2.1 Embeddings and decompositions

We first set out notation and conventions for various constructions associated to the set of embeddings of a totally real field F, which together with a prime p will be fixed throughout the article.

We assume that F has degree $d=[F:\mathbb {Q}]> 1$ , let ${\mathcal {O}}_F$ denote its ring of integers, ${\mathfrak {d}}$ its different and $\Sigma $ the set of embeddings $F \to {\overline {{\mathbb {Q}}}}$ , where ${\overline {{\mathbb {Q}}}}$ is the algebraic closure of ${\mathbb {Q}}$ in ${\mathbb {C}}$ .

We also fix an embedding ${\overline {{\mathbb {Q}}}} \to \overline {{\mathbb {Q}}}_p$ . We let $S_p$ denote the set of primes of ${\mathcal {O}}_F$ dividing p and identify $\Sigma $ with $\coprod _{{\mathfrak {p}} \in S_p} \Sigma _{{\mathfrak {p}}}$ under the natural bijection, where $\Sigma _{{\mathfrak {p}}}$ denotes the set of embeddings $F_{{\mathfrak {p}}} \to \overline {{\mathbb {Q}}}_p$ .

For each ${\mathfrak {p}} \in S_p$ , we let $F_{{\mathfrak {p}},0}$ denote the maximal unramified subextension of $F_{\mathfrak {p}}$ , which we identify with the field of fractions of $W({\mathcal {O}}_F/{\mathfrak {p}})$ . We also let $f_{\mathfrak {p}}$ denote the residue degree $[F_{{\mathfrak {p}},0}:{\mathbb {Q}}_p]$ , $e_{\mathfrak {p}}$ the ramification index $[F_{\mathfrak {p}}:F_{{\mathfrak {p}},0}]$ , and $\Sigma _{{\mathfrak {p}},0}$ the set of embeddings $F_{{\mathfrak {p}},0} \to \overline {{\mathbb {Q}}}_p$ , which we may identify with the set of embeddings ${\mathcal {O}}_F/{\mathfrak {p}} \to \overline {{\mathbb {F}}}_p$ or homomorphisms $W({\mathcal {O}}_F/{\mathfrak {p}}) \to W(\overline {{\mathbb {F}}}_p)$ . 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 $\overline {{\mathbb {F}}}_p$ (or $W(\overline {{\mathbb {F}}}_p)$ or its field of fractions), so that $\Sigma _{{\mathfrak {p}},0} = \{\tau _{{\mathfrak {p}},1},\tau _{{\mathfrak {p}},2},\ldots ,\tau _{{\mathfrak {p}},f_{{\mathfrak {p}}}}\}$ . We also let $\Sigma _0 = \coprod _{{\mathfrak {p}} \in S_p} \Sigma _{{\mathfrak {p}},0}$ . Letting ${\mathfrak {q}} = \prod _{{\mathfrak {p}} \in S_p}{\mathfrak {p}}$ denote the radical of p in ${\mathcal {O}}_F$ , note that $\Sigma _0$ may also be identified with the set of ring homomorphisms ${\mathcal {O}}_F/{\mathfrak {q}} \to \overline {{\mathbb {F}}}_p$ (or, indeed, ${\mathcal {O}}_F \to \overline {{\mathbb {F}}}_p$ ).

For each $\tau = \tau _{{\mathfrak {p}},i} \in \Sigma _0$ , we let $\Sigma _\tau \subset \Sigma _{\mathfrak {p}}$ denote the set of embeddings restricting to $\tau $ , for which we choose an ordering $\theta _{{\mathfrak {p}},i,1},\theta _{{\mathfrak {p}},i,2},\ldots ,\theta _{{\mathfrak {p}},i,e_{\mathfrak {p}}}$ , so that

$$ \begin{align*}\Sigma = \coprod_{\tau \in \Sigma_0} \Sigma_\tau = \{\,\theta_{{\mathfrak{p}},i,j}\,|\,{\mathfrak{p}} \in S_p, i \in \mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z}, 1 \le j \le e_{\mathfrak{p}}\,\}.\\[-18pt]\end{align*} $$

We also define a permutation $\sigma $ of $\Sigma $ whose restriction to each $\Sigma _{\mathfrak {p}}$ is the $e_{\mathfrak {p}} f _{\mathfrak {p}}$ -cycle corresponding to the right shift of indices with respect to the lexicographic ordering; that  is,

$$ \begin{align*}\begin{array}{ccccccccc} (1,1)& \mapsto &(1,2) &\mapsto &\cdots &\mapsto &(1,e_{\mathfrak{p}}) & \mapsto &\\ (2,1) & \mapsto & (2,2) & \mapsto & \cdots &\mapsto &(2,e_{\mathfrak{p}}) &\mapsto & \\ &&&&\vdots&&&&\\ (f_{\mathfrak{p}},1)& \mapsto&(f_{\mathfrak{p}},2)&\mapsto & \cdots &\mapsto &(f_{\mathfrak{p}},e_{\mathfrak{p}}) &\mapsto &(1,1) .\end{array}\end{align*} $$

Let $E \subset {\overline {{\mathbb {Q}}}}$ be a number field containing the image of $\theta $ for all $\theta \in \Sigma $ , let ${\mathcal {O}}$ be the completion of ${\mathcal {O}}_E$ at the prime determined by the choice of ${\overline {{\mathbb {Q}}}} \to \overline {{\mathbb {Q}}}_p$ and let $\mathbb {F}$ be its residue field. For any ${\mathcal {O}}_{F,p} = {\mathcal {O}}_F \otimes \mathbb {Z}_p$ -module M, we write $M = \bigoplus _{{\mathfrak {p}} \in S_p} M_{\mathfrak {p}}$ for the decomposition obtained from that of

$$ \begin{align*}{\mathcal{O}}_{F,p} \cong \prod_{{\mathfrak{p}} \in S_p} {\mathcal{O}}_{F,{\mathfrak{p}}}.\end{align*} $$

Similarly, for any $W({\mathcal {O}}_F/{\mathfrak {p}}) \otimes _{\mathbb {Z}_p} {\mathcal {O}}$ -module N, we have a decomposition $N = \bigoplus _{\tau \in \Sigma _{{\mathfrak {p}},0}} N_\tau $ obtained from

$$ \begin{align*}W({\mathcal{O}}_F/{\mathfrak{p}}) \otimes_{\mathbb{Z}_p} {\mathcal{O}} \cong \prod_{\tau \in \Sigma_{{\mathfrak{p}},0}} {\mathcal{O}}.\end{align*} $$

In particular, for any ${\mathcal {O}}_F\otimes {\mathcal {O}}$ -module M, we have the decomposition

$$ \begin{align*}M = \bigoplus_{{\mathfrak{p}} \in S_p} M_{\mathfrak{p}} = \bigoplus_{\tau \in \Sigma_0} M_\tau,\end{align*} $$

where we simply write $M_\tau $ for $M_{{\mathfrak {p}},\tau }$ . We also write $M_{{\mathfrak {p}},i}$ for $M_\tau $ if $\tau = \tau _{{\mathfrak {p}},i}$ ; thus, $M_{{\mathfrak {p}},i}$ is the summand of the ${\mathcal {O}}_{F,{\mathfrak {p}}}\otimes _{\mathbb {Z}_p} {\mathcal {O}}$ -module $M_{\mathfrak {p}}$ on which $W({\mathcal {O}}_F/{\mathfrak {p}})$ acts via $\tau _{{\mathfrak {p}},i}$ .

We also fix a choice of uniformiser $\varpi _{\mathfrak {p}}$ for each ${\mathfrak {p}} \in S_p$ . We let $f_{\mathfrak {p}}(u)$ denote the minimal polynomial of $\varpi _{\mathfrak {p}}$ over $W({\mathcal {O}}_F/{\mathfrak {p}})$ and let $f_\tau $ denote its image in ${\mathcal {O}}[u]$ for each $\tau \in \Sigma _{{\mathfrak {p}},0}$ ; thus, $u \mapsto \varpi _{\mathfrak {p}} \otimes 1$ induces an isomorphism

$$ \begin{align*}{\mathcal{O}}[u]/(f_\tau(u)) \stackrel{\sim}{\longrightarrow} {\mathcal{O}}_{F,{\mathfrak{p}}} \otimes_{W({\mathcal{O}}_F/{\mathfrak{p}}),\tau} {\mathcal{O}}.\end{align*} $$

Furthermore, we have $f_\tau (u) = \prod _{\theta \in \Sigma _\tau } (u - \theta (\varpi _{\mathfrak {p}}))$ , and we define elements

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

of ${\mathcal {O}}[u]/(f_\tau (u))$ for $j=0,\ldots ,e_{\mathfrak {p}}$ (with the obvious convention that $s_{\tau ,0} = t_{\tau ,e_{\mathfrak {p}}} = 1$ ). Note that each of the ideals $(s_{\tau ,j})$ and $(t_{\tau ,j})$ is the other’s annihilator; furthermore, the quotients of ${\mathcal {O}}[u]/(f_\tau (u))$ by these ideals are free over ${\mathcal {O}}$ , and the corresponding ideals in ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W({\mathcal {O}}_F/{\mathfrak {p}}),\tau } {\mathcal {O}}$ may be described as kernels of projection maps to products of copies of ${\mathcal {O}}$ , and hence depend only on j and the ordering of embeddings and not on the choice of uniformiser $\varpi _{\mathfrak {p}}$ .

For an invertible ${\mathcal {O}}_F$ -module L and an embedding $\theta = \theta _{{\mathfrak {p}},i,j} \in \Sigma _\tau $ , we define $L_\theta $ to be the free rank 1 ${\mathcal {O}}$ -module

(2) $$ \begin{align} L_\theta = t_{\tau,j}(L\otimes{\mathcal{O}})_\tau \otimes_{{\mathcal{O}}[u],\theta} {\mathcal{O}}. \end{align} $$

Note that $L_\theta $ is not to be identified with $L\otimes _{{\mathcal {O}}_F,\theta } {\mathcal {O}}$ ; rather, there is a canonical map $L_\theta \to L\otimes _{{\mathcal {O}}_F,\theta } {\mathcal {O}}$ which is an isomorphism if and only if $j = e_{\mathfrak {p}}$ . If L and $L^{\prime }$ are invertible ${\mathcal {O}}_F$ -modules, we will write $LL^{\prime }$ for $L\otimes _{{\mathcal {O}}_F}L^{\prime }$ and $L^{-1}$ for ${\operatorname {Hom}\,}_{{\mathcal {O}}_F}(L,{\mathcal {O}}_F)$ . Note that there are natural maps $L_\theta \otimes _{{\mathcal {O}}} L^{\prime }_\theta \to (LL^{\prime })_\theta $ and $(L^{-1})_\theta \to {\operatorname {Hom}\,}_{{\mathcal {O}}}(L_\theta ,{\mathcal {O}})$ but, again, these are isomorphisms if and only if $j = e_{\mathfrak {p}}$ .

2.2 Pappas–Rapoport models

In this section we recall the description of the Hilbert modular variety as a coarse moduli space for abelian varieties with additional structure, along with the construction by Pappas and Rapoport of a smooth integral model (see [Reference Pappas and Rapoport29] and [Reference Sasaki33]).

Let $G = {\operatorname {Res}}_{F/\mathbb {Q}} \operatorname {GL}_2$ and let U be an open compact subgroup of $\operatorname {GL}_2(\widehat {{\mathcal {O}}}_F) \subset \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}})= G({\mathbb {A}}_{\mathbf {f}})$ of the form $U_pU^p$ , where $U_p = \operatorname {GL}_2({\mathcal {O}}_{F,p})$ and $U^p \subset \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ is sufficiently small, in a sense to be specified below.

We 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 an embedding such that $(s_*\Omega ^1_{A/S})_{\mathfrak {p}}$ is, locally on S, free of rank $e_{\mathfrak {p}}$ over $W({\mathcal {O}}_F/{\mathfrak {p}})\otimes _{\mathbb {Z}_p} {\mathcal {O}}_S $ for each ${\mathfrak {p}} \in S_p$ ;

• • $\lambda $ is an ${\mathcal {O}}_F$ -linear quasi-polarisation 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, for a choice of geometric point $\overline {s}_i$ on each connected component $S_i$ of S, the data of a $\pi _1(S_i,\overline {s}_i)$ -invariant $U^p$ -orbit of $\widehat {{\mathcal {O}}}_F^{(p)} = {\mathcal {O}}_F\otimes \widehat {{\mathbb {Z}}}^{(p)}$ -linear isomorphismsFootnote ⁵

  $$ \begin{align*}\eta_i :(\widehat{{\mathcal{O}}}_F^{(p)})^2 \to {\mathfrak{d}} \otimes_{{\mathcal{O}}_F} T^{(p)}(A_{\overline{s}_i}),\end{align*} $$
  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 ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W({\mathcal {O}}_F/{\mathfrak {p}}),\tau } {\mathcal {O}}_S$ -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
  $$ \begin{align*}{{\mathcal{L}}}_{{\mathfrak{p}},i,j} := {\mathcal{F}}_\tau^{(j)}/{\mathcal{F}}_\tau^{(j-1)}\end{align*} $$
  is a line bundle on S on which ${\mathcal {O}}_F$ acts via $\theta _{{\mathfrak {p}},i,j}$ .

The proof of [Reference Diamond and Sasaki12, Lemma 2.4.1] does not assume p is unramified in F and shows that if $U^p$ is sufficiently small and $\alpha $ is an automorphism of a triple $(A,\iota ,\eta )$ over a connected scheme S, then $\alpha = \iota (\mu )$ for some $\mu \in U \cap {\mathcal {O}}_F^\times $ . If we assume further that $-1 \not \in U\cap {\mathcal {O}}_F^\times $ , then it follows from standard arguments that the functor above is representable by an infinite disjoint union of quasi-projective schemes over ${\mathcal {O}}$ , which we denote by $\widetilde {Y}_U$ , and the argument in the proof of [Reference Sasaki33, Prop. 6] shows that $\widetilde {Y}_U$ is smooth of relative dimension d over ${\mathcal {O}}$ . Furthermore, defining an action of ${\mathcal {O}}_{F,(p),+}^\times $ on $\widetilde {Y}_U$ by

$$ \begin{align*}\nu\cdot(A,\iota,\lambda,\eta,{\mathcal{F}}^\bullet) = (A,\iota,\nu\lambda,\eta,{\mathcal{F}}^\bullet)\end{align*} $$

(as in [Reference Diamond, Kassaei and Sasaki11, §2.1.3]), we see that the resulting action of ${\mathcal {O}}_{F,(p),+}^\times /(U\cap {\mathcal {O}}_F^\times )^2$ is free and the quotient is representable by a smooth quasi-projective scheme over ${\mathcal {O}}$ , which we denote by $Y_U$ .

We also have a natural right action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on the inverse system of schemes $Y_U$ induced by pre-composing the level structure $\eta $ with right multiplication by $g^{-1}$ . More precisely, suppose that $U_1$ and $U_2$ are as above (with $U_1^p$ and $U_2^p$ sufficiently small) and $g \in \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ is such that $g^{-1}U_1g \subset U_2$ . Letting $(A,\iota ,\lambda ,\eta ,{\mathcal {F}}^\bullet )$ denote the universal object over $\widetilde {Y}_{U_1}$ , there is a prime-to-p quasi-isogeny $A \to A^{\prime }$ of abelian varieties with ${\mathcal {O}}_F$ -action inducing isomorphisms ${\mathfrak {d}}\otimes _{{\mathcal {O}}_F} T^{(p)}(A^{\prime }_{\overline {s}_i}) \cong \eta _i((\widehat {{\mathcal {O}}}_{F}^{(p)})^2g^{-1})$ for each i, from which we obtain a level $U_2$ -structure $\eta ^{\prime } = \eta \circ r_{g^{-1}}$ on $A^{\prime }$ (where $r_{g^{-1}}$ denotes right multiplication by $g^{-1}$ ). Together with the other data inherited from A, we obtain an object $(A^{\prime },\iota ^{\prime },\lambda ^{\prime },\eta ^{\prime },{\mathcal {F}}^{\prime \bullet })$ corresponding to a morphism $\widetilde {\rho }_g: \widetilde {Y}_{U_1} \to \widetilde {Y}_{U_2}$ and descending to a morphism $Y_{U_1} \to Y_{U_2}$ which we denote $\rho _g$ . These morphisms satisfy the evident compatibility $\rho _{g_2}\circ \rho _{g_1} = \rho _{g_1g_2}$ whenever $g_1^{-1}U_1g_1 \subset U_2$ and $g_2^{-1}U_2g_2 \subset U_3$ .

