2.1 The PEL datum

We work with the integral PEL datum, in the sense of [Reference Kottwitz33, Chapter 5], defined as follows:

1. (1) The simple ${\mathbb Q}$ -algebra B is E, with ${\mathcal O_E}$ as its maximal ${\mathbb Z}$ -order.

2. (2) The positive involution ${}^\ast $ on E is the complex conjugation. The fixed field $F_0$ is ${\mathbb Q}$ .

3. (3) We let $V \cong E^n$ and $\Lambda = \mathcal {O}_E^n$ , the canonical ${\mathcal O_E}$ -lattice, with canonical basis $\{e_1, e_2, \dots , e_n\} \subset \Lambda $ .

4. (4) Take the Hermitian pairing $\left (\cdot , \cdot \right ) \colon V \times V \to E$ of signature $(n-1, 1)$ given by the diagonal matrix $I_{n-1,1}= {\mathrm {diag}}(1, \dots , 1, -1)$ , with respect to the canonical basis on V. It restricts to a perfect pairing $\left (\cdot , \cdot \right ) \colon \Lambda \times \Lambda \to {\mathcal O_E}$ . Then, $ \left < \cdot , \cdot \right> {:=}q {\mathrm {T}}_{E/{\mathbb Q}}(\delta _{E/{\mathbb Q}}^{-1}(\cdot , \cdot ))$ is a perfect alternating ${\mathbb Q}$ -linear pairing such that $\left <\alpha u, v\right> =\left <u, \overline {\alpha }v\right>,\, u, v \in V, \alpha \in {\mathcal O_E}$ , whose restriction to $\Lambda $ induces a perfect ${\mathbb Z}$ -linear pairing $\Lambda \times \Lambda \to {\mathbb Z}$ . By adjunction, $\left ( \cdot , \cdot \right )$ defines an involution ${}^\ast $ of $\operatorname {\mathrm {End}}_E(V)$ , which restricts to an involution of $\operatorname {\mathrm {End}}_{\mathcal O_E}(\Lambda )$ , compatible with the conjugation on $E \subset \operatorname {\mathrm {End}}_E(V)$ . With respect to the canonical basis, this involution is given by $M \mapsto I_{n-1,1}{}^t\overline {M}I_{n-1,1}$ .

5. (5) We have isomorphisms $\operatorname {\mathrm {End}}_E(V) \otimes _{\mathbb Q} {\mathbb R} \cong \operatorname {\mathrm {End}}_{E\otimes _{\mathbb Q} {\mathbb R}}(V\otimes _{\mathbb Q} {\mathbb R}) \cong {\mathrm {M}}_n({\mathbb C}),$ afforded by the canonical basis and $\sigma $ . We take $h \colon {\mathbb C} \to \operatorname {\mathrm {End}}_E(V) \otimes _{\mathbb Q} {\mathbb R}$ , via these isomorphisms, to be the map of ${\mathbb R}$ -algebras defined by $z \mapsto {\mathrm {diag}}(z, \dots , z, \overline {z})$ . Then, $(\Lambda , h, \left <\cdot , \cdot \right>)$ is an integral polarized Hodge structure.

These data satisfy some properties which we now recall. The Hodge structure on ${V_{\mathbb C} = V \otimes _{\mathbb Q} {\mathbb C} = V_1 \oplus V_2}$ induced by h can be described explicitly. Consider the primitive idempotents

$$\begin{align*}e_\sigma = \frac{1}{2}( 1\otimes 1 + \delta_{E/{\mathbb Q}} \otimes \delta_{E/{\mathbb Q}}^{-1}), \quad e_{\overline{\sigma}} = \frac{1}{2}( 1 \otimes 1 - \delta_{E/{\mathbb Q}} \otimes \delta_{E/{\mathbb Q}}^{-1}), \end{align*}$$

in the ring $E \otimes _{\mathbb Q} E \subset E \otimes _{\mathbb Q} {\mathbb C}$ . Then

$$ \begin{align*} V_1 = {\mathrm{Span}}_{\mathbb C}\left < e_\sigma\cdot e_1, \dots, e_\sigma\cdot e_{n-1}, e_{\overline{\sigma}}\cdot e_n \right>, V_2 = {\mathrm{Span}}_{\mathbb C}\left < e_{\overline{\sigma}}\cdot e_1, \dots, e_{\overline{\sigma}}\cdot e_{n-1}, e_\sigma \cdot e_n \right>. \end{align*} $$

Both $V_1$ and $V_2$ are defined over E, in the sense of [Reference Lan34, Definition 1.1.2.7]. In fact, they are integral: $e_\sigma , e_{\overline {\sigma }} \in {\mathcal O_E} \otimes _{\mathbb Z} {\mathcal {O}}_{E, {\mathrm {ur}}}$ and we can decompose $\Lambda \otimes _{{\mathbb Z}} {\mathcal {O}}_{E, {\mathrm {ur}}} \cong \Lambda _1 \oplus \Lambda _2$ , with $\Lambda _i \otimes _{{\mathcal {O}}_{E, {\mathrm {ur}}}} E \cong V_i, \,i=1,2$ . This shows that E is reflex field of the datum $(E, {}^\ast , V, \left <\cdot ,\cdot \right>, h)$ (see [Reference Kottwitz33, Chapter 5]).

Consider the action of any $\alpha \in {\mathcal O_E}$ on $\Lambda _1$ and $V_1$ . We have the characteristic polynomial

$$\begin{align*}p_\alpha(X) {:=}q \det(X-\alpha|_{V_1}) = (X - \sigma(\alpha))^{n-1} (X - \overline{\sigma}(\alpha)) \in {\mathcal O_E}[X]. \end{align*}$$

These polynomials are necessary to express Kottwitz’s determinant condition (see [Reference Kottwitz33, Chapter 5] and [Reference Lan34, Section 1.3.4]).

Remark 2.1 (CM decomposition)

We have decompositions of $V \otimes _{\mathbb Q} E$ and $\Lambda \otimes _{\mathbb Z} {{\mathcal {O}}_{E, {\mathrm {ur}}}}$ given by $e_\sigma , e_{\overline {\sigma }}$ ,

$$ \begin{align*} V \otimes_{\mathbb Q} E &\cong V_\sigma \oplus V_{\overline{\sigma}} {:=}q e_\sigma (V \otimes_{\mathbb Q} E) \oplus e_{\overline{\sigma}} (V \otimes_{\mathbb Q} E),\\ \Lambda \otimes_{\mathbb Z} {{\mathcal{O}}_{E, {\mathrm{ur}}}} &\cong \Lambda_\sigma \oplus \Lambda_{\overline{\sigma}} {:=}q e_\sigma(\Lambda \otimes_{\mathbb Z} {{\mathcal{O}}_{E, {\mathrm{ur}}}}) \oplus e_{\overline{\sigma}}(\Lambda \otimes_{\mathbb Z} {{\mathcal{O}}_{E, {\mathrm{ur}}}}). \end{align*} $$

We call this the CM decomposition of V and $\Lambda $ . More generally, we have CM decompositions for ${\mathcal O_E} \otimes _{\mathbb Z} {\mathcal {O}}_{E, {\mathrm {ur}}}$ -module or sheaves over ${\mathcal {O}}_{E, {\mathrm {ur}}}$ -schemes with a linear ${\mathcal O_E}$ -action.

2.2 The reductive group

From $({\mathcal O_E}, \overline {\cdot }, \Lambda , \left <\cdot , \cdot \right>, h)$ , we obtain a group scheme $\mathbf {G}$ over ${\mathbb Z}$ whose R-points, for R a commutative ring, are

$$ \begin{align*} \mathbf{G}(R) &= \mathbf{GU}(\Lambda, \left(\cdot, \cdot\right))(R) {:=}q \{g \in \operatorname{\mathrm{End}}_{{\mathcal{O}}_{E} \otimes R}(\Lambda \otimes_{{\mathbb Z}} R) \mid gg^\ast = \nu(g) \in R^\times \} \\ &= \{(g, \nu(g)) \in \operatorname{\mathrm{End}}_{{\mathcal{O}}_{E} \otimes R}(\Lambda_R) \times R^\times \mid \left(gu, gv\right) = \nu(g) \left(u, v\right),\, \forall u, v \in \Lambda_R\}. \end{align*} $$

The morphism $\nu \colon \mathbf {G} \to {\mathbb {G}}_m$ is the similitude factor. The kernel of $\nu $ is denoted $\mathbf {G}_1$ . We have the following.

Lemma 2.2 Let $s \colon \operatorname {\mathrm {Spec}} k \to \operatorname {\mathrm {Spec}} {{\mathbb Z}[1/2D]}$ be a morphism with k an algebraically closed field. We have natural isomorphisms of group schemes:

$$\begin{align*}\mathbf{G}_{{\mathcal{O}}_{E, {\mathrm{ur}}}} \cong {\mathrm{GL}}_{n} \times_{{\mathcal{O}}_{E, {\mathrm{ur}}}} \mathbb{G}_{m}, \, \mathbf{G}_{1, {\mathcal{O}}_{E, {\mathrm{ur}}}} \cong {\mathrm{GL}}_{n} \end{align*}$$

and similarly for $\mathbf {G}_s$ and $\mathbf {G}_{1,s}$ . In particular, $\mathbf {G}$ and $\mathbf {G}_1$ are reductive over $\operatorname {\mathrm {Spec}} {{\mathbb Z}[1/2D]}$ .

Moreover, under the isomorphism $\mathbf {G}_{{\mathcal {O}}_{E, {\mathrm {ur}}}} \cong {\mathrm {GL}}_{n} \times \mathbb {G}_{m}$ , the Levi subgroup H of the parabolic fixing the filtration $\Lambda _1 \subseteq \Lambda \otimes _{\mathbb Z} {\mathcal {O}}_{E, {\mathrm {ur}}} \cong \Lambda _1 \oplus \Lambda _2$ corresponds to ${\mathrm {GL}}_{n-1} \times {\mathrm {GL}}_1 \times \mathbb {G}_{m}$ . The similarly defined Levi $H_1$ of $\mathbf {G}_1$ is isomorphic after base change to ${\mathrm {GL}}_{n-1} \times {\mathrm {GL}}_1$ .

The real points of $\mathbf {G}, \mathbf {G}_1$ give the classical unitary groups $\mathbf {G}({\mathbb R}) = {\mathrm {GU}}(n-1, 1), \mathbf {G}_1({\mathbb R}) = {\mathrm {U}}(n-1,1)$ and one can see that $H({\mathbb R}) = G(U(n-1) \times U(1)), H_1({\mathbb R}) = U(n-1) \times U(1)$ .

We can restrict h to a morphism of algebraic groups $h \colon {\mathbb {S}} \to \mathbf {G}_{\mathbb R}$ , denoted again by h. Then, $(\mathbf {G}_{\mathbb Q}, h)$ is a Shimura datum, as defined in [Reference Deligne10, Section 1.5] and [Reference Deligne11, Section 2.1.1].

2.3 The moduli problem

We formulate the PEL moduli problem we are interested in. We work with a neat, p-hyperspecial level $K \subseteq \mathbf {G}({\mathbb {A}}^\infty )$ (see [Reference Lan34, Definition 1.4.1.8, Theorem 1.4.1.12]).

Let S be a scheme defined over $\mathcal {O}_{E, (p)} {:=}q {\mathcal O_E} \otimes _{\mathbb Z} {\mathbb Z}_{(p)}$ . We denote the category of such schemes by ${\underline {{\mathrm {Sch}}}}_{\mathcal {O}_{E, (p)}}$ . Recall that we assume all schemes to be locally Noetherian. To $S,$ we associate a quadruple ${\underline {A}} = (A, \lambda , \iota , \eta _K)$ :

1. (1) $A \to S$ is an abelian scheme.

2. (2) $\lambda \colon A \to A^\vee $ is a prime-to-p polarization.

3. (3) $\iota \colon {\mathcal O_E} \to \operatorname {\mathrm {End}}_S(A)$ is a ring homomorphism such that:

   1. (a) The Rosati relation $\lambda \iota (\overline {\alpha }) = \iota (\alpha )^\vee \lambda , \, \alpha \in {\mathcal O_E} $ holds.

   2. (b) For any $\alpha \in {\mathcal O_E}$ , we have $\det (X-\iota (\alpha )|_{{\mathrm {Lie}}(A/S)}) = p_\alpha (X) \in {\mathcal O_E}[X] \subset \mathcal {O}_S[X].$

4. (4) $\eta _K$ is an ${\mathcal O_E}$ -linear integral level K-structure (see [Reference Lan34, Section 1.3.7, Definition 1.3.7.8]).

An isomorphism of two such quadruples is an isomorphism of the underlying abelian schemes compatible with the remaining data in the natural way. Under our assumptions on K, the functor

$$ \begin{align*} {\underline{{\mathrm{Sch}}}}_{\mathcal{O}_{E, (p)}} & \longrightarrow {\underline{{\mathrm{Set}}}},\\ S & \longmapsto \{( A, \lambda, \iota, \eta_K )\}/_{\cong}, \end{align*} $$

is represented by a smooth, quasi-projective scheme $\mathcal {S}_K \in {\underline {{\mathrm {Sch}}}}_{\mathcal {O}_{E, (p)}}$ of relative dimension $n-1$ (see [Reference Lan34, Chapter 2]).

2.4 Hodge and determinant sheaves

For $f \colon G \to S$ a group scheme, we write

$$\begin{align*}{\underline{\omega}}_{G/S} {:=}q e^\ast(\Omega^1_{G/S}), \end{align*}$$

where e is the identity of G. We call the sheaf ${\underline {\omega }}_{G/S}$ the Hodge sheaf of $G/S$ . If G and S are clear from the context, we simply write ${\underline {\omega }}$ . If $G=A$ is an abelian scheme, then ${\underline {\omega }}_{A/S} \cong f_\ast (\Omega ^1_{A/S})$ .

