $\alpha = \pm p$ or $F(\alpha )$ is a CM-extension of F, it follows that $|\theta (\beta )| \le 4p$ for all embeddings $\theta :F \hookrightarrow \mathbb {R}$ . If ${\mathfrak {n}}$ is such that $N_{F/\mathbb {Q}}({\mathfrak {n}})> (4p)^{[F:\mathbb {Q}]}$ , this implies that $\beta = 0$ , and hence, $\alpha = p$ .

We have now proved that $\widetilde {h}$ is injective on geometric points. The injectivity on tangent spaces is immediate from the Grothendieck–Messing Theorem, a version of which we recall below for convenience and for later use.

Let S be a scheme and $i:S \hookrightarrow T$ a nilpotent divided power thickening. If $t: B \to T$ is an abelian scheme of dimension d, then the restriction over S of the crystal $R^1t_{\mathrm {crys},*}{\mathcal {O}}_{B,\mathrm {crys}}$ is canonically identified with $R^1{\overline {t}}_{\mathrm {crys},*}{\mathcal {O}}_{\overline {B},\mathrm {crys}}$ , where $\overline {t}: \overline {B} \to S$ is the base-change of $t:B \to T$ . The image of $t_*\Omega ^1_{B/T}$ under the resulting isomorphism

(2.3) $$ \begin{align} {\mathcal{H}}^1_{\operatorname{dR}}(B/T) \stackrel{\sim}{\longrightarrow} (R^1t_{\mathrm{crys},*}{\mathcal{O}}_{B,\mathrm{crys}})_{T} \stackrel{\sim}{\longrightarrow} (R^1\overline{t}_{\mathrm{crys},*}{\mathcal{O}}_{\overline{B},\mathrm{crys}})_{T}\end{align} $$

is thus an ${\mathcal {O}}_T$ -subbundle ${\mathcal {V}}_B$ of $(R^1\overline {t}_{\mathrm {crys},*}{\mathcal {O}}_{\overline {B},\mathrm {crys}})_{T}$ whose restriction to S corresponds to $\overline {t}_*\Omega ^1_{\overline {B}/S}$ under the canonical isomorphism

$$ \begin{align*}{\mathcal{H}}^1_{\operatorname{dR}}(\overline{B}/S) \stackrel{\sim}{\longrightarrow} (R^1\overline{t}_{\mathrm{crys},*}{\mathcal{O}}_{\overline{B},\mathrm{crys}})_{S} \longrightarrow i^{*}(R^1\overline{t}_{\mathrm{crys},*}{\mathcal{O}}_{\overline{B},\mathrm{crys}})_{T}.\end{align*} $$

The Grothendieck–Messing Theorem (as in [Reference GrothendieckGro74, Ch. V, §4]) states that the functor sending B to $(\overline {B},{\mathcal {V}}_B)$ defines an equivalence between the categories of abelian schemes over T and that of pairs $(A,{\mathcal {V}})$ , where $s: A \to S$ is an abelian scheme and ${\mathcal {V}}$ is an ${\mathcal {O}}_T$ -subbundle of $(R^1s_{\mathrm {crys},*}{\mathcal {O}}_{A,\mathrm {crys}})_{T}$ such that $i^{*}{\mathcal {V}}$ corresponds to $s_*\Omega ^1_{A/S}$ under the canonical isomorphism of $i^{*}(R^1s_{\mathrm {crys},*}{\mathcal {O}}_{A,\mathrm {crys}})_{T}$ with ${\mathcal {H}}^1_{\operatorname {dR}}(A/S)$ . A morphism of pairs $(A,{\mathcal {V}}_1) \to (A',{\mathcal {V}}')$ being a pair of morphisms $(A \to A', {\mathcal {V}}' \to {\mathcal {V}})$ satisfying the evident compatibility, it follows that if $t:B \to T$ and $t':B' \to T$ are abelian schemes and $\psi :\overline {B} \to \overline {B}'$ is a morphism of their base-changes to S, then $\psi $ extends (necessarily uniquely) to a morphism $B \to B'$ if and only if the morphism

$$ \begin{align*}\psi_{\mathrm{crys}}^{*}: (R^1\overline{t}^{\prime}_{\mathrm{crys},*}{\mathcal{O}}_{\overline{B}',\mathrm{crys}})_{T} \longrightarrow (R^1\overline{t}_{\mathrm{crys},*}{\mathcal{O}}_{\overline{B},\mathrm{crys}})_{T}\end{align*} $$

sends ${\mathcal {V}}'$ to ${\mathcal {V}}$ . In particular, the functor sending B to $\overline {B}$ is faithful, which is all that is needed in the proof of Proposition 2.1.

For later reference, we note if the data $\underline {A} = (A,\iota ,\lambda ,\eta ,{\mathcal {F}}^\bullet )$ corresponds to an element of $\widetilde {Y}_U(S)$ , then to give a lift to an element of $\widetilde {Y}_U(T)$ is equivalent to giving a lift ${\mathcal {E}}^\bullet $ of the Pappas–Rapoport filtrations to $(R^1s_{\mathrm {crys},*}{\mathcal {O}}_{A,\mathrm {crys}})_{T}$ , by which we mean a collection of ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W(\mathbb {F}_{{\mathfrak {p}}}),\tau } {\mathcal {O}}_T$ -submodules

$$ \begin{align*}0 = {\mathcal{E}}_\tau^{(0)} \subset {\mathcal{E}}_\tau^{(1)} \subset \cdots \subset {\mathcal{E}}_\tau^{(e_{{\mathfrak{p}}} - 1)} \subset {\mathcal{E}}_\tau^{(e_{{\mathfrak{p}}})}\end{align*} $$

of $(R^1s_{\mathrm {crys},*}{\mathcal {O}}_{A,\mathrm {crys}})_{T,\tau }$ for each $\tau = \tau _{{\mathfrak {p}},i}$ such that

• • ${\mathcal {E}}_\tau ^{(j)}/{\mathcal {E}}_\tau ^{(j-1)}$ is a line bundle on T on which ${\mathcal {O}}_F$ acts via $\theta _{{\mathfrak {p}},i,j}$ .

• • $\iota ^{*}{\mathcal {E}}_\tau ^{(j)}$ corresponds to ${\mathcal {F}}_\tau ^{(j)}$ under the canonical isomorphism

  $$ \begin{align*}\iota^{*}(R^1s_{\mathrm{crys},*}{\mathcal{O}}_{A,\mathrm{crys}})_{T} \cong (R^1s_{\mathrm{crys},*}{\mathcal{O}}_{A,\mathrm{crys}})_{S} \cong {\mathcal{H}}^1_{\operatorname{dR}}(A/S)\end{align*} $$

for $j=1,\ldots ,e_{{\mathfrak {p}}}$ . The bijection is defined by sending the data of a lift $\underline {\widetilde {A}} = (\widetilde {A},\widetilde {\iota },\widetilde {\lambda },\widetilde {\eta },\widetilde {{\mathcal {F}}}^\bullet )$ to the lift of filtrations corresponding to $\widetilde {{\mathcal {F}}}^\bullet $ under the canonical isomorphism

$$ \begin{align*}{\mathcal{H}}^1_{\operatorname{dR}}(\widetilde{A}/T) \cong (R^1\widetilde{s}_{\mathrm{crys},*}{\mathcal{O}}_{\widetilde{A},\mathrm{crys}})_{T} \cong (R^1s_{\mathrm{crys},*}{\mathcal{O}}_{A,\mathrm{crys}})_{T}.\end{align*} $$

The injectivity of the map is a straightforward consequence of the Grothendieck–Messing Theorem; for the surjectivity, one needs also to know that the quasi-polarization $\lambda $ extends to the lift of A associated to $(A,{\mathcal {V}})$ with ${\mathcal {V}} = \oplus _\tau {\mathcal {E}}_\tau ^{(e_{{\mathfrak {p}}})}$ , which is ensured by [Reference VollaardVol05, Prop. 2.10]. Since the theorem provides an equivalence of categories, it follows also that if a triple $(\underline {A}_1,\underline {A}_2,\psi )$ corresponds to an element of $\widetilde {Y}_{U_0({\mathfrak {P}})}(S)$ , then to give a lift to an element of $\widetilde {Y}_{U_0({\mathfrak {P}})}(T)$ is equivalent to giving lifts ${\mathcal {E}}_i^\bullet $ of the Pappas–Rapoport filtrations ${\mathcal {F}}_i^\bullet $ to $(R^1s_{i,\mathrm {crys},*}{\mathcal {O}}_{A_i,\mathrm {crys}})_{T}$ for $i=1,2$ , such that ${\mathcal {E}}_1^\bullet $ and ${\mathcal {E}}_2^\bullet $ are compatible with

$$ \begin{align*}\psi_{\mathrm{crys},T}^{*}: (R^1s_{2,\mathrm{crys},*}{\mathcal{O}}_{A_2,\mathrm{crys}})_{T} \longrightarrow (R^1s_{1,\mathrm{crys},*}{\mathcal{O}}_{A_1,\mathrm{crys}})_{T}.\end{align*} $$

2.5 Local structure: an example

In the next section, we will recall the analysis of the local structure of $\widetilde {Y}_{U_0({\mathfrak {P}})}$ . First, however, at the suggestion of the referee, we consider the case where F is a quadratic extension ramified at p. This will already illustrate the key ideas and techniques; the general case is then mainly a matter of transforming them into an inductive argument.

To further fix ideas and make the analysis more concrete, let $F = \mathbb {Q}(\sqrt {p})$ and ${\mathfrak {P}} = {\mathfrak {p}} = \varpi _{{\mathfrak {p}}}{\mathcal {O}}_F$ , where $\varpi _{{\mathfrak {p}}} = \sqrt {p}$ . We also let ${\mathcal {O}} = \mathbb {Z}_p[\varpi ]$ and $\Sigma = \{\theta _1,\theta _2\}$ , where $\varpi ^2 = p$ , $\theta _1(\varpi _{\mathfrak {p}}) = \varpi $ and $\theta _2(\varpi _{\mathfrak {p}}) = -\varpi $ , so that ${\mathcal {O}}_F \otimes {\mathcal {O}} \cong {\mathcal {O}}[u]/(u^2-p)$ , in which $s_1 = u - \varpi $ and $t_1 = u + \varpi $ in the notation of §2.1.

Suppose now that $y \in \widetilde {Y}_{U_0({\mathfrak {p}})}({\mathbb {F}}_p)$ , and let S denote the local ring ${\mathcal {O}}_{\widetilde {Y}_{U_0({\mathfrak {p}})},y}$ and $(\underline {A},\underline {A}',\psi )$ denote the corresponding triple over S. Thus, the $S[u]/(u^2-p)$ -module $H^0(A,\Omega ^1_{A/S})$ is free of rank two over S, equipped with a filtration of S-modules

$$ \begin{align*}0 = F^{(0)} \subset F^{(1)} \subset F^{(2)} = H^0(A,\Omega^1_{A/S})\end{align*} $$

such that $(u-\varpi )F^{(1)} = 0$ , $(u+\varpi )F^{(2)} \subset F^{(1)}$ , and $L_1:=F^{(1)}$ and $L_2:= F^{(2)}/F^{(1)}$ are each free of rank one. Viewing $H^0(A,\Omega ^1_{A/S})$ as a submodule of the free rank two $S[u]/(u^2-p)$ -module $H^1_{\mathrm {dR}}(A/S)$ , and letting $G^{(1)} = (u+\varpi )H^1_{\mathrm {dR}}(A/S)$ (or equivalently, the kernel of $u-\varpi $ on $H^1_{\operatorname {dR}}(A/S)$ ) and $G^{(2)} = (u+\varpi )^{-1}F^{(1)}$ (i.e., the preimage of $F^{(1)}$ in $H^1_{\operatorname {dR}}(A/S)$ under $u+\varpi $ ), we obtain inclusions of free S-modules

such that each successive quotient is free of rank one over S.

