2.1 Some notation

Let $L_0$ be a free $\mathcal {O}_{\mathcal {K}}$-module of rank $2$ with basis $\mathtt {w}_1, \mathtt {w}_2$, and we equip $L_0\otimes _{\mathbb {Z}}\mathbb {Q}$ with a skew-Hermitian form $\left <\,,\,\right>_{L_0}$ whose matrix with respect to the basis $\mathtt {w}_1, \mathtt {w}_2$ is given by a matrix $\zeta _{0} \in \text {skew-}\operatorname {\mathrm {Her}}_2(\mathcal {O}_{\mathcal {K}})$ with $\delta \zeta _0$ positive definite. Let $X, Y$ be free $\mathcal {O}_{\mathcal {K}}$-modules of rank $1$ with bases $\mathtt {x}_1, \mathtt {y}_1$. Let $X^{\vee }=\mathfrak {d}^{-1}_{\mathcal {K}/\mathbb {Q}}\cdot \mathtt {x}_1$ and $L=X^{\vee }\oplus L_0\oplus Y$. Equip $L\otimes _{\mathbb {Z}}\mathbb {Q}$ with the skew-Hermitian form $\left <\,,\,\right>_{L}$ whose matrix with respect to the basis $\mathtt {x}_1, \mathtt {w}_1, \mathtt {w}_2, \mathtt {y}_1$ is given by ${\zeta }=\begin {pmatrix}&&1\\&\zeta _0\\-1\end {pmatrix}.$

Define the similitude unitary groups ${G'}=\operatorname {\mathrm {GU}}(2)$ and ${G}=\operatorname {\mathrm {GU}}(3,1)$ (over $\mathbb {Z}$) as: for all $\mathbb {Z}$-algebra R

(2.1.1)$$ \begin{align} \begin{aligned} \operatorname{\mathrm{GU}}(2)(R)&=\{(g,\nu)\in \operatorname{\mathrm{GL}}_{\mathcal{O}_{\mathcal{K}}\otimes_{\mathbb{Z}} R}(L_0\otimes_{\mathbb{Z}} R)\times R^{\times}\,:\,\left<gv_1,gv_2\right>_{L_0}=\nu\left<v_1,v_2\right>_{L_0}\},\\ \operatorname{\mathrm{GU}}(3,1)(R)&=\{(g,\nu)\in \operatorname{\mathrm{GL}}_{\mathcal{O}_{\mathcal{K}}\otimes_{\mathbb{Z}} R}(L\otimes_{\mathbb{Z}} R)\times R^{\times}\,:\,\left<gv_1,gv_2\right>_{L}=\nu\left<v_1,v_2\right>_{L}\}, \end{aligned}\\[-15pt]\nonumber \end{align} $$

and the unitary groups $\operatorname {\mathrm {U}}(2)$ (resp. $\operatorname {\mathrm {U}}(3,1)$) as the subgroup of $\operatorname {\mathrm {GU}}(2)$ (resp. $\operatorname {\mathrm {GU}}(3,1)$) consisting of elements with $\nu =1$.

2.2 Shimura variety

We fix an open compact subgroup $K^p_f\subset G(\mathbb {A}^p_{\mathbb {Q},f})$, assumed throughout to be neat. Let $2\pi i\left <\,,\,\right>:L\times L\rightarrow \mathbb {Z}(1)$ be the alternating pairing $2\pi i\cdot \mathrm {Tr}_{\mathcal {K}/\mathbb {Q}}\circ \left <\,,\,\right>_L$ and $h:\mathbb {C}\rightarrow \mathrm {End}_{\mathcal {O}_{\mathcal {K}}\otimes _{\mathbb {Z}}\mathbb {R}}(L\otimes _{\mathbb {Z}}\mathbb {R})$ be the homomorphism given by

$$\begin{align*}h(u+iv):\left(\mathtt{x}_1,\mathtt{w}_1,\mathtt{w}_2,\mathtt{y}_1\right)\mapsto \left(\mathtt{x}_1,\mathtt{w}_1,\mathtt{w}_2,\mathtt{y}_1\right) \begin{pmatrix} 1\otimes u&&&-1\otimes v\\ &1\otimes u+\delta\otimes\frac{v}{\sqrt{\delta\bar{\delta}}}\\ &&1\otimes u+\delta\otimes\frac{v}{\sqrt{\delta\bar{\delta}}}\\ 1\otimes v&&& 1\otimes u \end{pmatrix}.\\[-15pt] \end{align*}$$

Then the tuple $\left (\mathcal {O}_{\mathcal {K}},c,L,2\pi i\left <\,,\,\right>,h\right )$ defines a Shimura datum of polarization, endomorphism and level structure (PEL) type with reflex field $\mathcal {K}$.

Consider the moduli problem sending every locally Noetherian connected $\mathcal {O}_{\mathcal {K},(p)}$-scheme S to the set of isomorphism classes of tuples $(A,\lambda ,i,\alpha ^p)$ with:

• • A an abelian scheme of relative dimension $4$ over S;

• • $\lambda :A\rightarrow A^{\vee }$ a $\mathbb {Z}^{\times }_{(p)}$-polarization;

• • $i:\mathcal {O}_{\mathcal {K}}\hookrightarrow \mathrm {End}_S A$ an embedding such that the induced $\mathcal {O}_{\mathcal {K}}$-action on $\operatorname {\mathrm {Lie}} A_{/S}$ satisfies the determinant condition, that is,

  $$ \begin{align*} \det\left(X-i(b)|\operatorname{\mathrm{Lie}} A_{/S}\right)&=(X-b)^3(X-\bar{b})\\[-15pt] \end{align*} $$
  for all $b\in \mathcal {O}_{\mathcal {K}}$;

• • $\alpha ^p$ an (integral) $K^p_f$-level structure on $(A, \lambda , i)$ of type $\left (L\otimes \widehat {\mathbb {Z}}^{(p)},\mathrm {Tr}_{\mathcal {K}/\mathbb {Q}}\circ \left <\,,\,\right>_L\right )$; that is, a $\pi _1(S,\bar {s})$-invariant $K^p_f$-orbit of $\mathcal {O}_{\mathcal {K}}$-module isomorphisms $L\otimes \widehat {\mathbb {Z}}^{(p)}\rightarrow T^{(p)}A_{\bar {s}}$, where $\bar {s}$ is a geometric point of S and $T^{(p)}A_{\bar {s}}$ is the prime-to-p Tate module of $A_{\bar {s}}$, together with an isomorphism $\widehat {\mathbb {Z}}^{(p)}(1)\stackrel {\sim }{\rightarrow }\mathbb {G}_{m,\bar {s}}$ making the following diagram commute:

  

(See [Reference LanLan13, Def. 1.4.1.4].)

Since $K^p_f$ is neat, the above moduli problem is represented by a smooth quasi-projective scheme ${\mathscr {S}}$ over $\mathcal {O}_{\mathcal {K},(p)}$ (see [Reference LanLan13, Thm. 1.4.1.2, Cor. 7.2.3.10]).

Denote by ${\mathscr {S}^{\mathrm {tor}}}$ the toroidal compactification of $\mathscr {S}$, which is a proper smooth scheme over $\mathcal {O}_{\mathcal {K},(p)}$ containing $\mathscr {S}$ as an open dense subscheme with complement being a relative Cartier divisor with normal crossings. (In our special case $\operatorname {\mathrm {GU}}(3,1)$ here, there is a unique choice of the polyhedral cone decomposition for the toroidal compactification.) We denote by ${\mathcal {I}_{\mathscr {S}^{\mathrm {tor}}}}$ the ideal sheaf of the boundary of $\mathscr {S}^{\mathrm {tor}}$. By [Reference LanLan13, Thm. 6.4.1.1], the universal family $(\mathscr {A},\lambda ,i,\alpha ^p)$ over $\mathscr {S}$ extends to a degenerating family $(\mathscr {G},\lambda ,i,\alpha ^p)$ over $\mathscr {S}^{\mathrm {tor}}$. Moreover, the base change of $\mathscr {S}$ (resp. $\mathscr {S}^{\mathrm {tor}}$) to $\mathcal {K}$ agrees with the Shimura variety over $\mathcal {K}$ (resp. its toroidal compactification) representing the moduli problem with full level structure at p (see [Reference LanLan15, (A.4.17), (A.4.18)]).

2.3 Hasse invariant

Set ${\underline {\omega }}:=e^*\Omega ^1_{\mathscr {G}/\mathscr {S}^{\mathrm {tor}}}$, where $e:\mathscr {S}^{\mathrm {tor}}\rightarrow \mathscr {G}$ is the zero section of the semiabelian scheme $\mathscr {G}$ over $\mathscr {S}^{\mathrm {tor}}$. Let ${\omega }$ be the line bundle $\det \underline {\omega }=\wedge ^{\mathrm {top}}\underline {\omega }$. The minimal compactification of $\mathscr {S}$ is defined as

$$\begin{align*}{\mathscr{S}^{\min}} =\mathrm{Proj}\Biggl(\bigoplus\limits_{k\geqslant 0}H^0(\mathscr{S}^{\mathrm{tor}},\omega^k)\Biggr). \end{align*}$$

Let $\pi :\mathscr {S}^{\mathrm {tor}}\rightarrow \mathscr {S}^{\min }$ be the canonical projection. The push-forward $\pi _*\omega $ is an ample line bundle, and $\pi ^*\pi _*\omega \cong \omega $ (see [Reference LanLan13, Thm. 7.2.4.1]).

In the following, with a slight abuse of notation, we also denote by $\mathscr {S}^{\mathrm {tor}}$ and $\mathscr {S}^{\min }$ their base change to $\mathbb {Z}_p$ via the map $\mathcal {O}_{\mathcal {K},(p)}\rightarrow \mathbb {Z}_p$ induced by our fixed embedding $\iota _p$, and let $\mathscr {S}^{\mathrm {tor}}_{/\mathbb {F}_p}$ and $\mathscr {S}^{\min }_{/\mathbb {F}_p}$ be their corresponding special fibers.

Let ${\mathrm {Ha}}\in H^0\left (\mathscr {S}^{\mathrm {tor}}_{/\mathbb {F}_p},\omega ^{p-1}\right )$ be the Hasse invariant defined as in [Reference LanLan18, §6.3.1]. In particular, for each geometric point $\bar {s}$ of $\mathscr {S}^{\mathrm {tor}}_{/\mathbb {F}_p}$, the Hasse invariant of the corresponding semiabelian scheme $\mathscr {G}_{\bar {s}}$ is nonzero if and only if the abelian part of $\mathscr {G}_{\bar {s}}$ is ordinary. Because $\pi _*\omega $ is ample, for some $t_E> 0$, there exists an element in $H^0\left (\mathscr {S}^{\min },(\pi _*\omega )^{t_E(p-1)}\right )$ lifting the $t_E$-th power of the push-forward of $\mathrm {Ha}$; we denote by ${E}$ the pullback under $\pi $ of any such lift, which (because $\pi ^*\pi _*\omega \cong \omega $) defines an element $E\in H^0\left (\mathscr {S}^{\mathrm {tor}},\omega ^{t_E(p-1)}\right )$.

2.4 Some groups

Before moving on, we need to introduce some more group-theoretic notations. Given a matrix $a\in \operatorname {\mathrm {GL}}_4(\mathcal {K}\otimes _{\mathbb {Q}} \mathbb {Q}_p)$, we define $a^+,a^-\in \operatorname {\mathrm {GL}}_4(\mathbb {Q}_p)$ as

$$ \begin{align*} a^+=\varrho_{\mathfrak{p}}(a),\quad a^-=\varrho_{\bar{\mathfrak{p}}}(a), \end{align*} $$

where $(\varrho _{\mathfrak {p}},\varrho _{\bar {\mathfrak {p}}}):\mathcal {K}\otimes _{\mathbb {Q}}\mathbb {Q}_p\rightarrow \mathbb {Q}_p\times \mathbb {Q}_p$ are defined as in Notation, and we view an element $g\in G(\mathbb {Q}_p)$ as a matrix inside $\operatorname {\mathrm {GL}}_4(\mathcal {K}\otimes _{\mathbb {Q}} \mathbb {Q}_p)$ via the basis $(\mathtt {x}_1,\mathtt {w}_1,\mathtt {w}_2,\mathtt {y}_1)$. Then the corresponding matrices $g^+,g^-\in \operatorname {\mathrm {GL}}_4(\mathbb {Q}_p)$ satisfy

In the following, we will often write $g\in G(\mathbb {Q}_p)$ as $(g^+,g^-)$. Note that the map

(2.4.1)$$ \begin{align} \begin{aligned} G(\mathbb{Q}_p)&\rightarrow \operatorname{\mathrm{GL}}_4(\mathbb{Q}_p)\times\mathbb{Q}^{\times}_p\\ g&\mapsto (g^+,\nu(g)) \end{aligned} \end{align} $$

is an isomorphism.