Let ${\underline {A}} = (A, \lambda , \iota , \eta _{N}) \in \mathcal {S}_K(S)$ , for $S \in {\underline {{\mathrm {Sch}}}}_{{\mathcal {O}}_{E,(p)}}$ . Then, ${\underline {\omega }}_{A/S}$ is locally free of rank n with an action of ${\mathcal O_E}$ of signature $(n-1, 1)$ , by which we mean that

$$\begin{align*}{\underline{\omega}}_{A/S, \sigma} {:=}q e_\sigma \cdot {\underline{\omega}}_{A/S}, \quad {\underline{\omega}}_{A/S, \overline{\sigma}} {:=}q e_{\overline{\sigma}} \cdot {\underline{\omega}}_{A/S}, \end{align*}$$

are locally free $\mathcal {O}_{\mathcal {S}_K}$ -sheaves of ranks $n-1$ and $1$ , respectively. This is the CM decomposition of ${\underline {\omega }}_{A/S}$ (see Remark 2.1). We call ${\underline {\omega }}_{A/S, \sigma }$ , ${\underline {\omega }}_{A/S, \overline {\sigma }}$ the $\sigma $ , ${\overline {\sigma }}$ -components of the Hodge sheaf, respectively.

We recall some basic facts concerning the de Rham cohomology of abelian schemes, following [Reference Berthelot, Breen and Messing4, 2.5.1]. For $f \colon X \to S$ a smooth morphism of schemes of relative dimension g, we have the de Rham complex $(\Omega _{X/S}^{\bullet }, d).$ The i-th de Rham cohomology of $X/S$ is defined as $H^i_{\mathrm {dR}}(X/S) {:=}q R^i f_\ast \Omega _{X/S}^{\bullet }$ . We have the following.

We are interested in the case $p+q=1$ of Proposition 2.3(2) (see [Reference Berthelot, Breen and Messing4, Lemma 2.5.3]). By [Reference Berthelot, Breen and Messing4, Section 5.1.1], $R^1f_\ast {\mathcal {O}}_A \cong {\underline {\omega }}_{A^\vee /S}^\vee {:=}q {\underline {{\mathrm {Hom}}}}_{{\mathcal {O}}_S}({\underline {\omega }}_{A^\vee /S}, {\mathcal {O}}_S).$ In particular, we have the short exact sequence

(2.1) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{A/S} \longrightarrow H^1_{\mathrm{dR}}(A/S) \longrightarrow {\underline{\omega}}_{A^\vee/S}^\vee \longrightarrow 0. \end{align} $$

From this and Proposition 2.3(1), we deduce that $H^1_{\mathrm {dR}}(A/S)$ has rank $2n$ . Using the prime-to-p polarization $\lambda $ , we can identify $\omega _{A^\vee /S}^\vee $ with $\omega _{A/S}^\vee $ to get

(2.2) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{A/S} \longrightarrow H^1_{\mathrm{dR}}(A/S) \longrightarrow {\underline{\omega}}_{A/S}^\vee \longrightarrow 0. \end{align} $$

We call (2.1) the natural Hodge filtration, as opposed to (2.2), which we call the polarized Hodge filtration. We simply call the short exact sequences (2.1) and (2.2) the Hodge filtration when it is clear from the context to which one we are referring. The action of ${\mathcal O_E}$ splits the Hodge filtration according to the CM decomposition of its terms, giving

(2.3) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{A/S, \sigma} \longrightarrow H^1_{\mathrm{dR}}(A/S)_{\sigma} \longrightarrow{\underline{\omega}}_{A^\vee/S,{\sigma}}^\vee \longrightarrow 0, \end{align} $$

(2.4) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{A/S, \overline{\sigma}} \longrightarrow H^1_{\mathrm{dR}}(A/S)_{\overline{\sigma}} \longrightarrow {\underline{\omega}}_{A^\vee/S, {\overline{\sigma}}}^\vee \longrightarrow 0, \end{align} $$

where the action of ${\mathcal O_E}$ on ${\underline {\omega }}_{A^\vee /S}$ is induced by the action of ${\mathcal O_E}$ on $A^\vee $ given by $a \mapsto \iota (a)^\vee $ . By the Rosati relation, $\lambda $ gives isomorphisms ${\underline {\omega }}_{A^\vee /S, \sigma } \cong {\underline {\omega }}_{A/S, \overline {\sigma }}, {\underline {\omega }}_{A^\vee /S, \overline {\sigma }} \cong {\underline {\omega }}_{A/S, {\sigma }}$ . In particular, splitting (2.2) according to the ${\mathcal O_E}$ -action, we obtain

(2.5) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{A/S, \sigma} \longrightarrow H^1_{\mathrm{dR}}(A/S)_\sigma \longrightarrow {\underline{\omega}}_{A/S,\overline{\sigma}}^\vee \longrightarrow 0, \end{align} $$

(2.6) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{A/S, \overline{\sigma}} \longrightarrow H^1_{\mathrm{dR}}(A/S)_{\overline{\sigma}} \longrightarrow {\underline{\omega}}_{A/S, {\sigma}}^\vee \longrightarrow 0. \end{align} $$

We call (2.3) and (2.4) the $\sigma $ and $\overline {\sigma }$ -components, respectively, of the natural Hodge filtration. Similarly, we call (2.5) and (2.6) the $\sigma $ and $\overline {\sigma }$ -components, respectively, of the polarized Hodge filtration. From (2.5) and (2.6), given that ${\underline {\omega }}_{A/S, \sigma }$ and ${\underline {\omega }}_{A/S, \overline {\sigma }}$ have ranks $n-1$ and $1$ , respectively, we can deduce that $H^1_{\mathrm {dR}}(A/S)_\sigma $ and $H^1_{\mathrm {dR}}(A/S)_{\overline {\sigma }}$ both have rank n. From the Hodge filtration (2.2), we deduce the isomorphism $\det H^1_{\mathrm {dR}}(A/S) \cong {\mathcal {O}}_S$ , which depends on the choice of the polarization $\lambda $ . We call $\delta _{A/S} {:=}q \det H^1_{\mathrm {dR}}(A/S)$ the determinant sheaf. Write $\delta _{A/S, \sigma } {:=}q \det (H^1_{\mathrm {dR}}(A/S)_\sigma ), \delta _{A/S, \overline {\sigma }} {:=}q \det (H^1_{\mathrm {dR}}(A/S)_{\overline {\sigma }}).$ Notice that $\delta _{A/S, \sigma } \otimes \delta _{A/S, \overline {\sigma }} \cong \delta _{A/S}$ . In particular, the choice of a polarization will induce an isomorphism $\delta _{A/S, \sigma }^{-1} \cong \delta _{A/S, \overline {\sigma }}$ . Taking determinants of (2.3) and (2.4), we deduce

(2.7) $$ \begin{align} \det {\underline{\omega}}_{A/S, \sigma} \cong \delta_{A/S,\sigma} \otimes {\underline{\omega}}_{A^\vee/S, \sigma}, \quad {\underline{\omega}}_{A/S, \overline{\sigma}} \cong \delta_{A/S,\overline{\sigma}} \otimes \det {\underline{\omega}}_{A^\vee/S, \overline{\sigma}}, \end{align} $$

identifications which do not involve the polarization. By making use of $\lambda $ instead, one can deduce similar isomorphisms from (2.5) and (2.6). We call $\delta _{A/S, \sigma }$ and $\delta _{A/S, \overline {\sigma }}$ the $\sigma $ and $\overline {\sigma }$ -determinant sheaf, respectively.

2.5 The Weyl modules of ${\mathrm {GL}}_m$

We recall the description of a natural class of algebraic representations ${\underline {{\mathrm {Re}}}}{\mathrm {p}}_R({\mathrm {GL}}_m)$ of ${\mathrm {GL}}_{m, R}$ , $m \geq 2$ , into locally free finite rank R-modules, for R a Dedekind domain. Our main reference will be [Reference Jantzen25]. Until the end of this discussion, we write $G = {\mathrm {GL}}_{m, R}$ , to ease notations.

Let $B \subset G$ be the Borel subgroup of upper-triangular matrices, $T \subset B$ the diagonal torus, and $U \subset B$ the unipotent radical of B. We denote by ${B}^- \supset {U}^-$ the Borel of G opposite to B with respect to T, with its unipotent radical. Write $X(T) {:=}q {\mathrm {Hom}}(T, {\mathbb {G}}_m)$ for the group of characters of T. We are interested in representations associated with dominant weights of G with respect to $B \supset T$ , which form the submonoid $X(T)_+ \subset X(T)$ . Through the usual isomorphism $X(T) \cong {\mathbb Z}^m$ , dominant weights correspond to tuples $(k_1, k_2, \dots , k_m) \in {\mathbb Z}^m$ such that $k_i \geq k_{i+1}$ . We write ${\underline {k}} = (k_1, k_2, \dots , k_m) \in X(T)$ for short. The projection $B^- \to T$ allows us to consider ${\underline {k}}$ a degree one representation of $B^-$ .

By [Reference Jantzen25, Section I.5.8], we can associate to any $B^-$ -module M a sheaf ${\mathcal {L}}(M)$ on the quotient $G/B^-$ . In particular, we can apply this construction to any ${\underline {k}} \in X(T)$ . By [Reference Jantzen25, Proposition I.5.12], $H^0({G}/B^-, {\mathcal {L}}_{{\underline {k}}}) \cong {\mathrm {ind}}_{B^-}^{G}({\underline {k}})$ . To ease notations, like [Reference Jantzen25], we write $H^i({\underline {k}}) {:=}q H^i(G/B^-, {\mathcal {L}}_{{\underline {k}}})$ . For ${\underline {k}}$ dominant with $k_m \geq 0$ , we define a Schur functor

$$\begin{align*}\mathbb{S}^{{\underline{k}}} \colon {\underline{{\mathrm{Mod}}}}_R \to {\underline{{\mathrm{Mod}}}}_R \end{align*}$$

from the category of R-modules to itself (or, more generally, the category of quasi-coherent sheaves over a scheme S). See, for instance, [Reference Akin, Buchsbaum and Weyman1] or [Reference Fulton20, Chapter 8]. Let ${\underline {l}}$ denote the partition of $\lvert {\underline {k}} \rvert = \sum _i k_i$ conjugate to ${\underline {k}}$ , that is, $l_j {:=}q \# \{ i \mid k_i \geq j \}$ , for $j \geq 1$ . For $M \in {\underline {{\mathrm {Mod}}}}_R$ , we have

$$\begin{align*}\mathbb{S}^{{\underline{k}}}(M) = {{\mathrm{im}}}( \otimes_j \wedge^{l_j}(M) \to \otimes_j M^{\otimes l_j} \to M^{\otimes \lvert k \rvert} \to \otimes_i M^{\otimes k_i} \to \otimes_i {\mathrm{Sym}}^{k_i}(M)), \end{align*}$$

where the first map is induced by $\wedge ^l M \to M^{\otimes l}, m_1 \wedge m_2 \wedge \cdots \wedge m_l \mapsto \sum _{\nu \in {\mathrm {S}}_l} {\mathrm {sgn}}(\nu ) m_{\nu (1)} \otimes m_{\nu (2)} \otimes \dots m_{\nu (l)}$ , the maps in the middle are identifications of $M^{\otimes \lvert k \rvert }$ , grouping its factors first in terms of the columns and then in terms of the rows of the Young diagram for ${\underline {k}}$ , and the last map is the natural quotient. In particular, if M is free of rank r with ordered basis $m_1, m_2, \dots , m_r \in M$ , so is $\mathbb {S}^{{\underline {k}}}(M)$ , with basis elements corresponding with semi-standard Young tableaux of diagram ${\underline {k}}$ and entries in $\{1, 2, \dots , r\}$ .

We call the G-module $H^0({\underline {k}})$ the (dual) Weyl module associated with the dominant weight ${\underline {k}}$ . Let ${\underline {k}}, {\underline {k}}' \in X(T)_+$ . Then ${\underline {k}} + {\underline {k}}' \in X(T)_+$ and we have a morphism of G-representations

(2.8) $$ \begin{align} H^0({\underline{k}}) \otimes_R H^0({\underline{k}}') \longrightarrow H^0({\underline{k}}+{\underline{k}}') \end{align} $$

defined by multiplication of global sections. This lets us define corresponding morphisms of twisted modules.

2.6 Twist of a sheaf by a representation

Let R be a Dedekind domain, $S \in {\underline {{\mathrm {Sch}}}}_R$ , $\rho \in {\underline {{\mathrm {Re}}}}{\mathrm {p}}_R({\mathrm {GL}}_m)$ and take ${\mathcal {F}}$ a locally free ${\mathcal {O}}_S$ -sheaf of rank m over S. Consider the contracted product

$$\begin{align*}F^{\rho} = {\underline{{\mathrm{Isom}}}}({\mathcal{O}}_S^m, {\mathcal{F}}) \times^{{\mathrm{GL}}_m} \rho. \end{align*}$$

This is a vector bundle $F^\rho \to S$ (see [Reference Grothendieck and Raynaud23, Exposé XI, Section 4]). The sheaf of sections ${\mathcal {F}}^\rho $ of $F^\rho $ is the twist of ${\mathcal {F}}$ by $\rho $ .