Furthermore, the free rank two S-modules $P_1:= G^{(1)}$ and $P_2:= G^{(2)}/F^{(1)}$ are equipped with perfect alternating pairings $\langle \cdot ,\cdot \rangle _i$ whose construction we briefly recall (see [Reference DiamondDia23, §3.1] for more details). Firstly, Poincaré duality and the polarization on A yield an isomorphism

$$ \begin{align*}H^1_{\operatorname{dR}}(A/S) \stackrel{\sim}{\longrightarrow} {\operatorname{Hom}\,}_S({\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_F}H^1_{\operatorname{dR}}(A/S),S) \stackrel{\sim}{\longleftarrow} {\operatorname{Hom}\,}_{{\mathcal{O}}_F\otimes S} (H^1_{\operatorname{dR}}(A/S),{\mathcal{O}}_F\otimes S),\end{align*} $$

and hence a perfect alternating pairing $\langle \cdot ,\cdot \rangle _{{\operatorname {dR}}}$ on $H^1_{\operatorname {dR}}(A/S)$ over ${\mathcal {O}}_F\otimes S = S[u]/(u^2 - p)$ . Furthermore, one finds that this induces an isomorphism

$$ \begin{align*}G^{(1)} \stackrel{\sim}{\longrightarrow} {\operatorname{Hom}\,}_S (H^1_{\operatorname{dR}}(A/S)/(u-\varpi)H^1_{\operatorname{dR}}(A/S), (u+\varpi)({\mathcal{O}}_F\otimes S)),\end{align*} $$

and our perfect pairing on $P_1 = G^{(1)}$ (over S) is then obtained from the isomorphisms

$$ \begin{align*}H^1_{\operatorname{dR}}(A/S)/(u-\varpi)H^1_{\operatorname{dR}}(A/S)\stackrel{\sim}{\longrightarrow} G^{(1)} \quad\mbox{and}\quad S\stackrel{\sim}{\longrightarrow} (u+\varpi)({\mathcal{O}}_F\otimes S)\end{align*} $$

defined by multiplication by $u+\varpi $ . However, one finds that $F^{(1)}$ and $G^{(2)}$ are orthogonal complements, and the perfect pairing on $P_2 = G^{(2)}/F^{(1)}$ is then obtained from the resulting isomorphism

$$ \begin{align*}G^{(2)}/F^{(1)} \stackrel{\sim}{\longrightarrow} {\operatorname{Hom}\,}_S (G^{(2)}/F^{(1)}, (u-\varpi)({\mathcal{O}}_F\otimes S)).\end{align*} $$

Similarly, we have the filtration

$$ \begin{align*}0 = F^{\prime(0)} \subset F^{\prime(1)} \subset F^{\prime(2)} = H^0(A',\Omega^1_{A'/S}),\end{align*} $$

which we use to define S-modules $L_i^{\prime }\subset P_i^{\prime }$ for $i=1,2$ such that $L_i^{\prime }$ and $P_i^{\prime }/L_i^{\prime }$ , free of rank one and $P_i^{\prime }$ is equipped with a perfect alternating pairing. Furthermore, the $S[u]/(u^2-\varpi )$ -linear map $\psi _{\operatorname {dR}}^{*}:H^1_{\operatorname {dR}}(A'/S) \to H^1_{\operatorname {dR}}(A/S)$ induces morphisms $\psi _i^{*}:P^{\prime }_i \to P_i$ restricting to $L_i^{\prime } \to L_i$ , and the equation $\psi \circ \lambda '\circ \psi ^\vee = \varpi _{{\mathfrak {p}}}\lambda $ (where $\lambda $ and $\lambda '$ are the polarizations on A and $A'$ ) implies that

$$ \begin{align*} \langle \psi_{\operatorname{dR}}^{*}(x),\psi_{\operatorname{dR}}^{*}(z) \rangle_{{\operatorname{dR}}} = u \langle x,z \rangle^{\prime}_{{\operatorname{dR}}},\end{align*} $$

and hence, $\langle \psi _i^{*}(x),\psi _i^{*}(z)\rangle _i = \theta _i(\varpi _{\mathfrak {p}})\langle x,z\rangle ^{\prime }_i = \pm \varpi \langle x,z\rangle ^{\prime }_i$ for $i=1,2$ and $x,z\in P_i^{\prime }$ .

Consider the behavior of the maps $\psi _i^{*}$ at the closed point (i.e., $\psi _{i,0}^{*}:{P}^{\prime }_{i,0} \to {P}_{i,0}$ ), where we use ${\cdot }_0$ to denote $\cdot \otimes _S{\mathbb {F}}_p$ . We find that ${\psi }_{i,0}^{*}$ has rank one (see the argument in the general case), and since ${\psi }_{i,0}^{*}({L}_{i,0}') \subset {L}_{i,0}$ , it follows that ${\psi }_{i,0}^{*}({L}_{i,0}') = 0$ or ${\psi }_{i,0}^{*}({P}_{i,0}') = {L}_{i,0}$ . To fix ideas even further, let us suppose that both these hold for $i = 1$ , and that only the first holds for $i=2$ . We will prove that in this case, the completion of S at its maximal ideal ${\mathfrak {n}}$ is isomorphic to the ${\mathcal {O}}$ -algebra

$$ \begin{align*}\widehat{R} := {\mathcal{O}}[[X_1,X_1^{\prime},X_2]]/(X_1X_1^{\prime}+\varpi).\end{align*} $$

In order to do so, we will construct a homomorphism $\widehat {R} \to \widehat {S}$ using a parametrization of the S-lines $L_i \subset P_i$ and $L_i^{\prime } \subset P_i^{\prime }$ with respect to suitable choices of bases for the ambient S-planes. We will then use the Grothendieck–Messing Theorem to prove inductively that the resulting homomorphism $R_n \to S_n$ is an isomorphism for all $n\ge 1$ , where $S_1 = S/(\varpi ,{\mathfrak {n}})$ , $S_n = S/{\mathfrak {n}}^n$ for $n\ge 2$ , and $R_n$ is defined similarly.

First, note that our assumptions on the $\psi _{i,0}^{*}$ imply that $F_{0}^{\prime (2)} = uH^1_{\operatorname {dR}}(A_0^{\prime }/{\mathbb {F}}_p)$ . Indeed, we have

$$ \begin{align*} {F}_0^{\prime (1)} \subset \ker(\psi_{{\operatorname{dR}},0}^{*}) \quad\mbox{and}\quad {\psi}_{{\operatorname{dR}},0}^{*}({F}_0^{\prime(2)}) \subset {F}_0^{(1)} = {\psi}_{{\operatorname{dR}},0}^{*}({G}_0^{\prime(1)}),\end{align*} $$

so that ${F}_0^{\prime (2)} \subset {G}_0^{\prime (1)}$ , and comparing dimensions gives equality. However,

$$ \begin{align*}{F}_0^{(1)} \subset {\psi}_{{\operatorname{dR}},0}^{*}({G}_0^{\prime(2)}) = \psi_{{\operatorname{dR}},0}^{*} (u^{-1}{F}_0^{\prime(1)}) \subset {G}_0^{(1)},\end{align*} $$

and comparing dimensions implies ${\psi }_{{\operatorname {dR}},0}^{*}({G}_0^{\prime (2)})= {G}_0^{(1)}$ , but ${\psi }_{{\operatorname {dR}},0}^{*}({G}_0^{\prime (2)})\not \subset {F}_0^{(2)}$ , so ${F}_0^{(2)} \neq {G}_0^{(1)}$ . It follows that ${F}_0^{(2)}$ is a cyclic ${\mathbb {F}}_p[u]/(u^2)$ -module and ${F}_0^{(1)} = u{F}_0^{(2)}$ . We may therefore choose bases $(x,z)$ for $H^1_{\operatorname {dR}}(A_0/{\mathbb {F}}_p)$ and $(x',z')$ for $H^1_{\operatorname {dR}}({A}_0^{\prime }/{\mathbb {F}}_p)$ as ${\mathbb {F}}_p[u]/(u^2)$ -modules so that

$$ \begin{align*}{F}_0^{(1)} = \langle ux \rangle,\, {F}_0^{(2)} = \langle x \rangle,\, {F}_0^{\prime(1)} = \langle ux' \rangle \,\,\,\mbox{and}\,\,\, {F}_0^{\prime(2)} = \langle ux',uz' \rangle \end{align*} $$

(where $\langle \cdot \rangle $ denotes generation as ${\mathbb {F}}_p[u]/(u^2)$ -modules). Elementary manipulations show that we can furthermore choose these bases so that the matrix of $\psi _{{\operatorname {dR}},0}^{*}$ is ${{\scriptsize {\left ( \begin {array}{cc} {{0}} & {{1}} \\ {{-u}} & {{0}} \end {array} \right )}}}$ and $\langle x,z \rangle _{{\operatorname {dR}},0} = \langle x',z' \rangle ^{\prime }_{{\operatorname {dR}},0} = 1$ .

We then have bases for $P_{1,0}$ and $P_{1,0}'$ defined by

$$ \begin{align*}e_{1,0} = ux,\,\, f_{1,0} = uz, \quad\mbox{and}\quad e^{\prime}_{1,0} = ux',\,\, f^{\prime}_{1,0} = uz',\end{align*} $$

so that $L_{1,0} = {\mathbb {F}}_p e_{1,0}$ , $L_{1,0}' = {\mathbb {F}}_p e_{1,0}'$ , $\langle e_{1,0},f_{1,0} \rangle _{1,0} = \langle e^{\prime }_{1,0},f^{\prime }_{1,0} \rangle ^{\prime }_{1,0}=1$ , and the matrix of $\psi _{1,0}^{*}$ takes the form ${ {\scriptsize {\left ( \begin {array}{cc} {{0}} & {{1}} \\ {{0}} & {{0}} \end {array} \right )}}}$ . Consider the isomorphisms $P_{1,0} \otimes _{{\mathbb {F}}_p} S_1 \cong P_{1,1}$ and $P^{\prime }_{1,0} \otimes _{{\mathbb {F}}_p} S_1 \cong P^{\prime }_{1,1}$ obtained from the canonical isomorphisms

$$ \begin{align*}H^1_{{\operatorname{dR}}}(A_0/{\mathbb{F}}_p)\otimes_{{\mathbb{F}}_p} S_1 \stackrel{\alpha}{\longrightarrow} H^1_{{\operatorname{dR}}}(A_1/S_1) \quad\mbox{and}\quad H^1_{{\operatorname{dR}}}(A^{\prime}_0/{\mathbb{F}}_p)\otimes_{{\mathbb{F}}_p} S_1 \stackrel{\alpha'}{\longrightarrow} H^1_{{\operatorname{dR}}}(A^{\prime}_1/S_1)\end{align*} $$

(systematically using $\cdot _n$ for $\otimes _S S_n$ ). Their functoriality properties further ensure that the matrix of $\psi _{1,1}^{*}$ has the same form as above with respect to the corresponding bases for $P_{1,1}$ and $P_{1,1}'$ , which we denote $(e_{1,1},f_{1,1})$ and $(e_{1,1}',f_{1,1}')$ , and that $\langle e_{1,1},f_{1,1} \rangle _{1,1} = \langle e^{\prime }_{1,1},f^{\prime }_{1,1} \rangle ^{\prime }_{1,1}=1$ . However $e_{1,1}$ and $e^{\prime }_{1,1}$ are no longer bases for $L_{1,1}$ and $L_{1,1}'$ ; instead we have $L_{1,1} = S_1(e_{1,1} - s_{1,1}f_{1,1})$ and $L_{1,1}' = S_1(e_{1,1}'-s_{1,1}'f_{1,1})$ for some (unique) $s_{1,1},s_{1,1}' \in {\mathfrak {n}} S_1$ . An elementary matrix calculation then shows that we may lift the chosen bases $(e_1,f_1)$ for $P_1$ and $(e_1^{\prime },f_1^{\prime })$ for $P_1^{\prime }$ over S so that $\langle e_1,f_1 \rangle _1 = \langle e_1^{\prime },f_1^{\prime } \rangle _1^{\prime } = 1$ and the resulting matrix of $\psi _1^{*}$ is ${{\scriptsize {\left ( \begin {array}{cc} {{0}} & {{1}} \\ {{-\varpi }} & {{0}} \end {array} \right )}}}$ . We then have $L_1 = S(e_1 - s_1 f_1)$ and $L_1^{\prime } = S(e_1^{\prime } - s_1^{\prime }f_1)$ for some uniquely determined $s_1,s_1^{\prime } \in {\mathfrak {n}}$ , and the fact that $\psi _1^{*}(L_1^{\prime }) \subset L_1$ means that

$$ \begin{align*}\left( \begin{array}{cc} {0} & {1} \\ {-\varpi} & {0} \end{array} \right)\left(\begin{array}{c}1\\-s_1^{\prime}\end{array}\right) = \left(\begin{array}{c}-s_1^{\prime}\\-\varpi\end{array}\right) = -s_1^{\prime}\left(\begin{array}{c}1\\-s_1\end{array}\right),\end{align*} $$

and therefore $s_1s_1^{\prime } = -\varpi $ .