There is a filtration ${\tt D} = \{\mathtt {D}^i\}_i$ of $L\otimes _{\mathbb {Z}}\mathbb {Z}_p$ given by

(2.4.2)$$ \begin{align} {\tt D}^{1}=0\;\subset\;{\tt D}^{0}=X^+_{p}\oplus L^+_{0,p}\oplus X^-_{p}\;\subset\; {\tt D}^{-1} = L\otimes_{\mathbb{Z}}\mathbb{Z}_p, \end{align} $$

where $X^+_p\subset X^{\vee }\otimes _{\mathbb {Z}}\mathbb {Z}_p=X\otimes _{\mathbb {Z}}\mathbb {Z}_p$ (resp. $X^-_p\subset X^{\vee }\otimes _{\mathbb {Z}}\mathbb {Z}_p$, $L^+_{0,p}\subset L_0\otimes _{\mathbb {Z}}\mathbb {Z}_p$) is the subspace on which $b\in \mathcal {O}_{\mathcal {K}}$ acts by $\iota _p(b)$ (resp. $\iota _p(\bar {b})$, $\iota _p(b)$). Note that $\mathtt {D}^0$ is isotropic with respect to $\mathrm {Tr}_{\mathcal {K}/\mathbb {Q}}\circ \left <\,,\,\right>_L$. For $R=\mathbb {Z}_p$ or $\mathbb {Q}_p$, we put

$$\begin{align*}{P_{\tt D}} (R)=\{g\in G(R)\,:\,g(\mathtt{D}^0)=\mathtt{D}^0\}; \end{align*}$$

that is, $P_{\mathtt {D}}(R)$ is the subgroup of $G(R)$ stabilizing the filtration $\mathtt {D}$ in (2.4.2). We see that $P_{\mathtt {D}}^{\pm }(R):=\{g^{\pm }\colon g\in P_{\mathtt {D}}(R)\}$ are the subgroups of $\operatorname {\mathrm {GL}}_4(R)$ given by

$$\begin{align*}P_{\mathtt{D}}^+=\left\{ \left(\begin{smallmatrix} \ast&\ast&\ast&\ast\\ \ast&\ast&\ast&\ast\\ \ast&\ast&\ast&\ast\\ &&&\ast \end{smallmatrix}\right) \right\},\quad P_{\mathtt{D}}^-=\left\{\left(\begin{smallmatrix} \ast&\ast&\ast&\ast\\ &\ast&\ast&\ast\\ &\ast&\ast&\ast\\ &\ast&\ast&\ast \end{smallmatrix}\right) \right\}. \end{align*}$$

Because $P_{\mathtt {D}}$ preserves both $X^+_p\oplus L^+_{0,p}$ and $X^-_p$, there is a natural projection $P_{\mathtt {D}}(R)\rightarrow \operatorname {\mathrm {GL}}_3(R)\times \operatorname {\mathrm {GL}}_1(R)$ which in terms of matrices is given by

$$ \begin{align*} g=(g^+,g^-)=\left(\left(\begin{smallmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ &&&a_{44}\end{smallmatrix}\right) ,\left(\begin{smallmatrix}b_{11}&b_{12}&b_{13}&b_{14}\\ &b_{22}&b_{23}&b_{24}\\ &b_{32}&b_{33}&b_{34}\\ &b_{42}&a_{43}&b_{44}\end{smallmatrix}\right) \right) &\mapsto \left(\left(\begin{smallmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{smallmatrix}\right) ,b_{11}\right). \end{align*} $$

We will consider the p-level subgroups given by

$$ \begin{align*} {K^{1}_{p,n}} &:=\left\{g\in G(\mathbb{Z}_p)\colon g^+\equiv\left(\begin{smallmatrix}\ast&\ast&\ast&\ast\\ p\ast&\ast&\ast&\ast\\ &&1&\ast\\ && &1\end{smallmatrix}\right) \quad\mod p^n\right\},\\ {K^{0}_{p,n}} &:=\left\{g\in G(\mathbb{Z}_p)\colon g^+\equiv\left(\begin{smallmatrix}\ast&\ast&\ast&\ast\\ p\ast&\ast&\ast&\ast\\ &&\ast&\ast\\ && &\ast\end{smallmatrix}\right)\quad \mod p^n\right\}, \end{align*} $$

where n is a positive integer.

2.5 Igusa towers

Let ${\mathscr {T}_{n}}$ be the ordinary locus of level $K^1_{p,n}$ attached to the PEL-type Shimura data introduced in §2.2 and the filtration $\mathtt {D}$ in (2.4.2), as constructed in [Reference LanLan18, Theorem 3.4.2.5]. By [Reference LanLan18, Prop. 3.4.6.3], $\mathscr {T}_{n}$ is a smooth quasi-projective scheme over $\mathbb {Z}_p$ representing a moduli problem for tuples $(A,\lambda ,i,\alpha ^p,\alpha _p)$, where $(A,\lambda ,i,\alpha ^p)$ is as in §2.2, and $\alpha _p$ is an ordinary $K^1_{p,n}$-level structure of A, that is, a $K^1_{p,n}$-orbit of group scheme embeddings $\mu _{p^n}\otimes \mathtt {D}^0\hookrightarrow A[p^n]$ with image isotropic for the $\lambda $-Weil pairing, compatible with the $\mathcal {O}_{\mathcal {K}}$-actions on $\mathtt {D}^0$ and $A[p^n]$ through i, as described in [Reference HidaHLTT16, §3.1.1].

Let ${\mathscr {T}^{\mathrm {tor}}_n}$ be the partial toroidal compactification of the ordinary locus $\mathscr {T}_{n}$ ([Reference LanLan18, Thm. 5.2.1.1]); it is obtained by gluing to $\mathscr {T}_{n}$ the toroidal boundary charts parameterizing degenerating families defined in [Reference LanLan18, Def. 3.4.2.0] (including an extensibility condition on the ordinary level structure). We note that, even though the generic fiber of $\mathscr {T}_{n}$ agrees with the Shimura variety of level $K^p_fK^1_{p,n}$ over $\mathbb {Q}_p$, the generic fiber of $\mathscr {T}^{\mathrm {tor}}_{n}$ is in general just an open subscheme of a toroidal compactification of the Shimura variety of level $K^p_fK^1_{p,n}$ over $\mathbb {Q}_p$.

Let ${\mathscr {T}^{\mathrm {tor}}_{n,m}}$ (resp. ${\mathscr {S}^{\mathrm {tor}}_{m}}$) be the base change of $\mathscr {T}_n^{\mathrm {tor}}$ (resp. $\mathscr {S}^{\mathrm {tor}}_{}$) to $\mathbb {Z}/p^m\mathbb {Z}$. By [Reference LanLan18, Lem. 6.3.2.7], $\mathscr {S}^{\mathrm {tor}}_m[1/E]$ agrees with the ordinary locus in [Reference LanLan18, Thm. 5.2.1.1] for full level at p, and by [Reference LanLan18, Cor. 5.2.2.3] the map $ \mathscr {T}^{\mathrm {tor}}_{n,m}\rightarrow \mathscr {S}^{\mathrm {tor}}_m[1/E]$ that forgets the ordinary $K^1_{p,n}$-level structure is finite étale. Concretely, we can describe $\mathscr {T}^{\mathrm {tor}}_{n,m}$ as

(2.5.1)$$ \begin{align} \underline{\operatorname{\mathrm{Isom}}}_{\mathscr{S}^{\mathrm{tor}}_m[1/E]}\left(\mu_{p^n}\otimes \mathtt{D}^0,\mathscr{G}[p^n]^{\mathrm{mult}}\right)/K^1_{p,n}\cap P_{\mathtt{D}}(\mathbb{Z}_p), \end{align} $$

where $P_{\mathtt {D}}(\mathbb {Z}_p)$ acts on an element $\alpha _p\in \underline {\operatorname {\mathrm {Isom}}}_{\mathscr {S}^{\mathrm {tor}}_m[1/E]}\left (\mu _{p^n}\otimes \mathtt {D}^0,\mathscr {G}[p^n]^{\mathrm {mult}}\right )$ by $(g\cdot \alpha _p)(v)=\alpha _p(g^{-1}v)$ for $v\in \mathtt {D}^0$. The fiber at a geometric point $\bar {s}\in \mathscr {S}^{\mathrm {tor}}_{n,m}$ parameterizes tuples $(\mathtt {F}^{\pm },\delta ^+_3,\delta ^-_1,\mathtt {F}^+_1[p])$, where ${{\tt F^\pm }}$ are filtrations

(2.5.2)$$ \begin{align} \begin{aligned} &\mathtt{F}^+:0=\mathtt{F}^+_0\subset \mathtt{F}^+_2\subset\mathtt{F}^+_3=\mathscr{G}_{\bar{s}}[p^n]^{\mathrm{mult}+}\\ &\mathtt{F}^-:0=\mathtt{F}^-_0\subset\mathtt{F}^-_1= \mathscr{G}_{\bar{s}}[p^n]^{\mathrm{mult} -} \end{aligned} \end{align} $$

compatible with the Weil pairing; $\delta ^+_3$ and $\delta ^-_1$ are isomorphisms

$$ \begin{align*} &{\delta^{+}_3} : \mu_{p^n}\cong \mathrm{Gr}^{\mathtt{F^+}}_3, &&{\delta^{-}_1}:\mu_{p^n}\cong \mathrm{Gr}^{\mathtt{F^-}}_1; \end{align*} $$

and $0\subset \mathtt {F}^+_1[p]\subset \mathtt {F}^+_2[p]$ is a two-step filtration of $\mathtt {F}^+_2[p]$.

2.6 p-adic forms

Define the space of mod $p^m$ automorphic forms on G of level n by

$$\begin{align*}{V_{n,m}} :=H^0\left(\mathscr{T}^{\mathrm{tor}}_{n,m},\mathcal{O}_{\mathscr{T}^{\mathrm{tor}}_{n,m}}\right). \end{align*}$$

Letting $\mathcal {I}_{\mathscr {T}^{\mathrm {tor}}_{n,m}}:=(\mathscr {T}^{\mathrm {tor}}_{n,m}\rightarrow \mathscr {S}^{\mathrm {tor}}_m[1/E])^*\mathcal {I}_{\mathscr {S}^{\mathrm {tor}}}$, we similarly define the space of mod $p^m$ cuspidal automorphic forms on G of level n by

$$\begin{align*}{V^0_{n,m}} :=H^0\left(\mathscr{T}^{\mathrm{tor}}_{n,m},\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_{n,m}}\right). \end{align*}$$

Passing to the limit, we obtain corresponding spaces of p-adic automorphic forms (with p-power torsion coefficients) by

Let ${T_{\mathrm {so}}}(\mathbb {Z}_p)=\mathbb {Z}_p^{\times }\times \mathbb {Z}_p^{\times }$. The map sending $(a_1,a_2)$ to $g\in G(\mathbb {Z}_p)$ with $g^+=\left (\begin {smallmatrix}1\\&1\\&&a_1\\&&&a^{-1}_2\end {smallmatrix}\right ) $and $\nu (g)=1$ identifies $T_{\mathrm {so}}(\mathbb {Z}/p^n)=(\mathbb {Z}/p^n\mathbb {Z})^{\times }\times (\mathbb {Z}/p^n\mathbb {Z})^{\times }$ with $K^0_{p,n}/K^1_{p,n}$. Hence, the group $T_{\mathrm {so}}(\mathbb {Z}_p)$ naturally acts on

making these spaces into $\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)]\!]$-modules.

By a p-adic weight (for a semiordinary form) we mean a $\overline {\mathbb {Q}}_p$-valued character of $T_{\mathrm {so}}(\mathbb {Z}_p)$, that is, a pair $(\tau ^+,\tau ^-)$, where $\tau ^{\pm }:\mathbb {Z}^{\times }_p\rightarrow \overline {\mathbb {Q}}^{\times }_p$ are continuous characters, and we say that a p-adic weight is arithmetic if it is of the form $(x,y)\mapsto \epsilon ^+(x)x^{t^+}\cdot \epsilon ^-(y)y^{t^-}$, where $\epsilon ^{\pm }$ is of finite order and $t^{\pm }\in \mathbb {Z}$. If $(\tau ^+,\tau ^-)$ is arithmetic, we put $\tau _{\mathrm {f}}^{\pm }:=\epsilon ^{\pm }$ and $\tau _{\mathrm {alg}}^{\pm }=t^{\pm }$. Given a p-adic weight $(\tau ^+,\tau ^-)$, we denote by $V_{n,m}[\tau ^+,\tau ^-]$ the subspace of $V_{n,m}\otimes _{\mathbb {Z}_p}\mathcal {O}_{\mathbb {Q}_p(\tau ^+,\tau ^-)}$ on which $T_{\mathrm {so}}(\mathbb {Z}_p)$ acts by the inverse of the character $(\tau ^+,\tau ^-)$. Similarly, we define the spaces $V^0_{n,m}[\tau ^+,\tau ^-]$, , .

2.7 Classical automorphic forms

The weights of classical holomorphic automorphic forms on G are indexed by tuples of integers ${\underline {t}} = (t^+_1,t^+_2,t^+_3;t^-_1)$ with $t^+_1\geq t^+_2\geq t^+_3$. (When $t^+_3\geq -t^-_1+4$, the Archimedean component of the corresponding automorphic representation is isomorphic to a holomorphic discrete series.)

Let ${W_{\underline {t}}}$ be the algebraic representation of $\operatorname {\mathrm {GL}}_3\times \operatorname {\mathrm {GL}}_1$ given by

$$ \begin{align*} W_{\underline{t}}:=W_{(t^+_1,t^+_2,t^+_3)}\boxtimes W_{t^-_1}.\\[-15pt] \end{align*} $$

Here, for any algebra R, letting $R[\underline {x},\det \underline {x}^{-1}]$ be the polynomial ring in the variables $x_{ij}$ ($1\leq i,j\leq 3$) and $\left (\det (x_{ij})_{1\leq i,j\leq 3}\right )^{-1}$, $W_{(t^+_1,t^+_2,t^+_3)}(R)$ is the R-submodule of $R[\underline {x},(\det \underline {x})^{-1}]$ spanned by

$$\begin{align*}x^{a_1}_{11}x^{a_2}_{12}x^{a_3}_{13}\det\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}^{b_1}\det\begin{pmatrix}x_{11}&x_{13}\\x_{21}&x_{23}\end{pmatrix}^{b_2}\det\begin{pmatrix}x_{12}&x_{13}\\x_{22}&x_{23}\end{pmatrix}^{b_3} \det\begin{pmatrix}x_{11}&x_{12}&x_{13}\\ x_{21}&x_{22}&x_{23}\\ x_{31}&x_{32}&x_{33}\end{pmatrix}^{t^+_3},\\[-15pt] \end{align*}$$

where $a_i,b_i\geq 0,\,a_1+a_2+a_3=t^+_1-t^+_2,\,b_1+b_2+b_3=t^+_2-t^+_3$, and $W_{t^-_1}(R)$ is the R-submodule of $R[x,x^{-1}]$ spanned by $x^{t^-_1}$.

The groups $\operatorname {\mathrm {GL}}_3$ and $\operatorname {\mathrm {GL}}_1$ act on $W_{\underline {t}}$ by right translation. One can check that the left translation of $\left (\begin {smallmatrix}a_1\\ \ast &a_2\\ \ast &\ast &a_3\end {smallmatrix}\right ) $on $W_{(t^+_1,t^+_2,t^+_3)}$ is by the scalar $a^{t^+_1}_1 a^{t^+_2}_2 a^{t^+_3}_3$; when $R=\mathbb {C}$, it is the irreducible algebraic representation of $\operatorname {\mathrm {GL}}_3(\mathbb {C})$ of highest weight $(t^+_1,t^+_2,t^+_3)$. Let $\mathfrak {e}_{\mathrm {can}}:W_{\underline {t}}\rightarrow \mathbb {A}^1$ be the linear functional defined by the evaluation at $(\mathbf {1}_3,\mathbf {1}_1)$.