Finally, we remark that the schemes $Y_U$ define smooth integral models over ${\mathcal {O}}$ for the Hilbert modular varieties associated to the group G (with the usual choice of Shimura datum), and their generic fibres and resulting $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ -action may be identified with those obtained from a system of canonical models. In particular, for any ${\mathcal {O}} \to \mathbb {C}$ , we have isomorphisms

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

compatible with the right action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on the inverse system and inducing a bijection between the set of geometric components of $Y_U$ and

$$ \begin{align*}{\mathbb{A}}_{F,\mathbf{f}}^\times/F_+^\times\det(U) \cong ({\mathbb{A}}_{F,\mathbf{f}}^{(p)})^\times/{\mathcal{O}}_{F,(p),+}^\times\det(U^p).\end{align*} $$

These isomorphisms arise in turn from ones of the form

$$ \begin{align*}\widetilde{Y}_U(\mathbb{C}) \cong \operatorname{SL}_2({\mathcal{O}}_{F,(p)})\backslash (\mathfrak{H}^\Sigma \times \operatorname{GL}_2({\mathbb{A}}^{(p)}_{F,\mathbf{f}})/U^p),\end{align*} $$

under which the set of geometric components of $\widetilde {Y}_U$ is described by $({\mathbb {A}}_{F,\mathbf {f}}^{(p)})^\times /\det (U^p)$ .

3.1 Pairings and duality

Before introducing the line bundles whose sections define the automorphic forms of interest in the article, we present a plethora of perfect pairings provided by Poincaré duality.

We fix a sufficiently small U as in Subsection 2.2 and consider the de Rham cohomology sheaves ${\mathcal {H}}^1_{\operatorname {dR}}(A/S) = \mathbb {R}^1s_*\Omega ^\bullet _{A/S}$ on the universal abelian scheme A over $S = \widetilde {Y}_U$ . Recall that these sheaves are locally free of rank 2 over ${\mathcal {O}}_F\otimes {\mathcal {O}}_S$ . Furthermore, Poincaré duality and the polarisation $\lambda $ induce an ${\mathcal {O}}_F\otimes {\mathcal {O}}_S$ -linear isomorphism

$$ \begin{align*}\begin{array}{ccccc} {\mathcal{H}}^1_{\operatorname{dR}}(A/S)& \stackrel{\sim}{\longrightarrow} & {\mathcal{H}om}_{{\mathcal{O}}_S}({\mathcal{H}}^1_{\operatorname{dR}}(A^\vee/S),{\mathcal{O}}_S)& \stackrel{\sim}{\longleftarrow}& {\mathcal{H}om}_{{\mathcal{O}}_S}({\mathcal{H}}^1_{\operatorname{dR}}({\mathfrak{c}}{\mathfrak{d}} \otimes_{{\mathcal{O}}_F} A)/S),{\mathcal{O}}_S)\\ &&&& \wr\parallel\\ &&&& {\mathcal{H}om}_{{\mathcal{O}}_S}({\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F}{\mathcal{H}}^1_{\operatorname{dR}}(A/S),{\mathcal{O}}_S)\end{array}\end{align*} $$

(where ${\mathfrak {c}}$ depends on the connected component of S and disappears from the last expression since it is prime to p). This in turn induces ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W({\mathcal {O}}_F/{\mathfrak {p}}),\tau } {\mathcal {O}}_S$ -linear isomorphisms

(3) $$ \begin{align} {\mathcal{H}}^1_{\operatorname{dR}}(A/S)_\tau \cong {\mathcal{H}om}_{{\mathcal{O}}_S}({\mathfrak{d}}_{\mathfrak{p}}^{-1} \otimes_{{\mathcal{O}}_{F,{\mathfrak{p}}}} {\mathcal{H}}^1_{\operatorname{dR}}(A/S)_\tau,{\mathcal{O}}_S) \end{align} $$

which we view as defining a perfect ${\mathcal {O}}_S$ -bilinear pairing $\langle \cdot ,\cdot \rangle _\tau ^0$ between ${\mathcal {H}}_\tau := {\mathcal {H}}^1_{\operatorname {dR}}(A/S)_\tau $ and ${\mathfrak {d}}_{{\mathfrak {p}}}^{-1}\otimes _{{\mathcal {O}}_{F,{\mathfrak {p}}}} {\mathcal {H}}_\tau $ for $\tau \in \Sigma _{{\mathfrak {p}},0}$ . Furthermore, the pairing is alternating in the sense that $\langle x, c \otimes y \rangle _\tau ^0 = - \langle y, c \otimes x\rangle _\tau ^0$ on sections. Alternatively, we may apply the canonical ${\mathcal {O}}_F\otimes R$ -linear isomorphism

$$ \begin{align*}{\operatorname{Hom}\,}_{{\mathcal{O}}_F\otimes R} (M, {\mathcal{O}}_F \otimes R) \stackrel{\sim}{\longrightarrow} {\operatorname{Hom}\,}_R({\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} M,R)\end{align*} $$

induced by the trace for any ${\mathcal {O}}_F \otimes R$ -module M to obtain an ${\mathcal {O}}_F \otimes {\mathcal {O}}_S$ -linear isomorphism

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

and hence a perfect alternating ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W({\mathcal {O}}_F/{\mathfrak {p}}),\tau } {\mathcal {O}}_S$ -bilinear pairing $\langle \cdot ,\cdot \rangle _\tau $ on ${\mathcal {H}}_\tau $ .

Note that ${\mathcal {H}}_\tau $ is locally free of rank 2 over ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W({\mathcal {O}}_F/{\mathfrak {p}}),\tau } {\mathcal {O}}_S$ and hence a vector bundle of rank $2e_{\mathfrak {p}}$ over ${\mathcal {O}}_S$ . Furthermore, $(s_*\Omega ^1_{A/S})_\tau $ is a subbundle of ${\mathcal {H}}_\tau $ of rank $e_{\mathfrak {p}}$ but is not locally free over ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W({\mathcal {O}}_F/{\mathfrak {p}}),\tau } {\mathcal {O}}_S$ if $e_{\mathfrak {p}}> 1$ (in which case the failure is on a closed subscheme of codimension 1) and, more generally, ${\mathcal {F}}_\tau ^{(j)}$ is a subbundle of rank j for $j=0,1,\ldots ,e_{\mathfrak {p}}$ .

Recall that for $j=0,1,\ldots ,e_{\mathfrak {p}}$ , we defined (see (1)) elements $s_{\tau ,j}$ and $t_{\tau ,j}$ of the ring ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _{W({\mathcal {O}}_F/{\mathfrak {p}}),\tau } {\mathcal {O}} \cong {\mathcal {O}}[u]/(f_\tau (u))$ , where $f_\tau $ is the image under $\tau $ of the Eisenstein polynomial associated to our choice of uniformiser $\varpi _{\mathfrak {p}}$ . In the following, we shall fix $\tau $ and omit the subscripts $\tau $ , ${\mathfrak {p}}$ and i to disencumber the notation; we also write simply W for $W({\mathcal {O}}_F/{\mathfrak {p}})$ . Note that since ${\mathcal {H}}$ is locally free over ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _W {\mathcal {O}}_S$ and ${\mathcal {F}}^{(j)}$ is annihilated by $s_j$ , we have that ${\mathcal {F}}^{(j)}$ is in fact a subbundle of $t_j{\mathcal {H}}$ .

For a subsheaf ${\mathcal {E}} \subset {\mathcal {H}}$ of ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _W {\mathcal {O}}_S$ -submodules, we define ${\mathcal {E}}^\perp $ to be its orthogonal complement under the pairing $\langle \cdot ,\cdot \rangle $ ; that is, the kernel of the morphism

$$ \begin{align*}{\mathcal{H}} \stackrel{\sim}{\longrightarrow} {\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S}({\mathcal{H}},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S) \longrightarrow {\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S}({\mathcal{E}},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S)\end{align*} $$

or, equivalently, the orthogonal complement of ${\mathfrak {d}}_{\mathfrak {p}}^{-1} \otimes _{{\mathcal {O}}_{F,{\mathfrak {p}}}} {\mathcal {E}}$ under the pairing $\langle \cdot ,\cdot \rangle ^0$ . Note from the latter description that if ${\mathcal {E}}$ is an ${\mathcal {O}}_S$ -subbundle of ${\mathcal {H}}$ , then so is ${\mathcal {E}}^\perp $ .

Proof. We prove the lemma by induction on j, the case of $j=0$ being obvious.

Suppose then that $1 \le j \le e$ and that $({\mathcal {F}}^{(j-1)})^\perp = t_{j-1}^{-1} {\mathcal {F}}^{(j-1)}$ . Note that $({\mathcal {F}}^{(j)})^\perp $ and $t_j^{-1} {\mathcal {F}}^{(j)}$ are both kernels of surjective morphisms from ${\mathcal {H}}$ to vector bundles of rank j on S, so each is a subbundle of rank $2e-j$ , and hence it suffices to prove the inclusion $t_j^{-1}{\mathcal {F}}^{(j)} \subset ({\mathcal {F}}^{(j)})^\perp $ . To do so, we may work locally on S and assume that ${\mathcal {F}}^{(j)}(V) = {\mathcal {F}}^{(j-1)}(V) \oplus R x_j$ where $V = {\operatorname {Spec}\,} R$ is a Noetherian open subscheme of $\widetilde {Y}_U$ such that ${\mathcal {H}}|_V$ is free over ${\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _W {\mathcal {O}}_V$ and $x_j \in {\mathcal {H}}(V)$ satisfies $(u-\theta _j(\varpi ))x_j \in {\mathcal {F}}^{(j-1)}(V)$ . In particular, $x_j = t_jy_j$ for some $y_j \in t_{j-1}^{-1}{\mathcal {F}}^{(j-1)}(V)$ , so that

$$ \begin{align*}t_j^{-1}{\mathcal{F}}^{(j)}(V) = t_j^{-1}{\mathcal{F}}^{(j-1)}(V) \oplus Ry_j\quad\text{and}\quad ({\mathcal{F}}^{(j)}(V))^\perp = ({\mathcal{F}}^{(j-1)}(V))^\perp \cap (Rx_j)^\perp.\end{align*} $$

Note that if $w \in t_j^{-1}{\mathcal {F}}^{(j)}(V)$ , then

$$ \begin{align*}t_{j-1}w = (u-\theta_j(\varpi)) t_j w \in (u-\theta_j(\varpi)){\mathcal{F}}^{(j)}(V) \subset {\mathcal{F}}^{(j-1)}(V),\end{align*} $$

so that $w \in t_{j-1}^{-1} {\mathcal {F}}^{(j-1)}(V) = ({\mathcal {F}}^{(j-1)})(V))^\perp $ . Furthermore, if $w \in t_j^{-1}{\mathcal {F}}^{(j-1)}(V)$ , then

$$ \begin{align*}\langle w,x_j \rangle = \langle w, t_jy_j \rangle = \langle t_jw, y_j \rangle\end{align*} $$

since $t_jw \in {\mathcal {F}}^{(j-1)}$ and $y_j \in t_{j-1}^{-1}{\mathcal {F}}^{(j-1)}(V) = ({\mathcal {F}}^{(j-1)}(V))^\perp $ . Finally, since the pairing is alternating, we have

$$ \begin{align*}\langle y_j , x_j \rangle = \langle y_j,t_jy_j \rangle = \langle t_jy_j,y_j \rangle = \langle x_j,y_j \rangle = - \langle y_j,x_j \rangle,\end{align*} $$

which implies that $\langle y_j,x_j \rangle = 0$ (since $\widetilde {Y}_U$ is flat over $\mathbb {Z}_2$ if $p=2$ ).

We have now shown that $t_j^{-1}{\mathcal {F}}^{(j)}(V) \subset ({\mathcal {F}}^{(j-1)}(V))^\perp \cap (Rx_j)^\perp = ({\mathcal {F}}^{(j)}(V))^\perp $ , as required.

We now define ${\mathcal {G}}^{(j)} = (u-\theta _j(\varpi ))^{-1}{\mathcal {F}}^{(j-1)}$ for $j=1,\ldots ,e$ . Thus, ${\mathcal {G}}^{(j)}$ is a rank $j+1$ subbundle of ${\mathcal {H}}$ , and we have inclusions of subbundles ${\mathcal {F}}^{(j-1)} \subset {\mathcal {F}}^{(j)} \subset {\mathcal {G}}^{(j)}$ , so that ${{\mathcal {L}}}_j := {\mathcal {F}}^{(j)}/{\mathcal {F}}^{(j-1)}$ is a rank 1 subbundle of the rank 2 vector bundle ${{\mathcal {P}}}_j := {\mathcal {G}}^{(j)}/{\mathcal {F}}^{(j-1)}$ . Furthermore, all of the inclusions are morphisms of ${\mathcal {O}}_{F,{\mathfrak {p}}}\otimes _W {\mathcal {O}}_S$ -modules, and ${\mathcal {O}}_F$ acts on ${{\mathcal {P}}}_j$ via $\theta _j$ .

Note that ${\mathcal {G}}^{(j)}$ is annihilated by $s_j$ , so that ${\mathcal {G}}^{(j)} \subset t_j{\mathcal {H}}$ , and we have $t_j^{-1}{\mathcal {G}}^{(j)} = t_{j-1}^{-1}{\mathcal {F}}^{(j-1)} = ({\mathcal {F}}^{j-1})^\perp $ by Lemma 3.1.1, from which it follows also that

$$ \begin{align*}t_j^{-1}{\mathcal{F}}^{(j-1)} = t_j^{-1}(t_j^{-1}{\mathcal{G}}^{(j)})^\perp = ({\mathcal{G}}^{(j)})^\perp.\end{align*} $$

(The last equality can be seen by arguing locally on sections or by noting that the diagram

commutes, where the left horizontal morphisms are defined by the pairing, the right by restriction and all of the vertical morphisms by $t_j$ . The kernel of the composite along the top is $({\mathcal {G}}^{(j)})^\perp $ , whereas $t_j^{-1}(t_j^{-1}{\mathcal {G}}^{(j)})^\perp $ is the kernel of the composite along the left and bottom. Since $t_j:t_j^{-1}{\mathcal {G}}^{(j)} \to {\mathcal {G}}^{(j)}$ is a surjective morphism of vector bundles, the leftmost vertical arrow is injective, so these kernels coincide.) Therefore, multiplication by $t_j$ defines an isomorphism $({\mathcal {F}}^{(j-1)})^\perp /({\mathcal {G}}^{(j)})^\perp \stackrel {\sim }{\longrightarrow } {\mathcal {G}}^{(j)}/{\mathcal {F}}^{(j-1)}$ , and composing its inverse with the isomorphism

$$ \begin{align*}({\mathcal{F}}^{(j-1)})^\perp/({\mathcal{G}}^{(j)})^\perp \stackrel{\sim}{\longrightarrow} {\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S}({\mathcal{G}}^{(j)}/{\mathcal{F}}^{(j-1)},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S)\end{align*} $$

induced by the pairing on ${\mathcal {H}}$ , we obtain an alternating $({\mathcal {O}}_{F,{\mathfrak {p}}} \otimes _W {\mathcal {O}}_S)$ -valued pairing $\langle \cdot , \cdot \rangle _j$ on ${{\mathcal {P}}}_j := {\mathcal {G}}^{(j)}/{\mathcal {F}}^{(j-1)}$ , whose description on sections is given in terms of (4) by

$$ \begin{align*}\langle t_j x, t_j y \rangle_j = \langle x, t_j y \rangle = \langle t_j x , y \rangle.\end{align*} $$

Note that since ${\mathcal {O}}_{F,{\mathfrak {p}}}$ acts via $\tau _j$ on ${{\mathcal {P}}}_j$ , we in fact have the identification

$$ \begin{align*}{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S}({{\mathcal{P}}}_j,{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_W{\mathcal{O}}_S) = {\mathcal{H}om}_{{\mathcal{O}}_S}({{\mathcal{P}}}_j,{\mathcal{I}}_j)\end{align*} $$

where ${\mathcal {I}}_j$ is the sheaf of ideals, and trivial rank 1 ${\mathcal {O}}_S$ -subbundle, of ${\mathcal {O}}_{F,{\mathfrak {p}}}\otimes _W{\mathcal {O}}_S$ generated by the global section $\prod _{j^{\prime }\neq j}(u-\tau _{j^{\prime }}(\varpi _{\mathfrak {p}}))$ . We thus obtain a trivialisation of $\wedge ^2_{{\mathcal {O}}_S} {{\mathcal {P}}}_j$ corresponding to a perfect ${\mathcal {O}}_S$ -valued pairing $\langle \cdot , \cdot \rangle _j^0$ , which an unravelling of definitions shows is given in terms of the original pairing $\langle \cdot , \cdot \rangle ^0$ of (3) by the formula

$$ \begin{align*}\langle t_j x , y \rangle_j^0 = \langle x, f^{\prime}(\varpi_{\pi_{\mathfrak{p}}})^{-1} \otimes y \rangle^0\end{align*} $$

on sections (where $f^{\prime }$ is the derivative of the Eisenstein polynomial f).