Similarly, using the bases for $P_{2,0}$ and $P_{2,0}^{\prime }$ defined by

$$ \begin{align*}e_{2,0} = x + F_0^{(1)},\,\,f_{2,0} = uz + F_0^{(1)},\quad \mbox{and}\quad e^{\prime}_{2,0} = uz' + F_0^{\prime(1)},\,\,f^{\prime}_{2,0} = -x' + F_0^{\prime(1)}\end{align*} $$

gives $L_{2,0} = {\mathbb {F}}_p e_{2,0}$ , $L_{2,0}^{\prime } = {\mathbb {F}}_p e_{2,0}^{\prime }$ , $\langle e_{2,0},f_{2,0} \rangle _{2,0} = \langle e^{\prime }_{2,0},f^{\prime }_{2,0} \rangle ^{\prime }_{2,0}=1$ , and the matrix of $\psi _{2,0}^{*}$ takes the form ${{\scriptsize {\left ( \begin {array}{cc} {{0}} & {{0}} \\ {{0}} & {{1}} \end {array} \right )}}}$ . Note however that the canonical isomorphism $\alpha $ does not (necessarily) send $L_{1,0} = F^{(1)}_0$ to $L_{1,1} = F^{(1)}_1$ , and therefore does not yield an isomorphism between $P_{2,0}\otimes _{{\mathbb {F}}_p}S_1$ and $P_{2,1}$ . However letting $S_{1/2} = S/I$ , where $I = (s_1,s_1^{\prime },{\mathfrak {n}}^2)\subset (\varpi ,{\mathfrak {n}}^2)$ , we find that $\alpha $ and $\alpha '$ still induce isomorphisms $P_{2,0}\otimes _{{\mathbb {F}}_p}S_{1/2}\cong P_{2,1/2}$ and $P^{\prime }_{2,0}\otimes _{{\mathbb {F}}_p}S_{1/2}\cong P^{\prime }_{2,1/2}$ . We may then lift the resulting bases for $P_{2,1/2}$ and $P^{\prime }_{2,1/2}$ to ones, say $(e_2,f_2)$ for $P_2$ and $(e_2^{\prime },f_2^{\prime })$ for $P_2^{\prime }$ , such that $\langle e_2,f_2 \rangle _2 = \langle e_2^{\prime },f_2^{\prime } \rangle _2^{\prime } = 1$ and the matrix for $\psi _2^{*}$ has the form ${{\scriptsize {\left ( \begin {array}{cc} {{-\varpi }} & {{0}} \\ {{0}} & {{1}} \end {array} \right )}}}$ . Defining $s_2,s_2^{\prime } \in {\mathfrak {n}}$ by $L_2 = S(e_2-s_2 f_2)$ and $L_2^{\prime } = (e_2^{\prime } - s_2^{\prime }f_2^{\prime })$ , the fact that $\psi _2^{*}(L_2^{\prime }) \subset L_2$ now translates into the equation $s_2^{\prime } = -\varpi s_2$ .

We now define the homomorphism $\rho : \widehat {R} \to \widehat {S}$ by $X_1 \mapsto s_1$ , $X_1^{\prime } \mapsto s_1^{\prime }$ and $X_2 \mapsto s_2$ , and we will sketch the proof that it is an isomorphism.

We first prove that $\rho $ is surjective, or equivalently, that the ${\mathbb {F}}_p$ -vector space ${\mathfrak {n}}/(\varpi ,{\mathfrak {n}}^2)$ is spanned by $s_{1,1}$ , $s_{1,1}'$ and $s_{2,1}$ , or equivalently, if $\delta :S_1 \to T:= {\mathbb {F}}_p[\epsilon ]/(\epsilon ^2)$ is an ${\mathbb {F}}_p$ -algebra homomorphism such that $\delta (s_{1,1}) = \delta (s_{1,1}') = \delta (s_{2,1}) = 0$ , then $\delta $ factors through ${\mathbb {F}}_p$ . This in turn is equivalent to the assertion that for such a $\delta $ , the triple $(\underline {\widetilde {A}},\underline {\widetilde {A}}',\widetilde {\psi }) := {\delta }^{*}(\underline {A}_1,\underline {A}_1^{\prime },\psi _1)$ , is isomorphic to the base-change (via ${\mathbb {F}}_p \hookrightarrow T$ ) of $(\underline {A}_{0},\underline {A}_{0}',\psi _{0})$ . By the Grothendieck–Messing Theorem,Footnote ⁷ this in turn is equivalent to the assertion that for $i = 1,2$ , the T-modules $ F_1^{(i)}\otimes _{S_1} T$ and $F_{1}^{\prime (i)}\otimes _{S_{1}} T$ correspond (respectively) to $F_0^{(i)}\otimes _{{\mathbb {F}}_p} T$ and $F_0^{\prime (i)}\otimes _{{\mathbb {F}}_p} T$ under the canonical isomorphisms

$$ \begin{align*}H^1_{{\operatorname{dR}}}(A_0/{\mathbb{F}}_p)\otimes_{{\mathbb{F}}_p} T \cong H^1_{{\operatorname{dR}}}(\widetilde{A}/T)\quad \mbox{and} \quad H^1_{{\operatorname{dR}}}(A^{\prime}_0/{\mathbb{F}}_p)\otimes_{{\mathbb{F}}_p} T \cong H^1_{{\operatorname{dR}}}(\widetilde{A}'/T)\end{align*} $$

(induced by $\alpha $ and $\alpha '$ ). Since $F^{(1)}_1 = L_{1,1} = S_1(e_{1,1}-s_{1,1}f_{1,1})$ and $\delta (s_{1,1}) = 0$ , we have that $ F_1^{(1)}\otimes _{S_1} T = T(e_{1,1}\otimes 1)$ , and by construction, $e_{1,1}$ corresponds to $e_{1,0} \otimes 1$ under $\alpha $ . Similarly, we see that $F_{1}^{\prime (1)}\otimes _{S_{1}} T$ corresponds to $F_0^{\prime (1)}\otimes _{{\mathbb {F}}_p} T$ . Note also that $\delta $ factors through $S_{1/2}$ , so the same argument shows that

$$ \begin{align*}(F_1^{(2)}/F_1^{(1)})\otimes_{S_1}T = L_{2,1}\otimes_{S_1} T = L_{2,1/2}\otimes_{S_{1/2}}T\end{align*} $$

corresponds to $L_{2,0}\otimes _{{\mathbb {F}}_p} T = (F_0^{(2)}/F_0^{(1)})\otimes _{{\mathbb {F}}_p}T $ under the isomorphism induced by $\alpha $ , and hence that $ F_1^{(2)}\otimes _{S_1} T$ corresponds to $ F_0^{(1)}\otimes _{{\mathbb {F}}_p} T$ . Finally, the fact that $L^{\prime }_{2,1/2}\otimes _{S} T = T(e^{\prime }_{2,0}\otimes 1)$ follows automatically from the description of $\psi _2^{*}$ , so we similarly obtain the desired conclusion for $F_1^{\prime (2)}\otimes _{S_1} T$ .

In order to prove that $\rho _1:R_1 \to S_1$ is an isomorphism, we could again consider the map on tangent spaces, but in order to be more indicative of the inductive step treating $\rho _n$ , we will interpret the argument as the construction of a surjective homomorphism $S_1 \to R_1$ . Note that if $\widetilde {y} \in \widetilde {Y}_{U_0({\mathfrak {p}})}(R_1)$ is a lift of y, then the induced map $S = {\mathcal {O}}_{\widetilde {Y}_{U_0({\mathfrak {p}})},y} \to R_1$ factors through $S_1$ . Furthermore, by the Grothendieck–Messing Theorem (with $i:{\operatorname {Spec}\,}({\mathbb {F}}_p) \hookrightarrow {\operatorname {Spec}\,}(R_1)$ ), to give such a lift is equivalent to giving lifts

$$ \begin{align*}\widetilde{F}^{(1)} \subset \widetilde{F}^{(2)} \subset H^1_{{\operatorname{dR}}}(A_0/{\mathbb{F}}_p)\otimes_{{\mathbb{F}}_p}R_1\quad\mbox{and}\quad \widetilde{F}^{\prime(1)} \subset \widetilde{F}^{\prime(2)} \subset H^1_{{\operatorname{dR}}}(A^{\prime}_0/{\mathbb{F}}_p)\otimes_{{\mathbb{F}}_p}R_1 \end{align*} $$

of the Pappas–Rapoport filtrations compatible with $\psi _{{\operatorname {dR}},0}^{*}\otimes 1$ . More precisely, we require that the $\widetilde {F}^{(i)}$ and $\widetilde {F}^{\prime (i)}$ (for $i=1,2$ ) be free $R_1$ -modulesFootnote ⁸ such that

• • $\widetilde {F}^{(i)}\otimes _{R_1}{\mathbb {F}}_p = F_0^{(i)}$ and $\widetilde {F}^{\prime (i)}\otimes _{R_1}{\mathbb {F}}_p = F_0^{\prime (i)}$ ;

• • $u\widetilde {F}^{(i)} \subset \widetilde {F}^{(i-1)}$ and $u\widetilde {F}^{\prime (i)} \subset \widetilde {F}^{\prime (i-1)}$ (where $\widetilde {F}^{(0)} = \widetilde {F}^{\prime (0)} = 0$ );

• • $(\psi _{{\operatorname {dR}},0}^{*}\otimes 1)(\widetilde {F}^{\prime (i)}) \subset \widetilde {F}^{(i)}$ .

We will write down such lifts in terms of the bases $(\widetilde {x},\widetilde {z})$ and $(\widetilde {x}',\widetilde {z}')$ , where $(x,z)$ and $(x',z')$ are the bases already chosen for $H^1_{\operatorname {dR}}(A_0/{\mathbb {F}}_p)$ and $H^1_{\operatorname {dR}}(A_0^{\prime }/{\mathbb {F}}_p)$ and $\widetilde {x} = x \otimes 1$ , etc. Letting $\widetilde {P}_1 = uH^1_{{\operatorname {dR}}}(A_0/{\mathbb {F}}_p)\otimes _{{\mathbb {F}}_p} R_1$ and $\widetilde {P}^{\prime }_1 = uH^1_{{\operatorname {dR}}}(A^{\prime }_0/{\mathbb {F}}_p)\otimes _{{\mathbb {F}}_p} R_1$ , note that the matrix of the $R_1$ -linear map $\widetilde {\psi }_1:\widetilde {P}_1^{\prime } \to \widetilde {P}_1$ induced by $\psi _{{\operatorname {dR}},0}^{*}$ with respect to the bases $(u\widetilde {x},u\widetilde {z})$ and $(u\widetilde {x}',u\widetilde {z}')$ is ${{\scriptsize {\left ( \begin {array}{cc} {{0}} & {{1}} \\ {{0}} & {{0}} \end {array} \right )}}}$ . It follows that

$$ \begin{align*}\widetilde{F}^{(1)} = R_1 u(\widetilde{x} - \overline{X}_1\widetilde{z}) \quad\mbox{and}\quad \widetilde{F}^{\prime(1)} = R_1 u(\widetilde{x}' - \overline{X}^{\prime}_1\widetilde{z}')\end{align*} $$

are free $R_1$ -modules such that $\widetilde {\psi }_1^{*}(\widetilde {F}^{\prime (1)}) \subset \widetilde {F}^{(1)}$ , where $\overline {X}_1$ denotes the image of $X_1$ in $R_1$ , etc. Letting $\widetilde {P}_2$ denote the free rank two $R_1$ -module $(u^{-1}\widetilde {F}^{(1)})/\widetilde {F}^{(1)}$ and similarly defining $\widetilde {P}^{\prime }_2$ , one finds that the matrix of the $R_1$ -linear map $\widetilde {\psi }_2:\widetilde {P}_2^{\prime } \to \widetilde {P}_2$ induced by $\psi _{{\operatorname {dR}},0}^{*}$ is ${{\scriptsize {\left ( \begin {array}{cc} {{0}} & {{0}} \\ {{0}} & {{1}} \end {array} \right )}}}$ with respect to the bases

$$ \begin{align*}(\widetilde{x} - \overline{X}_1\widetilde{z} + \widetilde{F}^{(1)}, u\widetilde{z} + \widetilde{F}^{(1)}) \quad\mbox{and}\quad (\overline{X}_1\widetilde{x}'+u\widetilde{z}' + \widetilde{F}^{\prime(1)}, -\widetilde{x}'+\overline{X}_1^{\prime}\widetilde{z}' + \widetilde{F}^{\prime(1)}).\end{align*} $$

(Note that $\overline {X}_1\widetilde {x}'+u\widetilde {z}' \in u^{-1}\widetilde {F}^{\prime (1)}$ since $u(\overline {X}_1\widetilde {x}'+u\widetilde {z}') = u\overline {X}_1(\widetilde {x}'-\overline {X}^{\prime }_1\widetilde {z}')$ .) We thus find that

$$ \begin{align*}\widetilde{F}^{(2)} = \langle \widetilde{x}- (\overline{X}_1+u\overline{X}_2)\widetilde{z} \rangle, \quad\mbox{and}\quad \widetilde{F}^{\prime(2)} = \langle u(\widetilde{x}'-\overline{X}^{\prime}_1\widetilde{z}'), \overline{X}_1\widetilde{x}'+u\widetilde{z}' \rangle\end{align*} $$

yields lifts of the Pappas–Rapoport filtrations with the desired properties (where $\langle \cdot \rangle $ here denotes generation as $R_1[u]/(u^2)$ -modules). As already explained, this yields a homomorphism $S_1 \to R_1$ . Furthermore our definition of the parameters $s_{1,1}$ , $s_{1,1}'$ and $s_{2,1}$ ensures that $s_{1,1} \mapsto \overline {X}_1$ , $s_{1,1}' \mapsto \overline {X}_1^{\prime }$ and $s_{2,1} \mapsto \overline {X}_2 \bmod (\overline {X}_1,\overline {X}_1^{\prime })$ , so the map is surjective. Since $R_1$ and $S_1$ are Artinian, and $\rho _1:R_1 \to S_1$ is also surjective, it follows that the lengths of $R_1$ and $S_1$ are the same, and therefore that $\rho _1$ is an isomorphism.