Let $\underline {\omega }^+$ (resp. $\underline {\omega }^-$) be the subsheaf of $\underline {\omega }$ on which $i(b)$ acts by b (resp. $\bar {b}$) for all $b\in \mathcal {O}_{\mathcal {K}}$. Because p is unramified in $\mathcal {K}$, $\underline {\omega }^+$ (resp. $\underline {\omega }^-$) is locally free of rank $3$ (resp. rank $1$) and $\underline {\omega }=\underline {\omega }^+\oplus \underline {\omega }^-$.

Set

$$ \begin{align*} \omega^+_{(t^+_1,t^+_2,t^+_3)}&=\underline{\operatorname{\mathrm{Isom}}}_{\mathscr{S}^{\mathrm{tor}}}(\mathcal{O}^{\oplus{3}}_{\mathscr{S}^{\mathrm{tor}}},\underline{\omega}^+)\times^{\operatorname{\mathrm{GL}}_3} W_{(t^+_1,t^+_2,t^+_3)},\\ \omega^+_{t^-_1}&=\underline{\operatorname{\mathrm{Isom}}}_{\mathscr{S}^{\mathrm{tor}}}(\mathcal{O}_{\mathscr{S}^{\mathrm{tor}}},\underline{\omega}^+)\times^{\operatorname{\mathrm{GL}}_1} W_{t^-_1}\cong (\underline{\omega}^-)^{\otimes t^-_1},\\[-15pt] \end{align*} $$

and put ${\omega _{\underline {t}}}=\omega ^+_{(t^+_1,t^+_2,t^+_3)}\otimes \omega ^-_{t^-_1}$.

Letting $F/\mathbb {Q}_p$ be a finite extension containing the values of the finite-order character $\epsilon ^{\pm }$ of $\mathbb {Z}^{\times }_p$, and $S^{\mathrm {tor}}_{K^p_fK^1_{p,n}}$ be the toroidal compactification of the Shimura variety of level $K^p_fK^1_{p,n}$ defined over $\mathcal {K}$, we have

$$\begin{align*}{M_{(0,0,t^+;t^-)}\left(K^p_fK^1_{p,n},\epsilon^+,\epsilon^-;F\right)} = \left(H^0\Big(S^{\mathrm{tor}}_{K^p_fK^1_{p,n}},\underline{\omega}_{(0,0,t^+;t^-)}\Big)\otimes_{\mathcal{K}}F\right)[\epsilon^+,\epsilon^-],\\[-15pt] \end{align*}$$

the space of classical automorphic forms on G of weight $(0,0,t^+;t^-)$, level $K^p_fK^1_{p,n}$, and nebentypus $(\epsilon ^+,\epsilon ^-)$ for the action of $K^0_{p,n}/K^1_{p,n}$. Here, F is viewed as a $\mathcal {K}$-algebra via $\mathcal {K}\stackrel {\iota _p}{\rightarrow } \mathbb {Q}_p\rightarrow F$. Similarly, we have the space of classical cuspidal automorphic forms

$$\begin{align*}{M^0_{(0,0,t^+;t^-)}\left(K^p_fK^1_{p,n},\epsilon^+,\epsilon^-;F\right)} =\left(H^0\Big(S^{\mathrm{tor}}_{K^p_fK^1_{p,n}},\underline{\omega}_{(0,0;t^+;t^-)}\otimes\mathcal{I}_{\mathscr{S}^{\mathrm{tor}}}\Big)\otimes_{\mathcal{K}}F\right)[\epsilon^+,\epsilon^-].\\[-15pt] \end{align*}$$

There are classical embeddings

(2.7.1)

induced by the trivialization of $\underline {\omega }$ over the Igusa tower and the canonical functional $\mathfrak {e}_{\mathrm {can}}$. More precisely, the trivialization of $\underline {\omega }$ arises from the Hodge–Tate map $(\mathcal {G}_{/S}[p^n]^{\mathrm {mult}})^D\otimes _{\mathbb {Z}}\mathcal {O}_S\stackrel {\sim }{\rightarrow } e^*\Omega ^1_{\mathcal {G}/S}\otimes _{\mathbb {Z}}\mathbb {Z}/p^n\mathbb {Z}$, where $\mathcal {G}_{/S}$ is an ordinary semiabelian variety over S, the superscript $^D$ denotes the Cartier dual, and $e:S\rightarrow \mathcal {G}$ is the zero section.

Similarly, we have embeddings

(2.7.2)

2.8 The $\mathbb {U}_p$-operator

Given a tuple $(A,\lambda ,i,\alpha ^p,\alpha _p)$ parameterized by $\mathscr {T}_{n,m}$, where $\alpha _p:\mu _{p^n}\otimes \mathtt {D}^0\rightarrow A[p^n]^{\mathrm {mult}}$ is an isomorphism up to $K^1_{p,n}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$, the corresponding filtration $\mathtt {F^{\pm }}$ of $A[p^n]^{\mathrm {mult}\pm }$ is

$$ \begin{align*} \mathtt{F}^+&\colon\; 0=\mathtt{F}^+_0\subset\mathtt{F}^+_2=\{e^+_1,e^+_2\}\subset\mathtt{F}^+_3=\{e^+_1,e^+_2,e^+_3\}=A[p^n]^{\mathrm{mult}+},\\ \mathtt{F}^-&\colon\; 0=\mathtt{F}^-_0\subset\mathtt{F}^-_1=\{e^-_1\}=A[p^n]^{\mathrm{mult}-}, \end{align*} $$

where $(e^+_1,e^+_2,e^+_3;e^-_1)=\alpha _p(\mathtt {x}^+_1,\mathtt {w}^+_1,\mathtt {w}^+_2;\mathtt {x}^-_1)$.

Now, we define three $\mathbb {U}_p$-operators $U^+_{p,2},U^+_{p,3},U^-_{p,1}$.

2.8.1 $\mathbb {U}_p$-operators on $V_{n,m}$

For $j=2,3$, let ${\mathscr {C}^{+}_{j,n,m}}$ denote the solution to the moduli problem classifying tuples $(A,\lambda ,i,\alpha ^p,\alpha _p,C)$ with $C\subset A[p^2]$ a Lagrangian subgroup, (that is, maximal isotropic for the $\lambda $-Weil pairing) stable under the $\mathcal {O}_{\mathcal {K}}$-action through i such that $A[p]=C[p]\oplus \mathtt {F}^+_j[p]$. With $e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4$ a basis of $A[p^n]$ corresponding to $\alpha _p$, such a C is spanned by $(e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4)\cdot p^n\gamma _{C,p}^{\pm }$, where for $j=2$

(2.8.1)$$ \begin{align} \begin{aligned} \gamma^+_{C,p}&= \begin{pmatrix}1&&u_1&\ast\\&1&u_2&\ast\\&&1&\\&&&1\end{pmatrix}\begin{pmatrix}1&&&\\&1&&\\&&\frac{1}{p}&\\&&&\frac{1}{p}\end{pmatrix},\\ \gamma^-_{C,p}&= \begin{pmatrix}1\\&\iota_p(\bar{\zeta}_0)^{-1}\\&&1\end{pmatrix}\begin{pmatrix}1&\ast&&\ast\\&1\\&-u_2&1&u_1\\&&&1\end{pmatrix}\begin{pmatrix}\frac{1}{p}&&&\\&\frac{1}{p^2}&&\\&&\frac{1}{p}&\\&&&\frac{1}{p^2}\end{pmatrix}\begin{pmatrix}1\\&\iota_p(\bar{\zeta}_0)\\&&1\end{pmatrix} \end{aligned} \end{align} $$

with $u_1,u_2,\ast \in \mathbb {Z}_p$, and for $j=3$,

(2.8.2)$$ \begin{align} \begin{aligned} \gamma^+_{C,p}&=\begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}1&&&\\&1&&\\&&1&\\&&&\frac{1}{p}\end{pmatrix}, &\gamma^-_{C,p}&= \begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}\frac{1}{p}&&\\&\frac{1}{p^2}\\&&\frac{1}{p^2}\\&&&\frac{1}{p^2}\end{pmatrix} \end{aligned} \end{align} $$

with $\ast \in \mathbb {Z}_p$.

Similarly, let ${\mathscr {C}^{-}_{1,n,m}}$ be the moduli space classifying tuples $(A,\lambda ,i,\alpha ^p,\alpha _p,C)$ with $C\subset A[p^2]$ a Lagrange subgroup stable under the $\mathcal {O}_{\mathcal {K}}$-action through i such that $A[p]=C[p]\oplus \mathtt {F}^-_1[p]$. Then such a C is spanned by $(e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4)\cdot p^n\gamma ^{\pm }_{C,p}$ with

(2.8.3)$$ \begin{align} \begin{aligned} \gamma^+_{C,p}&=\begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}\frac{1}{p}&&&\\&\frac{1}{p}&&\\&&\frac{1}{p}&\\&&&\frac{1}{p^2}\end{pmatrix}, &\gamma^-_{C,p}&= \begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}1&&\\&\frac{1}{p}\\&&\frac{1}{p}\\&&&\frac{1}{p}\end{pmatrix}. \end{aligned} \end{align} $$

For $(\bullet ,j)=(+,2),(+,3),(-,1)$, we consider the diagram

(2.8.4)

with $p_1,p_2$ the projections given by

$$ \begin{align*} p_1:(A,\lambda,i,\alpha^p,\alpha_p,C)&\longmapsto(A,\lambda,i,\alpha^p,\alpha_p),\\ p_2:(A,\lambda,i,\alpha^p,\alpha_p,C)&\longmapsto(A',\lambda',i',(\alpha^p)',\alpha_p'), \end{align*} $$

where $A'=A/C$, $\lambda '$ is such that $\pi ^{\vee }\circ \lambda \circ \pi =p^2\lambda $ with $\pi :A\rightarrow A'$ the natural projection, $i'$ is the $\mathcal {O}_{\mathcal {K}}$-action i on A descended to $A'$, $(\alpha ^{p})'=\pi \circ \alpha ^p$, and $\alpha _p'$ given by $(e^{+\prime }_1,e^{+\prime }_2,e^{+\prime }_3;e^{-\prime }_1)$ defined as

$$\begin{align*}(e^{+\prime}_1,e^{+\prime}_2,e^{+\prime}_3;e^{-\prime}_1)=\begin{cases} \pi\left((e^+_1,e^+_2,e^+_3;e^-_1)\left(\begin{smallmatrix}1&&u_1\\&1&u_2\\&&1&\\&&&1\end{smallmatrix}\right) \left(\begin{smallmatrix}1\\&1\\&&p^{-1}\\&&&p^{-1}\end{smallmatrix}\right) \right), &\bullet=+,\, j=2,\\[9pt] \pi\left((e^+_1,e^+_2, e^+_3;e^-_1)\left(\begin{smallmatrix}1\\&1\\&&1\\&&&p^{-1}\end{smallmatrix}\right) \right),&\bullet=+,\,j=3,\\[9pt] \pi\left(e^+_1,e^+_2, e^+_3;e^-_1)\left(\begin{smallmatrix}p^{-1}\\&p^{-1}\\&&p^{-1}\\&&&1\end{smallmatrix}\right) \right), &\bullet=-,\,j=1. \end{cases} \end{align*}$$

One can check that if $(e^{+}_1,e^{+}_2,e^{+}_3;e^{-}_1)$ is up to $K^1_{p,n}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$, then $(e^{+\prime }_1,e^{+\prime }_2,e^{+\prime }_3;e^{-\prime }_1)$ is well-defined up to $K^1_{p,n}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$.

Define

(2.8.5)$$ \begin{align} {U^+_{p,j}}\colon\; H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}) \stackrel{p^*_2}{\longrightarrow} H^0(\mathscr{C}^+_{j,n,m},\mathcal{O}_{\mathscr{T}_{n,m}})\xrightarrow{p^{-j}\mathrm{Tr}\,p_1} H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}), \quad(j=2,3) \end{align} $$

(2.8.6)$$ \begin{align} {U^-_{p,1}}\colon\; H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}) \stackrel{p^*_2}{\longrightarrow} H^0(\mathscr{C}^-_{1,n,m},\mathcal{O}_{\mathscr{T}_{n,m}})\xrightarrow{p^{-3}\mathrm{Tr}\,p_1} H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}). \end{align} $$

The normalization factors $p^{-j}$ and $p^{-3}$ are the inverse of the pure inseparability degree of the corresponding projection $p_1$; they are the optimal normalization to preserve integrality [Reference HidaHid02, p.71]. By computing the effect of equations (2.8.5) and (2.8.6) on Fourier–Jacobi expansions, one can check that the operators preserve $V_{n,m}$ and $V^0_{n,m}$.

2.8.2 ${ \mathbb {U}}_p$-operators on $H^0 \left ({ \mathscr {T}}^{0,\mathrm {tor}}_{n,m}, \omega _{ \underline {t}}\right )$

Let $\mathscr {T}^{0,\mathrm {tor}}_{n,m}$ be the quotient of $\mathscr {T}^{\mathrm {tor}}_{n,m}$ by $K^0_{p,n}$, which can be described as

$$\begin{align*}\underline{\operatorname{\mathrm{Isom}}}_{\mathscr{S}^{\mathrm{tor}}_m[1/E]}\left(\mu_{p^n}\otimes \mathtt{D}^0,\mathscr{G}[p^n]^{\mathrm{mult}}\right)/K^0_{p,n}.\\[-9pt] \end{align*}$$

Compared to $\mathscr {T}^{\mathrm {tor}}_{n,m}$, the level structure at p for $\mathscr {T}^{0,\mathrm {tor}}_{n,m}$ forgets $\delta ^+_3,\delta ^-_1$ and parameterizes $(\mathtt {F}^{\pm },\mathtt {F}^+_1[p])$. We can define $\mathbb {U}_p$-operators on $H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\right )$. In order to see that they increase the level, we need to introduce Igusa towers of more general level structure at p.