The construction ${\mathcal {F}} \mapsto {\mathcal {F}}^\rho $ , for a fixed $\rho $ , is compatible with isomorphisms ${\mathcal {F}} \cong {\mathcal {F}}'$ : from every isomorphism $\phi \colon {\mathcal {F}} \to {\mathcal {F}}'$ of locally free sheaves of rank m over S, we obtain an isomorphism $\phi ^\rho \colon {\mathcal {F}}^\rho \to {\mathcal {F}}^{\prime \rho }$ . Similarly, ${\mathcal {F}} \mapsto {\mathcal {F}}^\rho $ is compatible with twists by Frobenius, when S has positive characteristic.

The twist ${\mathcal {F}}^\rho $ is functorial in $\rho $ and gives rise to a tensor functor $\rho \otimes \rho ' \rightsquigarrow {\mathcal {F}}^{\rho } \otimes {\mathcal {F}}^{\rho '}.$ In particular, from Equation (2.8), for ${\underline {k}}, {\underline {k}}' \in X(T)_+$ , we obtain a morphism of sheaves ${\mathcal {F}}^{{\underline {k}}} \otimes {\mathcal {F}}^{{\underline {k}}'} \longrightarrow {\mathcal {F}}^{{\underline {k}}+{\underline {k}}'}$ . We often use this morphism implicitly in our constructions.

2.7 The automorphic sheaves

Take $R = {\mathcal {O}}_{E, (p)}$ . By Lemma 2.2, we have $\mathbf {G}_{1,R} \cong {\mathrm {GL}}_{n, R}$ and $H_{1, R} \cong {\mathrm {GL}}_{n-1} \times {\mathrm {GL}}_1$ . Let $S \in {\underline {{\mathrm {Sch}}}}_R$ and consider ${\underline {A}} = (A, \lambda , \iota , \eta _K) \in {\mathcal {S}}_K(S)$ . For ${\underline {k}}$ a dominant weight for ${\mathrm {GL}}_{n-1}$ and $w \in {\mathbb Z}$ , we define the sheaves

$$\begin{align*}{\underline{\omega}}_{A/S}^{{\underline{k}}, w} {:=}q {\underline{\omega}}_{A/S, \sigma}^{{\underline{k}}} \otimes \delta_{A/S, {\sigma}}^{ w}. \end{align*}$$

This construction is compatible with base change, thus we work mainly with the automorphic sheaves of weights $({\underline {k}}, w)$ associated with ${\underline {A}}$ in the universal class of ${\mathcal {S}}_K$ , or some base change of it. We denote these sheaves by ${\underline {\omega }}^{{\underline {k}}, w}$ . Our convention for ${\underline {\omega }}^{{\underline {k}}, w}$ has the advantage of allowing us to define Hecke-equivariant theta operators. For R any ${\mathcal {O}}_{E, (p)}$ -algebra, let $B = \operatorname {\mathrm {Spec}} R$ and consider ${\mathcal {S}}_{K, B}$ the base change of ${\mathcal {S}}_K$ .

Remark 2.8 (Algebraic Koecher principle)

In our setting, the results of [Reference Lan35] apply, since $\dim ({\mathcal {S}}_K^{\mathrm {min}} \setminus {\mathcal {S}}_K) = n-1 \geq 2$ , and we have, for a smooth, projective, toroidal compactification ${\mathcal {S}}_K^{\mathrm {tor}}$ and the minimal compactification ${\mathcal {S}}_K^{\mathrm {min}}$ , that

$$\begin{align*}H^0({\mathcal{S}}_{K,B}, {\underline{\omega}}^{{\underline{k}}, w}_B) = H^0({\mathcal{S}}^{\mathrm{tor}}_{K, B}, {\underline{\omega}}^{{\mathrm{can}}, {\underline{k}}, w}_B) = H^0({\mathcal{S}}^{\mathrm{min}}_{K, B}, {\underline{\omega}}^{{\mathrm{min}}, {\underline{k}}, w}_B). \end{align*}$$

This is called the Koecher principle. This allows us to carry out most of our constructions and computations on ${\mathcal {S}}_{K, B}$ , while still being able to deduce interesting consequences that make use of the geometry of ${\mathcal {S}}_{K, B}^{{\mathrm {tor}}}$ and ${\mathcal {S}}_{K, B}^{{\mathrm {min}}}$ and the extensions of the bundles ${\underline {\omega }}^{{\underline {k}}, w}_B$ .

4.1 Generalities on filtrations

We record here general, basic facts concerning filtrations.

4.1.1 Tensor product of filtrations

Let S be a scheme and consider $({\mathcal {F}}, F^{\bullet }({\mathcal {F}})), ({\mathcal {G}}, F^{\bullet }({\mathcal {G}}))$ finite locally free sheaves over S endowed with finite, separated, exhaustive, descending filtrations

$$ \begin{align*} {\mathcal{F}} &= F^0({\mathcal{F}}) \supseteq F^1({\mathcal{F}}) \supseteq \cdots \supseteq F^r({\mathcal{F}}) \supseteq F^{r+1}({\mathcal{F}}) = 0, \\ {\mathcal{G}} &= F^0({\mathcal{G}}) \supseteq F^1({\mathcal{G}}) \supseteq \cdots \supseteq F^s({\mathcal{G}}) \supseteq F^{s+1}({\mathcal{G}}) = 0, \end{align*} $$

by finite locally free subsheaves $F^i({\mathcal {F}}), F^j({\mathcal {G}})$ , such that the graded pieces ${\mathrm {gr}}^i {\mathcal {F}}, {\mathrm {gr}}^j {\mathcal {G}}$ , are themselves finite locally free. Then we can defined a tensor product filtration on ${\mathcal {F}} \otimes _{{\mathcal {O}}_S} {\mathcal {G}}$ with the same properties by setting

$$\begin{align*}F^k({\mathcal{F}} \otimes {\mathcal{G}}) {:=}q \sum_{i+j = k} F^i({\mathcal{F}}) \otimes F^j({\mathcal{G}}). \end{align*}$$

We write $P({\mathcal {F}} \otimes {\mathcal {G}}) = F^{r+s-1}({\mathcal {F}} \otimes {\mathcal {G}})$ to denote the penultimate step of the filtration. The graded pieces of $F^{\bullet }({\mathcal {F}} \otimes {\mathcal {G}})$ are ${\mathrm {gr}}^k({\mathcal {F}} \otimes {\mathcal {G}}) \cong \oplus _{i+j = k} {\mathrm {gr}}^i({\mathcal {F}}) \otimes {\mathrm {gr}}^j({\mathcal {G}}).$ If we have a finite family of finite locally free sheaves $({\mathcal {F}}_i, F^{\bullet }({\mathcal {F}}_{i}))_{1 \leq i \leq m}$ with filtrations subject to the same conditions as above, we can define by induction a filtration on their tensor product. Again, we denote by $P(\otimes _{i=1}^m {\mathcal {F}}_i)$ the penultimate step of the filtration.

4.1.3 Koszul filtration

Let S be as above and consider a short exact sequence of finite locally free sheaves over S of constant rank

$$\begin{align*}0 \to {\mathcal{F}}' \longrightarrow {\mathcal{F}} \longrightarrow {\mathcal{F}}" \to 0. \end{align*}$$

The Koszul filtration of $\wedge ^j {\mathcal {F}}$ , for $0 \leq j \leq {\mathrm {rk}}({\mathcal {F}})$ , is defined by

$$\begin{align*}K^i(\wedge^j {\mathcal{F}}) {:=}q {{\mathrm{im}}}(\wedge^i {\mathcal{F}}' \otimes_{{\mathcal{O}}_S} \wedge^{j - i} {\mathcal{F}} \longrightarrow \wedge^j {\mathcal{F}}),\quad 0 \leq i \leq j. \end{align*}$$

The penultimate step of this filtration is $P(\wedge ^j {\mathcal {F}}) = K^{j-1}(\wedge ^j {\mathcal {F}})$ . This filtration can also obtained by taking quotients from the tensor product filtration on ${\mathcal {F}}^{\otimes j}$ . We extend this filtration by setting $K^i(\wedge ^j {\mathcal {F}}) = \wedge ^j {\mathcal {F}}$ for $i<0$ and $K^i(\wedge ^j {\mathcal {F}}) = 0$ for $i>j$ . We have $K^i \wedge K^{i'} \subseteq K^{i+i'}(\wedge ^{\bullet } {\mathcal {F}})$ . The graded terms of the filtration are

$$\begin{align*}{\mathrm{gr}}^i(\wedge^j {\mathcal{F}}) = K^i/K^{i+1} \cong \wedge^{i} {\mathcal{F}}' \otimes_{{\mathcal{O}}_S} \wedge^{j-i} {\mathcal{F}}", \quad 0 \leq i \leq j \leq {\mathrm{rk}}({\mathcal{F}}). \end{align*}$$

4.1.4 Filtration of the symmetric algebra

Let S be as above. Let ${\mathcal {G}}$ be a finite locally free sheaf. For $j<0$ , we set ${\mathrm {Sym}}^j({\mathcal {G}}) {:=}q \left ( {\mathrm {Sym}}^{-j}({\mathcal {G}}) \right )^\vee $ and whenever ${\mathrm {Sym}}^{-j}({\mathcal {G}})$ is endowed with a filtration, ${\mathrm {Sym}}^j({\mathcal {G}})$ is endowed with the dual filtration. One can consider $({\mathcal {F}}, F^{\bullet }({\mathcal {F}}))$ as above. By taking the quotient of the filtration on ${\mathcal {F}}^{\otimes j}$ , we get a filtration on ${\mathrm {Sym}}^j {\mathcal {F}}$ .

Remark 4.1 We will consider a filtration on ${\mathbb {S}}^{{\underline {k}}}(H_\sigma )$ , where ${\underline {k}} = (k_1 \geq k_2 \geq \cdots \geq k_{n-1}) \in {\mathbb Z}^{n-1}_{\geq 0}$ and $H = H^1_{\mathrm {dR}}(A/S)$ , for $A/S$ an abelian scheme with ${\mathcal O_E}$ -action. To construct it, first, we consider the Koszul filtration $K^{\bullet }(\wedge ^j H_\sigma )$ induced by the Hodge filtration (2.5). Taking tensor products, we obtain a filtration on $\otimes _j \wedge ^{l_j}(H_\sigma )$ , ${\underline {l}}$ conjugate to ${\underline {k}}$ . Similarly, we construct filtrations on $H_\sigma ^{\lvert {\underline {k}} \rvert }$ and $\otimes _i {\mathrm {Sym}}^{k_i}(H_\sigma )$ . Finally, we notice that the construction of ${\mathbb {S}}^{{\underline {k}}}(H_\sigma )$ is compatible with these filtrations. In particular, the penultimate step of this filtration is given by the image of

$$\begin{align*}\bigoplus_j \bigg(\big(\bigotimes_{j' \neq j} \wedge^{l_{j'}}{\underline{\omega}}_\sigma\big) \bigotimes (\wedge^{l_j-1} {\underline{\omega}}_\sigma \otimes H_\sigma)\bigg) \end{align*}$$

in ${\mathbb {S}}^{{\underline {k}}}(H_\sigma )$ . When constructing generalized theta operators, we will consider more complicated filtrations, but they will be built following the same ideas.

4.2 Generalities on connections

We fix for the rest of this section the following data, a special case of [Reference Katz28, Section 1]: $X \overset {f}{\longrightarrow } S \overset {g}{\longrightarrow } B, $ where f is smooth and proper and g is smooth. In particular, the sheaves $H_{\mathrm {dR}}^i(X/S)$ are finite locally free. We are mainly interested in the case $X = A,$ where ${\underline {A}}$ is the universal object over $S={\mathcal {S}}_{K, B}$ , for some $B \in {\underline {{\mathrm {Sch}}}}_{{\mathcal {O}}_{E,(p)}}$ .

Definition 4.1 (Connection)

Let ${\mathcal {F}}$ be a quasi-coherent sheaf on S. A B-linear connection on ${\mathcal {F}}$ is a morphism $\nabla \colon {\mathcal {F}} \longrightarrow {\mathcal {F}} \otimes _{{\mathcal {O}}_S} \Omega ^1_{S/B}$ of abelian sheaves satisfying the Leibniz rule

$$\begin{align*}\nabla(h s) = h\nabla(s) + s \otimes d_{S/B}(h), \end{align*}$$

where h and s are sections of ${\mathcal {O}}_S$ and ${\mathcal {F}}$ , respectively, over some open $U \subset S$ and $d_{S/B} \colon {\mathcal {O}}_S \to \Omega ^1_{S/B}$ is the exterior differential.

We can derive from $\nabla $ , for $({\mathcal {F}}, \nabla )$ a sheaf with connection, a family of morphisms of abelian sheaves

$$\begin{align*}\nabla_i \colon {\mathcal{F}} \otimes \Omega^i_{S/B} \longrightarrow {\mathcal{F}} \otimes \Omega^{i+1}_{S/B} \end{align*}$$

defining it on sections s and $\omega $ of ${\mathcal {F}}$ and $\Omega ^i_{S/B}$ via a generalized Leibniz rule

$$\begin{align*}s \otimes \omega \longmapsto \nabla(s) \wedge \omega + s \otimes d\omega \end{align*}$$

and then extending by linearity. The curvature of the connection $\nabla $ is defined as $K(\nabla ) {:=}q \nabla _1 \circ \nabla \colon {\mathcal {F}} \to {\mathcal {F}} \otimes \Omega ^2_{S/B}$ , which is an ${\mathcal {O}}_S$ -linear morphism. A connection is said to be integrable if $K(\nabla ) = 0$ .