Suppose now that $n \ge 1$ and that $\rho _n:R_n \to S_n$ is an isomorphism. In order to prove that $\rho _{n+1}$ is an isomorphism, it suffices (arguing as above) to showFootnote ⁹ that the composite $S \twoheadrightarrow S_n \stackrel {\rho _n^{-1}}{\longrightarrow }R_n$ lifts to a surjective homomorphism $S \to R_{n+1}$ . Using the thickening ${\operatorname {Spec}\,}(S_n) \hookrightarrow {\operatorname {Spec}\,}(R_{n+1})$ , with $H^1_{\mathrm {crys}}(A_n/R_{n+1})$ instead of $H^1_{\operatorname {dR}}(A_0/{\mathbb {F}}_p)\otimes _{{\mathbb {F}}_p} R_1$ , the construction of lifts of the Pappas–Rapoport filtrations on $H^1_{\operatorname {dR}}(A_n/S_n)$ is similar to the one above, once one has abstracted the argument establishing the existence of suitable bases for $\widetilde {P}_i$ and $\widetilde {P}^{\prime }_i$ (see the proof in the general case). While the isomorphism $H^1_{\mathrm {crys}}(A_0/R_1) \cong H^1_{\operatorname {dR}}(A_0/{\mathbb {F}}_p)\otimes _{{\mathbb {F}}_p} R_1$ was implicitly used in the proof of the surjectivity of the resulting homomorphism $S_1 \to R_1$ , no such isomorphism is available relative to the thickening ${\operatorname {Spec}\,}(S_n) \hookrightarrow {\operatorname {Spec}\,}(R_{n+1})$ , nor is it needed, since the surjectivity of the resulting homomorphism $S_{n+1} \to R_{n+1}$ is immediate from that of its composite with $R_{n+1} \to R_n$ .

2.6 Local structure: in general

We now proceed to analyze the local structure of $\widetilde {Y}_0({\mathfrak {P}})$ for arbitrary F and ${\mathfrak {P}}$ , provingFootnote ¹⁰ in particular that the schemes are reduced and syntomic (i.e., flat local complete intersections, over ${\mathcal {O}}$ ).

Let y be a closed point of $\widetilde {Y}_{U_0({\mathfrak {P}})}$ in characteristic p. We let S denote the local ring ${\mathcal {O}}_{\widetilde {Y}_{U_0({\mathfrak {P}})},y}$ , ${\mathfrak {n}}$ its maximal ideal, $S_0$ the residue field, $S_1 = S/({\mathfrak {n}}^2,\varpi )$ and $S_n = S/{\mathfrak {n}}^n$ for $n> 1$ . We let $(\underline {A},\underline {A}',\psi )$ denote the corresponding triple over S, and similarly write $(\underline {A}_n,\underline {A}_n^{\prime },\psi _n)$ for the triple over $S_n$ for $n \ge 0$ . For each $\tau = \tau _{{\mathfrak {p}},i} \in \Sigma _0$ , we view the Pappas–Rapoport filtrations as being defined by $S[u]/(E_\tau )$ -submodules $F_\tau ^{(j)} \subset H^0(A,\Omega ^1_{A/S})_\tau $ and their quotients $F_{\tau ,n}^{(j)} \subset H^0(A_n,\Omega ^1_{A_n/S_n})_\tau $ over $S_n[u]/(E_\tau )$ for $n \ge 0$ , and similarly with A replaced by $A'$ . For each $\theta = \theta _{{\mathfrak {p}},i,j} \in \Sigma $ , we let $L_{\theta } = F_\tau ^{(j)}/F_\tau ^{(j-1)}$ and $P_{\theta } = G_\tau ^{(j)}/F_\tau ^{(j-1)}$ , where $G_\tau ^{(j)}$ denotes the preimage of $F_\tau ^{(j-1)}$ under multiplication by $u - \theta (\varpi _{{\mathfrak {p}}})$ on $H^1_{{\operatorname {dR}}}(A/S)_\tau $ , so that $L_\theta $ is a rank one summand of the free rank two S-module $P_\theta $ . We similarly define $L_\theta ^{\prime }$ and $P_\theta ^{\prime }$ , and systematically use $\cdot _n$ to denote $\cdot \otimes _S S_n$ .

By definition, the isogeny $\psi :A \to A'$ induces morphisms $\psi _\theta ^{*}:P_\theta ^{\prime } \to P_\theta $ for $\theta \in \Sigma $ such that $\psi _\theta ^{*}(L_\theta ^{\prime }) \subset L_\theta $ . Note that $\psi _\theta ^{*}$ is an isomorphism for $\theta \not \in \Sigma _{\mathfrak {P}}$ . Recall also from [Reference DiamondDia23, §3.1] that Poincaré duality and the quasi-polarizations give rise to perfect alternating pairings on $P_\theta $ and $P_\theta ^{\prime }$ , which we denote by $\langle \cdot ,\cdot \rangle _\theta $ and $\langle \cdot ,\cdot \rangle _\theta ^{\prime }$ . Furthermore, the compatibility of $\psi $ with the quasi-polarizations implies the commutativity of the resulting diagram

Note in particular if $\theta \in \Sigma _{\mathfrak {P}}$ , then the $S_0$ -linear map $\psi _{\theta ,0}^{*}:P_{\theta ,0}' \to P_{\theta ,0}$ is not invertible (since $\wedge ^2_{S_0}\psi _{\theta ,0}^{*} = 0$ ); we will show that it has rank one. To that end, let $W = W(S_0)$ and consider the free rank two ${\mathcal {O}}_F\otimes W$ -module $D:=H^1_{\operatorname {crys}}(A_0/W)$ . Thus, D decomposes as $\oplus _\tau D_\tau $ where each $D_\tau $ is free of rank two over $W[u]/(E_\tau )$ , and we let $\widetilde {G}_\tau ^{(j)}$ denote the preimage of $G_{\tau ,0}^{(j)} = u^{-1}F_{\tau ,0}^{(j-1)}$ in $D_\tau $ under the canonical projection to $D_\tau /pD_\tau \cong H^1_{\operatorname {dR}}(A_0/S_0)_\tau $ . Note that if $\tau = \tau _{{\mathfrak {p}},i}$ , then we have a sequence of inclusions

$$ \begin{align*}u^{e_{\mathfrak{p}} - 1}D_\tau = \widetilde{G}_{\tau}^{(1)} \subset \widetilde{G}_\tau^{(2)} \subset \cdots \subset \widetilde{G}_\tau^{(e_{\mathfrak{p}}-1)}\subset \widetilde{G}_\tau^{(e_{\mathfrak{p}})}\end{align*} $$

such that $\dim _{S_0}(\widetilde {G}_\tau ^{(j)}/pD_\tau ) = \dim _{S_0}(G_{\tau ,0}^{(j)}) = j+1$ and the projection to $H^1_{\operatorname {dR}}(A_0/S_0)_\tau $ identifies $\widetilde {G}_\tau ^{(j)}/u\widetilde {G}_\tau ^{(j)}$ with $P_{\theta ,0}$ for $j=1,\ldots ,e_{\mathfrak {p}}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ . Similarly, decomposing $D':=H^1_{\operatorname {crys}}(A^{\prime }_0/W)$ as $\oplus _\tau D^{\prime }_\tau $ , and defining $\widetilde {G}_\tau ^{\prime (j)}$ as the preimage of $G_{\tau ,0}^{\prime (j)}$ in $D^{\prime }_\tau $ , the isogeny $\psi _0:A_0 \to A_0^{\prime }$ induces an injective ${\mathcal {O}}_F\otimes W$ -linear homomorphism $D' \to D$ whose cokernel is killed by u and has dimension $f_{\mathfrak {P}}$ over $S_0$ . Furthermore, the restrictions $\widetilde {\psi }_\tau ^{*} :D^{\prime }_\tau \to D_\tau $ are isomorphisms for $\tau \not \in \Sigma _{{\mathfrak {P}},0}$ , and restrict to homomorphisms $\widetilde {G}_\tau ^{\prime (j)} \to \widetilde {G}_\tau ^{(j)}$ whose cokernel projects onto that of $\psi _{\theta ,0}^{*}$ for each $\tau = \tau _{{\mathfrak {p}},i} \in \Sigma _{{\mathfrak {P}},0}$ , $\theta = \theta _{{\mathfrak {p}},i,j}$ . Therefore, it suffices to prove that the cokernel of each $\widetilde {G}_\tau ^{\prime (j)} \to \widetilde {G}_\tau ^{(j)}$ , necessarily nontrivial if $\tau \in \Sigma _{{\mathfrak {P}},0}$ , has length (at most) one over W. For $j=1$ , this follows from the identifications $u^{e_{\mathfrak {p}} - 1}D_\tau = \widetilde {G}_{\tau }^{(1)}$ and $u^{e_{\mathfrak {p}} - 1}D^{\prime }_\tau = \widetilde {G}_{\tau }^{\prime (1)}$ , which implies that the sum over $\tau $ of the lengths of the cokernels is $f_{\mathfrak {P}}$ . For $j> 1$ , it then follows by induction from the injectivity of $\widetilde {\psi }_\tau ^{*}$ and the fact that $\widetilde {G}_\tau ^{(j)}/\widetilde {G}_\tau ^{(j-1)}$ and $\widetilde {G}_\tau ^{\prime (j)}/\widetilde {G}_\tau ^{\prime (j-1)}$ each have length one.

We then define

$$ \begin{align*}\Sigma_y = \{\,\theta \in \Sigma_{\mathfrak{P}}\,|\, {\operatorname{im}}(\psi^{*}_{\theta,0}) = L_{\theta,0}\,\}\quad\mbox{and}\quad \Sigma^{\prime}_y = \{\,\theta \in \Sigma_{\mathfrak{P}}\,|\, \ker(\psi^{*}_{\theta,0}) = L^{\prime}_{\theta,0}\,\}.\end{align*} $$

Note that $\Sigma _{\mathfrak {P}} = \Sigma _y \cup \Sigma ^{\prime }_y$ , and define

$$ \begin{align*}\begin{array}{c} R = {\mathcal{O}} [X_\theta,X_\theta^{\prime}]_{\theta \in \Sigma}/ (g_\theta)_{\theta\in \Sigma}, \\ \ \\ \mbox{where}\quad g_\theta = \left\{\begin{array}{ll} X_\theta - \theta(\varpi_{\mathfrak{P}}) X_\theta^{\prime}, & \mbox{if } \theta\in \Sigma \setminus \Sigma_y^{\prime},\\ X^{\prime}_\theta - \theta(\varpi_{\mathfrak{P}}) X_\theta, & \mbox{if } \theta\in \Sigma^{\prime}_y \setminus \Sigma_y,\\ X_\theta X^{\prime}_\theta + \theta(\varpi_{\mathfrak{P}}), & \mbox{if } \theta\in \Sigma_y \cap \Sigma^{\prime}_y. \end{array} \right. \end{array}\end{align*} $$

We will show that a suitable parametrization of the lines $L_\theta $ and $L_\theta ^{\prime }$ by the variables $X_\theta $ and $X_\theta ^{\prime }$ defines an ${\mathcal {O}}$ -algebra homomorphism $R \to S$ inducing an isomorphism $W \otimes _{W(k)} \widehat {R}_{\mathfrak {m}} \stackrel {\sim }{\to } \widehat {S}_{\mathfrak {n}}$ , where ${\mathfrak {m}}$ is the maximal ideal of R generated by $\varpi $ and the variables $X_\theta $ and $X_\theta ^{\prime }$ for $\theta \in \Sigma $ . (Recall that k is the residue field of ${\mathcal {O}}$ , so that $R/{\mathfrak {m}} = k$ , and that $W(k)$ denotes the ring of Witt vectors of k.)