Given $1\leq n_1,n_2\leq n_3\leq n_1+n_2$, define the level group

$$\begin{align*}K^0_{p,n_1,n_2,n_3}=\left\{g\in G(\mathbb{Z}_p): g^+=\left(\begin{smallmatrix} \ast & \ast & \ast & \ast\\ p\ast & \ast & \ast & \ast\\ p^{n_1}\ast & p^{n_1}\ast &\ast & \ast\\ p^{n_3}\ast & p^{n_3}\ast & p^{n_2}\ast & \ast \end{smallmatrix}\right) \right\}.\\[-9pt] \end{align*}$$

Define $\mathscr {T}^0_{n_1,n_2,n_3,m}$ as the quotient of $\mathscr {T}_{n,m}$, $n\geq n_1,n_2,n_3$, by $K^0_{p,n_1,n_2,n_3}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$, that is, the corresponding level structure at p parameterizes isomorphism $\alpha _p:\mu _{p^n}\otimes \mathtt {D}^0\rightarrow A[p^n]^{\mathrm {mult}}$ up to $K^0_{p,n_1,n_2,n_3}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$.

Like above, for $(\bullet ,j)=(+,2),(+,3),(-,1)$, let $\mathscr {C}^{0,\bullet }_{j,n_1,n_2,n_3,m}$ be the moduli space parameterizing tuples $(A,\lambda ,i,\alpha ^p,\alpha _p,C)$ with $C\subset A[p^2]$ a Lagrangian subgroup stable under the $\mathcal {O}_{\mathcal {K}}$-action through i such that $A[p]=C[p]\oplus \mathtt {F}^{\bullet }_j$.

Consider the diagrams

where $p_1,p_2$ are defined in the same way as in the diagram (2.8.4). This time, for $(e^{+}_1,e^{+}_2,e^{+}_3;,e^{-}_1)$ up to $K^0_{p,n_1,n_2,n_3}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$, the $(e^{+\prime }_1,e^{+\prime }_2,e^{+\prime }_3;,e^{-\prime }_1)$ is well defined up to $K^0_{p,n_1+1,n_2,n_3+1}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$ in case $\mathscr {C}^{0,+}_{2,n,m}$, and well defined up to $K^0_{p,n_1,n_2+1,n_3+1}\cap P_{\mathtt {D}}(\mathbb {Z}_p)$ in case $\mathscr {C}^{0,+}_{3,n,m}$ and case $\mathscr {C}^{0,-}_{1,n_1,n_2,n_3,m}$.

In order to define the $\mathbb {U}_p$-operators on the global sections of vector bundles, we also need maps

$$ \begin{align*} \pi^*&:H^0\left(\mathscr{C}^{0,-}_{1,n_1+1,n_2,n_3+1,m},p^*_2\omega_{\underline{t}}\right)\longrightarrow H^0\left(\mathscr{C}^{0,-}_{1,n_1,n_2,n_3,m},p^*_1\omega_{\underline{t}}\right)\\ \pi^*&:H^0\left(\mathscr{C}^{0,-}_{1,n_1,n_2+1,n_3+1,m},p^*_2\omega_{\underline{t}}\right)\longrightarrow H^0\left(\mathscr{C}^{0,-}_{1,n_1,n_2,n_3,m},p^*_1\omega_{\underline{t}}\right). \end{align*} $$

Suppose that $\underline {\varepsilon }_A$ (resp. $\underline {\varepsilon }_{A/C}$) is a basis of $\underline {\omega }_A$ (resp. $\underline {\omega }_{A/C}$) compatible with $\alpha _p$ (resp. $\alpha ^{\prime }_p$). We have $\underline {\varepsilon }_A=(\pi ^*\underline {\omega }_{A/C})g$ with

$$\begin{align*}g=\begin{cases} h\left(\begin{smallmatrix}1\\&1\\&&p^{-1}\\&&&p^{-1}\end{smallmatrix}\right) h', &\bullet=+,\, j=2,\\ h\left(\begin{smallmatrix}1\\&1\\&&1\\&&&p^{-1}\end{smallmatrix}\right) h', &\bullet=+,\, j=3,\\ h\left(\begin{smallmatrix}p^{-1}\\&p^{-1}\\&&p^{-1}\\&&&1\end{smallmatrix}\right) h', &\bullet=-,\, j=1, \end{cases} \end{align*}$$

for some $h,h'=\left (\begin {smallmatrix}\ast & p^n\ast & p^n\ast \\\ast & \ast & p\ast \\ \ast & \ast & \ast \\ &&&\ast \end {smallmatrix}\right ) \in \operatorname {\mathrm {GL}}_4(\mathbb {Z}_p)$. In all the cases, $g^{-1}$ belongs to the semigroup

$$\begin{align*}\Delta_+=\left\{\left(h\left(\begin{smallmatrix}a_1\\&a_2\\&&a_3\end{smallmatrix}\right) h',a_4\right)\,:\,h,h'=\left(\begin{smallmatrix}\ast & p\ast & p\ast\\\ast & \ast & p\ast\\\ast & \ast & \ast\end{smallmatrix}\right) \in\operatorname{\mathrm{GL}}_3(\mathbb{Z}_p),\,a^{-1}_1a_2,a^{-1}_2a_3\in\mathbb{Z}_p \right\}. \end{align*}$$

We make $\Delta _+$ act on $W_{\underline {t}}$ by

(2.8.7)$$ \begin{align} \begin{aligned} &\left(h\begin{pmatrix}a_1\\&a_2\\&&a_3\end{pmatrix}h',a_4\right)\cdot q(\underline{x},y)\\ =&\,q\left(\begin{pmatrix}p^{-v_p(a_1)}\\&p^{-v_p(a_2)}\\&&p^{-v_p(a_3)}\end{pmatrix}\underline{x}h\begin{pmatrix}a_1\\&a_2\\&&a_3\end{pmatrix}h',p^{-v_p(a_4)y}a_4 \right), \end{aligned} \end{align} $$

and define $\pi ^*$ as

$$ \begin{align*} \pi^*\vec{f}(A,\lambda,i,\alpha^p,\alpha_p,C,\underline{\varepsilon}_A)&=\vec{f}(A,\lambda,i,\alpha^p,\alpha_p,C,\underline{\varepsilon}_A,(\pi^*\underline{\varepsilon}_{A/C})g)\\ &=g^{-1}\cdot \vec{f}(A,\lambda,i,\alpha^p,\alpha_p,C,\underline{\varepsilon}_{A/C}). \end{align*} $$

The $\mathbb {U}_p$-operators are defined as

$$ \begin{align*} \begin{aligned} {U^{+}_{p,2}} &:H^0\left(\mathscr{T}^0_{n_1+1,n_2,n_3+1,m},\omega_{\underline{t}}\right) \stackrel{p^*_2}{\longrightarrow} H^0\left(\mathscr{C}^{0,+}_{2,n_1,n_2,n_3,m},p^*_2\omega_{\underline{t}}\right)\\ &\hspace{3em}\stackrel{\pi^*}{\longrightarrow} H^0\left(\mathscr{C}^{0,+}_{2,n_1,n_2,n_3,m},p^*_1\omega_{\underline{t}}\right) \xrightarrow{p^{-2}\mathrm{Tr}\,p_1} H^0(\mathscr{T}^0_{n_1,n_2,n_3,m},\omega_{\underline{t}}),\\ {U^{+}_{p,3}} &:H^0\left(\mathscr{T}^0_{n_1,n_2+1,n_3+1,m},\omega_{\underline{t}}\right) \stackrel{p^*_2}{\longrightarrow} H^0\left(\mathscr{C}^{0,+}_{3,n_1,n_2,n_3,m},p^*_2\omega_{\underline{t}}\right)\\ &\hspace{3em}\stackrel{\pi^*}{\longrightarrow} H^0\left(\mathscr{C}^{0,+}_{3,n_1,n_2,n_3,m},p^*_1\omega_{\underline{t}}\right) \xrightarrow{p^{-3}\mathrm{Tr}\,p_1} H^0(\mathscr{T}^0_{n_1,n_2,n_3,m},\omega_{\underline{t}}),\\ {U^{-}_{p,1}} &:H^0\left(\mathscr{T}^0_{n_1,n_2+1,n_3+1,m},\omega_{\underline{t}}\right) \stackrel{p^*_2}{\longrightarrow} H^0\left(\mathscr{C}^{0,-}_{1,n_1,n_2,n_3,m},p^*_2\omega_{\underline{t}}\right)\\ &\hspace{3em}\stackrel{\pi^*}{\longrightarrow} H^0\left(\mathscr{C}^{0,-}_{1,n_1,n_2,n_3,m},p^*_1\omega_{\underline{t}}\right) \xrightarrow{p^{-2}\mathrm{Tr}\,p_1} H^0(\mathscr{T}^0_{n_1,n_2,n_3,m},\omega_{\underline{t}}). \end{aligned} \end{align*} $$

Similarly as $V_{n,m}$, one can check that these $\mathbb {U}_p$-operators preserve the spaces $H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\right )$ and $H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{n,m}}\right )$. It is also easy to see that $U^+_{p,2}U^+_{p,3}U^-_{p,1}$ maps $H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\right )$ (resp. $H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{n,m}}\right )$) into $H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n-1,m},\omega _{\underline {t}}\right )$ (resp. $H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n-1,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{n-1,m}}\right )$), that is, the operator $U^+_{p,2}U^+_{p,3}U^-_{p,1}$ increases the level.

2.8.3 Adelic $\mathbb {U}_p$-operators

If we identify $G(\mathbb {Q}_p)$ with $\operatorname {\mathrm {GL}}_4(\mathbb {Q}_p)\times \operatorname {\mathrm {GL}}_1(\mathbb {Q}_p)$ via equation (2.4.1), the $\mathbb {U}_p$-operators defined above acting on classical automorphic forms on G of weight $(t^+_1,t^+_2.t^+_3;t^-_1)$ correspond to the following adelic operators (up to the action of the center of $G(\mathbb {Q}_p)$):

(2.8.8)$$ \begin{align} \begin{aligned} {U^{+}_{p,2}}:&\quad p^{\left<(t^+_1,t^+_2,t^+_3;-t^-_1)+2\rho_{\mathrm{c}},(1,1,0;0)\right>}\int_{N(\mathbb{Z}_p)}R \left(u\left(\begin{smallmatrix}p\\&p\\&&1\\&&&1\end{smallmatrix}\right) ,1\right)\,du; && \\ U^{+}_{p,3} :&\quad p^{\left<(t^+_1,t^+_2,t^+_3;-t^-_1)+2\rho_{\mathrm{c}},(1,1,1;0)\right>}\int_{N(\mathbb{Z}_p)}R \left(u\left(\begin{smallmatrix}p\\&p\\&&p\\&&&1\end{smallmatrix}\right) ,1\right)\,du; && \\ U^{-}_{p,1}:&\quad p^{\left<(t^+_1,t^+_2,t^+_3;-t^-_1)+2\rho_{\mathrm{c}},(0,0,0;-1)\right>}\int_{N(\mathbb{Z}_p)}R \left(u\left(\begin{smallmatrix}1\\&1\\&&1\\&&&p^{-1}\end{smallmatrix}\right) ,1\right)\,du, && \end{aligned}\\[-8pt]\nonumber \end{align} $$

where $N(\mathbb {Z}_p)=\left \{\left (\begin {smallmatrix}1&\ast & \ast & \ast \\&1&\ast & \ast \\&&1&\ast \\&&&1\end {smallmatrix}\right ) \in \operatorname {\mathrm {GL}}_4(\mathbb {Z}_p)\right \}$, $R(-)$ denotes the right translation, and $2\rho _{\mathrm {c}}=(2,0,-2;0)$ is the sum of the compact positive roots of G.

2.9 Statement of main theorem

Let $T_{\mathrm {so}}(\mathbb {Z}_p)^{\circ }$ be the connected component of $T_{\mathrm {so}}(\mathbb {Z}_p)$ containing $\mathbf {1}_2$, that is, $T_{\mathrm {so}}(1+p\mathbb {Z}_p)$, and put ${\Lambda _{\mathrm {so}}} =\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)^{\circ }]\!]$.

4.1 Cusp labels

Following [Reference LanLan13], a cusp label is a $K_f$-orbit of triples ${({\tt Z},\Phi ,\delta )}$, with:

• • $\mathtt {Z}:0\subset \mathtt {Z}_{-2}\subset \mathtt {Z}_{-1}=\mathtt {Z}_{-2}^\bot \subset L\otimes \hat {\mathbb {Z}}$ a fully symplectic admissible filtration.

• • $\Phi =(\mathtt {X},\mathtt {Y},\phi ,\varphi _{-2},\varphi _0)$ a torus argument, where $\phi :\mathtt {Y}\rightarrow \mathtt {X}$ is an $\mathcal {O}_{\mathcal {K}}$-linear embedding of locally free $\mathcal {O}_{\mathcal {K}}$-modules with finite cokernel, and $\varphi _{-2}:\mathrm {Gr}^{\mathtt { Z}}_{-2}\stackrel {\sim }{\rightarrow }\operatorname {\mathrm {Hom}}\left (\mathtt {X}\otimes \hat {\mathbb {Z}},\hat {\mathbb {Z}}(1)\right )$ and $\varphi _0:\mathrm {Gr}^{\mathtt {Z}}_0\stackrel {\sim }{\rightarrow }\mathtt {Y}\otimes \hat {\mathbb {Z}}$, with $\mathrm {Gr}^{\mathtt {Z}}_{-i}=\mathtt {Z}_{-i}/\mathtt {Z}_{-i-1}$, are isomorphisms such that $\left <v,w\right>=\varphi _0(v)\Big (\phi \big (\varphi _{-2}(w)\big )\Big )$,

• • $\delta :\mathrm {Gr}^{\mathtt {Z}}=\mathrm {Gr}^{\mathtt {Z}}_{-2}\oplus \mathrm {Gr}^{\mathtt {Z}}_{-1}\oplus \mathrm {Gr}^{\mathtt {Z}}_{0}\stackrel {\sim }{\rightarrow } L\otimes \hat {\mathbb {Z}}$ is an $\mathcal {O}_{\mathcal {K}}\otimes \hat {\mathbb {Z}}$-equivariant splitting.