Given two quasi-coherent sheaves with connections $({\mathcal {F}}, \nabla _{{\mathcal {F}}}), ({\mathcal {G}}, \nabla _{{\mathcal {G}}}),$ we can use the Leibniz rule to define the tensor product connection $({\mathcal {F}} \otimes {\mathcal {G}}, \nabla _{{\mathcal {F}} \otimes {\mathcal {G}}})$ , obtained on local sections as

$$ \begin{align*} \nabla_{{\mathcal{F}} \otimes {\mathcal{G}}} \colon {\mathcal{F}} \otimes_{{\mathcal{O}}_S} {\mathcal{G}} &\longrightarrow {\mathcal{F}} \otimes {\mathcal{G}} \otimes \Omega^1_{S/B}, \\ f \otimes g & \longmapsto \nabla(f) \otimes g + f \otimes \nabla(g) \end{align*} $$

and then extending by linearity, where we identify ${\mathcal {F}} \otimes \Omega ^1 \otimes {\mathcal {G}} \cong {\mathcal {F}} \otimes {\mathcal {G}} \otimes \Omega ^1$ . Thus, from $({\mathcal {F}}, \nabla ),$ we can define a natural connection $\nabla \colon {\mathcal {F}}^{\otimes j} \to {\mathcal {F}}^{\otimes j} \otimes \Omega ^1_{S/B}$ , for $j \geq 0$ , still denoted by $\nabla $ . From this, taking quotients, we get well-defined connections on $\wedge ^j {\mathcal {F}}$ and ${\mathrm {Sym}}^j {\mathcal {F}}$ .

Lemma 4.2 (Koszul transversality)

Consider $({\mathcal {F}}, F^{\bullet }({\mathcal {F}}))$ a finite locally free sheaf finitely filtered by finite locally free sheaves with finite locally free quotients. Suppose that ${\mathcal {F}}$ is endowed with a connection $\nabla $ such that $\nabla (F^i({\mathcal {F}})) \subseteq F^{i-1}({\mathcal {F}})$ . Then, for any $ j\geq 1$ , the induced filtration $({\mathcal {F}}^{\otimes j}, F^{\bullet })$ and connection $({\mathcal {F}}^{\otimes j}, \nabla )$ satisfy

$$\begin{align*}\nabla(F^i({\mathcal{F}}^{\otimes j})) \subseteq F^{i-1}({\mathcal{F}}^{\otimes j}) \otimes \Omega^1_{S/B}, \end{align*}$$

for $i \geq 1$ . Similarly,

$$\begin{align*}\nabla(K^i(\wedge^j {\mathcal{F}})) \subseteq K^{i-1}(\wedge^j {\mathcal{F}}) \otimes \Omega^1_{S/B},\nabla(F^i({\mathrm{Sym}}^j{\mathcal{F}})) \subseteq F^{i-1}({\mathrm{Sym}}^j({\mathcal{F}})) \otimes \Omega^1_{S/B}. \end{align*}$$

4.3 The Gauss–Manin connection

For more details, see [Reference Katz and Oda32, Section 2]. The first exact sequence of $X/S/B$ is

(4.1) $$ \begin{align} 0 \longrightarrow f^\ast \Omega^1_{S/B} \longrightarrow \Omega^1_{X/B} \longrightarrow \Omega^1_{X/S} \longrightarrow 0. \end{align} $$

The corresponding Koszul filtration of $\Omega ^j_{X/B}$ has graded pieces $ {\mathrm {gr}}^i (\Omega ^j_{X/B}) = \Omega ^{j-i}_{X/S} \otimes _{{\mathcal {O}}_X} f^\ast \Omega ^i_{S/B}. $ Since $K^i(\Omega ^{\bullet }_{X/B})$ is a sub-complex of $\Omega ^{\bullet }_{X/B}$ , the finite filtration $K^{\bullet }(\Omega ^{\bullet }_{X/B})$ gives a convergent spectral sequence whose first page is

$$\begin{align*}E_1^{i, j} {:=}q R^{i+j}f_\ast({\mathrm{gr}}^i(\Omega^{\bullet}_{X/B})) \cong H^j_{\mathrm{dR}}({X/S}) \otimes_{{\mathcal{O}}_S} \Omega^i_{S/B}. \end{align*}$$

Consider $\nabla = \nabla ^j = d_1^{0, j}$ , the differentials of this spectral sequence. We have the following.

Since the GM connection and the ones derived from it are essentially the only connections we will discuss, we reserve the notation $\nabla $ for them, from now on.

Proposition 4.4 (Griffiths transversality)

Let $F^{\bullet } = F^{\bullet }(H^{\bullet }_{\mathrm {dR}}(X/S))$ denote the Hodge filtration. The GM connection satisfies

$$\begin{align*}\nabla(F^i(H^j_{\mathrm{dR}}(X/S))) \subseteq F^{i-1}(H^j_{\mathrm{dR}}(X/S)) \otimes \Omega^1_{X/S}. \end{align*}$$

In particular, the GM connection induces a mapping between graded terms.

For more details, see [Reference Katz28, Section 1.4.1, Proposition 1.4.1.6].

4.3.1 Base change

Let $f' \colon X' \to S$ be a proper, smooth morphism and let $\phi \colon X \to X'$ be a morphism of S-schemes. The construction of the GM connection is functorial in the following sense.

Lemma 4.5 We have the natural commutative diagram

Let T be a smooth scheme over B and $u \colon T \to S$ a morphism of B-schemes. The base change map $u^\ast H^i_{\mathrm {dR}}(X/S) \longrightarrow H^i_{\mathrm {dR}}(X_T/T) $ is an isomorphism for every u. The construction of $\nabla $ immediately implies the following.

Lemma 4.6 With $X, S, B,$ and $u \colon T \to S$ as above, we have the commutative diagram

where the diagonal map is the pullback connection induced by base change $u^\ast H^i_{\mathrm {dR}}(X/S) \cong H^i(X_T/T)$ and the natural map $d u \colon u^\ast \Omega ^1_{S/B} \to \Omega ^1_{T/B}$ .

Remark 4.7 Consider the case $X=A$ , for ${\underline {A}} \in {\mathcal {S}}_{K, B}(S)$ , $B \in {\underline {{\mathrm {Sch}}}}_{{\mathcal {O}}_{E,(p)}}$ . Lemma 4.5 implies that the GM connection respects the ${\mathcal O_E}$ -action on $H^i_{\mathrm {dR}}(A/S)$ , that is, for $H=H^1_{\mathrm {dR}}(A/S)$ , we have restrictions

$$\begin{align*}\nabla \colon H_\sigma \longrightarrow H_\sigma \otimes_{{\mathcal{O}}_S} \Omega^1_{S/B}, \nabla \colon H_{\overline{\sigma}} \longrightarrow H_{\overline{\sigma}} \otimes_{{\mathcal{O}}_S} \Omega^1_{S/B}. \end{align*}$$

Similarly, if B is of characteristic $p,$ we can consider the isogenies Frobenius and Verschiebung and take their pullbacks $F\colon H^{(p)} \to H$ and $V\colon H \to H^{(p)}$ . Lemma 4.5 tells us that $\nabla $ commutes with $F, V,$ and therefore it respects kernels and images of both. This will be relevant when defining generalized theta operators.

4.4 The Kodaira–Spencer map

The short exact sequence (4.1) gives a natural section

$$\begin{align*}{\underline{{\mathrm{KS}}}} \in H^0(S, R^1f_\ast({\underline{{\mathrm{Der}}}}(X/S) \otimes_{{\mathcal{O}}_X} f^\ast \Omega^1_{S/B})) \cong {\mathrm{Hom}}_{S}({\underline{{\mathrm{Der}}}}(S/B), R^1f_\ast({\underline{{\mathrm{Der}}}}(X/S))), \end{align*}$$

called the Kodaira–Spencer section.

Let $X=A$ be an abelian scheme over S. We are interested in the special case of Proposition 4.8 for $A/S/B$ . Let $H = H^1_{\mathrm {dR}}(A/S)$ . We have, on the one hand, a morphism:

$$\begin{align*}{\underline{\omega}}_{A/S} \subset H \overset{\nabla}{\longrightarrow} H \otimes \Omega^1_{S/B} \twoheadrightarrow {\underline{\omega}}_{A^\vee/S}^\vee \otimes_{{\mathcal{O}}_S} \Omega^1_{S/B} \end{align*}$$

whence, equivalently, a morphism

$$\begin{align*}{\underline{\omega}}_{A/S} \otimes_{{\mathcal{O}}_S} {\underline{\omega}}_{A^\vee/S} {\longrightarrow} \Omega^1_{S/B}. \end{align*}$$

On the other hand, $R^1f_\ast {\underline {{\mathrm {Der}}}}(A/S) \cong {\underline {\omega }}_{A^\vee /S}^\vee \otimes _{{\mathcal {O}}_S} {\underline {\omega }}_{A/S}^\vee $ . By duality, the KS morphism is equivalent to

$$\begin{align*}{\underline{{\mathrm{KS}}}} \colon {\underline{\omega}}_{A/S} \otimes_{{\mathcal{O}}_S} {\underline{\omega}}_{A^\vee/S} {\longrightarrow} \Omega^1_{S/B}. \end{align*}$$

Then, Proposition 4.8 tells us that these two morphisms ${\underline {\omega }}_{A/S} \otimes _{{\mathcal {O}}_S} {\underline {\omega }}_{A^\vee /S} {\to } \Omega ^1_{S/B}$ are the same.

Assume now that $A \to S ={\mathcal {S}}_{K,B}$ is the universal object, $B \to \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E, (p)}$ . Then, using the identification,

$$\begin{align*}{\underline{\omega}}_{A/S, \sigma} \otimes_{{\mathcal{O}}_S} {\underline{\omega}}_{A^\vee/S, \sigma} \cong {\underline{\omega}}_{A/S, \sigma} \otimes_{{\mathcal{O}}_S} \det {\underline{\omega}}_{A/S, \sigma} \otimes \delta_{A/S, \sigma}^{-1}, \end{align*}$$

which follows from Equation (2.7), we see that ${\underline {{\mathrm {KS}}}}$ induces a morphism

$$\begin{align*}{\underline{{\mathrm{ks}}}} = {\underline{{\mathrm{ks}}}}_{A/S/B} {:=}q {\underline{{\mathrm{KS}}}}_\sigma \colon {\underline{\omega}}_{A/S, \sigma} \otimes_{{\mathcal{O}}_S} \det {\underline{\omega}}_{A/S, \sigma} \otimes \delta_{A/S, \sigma}^{-1} {\longrightarrow} \Omega^1_{S/B}. \end{align*}$$

Then we have the following.

Proof The proof in [Reference Lan34, Proposition 2.3.4.2] is given for $B = \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E, (p)}$ but the construction of ${\underline {{\mathrm {ks}}}}$ is compatible with base change, so the statement over a general base $B \to \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E, (p)}$ follows. Notice that, in the notation of [Reference Lan34, Definition 2.3.4.1], the injection ${\underline {\omega }}_{A/S, \sigma } \otimes _{{\mathcal {O}}_S} {\underline {\omega }}_{A^\vee /S, \sigma } \to {\underline {\omega }}_{A/S} \otimes _{{\mathcal {O}}_S} {\underline {\omega }}_{A^\vee /S}$ composed with the projection

$$\begin{align*}&{\underline{\omega}}_{A/S} \otimes_{{\mathcal{O}}_S} {\underline{\omega}}_{A^\vee/S} \to {\underline{\omega}}_{A/S} \otimes_{{\mathcal{O}}_S} {\underline{\omega}}_{A^\vee/S}/\\&\quad\big(\lambda^\ast(y)\otimes z - \lambda^\ast(z)\otimes y, (\alpha \cdot x) \otimes y - x \otimes (\alpha \cdot y) \big) \end{align*}$$

is an isomorphism, where $x \in {\underline {\omega }}_{A/S}(U), y, z \in {\underline {\omega }}_{A^\vee /S}(U), \alpha \in {\mathcal O_E}$ are sections over $U \subset S$ some open.

Equivalently, from [Reference Faltings and Chai19, Section III.9], one can deduce a description of the KS map in terms of the universal vector extension. This and an argument using Grothendieck–Messing theory, as explained, for instance, in [Reference Grothendieck22, pp. 116–118], also gives a proof of the proposition: see [Reference Bellaïche3, Proposition 2.1.5] for the case $n=3$ , which can be easily generalized to our setting. See also the proof of Proposition 5.5.

5.1 The standard Dieudonné modules

Following [Reference Wooding39, Section 3.5], for a given point $s \in S_{K, w_r}(k)$ , where $k/{\mathbb {F}}$ is an algebraically closed extension and $1 \leq r \leq n$ , we can describe the structure of the Dieudonné module $D = {\mathbb {D}}(A_s[p])$ using standard objects, defined in [Reference Moonen36, Section 4.9]. In fact, from [Reference Moonen36, Theorem 4.7], we deduce the following.