Let $\alpha $ denote the composite of the canonical isomorphisms

$$ \begin{align*}H^1_{\operatorname{dR}}(A_0/S_0) \otimes_{S_0} S_1 \stackrel{\sim}{\longrightarrow} H^1_{\operatorname{crys}}(A_0/S_1) \stackrel{\sim}{\longrightarrow} H^1_{\operatorname{dR}}(A_1/S_1)\end{align*} $$

(where we write $H^1_{\operatorname {crys}}(A_0/S_1)$ for $(R^1s_{0,\mathrm {crys},*}{\mathcal {O}}_{A_0,\mathrm {crys}})(S_1)$ ), and similarly define

$$ \begin{align*}\alpha': H^1_{\operatorname{dR}}(A_0^{\prime}/S_0) \otimes_{S_0} S_1 \stackrel{\sim}{\longrightarrow} H^1_{\operatorname{dR}}(A_1^{\prime}/S_1).\end{align*} $$

Thus, $\alpha $ and $\alpha '$ are ${\mathcal {O}}_F\otimes S_1$ -linear and compatible with $\psi _0^{*}$ and $\psi _1^{*}$ . It follows that for each $\tau = \tau _{{\mathfrak {p}},i} \in \Sigma _0$ , $\alpha $ and $\alpha '$ decompose as direct sums of $S_1[u]/u^{e_{{\mathfrak {p}}}}$ -linear isomorphisms $\alpha _\tau $ and $\alpha ^{\prime }_{\tau }$ such that the diagrams

commute. We then consider the chain of ideals

$$ \begin{align*}({\mathfrak{n}}^2,\varpi) = I_\tau^{(0)} \subset I_\tau^{(1)} \subset \cdots \subset I_\tau^{(e_{{\mathfrak{p}}} -1)} \subset I_\tau^{(e_{{\mathfrak{p}}})} \subset {\mathfrak{n}}\end{align*} $$

where $I_\tau ^{(j)}$ is defined by the vanishing of the maps

$$ \begin{align*}F_{\tau,0}^{(\ell)}\otimes_{S_0} S_1 \longrightarrow H^1_{\operatorname{dR}}(A_1/S_1)_\tau / F_{\tau,1}^{(\ell)} \quad\mbox{and}\quad F_{\tau,0}^{\prime(\ell)}\otimes_{S_0} S_1 \longrightarrow H^1_{\operatorname{dR}}(A^{\prime}_1/S_1)_\tau / F_{\tau,1}^{\prime(\ell)}\end{align*} $$

induced by $\alpha _\tau $ and $\alpha _\tau '$ for $\ell = 1,\ldots ,j$ . For $\theta = \theta _{{\mathfrak {p}},i,j}$ , we let $S_\theta = S/I_\tau ^{(j-1)}$ , so that $\alpha _\tau $ and $\alpha _\tau '$ restrict in particular to $S_\theta [u]$ -linear isomorphisms

$$ \begin{align*}F_{\tau,0}^{(j-1)} \otimes_{S_0} S_\theta \stackrel{\sim}{\longrightarrow} F_\tau^{(j-1)} \otimes_S S_\theta \quad\mbox{and}\quad F_{\tau,0}^{\prime(j-1)} \otimes_{S_0} S_\theta \stackrel{\sim}{\longrightarrow} F_\tau^{\prime(j-1)} \otimes_S S_\theta,\end{align*} $$

and hence to $S_\theta [u]$ -linear isomorphisms

$$ \begin{align*}G_{\tau,0}^{(j)} \otimes_{S_0} S_\theta \stackrel{\sim}{\longrightarrow} G_\tau^{(j)} \otimes_S S_\theta \quad\mbox{and}\quad G_{\tau,0}^{\prime(j)} \otimes_{S_0} S_\theta \stackrel{\sim}{\longrightarrow} G_\tau^{\prime(j)} \otimes_S S_\theta.\end{align*} $$

We thus obtain $S_\theta $ -linear isomorphisms $\alpha _\theta $ and $\alpha _\theta ^{\prime }$ such that the diagram

commutes. Furthermore, $\alpha _\theta $ is compatible with the pairings $\langle \cdot ,\cdot \rangle _{\theta ,0}$ and $\langle \cdot ,\cdot \rangle _\theta $ , and similarly, $\alpha _\theta ^{\prime }$ is compatible with $\langle \cdot ,\cdot \rangle ^{\prime }_{\theta ,0}$ and $\langle \cdot ,\cdot \rangle ^{\prime }_\theta $ .

We claim that bases ${\mathcal {B}}_\theta = (e_\theta ,f_\theta )$ for $P_\theta $ and ${\mathcal {B}}_\theta ^{\prime } = (e_\theta ^{\prime },f_\theta ^{\prime })$ for $P_\theta ^{\prime }$ may be chosen so that

• • $e_\theta \otimes _S S_\theta = \alpha _\theta (L_{\theta ,0} \otimes _{S_0} S_\theta )$ and $e^{\prime }_\theta \otimes _S S_\theta = \alpha ^{\prime }_\theta (L^{\prime }_{\theta ,0} \otimes _{S_0} S_\theta )$ ;

• • $\langle e_\theta ,f_\theta \rangle _\theta = \langle e^{\prime }_\theta , f^{\prime }_\theta \rangle ^{\prime }_\theta = 1$ ;

• • the matrix of $\psi _\theta ^{*}$ with respect to ${\mathcal {B}}_\theta $ and ${\mathcal {B}}_\theta ^{\prime }$ is

  (2.4) $$ \begin{align} \left( \begin{array}{cc} {1} & {0} \\ {0} & {\theta(\varpi_{\mathfrak{P}})} \end{array} \right),\quad \left( \begin{array}{cc} {\theta(\varpi_{\mathfrak{P}})} & {0} \\ {0} & {1} \end{array} \right)\quad\mbox{or}\quad\left( \begin{array}{cc} {0} & {1} \\ {-\theta(\varpi_{\mathfrak{P}})} & {0} \end{array} \right) \end{align} $$
  according to whetherFootnote ¹¹ $\theta \in \Sigma \setminus \Sigma _y^{\prime }$ , $\Sigma _y^{\prime } \setminus \Sigma _y$ or $\Sigma _y \cap \Sigma _y^{\prime }$ .

Indeed, first choose bases ${\mathcal {B}}_{\theta ,0} = (e_{\theta ,0},f_{\theta ,0})$ for $P_{\theta ,0}$ and ${\mathcal {B}}_{\theta ,0}' = (e_{\theta ,0}',f_{\theta ,0}')$ for $P_{\theta ,0}'$ so that $L_{\theta ,0} = S_0e_{\theta ,0}$ , $L_{\theta ,0}' = S_0e_{\theta ,0}'$ , $\langle e_{\theta ,0},f_{\theta ,0} \rangle _{\theta ,0} = \langle e^{\prime }_{\theta ,0},f^{\prime }_{\theta ,0} \rangle ^{\prime }_{\theta ,0} = 1$ and $\psi _{\theta ,0}^{*}$ has the required form ( $\!\bmod \,{\mathfrak {n}}$ ), and lift the bases $(\alpha _\theta (e_{\theta ,0} \otimes 1),\alpha _\theta (f_{\theta ,0}\otimes 1))$ and $(\alpha ^{\prime }_\theta (e^{\prime }_{\theta ,0} \otimes 1),\alpha ^{\prime }_\theta (f^{\prime }_{\theta ,0}\otimes 1))$ arbitrarily to bases ${\mathcal {B}}_\theta $ and ${\mathcal {B}}_\theta ^{\prime }$ over S satisfying the condition on the pairings. The matrix T of $\psi _\theta ^{*}$ with respect to ${\mathcal {B}}_\theta $ and ${\mathcal {B}}_\theta ^{\prime }$ then has the required form mod $I_\tau ^{(j-1)}$ and satisfies $\det (T) = \theta (\varpi _{\mathfrak {P}})$ . We may then replace ${\mathcal {B}}_\theta $ and ${\mathcal {B}}^{\prime }_\theta $ by ${\mathcal {B}}_\theta U$ and ${\mathcal {B}}_\theta ^{\prime } U'$ for some $ U,\,U'\,\in \ker (\operatorname {SL}_2(S) \to \operatorname {SL}_2(S_\theta ))$ so that the resulting matrix $U^{-1}TU'$ has the required form.

We now have $L_\theta = S(e_\theta - s_\theta f_\theta )$ and $L^{\prime }_\theta = S(e^{\prime }_\theta - s^{\prime }_\theta f^{\prime }_\theta )$ for unique $s_\theta , s^{\prime }_\theta \in {\mathfrak {n}}$ , and the fact that $\psi _\theta ^{*}(L_\theta ^{\prime }) \subset L_\theta $ means that

• • $s_\theta = \theta (\varpi _{\mathfrak {P}})s^{\prime }_\theta $ if $\theta \in \Sigma \setminus \Sigma _y^{\prime }$ ;

• • $s^{\prime }_\theta = \theta (\varpi _{\mathfrak {P}})s_\theta $ if $\theta \in \Sigma _y^{\prime } \setminus \Sigma _y$ ;

• • $s_\theta s^{\prime }_\theta = - \theta (\varpi _{\mathfrak {P}})$ if $\theta \in \Sigma _y \cap \Sigma ^{\prime }_y$ .

Thus, the ${\mathcal {O}}$ -algebra homomorphism ${\mathcal {O}} [X_\theta ,X_\theta ^{\prime }]_{\theta \in \Sigma } \longrightarrow S$ defined by $X_\theta \mapsto s_\theta $ , $X^{\prime }_\theta \mapsto s^{\prime }_\theta $ factors through R. We will prove that the resulting morphism $\rho : W\otimes _{W(k)} \widehat {R}_{{\mathfrak {m}}} \to \widehat {S}_{\mathfrak {n}}$ is an isomorphism.

We let $R_1 = W \otimes _{W(k)} R/({\mathfrak {m}}^2,\varpi )$ and $R_n = W \otimes _{W(k)} R/{\mathfrak {m}}^n$ , so we obtain morphisms $\rho _n: R_n \to S_n$ . It suffices to prove that each of these is an isomorphism, which we achieve by induction on n.

We start by showing that $\rho _1$ is surjective (i.e., that the homomorphism induced by $\rho $ on tangent spaces is injective). First, note that $I_\tau ^{(j)} = (I_\tau ^{(j-1)},s_\theta ,s^{\prime }_\theta )$ for all $\tau = \tau _{{\mathfrak {p}},i}$ , $\theta = \theta _{{\mathfrak {p}},i,j}$ . Indeed, by construction, the ideal $I_\tau ^{(j)}/I_\tau ^{(j-1)}$ of $S_\theta $ is defined by the vanishing of the maps

$$ \begin{align*}L_{\theta,0}\otimes_{S_0} S_\theta \longrightarrow (P_\theta/L_\theta) \otimes_S S_\theta \quad\mbox{and}\quad L^{\prime}_{\theta,0}\otimes_{S_0} S_\theta \longrightarrow (P^{\prime}_\theta/L^{\prime}_\theta) \otimes_S S_\theta\end{align*} $$

induced by $\alpha _\theta $ and $\alpha ^{\prime }_\theta $ , and these are precisely $s_\theta $ and $s^{\prime }_\theta $ ( $\bmod \, I_{\tau }^{(j-1)}$ ) relative to the bases $\alpha _\theta ^{-1}(e_\theta \otimes 1)$ , $(f_\theta + L_\theta ) \otimes 1$ , $\alpha _\theta ^{\prime -1}(e_\theta ^{\prime }\otimes 1)$ and $(f_\theta ^{\prime }+L^{\prime }_\theta ) \otimes 1$ . It follows that the image of $\rho _1$ contains each $I_\tau ^{(e_{{\mathfrak {p}}})}$ , so letting T denote the quotient of S by the ideal generated by $I_\tau ^{(e_{{\mathfrak {p}}})}$ for all $\tau \in \Sigma _0$ , it suffices to prove the projection $T \to S_0$ is an isomorphism (i.e., that the triple $(\underline {A},\underline {A}', \psi ) \otimes _{S} T$ is isomorphic to $(\underline {A}_0,\underline {A}^{\prime }_0, \psi _0)\otimes _{S_0} T$ ). By the definition of T, we have that $F_{\tau ,0}^{(j)} \otimes _{S_0} T$ corresponds to $F_{\tau ,1}^{(j)} \otimes _{S_1} T$ under $\alpha _\tau \otimes _{S_1} T$ for all $\tau $ and j, and similarly for $F_{\tau ,0}^{\prime (j)}$ , $F_{\tau ,1}^{\prime (j)}$ and $\alpha _\tau '$ , so the conclusion is immediate from the equivalence established at the end of §2.4.