Following [Reference LanLan18, Def. 3.2.3.1], an ordinary cusp label of tame level $K_f^p$ is a $K^p_fP_{\mathtt {D}}(\mathbb {Z}_p)$-orbit of triples $(\mathtt {Z},\Phi ,\delta )$ as above compatible with $\mathtt {D}$, in the sense that

$$ \begin{align*} \mathtt{Z}_{-2}\otimes_{\hat{\mathbb{Z}}}\mathbb{Z}_p\subset \mathtt{D}\subset \mathtt{Z}_{-1}\otimes_{\hat{\mathbb{Z}}}\mathbb{Z}_p. \end{align*} $$

There is a unique cusp label for which $\mathtt {Z}_{-2}=0$, and in our case, all other cusp labels have $\mathtt {Z}_{-2}$ with rank $1$. As explained in [Reference HidaHLTT16, B.2, B.3, B.11], the latter are parametrized by a certain double coset space. To recall this, we have the filtration

(4.1.1)$$ \begin{align} X^{\vee}\subset X^{\vee}\oplus L_0\subset L=X^{\vee}\oplus L_0\oplus Y. \end{align} $$

Let ${P}\subset G$ be the parabolic subgroup preserving this filtration, and ${P'}\subset P$ be the kernel of the natural projection $P\rightarrow \operatorname {\mathrm {GL}}(Y)$. Define the triple ${({\tt Z}^{(1)},\Phi ^{(1)},\delta ^{(1)})}$ as

• • $\mathtt {Z}^{(1)}$: $\mathtt {Z}_{-2}=X^{\vee }\otimes \hat {\mathbb {Z}}$, $\mathtt {Z}_{-1}=(X^{\vee }\oplus L_0)\otimes \hat {\mathbb {Z}}$.

• • $\Phi ^{(1)}=(\mathtt {X}^{(1)},\mathtt {Y}^{(1)},\phi ^{(1)},\varphi ^{(1)}_{-2},\varphi ^{(1)}_0)$ with $\mathtt {X}^{(1)}=\operatorname {\mathrm {Hom}}_{\mathcal {O}_{\mathcal {K}}}\left (X^{\vee },\mathcal {O}_{\mathcal {K}}(1)\right )$, $\mathtt {Y}^{(1)}=Y$, $\phi ^{(1)}:Y\rightarrow \operatorname {\mathrm {Hom}}_{\mathcal {O}_{\mathcal {K}}}\left (\mathfrak {d}^{-1}_{\mathcal {K}/\mathbb {Q}}\cdot \mathtt {x}_1,\mathcal {O}_{\mathcal {K}}(1)\right )$ induced by our fixed basis of $\mathtt {x}_1$ and $\mathtt {y}_1$ of X and Y and the pairing $2\pi i\cdot \mathrm {Tr}_{\mathcal {K}/\mathbb {Q}}\circ \left <\,,\,\right>_L$, $\varphi ^{(1)}_{-2}=2\pi \sqrt {-1}\cdot \mathrm {id}$ and $\varphi ^{(1)}_{0}=\mathrm {id}$.

• • $\delta ^{(1)}=\mathrm {id}$.

Denote by $\mathtt {V}\colon 0\subset \mathtt {V}_{-2}\subset \mathtt {V}_{-1}\subset L\otimes \mathbb {Q}$ the filtration obtained as equation (4.1.1) tensored with $\mathbb {Q}$. For every $g\in G(\mathbb {A}_{\mathbb {Q},f})$, define ${({\tt Z}^{(g)},\Phi ^{(g)},\delta ^{(g)})}$ by

• • $\mathtt {Z}^{(g)}_j=\left (g^{-1}(\mathtt {V}_j\otimes \mathbb {A}_{\mathbb {Q},f})\right )\cap \left (L\otimes \hat {\mathbb {Z}}\right )$.

• • $\Phi ^{(g)}=(\mathtt {X}^{(g)},\mathtt {Y}^{(g)},\phi ^{(g)},\varphi ^{(g)}_{-2},\varphi ^{(g)}_0)$ with $(\mathtt {X}^{(g)},\mathtt {Y}^{(g)},\phi ^{(g)})=(\mathtt {X}^{(1)},\mathtt {Y}^{(1)},\phi ^{(1)})$, $\varphi ^{(g)}_{-2}$ (resp. $\varphi ^{(g)}_{0}$) is the composition of $\varphi ^{(1)}_{-2}$ (resp. $\varphi ^{(1)}_{0}$) with $g:\mathrm {Gr}^{\mathtt {Z}^{(g)}}_{-2}\rightarrow \mathrm {Gr}^{\mathtt {Z}}_{-2}$ (resp. $g:\mathrm {Gr}^{\mathtt {Z}^{(g)}}_{0}\rightarrow \mathrm {Gr}^{\mathtt {Z}}_{0}$).

• • $\delta ^{(g)}$ is $\mathrm {Gr}^{\mathtt {Z}^{(g)}}\stackrel {\mathrm {Gr}(g)}{\rightarrow }\mathrm {Gr}^{\mathtt {Z^{(1)}}}\stackrel {\delta ^{(1)}}{\rightarrow } L\otimes \hat {\mathbb {Z}}\stackrel {g^{-1}}{\rightarrow }L\otimes \hat {\mathbb {Z}}$.

Then the map that sends g to (the cusp label represented by) $(\mathtt {Z}^{(g)},\Phi ^{(g)},\delta ^{(g)})$ sets up a bijection between the set of cusp labels with $\mathtt {Z}_{-2}$ rank $1$ and the double coset space

$$ \begin{align*} {C(K^p_f)} &= \operatorname{\mathrm{GL}}(X\otimes\mathbb{Q})\ltimes P'(\mathbb{A}^p_{\mathbb{Q},f})\backslash G(\mathbb{A}^p_{\mathbb{Q},f})/K^p_f. \end{align*} $$

(See [Reference LanLan15, Prop. A.5.9].)

Similarly, the ordinary cusp labels with $\mathtt {Z}_{-2}$ of rank $1$ are parameterized by

(4.1.2)$$ \begin{align} {C(K^p_fK^1_{p,n})_{\mathrm{ord}}} =\mathrm{Im}\left( G(\mathbb{A}^p_{\mathbb{Q},f}) P_{\mathtt{D}}(\mathbb{Q}_p)\longrightarrow\operatorname{\mathrm{GL}}(X\otimes\mathbb{Q})\ltimes P'(\mathbb{A}_{\mathbb{Q},f})\backslash G(\mathbb{A}_{\mathbb{Q},f})/K^p_fK^1_{p,n}\right), \end{align} $$

which is in a natural bijection with

(4.1.3)$$ \begin{align} \begin{aligned} &\left.\left(\operatorname{\mathrm{GL}}(X_{(p)})\ltimes (P'(\mathbb{A}^p_{\mathbb{Q},f})\times P'(\mathbb{Z}_p))\right)\middle\backslash G(\mathbb{A}^p_{\mathbb{Q},f})\times P_{\mathtt{D}}(\mathbb{Z}_p)\right/K^p_f(K^1_{p,n}\cap P_{\mathtt{D}}(\mathbb{Z}_p))\\ \cong & \,\,C(K^p_f)\times \left(U_{\mathcal{K},K^p_f}\ltimes P'(\mathbb{Z}_p)\backslash P_{\mathtt{D}}(\mathbb{Z}_p)/K^1_{p,n}\cap P_{\mathtt{D}}(\mathbb{Z}_p)\right), \end{aligned} \end{align} $$

where . By abuse of notation, we shall denote by g both elements in $G(\mathbb {A}^p_{\mathbb {Q},f})P_{\mathtt {D}}(\mathbb {Q}_p)$ and in its quotient $C(K^p_fK^1_{p,n})_{\mathrm {ord}}$. The context should eliminate any possible confusions.

4.2 The formal completion along boundary strata

Let $\mathscr {Z}_{g,n}$ be the stratum associated to $g\in C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ in the partial toroidal compactification $\mathscr {T}^{\mathrm {tor}}_n$. Then we have

$$\begin{align*}\mathscr{T}^{\mathrm{tor}}_n=\mathscr{T}_{n}\sqcup\bigsqcup\limits_{g\in C(K^p_fK^1_{p,n})_{\mathrm{ord}}} \mathscr{Z}_{g,n}. \end{align*}$$

(The choice of a polyhedral cone decomposition is unique in our special case $\operatorname {\mathrm {GU}}(3,1)$ because with $\Big (\mathbf {S}_{\Phi ^{(g)}_{K^p_fK^1_{p,n}}}\Big )$ defined as below and $\mathbf {P}_{\Phi ^{(g)}_{K^p_fK^1_{p,n}}}$ the subset of $\operatorname {\mathrm {Hom}}_{\mathbb {Z}}\Big (\mathbf {S}_{\Phi ^{(g)}_{K^p_fK^1_{p,n}}},\mathbb {Z}\Big )\otimes \mathbb {R}$ consisting of positive semidefinite Hermitian forms, we have $\mathbf { P}_{\Phi ^{(g)}_{K^p_fK^1_{p,n}}}=\mathbb {R}_{\geq 0}$.) Let ${\mathscr {T}^{\min }_n}$ be the partial minimal compactification of $\mathscr {T}_n$ (constructed in [Reference LanLan18, Theorem 6.2.1.1]). Denoting by ${\mathscr {Y}_{g,n}}$ the stratum associated to $g\in C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ in $\mathscr {T}^{\min }_n$, we similarly have

$$\begin{align*}\mathscr{T}^{\min}_n=\mathscr{T}_{n}\sqcup\bigsqcup\limits_{g\in C(K^p_fK^1_{p,n})_{\mathrm{ord}}} \mathscr{Y}_{g,n}. \end{align*}$$

The stratum $\mathscr {Y}_{g,n}$ can be identified with the $0$-dimensional Shimura variety ${\mathscr {S}_{G',K^{\prime }_{f,g,n}}}$, where

(4.2.1)$$ \begin{align} {K^{\prime}_{f,g,n}} &= \mathrm{Im} \left(P(\mathbb{A}_{\mathbb{Q},f})\cap g K^p_f K^1_{p,n} g^{-1}\rightarrow G'(\mathbb{A}_{\mathbb{Q},f})\right). \end{align} $$

We have the following diagram:

(4.2.2)

where ${\mathscr {C}^{\mathrm {ord}}_{g,n}}\rightarrow \mathscr {S}_{G',K^{\prime }_{f,g,n}}$ is a torsor of an abelian scheme quasi-isogenous to $\underline {\operatorname {\mathrm {Hom}}}_{\mathcal {O}_{\mathcal {K}}}(\mathtt {X}^{(g)},\mathcal {A})$ with $\mathcal {A}$ the universal abelian scheme over $\mathscr {S}_{G',K^{\prime }_{f,g,n}}$, and $\Xi ^{\mathrm {ord}}_{g,n}\rightarrow \mathscr {C}^{\mathrm {ord}}_{g,n}$ is a torsor of the torus with character group $\mathbf {S}_{\Phi ^{(g)}_{K^p_fK^1_{p,n}}}$, which is a finite index subgroup of the free quotient of

$$\begin{align*}\left.\left(\frac{1}{N} \mathtt{Y}^{(g)}\otimes_{\mathbb{Z}} \mathtt{X}^{(g)}\right)\middle/ \begin{pmatrix} y\otimes\phi^{(g)}(y')-y'\otimes\phi^{(g)}(y)\\ (b\frac{1}{N}y)\otimes x-(\frac{1}{N}y\otimes (b^cx) \end{pmatrix}_{y,y'\in \mathtt{Y}^{(g)},x\in \mathtt{X}^{(g)},b\in\mathcal{O}_{\mathcal{K}}} \right. \end{align*}$$

with N sufficiently large with respect to the level $K^p_fK^1_{p,n}$; and $\Xi ^{\mathrm {ord}}_{g,n}\hookrightarrow \overline {\Xi }^{\mathrm {ord}}_{g,n}$ is the torus embedding with respect to the unique cone decomposition.

Denote by ${\mathfrak {X}_{g,n}}$ the formal completion of $\mathscr {T}^{\mathrm {tor}}_n$ along the stratum $\mathscr {Z}_{g,n}$. Then $\mathfrak {X}_{g,n}$ is canonically isomorphic to the quotient by $\Gamma _g$ of the formal completion of $\overline {\Xi }^{\mathrm {ord}}_{g,n}$ along the boundary $\overline {\Xi }^{\mathrm {ord}}_{g,n}-\Xi ^{\mathrm {ord}}_{g,n}$. Here, $\Gamma _g$ is the (finite, in our case) subgroup of $\operatorname {\mathrm {GL}}(\mathtt {X}^{(g)})\times \operatorname {\mathrm {GL}}(\mathtt {Y}^{(g)})$ preserving $\varphi ^{(g)}_{-2}$ and $\varphi ^{(g)}_{0}$ (as orbits).

4.3 The subset $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ of $C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ and the subspace of

By a generic point, denoted by ${\eta }$, of the formal completion $\mathfrak {X}_{g,n}$, we mean a generic point of $\operatorname {\mathrm {Spec}}(\Gamma (U,\mathcal {O}_U))$ with U an affine open subscheme of $\mathfrak {X}_{g,n}$. Let $(\mathcal {G},\lambda ,i,\alpha ^p,\alpha _p)$ be the pullback to $\operatorname {\mathrm {Spec}}(\Gamma (U,\mathcal {O}_U))$ of the family of semiabelian varieties over $\mathscr {T}^{\mathrm {tor}}_n$, which is a degenerating family obtained by Mumford’s construction. Let

$$\begin{align*}0\longrightarrow{\mathcal{T}} \longrightarrow {\mathcal{G}^{\natural}} \longrightarrow \mathcal{A}\rightarrow 0 \end{align*}$$

be the Raynaud extension of $\mathcal {G}$. Then, for any positive integer N, the group $\mathcal {T}[N]$ (resp. $\mathcal {G}^\natural [N]$) is canonically isomorphic to a subgroup of $\mathcal {G}[N]$, which we denote by ${(\mathcal {G}[N])^\mu }$ (resp. ${(\mathcal {G}[N])^{\mathrm {f}}}$). We have $(\mathcal {G}[N])^\mu \subset (\mathcal {G}[N])^{\mathrm {f}}$ and the quotient ${(\mathcal {G}[N])^{\mathrm {ab}}} =(\mathcal {G}[N])^{\mathrm {f}}/(\mathcal {G}[N])^\mu $ is canonically isomorphic to $\mathcal {A}[N]$. (See also [Reference LanLan18, §3.4.2].)