3.2 Automorphic line bundles

Recall that the Pappas–Rapoport model $\widetilde {Y}_U$ is equipped with line bundlesFootnote ⁶ ${\mathcal {L}}_j$ , which we described in Subsection 3.1 as subbundles of the rank 2 vector bundles ${\mathcal {P}}_j$ . It is natural and convenient to consider also the twists of ${\mathcal {L}}_j$ by powers of the determinant bundle of  ${\mathcal {P}}_j$ :

$$ \begin{align*}{{\mathcal{N}}}_j = \wedge^2_{{\mathcal{O}}_S}{{\mathcal{P}}}_j \cong {{\mathcal{L}}}_j \otimes_{{\mathcal{O}}_S} {{\mathcal{M}}}_j,\end{align*} $$

where ${{\mathcal {M}}}_j$ is the line bundle ${{\mathcal {P}}}_j/{{\mathcal {L}}}_j$ . Note that the pairing $\langle \cdot , \cdot \rangle ^0_j$ defines an isomorphism ${{\mathcal {M}}}_j\stackrel {\sim }{\longrightarrow } {{\mathcal {L}}}_j^{-1}$ and a trivialisation ${\mathcal {O}}_S \stackrel {\sim }{\longrightarrow } {{\mathcal {N}}}_j$ (which depends on the choice of $\varpi $ ).

As we will now consider these bundles for varying $\tau $ , we resume writing the indicative subscripts; thus, for $\tau = \tau _{{\mathfrak {p}},i}$ , we will denote ${\mathcal {G}}^{(j)}$ by ${\mathcal {G}}_\tau ^{(j)}$ , ${{\mathcal {P}}}_j$ by ${{\mathcal {P}}}_{{\mathfrak {p}},i,j}$ and similarly for ${{\mathcal {M}}}_j$ and ${{\mathcal {N}}}_j$ . We also freely replace the subscript ‘ ${\mathfrak {p}},i,j$ ’ by $\theta $ , where $\theta = \theta _{{\mathfrak {p}},i,j}$ , so that for each $\theta \in \Sigma $ , we have now defined a rank 2 vector bundle ${{\mathcal {P}}}_\theta $ and line bundles ${{\mathcal {L}}}_\theta $ , ${{\mathcal {M}}}_\theta $ , ${{\mathcal {N}}}_\theta $ on $S = \widetilde {Y}_U$ , along with exact sequences

(5) $$ \begin{align} 0 \longrightarrow {{\mathcal{L}}}_\theta \longrightarrow {{\mathcal{P}}}_\theta \longrightarrow {{\mathcal{M}}}_\theta \longrightarrow 0 \end{align} $$

and a trivialisation of ${{\mathcal {N}}}_\theta = \wedge ^2_{{\mathcal {O}}_S} {{\mathcal {P}}}_\theta \cong {\mathcal {L}}_\theta \otimes _{{\mathcal {O}}_S} {{\mathcal {M}}}_\theta $ . Furthermore, the bundles ${{\mathcal {P}}}_\theta $ , ${{\mathcal {L}}}_\theta $ and ${{\mathcal {M}}}_\theta $ are ${\mathcal {O}}_F\otimes {\mathcal {O}}_S$ -subquotients of ${\mathcal {H}}^1_{\operatorname {dR}}(A/S)$ on which ${\mathcal {O}}_F$ acts via $\theta $ .

Recall that we have an action of ${\mathcal {O}}_{F,(p),+}^\times $ on $\widetilde {Y}_U$ defined by multiplication on the quasi-polarisation. In particular, if $\nu \in {\mathcal {O}}_{F,(p),+}^\times $ , then the identification of $\nu ^*A$ with A induces an ${\mathcal {O}}_F\otimes {\mathcal {O}}_S$ -linear isomorphism $\nu ^*{\mathcal {H}}^1_{\operatorname {dR}}(A/S) \cong {\mathcal {H}}^1_{\operatorname {dR}}(A/S)$ under which $\nu ^*{\mathcal {F}}^\bullet $ corresponds to ${\mathcal {F}}^\bullet $ , and we thus obtain isomorphisms $\alpha _\nu : \nu ^*{{\mathcal {P}}}_\theta \stackrel {\sim }{\longrightarrow } {{\mathcal {P}}}_\theta $ compatible with (5) and satisfying $\alpha _{\nu ^{\prime }\nu } = \alpha _\nu \circ \nu ^*(\alpha _{\nu ^{\prime }})$ (for $\nu ,\nu ^{\prime } \in {\mathcal {O}}_{F,(p),+}^\times $ ). Recall also that the action of ${\mathcal {O}}_{F,(p),+}^\times $ on $\widetilde {Y}_U$ factors through ${\mathcal {O}}_{F,(p),+}^\times /(U\cap {\mathcal {O}}_F^\times )^2$ , the isomorphism ${\underline {A}} \stackrel {\sim }{\longrightarrow } \nu ^*{\underline {A}}$ being defined by $\iota (\mu ^{-1})$ if $\nu = \mu ^2$ for $\mu \in U\cap {\mathcal {O}}_F^\times $ , and one finds that the automorphism of ${{\mathcal {P}}}_\theta $ obtained from $\alpha _\nu $ is multiplication by $\theta (\mu )$ , so the natural action of ${\mathcal {O}}_{F,(p),+}^\times $ on the bundles fails to define descent data with respect to the cover $\widetilde {Y}_U \to Y_U$ . We do, however, obtain descent data after taking suitable tensor products or base changes of these bundles, which we now consider.

For any ${\mathcal {O}}$ -algebra R, we will use $\cdot _R$ to denote the base change to R of an ${\mathcal {O}}$ -scheme X, as well as the pullback to $X_R$ of a quasi-coherent sheaf on X. Let $\{\,{\mathbf {e}}_\theta \,|\,\theta \in \Sigma \,\}$ denote the standard basis of $\mathbb {Z}^\Sigma $ . For $\mathbf {k} = \sum k_\theta {\mathbf {e}}_{\theta }$ and $\mathbf {l} = \sum l_\theta {\mathbf {e}}_\theta \in \mathbb {Z}^\Sigma $ , we define the line bundle

$$ \begin{align*}\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l}} = \bigotimes_{\theta\in \Sigma} \left( {{\mathcal{L}}}_\theta^{\otimes k_\theta} \otimes {{\mathcal{N}}}_\theta^{\otimes l_\theta} \right) \cong \bigotimes_{\theta\in \Sigma} \left( {{\mathcal{L}}}_\theta^{\otimes k_\theta + l_\theta} \otimes {{\mathcal{M}}}_\theta^{\otimes l_\theta} \right)\end{align*} $$

on $S = \widetilde {Y}_U$ , where all tensor products are over ${\mathcal {O}}_S$ . For $\mathbf {n} = \sum n_\theta {\mathbf {e}}_\theta \in \mathbb {Z}^\Sigma $ , we let $\chi _{\mathbf {n}}:{\mathcal {O}}_F^\times \to {\mathcal {O}}^\times $ denote the character defined by $\chi _{\mathbf {n}}(\mu ) = \prod _{\theta } \theta (\mu )^{n_\theta }$ , and we let $\chi _{\mathbf {n},R}$ denote the associated $R^\times $ -valued character. If $\mathbf {k}$ , $\mathbf {l}$ , R and U are such that $\chi _{\mathbf {k}+2\mathbf {l},R}$ is trivial on ${\mathcal {O}}_F^\times \cap U$ , then the action of ${\mathcal {O}}_{F,(p),+}^\times $ on $\widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {l},R}$ (over its action on $\widetilde {Y}_{U,R}$ ) factors through ${\mathcal {O}}_{F,(p),+}^\times /(U\cap {\mathcal {O}}_F^\times )^2$ and hence defines descent data, in which case we denote the resulting line bundle on $Y_{U,R}$ by ${\mathcal {A}}_{\mathbf {k},\mathbf {l},R}$ .

Definition 3.2.1. For $\mathbf {k}$ , $\mathbf {l}$ , U and R as above, we call ${\mathcal {A}}_{\mathbf {k},\mathbf {l},R}$ the automorphic line bundle of weight $(\mathbf {k},\mathbf {l})$ on $Y_{U,R}$ , and we define the space of Hilbert modular forms of weight $(\mathbf {k},\mathbf {l})$ and level U with coefficients in R to be

$$ \begin{align*}M_{\mathbf{k},\mathbf{l}}(U;R) : = H^0(Y_{U,R} ,{\mathcal{A}}_{\mathbf{k},\mathbf{l},R}).\end{align*} $$

We note some general situations in which this space is defined:

• • The paritious setting: if $w=k_\theta + 2l_\theta $ is independent of $\theta $ , then $\chi _{\mathbf {k}+2\mathbf {l}}(\mu ) = \mathrm {Nm}_{F/\mathbb {Q}}(\mu )^w = 1$ for all $\mu \in U \cap {\mathcal {O}}_F^\times $ (assuming only U is small enough that $U\cap {\mathcal {O}}_F^\times $ has no elements of norm $-1$ if w is odd).

• • The mod p setting: if R is any $\mathbb {F} = {\mathcal {O}}/{\mathfrak {m}}_{\mathcal {O}}$ -algebra and U is sufficiently small that $\mu \equiv 1 \bmod {\mathfrak {p}}$ for all $\mu \in U \cap {\mathcal {O}}_F^\times $ and ${\mathfrak {p}} \in S_p$ , then $\theta (\mu ) \equiv 1 \bmod {\mathfrak {m}}_{\mathcal {O}}$ for all $\theta \in \Sigma $ , so $\chi _{\mathbf {k}+2\mathbf {l},\mathbb {F}}$ is trivial on $U \cap {\mathcal {O}}_F^\times $ , and hence so is $\chi _{\mathbf {k}+2\mathbf {l},R}$ .

• • The torsion setting: if R is an ${\mathcal {O}}/p^N{\mathcal {O}}$ -algebra and U is sufficiently small that $\mu \equiv 1 \bmod p^N{\mathcal {O}}_F$ for all $\mu \in U \cap {\mathcal {O}}_F^\times $ , then $\chi _{\mathbf {k}+2\mathbf {l},{\mathcal {O}}/p^N{\mathcal {O}}}$ is trivial on $U \cap {\mathcal {O}}_F^\times $ , and hence so is $\chi _{\mathbf {k}+2\mathbf {l},R}$ .

We also have a natural left action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on the direct limit over U of the spaces $M_{\mathbf {k},\mathbf {l}}(U;R)$ . More precisely, suppose that $U_1$ and $U_2$ are as above and $g \in \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ is such that $g^{-1}U_1g \subset U_2$ , in which case recall that in Subsection 2.2 we defined a morphism $\widetilde {\rho }_g: \widetilde {Y}_{U_1} \to \widetilde {Y}_{U_2}$ descending to a morphism $\rho _g:Y_{U_1} \to Y_{U_2}$ . Furthermore, the morphism $\widetilde {\rho }_g$ is obtained from a prime-to-p quasi-isogeny $A \to A^{\prime }$ where A is the universal abelian scheme over $\widetilde {Y}_{U_1}$ and $A^{\prime }$ is the pullback of the universal abelian scheme. We thus obtain isomorphisms $\widetilde {\rho }_g^*{{\mathcal {P}}}_{\theta ,2} \to {{\mathcal {P}}}_{\theta ,1}$ compatible with (5) and the action of ${\mathcal {O}}_{F,(p),+}^\times $ (augmenting the notation for the automorphic bundles on $\widetilde {Y}_{U_i}$ and $Y_{U_i,R}$ with the subscript i). Note that $U_1 \cap {\mathcal {O}}_F^\times \subset U_2 \cap {\mathcal {O}}_F^\times $ , so if $\widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {l},R,2}$ descends to $Y_{U_2,R}$ , then $\widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {l},R,1}$ descends to $Y_{U_1,R}$ , and we obtain an isomorphism $\rho _g^*{\mathcal {A}}_{\mathbf {k},\mathbf {l},R,2} \cong {\mathcal {A}}_{\mathbf {k},\mathbf {l},R,1}$ . We then define $[g] = [g]_{U_1,U_2}: M_{\mathbf {k},\mathbf {l}}(U_2;R) \to M_{\mathbf {k},\mathbf {l}}(U_1;R)$ as the composite

$$ \begin{align*}H^0(Y_{U_2}, {\mathcal{A}}_{\mathbf{k},\mathbf{l},R,2}) \stackrel{\rho_g^*}{\longrightarrow} H^0(Y_{U_1}, \rho_g^*{\mathcal{A}}_{\mathbf{k},\mathbf{l},R,2}) \stackrel{\sim}{\longrightarrow} H^0(Y_{U_1}, {\mathcal{A}}_{\mathbf{k},\mathbf{l},R,1}).\end{align*} $$

These maps satisfy the obvious compatibility, namely, that

$$ \begin{align*}[g_1]_{U_1,U_2} \circ [g_2]_{U_2,U_3} = [g_1g_2]_{U_1,U_3}\end{align*} $$

whenever $g_1^{-1}U_1g_1 \subset U_2$ and $g_2^{-1}U_2g_2 \subset U_3$ , and hence define an action of $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ on

(6) $$ \begin{align} M_{\mathbf{k},\mathbf{l}}(R) := \varinjlim_U M_{\mathbf{k},\mathbf{l}}(U;R)\end{align} $$

(where the limit is over sufficiently small open compact U with respect to the maps $[1]_{U_1,U_2}$ ). For paritious $\mathbf {k}$ , $\mathbf {l}$ and any choice of ${\mathcal {O}} \to \mathbb {C}$ , we may identify the spaces $M_{\mathbf {k},\mathbf {l}}(U;\mathbb {C})$ with those of holomorphic Hilbert modular forms, compatibly with the usual action (up to normalisation by a factor of $||\det g||$ depending on conventions).