To deduce that $\rho _1$ is an isomorphism, it suffices to prove that $\lg (R_1) \le \lg (S_1)$ , which will follow from the existence of a surjection $S \to R_1$ . We construct such a surjection by defining suitable lifts of the Pappas–Rapoport filtrations $F_{0}^\bullet $ and $F_{0}^{\prime \bullet }$ to $H^1_{\mathrm {crys}}(A_0/R_1)$ and $H^1_{\mathrm {crys}}(A_0^{\prime }/R_1)$ . For $\tau = \tau _{{\mathfrak {p}},i}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ , we let $R_\theta = R_1/J_\tau ^{(j-1)}$ where the ideals $J_\tau ^{(j)}$ are defined inductively by $J_\tau ^{(0)} = (0)$ and $J_\tau ^{(j)} = (J_\tau ^{(j-1)}, X_\theta , X^{\prime }_\theta )$ . We will inductively define chains of $R_1[u]/u^{e_{{\mathfrak {p}}}}$ -submodules

$$ \begin{align*}\begin{array}{ll} & 0 = \widetilde{F}_\tau^{(0)} \subset \widetilde{F}_\tau^{(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{(e_{{\mathfrak{p}}})} \subset H^1_{\mathrm{crys}}(A_0/R_1)_\tau \\ \mbox{and}\quad & 0 = \widetilde{F}_\tau^{\prime(0)} \subset \widetilde{F}_\tau^{\prime(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{\prime(e_{{\mathfrak{p}}})} \subset H^1_{\mathrm{crys}}(A_0^{\prime}/R_1)_\tau\end{array}\end{align*} $$

such that the following hold for $j = 1,\ldots ,e_{{\mathfrak {p}}}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ :

• • $\widetilde {F}_\tau ^{(j)}$ and $\widetilde {F}_\tau ^{\prime (j)}$ are free (of rank j) over $R_1$ ;

• • $u\widetilde {F}_\tau ^{(j)} \subset \widetilde {F}_\tau ^{(j-1)}$ , $u\widetilde {F}_\tau ^{\prime (j)} \subset \widetilde {F}_\tau ^{\prime (j-1)}$ and $\psi _{0,\mathrm {crys}}^{*}(\widetilde {F}_\tau ^{\prime (j)}) \subset \widetilde {F}_\tau ^{(j)}$ ;

• • $\widetilde {F}_\tau ^{(j-1)} \otimes _{R_1} R_\theta $ corresponds to $F_{\tau ,0}^{(j-1)} \otimes _{S_0} R_\theta $ and $\widetilde {F}_\tau ^{\prime (j-1)} \otimes _{R_1} R_\theta $ corresponds to $F_{\tau ,0}^{\prime (j-1)} \otimes _{S_0} R_\theta $ under the canonical isomorphisms $H^1_{\operatorname {dR}}(A_0/S_0)\otimes _{S_0} R_\theta \cong H^1_{\mathrm {crys}}(A_0/R_1)\otimes _{R_1} R_\theta $ and $H^1_{\operatorname {dR}}(A^{\prime }_0/S_0)\otimes _{S_0} R_\theta \cong H^1_{\mathrm {crys}}(A^{\prime }_0/R_1)\otimes _{R_1} R_\theta $ , which therefore induce injective $R_\theta $ -linear homomorphisms

  $$ \begin{align*}\begin{array}{ll} & \beta_\theta: P_{\theta,0} \otimes_{S_0} R_\theta \longrightarrow (H^1_{\mathrm{crys}}(A_0/R_1)_\tau/\widetilde{F}_\tau^{(j-1)}) \otimes_{R_1} R_\theta\\ \mbox{and}\quad & \beta_\theta^{\prime} : P_{\theta,0}' \otimes_{S_0} R_\theta \longrightarrow (H^1_{\mathrm{crys}}(A^{\prime}_0/R_1)_\tau/\widetilde{F}_\tau^{\prime(j-1)}) \otimes_{R_1} R_\theta; \end{array}\end{align*} $$

• • $(\widetilde {F}_\tau ^{(j)}/\widetilde {F}_\tau ^{(j-1)}) \otimes _{R_1} R_\theta $ is generated over $R_\theta $ by $\beta _\theta (e_{\theta ,0} \otimes 1 - f_{\theta ,0} \otimes X_\theta )$ and $(\widetilde {F}_\tau ^{\prime (j)}/\widetilde {F}_\tau ^{\prime (j-1)}) \otimes _{R_1} R_\theta $ is generated over $R_\theta $ by $\beta ^{\prime }_\theta (e^{\prime }_{\theta ,0} \otimes 1 - f^{\prime }_{\theta ,0} \otimes X^{\prime }_\theta )$ .

Suppose then that $1 \le j \le e_{{\mathfrak {p}}}$ and that

$$ \begin{align*}0 = \widetilde{F}_\tau^{(0)} \subset \widetilde{F}_\tau^{(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{(j-1)} \quad \mbox{and}\quad 0 = \widetilde{F}_\tau^{\prime(0)} \subset \widetilde{F}_\tau^{\prime(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{\prime(j-1)}\end{align*} $$

have been constructed as above. We let $\widetilde {G}_\tau ^{(j)} = u^{-1}\widetilde {F}_{\tau }^{(j-1)}$ , that is, the preimage of $\widetilde {F}_{\tau }^{(j-1)}$ in $H^1_{\mathrm {crys}}(A_0/R_1)_\tau $ under u, and $\widetilde {P}_\theta = \widetilde {G}_\tau ^{(j)}/\widetilde {F}_{\tau }^{(j-1)}$ . Since $H^1_{\mathrm {crys}}(A_0/R_1)_\tau $ is free of rank two over $R_1[u]/(u^{e_{\mathfrak {p}}})$ , we see that $\widetilde {F}_{\tau }^{(j-1)} \subset u H^1_{\mathrm {crys}}(A_0/R_1)_\tau $ , so that $\widetilde {G}_\tau ^{(j)}$ and $\widetilde {P}_\theta $ are free (of ranks $j+1$ and $2$ , respectively) over $R_1$ . Similarly, we let $\widetilde {G}_\tau ^{\prime (j)} = u^{-1}\widetilde {F}_\tau ^{\prime (j-1)}$ and $\widetilde {P}_\theta ^{\prime } = \widetilde {G}_\tau ^{\prime (j)}/\widetilde {F}_\tau ^{\prime (j-1)}$ . The conditions on $\widetilde {F}_\tau ^{(j-1)}$ and $\widetilde {F}_\tau ^{\prime (j-1)}$ imply that the third bullet holds, that $\psi _{0,\mathrm {crys}}^{*}$ induces an $R_1$ -linear homomorphism $\widetilde {\psi }^{*}_\theta : \widetilde {P}_\theta ^{\prime } \to \widetilde {P}_\theta $ , and that $\beta _\theta $ and $\beta _\theta ^{\prime }$ define isomorphisms $P_{\theta ,0} \otimes _{S_0} R_\theta \stackrel {\sim }{\longrightarrow } \widetilde {P}_\theta \otimes _{R_1} R_\theta $ and $P^{\prime }_{\theta ,0} \otimes _{S_0} R_\theta \stackrel {\sim }{\longrightarrow } \widetilde {P}^{\prime }_\theta \otimes _{R_1} R_\theta $ compatible with $\psi ^{*}_{\theta ,0}$ and $\widetilde {\psi }^{*}_\theta $ . Furthermore, the construction of $\widetilde {F}_\tau ^{(j)}$ and $\widetilde {F}_\tau ^{\prime (j)}$ satisfying the remaining conditions is equivalent to that of invertible $R_1$ -submodules $\widetilde {L}_\theta \subset \widetilde {P}_\theta $ and $\widetilde {L}^{\prime }_\theta \subset \widetilde {P}^{\prime }_\theta $ such that

• • $\widetilde {\psi }_\theta ^{*}(\widetilde {L}_\theta ^{\prime }) \subset \widetilde {L}_\theta $ ;

• • $\widetilde {L}_\theta \otimes _{R_1} R_\theta = \beta _\theta (e_{\theta ,0} \otimes 1 - f_{\theta ,0} \otimes X_\theta )R_\theta $ ;

• • $\widetilde {L}^{\prime }_\theta \otimes _{R_1} R_\theta = \beta ^{\prime }_\theta (e^{\prime }_{\theta ,0} \otimes 1 - f^{\prime }_{\theta ,0} \otimes X^{\prime }_\theta )R_\theta $ .

Exactly as for de Rham cohomology (see [Reference DiamondDia23, §3.1]), we obtain a perfect pairing

$$ \begin{align*}H^1_{\mathrm{crys}}(A_0/R_1)_\tau \stackrel{\sim}{\longrightarrow} {\operatorname{Hom}\,}_{R_1[u]/u^{e_{{\mathfrak{p}}}}}(H^1_{\mathrm{crys}}(A_0/R_1)_\tau, R_1[u]/(u^{e_{\mathfrak{p}}}))\end{align*} $$

from the homomorphisms induced by the quasi-polarization on $A_0$ and Poincaré duality on crystalline cohomology. Furthermore, the pairing is compatible with the corresponding one on $H^1_{\operatorname {dR}}(B/R_1)_\tau $ , where B and the resulting quasi-polarization are associated to an arbitrarily chosen extension of

$$ \begin{align*}0 \subset \widetilde{F}_\tau^{(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{(j-1)}\end{align*} $$

to a lift $\widetilde {F}^\bullet $ of the Pappas–Rapoport filtrations to $H^1_{\mathrm {crys}}(A_0/R_1)$ . We may therefore apply [Reference DiamondDia23, Lemma 3.1.1] to conclude that $(\widetilde {F}_\tau ^{(j-1)})^\perp = u^{j+1-e_{{\mathfrak {p}}}} \widetilde {F}_\tau ^{(j-1)} = u^{j-e_{{\mathfrak {p}}}}\widetilde {G}_\tau ^{(j)}$ , and hence obtain a perfect alternating $R_1$ -valued pairing on $\widetilde {P}_\theta $ , compatible via $\beta _\theta $ with $\langle \cdot , \cdot \rangle _{\theta ,0}$ . We similarly obtain a perfect alternating $R_1$ -valued pairing on $\widetilde {P}_\theta ^{\prime }$ compatible via $\beta _\theta ^{\prime }$ with $\langle \cdot , \cdot \rangle ^{\prime }_{\theta ,0}$ . Furthermore, the compatibility of $\psi _0$ with the quasi-polarizations again implies that $\det \widetilde {\psi }^{*}_\theta = \theta (\varpi _{\mathfrak {P}})\, (=0)$ with respect to the pairings.

The same argument as in the construction of the bases ${\mathcal {B}}_\theta $ and ${\mathcal {B}}_\theta ^{\prime }$ then yields lifts of

$$ \begin{align*}\beta_\theta(e_{\theta,0} \otimes 1,f_{\theta,0} \otimes 1)\quad\mbox{and}\quad \beta_\theta^{\prime}(e^{\prime}_{\theta,0} \otimes 1,f^{\prime}_{\theta,0} \otimes 1)\end{align*} $$

to bases $(\widetilde {e}_\theta ,\widetilde {f}_\theta )$ for $\widetilde {P}_\theta $ and $(\widetilde {e}^{\prime }_\theta ,\widetilde {f}^{\prime }_\theta )$ for $\widetilde {P}^{\prime }_\theta $ with respect to which the matrix of $\widetilde {\psi }_\theta ^{*}$ is as in (2.4). Defining $\widetilde {L}_\theta = (\widetilde {e}_\theta - X_\theta \widetilde {f}_\theta )R_1$ and $\widetilde {L}^{\prime }_\theta = (\widetilde {e}^{\prime }_\theta - X^{\prime }_\theta \widetilde {f}^{\prime }_\theta )R_1$ then completes the construction of $\widetilde {F}_\tau ^{(j)}$ and $\widetilde {F}_\tau ^{\prime (j)}$ satisfying the desired properties.

We may now apply the Grothendieck–Messing Theorem (as described at the end of §2.4) to obtain a triple $(\widetilde {\underline {A}},\widetilde {\underline {A}}',\widetilde {\psi })$ over $R_1$ lifting $(\underline {A}_0,\underline {A}_0^{\prime },\psi _0)$ . Furthermore, the properties of the filtrations $\widetilde {F}^\bullet $ and $\widetilde {F}^{\prime \bullet }$ ensure, by induction on j for each $\tau $ , that the resulting morphism $S \to R_1 \to R_\theta $ factors through $S_\theta $ and induces isomorphisms

$$ \begin{align*}\begin{array}{ccccccc} P_\theta \otimes_S R_\theta & \stackrel{\sim}{\longrightarrow}& \widetilde{P}_\theta \otimes_{R_1} R_\theta&& P_\theta^{\prime} \otimes_S R_\theta & \stackrel{\sim}{\longrightarrow}& \widetilde{P}^{\prime}_\theta \otimes_{R_1} R_\theta\\ \cup&&\cup& \mbox{and} & \cup &&\cup \\ L_\theta \otimes_S R_\theta & \stackrel{\sim}{\longrightarrow}& \widetilde{L}_\theta \otimes_{R_1} R_\theta&& L_\theta^{\prime} \otimes_S R_\theta & \stackrel{\sim}{\longrightarrow}& \widetilde{L}^{\prime}_\theta \otimes_{R_1} R_\theta\end{array}\end{align*} $$

sending $(e_\theta \otimes 1,f_\theta \otimes 1)$ to $(\widetilde {e}_\theta \otimes 1,\widetilde {f}_\theta \otimes 1)$ and $(e^{\prime }_\theta \otimes 1,f^{\prime }_\theta \otimes 1)$ to $(\widetilde {e}^{\prime }_\theta \otimes 1,\widetilde {f}^{\prime }_\theta \otimes 1)$ , so that $s_\theta \mapsto X_\theta $ and $s^{\prime }_\theta \mapsto X^{\prime }_\theta $ ( $\!\bmod \, J_\tau ^{(j-1)}$ ). We therefore conclude that the morphism $S \to R_1$ is surjective, and hence that $\rho _1:R_1 \to S_1$ is an isomorphism.

Suppose then that $n \ge 1$ and that $\rho _n:R_n \to S_n$ is an isomorphism. The surjectivity of $\rho _{n+1}$ is already immediate from that of $\rho _n$ , so it suffices to prove that $\lg (R_{n+1}) \le \lg (S_{n+1})$ , which will follow from the existence of a lift of $\rho _n^{-1}$ to morphism $S \to R_{n+1}$ , which necessarily factors through a surjective morphism from $S_{n+1}$ .