Let ${\bar {\eta }}$ be a geometric point over $\eta $. The ordinary level structure $\alpha _{p,\bar {\eta }}$ of $\mathcal {G}_{\bar {\eta }}$ gives rise to a filtration

$$\begin{align*}\mathtt{F}^+_{\bar{\eta}}\colon 0\subset \mathtt{F}^+_{2,\bar{\eta}}\subset \mathtt{F}^+_{3,\overline{\eta}}\subset \mathcal{G}_{\bar{\eta}}[p^n]^{\mathrm{mult} +} \end{align*}$$

(refer to text around equation (2.5.2)). We define a subset $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}\subset C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ in terms of the relative positions of $\mathtt {F}^+_{2,\overline {\eta }}$ and $(\mathcal {G}_{\bar {\eta }}[p])^\mu $.

Definition 4.3.1. Define $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ as the subset of $C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ consisting of g such that

$$\begin{align*}\mathtt{F}^+_{2,\bar{\eta}}\cap (\mathcal{G}_{\bar{\eta}}[p])^\mu=0 \end{align*}$$

for a geometric point $\bar {\eta }$ over a generic point $\eta $ of $\mathfrak {X}_{g,n}$.

One can check that the preceding definition does not depend on the choice of $\bar {\eta }$. Let

$$ \begin{align*} {P^{\flat}_{\tt D}(\mathbb{Q}_p)} & =\left\{g\in P_{\mathtt{D}}(\mathbb{Q}_p)\,\middle|\, g^+=\left(\begin{smallmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ &&&a_{44}\end{smallmatrix}\right) \text{ with }\left(\begin{smallmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{smallmatrix}\right) ^{-1}\left(\begin{smallmatrix}a_{23}\\ a_{33}\end{smallmatrix}\right) \in \mathbb{Z}^2_p \right\} \end{align*} $$

and

$$ \begin{align*} P^{\flat}_{\mathtt{D}}(\mathbb{Z}_p)&=\left\{g_p\in P_{\mathtt{D}}(\mathbb{Z}_p)\,\middle|\, g^+_p=\left(\begin{smallmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ &&&a_{44}\end{smallmatrix}\right) \text{ with }\left(\begin{smallmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{smallmatrix}\right) \in \operatorname{\mathrm{GL}}_2(\mathbb{Z}_p) \right\}\\ &=P'(\mathbb{Z}_p) \left(\begin{smallmatrix}&&1\\1\\&1\\&&&1\end{smallmatrix}\right) \left(K^0_{p,n} \cap P_{\mathtt{D}}(\mathbb{Z}_p)\right). \end{align*} $$

Proposition 4.3.2. We have

(4.3.1)$$ \begin{align} C(K^p_fK^1_{p,n})^{\flat}_{\mathrm{ord}}= \mathrm{Im}\left( G(\mathbb{A}^p_{\mathbb{Q},f}) P^{\flat}_{\mathtt{D}}(\mathbb{Q}_p)\longrightarrow\operatorname{\mathrm{GL}}(X\otimes\mathbb{Q})\ltimes P'(\mathbb{A}_{\mathbb{Q},f})\backslash G(\mathbb{A}_{\mathbb{Q},f})/K^p_fK^1_{p,n}\right), \end{align} $$

which is in a natural bijection with

(4.3.2)$$ \begin{align} C(K^p_f)\times \left(U_{\mathcal{K},K^p_f}\ltimes P'(\mathbb{Z}_p)\backslash P^{\flat}_{\mathtt{D}}(\mathbb{Z}_p)/K^1_{p,n}\cap P_{\mathtt{D}}(\mathbb{Z}_p)\right). \end{align} $$

Proof. Given $g_p\in P_{\mathtt {D}}(\mathbb {Z}_p)$ with

$$ \begin{align*} g_p^+&=\left(\begin{smallmatrix}a_{11}&a_{12}&a_{13}&\ast\\a_{21}&a_{22}&a_{23}&\ast\\ a_{31}&a_{32}&a_{33}&\ast\\ &&&\ast\end{smallmatrix}\right) , &(g^{-1}_p)^+&=\left(\begin{smallmatrix}c_{11}&c_{12}&c_{13}&\ast\\c_{21}&c_{22}&c_{23}&\ast\\ c_{31}&c_{32}&c_{33}&\ast\\ &&&\ast\end{smallmatrix}\right) , &(g^{-1}_p)^-&=\left(\begin{smallmatrix}d_{11}&\ast & \ast & \ast\\ &\ast & \ast & \ast\\&\ast & \ast &\ast\\ &\ast & \ast & \ast\end{smallmatrix}\right) , \end{align*} $$

the corresponding $\mathtt {F}_{2,\bar {\eta }}$ is spanned by the image of $x^+_1,w^+_1$ under $\alpha _{p,\bar {\eta }}$, and $\mathcal {T}_{\bar {\eta }}[p]$ is spanned by the image under $\alpha _{p,\bar {\eta }}$ of

(4.3.3)$$ \begin{align} (\mathtt{x}^+_1,\mathtt{w}^+_1,\mathtt{w}^+_2;\mathtt{x}^-_1)\begin{pmatrix}c_{11}&c_{12}&c_{13}\\c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\\&&&d_{11}\end{pmatrix}\begin{pmatrix}1&0\\0&0\\0&0\\0&1\end{pmatrix}. \end{align} $$

Therefore, the condition $\mathtt {F}^+_{2,\overline {\eta }}\cap (\mathcal {G}_{\bar {\eta }}[p])^\mu =0$ is equivalent to the condition that $c_{11}\mathtt {x}^+_1+c_{21}\mathtt {w}^+_1+c_{31}\mathtt {w}^+_2$ does not belong to $\mathrm {Span}_{\mathbb {F}_p}\{\mathtt {x}^+_1,\mathtt {w}^+_1\}$, which is equivalent to $c_{31}\in \mathbb {Z}^{\times }_p$. The condition $c_{31}\in \mathbb {Z}^{\times }_p$ for $(g^{-1}_p)^+$ is equivalent to the condition $\left (\begin {smallmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end {smallmatrix}\right ) \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}_p)$ for $g^+_p$. This shows that $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ equals equation (4.3.2). It is also easy to see that there is a natural bijection between equations (4.3.1) and (4.3.2).

Definition 4.3.3. Define

$$\begin{align*}{V^{\flat}_{n,m}} = \left\{f\in V_{n,m}\;\colon\; f\vert_{\mathscr{Z}_{g,n}}= 0\quad \text{for all } g\in C(K^p_fK^1_{p,n})_{\mathrm{ord}}- C(K^p_fK^1_{p,n})^{\flat}_{\mathrm{ord}}\right\}, \end{align*}$$

that is, the subspace of $V_{n,m}$ consisting of forms vanishing along all the boundary strata of $\mathscr {T}_n^{\mathrm {tor}}$ indexed by ordinary cusp labels outside $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, and define

4.4 The exact sequence for $V^{\flat }_{n,m}$

When $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, the level $K^{\prime }_{f,g,n}$ is independent of n, and we denote it by $K^{\prime }_{f,g}$. We have

(4.4.1)$$ \begin{align} \begin{aligned} K^{\prime p}_{f,g}&= \mathrm{Im} \left(P(\mathbb{A}^p_{\mathbb{Q},f})\cap g K^p_fg^{-1}\longrightarrow G'(\mathbb{A}^p_{\mathbb{Q},f})\right),\\ {K^{\prime}_{p,g}} &=\left\{h\in G'(\mathbb{Z}_p):h^+\equiv \begin{pmatrix}\ast &\ast\\ &\ast\end{pmatrix}\quad\mod p\right\}. \end{aligned} \end{align} $$

Let ${M_{(0,0)}(K^{\prime }_{f,g};\mathbb {Z}/p^m\mathbb {Z})}$ be the space of classical automorphic forms on $G'$ valued in $\mathbb {Z}/p^m\mathbb {Z}$ of weight $(0,0)$ and level $K^{\prime }_{f,g}$ (which are functions valued in $\mathbb {Z}/p^m\mathbb {Z}$ on the finite set $G'(\mathbb {Q})\backslash G'(\mathbb {A}_{\mathbb {Q},f})/K^{\prime }_{f,g}$), and $M_{(0,0)}(K^{\prime }_{f,g};\mathbb {Q}_p/\mathbb {Z}_p)=\mathop {\varinjlim }\limits _mM_{(0,0)}(K^{\prime }_{f,g};\mathbb {Z}/p^m\mathbb {Z})$.

Proposition 4.4.1. We have the exact sequences

(4.4.2)$$ \begin{align} 0\longrightarrow V^0_{n,m} \longrightarrow V^{\flat}_{n,m}\xrightarrow{\oplus\Phi_{g}} \bigoplus_{g\in C(K^p_f)} M_{(0,0)}(K^{\prime}_{f,g};\mathbb{Z}/p^m\mathbb{Z})\otimes\mathbb{Z}_p[\![ T_{\mathrm{so}}(\mathbb{Z}_p)/U_{\mathcal{K},K^p_f}]\!] \longrightarrow 0 \end{align} $$

and

(4.4.3)

where the map ${\Phi _g}$ is obtained by restriction to the stratum $\mathscr {Z}_{g,n}$ (whose global sections are the same as the global sections of $\mathscr {Y}_{g,n}$, which can be identified with the $0$-dimensional Shimura variety $\mathscr {S}_{G',K^{\prime }_{f,g}}$), and is sometimes called a Siegel operator.

Proof. Let $\pi _n:\mathscr {T}^{\mathrm {tor}}_{n}\rightarrow \mathscr {T}^{\min }_{n}$ be the natural map. By [Reference LanLan18, Lem. 8.2.2.10], we have

$$ \begin{align*} \left(\pi_{n,*}\mathcal{O}_{\mathscr{T}^{\mathrm{tor}}_{n,m}}/\pi_{n,*}\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_{n,m}}\right)^\wedge_{\mathscr{Y}_{g,n}}\simeq h_*\mathcal{O}_{\mathscr{C}^{\mathrm{ord}}_{g,n}}\otimes\mathbb{Z}/p^m\mathbb{Z} = \mathcal{O}_{\mathscr{S}_{G',K^{\prime}_{f,g}}}\otimes\mathbb{Z}/p^m\mathbb{Z}. \end{align*} $$

(Here, h is the map in the diagram (4.2.2).) Taking global sections over the affine scheme $\mathscr {T}^{\min }_{n,m}$, we get

(4.4.4)$$ \begin{align} V_{n,m}/V^0_{n,m}=\bigoplus_{g\in C(K^p_fK^1_{p,n})_{\mathrm{ord}}} M_{(0,0)}(K^{\prime}_{f,g,n};\mathbb{Z}/p^m\mathbb{Z}). \end{align} $$

By definition, an element in $V_{n,m}$ belongs to $V^{\flat }_{n,m}$ if and only if it vanishes along the strata associated to the cusp labels outside $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, so

(4.4.5)$$ \begin{align} V^{\flat}_{n,m}/V^0_{n,m}=\bigoplus_{g\in C(K^p_fK^1_{p,n})^{\flat}_{\mathrm{ord}}} M_{(0,0)}(K^{\prime}_{f,g};\mathbb{Z}/p^m\mathbb{Z}). \end{align} $$

The action of $T_{\mathrm {so}}(\mathbb {Z}_p)$ on $V_{n,m}$ preserves $V^0_{n,m}$, so it descends to $V_{n,m}/V^0_{n,m}$. This action permutes the direct summands of the right-hand side of equation (4.4.4). More precisely, the corresponding action of $(a_1,a_2)\in T_{\mathrm {so}}(\mathbb {Z}_p)$ on the index set

$$\begin{align*}C(K^p_fK^1_{p,n})_{\mathrm{ord}}=C(K^p_f)\times\left(U_{\mathcal{K},K^p_f}\ltimes P'(\mathbb{Z}_p)\backslash P_{\mathtt{D}}(\mathbb{Z}_p)/K^1_{p,n}\cap P_{\mathtt{D}}(\mathbb{Z}_p)\right) \end{align*}$$

sends $g=g^pg_p$ to $g^pg_p t_{a_1,a_2}$, where $t_{a_1,a_2}\in G(\mathbb {Z}_p)$ with $t^+_{a_1,a_2}=\left (\begin {smallmatrix}1\\&1\\&&a_1\\&&&a^{-1}_2\end {smallmatrix}\right ) $. It is easy to see that $T_{\mathrm {so}}(\mathbb {Z}_p)$-action on $C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ preserves $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, so the $T_{\mathrm {so}}(\mathbb {Z}_p)$ on $V_{n,m}$ preserves $V^{\flat }_{n,m}$ and induces an action on $V^{\flat }_{n,m}/V^0_{n,m}$.

For given $g\in C(K^p_f)$, set

(4.4.6)$$ \begin{align} \begin{aligned} C(K^p_f K^1_{p,n})_{\mathrm{ord},g}&=\{\text{elements in } C(K^p_f K^1_{p,n})_{\mathrm{ord}} \text{ whose projection to } C(K^p_f) \text{ is } g\},\\ C(K^p_f K^1_{p,n})^{\flat}_{\mathrm{ord},g}&=\{\text{elements in } C(K^p_f K^1_{p,n})^{\flat}_{\mathrm{ord}} \text{ whose projection to } C(K^p_f) \text{ is } g\}. \end{aligned} \end{align} $$

The $\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)]\!]$-module structures of $V_{n,m}/V^0_{n,m}$ (resp. $V^{\flat }_{n,m}/V^0_{n,m}$) are determined by the $\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)]\!]$-actions on $C(K^p_fK^1_{p,n})_{\mathrm {ord},g}$ (resp. $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord},g}$). The $T_{\mathrm {so}}(\mathbb {Z}_p)$-action on $C(K^p_fK^1_{p,n})_{\mathrm {ord},g}$ has too many orbits, and the number of orbits grows with n, so $V_{n,m}/V^0_{n,m}$ is not very nice as a $\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)]\!]$-module. In contrast, the $T_{\mathrm {so}}(\mathbb {Z}_p)$-action on $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord},g}$ is transitive, factoring through $\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p/p^n\mathbb {Z})/U_{\mathcal {K},K^p_f}]\!]$, and the action of this quotient is free. Therefore, by fixing an element in $P^{\flat }_{\mathtt {D}}(\mathbb {Z}_p)$, which we will always choose to be $\left (\begin {smallmatrix}&&1\\1\\&1\\&&&1\end {smallmatrix}\right ) $, we get the isomorphism

(4.4.7)$$ \begin{align} \text{Right hand side of}\ (4.4.5)\cong M_{(0,0)}(K^{\prime}_{f,g};\mathbb{Z}/p^m\mathbb{Z})\otimes\mathbb{Z}_p[\![ T_{\mathrm{so}}(\mathbb{Z}_p)/U_{\mathcal{K},K^p_f}]\!], \end{align} $$

from which the exact sequence (4.4.2) follows. The exact sequence (4.4.3) follows from equation (4.4.2) by taking the direct limit.