Finally, we remark that the action of $\nu \in {\mathcal {O}}_{F,(p),+}^\times $ on the trivialisation of ${{\mathcal {N}}}_\theta $ is given by multiplication by $\theta (\nu )$ , so their products do not descend to trivialisations of line bundles on $Y_{U,R}$ . However, since the stabiliser of each geometric connected component of $\widetilde {Y}_U$ is ${\mathcal {O}}_{F,+}^\times \cap \det (U)$ , we can obtain a (noncanonical) trivialisation of ${\mathcal {A}}_{{\mathbf {0}},\mathbf {l},R}$ as in [Reference Diamond and Sasaki12, Prop. 3.6.1], provided $\chi _{\mathbf {l},R}$ is trivial on ${\mathcal {O}}_{F,+}^\times \cap \det (U)$ and the geometric connected components of $Y_U$ are defined over R. Furthermore, the same argument as in the proof of [Reference Diamond and Sasaki12, Lemma 4.5.1] shows the following.

3.3 The Kodaira–Spencer filtration

In this section we define a filtration on $\Omega ^1_{Y_U/{\mathcal {O}}}$ whose pieces are described by automorphic bundles with weight components $k_\theta = 2$ , $l_\theta = -1$ . The construction of the filtration is due to Reduzzi and Xiao (see [Reference Reduzzi and Xiao32, §2.8]), but their presentation is complicated by the fact that they wish to prove smoothness simultaneously, and it obscures the fact that the bundles we denoted ${\mathcal {G}}^{(j)}$ automatically satisfy the orthogonality condition appearing in the definition of their counterparts in [Reference Reduzzi and Xiao32]. We will show below that, with smoothness already established, one can give a more direct conceptual description of the filtration and its properties.Footnote ⁷ Furthermore, in the case $p=2$ , the argument in [Reference Reduzzi and Xiao32] appeals to a very general flatness assertion for divided power envelopes for which we could not find a proof or reference, so it is not used here.

Theorem 3.3.1 Reduzzi–Xiao

There exists a decomposition

$$ \begin{align*}\Omega^1_{Y_U/{\mathcal{O}}} = \bigoplus_{{\mathfrak{p}} \in S_p} \bigoplus_{i = 1}^{f_{\mathfrak{p}}} \Omega^1_{Y_U/{\mathcal{O}},{\mathfrak{p}},i},\end{align*} $$

together with an increasing filtration

$$ \begin{align*}\begin{array}{rcccl} 0 = {\operatorname{Fil}\,}^0 (\Omega^1_{Y_U/{\mathcal{O}},{\mathfrak{p}},i} ) &\subset& {\operatorname{Fil}\,}^1 (\Omega^1_{Y_U/{\mathcal{O}},{\mathfrak{p}},i}) &\subset& \cdots\\ &\subset & {\operatorname{Fil}\,}^{e_{\mathfrak{p}} - 1}( \Omega^1_{Y_U/{\mathcal{O}},{\mathfrak{p}},i} )& \subset & {\operatorname{Fil}\,}^{e_{\mathfrak{p}}}( \Omega^1_{Y_U/{\mathcal{O}},{\mathfrak{p}},i} ) = \Omega^1_{Y_U/{\mathcal{O}},{\mathfrak{p}},i}\end{array}\end{align*} $$

for each ${\mathfrak {p}} \in S_p$ and $i=1,\ldots ,f_{\mathfrak {p}}$ , such that for each $j=1,\ldots ,e_{\mathfrak {p}}$ , ${\operatorname {gr}\,}^j (\Omega ^1_{Y_U/{\mathcal {O}},{\mathfrak {p}},i} )$ is isomorphic to the automorphic bundle ${\mathcal {A}}_{2{\mathbf {e}}_\theta ,-{\mathbf {e}}_\theta ,{\mathcal {O}}}$ , where $\theta = \theta _{{\mathfrak {p}},i,j}$ .

Proof. As usual, we first prove the analogous result for $S: = \widetilde {Y}_U$ and then descend to $Y_U$ .

We let $\delta _0: S \stackrel {\sim }{\longrightarrow } \Delta \hookrightarrow Z_0$ denote the first infinitesimal thickening of the diagonal embedding, and we view $\Omega ^1_{S/{\mathcal {O}}}$ as $\delta _0^*{\mathcal {I}}$ , where ${\mathcal {I}}$ denotes the sheaf of ideals defining $\Delta $ in $Z_0$ . Letting $s:A \to S$ denote the universal abelian scheme, the transition maps for the crystal $R^1s_{{\operatorname {cris}},*}{\mathcal {O}}_{A/{\mathbb {Z}}_p}$ and canonical isomorphisms with de Rham cohomology yield an ${\mathcal {O}}_F \otimes {\mathcal {O}}_{Z_0}$ -linear isomorphism

$$ \begin{align*}\alpha: p_0^*{\mathcal{H}}^1_{\operatorname{dR}}(A/S) \stackrel{\sim}{\longrightarrow} q_0^*{\mathcal{H}}^1_{\operatorname{dR}}(A/S)\end{align*} $$

extending the identity on $S \cong \Delta $ , where $p_0,q_0: Z_0 \to S$ are the two projection maps $Z_0 \to S$ . Since $\alpha $ is ${\mathcal {O}}_F$ -linear, it follows from the definition of ${\mathcal {P}}_{\tau ,1} = {\mathcal {G}}_\tau ^{(1)}$ that $\alpha $ restricts to an isomorphism

$$ \begin{align*}\alpha_{\tau,1}: p_0^*{\mathcal{P}}_{\tau,1} \stackrel{\sim}{\longrightarrow} q_0^*{\mathcal{P}}_{\tau,1}\end{align*} $$

for each ${\mathfrak {p}} \in S_p$ and $\tau \in \Sigma _{{\mathfrak {p}},0}$ . Furthermore, since the composite

$$ \begin{align*}p_0^*{\mathcal{L}}_{\tau,1} \hookrightarrow p_0^*{\mathcal{P}}_{\tau,1} \stackrel{\sim}{\longrightarrow} q_0^*{\mathcal{P}}_{\tau,1} \twoheadrightarrow q_0^*{\mathcal{M}}_{\tau,1}\end{align*} $$

has trivial pullback to $S \cong \Delta $ , it factors through a morphism

$$ \begin{align*}\delta_{0,*} {\mathcal{L}}_{\tau,1} = p_0^*{\mathcal{L}}_{\tau,1} \otimes_{{\mathcal{O}}_{Z_0}} ({\mathcal{O}}_{Z_0}/{\mathcal{I}}) \longrightarrow q_0^*{\mathcal{M}}_{\tau,1} \otimes_{{\mathcal{O}}_{Z_0}} {\mathcal{I}} = \delta_{0,*} {\mathcal{M}}_{\tau,1}\otimes_{{\mathcal{O}}_{Z_0}} {\mathcal{I}}\end{align*} $$

and hence induces a morphism

$$ \begin{align*}\beta_{\tau,1}: \delta_{0,*} ({\mathcal{L}}_{\tau,1}\otimes_{{\mathcal{O}}_S} {\mathcal{M}}_{\tau,1}^{-1}) \longrightarrow {\mathcal{I}}.\end{align*} $$

We then define the sheaf of ideals ${\mathcal {I}}_{\tau ,1}$ on $Z_0$ to be the image of $\beta _{\tau ,1}$ , and we let $Z_{\tau ,1}$ denote the subscheme of $Z_0$ defined by ${\mathcal {I}}_{\tau ,1}$ , and $p_{\tau ,1}$ and $q_{\tau ,1}$ the resulting projection maps $Z_{\tau ,1} \to S$ . By construction, the pullback of $\beta _{\tau ,1}$ to $Z_{\tau ,1}$ is trivial, and hence so is that of the morphism $p_0^*{\mathcal {L}}_{\tau ,1} \to q_0^*{\mathcal {M}}_{\tau ,1}$ , which implies that the pullback of $\alpha $ maps $p_{\tau ,1}^*{\mathcal {L}}_{\tau ,1} = p_{\tau ,1}^*{\mathcal {F}}_\tau ^{(1)}$ isomorphically to $q_{\tau ,1}^*{\mathcal {L}}_{\tau ,1} = q_{\tau ,1}^*{\mathcal {F}}_\tau ^{(1)}$ . It follows from ${\mathcal {O}}_F$ -linearity that $\alpha $ induces an isomorphism $p_{\tau ,1}^*{\mathcal {G}}_\tau ^{(2)} \stackrel {\sim }{\longrightarrow } q_{\tau ,1}^*{\mathcal {G}}_\tau ^{(2)}$ (if $e_{\mathfrak {p}}> 1$ ), and hence an isomorphism

$$ \begin{align*}\alpha_{\tau,2}: p_{\tau,1}^*{\mathcal{P}}_{\tau,2} \stackrel{\sim}{\longrightarrow} q_{\tau,1}^*{\mathcal{P}}_{\tau,2}.\\[-17pt]\end{align*} $$

The same argument as above now yields a morphism

$$ \begin{align*}\beta_{\tau,2}: \delta_{0,*} ({\mathcal{L}}_{\tau,2}\otimes_{{\mathcal{O}}_S} {\mathcal{M}}_{\tau,2}^{-1}) \longrightarrow {\mathcal{I}}/{\mathcal{I}}_{\tau,1},\\[-17pt]\end{align*} $$

whose image is that of a sheaf of ideals on $Z_0$ we denote by ${\mathcal {I}}_{\tau ,2}$ .

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

$$ \begin{align*}0 = {\mathcal{I}}_{\tau,0} \subset {\mathcal{I}}_{\tau,1} \subset \cdots \subset {\mathcal{I}}_{\tau,e_{\mathfrak{p}}}\\[-17pt]\end{align*} $$

on $Z_0$ such that $\alpha $ induces

• • isomorphisms $p_{\tau ,j}^*{\mathcal {F}}_\tau ^{(j)} \stackrel {\sim }{\longrightarrow } q_{\tau ,j}^* {\mathcal {F}}_\tau ^{(j)}$

• • and surjections $\delta _{0,*} ({\mathcal {L}}_{\tau ,j}\otimes _{{\mathcal {O}}_S} {\mathcal {M}}_{\tau ,j}^{-1}) \twoheadrightarrow {\mathcal {I}}_{\tau ,j}/{\mathcal {I}}_{\tau ,j-1}$ ,

for $j=1,\ldots ,e_{\mathfrak {p}}$ , where $Z_{\tau ,j}$ denotes the closed subscheme of $Z_0$ defined by ${\mathcal {I}}_{\tau ,j}$ and $p_{\tau ,j}$ , $q_{\tau ,j}$ are the projections $Z_{\tau ,j} \to S$ .

Furthermore, we claim that the map

$$ \begin{align*}\bigoplus_{{\mathfrak{p}} \in S_p} \bigoplus_{\tau \in \Sigma_{{\mathfrak{p}},0}} {\mathcal{I}}_{\tau,e_{\mathfrak{p}}} \to {\mathcal{I}}\\[-17pt]\end{align*} $$

is surjective. Indeed, let ${\mathcal {J}}$ denote the image and let T denote the corresponding closed subscheme of $Z_0$ , so T is the scheme-theoretic intersection of the $Z_{\tau ,e_{\mathfrak {p}}}$ , and letFootnote ⁸ p, $q: T \to S$ denote the projection maps. By construction, $\alpha $ pulls back to an isomorphism $p^*{\mathcal {H}}^1_{{\operatorname {dR}}}(A/S) \stackrel {\sim }{\longrightarrow } q^*{\mathcal {H}}^1_{\operatorname {dR}}(A/S)$ under which $p^*{\mathcal {F}}_\tau ^{(j)} \stackrel {\sim }{\longrightarrow } q^*{\mathcal {F}}_\tau ^{(j)}$ for all $\tau $ and j. In particular, $t_*\Omega ^1_{p^*A/T} = p^*(s_*\Omega ^1_{A/S} ) \stackrel {\sim }{\longrightarrow } q^*(s_*\Omega ^1_{A/S}) = u_*\Omega ^1_{q^*A/T}$ (where $t:p^*A \to T$ and $u:q^*A \to T$ are the structure morphisms), which the Grothendieck–Messing theorem implies is induced by an isomorphism $p^*A \cong q^*A$ of abelian schemes lifting the identity over S. Since the isomorphism respects the filtrations ${\mathcal {F}}^\bullet $ and the lifts of the universal auxiliary structures $\iota $ , $\lambda $ and $\eta $ over T are unique, it follows that $p^*\underline {A} \cong q^*\underline {A}$ , which means that $p=q \in S(T)$ , so $T = \Delta $ .

Now defining $\Omega ^1_{S/{\mathcal {O}},{\mathfrak {p}},i} = \delta _0^*{\mathcal {I}}_{\tau _{{\mathfrak {p}},i},e_{\mathfrak {p}}}$ and ${\operatorname {Fil}\,}^j( \Omega ^1_{S/{\mathcal {O}},{\mathfrak {p}},i} )= \delta _0^*{\mathcal {I}}_{\tau _{{\mathfrak {p}},i},j}$ for ${\mathfrak {p}} \in S_p$ , $1 \le i \le f_{\mathfrak {p}}$ , $1 \le j \le e_{\mathfrak {p}}$ , we obtain surjective morphisms

$$ \begin{align*}{\mathcal{L}}_{\tau_{{\mathfrak{p}},i},j}\otimes_{{\mathcal{O}}_S} {\mathcal{M}}_{\tau_{{\mathfrak{p}},i},j}^{-1} \twoheadrightarrow {\operatorname{gr}\,}^j (\Omega^1_{S/{\mathcal{O}},{\mathfrak{p}},i} ) \qquad \text{and}\qquad \bigoplus_{{\mathfrak{p}},i} \Omega^1_{S/{\mathcal{O}},{\mathfrak{p}},i} \twoheadrightarrow \Omega^1_{S/{\mathcal{O}}}.\\[-17pt]\end{align*} $$

Since the ${\mathcal {L}}_{\tau _{{\mathfrak {p}},i},j}\otimes _{{\mathcal {O}}_S} {\mathcal {M}}_{\tau _{{\mathfrak {p}},i},j}^{-1}$ are line bundles and $\Omega ^1_{S/{\mathcal {O}}}$ is locally free of rank d, it follows that all of the maps are isomorphisms.

Finally, the constructions above are independent of the polarisation $\lambda $ and hence are compatible with the action of $\nu \in {\mathcal {O}}_{F,(p),+}^\times $ on $S = \widetilde {Y}_U$ . More precisely, the pullback of $\alpha $ via the diagonal map $(\nu ,\nu )$ is compatible with the canonical isomorphism $\nu ^*{\mathcal {H}}^1_{\operatorname {dR}}(A/S) \cong {\mathcal {H}}^1_{\operatorname {dR}}(A/S)$ induced by the identification of $\nu ^*A$ with A, from which it follows easily that the morphisms in the construction of the filtration are invariant under the action of ${\mathcal {O}}_{F,(p),+}^\times $ and hence descend to give the decomposition, filtrations and isomorphisms in the statement of the theorem.