Proposition 5.3 Let $D = D_\sigma \oplus D_{\overline {\sigma }}$ be the CM decomposition of D. There are k-bases $\{e_i\}_{1 \leq i \leq n}$ , $\{f_i\}_{1 \leq i \leq n}$ of $D_\sigma $ , $D_{\overline {\sigma }}$ , respectively, such that $\left <e_i, f_j\right>= \delta _{ij}$ and the semilinear Frobenius $F \colon D \to D$ and Verschiebung $V \colon D \to D$ are described as follows:

(5.1) $$ \begin{align} F(e_i) = \left \{ \begin{array}{@{}cc} 0 & \text{ if } i\neq r, \\ e_1 & \text{ if } i=r, \end{array} \right. F(f_i) = \left \{ \begin{array}{@{}cc} f_{i+1} & \text{ if } 1 \leq i \leq r-1, \\ 0 & \text{ if } i=r, \\ f_{i} & \text{ if } r+1 \leq i \leq n, \end{array} \right. \end{align} $$

(5.2) $$ \begin{align} V(e_i) = \left \{ \begin{array}{@{}cc} 0 & \text{ if } i=1, \\ e_{i-1} & \text{ if } 2 \leq i \leq r, \\ e_{i} & \text{ if } r+1 \leq i \leq n, \end{array} \right. V(f_i) = \left \{ \begin{array}{@{}cc} f_r & \text{ if } i=1, \\ 0 & \text{ otherwise}. \end{array} \right. \end{align} $$

In particular, we have

$$ \begin{align*} D[F] &= \ker F = {\mathrm{Span}}_{k} \left<e_1, \dots, e_{r-1}, f_r, e_{r+1}, \dots, e_n\right>,\\ D[V] &= \ker V = {\mathrm{Span}}_{k} \left<e_1, f_2, \dots, f_n\right>, \end{align*} $$

and:

1. (1) if $r=1$ , the p-rank of $A_s[p]$ is n and that of $A_s[p]_{\overline {\sigma }}$ is $n-1$ ;

2. (2) if $r>1$ , the p-rank of $A_s[p]$ and $A_s[p]_{\overline {\sigma }}$ is $n-r$ .

Thus, the EO stratification coincides with the p-rank stratification for the $\overline {\sigma }$ -component of the p-torsion.

5.2 Partial Hasse invariants

We are interested in the action of V on ${\underline {\omega }}_{A_s/k} \cong D[F]$ for $s \in S_{K, w_r}(k)$ , $D={\mathbb {D}}(A_s[p])$ , $k/{\mathbb {F}}$ algebraically closed, and $1 \leq r \leq n-1$ . Consider the sheaf

$$\begin{align*}{\underline{\omega}}_{0, r} {:=}q \ker( V^{r-1} \colon {\underline{\omega}}_{A/\overline{S}_{K, w_r}, \sigma} \to {\underline{\omega}}_{A/\overline{S}_{K, w_r}, \sigma}^{(p^{r-1})}) \end{align*}$$

on $\overline {S}_{K, w_r}$ . By Proposition 5.3, the rank of ${\underline {\omega }}_{0, r}$ is constant and equal to $r-1$ . Therefore, ${\underline {\omega }}_{0, r}$ locally free of rank $r-1$ . The quotient sheaf ${\underline {\omega }}_{\mu , r} {:=}q {\underline {\omega }}_{\sigma }/{\underline {\omega }}_{0, r}$ is also locally free, of rank $n-r$ , over $\overline {S}_{K, w_r}$ . Over $S_{K, w_r}$ , ${\underline {\omega }}_{0, r}$ and ${\underline {\omega }}_{\mu , r}$ correspond to the local–local and multiplicative part of the canonical filtration of the p-torsion of A, hence the names. In particular, the quotient morphism $V \colon {\underline {\omega }}_{\mu , r} \to {\underline {\omega }}_{\mu , r}^{(p)}$ is an isomorphism over the dense open ${S}_{K, w_r} \subseteq \overline {S}_{K, w_r}$ and has a kernel of rank $1$ over $\overline {S}_{K, w_{r+1}} = \overline {S}_{K, w_r} \setminus {S}_{K, w_r}$ . This is because over $\overline {S}_{K, w_{r+1}}$ , $\ker (V^r\colon {\underline {\omega }}_\sigma \to {\underline {\omega }}_{\sigma }^{(p^r)})$ has constant rank r, by Proposition 5.3, and $\ker (V \colon {\underline {\omega }}_{\mu , r} \to {\underline {\omega }}_{\mu , r}^{(p)}) \cong {{\underline {\omega }}_{0, r+1}}/{{\underline {\omega }}_{0, r}} $ has constant rank $1$ . Consider the corresponding section $A_r = \det V \in H^0(\overline {S}_{K, w_r}, (\det {\underline {\omega }}_{\mu , r})^{p-1})$ .

We call the $A_r$ ’s (generalized) partial Hasse invariants. They are factors of the generalized Hasse invariants of [Reference Boxer5].

5.3 Smoothness of closures in the EO stratification

We establish a geometric property of the KS morphism on lower EO strata. Consider $2 \leq r \leq n-1$ . We show in particular that $\overline {S}_{K, w_r}$ is smooth. We write

$$\begin{align*}\nabla_r \colon H^1_{\mathrm{dR}}(A/\overline{S}_{K, w_r}) \longrightarrow H^1_{\mathrm{dR}}(A/\overline{S}_{K, w_r}) \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}} \end{align*}$$

to denote the Gauss–Manin connection relative to $A/\overline {S}_{K, w_r}/{\mathbb {F}}$ . We use the same notation for the restriction of $\nabla _r$ to $S_{K, w_r} \subseteq \overline {S}_{K, w_r}$ . Recall that $\nabla _r$ is related to $\nabla \colon H^1_{\mathrm {dR}}(A/S_K) \to H^1_{\mathrm {dR}}(A/S_K) \otimes \Omega ^1_{S_K/{\mathbb {F}}}$ via base change (see Lemma 4.6). We write $u \colon S_{K, w_r} \to S_{K}$ and $\overline {u} \colon \overline {S}_{K, w_r} \to S_K$ for the natural immersions. To ease notations, we set $\tilde {\delta } = \det {\underline {\omega }}_{A/S, \sigma } \otimes \delta _{\sigma }^{-1}$ , with S depending on the context. We also write ${\mathcal {C}}_{\overline {S}_{K, w_r}/S_K}$ to denote the conormal sheaf of $\overline {S}_{K, w_r}$ in $S_K$ .

Proposition 5.5 We have a commutative diagram with exact rows

(5.3)

whose vertical arrows are isomorphisms. In particular, $\overline {S}_{K, w_r}$ is smooth and $\nabla _r({\underline {\omega }}_0) \subseteq {\underline {\omega }}_0 \otimes \Omega ^1_{\overline {S}_{K, w_r}/{\mathbb {F}}}.$

Proof We prove this via Grothendieck–Messing theory. For what we intend to prove the level K is irrelevant, so we drop it from the notation. We write $S_r$ (resp. $\overline {S}_r$ ) instead of $S_{K, w_r}$ (resp. $\overline {S}_{K, w_r}$ ).

Over $S_r,$ we have $\nabla _r({\underline {\omega }}_0) \subseteq {\underline {\omega }}_0 \otimes \Omega ^1_{S_r/{\mathbb {F}}}$ , by Lemma 4.5, Remark 4.7, and the fact that ${\underline {\omega }}_0 = \ker (V^{r-1} \colon H_{{\mathrm {dR}}, \sigma } \to H_{{\mathrm {dR}}, \sigma }^{(p^{r-1})})$ , by Proposition 5.3. By Lemma 4.6 and Proposition 4.8, this means that, over $S_r,$ the composition

is zero. By density of $S_r \subseteq \overline {S}_r$ , we deduce the existence of the commutative diagram (5.3) and the morphism ${\underline {{\mathrm {ks}}}}_{\mu }$ . We now show that its vertical arrows are isomorphism and that the second row is a short exact sequence.

Consider a point $s \in \overline {S}_r \subseteq S$ and write $k = k(s)$ . We want to describe the Zariski tangent spaces

$$\begin{align*}T_{\overline{S}_r, s} = T_{\overline{S}_r/\mathbb{F}, s}, \quad T_{S, s} = T_{S/\mathbb{F}, s}. \end{align*}$$

These are k-vector spaces whose elements v correspond to lifts of $s \colon \operatorname {\mathrm {Spec}} k \to \overline {S}_r$ to $v \colon \operatorname {\mathrm {Spec}} k[\epsilon ] \to \overline {S}_r$ (resp. S), where $k[\epsilon ] {:=}q k[T]/(T^2)$ . By definition of S, resp. $\overline {S}_r$ , any $v \in T_{S, s}$ , resp. $v \in T_{\overline {S}_r, s}$ , corresponds to an abelian scheme with PEL structure ${\underline {A}}_v \in S(k[\epsilon ])$ , resp. ${\underline {A}}_v \in \overline {S}_r(k[\epsilon ])$ , lifting the object ${\underline {A}} \in \overline {S}_r(k)$ corresponding to s. Grothendieck–Messing theory tells us that the category of such lifts ${\underline {A}}_v$ is equivalent to the category of lifts of the k-subspace ${\underline {\omega }} = {\underline {\omega }}_{A/k}$ of $H = H^1_{\mathrm {dR}}(A/k)$ to sub- $k[\epsilon ]$ -modules $\tilde{\underline {\omega }}$ of $H[\epsilon ] = H \otimes _k k[\epsilon ]$ , the trivial lift of H, such that:

1. (1) $\tilde {\underline {\omega }}$ is a free direct summand of $H[\epsilon ]$ of rank n.

2. (2) $\tilde {\underline {\omega }}$ is ${\mathcal {O}}_E$ -stable (of type necessarily $(n-1, 1)$ ).

3. (3) $\tilde {\underline {\omega }}$ is maximal isotropic for the perfect alternating pairing on $H[\epsilon ]$ induced by the de Rham pairing $\left <\cdot , \cdot \right>^\lambda _{\mathrm {dR}}$ on H.

Moreover, for $v \in T_{S, s}$ , we have that:

1. (4) $v \in T_{\overline {S}_r, s}$ if and only if the lift ${\underline {A}}_v$ corresponds to $\tilde {\underline {\omega }}$ such that $\tilde {\underline {\omega }}_\sigma [V^{r-1}]$ is free of rank $r-1$ .

Condition (4) follows from Proposition 5.3. By [Reference de Jong6, Section 2], the BT₁ with extra structure $A_v[F]$ is determined by $(\tilde {\underline {\omega }}, V\colon \tilde {\underline {\omega }} \to \tilde {\underline {\omega }}^{(p)})$ , with the extra structures detailed at the points (1–3) above. Therefore, for $r \neq 1$ , $\tilde {\underline {\omega }}[V^{r-1}]$ is free of rank $r-1$ if and only if $V^{r-1}(A_v[p]_{\overline {\sigma }}) \subseteq A_v[F]_{\overline {\sigma }}$ exists and is finite locally free of rank $r-1$ , which happens if and only if $v \in \overline {S}_r(k[\epsilon ])$ , by the structure of the canonical filtration on $S_j, j \geq r$ (see Proposition 5.3 and the beginning of Section 5.2). The case $r = 1$ is trivial.

Such a $\tilde {\underline {\omega }}$ corresponds to a k-linear morphism $h \colon {\underline {\omega }} \to H/{\underline {\omega }} \cong {\underline {\omega }}_{A^\vee /k}^\vee \underset {\lambda ^\ast }{\cong } {\underline {\omega }}^\vee $ subject to the conditions:

1. (1) The pairing $\left <\cdot , h(\cdot )\right>^\lambda _{\mathrm {dR}}$ on ${\underline {\omega }}$ is symmetric.

2. (2) h is ${\mathcal {O}}_E$ -linear.

The first condition corresponds to $\tilde {\underline {\omega }}$ being maximal isotropic, the second to the fact that $\tilde {\underline {\omega }}$ is ${\mathcal {O}}_E$ -stable. Furthermore, the extra condition that $\tilde {\underline {\omega }}_\sigma [V^{r-1}]$ is free of rank $r-1$ corresponds to:

1. (3) h is identically $0$ on ${\underline {\omega }}_\sigma [V^{r-1}] \subseteq {\underline {\omega }}_\sigma $ .

Notice that the first two conditions together imply that h is determined by

$$\begin{align*}h_\sigma \colon {\underline{\omega}}_\sigma \longrightarrow (H/{\underline{\omega}})_\sigma \cong {\underline{\omega}}_{A^\vee/k, \sigma}^\vee \underset{\lambda^\ast}{\cong} {\underline{\omega}}_{\overline{\sigma}}^\vee. \end{align*}$$

Therefore, the deformations $T_{S, s}$ of A are in correspondence with ${\mathrm {Hom}}_k({\underline {\omega }}_\sigma , {\underline {\omega }}_{\overline {\sigma }}^\vee ) \cong {\underline {\omega }}_\sigma ^\vee \otimes _k {\underline {\omega }}_{\overline {\sigma }}^\vee $ , and $T_{\overline {S}_r, s} \subset T_{S, s}$ corresponds to

$$\begin{align*}{\mathrm{Hom}}_k({\underline{\omega}}_\mu, \omega_{\overline{\sigma}}^\vee) = \{h \colon {\underline{\omega}}_\sigma \to {\underline{\omega}}_{\overline{\sigma}}^\vee \mid h|_{{\underline{\omega}}_\sigma[V^{r-1}]} = 0\} \subset {\mathrm{Hom}}_k({\underline{\omega}}_\sigma, {\underline{\omega}}_{\overline{\sigma}}^\vee). \end{align*}$$

As discussed in [Reference Bellaïche3, p. 40], this correspondence is given by isomorphisms of k-vector spaces and, as noted in Proposition 4.9, the isomorphism $T_{S, s} \to {\underline {\omega }}_\sigma ^\vee \otimes _k {\underline {\omega }}_{\overline {\sigma }}^\vee $ is the dual of the fiber at s of the KS morphism. Hence, we have a commutative diagram

where the vertical arrows are isomorphisms and ${\mathcal {N}}_{\overline {S}_r/S, s}$ denotes the normal sheaf at s of $\overline {S}_r$ in S. This proves at once that $\overline {S}_r$ is smooth and by duality that the diagram of exact sequences (5.3) is exact and its vertical arrows are isomorphisms.

Finally, from Lemma 4.5 with $\phi = V^{r-1}$ , we deduce that $\nabla _r({\underline {\omega }}_0) \subseteq {\underline {\omega }}_0 \otimes \Omega ^1_{\overline {S}_r/{\mathbb {F}}}$ .