4.5 Three propositions on the $\mathbb {U}_p$-action on and .

We introduce some setting for the proof of the following propositions. Given $g\in C(K^p_fK^1_{p,n})_{\mathrm {ord}}$, recall that $\eta $ denotes a generic point of $\mathfrak {X}_{g,n,m}$, and $\bar {\eta }$ denotes a geometric point over $\eta $. Let $(\mathcal {G}_{\bar {\eta }},i_{\bar {\eta }},\lambda _{\bar {\eta }},\alpha ^p_{\bar {\eta }},\alpha _{p,\bar {\eta }})$ be the pullback to $\bar {\eta }$ of the family of semiabelian schemes over $\mathscr {T}^{\mathrm {tor}}_{n,m}$. Recall that certain Lagrangian subgroups $C\subset \mathcal {G}_{\bar {\eta }}[p^2]$ are used in the construction of $\mathscr {C}^{\bullet }_{j,n,m}$ for defining $U^{\bullet }_{p,j}$ in §2.8. (Here, $(\bullet ,j)=(+,2),\,(+,3)$ or $(-,1)$.) Let

(4.5.1)$$ \begin{align} (\mathcal{G}^{\prime}_{\bar{\eta}}=\mathcal{G}_{\bar{\eta}}/C,i^{\prime}_{\bar{\eta}}, \lambda^{\prime}_{\bar{\eta}},(\alpha^p_{\bar{\eta}})',\alpha^{\prime}_{p,\bar{\eta}}) \end{align} $$

be the tuple defined as in the construction of $\mathscr {C}^{\bullet }_{j,n,m}$. If $(e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4)$ is a basis of $\mathcal {G}_{\bar {\eta }}[p^n]$ compatible with $\alpha _{p,\bar {\eta }}$, then C is spanned by $(e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4)\cdot p^n \gamma ^{\pm }_{C,p}$ with $\gamma _{C,p}$ given as in equations (2.8.1), (2.8.2) and (2.8.3). From Mumford’s construction of the degenerating family over $\mathfrak {X}_{g,n,m}$, we see that the tuple (4.5.1), corresponds to a geometric point $\bar {\eta }'$ over a generic point $\eta '$ of $\mathfrak {X}_{g',n,m}$ with $g'=g\cdot \gamma _{C,p}$.

Proof. We shall show that $g'=g\cdot \gamma _{C,p}\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ only if $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. By the definition of the $\mathbb {U}_p$-operators, this will imply that if vanishes along the strata outside $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, then so does $U^{\bullet }_{p,j}(f)$. We use the description (4.3.1) for $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. Consider the case $(\bullet ,j)=(+,2)$. Write $g^+_p=\left (\begin {smallmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ &&&a_{44}\end {smallmatrix}\right ) $ and $g^{\prime +}_p=g^+_p \cdot \gamma ^+_{C,p}= \left (\begin {smallmatrix}a^{\prime }_{11}&a^{\prime }_{12}&a^{\prime }_{13}&a^{\prime }_{14}\\ a^{\prime }_{21}&a^{\prime }_{22}&a^{\prime }_{23}&a^{\prime }_{24}\\ a^{\prime }_{31}&a^{\prime }_{32}&a^{\prime }_{33}&a^{\prime }_{34}\\ &&&a^{\prime }_{44}\end {smallmatrix}\right ) $ with $\gamma ^+_{C,p}$ as in equation (2.8.1). We have

(4.5.2)$$ \begin{align} \begin{pmatrix}a^{\prime}_{11}&a^{\prime}_{12}&a^{\prime}_{13}&a^{\prime}_{14}\\ a^{\prime}_{21}&a^{\prime}_{22}&a^{\prime}_{23}&a^{\prime}_{24}\\ a^{\prime}_{31}&a^{\prime}_{32}&a^{\prime}_{33}&a^{\prime}_{34}\\ &&&a^{\prime}_{44}\end{pmatrix} = \begin{pmatrix}a_{11}&a_{12}&\frac{a_{13}+u_1a_{11}+u_2a_{12}}{p}&\ast\\ a_{21}&a_{22}&\frac{a_{23}+u_1a_{21}+u_2a_{22}}{p}&\ast\\ a_{31}&a_{32}&\frac{a_{33}+u_1a_{31}+u_2a_{32}}{p}&\ast\\ &&&\frac{a_{44}}{p}\end{pmatrix} \end{align} $$

and

(4.5.3)$$ \begin{align} \begin{pmatrix}a^{\prime}_{21}&a^{\prime}_{22}\\a^{\prime}_{31}&a^{\prime}_{32}\end{pmatrix}^{-1}\begin{pmatrix}a^{\prime}_{23}\\a^{\prime}_{33}\end{pmatrix} =p^{-1}\left[\begin{pmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}^{-1}\begin{pmatrix}a_{23}\\a_{33}\end{pmatrix}+\begin{pmatrix}u_1\\u_2\end{pmatrix}\right]. \end{align} $$

Since $u_1,u_2\in \mathbb {Z}_p$, the right-hand side can be integral only if $\left (\begin {smallmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end {smallmatrix}\right ) ^{-1}\left (\begin {smallmatrix}a_{23}\\a_{33}\end {smallmatrix}\right ) \in \mathbb {Z}^2_p$. Therefore, $g'=g\cdot \gamma _{C,p}\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ only if $g \in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. The argument for the other two cases is similar.

Proposition 4.5.2. There exists a positive integer N such that the exact sequences in Proposition 4.4.1 are $(U^{\bullet }_{p,j})^N$-equivariant for $(\bullet ,j)=(+,2),(+,3),(-,1)$ in the sense that for all $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ and , we have

$$\begin{align*}\Phi_{g}\left((U^{\bullet}_{p,j})^N f\right)= \Phi_{g}(f). \end{align*}$$

Proof. Take $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, and we use the setting described at the beginning of this subsection. In order to look at the image of $U^{\bullet }_{p,j}f$ under the Siegel operator $\Phi _g$, we consider the abelian quotients $\mathcal {A}_{\bar {\eta }}$ and $\mathcal {A}_{\bar {\eta }'}$ of the Raynaud extensions

$$ \begin{align*} &0\longrightarrow\mathcal{T}_{\bar{\eta}}\longrightarrow \mathcal{G}^\natural_{\bar{\eta}}\longrightarrow\mathcal{A}_{\bar{\eta}}\longrightarrow 0, &0\longrightarrow\mathcal{T}_{\bar{\eta}'}\longrightarrow \mathcal{G}^\natural_{\bar{\eta}'}\longrightarrow\mathcal{A}_{\bar{\eta}'}\longrightarrow 0. \end{align*} $$

Let

(4.5.4)$$ \begin{align} \begin{aligned} {C^{\mathrm{f}}} &=\big(C+(\mathcal{G}_{\bar{\eta}}[p^n])^\mu\big)\cap (\mathcal{G}_{\bar{\eta}}[p^n])^{\mathrm{f}}, &\quad\quad\quad {C^{\mathrm{ab}}}&=C^{\mathrm{f}}/(\mathcal{G}_{\bar{\eta}}[p^n])^\mu. \end{aligned}\\[-15pt]\nonumber \end{align} $$

(See the beginning of §4.3 for the definition of $(\mathcal {G}_{\bar {\eta }}[p^n])^\mu $ and $(\mathcal {G}_{\bar {\eta }}[p^n])^{\mathrm {f}}$.) Then we have

$$\begin{align*}\mathcal{A}_{\bar{\eta}'}=\mathcal{A}_{\bar{\eta}}/C^{\mathrm{ab}}.\\[-15pt] \end{align*}$$

We use the description (4.1.3) and equation (4.3.2) for $C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ and $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. Write $g=g^pg_p$ with $g_p\in P_{\mathtt {D}}(\mathbb {Z}_p)$ and

(4.5.5)$$ \begin{align} \begin{aligned} (g^{-1}_p)^+&=\left(\begin{smallmatrix}c_{11}&c_{12}&c_{13}&\ast\\c_{21}&c_{22}&c_{23}&\ast\\ c_{31}&c_{32}&c_{33}&\ast\\&&&\ast\end{smallmatrix}\right) , &(g^{-1}_p)^-&=\left(\begin{smallmatrix}d_{11}&\ast & \ast & \ast\\ &\ast & \ast & \ast\\&\ast & \ast & \ast\\&\ast & \ast & \ast\end{smallmatrix}\right). \end{aligned}\\[-15pt]\nonumber \end{align} $$

The condition $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ implies that $c^{-1}_{31}c_{11},c^{-1}_{31}c_{21}\in \mathbb {Z}_p$. Thus, we see that $(\mathcal {G}_{\overline {\eta }}[p^n])^\mu $ is spanned by

(4.5.6)$$ \begin{align} c^{-1}_{31}c_{11} e^+_1+c^{-1}_{31}c_{21}e^+_2+e^+_3,\quad e^-_1\\[-15pt]\nonumber \end{align} $$

(cf. equation (4.3.3)), and $(\mathcal {G}_{\overline {\eta }}[p^n])^{\mathrm {f}}$ is spanned by

(4.5.7)$$ \begin{align} e^+_1,\quad e^+_2,\quad e^+_3,\quad e^-_1,\quad (e^-_2,e^-_3,e^-_4)\begin{pmatrix}-\iota_p(\bar{\zeta}_0)\\ &1\end{pmatrix}\begin{pmatrix}0&1\\c^{-1}_{31}c_{11}&c^{-1}_{31}c_{21}\\1 &0\end{pmatrix}.\\[-15pt]\nonumber \end{align} $$

We first consider the operator $U^+_{p,3}$. By a direct computation, we see that for $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, the p-part of $g'\in C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ is

(4.5.8)$$ \begin{align} g^{\prime+}_p=g^+_p\cdot\gamma^+_{C,p}=\left(\begin{smallmatrix}1&&&\ast\\ &1&&\ast\\ &&1&\ast\\&&&p^{-1} \end{smallmatrix}\right) g^+_p,\\[-15pt]\nonumber \end{align} $$

where $\ast \in \mathbb {Q}_p$. This shows that the cusp label $g'$ is independent of C and belongs to $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. For given C spanned by $(e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4)\cdot p^{n-2}\gamma ^{\pm }_{C,p}$ with $\gamma _{C,p}^{\pm }$ as in (2.8.2), from the definition (4.5.4) and the basis (4.5.7) of $(\mathcal {G}_{\overline {\eta }}[p^n])^{\mathrm {f}}$, we see that $C^{\mathrm {f}}$ is spanned by

$$\begin{align*}c^{-1}_{31}c_{11} e^+_1+c^{-1}_{31}c_{21}e^+_2+e^+_3,\quad e^-_1, \quad (e^-_2,e^-_3,e^-_4)\cdot p^{n-2}\begin{pmatrix}-\iota_p(\bar{\zeta}_0)\\ &1\end{pmatrix}\begin{pmatrix}0&1\\c^{-1}_{31}c_{11}&c^{-1}_{31}c_{21}\\1 &0\end{pmatrix}.\\[-15pt] \end{align*}$$

Its quotient $C^{\mathrm {ab}}$ is independent of the choice of C and is spanned by $(\epsilon ^-_1,\epsilon ^-_2)\cdot p^{n-2}$, with $(\epsilon ^-_1,\epsilon ^-_2)$ the image of $(e^-_2,e^-_3,e^-_4)\begin {pmatrix}-\iota _p(\bar {\zeta }_0)\\ &1\end {pmatrix}\left (\begin {smallmatrix}0&1\\c^{-1}_{31}c_{11}&c^{-1}_{31}c_{21}\\1 &0\end {smallmatrix}\right ) $under the quotient map $\mathcal {G}^\natural _{\bar {\eta }}\rightarrow \mathcal {A}_{\bar {\eta }}$ which is a basis of $\mathcal {A}_{\bar {\eta }}[p^n]^-$. (Here, we identify $(\mathcal {G}_{\bar {\eta }}[p^n])^{\mathrm {f}}$ with $\mathcal {G}^\natural _{\bar {\eta }}[p^n]$ by the canonical isomorphism between them.) Thus, we get

(4.5.9)$$ \begin{align} \mathcal{A}_{\bar{\eta}'}=\mathcal{A}_{\bar{\eta}}/\mathcal{A}_{\bar{\eta}}[p^2]^-.\\[-15pt]\nonumber \end{align} $$

Denote by $\left <p\right>^-$ the operator on $M_{(0,0)}(K^{\prime }_{f,g};\mathbb {Z}/p^m\mathbb {Z})$ induced by taking the quotient by $\mathcal {A}[p^2]^-$. Combining equations (4.5.8) and (4.5.9), we see that the trace cancels the normalizer in equation (2.8.5) due to the independence of the choice of C, and

$$\begin{align*}\Phi_g(U^+_{p,3}f) =(\left<p\right>^-)^2 \Phi_{g'}(f). \end{align*}$$

With $K^p_f$ fixed, there exists $N_1\geq 1$, such that $\left (\begin {smallmatrix} 1&&&\ast \\&1&&\ast \\&&1&\ast \\&&&p^{-N_1}\end {smallmatrix}\right ) _pg$ and g represent the same element in the double coset $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, and $(\left <p\right>^-)^{2N_1}=1$ on $M_{(0,0)}(K^{\prime }_{f,g};\mathbb {Z}/p^m\mathbb {Z})$. For such $N_1$, we deduce that

$$\begin{align*}\Phi_g\left((U^+_{p,3})^{N_1}f\right) = \Phi_{g}(f)\\[-9pt] \end{align*}$$

for all and $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. A similar calculation shows that that there exists $N_2\geq 1$ such that

$$\begin{align*}\Phi_g\left((U^-_{p,1})^{N_2}f\right) = \Phi_{g}(f)\\[-9pt] \end{align*}$$

for all and $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$.