Let us also note the interpretation of the Kodaira–Spencer filtration in terms of tangent spaces. For a closed point y of S corresponding to the data $\underline {A}_0 = (A_0,\iota _0,\lambda _0,\eta _0,{\mathcal {F}}_0^\bullet )$ over a finite extension k of the residue field of ${\mathcal {O}}$ , the fibre $T_y(S)$ of ${\mathcal {H}om}_{{\mathcal {O}}_S}(\Omega ^1_{S/{\mathcal {O}}},{\mathcal {O}}_S)$ is canonically identified with the set of isomorphism classes of data $\underline {A}_1$ over $k[\epsilon ]$ lifting $\underline {A}_0$ , and the decomposition and filtrations of the theorem yield dual decompositions of $T_y(S)$ into components $T_y(S)_{\tau }$ with decreasing filtrations ${\operatorname {Fil}\,}^j( T_y(S)_{\tau })$ . From the proof of the theorem one sees immediately that $\bigoplus _{\tau } {\operatorname {Fil}\,}^{j_\tau }( T_y(S)_\tau )$ corresponds to the set of $(A_1,\iota _1,\lambda _1,\eta _1,{\mathcal {F}}_1^\bullet )$ such that ${\mathcal {F}}_{1,\tau }^{(j)}$ is the image of ${\mathcal {F}}_{0,\tau }^{(j)} \otimes _k k[\epsilon ]$ for all $\tau $ and $j \le j_\tau $ under the canonical isomorphism

$$ \begin{align*}H^1_{\operatorname{dR}}(A_1/k[\epsilon]) \cong H^1_{\operatorname{cris}}(A_0/k[\epsilon]) \cong H^1_{\operatorname{dR}}(A_0/k) \otimes_k k[\epsilon].\end{align*} $$

We note also that the theorem yields a canonical (Kodaira–Spencer) isomorphism

$$ \begin{align*}\Omega_{Y_U/{\mathcal{O}}}^d \cong \wedge_{{\mathcal{O}}_{Y_U}}^d \Omega^1_{Y_U/{\mathcal{O}}} \cong {\mathcal{A}}_{\mathbf{{2}},\mathbf{{-1}},{\mathcal{O}}}\end{align*} $$

(writing $\sum m{\mathbf {e}}_\theta $ as $\mathbf {m}$ for $m \in \mathbb {Z}$ ). Furthermore, the decomposition, filtrations and isomorphisms of the theorem (and hence also the Kodaira–Spencer isomorphism) are Hecke-equivariant in the obvious sense. More precisely, the same argument as for the compatibility with the ${\mathcal {O}}_{F,(p),+}^\times $ -action, but using the quasi-isogeny in the construction of $\rho _g$ , shows that if $U_1$ , $U_2$ and $g \in \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ are such that $g^{-1}U_1g \subset U_2$ , then $\rho _g^* {\operatorname {Fil}\,}^j (\Omega ^1_{Y_{U_2}/{\mathcal {O}},{\mathfrak {p}},i})$ corresponds to ${\operatorname {Fil}\,}^j (\Omega ^1_{Y_{U_1}/{\mathcal {O}},{\mathfrak {p}},i})$ for all ${\mathfrak {p}},i,j$ under the canonical isomorphism $\rho _g^*\Omega ^1_{Y_{U_2}/{\mathcal {O}}} \stackrel {\sim }{\longrightarrow } \Omega ^1_{Y_{U_1}/{\mathcal {O}}}$ , and the resulting diagrams

commute (where the top arrow is defined in the discussion preceding (6)).

4.1 Construction of $H_\theta $ and $G_\theta $

We now recall the definition, due to Reduzzi and Xiao [Reference Reduzzi and Xiao32], of generalised partial Hasse invariants on Pappas–Rapoport models. These will be, for each $\theta = \theta _{{\mathfrak {p}},i,j} \in \Sigma $ , a Hilbert modular form $H_\theta $ of weight $(\mathbf {h}_\theta ,\mathbf {0})$ with coefficients in $\mathbb {F} = {\mathcal {O}}/{\mathfrak {m}}_{\mathcal {O}}$ , where $\mathbf {h}_\theta : = n_\theta {\mathbf {e}}_{\sigma ^{-1}\theta } - {\mathbf {e}}_\theta $ , with $n_\theta = p$ if $j=1$ and $n_\theta = 1$ if $j> 1$ . We also define below a(n in)variant $G_\theta $ of weight $(\mathbf {0},\mathbf {h}_\theta )$ .

We will now be working in the mod p setting, so until further notice S will denote $\widetilde {Y}_{U,\mathbb {F}}$ , and $s:A \to S$ the universal abelian scheme over it. Thus, ${\mathcal {H}}^1_{\operatorname {dR}}(A/S)$ is a locally free sheaf of rank 2 over

$$ \begin{align*}{\mathcal{O}}_F\otimes {\mathcal{O}}_S \cong \bigoplus_{{\mathfrak{p}} \in S_p} \bigoplus_{\tau \in \Sigma_{{\mathfrak{p}},0}} {\mathcal{O}}_S [u] / (u^{e_{\mathfrak{p}}}),\end{align*} $$

where u acts via $\iota (\varpi _{\mathfrak {p}})^*$ on the ${\mathfrak {p}}$ -component of

$$ \begin{align*}{\mathcal{H}}^1_{\operatorname{dR}}(A/S) = \bigoplus_{{\mathfrak{p}} \in S_p} {\mathcal{H}}_{\mathfrak{p}} = \bigoplus_{{\mathfrak{p}} \in S_p} \bigoplus_{\tau \in \Sigma_{{\mathfrak{p}},0}} {\mathcal{H}}_\tau.\end{align*} $$

We will also now be working with a fixed ${\mathfrak {p}}$ and omit the subscript from the notation, so that

$$ \begin{align*}{\mathcal{H}} = {\mathcal{H}}^1_{\operatorname{dR}}(A/S)_{\mathfrak{p}} = \bigoplus _{\tau \in \Sigma_{{\mathfrak{p}},0}} {\mathcal{H}}_\tau = \bigoplus_{i \in \mathbb{Z}/f\mathbb{Z}} {\mathcal{H}}_i\end{align*} $$

with each ${\mathcal {H}}_i$ locally free of rank 2 over ${\mathcal {O}}_S[u]/(u^e)$ (where we have also abbreviated the subscript $\tau _{{\mathfrak {p}},i}$ by i). Furthermore, for each $i\in \mathbb {Z}/f\mathbb {Z}$ , we have a filtration

$$ \begin{align*}0 = {\mathcal{F}}_i^{(0)} \subset {\mathcal{F}}_i^{(1)}\subset \cdots \subset {\mathcal{F}}_i^{(e-1)} \subset {\mathcal{F}}_i^{(e)} = (s_*\Omega^1_{A/S})_i\end{align*} $$

by sheaves of ${\mathcal {O}}_S[u]/(u^e)$ -modules such that the quotients ${\mathcal {L}}_{i,j} = {\mathcal {F}}_i^{(j)}/{\mathcal {F}}_i^{(j-1)}$ are line bundles annihilated by u.

Firstly, note that if $j> 1$ , then $u: {\mathcal {F}}_i^{(j)} \to {\mathcal {F}}_i^{(j-1)}$ induces a morphism ${\mathcal {L}}_{i,j} \to {\mathcal {L}}_{i,j-1}$ . On the other hand, if $j=1$ , then the Verschiebung morphism $\phi _S^*A \to A$ over S induces ${\mathcal {O}}_S[u]/(u^e)$ -linear morphisms

$$ \begin{align*}\operatorname{Ver}_i^*: {\mathcal{H}}_i = {\mathcal{H}}_{\tau_i} \longrightarrow {\mathcal{H}}^1_{\operatorname{dR}}(\phi_S^*(A)/S)_{\tau_i} \cong \phi_S^*({\mathcal{H}}_{\tau_{i-1}}) = \phi_S^*({\mathcal{H}}_{i-1})\end{align*} $$

with image $\phi _S^*({\mathcal {F}}_{i-1}^{(e)})$ (where $\phi _S$ denotes the absolute Frobenius on S). Note that ${\mathcal {L}}_{i,1} = {\mathcal {F}}_i^{(1)} \subset u^{e-1} {\mathcal {H}}_i$ , so that $u^{e-1}$ defines an isomorphism

$$ \begin{align*}u^{1-e}{\mathcal{L}}_{i,1}/u {\mathcal{H}}_i \stackrel{\sim}{\longrightarrow} {\mathcal{F}}_i^{(1)}\end{align*} $$

and that $\operatorname {Ver}_i^*(u{\mathcal {H}}_i) = u\phi _S^*({\mathcal {F}}_{i-1}^{(e)}) \subset \phi _S^*({\mathcal {F}}_{i-1}^{(e-1)})$ , so we obtain a well-defined ${\mathcal {O}}_S$ -linear morphism ‘ $\operatorname {Ver}_i^*\circ u^{1-e}$ ’

$$ \begin{align*}{\mathcal{L}}_{i,1} \stackrel{\sim}{\longleftarrow} u^{1-e}{\mathcal{L}}_{i,1} /u {\mathcal{H}}_i \longrightarrow \phi_S^*( {\mathcal{L}}_{i-1,e}) \cong {\mathcal{L}}_{i-1,e}^{\otimes p}.\end{align*} $$

We have now defined a morphism ${\mathcal {L}}_\theta \longrightarrow {\mathcal {L}}_{\sigma ^{-1}\theta }^{\otimes n_\theta }$ for all $\theta $ and hence a section of $\widetilde {{\mathcal {A}}}_{\mathbf {h}_\theta ,\mathbf {0},\mathbb {F}}$ over S. Furthermore, it is straightforward to check that the section is invariant under the action of ${\mathcal {O}}_{F,(p),+}^\times $ and therefore descends to an element

$$ \begin{align*}H_\theta \in M_{\mathbf{h}_\theta,\mathbf{0}} (U; \mathbb{F}) = H^0(Y_{U,\mathbb{F}} , {\mathcal{A}}_{\mathbf{h}_\theta,\mathbf{0},\mathbb{F}}),\end{align*} $$

which we call the partial Hasse invariant (indexed by $\theta $ ). Furthermore, the partial Hasse invariants are stable under the Hecke action, in the sense that if $U_1$ , $U_2$ and $g \in \operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ are such that $g^{-1}U_1g \subset U_2$ , then $[g]H_{\theta ,2} = H_{\theta ,1}$ . (Note also that the partial Hasse invariants are dependent on the choice of uniformiser $\varpi = \varpi _{\mathfrak {p}}$ only up to a scalar in $\mathbb {F}^\times $ : if $\varpi $ is replaced by $a \varpi $ for some $a \in {\mathcal {O}}_{F,{\mathfrak {p}}}^\times $ , then $H_\theta $ is replaced by $\tau (a) H_\theta $ if $j>1$ and by $\tau (a)^{1-e} H_\theta $ if $j=1$ .)

We remark also that the line bundles ${\mathcal {A}}_{\mathbf {{0},\mathbf {h}_\theta ,\mathbb {F}}}$ have canonical trivialisations. Indeed, for each $i \in \mathbb {Z}/f\mathbb {Z}$ and $j=2,\ldots ,e$ , we have the exact sequence

$$ \begin{align*}0 \to {\mathcal{G}}_i^{(j-1)}/{\mathcal{F}}_i^{(j-1)} \longrightarrow {\mathcal{G}}_i^{(j)}/{\mathcal{F}}_i^{(j-1)} \stackrel{u}{\longrightarrow} {\mathcal{F}}_i^{(j-1)}/{\mathcal{F}}_i^{(j-2)} \to 0\end{align*} $$

over S; that is, $0 \to {\mathcal {M}}_{i,j-1} \to {\mathcal {P}}_{i,j} \to {\mathcal {N}}_{i,j-1} \to 0$ , inducing an isomorphism

$$ \begin{align*}{\mathcal{N}}_{i,j-1} = {\mathcal{L}}_{i,j-1}\otimes_{{\mathcal{O}}_S} {\mathcal{M}}_{i,j-1} \cong \wedge^2_{{\mathcal{O}}_S} {\mathcal{P}}_{i,j} = {\mathcal{N}}_{i,j}\end{align*} $$

and hence ${\mathcal {O}}_S \simeq {\mathcal {N}}_{i,j-1}^{-1}\otimes _{{\mathcal {O}}_S}{\mathcal {N}}_{i,j} = \widetilde {{\mathcal {A}}}_{\mathbf {{0},\mathbf {h}_\theta ,\mathbb {F}}}$ for $\theta = \theta _{{\mathfrak {p}},i,j}$ , which it is straightforward to check descends to $Y_{U,\mathbb {F}}$ . Similarly, we have the exact sequence

$$ \begin{align*}0 \to\phi_S^* ({\mathcal{G}}_{i-1}^{(e)}/{\mathcal{F}}_{i-1}^{(e)}) \stackrel{{\operatorname{Frob}}_A^*}{\longrightarrow} {\mathcal{G}}_{i}^{(1)} \stackrel{\operatorname{Ver}_A^*u^{1-e}}{\longrightarrow} \phi_S^*({\mathcal{F}}_{i-1}^{(e)}/{\mathcal{F}}_{i-1}^{(e-1)}) \to 0\end{align*} $$

inducing an isomorphism $\phi _S^*({\mathcal {N}}_{i-1,e}) \cong {\mathcal {N}}_{i,1}$ and hence ${\mathcal {O}}_S \simeq \widetilde {{\mathcal {A}}}_{\mathbf {{0},\mathbf {h}_\theta ,\mathbb {F}}}$ for $\theta = \theta _{{\mathfrak {p}},i,1}$ descending to $Y_{U,\mathbb {F}}$ . Furthermore, these isomorphisms are Hecke-equivariant in the usual sense, but note that they depend via u on the choice of $\varpi _{\mathfrak {p}}$ . For each $\theta $ , we let $G_\theta \in M_{\mathbf {{0},\mathbf {h}_\theta }}(U;\mathbb {F})$ denote the canonical trivialising section.

4.2 Stratification

We also recall how the partial Hasse invariants define a stratification of the Hilbert modular variety in characteristic p. For any $\theta \in \Theta $ , we define $\widetilde {Z}_\theta $ (respectively $Z_\theta $ ) to be the closed subscheme of $S = \widetilde {Y}_{U,\mathbb {F}}$ (respectively $Y_{U,\mathbb {F}}$ ) defined by the vanishing of $H_\theta $ , and for any subset $T \subset \Sigma $ , we let

$$ \begin{align*}\widetilde{Z}_T = \bigcap_{\theta \in T} \widetilde{Z}_\theta\quad \text{and} \quad Z_T = \bigcap_{\theta \in T} Z_\theta.\end{align*} $$

Note that the schemes $Z_T$ are stable under the Hecke action, in the strong sense that $Z_{T,1}$ is the pullback of $Z_{T,2}$ under $\rho _g:Y_{U_1} \to Y_{U_2}$ .

We then have the following consequence ([Reference Reduzzi and Xiao32, Thm. 3.10]) of the description of the Kodaira–Spencer filtration on tangent spaces at closed points; we give a proof here as some of the details are relevant to the construction of $\Theta $ -operators in Subsection 5.2.

Proof. We prove the result for $\widetilde {Z}_T$ , from which the result for $Z_T$ is immediate.

Let y be a closed point of S with local ring $R = {\mathcal {O}}_{S,y}$ , maximal ideal ${\mathfrak {m}}$ and residue field $k = R/{\mathfrak {m}}$ . For each $\theta \in \Sigma $ , choose a basis $b_\theta $ for ${{\mathcal {L}}}_{\theta ,y}$ over R and write $H_{\theta ,y} b_\theta = x_\theta b_{\sigma ^{-1}\theta }^{n_\theta }$ . Thus, if $y \in \widetilde {Z}_\theta $ , then $x_\theta \in {\mathfrak {m}}$ , and we let $\overline {x}_\theta $ denote its image in ${\mathfrak {m}}/{\mathfrak {m}}^2$ .

Identifying ${\mathfrak {m}}/{\mathfrak {m}}^2$ with the fibre of $\Omega ^1_{\widetilde {Y}_U/{\mathcal {O}}}$ at y and writing ${\operatorname {Fil}\,}^j({\mathfrak {m}}/{\mathfrak {m}}^2)_\tau $ for the subspaces obtained from the Kodaira–Spencer filtration, we claim that if $y \in Z_\theta $ , then

(7) $$ \begin{align} {\operatorname{Fil}\,}^j({\mathfrak{m}}/{\mathfrak{m}}^2)_\tau = k\overline{x}_\theta + {\operatorname{Fil}\,}^{j-1}({\mathfrak{m}}/{\mathfrak{m}}^2)_\tau, \end{align} $$

where $\tau = \tau _{{\mathfrak {p}},i}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ . Comparing dimensions, we see it suffices to prove the inclusion of the left-hand side in the right or, equivalently, that if

$$ \begin{align*}v \in T_y(S) = \bigoplus_{\tau^{\prime} \in \Sigma_0} T_y(S)_{\tau^{\prime}}\end{align*} $$

is such that its $\tau $ -component $v_\tau $ lies in ${\operatorname {Fil}\,}^{j-1} T_y(S)_\tau $ and v is orthogonal to $\overline {x}_\theta $ , then in fact $v_\tau \in {\operatorname {Fil}\,}^j(T_y(S)_\tau )$ (using the notation of the discussion following the proof of Theorem 3.3.1).