6.1 Unit-root splitting

We present here the classic unit-root splitting of the Hodge filtration (2.3) over the ordinary locus $S^\mu _K$ and then we show how one can obtain a similar splitting over lower EO strata. In our discussion, we will make implicit use of Proposition 5.3 and of technical Lemma 6.1, inspired by [Reference Eischen, Flander, Ghitza, Mantovan and McAndrew15, Section 3.3.1].

Let us introduce some notation. Let $\overline {S}$ be a scheme and consider ${\mathcal {F}}\subseteq {\mathcal {G}}$ finite locally free sheaves with constant rank over $\overline {S}$ , such that ${\mathcal {G}}/{\mathcal {F}}$ is also finite locally free. Let r denote the rank of ${\mathcal {F}}$ . Consider a tuple of integers ${\underline {k}} = (k_1 \geq k_2 \geq \cdots \geq k_r \geq 0)$ . From the results of Section 4.1, we obtain a natural descending filtration on ${\mathbb {S}}^{{\underline {k}}}({\mathcal {G}})$ , whose last nonzero step is ${\mathbb {S}}^{{\underline {k}}}({\mathcal {F}})$ . Let us denote by $P({\mathbb {S}}^{{\underline {k}}}({\mathcal {G}}))$ the penultimate step of this filtration (see also Remark 4.1).

Let ${\mathcal {H}}$ be another finite locally free sheaf on $\overline {S}$ , with the same rank r as ${\mathcal {F}}$ , and consider $\psi \colon {\mathcal {F}} \to {\mathcal {H}}$ a morphism. The adjugate morphism $ \psi ^{\mathrm {adj}} \colon {\mathcal {H}} \longrightarrow {\mathcal {F}} \otimes \det ({\mathcal {H}}) \otimes \det ({\mathcal {F}})^{-1}$ is defined as

$$\begin{align*}\psi^{\mathrm{adj}} \colon {\mathcal{H}} \cong (\wedge^{r-1} {\mathcal{H}})^\vee& \otimes \det({\mathcal{H}}) \xrightarrow{(\wedge^{r-1}\psi)^\vee \otimes {\mathrm{id}}_{\det({\mathcal{H}})}} (\wedge^{r-1}{{\mathcal{F}}})^\vee \\&\otimes \det({\mathcal{H}}) \cong {\mathcal{F}} \otimes {\mathcal{H}} \otimes \det({\mathcal{F}})^{-1}. \end{align*}$$

Lemma 6.1 Let $\overline {S}, {\mathcal {F}}, {\mathcal {G}}, {\mathcal {H}}$ , and ${\underline {k}}$ be as above. Suppose that we have a morphism

$$\begin{align*}\phi \colon {\mathcal{G}} \longrightarrow {\mathcal{H}} \end{align*}$$

of ${\mathcal {O}}_{\overline {S}}$ -sheaves such that the restriction $\phi |_{\mathcal {F}}$ of $\phi $ to ${\mathcal {F}}$ is an isomorphism on a dense open $S \subseteq \overline {S}$ . Write $d = \det \phi |_{\mathcal {F}} \in H^0(\overline {S}, \det {\mathcal {H}} \otimes (\det {\mathcal {F}})^{-1})$ for the determinant of that restriction. Then, the morphism

$$\begin{align*}{\mathcal{G}} \overset{\phi}{\longrightarrow} {\mathcal{H}} \overset{\phi^{-1}}{\longrightarrow} {\mathcal{F}} \overset{d}{\longrightarrow} {\mathcal{F}} \otimes \det {\mathcal{H}} \otimes (\det {\mathcal{F}})^{-1} \end{align*}$$

extends naturally from S to $\overline {S}$ . Similarly, the map

$$\begin{align*}P({\mathbb{S}}^{{\underline{k}}}({\mathcal{G}})) \overset{{\mathbb{S}}^{{\underline{k}}}(\phi)}{\longrightarrow} {\mathbb{S}}^{{\underline{k}}}({\mathcal{H}}) \overset{{\mathbb{S}}^{{\underline{k}}}(\phi^{-1})}{\longrightarrow} {\mathbb{S}}^{{\underline{k}}}({\mathcal{F}}) \overset{d}{\longrightarrow} {\mathbb{S}}^{{\underline{k}}}({\mathcal{F}}) \otimes \det {\mathcal{H}} \otimes \det {\mathcal{F}}^{-1} \end{align*}$$

extends from S to $\overline {S}$ .

Proof The key observation, due to [Reference Eischen, Flander, Ghitza, Mantovan and McAndrew15, Section 3.3.1], is that the morphism ${d\cdot \phi ^{-1} \colon {\mathcal {H}} \longrightarrow {\mathcal {F}} \otimes \det {\mathcal {H}} \otimes (\det {\mathcal {F}})^{-1}}$ is the adjugate of $\phi |_{\mathcal {F}}$ and thus it is naturally defined over all of $\overline {S}$ . Notice that on ${\mathcal {F}}$ the morphism $\phi |_{\mathcal {F}}^{\mathrm {adj}} \circ \phi $ is simply multiplication by d.

The statement is trivial when ${\underline {k}} = {\underline {0}}$ . We present the proof in the case ${{\underline {k}} = (k, 0, \dots , 0)}$ , where ${\mathbb {S}}^{{\underline {k}}}({\mathcal {G}}) = {\mathrm {Sym}}^k({\mathcal {G}})$ . The general case is proved similarly. One can consider the commutative diagram

with the vertical arrow being the morphism we want to extend. The diagonal arrow is simply ${\mathrm {id}} \otimes (\phi |_{\mathcal {F}}^{\mathrm {adj}} \circ \phi )$ , so it extends from S to $\overline {S}$ , and it clearly factors through $F^{k-1}({\mathrm {Sym}}^k({\mathcal {G}}))$ , so we conclude.

6.1.1 The ordinary splitting

On $S = S^\mu _K,$ the morphism $F \colon H^{(p)}_\sigma \to H_\sigma $ induces via (2.3) an isomorphism $ F \colon {\underline {\omega }}_{A^\vee /S, \sigma }^{\vee , p} \to {\underline {\omega }}_{A^\vee /S, \sigma }^{\vee }. $ Moreover, F kills the subsheaf ${\underline {\omega }}_{A/S, \sigma }^{(p)} \subset H^{(p)}_\sigma = H^{1, (p)}_{\mathrm {dR}}(A/S^\mu _K)$ and the composition

$$\begin{align*}{\underline{\omega}}_{A^\vee/S, \sigma}^{\vee} \overset{F^{-1}}{\longrightarrow} {\underline{\omega}}_{A^\vee/S, \sigma}^{\vee, p} \overset{F}{\longrightarrow} H_\sigma \end{align*}$$

gives a natural splitting of the Hodge filtration (2.3). We write ${{\mathcal {U}}_\sigma {\kern-1pt}={\kern-1pt} {{\mathrm {im}}}(F \colon H^{(p)}_\sigma {\kern-1pt}\to{\kern-1pt} H_\sigma )}$ , the locally free subsheaf giving the splitting. We have that ${\mathcal {U}}_\sigma = \ker (V \colon H_\sigma \to H_\sigma ^{(p)})$ . Moreover, $V \colon H_\sigma \to H_\sigma ^{(p)}$ has image ${\underline {\omega }}_{A/S, \sigma }^{(p)}$ and, over $S^\mu _K$ , it induces the isomorphism $V \colon {\underline {\omega }}_{A/S, \sigma } \to {\underline {\omega }}_{A/S, \sigma }^{(p)}$ . This implies that the composition

$$\begin{align*}p_{{\mathrm{ur}}, 1} \colon H_\sigma \overset{V}{\longrightarrow} {\underline{\omega}}_{A/S, \sigma}^{(p)} \overset{V^{-1}}{\longrightarrow} {\underline{\omega}}_{A/S, \sigma} \end{align*}$$

is the projection $H_\sigma \to {\underline {\omega }}_{A/S, \sigma }$ parallel to ${\mathcal {U}}_\sigma $ , or, in other words, the projection ${H_\sigma \to {\underline {\omega }}_{A/S, \sigma }}$ relative to the splitting of (2.3) given by ${\mathcal {U}}_\sigma $ . This is usually called the unit-root splitting. We have proved the following.

Lemma 6.2 (Ordinary unit-root splitting)

Over the ordinary locus $S^\mu _K$ , the Hodge filtration (2.3) admits a natural splitting induced by the Frobenius $F \colon H_\sigma ^{(p)} \to H_\sigma $ . The corresponding projection $p_{{\mathrm {ur}}, 1} \colon H_\sigma \to {\underline {\omega }}_\sigma $ is induced by V as described above.

The morphism $p_{{\mathrm {ur}}, 1}$ cannot be extended naturally to $S_K = \overline {S}^\mu _K$ , because the factor $V^{-1} \colon {\underline {\omega }}_{\sigma }^{(p)} \to {\underline {\omega }}_{\sigma }$ does not make sense outside $S^\mu _K$ . If we consider instead the map ${A_1 \cdot p_{{\mathrm {ur}}, 1}}$ , we see that it can be extended to the whole $S_K$ : the linear map

$$\begin{align*}A_1 \cdot V^{-1} \colon {\underline{\omega}}_{\sigma}^{(p)} \longrightarrow {\underline{\omega}}_{\sigma} \otimes (\det {\underline{\omega}}_\sigma)^{p-1} \end{align*}$$

is the adjugate to the morphism $V \colon {\underline {\omega }}_\sigma \to {\underline {\omega }}_\sigma ^{(p)}$ , which is defined over $S_K$ . In fact, from Lemma 6.1, we deduce the following.

Lemma 6.3 The morphism $A_1 \cdot p_{{\mathrm {ur}}, 1} \colon H_{\sigma } \to {\underline {\omega }}_{\sigma } \otimes (\det {\underline {\omega }}_\sigma )^{p-1}$ extends to $S_K$ . Similarly, for ${\underline {k}} = (k_1, \dots , k_{n-1})$ , with $k_1 \geq k_2 \geq \cdots \geq k_{n-1} \geq 0$ ,

$$\begin{align*}A_1 \cdot {\mathbb{S}}^{{\underline{k}}}(p_{{\mathrm{ur}}, 1})|_{P({\mathbb{S}}^{{\underline{k}}}(H_\sigma))} \colon P({\mathbb{S}}^{{\underline{k}}}(H_\sigma)) \longrightarrow {\mathbb{S}}^{{\underline{k}}}({\underline{\omega}}_\sigma) \otimes (\det {\underline{\omega}}_\sigma)^{p-1} \end{align*}$$

extends to $S_K$ .

6.1.2 Partial splittings

Fix some $2 \leq r \leq n-1$ . On $S=S_{K, w_r}$ , the map $F \colon {\underline {\omega }}_{A^\vee /S, \sigma }^{\vee , p} \longrightarrow {\underline {\omega }}_{A^\vee /S, \sigma }^{\vee }$ is zero. Nevertheless, we can still look at the surjection $V \colon H_\sigma \to {\underline {\omega }}_\sigma ^{(p)}$ . On S, the map $V \colon {\underline {\omega }}_\sigma \to {\underline {\omega }}_\sigma ^{(p)}$ is not surjective, but it induces an isomorphism between the multiplicative part ${\underline {\omega }}_\mu $ and its p-twist. Write $H_{\mu , r} {:=}q H_\sigma /{\underline {\omega }}_{0, r}$ . We have the composition

$$\begin{align*}p_{{\mathrm{ur}}, r} \colon H_{\mu, r} \overset{V}{\longrightarrow} {\underline{\omega}}_\mu^{(p)} \overset{V^{-1}}{\longrightarrow} {\underline{\omega}}_{\mu, r}. \end{align*}$$

This composition is a right inverse of the inclusion ${\underline {\omega }}_{\mu , r} \to H_{\mu , r}$ . That is, the kernel of $p_{{\mathrm {ur}}, r}$ gives a splitting of the short exact sequence

$$\begin{align*}0 \longrightarrow {\underline{\omega}}_{\mu, r} \longrightarrow H_{\mu, r} \longrightarrow {\underline{\omega}}_{A^\vee/S, \sigma}^\vee \longrightarrow 0. \end{align*}$$

We call this the (generalized) partial unit-root splitting. Recall that we write $A_{r} = \det (V \colon {\underline {\omega }}_\mu \to {\underline {\omega }}_\mu ^{(p)})$ . As in the ordinary case, we can prove the following using Lemma 6.1.

Lemma 6.4 The morphism $A_{r} \cdot p_{{\mathrm {ur}}, r} \colon H_{\mu , r} \longrightarrow {\underline {\omega }}_{\mu , r} \otimes (\det {\underline {\omega }}_{\mu , r})^{p-1}$ extends to $\overline {S}_{K, w_r}$ . Similarly, for ${\underline {k}} = (k_1, k_2, \dots , k_{n-r})$ , with $k_1 \geq k_2 \geq \cdots \geq k_{n-r} \geq 0$ ,

$$\begin{align*}A_{r} \cdot {\mathbb{S}}^{{\underline{k}}}(p_{{\mathrm{ur}}, r})|_{P({\mathbb{S}}^{{\underline{k}}}(H_{\mu, r}))} \colon P({\mathbb{S}}^{{\underline{k}}}(H_{\mu, r})) \longrightarrow {\mathbb{S}}^{{\underline{k}}}({\underline{\omega}}_{\mu, r}) \otimes (\det {\underline{\omega}}_{\mu, r})^{p-1} \end{align*}$$

extends to $\overline {S}_{K, w_r}$ .

In the process of constructing the generalized theta operators, we will give a mild generalization of Lemma 6.4, which again follows from Lemma 6.1.