As in the case $n=0$ , we proceed by defining suitable lifts of the Pappas–Rapoport filtrations $F^\bullet _{n}$ and $F^{\prime \bullet }_{n}$ to the crystalline cohomology of $A_n$ and $A_n^{\prime }$ evaluated on the thickening $R_{n+1} \to R_n \stackrel {\sim }{\to } S_n$ (with trivial divided power structure), but the argument is now simpler since the isomorphism on tangent spaces has already been established. More precisely, we inductively define chains of $R_{n+1}[u]/(E_\tau )$ -submodules

$$ \begin{align*}\begin{array}{ll} & 0 = \widetilde{F}_\tau^{(0)} \subset \widetilde{F}_\tau^{(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{(e_{{\mathfrak{p}}})} \subset H^1_{\mathrm{crys}}(A_n/R_{n+1})_\tau \\ \mbox{and}\quad & 0 = \widetilde{F}_\tau^{\prime(0)} \subset \widetilde{F}_\tau^{\prime(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{\prime(e_{{\mathfrak{p}}})} \subset H^1_{\mathrm{crys}}(A_n^{\prime}/R_{n+1})_\tau\end{array}\end{align*} $$

such that the following hold for $j = 1,\ldots ,e_{{\mathfrak {p}}}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ :

• • $\widetilde {F}_\tau ^{(j)}$ and $\widetilde {F}_\tau ^{\prime (j)}$ are free (of rank j) over $R_{n+1}$ ;

• • $(u - \theta (\varpi _{\mathfrak {P}}))\widetilde {F}_\tau ^{(j)} \subset \widetilde {F}_\tau ^{(j-1)}$ , $(u- \theta (\varpi _{\mathfrak {P}}))\widetilde {F}_\tau ^{\prime (j)} \subset \widetilde {F}_\tau ^{\prime (j-1)}$ and $\psi _{n,\mathrm {crys}}^{*}(\widetilde {F}_\tau ^{\prime (j)}) \subset \widetilde {F}_\tau ^{(j)}$ ;

• • $\widetilde {F}_\tau ^{(j)} \otimes _{R_{n+1}} S_n$ corresponds to $F_{\tau ,n}^{(j)}$ and $\widetilde {F}_\tau ^{\prime (j)} \otimes _{R_{n+1}} S_n$ to $F_{\tau ,n}^{\prime (j)}$ under the canonical isomorphisms $H^1_{\operatorname {dR}}(A_n/S_n) \cong H^1_{\mathrm {crys}}(A_n/R_{n+1})\otimes _{R_{n+1}} S_n$ and $H^1_{\operatorname {dR}}(A^{\prime }_n/S_n) \cong H^1_{\mathrm {crys}}(A^{\prime }_n/R_{n+1})\otimes _{R_{n+1}} S_n$ .

Suppose then that $1 \le j \le e_{{\mathfrak {p}}}$ and that

$$ \begin{align*}0 = \widetilde{F}_\tau^{(0)} \subset \widetilde{F}_\tau^{(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{(j-1)} \quad \mbox{and}\quad 0 = \widetilde{F}_\tau^{\prime(0)} \subset \widetilde{F}_\tau^{\prime(1)} \subset \cdots \subset \widetilde{F}_{\tau}^{\prime(j-1)}\end{align*} $$

have been constructed as above. Let $\widetilde {G}_\tau ^{(j)} = (u-\theta (\varpi _{\mathfrak {p}}))^{-1}\widetilde {F}_{\tau }^{(j-1)}$ , $\widetilde {P}_\theta = \widetilde {G}_\tau ^{(j)}/\widetilde {F}_{\tau }^{(j-1)}$ and similarly define $\widetilde {P}^{\prime }_\theta $ , so that $\widetilde {P}_\theta $ and $\widetilde {P}_\theta ^{\prime }$ are free of rank two over $R_{n+1}$ . Furthermore, we have canonical isomorphisms

$$ \begin{align*}\beta_\theta: P_{\theta,n} \stackrel{\sim}{\longrightarrow} \widetilde{P}_\theta \otimes_{R_{n+1}} S_n \quad\mbox{and}\quad \beta^{\prime}_\theta: P^{\prime}_{\theta,n} \stackrel{\sim}{\longrightarrow} \widetilde{P}^{\prime}_\theta \otimes_{R_{n+1}} S_n,\end{align*} $$

and $\psi _{n,\mathrm {crys}}^{*}$ induces an $R_{n+1}$ -linear homomorphism $\widetilde {\psi }^{*}_\theta : \widetilde {P}_\theta ^{\prime } \to \widetilde {P}_\theta $ compatible with $\psi _{\theta ,n}^{*}$ . The construction of the required $\widetilde {F}_\tau ^{(j)}$ and $\widetilde {F}_\tau ^{\prime (j)}$ is then equivalent to that of invertible $R_{n+1}$ -submodules $\widetilde {L}_\theta \subset \widetilde {P}_\theta $ and $\widetilde {L}^{\prime }_\theta \subset \widetilde {P}^{\prime }_\theta $ such that

• • $\widetilde {\psi }_\theta ^{*}(\widetilde {L}_\theta ^{\prime }) \subset \widetilde {L}_\theta $ ;

• • $\widetilde {L}_\theta \otimes _{R_{n+1}} S_n = \beta _\theta (e_{\theta ,n} - s_\theta f_{\theta ,n} )S_n$ ;

• • $\widetilde {L}^{\prime }_\theta \otimes _{R_{n+1}} S_n = \beta ^{\prime }_\theta (e^{\prime }_{\theta ,n} - s^{\prime }_\theta f^{\prime }_{\theta ,n} )S_n$ .

The same argument as for $n=0$ (with $t_j^{-1}$ generalizing $u^{j-e_{{\mathfrak {p}}}}$ ) yields perfect alternating $R_{n+1}$ -valued pairings on $\widetilde {P}_\theta $ on $\widetilde {P}^{\prime }_\theta $ compatible via $\beta _\theta $ and $\beta _\theta ^{\prime }$ with $\langle \cdot ,\cdot \rangle _{\theta ,n}$ and $\langle \cdot , \cdot \rangle ^{\prime }_{\theta ,n}$ , and with respect to which $\det \widetilde {\psi }^{*}_\theta = \theta (\varpi _{\mathfrak {P}})$ . We may thus lift $\beta _\theta (e_{\theta ,n},f_{\theta ,n})$ and $\beta _\theta ^{\prime }(e^{\prime }_{\theta ,n},f^{\prime }_{\theta ,n})$ to bases $(\widetilde {e}_\theta ,\widetilde {f}_\theta )$ for $\widetilde {P}_\theta $ and $(\widetilde {e}^{\prime }_\theta ,\widetilde {f}^{\prime }_\theta )$ for $\widetilde {P}^{\prime }_\theta $ with respect to which the matrix of $\widetilde {\psi }_\theta ^{*}$ is as in (2.4). Defining $\widetilde {L}_\theta = (\widetilde {e}_\theta - X_\theta \widetilde {f}_\theta )R_{n+1}$ and $\widetilde {L}^{\prime }_\theta = (\widetilde {e}^{\prime }_\theta - X^{\prime }_\theta \widetilde {f}^{\prime }_\theta )R_{n+1}$ thus yields the desired filtrations $\widetilde {F}^\bullet $ and $\widetilde {F}^{\prime \bullet }$ , and hence a lift of $(A_n,A_n^{\prime },\psi _n)$ to a triple over $R_{n+1}$ .

This completes the proof that $\rho $ is isomorphism, and hence that ${\mathcal {O}}^{\wedge }_{\widetilde {Y}_{U_0({\mathfrak {P}})},y}$ is isomorphic to

$$ \begin{align*}W\otimes_{W(k)} {\mathcal{O}} [[X_\theta,X_\theta^{\prime}]]_{\theta \in \Sigma}/ (g_\theta)_{\theta\in \Sigma}.\end{align*} $$

As this is a reduced complete intersection, flat of relative dimension d over ${\mathcal {O}}$ , and $\widetilde {Y}_{U_0({\mathfrak {P}}),K}$ is smooth of dimension d over K, it follows that $\widetilde {Y}_{U_0({\mathfrak {P}})}$ is reduced and syntomic of dimension d over ${\mathcal {O}}$ , and that the same holds for $Y_{U_0({\mathfrak {P}})}$ since $\widetilde {Y}_{U_0({\mathfrak {P}})} \to Y_{U_0({\mathfrak {P}})}$ is an étale cover.

3.1 The layers of a thickening

Recall that the Kodaira–Spencer isomorphism describes the dualizing sheaf of the smooth scheme $Y_U$ over ${\mathcal {O}}$ in terms of automorphic line bundles; more precisely we have a canonicalFootnote ¹² isomorphism

$$ \begin{align*}\delta^{-1} \otimes \omega^{\otimes 2} \stackrel{\sim}{\longrightarrow} {\mathcal{K}}_{Y_U/{\mathcal{O}}},\end{align*} $$

where $\omega $ and $\delta $ are the line bundles on $Y_U$ defined in §2.3 and ${\mathcal {K}}_{Y_U/{\mathcal {O}}} = \wedge ^d_{{\mathcal {O}}_{Y_U}} \Omega ^1_{Y_U/{\mathcal {O}}}$ (see [Reference DiamondDia23, §3.3]). We will prove an analogous result for $Y_{U_0({\mathfrak {P}})}$ , which by the results of the preceding section is Gorenstein over ${\mathcal {O}}$ , and hence has an invertible dualizing sheaf ${\mathcal {K}}_{Y_{U_0({\mathfrak {P}})}/{\mathcal {O}}}$ .

First, recall that the Kodaira–Spencer isomorphism for $Y_U$ was established in [Reference Reduzzi and XiaoRX17] by constructing a suitable filtration on the sheaf of relative differentials and relating its graded pieces to automorphic line bundles. In [Reference DiamondDia23, §3.3], we gave a version of the construction which can be viewed as peeling off layers of the first infinitesimal thickening of the diagonal in $\widetilde {Y}_U \times _{{\mathcal {O}}} \widetilde {Y}_U$ . We use the same approach here to describe the layers of the first infinitesimal thickening of $\widetilde {Y}_{U_0({\mathfrak {P}})}$ in $\widetilde {Y}_U \times _{{\mathcal {O}}} \widetilde {Y}_U$ along

$$ \begin{align*}\widetilde{h}: = (\widetilde{\pi}_1,\widetilde{\pi}_2): \widetilde{Y}_{U_0({\mathfrak{P}})} \to \widetilde{Y}_U \times_{{\mathcal{O}}} \widetilde{Y}_U,\end{align*} $$

which we recall is a closed immersion for sufficiently small U by Proposition 2.1. Indeed the Kodaira–Spencer filtration for $Y_U$ may be viewed as the particular case where ${\mathfrak {P}} = {\mathcal {O}}_F$ , for which the proof in [Reference DiamondDia23, §3.3] requires only minor modifications to establish the desired generalization.

We assume throughout this section that U is sufficiently small that Proposition 2.1 holds. To simplify notation and render it more consistent with that of [Reference DiamondDia23, §3.3], we will write S for $\widetilde {Y}_U$ , X for $S \times _{{\mathcal {O}}} S$ , Y for $\widetilde {Y}_{U_0({\mathfrak {P}})}$ , ${\mathcal {J}}$ for the sheaf of ideals of ${\mathcal {O}}_X$ defining the image of $\widetilde {h}:Y \hookrightarrow X$ , $\iota :Y \hookrightarrow Z = \mathbf {Spec}({\mathcal {O}}_X/{\mathcal {J}}^2)$ for the first infinitesimal thickening of Y in X, and ${\mathcal {I}} = {\mathcal {J}}/{\mathcal {J}}^2$ for the sheaf of ideals of ${\mathcal {O}}_Z$ defining the image of $\iota $ . Since X is smooth over ${\mathcal {O}}$ of dimension $2d$ and Y is syntomic over ${\mathcal {O}}$ of dimension d (all dimensions being relative), it follows that $\widetilde {h}$ is a regular immersion (see for example [StaX, §0638]) and its conormal bundle

$$ \begin{align*}{\widetilde{{\mathcal{C}}}}:= \widetilde{h}^{*}{\mathcal{J}} = \iota^{*}{\mathcal{I}}\end{align*} $$

is locally free of rank d on $Y = \widetilde {Y}_{U_0({\mathfrak {P}})}$ .