Let $\underline {A}_0 = (A_0,\iota _0,\lambda _0,\eta _0,{\mathcal {F}}_0^\bullet )$ denote the data corresponding to the point $y \in S(k)$ and $\underline {A}_1 = (A_1,\iota _1,\lambda _1,\eta _1,{\mathcal {F}}_1^\bullet )$ that of its lift $v \in S(k[\epsilon ])$ . With $\tau = \tau _{{\mathfrak {p}},i}$ fixed for now, we will suppress ${\mathfrak {p}}$ from the notation and replace the subscript $\tau _{p,i^{\prime }}$ by $i^{\prime }$ (for $i^{\prime } = i,i-1$ ). Recall the assumption that $v_i \in {\operatorname {Fil}\,}^{j-1}( T_y(S)_i)$ means that ${\mathcal {F}}_{1,i}^{(j^{\prime })}$ corresponds to ${\mathcal {F}}_{0,i}^{(j^{\prime })} \otimes _k k[\epsilon ]$ for $j^{\prime } = 1,\ldots ,j-1$ under the canonical isomorphism

(8) $$ \begin{align} H^1_{\operatorname{dR}}(A_1/k[\epsilon]) \cong H^1_{\operatorname{cris}}(A_0/k[\epsilon]) \cong H^1_{\operatorname{dR}}(A_0/k) \otimes_k k[\epsilon]. \end{align} $$

For $v_i$ to be orthogonal to $\overline {x}_\theta $ means that the morphism

(9) $$ \begin{align} {\mathcal{L}}_{1,\theta} \longrightarrow {\mathcal{L}}_{1,\sigma^{-1}\theta}^{\otimes n_\theta} \end{align} $$

induced by $H_\theta $ vanishes, and we need to show this implies that ${\mathcal {F}}_{1,i}^{(j)}$ is the image of ${\mathcal {F}}_{0,i}^{(j)} \otimes _k k[\epsilon ]$ .

Suppose first that $j> 1$ . Then (9) is simply

$$ \begin{align*}u: {\mathcal{F}}_{1,i}^{(j)}/{\mathcal{F}}_{1,i}^{(j-1)} \longrightarrow {\mathcal{F}}_{1,i}^{(j-1)}/{\mathcal{F}}_{1,i}^{(j-2)},\end{align*} $$

whose vanishing means ${\mathcal {F}}_{1,i}^{(j)} = u^{-1}{\mathcal {F}}_{1,i}^{(j-2)}$ . Since (8) sends ${\mathcal {F}}_{1,i}^{(j-2)}$ to ${\mathcal {F}}_{0,i}^{(j-2)} \otimes _k k[\epsilon ]$ and is compatible with u, it follows that it also sends ${\mathcal {F}}_{1,i}^{(j)}$ to ${\mathcal {F}}_{0,i}^{(j)} \otimes _k k[\epsilon ]$ .

On the other hand, if $j=1$ , then the vanishing of (9) means that $u^{1-e}{\mathcal {F}}_{1,i}^{(1)}$ is the pre-image of $\phi _1^*({\mathcal {F}}_{1,i-1}^{(e-1)})$ under $\operatorname {Ver}_i^*$ (where $\phi _1$ is the absolute Frobenius on $k[\epsilon ]$ , and $\phi _0$ will denote the absolute Frobenius on k). Since the diagram

commutes, where the vertical maps are induced by Verschiebung, the top arrow is (8) and the bottom one is given by the identification of $\phi _1^*A_1$ with $\phi _0^*A_0 \otimes _k k[\epsilon ]$ , so in particular identifies $\phi _1^*({\mathcal {F}}_{1,i-1}^{(e-1)})$ with $\phi _0^*({\mathcal {F}}_{0,i-1}^{(e-1)}) \otimes _k k[\epsilon ]$ , it follows that the top arrow sends $u^{1-e}{\mathcal {F}}_{1,i}^{(1)}$ to $u^{1-e}{\mathcal {F}}_{0,i}^{(1)} \otimes _k k[\epsilon ]$ , and hence ${\mathcal {F}}_{1,i}^{(1)}$ to ${\mathcal {F}}_{0,i}^{(1)} \otimes _k k[\epsilon ]$ . This completes the proof of the claim.

Now note that if $y \in \widetilde {Z}_T$ , then (7) implies that the elements $\overline {x}_\theta $ for $\theta \in T$ can be extended to a basis for ${\mathfrak {m}}/{\mathfrak {m}}^2$ over k and hence are linearly independent. Since R is regular of dimension $d = |\Sigma |$ , it follows that ${\mathcal {O}}_{\widetilde {Z}_T,y} = R/\langle x_\theta \rangle _{\theta \in T}$ is regular of dimension of $d - |T|$ and hence that $\widetilde {Z}_T$ is smooth over $\mathbb {F}$ of dimension $d - |T|$ .

Finally, we recall the definition of the minimal weight of a nonzero mod p Hilbert modular form. If $f \in M_{\mathbf {k},\mathbf {l}}(U;\mathbb {F})$ , then $\mathbf {k}_{\min }(f)$ is defined to be $\mathbf {k} - \sum _{\theta } m_\theta \mathbf {h}_\theta $ where $\sum _\theta m_\theta {\mathbf {e}}_\theta $ is the unique maximal element of the set

$$ \begin{align*}\left\{\,\left. \sum_\theta m_\theta {\mathbf{e}}_\theta \in \mathbb{Z}_{\ge 0}^\Sigma\,\right|\, f = g\prod_{\theta \in \Sigma} H_\theta^{m_\theta}\ \text{for some}\ g \in M_{\mathbf{k} - \sum_\theta m_\theta \mathbf{h}_\theta,\mathbf{l}}(U;\mathbb{F})\,\right\}.\\[-17pt]\end{align*} $$

By the main result of [Reference Diamond and Kassaei10], the minimal weight of f always lies in the cone

(10) $$ \begin{align} \Xi^{\min} := \left\{\, \left.\sum_\theta k_\theta {\mathbf{e}}_\theta\,\right|\ n_\theta k_\theta \ge k_{\sigma^{-1}\theta}\ \text{for all}\ \theta \in \Sigma\,\right\}. \\[-17pt]\nonumber\end{align} $$

Note that the result stated in [Reference Diamond and Kassaei10] applies to forms on a finite étale cover of $Y_{U,\mathbb {F}}$ , from which the analogous result for forms on $Y_{U,\mathbb {F}}$ is immediate.

5.1 Fundamental Hasse invariants

In order to define the partial $\Theta $ -operators (in Subsection 5.2), we first define a canonical factorisation of the partial Hasse invariants over a finite flat (Igusa) cover of the Hilbert modular variety over $\mathbb {F}$ .

We fix a sufficiently small U that the line bundles ${\mathcal {L}}_\theta $ , ${\mathcal {M}}_\theta $ , ${\mathcal {N}}_\theta $ (and hence $\widetilde {{\mathcal {A}}}_{\mathbf {k},\mathbf {l},\mathbb {F}}$ ) on $\widetilde {Y}_{U,\mathbb {F}}$ descend to $Y_{U,\mathbb {F}}$ for all $\theta \in \Sigma $ (and all $\mathbf {k},\mathbf {l}\in \mathbb {Z}^\Sigma $ ), and we write simply $\overline {Y}$ for $Y_{U,\mathbb {F}}$ and $\overline {{\mathcal {L}}}_{\tau ,j}$ , $\overline {{\mathcal {M}}}_{\tau ,j}$ and $\overline {{\mathcal {N}}}_{\tau ,j}$ for the line bundles on $\overline {Y}$ . For each ${\mathfrak {p}} \in S_p$ and $\tau \in \Sigma _{{\mathfrak {p}},0}$ , we let

$$ \begin{align*}H_\tau = \prod_{j=1}^{e_{\mathfrak{p}}} H_{\tau,j} \in H^0(\overline{Y}, \overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1}\otimes_{{\mathcal{O}}_{\overline{Y}}} \overline{{\mathcal{L}}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}}^{\otimes p}).\\[-17pt]\end{align*} $$

Viewing each $H_\tau $ as a morphism $(\overline {{\mathcal {L}}}_{\phi ^{-1}\circ \tau ,e_{\mathfrak {p}}}^{-1})^{\otimes p} \to \overline {{\mathcal {L}}}_{\tau ,e_{\mathfrak {p}}}^{-1}$ and $H_{\mathfrak {p}} := \prod _{\tau \in \Sigma _{{\mathfrak {p}},0}} H_\tau $ as a morphism $\otimes _{\tau \in \Sigma _{{\mathfrak {p}},0}} (\overline {{\mathcal {L}}}_{\tau ,e_{\mathfrak {p}}}^{-1})^{\otimes (p-1)} \to {\mathcal {O}}_{\overline {Y}}$ , we define the Igusa coverFootnote ⁹ of $\overline {Y}$ (of level ${\mathfrak {q}} = \prod {\mathfrak {p}}$ ) to be

$$ \begin{align*}{Y}^{\mathrm{Ig}} = \mathbf{Spec} \left(\mathrm{Sym}_{\mathcal{O}_{\overline{Y}}} (\bigoplus_{\tau \in \Sigma_0} \overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1})/\mathcal{I}\right),\\[-17pt]\end{align*} $$

where ${\mathcal {I}}$ is the sheaf of ideals generated by the ${\mathcal {O}}_{\overline {Y}}$ -submodules

$$ \begin{align*}(H_\tau - 1) \overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1}\ \text{for}\ \tau \in \Sigma_{0},\ \text{and}\ (H_{\mathfrak{p}} -1)\left(\bigotimes_{\tau \in \Sigma_{{\mathfrak{p}},0}} (\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1})^{\otimes (p-1)}\right)\ \text{for}\ {\mathfrak{p}} \in S_p\end{align*} $$

(where all tensor products are over ${\mathcal {O}}_{\overline {Y}}$ ). We then define an action of $({\mathcal {O}}_F/{\mathfrak {q}})^\times $ on ${Y}^{{\mathrm {Ig}}}$ over $\overline {Y}$ by letting $\alpha \in ({\mathcal {O}}_F/{\mathfrak {q}})^\times $ act on the structure sheaf by the automorphism of sheaves of ${\mathcal {O}}_{\overline {Y}}$ -algebras induced by multiplication by $\tau ({\alpha })^{-1}$ on $\overline {{\mathcal {L}}}_{\tau ,e_{\mathfrak {p}}}^{-1}$ for each $\tau $ . We then see, exactly as in the proof of parts (1) and (2) of [Reference Diamond and Sasaki12, Prop. 8.1.1], that the projection $\pi :{Y}^{{\mathrm {Ig}}} \to \overline {Y}$ is finite flat, generically étale and identifies $\overline {Y}$ with the quotient of ${Y}^{{\mathrm {Ig}}}$ by the action of $({\mathcal {O}}_F/{\mathfrak {q}})^\times $ .

For each $\tau \in \Sigma _{{\mathfrak {p}},0}$ , we let $h_{\tau ,e_{\mathfrak {p}}}$ denote the tautological section of $\pi ^*\overline {{\mathcal {L}}}_{\tau ,e_{\mathfrak {p}}}$ induced by the inclusion $\overline {{\mathcal {L}}}_{\tau ,e_{\mathfrak {p}}}^{-1} \hookrightarrow \pi _*{\mathcal {O}}_{{Y}^{\mathrm {Ig}}}$ . We also define the section

$$ \begin{align*}h_{\tau,j} = \pi^*(H_{\tau,j+1}\cdots H_{\tau,e_{\mathfrak{p}}})h_{\tau,e_{\mathfrak{p}}}\end{align*} $$

of $\pi ^*\overline {{\mathcal {L}}}_{\tau ,j}$ for $j=1,\ldots ,e_{{\mathfrak {p}}}-1$ . Note that since ${Y}^{\mathrm {Ig}}$ is reduced (or since $\prod _{\tau \in \Sigma _{{\mathfrak {p}},0}} h_{\tau ,e_{\mathfrak {p}}}^{p-1} = \pi ^*H_{\mathfrak {p}}$ by construction), the sections $h_{\tau ,e_{\mathfrak {p}}}$ are injective, and hence so are the $h_{\tau ,j}$ for all $\tau $ and j. We write simply $h_\theta $ for the section $h_{\tau ,j}$ of $\pi ^*\overline {{\mathcal {L}}}_\theta = \pi ^*\overline {{\mathcal {L}}}_{\tau ,j}$ , and we call the $h_\theta $ the fundamental Hasse invariant (indexed by $\theta $ ).

5.2 Construction of $\Theta _\tau $

We now explain how the construction of $\Theta $ -operators in [Reference Diamond and Sasaki12] directly generalises to the case where p is ramified in F, yielding an operator that shifts the weight $\mathbf {k}$ by $(1,1)$ in the final two components corresponding to embeddings with the same reduction; that is, $\theta _{{\mathfrak {p}},i,e_{{\mathfrak {p}}}-1}$ , $\theta _{{\mathfrak {p}},i,e_{\mathfrak {p}}}$ (and hence, by composing with multiplication by partial Hasse invariants, one can shift weights by $+1$ for any pair of embeddings with the same reduction).

Indeed, for each $\tau \in \Sigma _0$ , we define the operator $\Theta _{\tau }$ exactly as in [Reference Diamond and Sasaki12, §8] but using the morphism

$$ \begin{align*}KS_{\tau}: \Omega^1_{\overline{Y}/{\mathbb{F}}} \longrightarrow {\operatorname{gr}\,}^{e_{{\mathfrak{p}}}}(\Omega^1_{\overline{Y}/{\mathbb{F}}})_{\tau} \stackrel{\sim}{\longrightarrow} \overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}\otimes_{{\mathcal{O}}_{\overline{Y}}} \overline{{\mathcal{M}}}^{-1}_{\tau,e_{\mathfrak{p}}}\end{align*} $$

provided by Theorem 3.3.1 via projection to the top graded piece of the filtration of the $\tau $ -component of $\Omega ^1_{Y/{\mathbb {F}}}$ . More precisely, fix ${\mathfrak {p}}_0 \in S_p$ and $\tau _0 = \tau _{{\mathfrak {p}}_0,i}$ , let $\theta _0 = \theta _{{\mathfrak {p}}_0,i,e_{{\mathfrak {p}}_0}}$ and consider the morphism

$$ \begin{align*}KS_{\tau_0}^{{\mathrm{Ig}}}: \Omega^1_{{Y}^{\mathrm{Ig}}/{\mathbb{F}}}\otimes_{\mathcal{O}_{{Y}^{\mathrm{Ig}}}}\mathcal{F}^{\mathrm{Ig}} \cong \pi^*\Omega^1_{\overline{Y}/{\mathbb{F}}}\otimes_{\mathcal{O}_{{Y}^{\mathrm{Ig}}}} \mathcal{F}^{\mathrm{Ig}} \longrightarrow \pi^* ({\mathcal{A}}_{2{\mathbf{e}}_{\theta_0},-{\mathbf{e}}_{\theta_0},\mathbb{F}}) \otimes_{\mathcal{O}_{Y^{\mathrm{Ig}}}} \mathcal{F}^{\mathrm{Ig}}\end{align*} $$

induced by $KS_{\tau _0}$ , where ${\mathcal {F}}^{\mathrm {Ig}}$ is the sheaf of total fractions on ${Y}^{\mathrm {Ig}}$ . Suppose now that $f \in M_{\mathbf {k},\mathbf {l}}(U;{\mathbb {F}})$ , and write $h^{\mathbf {k}} = \prod _{\theta \in \Sigma } h_\theta ^{k_\theta }$ and $g^{\mathbf {l}} = \prod _{\theta \in \Sigma } g_\theta ^{l_\theta }$ for any choice of trivialisations $g_\theta $ of the line bundles $\overline {{\mathcal {N}}}_\theta $ on $\overline {Y}$ . We then define the section

$$ \begin{align*}\Theta_{\tau_0}^{\mathrm{Ig}}(f) := h^{\mathbf{k}} \pi^*(g^{\mathbf{l}} H_{\theta_0}) KS_{\tau_0}^{\mathrm{Ig}}(d(h^{-\mathbf{k}} \pi^*(g^{-\mathbf{l}}f))),\end{align*} $$

where

(11) $$ \begin{align} \mathbf{k}^{\prime} = \mathbf{k} + n_{\theta_0} {\mathbf{e}}_{\sigma^{-1}\theta_0} + {\mathbf{e}}_{\theta_0}\quad\text{and}\quad \mathbf{l}^{\prime} = \mathbf{l} + {\mathbf{e}}_{\theta_0}.\end{align} $$

Furthermore, the section is independent of the choices of $g_\theta $ and invariant under the action of $({\mathcal {O}}_F/{\mathfrak {q}})^\times $ and hence descends to a section of ${\mathcal {A}}_{\mathbf {k}^{\prime },\mathbf {l}^{\prime },{\mathbb {F}}} \otimes _{{\mathcal {O}}_{\overline {Y}}} {\mathcal {F}}$ , where ${\mathcal {F}}$ is the sheaf of total fractions on $\overline {Y}$ . Denoting the section $\Theta _{\tau _0}(f)$ , we have the following generalisation of [Reference Diamond and Sasaki12, Thm. 8.2.2].