6.2 Generalized theta operators

We prove here the main result of this article (Theorem 6.5). Let us start by recalling some notations. Pick $1 \leq r \leq n-1$ an integer and $({\underline {k}}, w)$ an automorphic weight with $k_{n-1}\geq 0$ . On $\overline {S}_{K, w_r}$ , we have the short exact sequence

$$\begin{align*}0 \longrightarrow {\underline{\omega}}_{0, r} \longrightarrow {\underline{\omega}}_{\sigma} \longrightarrow {\underline{\omega}}_{\mu, r} \longrightarrow 0, \end{align*}$$

where ${\underline {\omega }}_{0, r} = \ker (V^{r-1} \colon {\underline {\omega }}_\sigma \to {\underline {\omega }}_\sigma ^{(p^{r-1})})$ is a locally free sheaf of rank $r-1$ . This short exact sequence induces a filtration $F^{\bullet , r}$ on ${\underline {\omega }}^{{\underline {k}}, w}$ over $\overline {S}_{K, w_r}$ , which is trivial when $r=1$ (see Section 4.1 for more details on the definition of $F^{\bullet , r}$ ). The filtration $F^{\bullet , r}$ gives rise to the graded sheaf ${\mathrm {gr}}^{\bullet , r}({\underline {\omega }}^{{\underline {k}}, w})$ over $\overline {S}_{K, w_r}$ . Notice that the graded sheaf ${\mathrm {gr}}^{\bullet , r}({\underline {\omega }}^{{\underline {k}}, w})$ decomposes as a direct sum of sheaves

$$\begin{align*}{\mathrm{gr}}^{{\underline{a}}, r}({\underline{\omega}}^{{\underline{k}}, w}) {:=}q\, {{\mathrm{im}}}( \otimes_j \wedge^{l_j-b_j}({\underline{\omega}}_0) \otimes \wedge^{b_j}({\underline{\omega}}_\mu) \to \otimes_i {\mathrm{Sym}}^{k_i-a_i}({\underline{\omega}}_0) \otimes {\mathrm{Sym}}^{a_i}({\underline{\omega}}_\mu)), \end{align*}$$

where ${\underline {a}}, {\underline {b}} \in {\mathbb Z}^{n-1}_{\geq 0}$ are such that $a_i \leq k_i, a_i \geq a_{i+1}$ for all i, that is, ${\underline {a}}$ is a subtableau of ${\underline {k}}$ , and ${\underline {b}}$ is the transpose of ${\underline {a}}$ . In particular, given $f \in H^0(\overline {S}_{K, w_r}, \mathrm {gr}^{\bullet , r}({\underline {\omega }}^{{\underline {k}}, w}))$ , we can write it uniquely as $f = \sum _{{\underline {a}}} f_{{\underline {a}}}$ , where $f_{{\underline {a}}} \in H^0(\overline {S}_{K, w_r}, {\mathrm {gr}}^{{\underline {a}}, r}({\underline {\omega }}^{{\underline {k}}, w}))$ .

Theorem 6.5 Let $1 \leq r < n$ be an integer and $({\underline {k}}, w)$ an automorphic weight with ${k_{n-1}\geq 0}$ . There exists a differential operator

$$\begin{align*}\theta_r \colon {\mathrm{gr}}^{\bullet, r}({\underline{\omega}}^{{\underline{k}}, w}) \longrightarrow {\mathrm{gr}}^{\bullet, r}({\underline{\omega}}^{{\underline{k}}+{\underline{\Delta}}_r, w-1}), \end{align*}$$

defined on the (closure of the) Ekedahl–Oort stratum $\overline {S}_{K, w_r}$ , with

$$\begin{align*}{\underline{\Delta}}_r = (p+1, p, \ldots, p, 1, \ldots, 1), \end{align*}$$

where exactly the last $r-1$ entries are $1$ . The operator $\theta _r$ satisfies the following properties:

1. (1) The operator $\theta _r$ is $A_{r}$ -linear, that is, $\theta _r(A_{r}) = 0$ , where $A_{r}$ is the partial Hasse invariant defined above.

2. (2) The operator $\theta _r$ commutes with the action of prime-to-p Hecke operators.

3. (3) Let $f \in H^0(\overline {S}_{K, w_r}, \mathrm {gr}^{\bullet , r}({\underline {\omega }}^{{\underline {k}}, w}))$ and write it as $f = \sum _{{\underline {a}}} f_{{\underline {a}}}$ . If $r = n-1, n-2$ , then $\theta _r(f)$ is divisible by the Hasse invariant $A_r$ if and only if for each component $f_{{\underline {a}}}$ either $A_r \mid f_{{\underline {a}}}$ or $p \mid a_1$ .

We prove this theorem throughout this section.

6.2.1 Ordinary operator

Let $({\underline {k}}, w)$ be an automorphic weight such that $k_{n-1} \geq 0$ , with the corresponding automorphic sheaf ${\underline {\omega }}_{A/S_K}^{{\underline {k}},w}$ , which we denote simply by ${\underline {\omega }}^{{\underline {k}},w}$ . Consider $H = H^1_{\mathrm {dR}}(A/S_K)$ and write

$$\begin{align*}H^{{\underline{k}}, w} {:=}q {\mathbb{S}}^{\underline{k}}(H_\sigma) \otimes_{{\mathcal{O}}_S} \delta_{\sigma}^w. \end{align*}$$

The GM connection induces a morphism

$$\begin{align*}{\underline{\omega}}^{{\underline{k}}, w} \overset{\nabla}{\longrightarrow} H^{{\underline{k}}, w} \otimes \Omega^1_{S_K/{\mathbb{F}}}. \end{align*}$$

By Lemma 4.2, the image of this map lies in the penultimate step of the natural filtration $F^{\bullet }$ of $H^{{\underline {k}}, w}$ described in Remark 4.1. In particular, from Lemma 6.3, we deduce the following.

Lemma 6.7 [Reference Eischen, Flander, Ghitza, Mantovan and McAndrew15, Section 3.3.2]

Consider the composition

$$\begin{align*}\psi \colon {\underline{\omega}}^{{\underline{k}}, w} \overset{\nabla}{\longrightarrow} H^{{\underline{k}}, w} \otimes \Omega^1_{S/{\mathbb{F}}} {\longrightarrow} {\underline{\omega}}^{{\underline{k}}, w} \otimes (\det {\underline{\omega}}_\sigma)^{p-1} \otimes \Omega^1_{S_K/{\mathbb{F}}} \longrightarrow {\underline{\omega}}^{{\underline{k}}+{\underline{p-1}}, w} \otimes \Omega^1_{S_K/{\mathbb{F}}}, \end{align*}$$

where the second arrow is $A_1 \cdot ({\mathbb {S}}^{{\underline {k}}}(p_{{\mathrm {ur}},1}) \otimes {\mathrm {id}}_{\delta _\sigma } \otimes {\mathrm {id}}_{\Omega ^1})$ and $({\underline {p-1}}, 0)=((p-1, \ldots , p-1), 0)$ is the weight of $A_1$ . Then $\psi $ extends from $S_K^\mu $ to $S_K$ .

We may now define $\theta _1$ , the ordinary theta operator, as follows. Composing $\psi $ from Lemma 6.7 with ${\mathrm {id}} \otimes {\underline {{\mathrm {ks}}}}^{-1}$ , we obtain

(6.1) $$ \begin{align} {\underline{\omega}}^{{\underline{k}}, w} {\longrightarrow} {\underline{\omega}}^{{\underline{k}}+{\underline{p-1}}, w} \otimes {\underline{\omega}}_\sigma \otimes \det {\underline{\omega}}_\sigma \otimes \delta_\sigma^{-1}. \end{align} $$

To define $\theta _1$ , we further compose (6.1) with the natural morphism ${\underline {\omega }}^{{\underline {k}} + {\underline {p-1}}, w} \otimes {\underline {\omega }}_\sigma \otimes \det {\underline {\omega }}_\sigma \to {\underline {\omega }}^{{\underline {k}} + {{\underline {\Delta }}}_1, w}$ and obtain

$$\begin{align*}\theta_1 \colon {\underline{\omega}}^{{\underline{k}}, w} {\longrightarrow} {\underline{\omega}}^{{\underline{k}}+{\underline{\Delta}}_1, w-1}, \end{align*}$$

where ${\underline {\Delta }}_1 = (p+1, p, \dots , p)$ . This proves the existence part of Theorem 6.5 in the case $r=1$ . Note that this theta operator is already defined in [Reference Eischen, Flander, Ghitza, Mantovan and McAndrew15].

Proposition 6.8 The operator $\theta _1$ satisfies the following fundamental properties:

1. (1) For $f \in H^0(S_K, {\underline {\omega }}^{{\underline {k}}, w}), g \in H^0(S_K, {\underline {\omega }}^{{\underline {k}}', w'}),$ we have $\theta _1(fg) = f\theta _1(g) +\theta _1(f)g,$ that is, the operator $\theta _1$ is a derivation of the algebra of modular forms of level K with coefficients in ${\mathbb {F}}$ .

2. (2) The operator $\theta _1$ is $A_1$ -linear, that is, $\theta _1(A_1) = 0$ .

3. (3) The operator $\theta _1$ is Hecke-equivariant.

Proof

1. (1) This is clear from the definition of $\nabla $ and thus of $\theta _1$ .

2. (2) We can choose a small enough dense open $U \subseteq S^\mu _K$ such that ${\underline {\omega }}_\sigma $ and $H_\sigma $ are both free on U and the Hodge filtration (2.3) splits. In particular, we can pick a local basis $e_1, \dots , e_{n-1}$ of ${\underline {\omega }}_\sigma $ and complete it to a local basis of $H_\sigma $ with the addition of some $e_n$ . Then, in these local coordinates, the ordinary Hasse invariant is $A_1|_U = a \cdot (e_1 \wedge \cdots \wedge e_{n-1})^{p-1}$ , for some $a \in {\mathcal {O}}_{S_K}(U)$ . With these notations, from the definition of $A_1$ , one can compute that

   $$ \begin{align*} (\wedge^{p-1} V)(A_1)|_U &= a (Ve_1 \wedge \cdots \wedge Ve_{n-1})^{p-1} = a (a (e_1 \wedge \cdots \wedge e_{n-1})^{p})^{p-1} \\ &= a^p (e_1 \wedge \cdots \wedge e_{n-1})^{p(p-1)} = A_1^p. \end{align*} $$

   Therefore, from the definition of $p_{{\mathrm {ur}}, 1} = V|_{{\underline {\omega }}_\sigma }^{-1} \circ V_{\sigma }$ and Lemma 4.5, we have on U that

   $$ \begin{align*} (\wedge^{p-1}p_{{\mathrm{ur}}, 1}) \circ \nabla(A_1) &= (\wedge^{p-1}V^{-1})(\wedge^{p-1} V) \nabla(A_1)\\ &= (\wedge^{p-1} V^{-1}) \nabla((\wedge^{p-1} V) A_1) = (\wedge^{p-1} V^{-1}) \nabla(A_1^p) = 0. \end{align*} $$

   Hence, $\theta _1(A_1)|_U = 0$ . By density of $U \subset S_K$ , we deduce that $\theta _1(A_1) = 0$ .

3. (3) The construction of $\theta _1$ does not depend on the level structure and it is compatible with changes of base involved in the definition of the Hecke operators. Moreover, any prime-to-p quasi-isogeny $f \colon A \to A'$ , for $A, A' \in S_K(T)$ , $T \to S_K$ finite étale, induces an isomorphism $f^\ast \colon {\underline {\omega }}_{A'/T} \to {\underline {\omega }}_{A/T}$ such that

   
   is commutative. All of this, combined with Lemma 4.5, implies that $\theta _1$ is Hecke-equivariant. Notice that this relies on our convention for the definition of ${\underline {{\mathrm {ks}}}}$ : had the polarization been involved, the morphism ${\underline {{\mathrm {ks}}}}$ would have introduced a nontrivial Hecke-action (see [Reference Eischen, Flander, Ghitza, Mantovan and McAndrew15, Lemma 4.1.2]).

Remark 6.9 We can provide an alternative proof of point (2) of Proposition 6.8. From, for instance, the proof of [Reference Katz29, Proposition 4.1.1], we see that we can cover $S^\mu _K$ with maps $U \to S_K^\mu $ , finite étale of degree $\# {\mathrm {GL}}_{n-1}({\mathbb {F}}_p)$ , on which we can find a local basis $e_1, \dots , e_{n-1} \in {\underline {\omega }}_\sigma (U)$ , which extends to a basis $e_1, \dots , e_n \in H_\sigma (U)$ , such that $V(e_i) = e_i^{(p)} = e_i \otimes 1$ , for $i = 1, \dots , n-1,$ and $V(e_n) = 0.$ This follows from applying the proposition in loc. cit. with $H = {\underline {\omega }}^\vee _\sigma |_O$ , $S_1 = O$ , for some non-empty affine open $O\subseteq S^\mu _K$ , with notations H and $S_1$ as in the reference, and $F = V^{\vee }$ . In that case, we have

$$\begin{align*}A_1|_U = (e_1 \wedge \cdots \wedge e_{n-1})^{p-1}. \end{align*}$$

Moreover, by Lemmas 4.5 and 4.6, we have that on U

$$\begin{align*}0 = \nabla(e_i^{(p)}) = \nabla(V(e_i)) = V(\nabla(e_i)), \quad i = 1, 2, \dots, n-1, \end{align*}$$

since $dF_{U, {\mathrm {abs}}} = 0$ . This implies that $\nabla (e_i) = e_n \otimes \omega _i$ , for some sections $\omega _i \in \Omega ^1_{S_K/{\mathbb {F}}}(U),$ which form a local basis of $\Omega ^1_{S_K/{\mathbb {F}}}$ , by Proposition 4.9. With this description of the $\nabla (e_i)$ ’s, one can see that, over $U,$ we have $\theta _1(A_1)|_U = 0$ , thanks to a simple computation, using the definitions of $\nabla , p_{{\mathrm {ur}}, 1}$ and thus $\theta _1$ . Therefore, since we can cover $S^\mu _K$ with such finite étale U’s, $\theta _1(A_1)$ vanishes on the ordinary locus $S^\mu _K$ , and by density, we have $\theta _1(A_1) = 0$ on all of $S_K$ .