Theorem 3.1. There exists a decomposition $\widetilde {{\mathcal {C}}} = \bigoplus _{\tau \in \Sigma _0} \widetilde {{\mathcal {C}}}_\tau $ , together with an increasing filtration

$$ \begin{align*}0 = {\operatorname{Fil}\,}^0 (\widetilde{{\mathcal{C}}}_\tau) \subset {\operatorname{Fil}\,}^1 (\widetilde{{\mathcal{C}}}_\tau) \subset \cdots \subset {\operatorname{Fil}\,}^{e_{\mathfrak{p}} - 1}( \widetilde{{\mathcal{C}}}_\tau ) \subset {\operatorname{Fil}\,}^{e_{\mathfrak{p}}}( \widetilde{{\mathcal{C}}}_\tau) = \widetilde{{\mathcal{C}}}_\tau\end{align*} $$

for each $\tau = \tau _{{\mathfrak {p}},i}$ , such that ${\operatorname {gr}\,}^j(\widetilde {{\mathcal {C}}}_\tau ) \cong \widetilde {\pi }_1^{*}{\mathcal {M}}_\theta ^{-1} \otimes _{{\mathcal {O}}_Y} \widetilde {\pi }_2^{*}{\mathcal {L}}_\theta $ for each $j=1,\ldots ,e_{\mathfrak {p}}$ , where $\theta = \theta _{{\mathfrak {p}},i,j}$ .

Proof. Let A denote the universal abelian scheme over S, let $\psi :A_1 \to A_2$ denote the universal isogeny over Y, and let $B_i = q_i^{*}A$ for $i=1,2$ , where $q_i$ denotes the restriction to Z of the $i^{\mathrm {th}}$ projection $S \times _{{\mathcal {O}}} S \to S$ . Consider the ${\mathcal {O}}_F\otimes {\mathcal {O}}_Z$ -linear morphism

(3.1) $$ \begin{align} q_2^{*}{\mathcal{H}}_{\operatorname{dR}}(A/S) \cong {\mathcal{H}}^1_{\operatorname{dR}}(B_2/Z) \cong &(R^1 s_{2,\mathrm{crys},*} {\mathcal{O}}_{A_2,\mathrm{crys}})_Z\notag \\ &\qquad\quad \stackrel{-\psi_{\mathrm{crys}}^{*}}{\longrightarrow} (R^1 s_{1,\mathrm{crys},*} {\mathcal{O}}_{{A}_1,\mathrm{crys}})_Z \cong {\mathcal{H}}^1_{\operatorname{dR}}(B_1/Z) \cong q_1^{*}{\mathcal{H}}^1_{\operatorname{dR}}(A/S) \end{align} $$

extendingFootnote ¹³ $\iota ^{*}{\mathcal {H}}^1_{\operatorname {dR}}(B_2/Z) \cong {\mathcal {H}}^1_{\operatorname {dR}}(A_2/Y) \stackrel {-\psi ^{*}}{\longrightarrow } {\mathcal {H}}^1_{\operatorname {dR}}(A_1/Y) \cong \iota ^{*}{\mathcal {H}}^1_{\operatorname {dR}}(B_1/Z)$ , where the isomorphisms flanking $-\psi _{\mathrm {crys}}^{*}$ are the ones in (2.3), for $\iota :Y \hookrightarrow Z$ with $B=B_i$ . Fixing for the moment $\tau = \tau _{{\mathfrak {p}},i} \in \Sigma _0$ and $\theta = \theta _{{\mathfrak {p}},i,1}$ , it follows from the definition of ${\mathcal {P}}_{\theta } = {\mathcal {G}}_\tau ^{(1)}$ that (3.1) restricts to an ${\mathcal {O}}_Z$ -linear morphism $q_2^{*}{\mathcal {P}}_{\theta } \to q_1^{*}{\mathcal {P}}_{\theta }$ . Furthermore, the composite

$$ \begin{align*}q_2^{*}{\mathcal{L}}_{\theta} \hookrightarrow q_2^{*}{\mathcal{P}}_{\theta} \to q_1^{*}{\mathcal{P}}_{\theta} \twoheadrightarrow q_1^{*}{\mathcal{M}}_{\theta}\end{align*} $$

has trivial pull-back to Y, so it factors through a morphism

$$ \begin{align*}\iota_*\widetilde{\pi}_2^{*} {\mathcal{L}}_{\theta} = q_2^{*}{\mathcal{L}}_{\theta} \otimes_{{\mathcal{O}}_{Z}} ({\mathcal{O}}_{Z}/{\mathcal{I}}) \longrightarrow q_1^{*}{\mathcal{M}}_{\theta} \otimes_{{\mathcal{O}}_{Z}} {\mathcal{I}} = \iota_*\widetilde{\pi}_1^{*} {\mathcal{M}}_{\theta} \otimes_{{\mathcal{O}}_{Z}} {\mathcal{I}},\end{align*} $$

and hence induces a morphism

$$ \begin{align*}\Xi_{\theta}: \iota_* (\widetilde{\pi}_1^{*}{\mathcal{M}}^{-1}_{\theta}\otimes_{{\mathcal{O}}_Y} \widetilde{\pi}_2^{*}{\mathcal{L}}_{\theta}) \longrightarrow {\mathcal{I}}.\end{align*} $$

We then define the sheaf of ideals ${\mathcal {I}}_{\tau }^{(1)}$ on Z to be the image of $\Xi _\theta $ , and we let $Z_{\tau }^{(1)}$ denote the subscheme of Z defined by ${\mathcal {I}}_{\tau }^{(1)}$ , and $q^{(1)}_{\tau ,i}$ (for $i=1,2$ ) the restrictions of the projection maps to $Z_{\tau }^{(1)} \to S$ . By construction, the pull-back of $\Xi _\theta $ to $Z_{\tau }^{(1)}$ is trivial, and hence so is that of the morphism $q_2^{*}{\mathcal {L}}_{\theta } \to q_1^{*}{\mathcal {M}}_{\theta }$ , which implies that the pull-back of (3.1) maps $q_{\tau ,2}^{(1)*}{\mathcal {L}}_{\theta } = q_{\tau ,2}^{(1)*}{\mathcal {F}}_\tau ^{(1)}$ to $q_{\tau ,1}^{(1)*}{\mathcal {L}}_{\theta } = q_{\tau ,1}^{(1)*}{\mathcal {F}}_\tau ^{(1)}$ . It therefore follows from the definition of ${\mathcal {G}}_\tau ^{(2)}$ and the ${\mathcal {O}}_F\otimes {\mathcal {O}}_Z$ -linearity of (3.1) that its pull-back to $Z_\tau ^{(1)}$ restricts to an ${\mathcal {O}}_{Z_\tau ^{(1)}}$ -linear morphism $q_{\tau ,2}^{(1)*}{\mathcal {G}}_\tau ^{(2)} \to q_{\tau ,1}^{(1)*}{\mathcal {G}}_\tau ^{(2)}$ (if $e_{\mathfrak {p}}> 1$ ), and hence to a morphism

$$ \begin{align*}q_{\tau,2}^{(1)*}{\mathcal{P}}_{\theta'} {\longrightarrow} q_{\tau,1}^{(1)*}{\mathcal{P}}_{\theta'},\end{align*} $$

where $\theta ' = \theta _{{\mathfrak {p}},i,2}$ . The same argument as above now yields a morphism

$$ \begin{align*}\Xi_{\theta'}: \iota_* (\widetilde{\pi}_1^{*}{\mathcal{M}}^{-1}_{\theta'}\otimes_{{\mathcal{O}}_Y} \widetilde{\pi}_2^{*}{\mathcal{L}}_{\theta'}) \longrightarrow {\mathcal{I}}/{\mathcal{I}}_\tau^{(1)}\end{align*} $$

whose image is ${\mathcal {I}}_\tau ^{(2)}/{\mathcal {I}}_\tau ^{(1)}$ for some sheaf of ideals ${\mathcal {I}}_{\tau }^{(2)} \supset {\mathcal {I}}_{\tau }^{(1)}$ on Z.

Iterating the above construction thus yields, for each $\tau = \tau _{{\mathfrak {p}},i} \in \Sigma _{0}$ , a chain of sheaves of ideals

$$ \begin{align*}0 = {\mathcal{I}}_{\tau}^{(0)} \subset {\mathcal{I}}_{\tau}^{(1)} \subset \cdots \subset {\mathcal{I}}_{\tau}^{(e_{\mathfrak{p}})}\end{align*} $$

on Z such that for $j=1,\ldots ,e_{\mathfrak {p}}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ , the morphism (3.1) induces

• • ${\mathcal {O}}_F \otimes {\mathcal {O}}_{Z_\tau ^{(j)}}$ -linear morphisms $q_{\tau ,2}^{(j)*}{\mathcal {F}}_\tau ^{(j)} \longrightarrow q_{\tau ,1}^{(j)*} {\mathcal {F}}_\tau ^{(j)}$

• • and surjections $\Xi _\theta : \iota _* (\widetilde {\pi }_1^{*}{\mathcal {M}}_\theta ^{-1} \otimes _{{\mathcal {O}}_Y} {\mathcal {L}}_{\theta }) \twoheadrightarrow {\mathcal {I}}_{\tau }^{(j)}/{\mathcal {I}}_{\tau }^{(j-1)}$ ,

where $Z_{\tau }^{(j)}$ is the closed subscheme of Z defined by ${\mathcal {I}}_\tau ^{(j)}$ , and $q_{\tau ,1}^{(j)}$ , $q_{\tau ,2}^{(j)}$ are the projections $Z_{\tau }^{(j)} \to S$ .

Furthermore, we claim that the map

$$ \begin{align*}\bigoplus_{{\mathfrak{p}} \in S_p} \bigoplus_{\tau \in \Sigma_{{\mathfrak{p}},0}} {\mathcal{I}}_{\tau}^{(e_{\mathfrak{p}})} \to {\mathcal{I}}\end{align*} $$

is surjective. Indeed, let $Z'$ denote the closed subscheme of Z defined by the image, so $Z'$ is the scheme-theoretic intersection of the $Z_{\tau }^{(e_{\mathfrak {p}})}$ . For $i=1,2$ , let $q^{\prime }_i$ denote the projection map $Z' \to S$ and $s_i^{\prime }:B_i^{\prime } \to Z'$ the pull-back of $s:A\to S$ . By construction, (3.1) pulls back to a morphism $q_2^{\prime *}{\mathcal {H}}^1_{{\operatorname {dR}}}(A/S) \to q_1^{\prime *}{\mathcal {H}}^1_{\operatorname {dR}}(A/S)$ under which $q_2^{\prime *}{\mathcal {F}}_\tau ^{(j)}$ maps to $q_1^{\prime *}{\mathcal {F}}_\tau ^{(j)}$ for all $\tau $ and j. In particular, $s_{2,*}'\Omega ^1_{B_2^{\prime }/T} = q_2^{\prime *}(s_*\Omega ^1_{A/S})$ maps to $q_1^{\prime *}(s_*\Omega ^1_{A/S}) = s^{\prime }_{1,*} \Omega ^1_{B_1^{\prime }/T}$ , which the Grothendieck–Messing Theorem therefore implies is induced by an isogeny $\widetilde {\psi }: B^{\prime }_1 \to B_2^{\prime }$ of abelian schemes lifting $\psi $ , which is necessarily compatible with ${\mathcal {O}}_F$ -actions, quasi-polarizations and level structures on $\underline {B}_i^{\prime } : = q_i^{\prime *}\underline {A}$ . Since $\widetilde {\psi }$ also respects the Pappas–Rapoport filtrations $q_i^{\prime *}{\mathcal {F}}^\bullet $ , it follows that the triple $(\underline {B}_1^{\prime },\underline {B}_2^{\prime },\widetilde {\psi })$ corresponds to a morphism $r:Z' \to Y$ such that $\widetilde {h}\circ r$ is the closed immersion $Z' \hookrightarrow X$ . Since $\widetilde {h}$ is also a closed immersion, it follows that r is an isomorphism, yielding the desired surjectivity.

Now defining $\widetilde {{\mathcal {C}}}_\tau = \iota ^{*}{\mathcal {I}}_\tau ^{(e_{\mathfrak {p}})}$ and ${\operatorname {Fil}\,}^j(\widetilde {{\mathcal {C}}}_\tau )= \iota ^{*}{\mathcal {I}}_{\tau }^{(j)}$ for each $\tau = \tau _{{\mathfrak {p}},i}$ and $j=1,\ldots ,e_{\mathfrak {p}}$ , we obtain surjective morphisms

$$ \begin{align*}\bigoplus_{\tau \in \Sigma_0} \widetilde{{\mathcal{C}}}_\tau \twoheadrightarrow \widetilde{{\mathcal{C}}}, \qquad \mbox{and}\qquad \widetilde{\pi}_1^{*}{\mathcal{M}}_\theta^{-1} \otimes_{{\mathcal{O}}_Y} \widetilde{\pi}_2^{*}{\mathcal{L}}_{\theta} \twoheadrightarrow {\operatorname{gr}\,}^j (\widetilde{{\mathcal{C}}}_\tau ) \quad \mbox{for each } \theta = \theta_{{\mathfrak{p}},i,j}. \end{align*} $$

Since the $\widetilde {\pi }_1^{*}{\mathcal {M}}_\theta ^{-1} \otimes _{{\mathcal {O}}_Y} \widetilde {\pi }_2^{*}{\mathcal {L}}_{\theta }$ are line bundles and $\widetilde {{\mathcal {C}}}$ is a vector bundle of rank d, it follows that all the maps are isomorphisms.