Proof. We see exactly as in [Reference Diamond and Sasaki12] that $\Theta _{\tau _0}(f)$ is regular on the ordinary locus of $\overline {Y}$ ; that is, the complement of the divisor $\cup _{\theta \in \Sigma } Z_\theta $ (where $Z_\theta $ was defined in Subsection 4.2), so the theorem reduces to proving that if z is the generic point of an irreducible component of $Z_{\theta _1}$ for some $\theta _1 \in \Sigma $ , then

• • ${\operatorname {ord}}_z(\Theta _{\tau _0}(f)) \ge 0$ ,

• • if $\theta _1 = \theta _0$ , then ${\operatorname {ord}}_z(\Theta _{\tau _0}(f))> 0$ if and only if $p|k_{\theta _0}$ or ${\operatorname {ord}}_z(f)> 0$ .

Let R denote the discrete valuation ring ${\mathcal {O}}_{\overline {Y},z}$ , and for each $\tau \in \Sigma _{{\mathfrak {p}},0}$ and $\theta \in \Sigma _\tau $ , let $y_\theta = y_{\tau ,j}$ be a basis for the stalk $\overline {{\mathcal {L}}}_{\theta ,z} = \overline {{\mathcal {L}}}_{\tau ,j,z}$ over R (for $j = 1,\ldots ,e_{\mathfrak {p}}$ ). For each $\theta \in \Sigma $ , we may then write

$$ \begin{align*}H_\theta y_\theta = r_\theta y_{\sigma^{-1}\theta}^{n_\theta}\end{align*} $$

for some $r_\theta = r_{\tau ,j} \in R$ , and we let $r_\tau = \prod _{j=1}^{e_{\mathfrak {p}}} r_{\tau ,j}$ . By construction, we have $T: = (\pi _*{\mathcal {O}}_{{Y}^{{\mathrm {Ig}}}})_z = R[x_\tau ]_{\tau \in \Sigma _0} / I$ , where I is the ideal generated by

$$ \begin{align*}x_{\phi^{-1}\circ\tau}^p - r_\tau x_\tau\ \text{for}\ \tau \in \Sigma_0,\quad\text{and}\quad \prod_{\tau \in \Sigma_{{\mathfrak{p}},0}} x_\tau^{p-1} - \prod_{\tau \in \Sigma_{{\mathfrak{p}},0}}r_{\tau}^{p-1}\ \text{for}\ {\mathfrak{p}} \in S_p,\end{align*} $$

where each $x_\tau $ is the dual basis of $y_{\tau ,e_{\mathfrak {p}}}$ . We then have $h_{\tau ,e_{\mathfrak {p}}} = x_\tau y_{\tau ,e_{\mathfrak {p}}}$ (in $(\pi _*\pi ^*{\mathcal {L}}_{\tau ,e_{\mathfrak {p}}})_z$ ), from which it follows that

$$ \begin{align*}h_{\tau,j} = r_{\tau,j+1} r_{\tau,j+2} \cdots r_{\tau,e_{\mathfrak{p}}} x_\tau y_{\tau,j}\end{align*} $$

for $j=1,\ldots ,e_{{\mathfrak {p}}} - 1$ , and hence that $h^{\mathbf {k}} = \varphi _{\mathbf {k}} y^{\mathbf {k}}$ , where $y^{\mathbf {k}} = \prod _{\theta \in \Sigma } y_\theta ^{k_\theta }$ and

$$ \begin{align*}\varphi_{\mathbf{k}} = \prod_{{\mathfrak{p}} \in S_p} \prod_{\tau \in \Sigma_{{\mathfrak{p}},0}} \left( (r_\tau x_\tau)^{\sum_{\theta \in \Sigma_\tau} k_\theta} \prod_{j=1}^{e_{\mathfrak{p}}} r_{\tau,j}^{-\sum_{j^{\prime} = j}^{e_{\mathfrak{p}}} k_{\tau,j^{\prime}}} \right)\end{align*} $$

(writing $k_{\tau _{{\mathfrak {p}},i},j}$ for $k_{\theta _{{\mathfrak {p}},i,j}}$ as usual and working over the field of fractions of T).

Writing $f = \varphi _f y^{\mathbf {k}} g^{\mathbf {l}}$ , we see that

$$ \begin{align*}\Theta_{\tau_0}^{\mathrm{Ig}}(f) = KS_{\tau_0}^{\mathrm{Ig}}( r_{\theta_0} \varphi_{\mathbf{k}} d (\varphi_f\varphi_{\mathbf{k}}^{-1} ) ) y_{\theta_0}^{-1} y_{\sigma^{-1} \theta_0}^{n_{\theta_0}} y^{\mathbf{k}} g^{\mathbf{l}}.\end{align*} $$

Since $r_\tau x_\tau = x_{\phi ^{-1}\circ \tau }^p $ , we have $d(r_\tau x_\tau ) = 0$ and

(12) $$ \begin{align} \Theta_{\tau_0}(f) = KS_{\tau_0} \left(r_{\theta_0} d\varphi_f + r_{\theta_0} \varphi_f \sum_{\theta \in \Sigma} k_\theta^{\prime} \frac{dr_\theta}{r_\theta}\right) y_{\theta_0}^{-1} y_{\sigma^{-1} \theta_0}^{n_{\theta_0}} y^{\mathbf{k}} g^{\mathbf{l}}, \end{align} $$

where $k^{\prime }_\theta = k_{\tau ,j} + k_{\tau ,j+1} + \cdots k_{\tau ,e_{\mathfrak {p}}}$ if $\tau = \tau _{{\mathfrak {p}},i}$ and $\theta = \theta _{{\mathfrak {p}},i,j}$ . We are therefore reduced to showing that ${\operatorname {ord}}_z KS_{\tau _0} (dr_{\theta _1})> 0$ if and only if $\theta _1 = \theta _0$ . However, the proof of Proposition 4.2.1 shows that if y is a closed point of $Z_{\theta _1}$ , then $KS_{\tau _0}(dr_{\theta _1})$ vanishes at y if and only if $\theta _1 = \theta _0$ .

We call $\Theta _{\tau _0}$ the partial $\Theta $ -operator (indexed by $\tau _0$ ). It is immediate from its definition that the resulting map on $\mathbb {F}$ -algebras

$$ \begin{align*}\bigoplus_{\mathbf{k},\mathbf{l} \in \mathbb{Z}^\Sigma} M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}}) \longrightarrow \bigoplus_{\mathbf{k},\mathbf{l} \in \mathbb{Z}^\Sigma} M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}}),\end{align*} $$

given by the direct sum over all weights of the operators $\Theta _{\tau _0}$ , is an $\mathbb {F}$ -linear derivation; that is, that

$$ \begin{align*}\Theta_{\tau_0}(f_1f_2) = f_1 \Theta_{\tau_0}(f_2) + \Theta_{\tau_0}(f_1)f_2\end{align*} $$

for all $f_1,f_2$ in $\oplus M_{\mathbf {k},\mathbf {l}}(U;\mathbb {F})$ . It is also clear that $\Theta _{\tau _0}(H_\theta ) = 0$ for all $\theta \in \Sigma $ , and hence that $\Theta _{\tau _0}$ commutes with multiplication by partial Hasse invariants.

It is also straightforward to check that the operator $\Theta _{\tau _0}$ commutes with the Hecke action in the obvious sense and hence induces a $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ -equivariant map

$$ \begin{align*}\Theta_\tau: M_{\mathbf{k},\mathbf{l}}({\mathbb{F}}) \longrightarrow M_{\mathbf{k}^{\prime},\mathbf{l}^{\prime}}({\mathbb{F}})\end{align*} $$

where $M_{\mathbf {k},\mathbf {l}}({\mathbb {F}})$ (and $M_{\mathbf {k}^{\prime },\mathbf {l}^{\prime }}({\mathbb {F}})$ , with their $\operatorname {GL}_2({\mathbb {A}}_{F,\mathbf {f}}^{(p)})$ actions) is defined in (6) as direct limits over suitable open compact U.

Let us also make the effect of $\Theta _{\tau _0}$ on the weight $\mathbf {k}$ more explicit. Note that if $\mathbf {k} = \sum _\theta k_\theta {\mathbf {e}}_\theta $ and $\tau _0 = \tau _{{\mathfrak {p}}_0,i_0}$ , then $\mathbf {k}^{\prime } = \sum _\theta k^{\prime }_\theta {\mathbf {e}}_\theta $ , where

• • if $e_{{\mathfrak {p}}_0} = f_{{\mathfrak {p}}_0} = 1$ , then $k^{\prime }_\theta = \left \{\hspace{-4pt}\begin {array}{ll} k_\theta + p + 1, & \text {if}\ \theta = \theta _0 = \theta _{{\mathfrak {p}}_0,1,1},\\ k_\theta ,&{otherwise;}\end {array}\right .$

• • if $e_{{\mathfrak {p}}_0} = 1$ and $f_{{\mathfrak {p}}_0}> 1$ , then $k^{\prime }_\theta = \left \{\hspace{-4pt}\begin {array}{ll} k_\theta + 1, & \text {if}\ \theta = \theta _0 = \theta _{{\mathfrak {p}}_0,i_0,1},\\ k_\theta + p, & \text {if}\ \theta = \sigma ^{-1}\theta _0 = \theta _{{\mathfrak {p}}_0,i_0-1,1},\\ k_\theta ,&{otherwise;}\end {array}\right .$

• • if $e_{{\mathfrak {p}}_0}> 1$ , then $k^{\prime }_\theta = \left \{\hspace{-4pt}\begin {array}{ll} k_\theta + 1, & \text {if}\ \theta = \theta _0 = \theta _{{\mathfrak {p}}_0,i_0,e_{{\mathfrak {p}}_0}}\ \text {or}\ \theta = \sigma ^{-1}\theta _0 = \theta _{{\mathfrak {p}}_0,i_0,e_{{\mathfrak {p}}_0}-1},\\ k_\theta ,&{otherwise.}\end {array}\right .$

Remark 5.2.4. Considerations from the theory of Serre weights from the point of [Reference Diamond and Sasaki12] suggest that the above weight shifts are in a certain sense optimal. One can also define cruder partial $\Theta $ -operators by composing with multiplication by (products of) partial Hasse invariants. For example, the operator $H_{\tau _0,1}H_{\tau _0,2}\cdots H_{\tau _0,e_{{\mathfrak {p}}_0} - 1}\Theta _{\tau _0}$ is the one constructed in [Reference Deo, Dimitrov and Wiese8], and for any $j=1,\ldots ,e_{{\mathfrak {p}}_0}$ , the operator

$$ \begin{align*}H_{\tau_0,j}H^2_{\tau_0,j+1}\cdots H^2_{\tau_0,e_{{\mathfrak{p}}_0}-1}H_{\tau_0,e_{{\mathfrak{p}}_0}}\Theta_{\tau_0}\end{align*} $$

shifts the weight $\mathbf {k}$ by ${\mathbf {e}}_\theta + n_{\sigma ^{-1}\theta }{\mathbf {e}}_{\sigma ^{-1}\theta }$ , where $\theta = \theta _{{\mathfrak {p}}_0,i_0,j}$ .

6.1 Partial Frobenius endomorphisms

In order to define partial Frobenius operators on Hilbert modular forms (in Subsection 6.2), we first need to define partial Frobenius endomorphisms of Hilbert modular varieties over $\mathbb {F}$ .

Fix a prime ${\mathfrak {p}}$ dividing p and a level U, assumed as usual to be sufficiently small and prime to p. We will draw on ideas from [Reference Diamond, Kassaei and Sasaki11, §7.1] to construct an isogeny on the universal abelian variety $s: A \to S$ , where $S = \widetilde {Y}_{U,\mathbb {F}}$ .

We begin by associating Raynaud data to the line bundles ${{\mathcal {L}}}_{{\mathfrak {p}},i,e_{\mathfrak {p}}}$ over S, which we write simply as ${{\mathcal {L}}}_i$ for $i \in \mathbb {Z}/f\mathbb {Z} = \mathbb {Z}/f_{\mathfrak {p}}\mathbb {Z} $ (omitting the subscripts for the fixed ${\mathfrak {p}}$ and $j = e = e_{\mathfrak {p}}$ ). We define $f_i: {{\mathcal {L}}}^{\otimes p}_i \to {{\mathcal {L}}}_{i+1}$ to be zero, and we define $v_i:{{\mathcal {L}}}_{i+1}\to {{\mathcal {L}}}^{\otimes p}_i$ to be the morphism induced by the restriction of

$$ \begin{align*}\operatorname{Ver}^*_A: {\mathcal{H}} = {\mathcal{H}}^1_{\operatorname{dR}}(A/S)_{\mathfrak{p}} \to {\mathcal{H}}^1_{\operatorname{dR}}((\phi_S^*A)/S)_{\mathfrak{p}} = \phi_S^*{\mathcal{H}}^1_{\operatorname{dR}}(A/S)_{\mathfrak{p}} = \phi_S^*{\mathcal{H}}\end{align*} $$

to ${\mathcal {F}}_{i}^{(e)} = (s_*\Omega ^1_{A/S})_{i}$ (abbreviating subscripts $\tau _{{\mathfrak {p}},i}$ by i). Note that since the image of ${\mathcal {H}}_{i+1}$ under $\operatorname {Ver}_A^*$ is $\phi _S^*({\mathcal {F}}_{i}^{(e)})$ , the inclusions ${\mathcal {F}}_{i+1}^{(e-1)} \subset u {\mathcal {H}}_{i+1}$ and $u{\mathcal {F}}_i^{(e)} \subset {\mathcal {F}}_i^{(e-1)}$ ensure that $\operatorname {Ver}_A^*({\mathcal {F}}_{i+1}^{(e-1)}) \subset \phi _S^*({\mathcal {F}}_i^{(e-1)})$ , so the morphism $v_i$ is well-defined. We then let H denote the finite flat $({\mathcal {O}}_F/{\mathfrak {p}})$ -vector space scheme over S associated to the Raynaud data $({{\mathcal {L}}}_i, f_i, v_i)_{i \in \mathbb {Z}/f\mathbb {Z}}$ .