This shows that $\theta _1$ admits a relatively simple and explicit description étale-locally on the ordinary locus. In fact, one can prove that it is enough to take $U = I$ , with I being the (small) Igusa variety $I \to S^\mu _K$ of level $1$ and achieve the same results working globally. This is related to the approach taken in [Reference de Shalit and Goren9] to construct theta operators when p is inert. See, in particular, [Reference de Shalit and Goren9, Section 2.1]. See also [Reference Howe24].

One could use this approach more generally on $S_{K, w_r}$ , $2 \leq r \leq n-1$ , instead of $S_K^\mu $ , to obtain étale-local bases of ${\underline {\omega }}_{\mu , r}$ such that $V \colon {\underline {\omega }}_{\mu , r} \to {\underline {\omega }}_{\mu , r}^{(p)}$ admits a description relative to them analogous to the one given above. We do not pursue this direction here.

6.2.2 Generalized theta operators

Fix $2 \leq r \leq n-1$ . Let $({\underline {k}}, w)$ be an automorphic weight such that $k_{n-1} \geq 0$ and consider the sheaf ${\underline {\omega }}^{{\underline {k}},w}$ . We look in particular at the restriction of ${\underline {\omega }}^{{\underline {k}}, w}$ to $\overline {S}_{K, w_r}$ , where we have, as above, the short exact sequence

(6.2) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{0, r} \longrightarrow {\underline{\omega}}_\sigma \longrightarrow {\underline{\omega}}_{\mu, r} \longrightarrow 0. \end{align} $$

From this sequence, we get a Koszul filtration on $\wedge ^j {\underline {\omega }}_\sigma $ , for all $j=1, \dots , n-1$ , following the notations of Section 4.1.3. Similarly, we can define a Koszul filtration on $\wedge ^j H_\sigma $ , $j=1, \dots , n-1$ , from

(6.3) $$ \begin{align} 0 \longrightarrow {\underline{\omega}}_{0, r} \longrightarrow H_\sigma \longrightarrow H_{\mu, r} \longrightarrow 0. \end{align} $$

From Lemma 4.5 and Proposition 5.5, we deduce that the GM connection on $\overline {S}_{K, w_r}$ induces morphisms between the graded pieces of these two Koszul filtrations as follows:

In fact, we can be more precise. From the inclusion ${\underline {\omega }}_{\mu , r} \subset H_{\mu , r}$ , we obtain a third Koszul filtration on $\wedge ^{j-i} H_{\mu , r}$ , which we denote $K^{\bullet }(\wedge ^{j-i} H_{\mu , r})$ . Applying Lemma 4.2 to $K^{\bullet }$ , we see that actually

$$\begin{align*}\nabla_r({\mathrm{gr}}^i(\wedge^j {\underline{\omega}}_\sigma)) \subseteq \wedge^i {\underline{\omega}}_{0, r} \otimes K^{j-i-1}(\wedge^{j-i} H_{\mu, r}) \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}} \subseteq {\mathrm{gr}}^i(\wedge^j H_\sigma) \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}}. \end{align*}$$

Let $\psi $ be the following composition:

$$ \begin{align*} \psi \colon {\mathrm{gr}}^i(\wedge^j {\underline{\omega}}_\sigma) &\overset{\nabla_r}{\longrightarrow} \wedge^i {\underline{\omega}}_{0, r} \otimes K^{j-i-1}(\wedge^{j-i} H_{\mu, r}) \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}}\\ &\longrightarrow {\mathrm{gr}}^i(\wedge^j {\underline{\omega}}_\sigma) \otimes (\det {\underline{\omega}}_{\mu, r})^{p-1} \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}}, \end{align*} $$

considered over $S_{K, w_r}$ , where the second map acts as $A_{r} \cdot \wedge ^{j-i} p_{{\mathrm {ur}}, r}$ on the factor $K^{j-i-1}(\wedge ^{j-i} H_{\mu , r})$ and is the identity on the others. From Lemma 6.4, we deduce that $\psi $ extends to $\overline {S}_{K, w_r}$ .

This construction can be generalized. From the Koszul filtrations on $\wedge ^j {\underline {\omega }}_\sigma , \wedge ^j H_\sigma $ defined by the subsheaf ${\underline {\omega }}_{0, r}$ , we obtain filtrations on

$$\begin{align*}\otimes_j \wedge^{l_j}( {\underline{\omega}}_\sigma), \otimes_j \wedge^{l_j}(H_\sigma),\quad j=1, 2, \dots, n-1, \end{align*}$$

as explained in Section 4.1.4. From these, in turn, we obtain filtrations on ${\underline {\omega }}^{{\underline {k}}, w}$ and $H^{{\underline {k}}, w} = {\mathbb {S}}^{{\underline {k}}}(H_\sigma ) \otimes \delta _\sigma ^w$ , the filtration on $\delta _\sigma ^w$ being trivial. The morphism $p_{{\mathrm {ur}}, r} \colon H_{\mu , r} \to {\underline {\omega }}_{\mu , r}$ from Section 6.1.2, along with ${\mathrm {id}}_{{\underline {\omega }}_{0, r}}, {\mathrm {id}}_{\delta _\sigma }$ , lends naturally a morphism of sheaves

$$\begin{align*}{\mathrm{gr}}^{\bullet}(p_{{\mathrm{ur}}, r}) \colon {\mathrm{gr}}^{\bullet}(H^{{\underline{k}}, w}) \longrightarrow {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}, w}) \end{align*}$$

on $S_{K, w_r}$ . In fact, under the natural identification ${\mathrm {gr}}^{\bullet }({\mathbb {S}}^{{\underline {k}}}(H_\sigma )) = {\mathbb {S}}^{{\underline {k}}}({\mathrm {gr}}^{\bullet }(H_\sigma )) = {\mathbb {S}}^{{\underline {k}}}({\underline {\omega }}_{0,r} \oplus H_{\mu , r})$ , see (6.2) and (6.3), we have ${\mathrm {gr}}^{\bullet }(p_{{\mathrm {ur}}, r}) = {\mathbb {S}}^{{\underline {k}}}({\mathrm {id}}_{{\underline {\omega }}_{0,r}} \oplus p_{{\mathrm {ur}}, r}) \otimes {\mathrm {id}}_{\delta _\sigma }^w$ . From Lemmas 6.1 and 6.4, we deduce the following, which is analogous to Lemma 6.7.

Lemma 6.10 Consider the composition

$$\begin{align*}\psi \colon {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}, w}) \overset{\nabla_r}{\longrightarrow} {\mathrm{gr}}^{\bullet}(H^{{\underline{k}}, w}) \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}} {\longrightarrow} {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}, w}) \otimes (\det {\underline{\omega}}_\mu)^{p-1} \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}}, \end{align*}$$

where the second arrow is $A_{r} \cdot {\mathrm {gr}}^{\bullet }(p_{{\mathrm {ur}}, r}) \otimes {\mathrm {id}}_{\Omega ^1}$ . Then, $\psi $ , which a priori is only defined on $S_{K, w_r}$ , extends to $\overline {S}_{K, w_r}$ .

We can further compose $\psi $ with ${\mathrm {id}} \otimes {\underline {{\mathrm {ks}}}}_{\mu , r}^{-1}$ defined in Proposition 5.5 to get

$$ \begin{align*} {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}, w}) &\overset{\psi}{\longrightarrow} {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}, w}) \otimes (\det {\underline{\omega}}_{\mu, r})^{p-1} \otimes \Omega^1_{\overline{S}_{K, w_r}/{\mathbb{F}}} \\ & \overset{{\mathrm{id}} \otimes {\underline{{\mathrm{ks}}}}^{-1}_{\mu}}{\longrightarrow} {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}, w}) \otimes (\det {\underline{\omega}}_{\mu, r})^{p-1} \otimes {\underline{\omega}}_\mu \otimes \det {\underline{\omega}}_\sigma \otimes \delta_{\sigma}^{-1}\\ & \longrightarrow {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}+{\underline{\Delta}}_r, w-1}), \end{align*} $$

where the last map is given by the natural multiplication morphisms. Therefore, we obtain a morphism on ${\mathrm {gr}}^{\bullet }({\underline {\omega }}^{{\underline {k}}, w})$ , shifting the weight by ${\underline {\Delta }}_r =(p+1, p, \ldots , p, 1, \ldots , 1)$ . We call this composition

$$\begin{align*}\theta_r \colon {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}, w}) \longrightarrow {\mathrm{gr}}^{\bullet}({\underline{\omega}}^{{\underline{k}}+{\underline{\Delta}}_r, w-1}) \end{align*}$$

the generalized (partial) theta operator relative to the stratum $\overline {S}_{K, w_r}$ . This proves the existence part of Theorem 6.5 in the case $r \geq 2$ .

Proposition 6.11 (Basic properties of $\theta _r$ )

The operator $\theta _r$ satisfies the following fundamental properties:

1. (1) For $f \in H^0(\overline {S}_{K, w_r}, {\mathrm {gr}}^{\bullet }({\underline {\omega }}^{{\underline {k}}, w})), g \in H^0(\overline {S}_{K, w_r}, {\mathrm {gr}}^{\bullet }({\underline {\omega }}^{{\underline {k'}}, w'})),$ we have $\theta _r(fg) = f\theta _r(g) +\theta _r(f)g$ .

2. (2) The operator $\theta _r$ is $A_{r}$ -linear, that is, $\theta _r(A_{r}) = 0$ .

3. (3) The operator $\theta _r$ is Hecke-equivariant.

Proof

1. (1) This follows from the construction of $\theta _r$ and the properties of $\nabla _r$ .

2. (2) Consider $U \subseteq S_{K, w_r}$ a small enough dense open, so that we can choose a local basis $e_1, \dots , e_n$ of $H_\sigma $ over U such that $e_1, \dots , e_{r-1}$ is a basis of ${\underline {\omega }}_0$ , $e_1, \dots , e_{n-1}$ a basis of ${\underline {\omega }}_\sigma $ , and $e_n$ reduces to a basis of $H_\sigma /{\underline {\omega }}_\sigma $ . In particular, we see that

   $$\begin{align*}A_{r}|_U = a (\overline{e}_r \wedge \cdots \wedge \overline{e}_{n-1})^{p-1}, \end{align*}$$
   for some $a \in {\mathcal {O}}_{S_{K, w_r}}(U)$ , where $\overline {\cdot }$ denotes the reduction through $H_\sigma \to H_{\mu , r}$ . As in Proposition 6.8, one can see that $\theta _r(A_{r})|_U = 0$ , so that by density $\theta _r(A_{r}) = 0$ .

3. (3) The same arguments that we used for $\theta _1$ work.

Remark 6.12 Notice that multiplication by $A_r$ , for $1 \leq r \leq n-1$ , is also Hecke-equivariant. By this, we mean that it induces the commutative diagram

for any $i \geq 0$ , $g \in \mathbf {G}({\mathbb {A}}^{p, \infty })$ and automorphic weight $({\underline {k}}, w)$ , where ${\underline {w}}_r = (p-1, \dots , p-1, \dots , 0)$ is the weight shift produced by $A_r$ . This follows from the fact that any prime-to-p quasi-isogeny $f \colon A \to A'$ , for $A, A' \in \overline {S}_{K, w_r}(T)$ , $T \to \overline {S}_{K, w_r}$ finite étale, induces an isomorphism $f^\ast \colon {\underline {\omega }}_{A'/T} \to {\underline {\omega }}_{A/T}$ that respects the filtration ${\underline {\omega }}_{0, r} \subseteq {\underline {\omega }}_{\sigma }$ and sends $A_r(A'/T) \in H^0(T, (\det {\underline {\omega }}_{A'/T, \mu })^{p-1})$ to $A_r(A/T) \in H^0(T, (\det {\underline {\omega }}_{A/T, \mu })^{p-1})$ .

Remark 6.13 Notice that even though $\theta _r$ is not defined on ${\underline {\omega }}^{{\underline {k}}, w}$ , it is defined on the graded parts of a filtration on such sheaves. Since this filtration is Hecke-equivariant, so is the one it induces on $H^0(\overline {S}^{\mathrm {min}}_{K, w_r}, {\underline {\omega }}^{{\underline {k}}, w})$ and, therefore, every Hecke-eigensystem appearing in $H^0(\overline {S}_{K, w_r}, {\underline {\omega }}^{{\underline {k}}, w})$ will also appear in

$$\begin{align*}H^0(\overline{S}_{K, w_r}, {\mathrm{gr}}^{\bullet, r}({\underline{\omega}}^{{\underline{k}}, w})). \end{align*}$$

Furthermore, thanks to the work of [Reference Goldring and Koskivirta21], the Hecke-eigensystems found in $\{H^0(\overline {S}_{K, w_r}, {\underline {\omega }}^{{\underline {k}}, w})\}_{{\underline {k}}, w}$ are the same as those found in $\{H^0(\overline {S}_{K}, {\underline {\omega }}^{{\underline {k}}, w})\}_{{\underline {k}}, w}$ .