Recall that the Dieudonné crystal of $\ker ({\operatorname {Frob}}_A)$ is canonically isomorphic to $\Phi ^*(s_*\Omega ^1_{A/S})$ , with $F = 0$ and V induced by $\Phi ^*(\operatorname {Ver}_A^*)$ (in the notation of [Reference Berthelot, Breen and Messing2, §4.4.3]). On the other hand, the Dieudonné crystal of H is identified with $\Phi ^*(\oplus _i {{\mathcal {L}}}_i)$ with $F = \Phi ^*(\oplus _i f_i) = 0$ and $V = \Phi ^*(\oplus _i v_i)$ (as a simple special case of [Reference Diamond, Kassaei and Sasaki11, Prop. 7.1.3]). Therefore, the canonical projection $s_*\Omega ^1_{A/S} \to \oplus _{i} {{\mathcal {L}}}_i$ induces a surjective morphism of Dieudonné crystals ${\mathbb {D}}(\ker ({\operatorname {Frob}}_A)) \to {\mathbb {D}}(H)$ . As the base S is smooth over $\mathbb {F}$ , the exact contravariant functor ${\mathbb {D}}$ is fully faithful on finite flat p-group schemes over S ([Reference Berthelot and Messing3, Thm. 4.1.1]), so the surjection arises from a closed immersion $H \hookrightarrow \ker ({\operatorname {Frob}}_A)$ , and we  let

$$ \begin{align*}\alpha: A \longrightarrow A^{\prime} := A/H\end{align*} $$

denote the resulting isogeny of abelian varieties over S. Note that $A^{\prime }$ naturally inherits an ${\mathcal {O}}_F$ -action $\iota ^{\prime }$ from the action $\iota $ on A.

Let ${\mathcal {I}}$ denote the image of the morphism $\alpha ^*: {\mathcal {H}}^1_{\operatorname {dR}}(A^{\prime }/\widetilde {Y}_{U,\mathbb {F}})_{\mathfrak {p}} \to {\mathcal {H}}$ . By construction, we have the exact sequence

$$ \begin{align*}\begin{array}{ccccccc} {\mathbb{D}}(A^{\prime}[p])_{S} & \longrightarrow & {\mathbb{D}}(A[p])_{S} & \longrightarrow & {\mathbb{D}}(H)_{S} & \longrightarrow & 0\\ \parallel\wr&&\parallel\wr && \parallel\wr &&\\ {\mathcal{H}}^1_{\operatorname{dR}}(A^{\prime}/S) &\stackrel{\alpha^*}{\longrightarrow} &{\mathcal{H}}^1_{\operatorname{dR}}(A/S) & \stackrel{\operatorname{Ver}_A^*}{\longrightarrow} & \bigoplus_{i} \phi_S^*({\mathcal{F}}_\tau^{(e)}/{\mathcal{F}}_\tau^{(e-1)}) & \longrightarrow & 0, \end{array}\end{align*} $$

showing that ${\mathcal {I}} = \oplus _{i} {\mathcal {I}}_i$ , where ${\mathcal {I}}_i$ is the pre-image of $\phi _S^*({\mathcal {F}}_{i-1}^{(e)})$ under $\operatorname {Ver}_{A,i}^*:{\mathcal {H}}_i \to \phi _S^*({\mathcal {H}}_{i-1})$ . Note in particular that $u{\mathcal {H}}_i \subset {\mathcal {I}}_i$ for all i, so that $H \subset A[{\mathfrak {p}}]$ and there is a unique isogeny $\beta : {\mathfrak {p}} \otimes _{{\mathcal {O}}_F} A^{\prime } \longrightarrow A$ such that $\alpha \circ \beta $ is the canonical isogeny ${\mathfrak {p}}\otimes _{{\mathcal {O}}_F} A^{\prime } \to A^{\prime }$ .

We now equip $A^{\prime }$ with auxiliary data corresponding to an element of $\widetilde {Y}_{U,\mathbb {F}}(S)$ .

Since $\alpha $ induces isomorphisms $T^{(p)}(A_{\overline {s}}) \stackrel {\sim }{\longrightarrow } T^{(p)}(A^{\prime }_{\overline {s}})$ for all geometric points $\overline {s}$ of S, we immediately have a level $U^p$ structure $\eta ^{\prime }$ on $A^{\prime }$ inherited from A.

Next we claim that the quasi-polarisation $\lambda $ on A induces an isomorphismFootnote ¹⁰ ${\mathfrak {p}}{\mathfrak {c}}{\mathfrak {d}} \otimes _{{\mathcal {O}}_F} A^{\prime } \to (A^{\prime })^\vee $ or, equivalently, $A^{\prime } \to {\mathfrak {p}}^{-1}{\mathfrak {c}}^{-1}{\mathfrak {d}}^{-1} \otimes _{{\mathcal {O}}_F} (A^{\prime })^\vee $ , which amounts to the claim that H corresponds to the kernel of

$$ \begin{align*}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes \beta^\vee: {\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} A^\vee \longrightarrow {\mathfrak{p}}^{-1}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F}( A^{\prime})^\vee\end{align*} $$

under the isomorphism induced by $\lambda $ . Denoting this kernel by I, we have that H and I are finite flat group schemes over S of the same rank, so it suffices to prove that the composite

$$ \begin{align*}I \longrightarrow {\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} A^\vee[p] \stackrel{\sim}{\longrightarrow} A[p] \longrightarrow A^{\prime}[p]\end{align*} $$

is trivial. Taking Dieudonné modules, this in turn amounts to the vanishing of the composite

$$ \begin{align*}{\mathbb{D}}(A^{\prime}[p])_{S} \longrightarrow {\mathbb{D}}(A[p])_{S} \longrightarrow {\mathbb{D}}({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} A^\vee[p])_{S} \longrightarrow {\mathbb{D}}(I)_{S}.\end{align*} $$

We have already noted that the image of the first map has ${\mathfrak {p}}$ -component $\oplus _{i} {\mathcal {I}}_i$ ; on the other hand, the kernel of the last map is the image of the map

$$ \begin{align*}{\mathbb{D}}({\mathfrak{p}}^{-1}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} (A^{\prime})^\vee[p])_{S} \longrightarrow {\mathbb{D}}({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} A^\vee[p])_{S}\end{align*} $$

corresponding to the adjoint of

$$ \begin{align*}\beta^*: {\mathcal{H}}^1_{\operatorname{dR}}(A/S) \to {\mathcal{H}}^1_{\operatorname{dR}}(({\mathfrak{p}}\otimes_{{\mathcal{O}}_F} A^{\prime})/S) \cong {\mathfrak{p}}^{-1} \otimes_{{\mathcal{O}}_F} {\mathcal{H}}^1_{\operatorname{dR}}(A^{\prime}/S)\end{align*} $$

under the canonical isomorphisms

$$ \begin{align*}\begin{array}{l} {\mathbb{D}}({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} A^\vee[p])_S \cong {\mathcal{H}}^1_{\operatorname{dR}}(({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} A^\vee / S) \\ \qquad\qquad \qquad\cong {\mathcal{H}om}_{{\mathcal{O}}_S}({\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_F} {\mathcal{H}}^1_{\operatorname{dR}}(A/ S), {\mathcal{O}}_{S}) \cong {\mathcal{H}om}_{{\mathcal{O}}_F\otimes {\mathcal{O}}_{S}}({\mathcal{H}}^1_{\operatorname{dR}}(A/ S), {\mathcal{O}}_F \otimes {\mathcal{O}}_{S}) \end{array}\end{align*} $$

and, similarly,

$$ \begin{align*}{\mathbb{D}}({\mathfrak{p}}^{-1}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1} \otimes_{{\mathcal{O}}_F} (A^{\prime})^\vee[p])_{S } \cong {\mathcal{H}om}_{{\mathcal{O}}_F\otimes {\mathcal{O}}_{S}}({\mathfrak{p}}^{-1} \otimes_{{\mathcal{O}}_F}{\mathcal{H}}^1_{\operatorname{dR}}(A^{\prime}/S), {\mathcal{O}}_F \otimes {\mathcal{O}}_S)\end{align*} $$

obtained from duality. We are therefore reduced to proving that ${\mathcal {I}}_i$ is orthogonal to the kernel of $\beta _i^*$ for each $i \in \mathbb {Z}/f\mathbb {Z} $ under the pairing $\langle \cdot ,\cdot \rangle _i$ defined by (4). Note, however, that the kernel of $\beta _i^*$ is $u^{e-1} {\mathcal {I}}_i$ , as can be seen, for example, from the commutative diagram

of $W(\overline {{\mathbb {F}}}_p)[u]$ -modules for ${\overline {s}} \in S(\overline {{\mathbb {F}}}_p)$ . Finally, the orthogonality of ${\mathcal {I}}_i$ and $u^{e-1}{\mathcal {I}}_i$ is immediate from that of ${\mathcal {F}}_{i-1}^{(e-1)}$ and $u^{-1}{\mathcal {F}}_{i-1}^{(e-1)}$ provided by Lemma 3.1.1, completing the proof of the claim. We may then define the quasi-polarisation on $A^{\prime }$ by $\alpha ^*(\lambda ^{\prime }) = \delta \lambda $ for any totally positive generator $\delta = \delta _{\mathfrak {p}}$ of ${\mathfrak {p}}{\mathcal {O}}_{F,(p)}$ , so that $\lambda ^{\prime }$ induces an isomorphism ${\mathfrak {c}}^{\prime }{\mathfrak {d}}\otimes _{{\mathcal {O}}_F} A^{\prime } \stackrel {\sim }{\longrightarrow } (A^{\prime })^\vee $ where ${\mathfrak {c}}^{\prime } = \delta ^{-1}{\mathfrak {p}}{\mathfrak {c}}$ .

Finally, we define a Pappas–Rapoport filtration on ${\mathcal {F}}^{\prime }_\tau := (s^{\prime }_*\Omega ^1_{A^{\prime }/S})_\tau $ for all $\tau \in \Sigma _0$ .

First, note that if $\tau \not \in \Sigma _{{\mathfrak {p}},0}$ , then $\alpha ^*$ induces an isomorphism ${\mathcal {F}}^{\prime }_\tau \simeq {\mathcal {F}}_\tau ^{(e)}$ , and we define $({\mathcal {F}}_\tau ^{\prime })^{(j)}$ as the pre-image of ${\mathcal {F}}_\tau ^{(j)}$ .

Suppose now that $\tau = \tau _{{\mathfrak {p}},i}$ . Recall from the construction of $A^{\prime } = A/H$ that $\operatorname {Ver}_A^*({\mathcal {F}}_{i}^{(e-1)}) \subset \phi _S^*({\mathcal {F}}_{i-1}^{(e-1)})$ , so we have ${\mathcal {F}}_i^{(e-1)} \subset {\mathcal {I}}_i$ . It follows that $(\alpha _i^*)^{-1}({\mathcal {F}}_i^{(e-1)})$ is a subbundle of ${\mathcal {H}}^{\prime }_i := {\mathcal {H}}^1_{\operatorname {dR}}(A^{\prime })_i$ of the same rank as ${\mathcal {F}}^{\prime }_i$ , namely, e. Furthermore, we  have

$$ \begin{align*}\begin{array}{rl} \phi_S^*(\alpha^*({\mathcal{F}}_i^{\prime})) = (\phi_S^*(\alpha))^* ({\mathcal{F}}_i^{\prime}) = &(\phi_S^*(\alpha))^*(\operatorname{Ver}_{A^{\prime}}^*({\mathcal{H}}_{i+1}^{\prime}))\\ = &\operatorname{Ver}_A^*(\alpha^*({\mathcal{H}}_{i+1}^{\prime})) = \operatorname{Ver}_A^*({\mathcal{I}}_{i+1}) \subset \phi_S^*({\mathcal{F}}_i^{(e+1)}),\end{array}\\[-15pt]\end{align*} $$

so in fact ${\mathcal {F}}_i^{\prime } \subset (\alpha _i^*)^{-1}({\mathcal {F}}_i^{(e-1)})$ , and hence equality holds. We thus obtain an exact sequence

$$ \begin{align*}0 \to \ker(\alpha_i^*) \longrightarrow {\mathcal{F}}^{\prime}_i \stackrel{\alpha_i^*}{\longrightarrow} {\mathcal{F}}_i^{(e-1)} \to 0.\\[-15pt]\end{align*} $$

We may thus define a Pappas–Rapoport filtration on ${\mathcal {F}}_i^{\prime }$ by setting

$$ \begin{align*}({\mathcal{F}}_i^{\prime})^{(j)} = (\alpha_i^*)^{-1}({\mathcal{F}}_i^{(j-1)})\\[-15pt]\end{align*} $$

for $j=1,\ldots ,e$ , so, in particular, $({\mathcal {F}}_i^{\prime })^{(1)} = \ker (\alpha _i^*)$ .

We now define $\widetilde {\Phi }_{\mathfrak {p}}:\widetilde {Y}_{U,\mathbb {F}} = S \to \widetilde {Y}_{U,\mathbb {F}}$ to be the endomorphism corresponding to the data $(A^{\prime },\iota ^{\prime },\lambda ^{\prime },\eta ^{\prime },({\mathcal {F}}^{\prime })^\bullet )$ . Note that $\widetilde {\Phi }_{\mathfrak {p}}$ depends on the choice of $\delta $ in the definition of $\lambda ^{\prime }$ ; however, it is straightforward to check that $\widetilde {\Phi }_{\mathfrak {p}}$ is compatible with the ${\mathcal {O}}_{F,(p),+}^\times $ -action on S and descends to an endomorphism $\Phi _{\mathfrak {p}}$ of $\overline {Y}_U$ which is independent of this choice. We call $\widetilde {\Phi }_{\mathfrak {p}}$ (respectively $\Phi _{\mathfrak {p}}$ ) the partial Frobenius endomorphism (indexed by ${\mathfrak {p}}$ ) of $\widetilde {Y}_{U,\mathbb {F}}$ (respectively $\overline {Y}_U$ ); the terminology is justified by the next proposition.

For the statement of the proposition, we also define the endomorphism $\widetilde {\Phi }$ of $S = \widetilde {Y}_{U,\mathbb {F}}$ corresponding to the data $\phi _S^*({\underline {A}}) = (\phi _S^*A, \phi _S^*\iota ,\phi _S^*\lambda ,\phi _S^*\eta ,(\phi _S^*{\mathcal {F}})^\bullet )$ , where $(\phi _S^*{\mathcal {F}})^\bullet $ is the collection of filtrations on the vector bundles

$$ \begin{align*}((\phi_S^*s)_*(\Omega^1_{(\phi_S^*A)/S}))_\tau = (\phi_S^*(s_*\Omega^1_{A/S}))_\tau = \phi_S^*((s_*\Omega^1_{A/S})_{\phi^{-1}\circ\tau})\\[-15pt]\end{align*} $$

given by $(\phi _S^*{\mathcal {F}})_\tau ^{(j)} = \phi _S^*({\mathcal {F}}_{\phi ^{-1}\circ \tau }^{(j)})$ . Note that $\widetilde {\Phi }$ is not the absolute Frobenius $\phi _S$ on S (unless $\mathbb {F} = \mathbb {F}_p$ ), but we may write $\phi _S = \widetilde {\Phi }\circ \widetilde {\epsilon }$ where $\widetilde {\epsilon }$ is the isomorphism defined by the commutative diagram

where the square is Cartesian and $\varepsilon $ is the inverse of the isomorphism associated to $\phi ^*A = A \times _{\mathbb {F},\phi } \mathbb {F}$ with the evident auxiliary data. We thus have an isomorphism $\widetilde {\epsilon }^*A \cong A$ compatible with $\iota $ , $\lambda $ and $\eta $ and inducing $\widetilde {\epsilon }^*{\mathcal {F}}_\tau ^{(j)} \cong {\mathcal {F}}_{\phi \circ \tau }^{(j)}$ for all $\tau $ and j. (Note also that $\widetilde {\Phi }$ may be viewed as the base change of the absolute Frobenius on the descent of S to $\mathbb {F}_p$ defined by the diagram.)

The endomorphism $\widetilde {\Phi }$ is compatible with the ${\mathcal {O}}_{F,(p),+}$ -action on $S = \widetilde {Y}_{U,\mathbb {F}}$ , and we let $\Phi $ denote the resulting endomorphism of $\overline {Y}_U$ . Similarly, $\widetilde {\epsilon }$ descends to a $\phi $ -linear automorphism $\epsilon $ of $\overline {Y}_U$ such that the absolute Frobenius on $\overline {Y}_U$ is $\Phi \circ \epsilon $ .