It remains to consider $U^+_{p,2}$. Write $g^+_p=\left (\begin {smallmatrix}a_{11}&a_{12}&a_{13}&\ast \\a_{21}&a_{22}&a_{32}&\ast \\ a_{31}&a_{32}&a_{33}&\ast \\ &&&\ast \end {smallmatrix}\right ) $ and $g^{\prime +}_p=g^+_p \cdot \gamma ^+_{C,p}= \left (\begin {smallmatrix}a^{\prime }_{11}&a^{\prime }_{12}&a^{\prime }_{13}&a^{\prime }_{14}\\ a^{\prime }_{21}&a^{\prime }_{22}&a^{\prime }_{23}&a^{\prime }_{24}\\ a^{\prime }_{31}&a^{\prime }_{32}&a^{\prime }_{33}&a^{\prime }_{34}\\ &&&a^{\prime }_{44}\end {smallmatrix}\right ) $ with $\gamma _{C,p}^+$ as in (2.8.1). Then we have equations (4.5.2) and (4.5.3), and $g'\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ if and only if

(4.5.10)$$ \begin{align} \begin{pmatrix}u_1\\u_2\end{pmatrix}\equiv -\begin{pmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}^{-1}\begin{pmatrix}a_{23}\\a_{33}\end{pmatrix}\quad\mod p.\\[-9pt]\nonumber \end{align} $$

Now, suppose $g'\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, that is, the entries $u_1,u_2$ in $\gamma ^{\pm }_{C,p}$ (see (2.8.1)) satisfy the congruence (4.5.10). Then we have $ g^{\prime +}_p\in \left (\begin {smallmatrix}p^{-1}&\ast & \ast & \ast \\&1&&\ast \\&&1&\ast \\&&&p^{-1}\end {smallmatrix}\right ) g^+_p K^1_{p,n}, $and as an element in the double coset $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, $g'$ represents the same element as $\left (\begin {smallmatrix}p^{-1}&&&\\&1&&\\&&1&\\&&&p^{-1}\end {smallmatrix}\right ) g$ in $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. The congruence (4.5.10) implies that the $c_{ij}$’s in equation (4.5.5) satisfies

(4.5.11)$$ \begin{align} \begin{aligned} u_1&\equiv -c^{-1}_{31}c_{11} \quad\mod p, &u_2&\equiv -c^{-1}_{31}c_{21} \quad\mod p. \end{aligned}\\[-9pt]\nonumber \end{align} $$

From the definition (4.5.4) and the relation (4.5.11), by using the basis (4.5.7) of $(\mathcal {G}_{\overline {\eta }}[p^n])^{\mathrm {f}}$ and the basis of C in equation (2.8.1), we see that $C^{\mathrm {f}}$ is spanned by

$$\begin{align*}c^{-1}_{31}c_{11} e^+_1+c^{-1}_{31}c_{21}e^+_2+e^+_3,\quad e^-_1, \quad (e^-_2,e^-_3,e^-_4)\cdot p^{n-2}\begin{pmatrix}-\iota_p(\bar{\zeta}_0)^{-1}\\ &1\end{pmatrix}\begin{pmatrix}0&1\\c^{-1}_{31}c_{11}&c^{-1}_{31}c_{21}\\1 &0\end{pmatrix}. \end{align*}$$

Like in the case of $U^+_{p,3}$, we can deduce that $C^{\mathrm {ab}}=\mathcal {A}_{\bar {\eta }}[p^n]^-$, and

$$\begin{align*}\mathcal{A}_{\bar{\eta}'}=\mathcal{A}_{\bar{\eta}}/\mathcal{A}_{\bar{\eta}}[p^2]^-.\\[-9pt] \end{align*}$$

The independence of $C^{\mathrm {ab}}$ on the $\ast $’s in $\gamma ^+_{C,p}$ cancels the normalizer in equation (2.8.5), and we get

$$\begin{align*}\Phi_g(U^+_{p,2}f) =(\left<p\right>^-)^2 \Phi_{g'}(f). \end{align*}$$

For fixed $K^p_f$, take $N_3$ such that $ \left (\begin {smallmatrix}p^{-1}&&&\\&1&&\\&&1&\\&&&p^{-1}\end {smallmatrix}\right ) ^{N_3}_pg $ and g represent the same element in the double coset $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ and $(\left <p\right>^-)^{2N_3}=1$ on $M_{(0,0)}(K^{\prime }_{f,g};\mathbb {Z}/p^m\mathbb {Z})$. Then

$$\begin{align*}\Phi_g\left((U^+_{p,3})^{N_3}f\right) = \Phi_{g}(f) \end{align*}$$

for all and $g\in C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. By taking $N=N_1N_2N_3$, we conclude the proof.

Proof. We shall show that for all $g\in C(K^p_fK^1_{p,n})_{\mathrm {ord}}- C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ and $f\in V_{n,m}$,

(4.5.12)$$ \begin{align} \Phi_g\left(U^+_{p,2}f\right)\in pM_{(0,0)}(K^{\prime}_{f,g};\mathbb{Z}/p^m\mathbb{Z}), \end{align} $$

from which the proposition follows. For a given $g\in C(K^p_fK^1_{p,n})_{\mathrm {ord}}- C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, we use the setting described at the beginning of this subsection. As in the proof of Proposition 4.5.2, we consider $\mathcal {A}_{\bar {\eta }'}$, the abelian quotient of the Raynaud extension associated to $\mathcal {G}_{\bar {\eta }'}=\mathcal {G}_{\bar {\eta }}/C$. We have

$$\begin{align*}\mathcal{A}_{\bar{\eta}'}=\mathcal{A}_{\bar{\eta}}/C^{\mathrm{ab}}, \end{align*}$$

where $\mathcal {A}_{\bar {\eta }}$ is the abelian quotient of the Raynaud extension associated to $\mathcal {G}_{\bar {\eta }}$, and $C^{\mathrm {ab}}$ is defined in equation (4.5.4). Since the $\mathbb {U}_p$-operator considered here is $U^+_{p,2}$, the Lagrangian subgroup $C\subset \mathcal {G}_{\bar {\eta }}[p^2]$ is spanned by $(e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4)\cdot p^n \gamma ^{\pm }_{C,p}$ with $\gamma _{C,p}^{\pm }$ as in equation (2.8.1). We use the description (4.1.3) and equation (4.3.2) for $C(K^p_fK^1_{p,n})_{\mathrm {ord}}$ and $C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$. Write $(g^{-1}_p)^+=\left (\begin {smallmatrix}c_{11}&c_{12}&c_{13}&\ast \\c_{21}&c_{22}&c_{23}&\ast \\ c_{31}&c_{32}&c_{33}&\ast \\&&&\ast \end {smallmatrix}\right ) $, $(g^{-1}_p)^-=\left (\begin {smallmatrix}d_{11}&\ast & \ast & \ast \\ &\ast & \ast & \ast \\&\ast & \ast & \ast \\&\ast & \ast & \ast \end {smallmatrix}\right ) $(which are elements in $\operatorname {\mathrm {GL}}_4(\mathbb {Z}_p)$). The condition $g\notin C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$ implies that $c_{31}\in p\mathbb {Z}_p$ and $c_{11}$ or $c_{21}$ belongs to $\mathbb {Z}^{\times }_p$.

Suppose $c_{11}\in \mathbb {Z}^{\times }_p$. The group $(\mathcal {G}_{\bar {\eta }}[p^n])^\mu $ is spanned by

(4.5.13)$$ \begin{align} c_{11} e^+_1+c_{21}e^+_2+c_{31}e^+_3,\quad e^-_1, \end{align} $$

and $(\mathcal {G}_{\overline {\eta }}[p^n])^{\mathrm {f}}$ is spanned by

(4.5.14)$$ \begin{align} e^+_1,\quad e^+_2,\quad e^+_3,\quad e^-_1,\quad (e^-_2,e^-_3,e^-_4)\begin{pmatrix}-\iota^c_p(\zeta_0)^{-1}\\ &1\end{pmatrix}\begin{pmatrix}c_{11}&0\\0&c_{11}\\-c_{21}&c_{31} \end{pmatrix}. \end{align} $$

By the definition of $C^{\mathrm {f}}$ in equation (4.5.4), a direct computation shows that $C^{\mathrm {f}}$ is spanned by $(\mathcal {G}_{\bar {\eta }}[p^n])^\mu $ plus

(4.5.15)$$ \begin{align} \left((c_{11}u_2-c_{21}u_1)e^+_2+c_{11} e^+_3\right)\cdot p^{n-1}, \quad (e^-_2,e^-_3,e^-_4)\begin{pmatrix}-\iota^c_p(\zeta_0)^{-1}\\ &1\end{pmatrix}\begin{pmatrix}c_{11}&0\\0&c_{11}\\-c_{21}&c_{31} \end{pmatrix} \begin{pmatrix}c_{11}&0\\c_{11}u_2-c_{21}u_1&p\end{pmatrix}. \end{align} $$

Hence, $C^{\mathrm {ab}}$ is spanned by the image of equation (4.5.15) modulo $(\mathcal {G}_{\bar {\eta }}[p^n])^\mu $. Let $u^{\prime }_2=c_{11}u_2-c_{21}u_1$. The trace in equation (2.8.5) corresponds to a sum with $u_1,u^{\prime }_2$ and the two $\ast $’s in $\gamma _{C,p}$ varying in $\mathbb {Z}/p\mathbb {Z}$. Since equation (4.5.15) depends only on $u^{\prime }_2$, we see that $C^{\mathrm {ab}}$, and hence $\mathcal {A}_{\bar {\eta }'}$ depends only on $u^{\prime }_2$. Its independence of the two $\ast $’s implies that the sum over the two $\ast $’s cancels the normalization factor in equation (2.8.5). Its independence of $u_1$ implies that the sum over $u_1$ contributes a factor p. Therefore, the evaluation of $\Phi _g\left (U^+_{p,2}f\right )$ at $\bar {\eta }'$ is divisible by p. This shows the inclusion (4.5.12), hence the result in this case. The case $c_{21}\in \mathbb {Z}^{\times }_p$ can be similarly treated.

4.6 The ordinary projection on and the fundamental exact sequence

In this section, we use the exact sequence in Proposition 4.4.1 and the three propositions proved in §4.5 to show that $\lim \limits _{r\rightarrow \infty }(U_p)^{r!}f$ converges for all and that is a free $\Lambda _{\mathrm {so}}$-module of finite rank and deduce the fundamental exact sequence in part (4) of Theorem 2.9.1.

Proof. Given $f\in V_{n,m}$, we define the following finiteness property for f:

(F)$$ \begin{align} \text{The submodule generated by } \left(U_p\right)^{r}f, r\geq 0, \text{ is finitely generated over } \mathbb{Z}/p^m. \end{align} $$

It is easy to see that the convergence of $\lim \limits _{r\rightarrow \infty }(U_p)^{r!}f$ follows from the property (F) for f. By Proposition 4.5.3, in order to show that all $f\in V_{n,m}$ satisfy (F), it suffices to show that all $f\in V^{\flat }_{n,m}$ satisfy (F). Given $f\in V^{\flat }_{n,m}$, it follows from Proposition 4.5.2 that $f'=U^N_pf-f$ belongs to $V^0_{n,m}$. Proposition 3.0.2 implies that there exists $M\geq 0$ such that

$$\begin{align*}U^{M+1}_p f'\in \mathrm{Span}_{\mathbb{Z}}\left\{f',U_pf',U^2_pf',\dots,U^M_pf'\right\}. \end{align*}$$

Then

$$\begin{align*}U^{M+N+1}_p f\in \mathrm{Span}_{\mathbb{Z}}\left\{f,U_pf,U^2_pf,\dots,U^{M+N}_pf\right\}, \end{align*}$$

from which it follows that for all $r\geq 0$,

$$\begin{align*}U^r_p f\in \mathrm{Span}_{\mathbb{Z}}\left\{f,U_pf,U^2_pf,\dots,U^{M+N}_pf\right\}. \end{align*}$$

Hence, (F) holds for f. Therefore, we have proved that $\lim \limits _{r\rightarrow \infty }(U_p)^{r!}f$ converges for all , and we can define the semiordinary projector $e_{\mathrm {so}}$ on as

(4.6.1)$$ \begin{align} e_{\mathrm{so}}=\lim\limits_{r\rightarrow\infty}(U_p)^{r!}. \end{align} $$

By Proposition 4.5.3, we have

(4.6.2)

Applying $e_{\mathrm {so}}$ to the Pontryagin dual of the exact sequence in Proposition 4.4.1 and using equation (4.6.2), we obtain the exact sequence

(4.6.3)

We know that $M_{(0,0)}(K^{\prime }_{f,g};\mathbb {Z}_p)$ is a free $\mathbb {Z}_p$-module of finite rank, so the leftmost term in equation (4.6.3) is a free $\Lambda _{\mathrm {so}}$-module of finite rank. The rightmost term is also a free $\Lambda _{\mathrm {so}}$-module of finite rank by Theorem 3.0.7. Since $\mathrm {Ext}^1_{\Lambda _{\mathrm {so}}}(M,N)$ vanishes for free $\Lambda _{\mathrm {so}}$-modules M, we deduce that , as a $\Lambda _{\mathrm {so}}$-module, is isomorphic to the direct sum of the terms on the two ends of equation (4.6.3) and therefore is a free $\Lambda _{\mathrm {so}}$-module of finite rank.

Part (2) of Theorem 2.9.1 for noncuspidal semiordinary families can be proved in the same way as the cuspidal semiordinary families. The freeness of implies that applying $\operatorname {\mathrm {Hom}}_{\Lambda _{\mathrm {so}}}(\,\cdot \,,\Lambda _{\mathrm {so}})$ preserves the exactness of equation (4.6.3), and part (4) of Theorem 2.9.1 follows.

4.7 The Fourier–Jacobi expansion

We introduce the Fourier–Jacobi expansions of p-adic forms on $\operatorname {\mathrm {GU}}(3,1)$, which will be used in §7 for analyzing the Klingen Eisenstein family on $\operatorname {\mathrm {GU}}(3,1)$ constructed in §5.

In §4.2, for an ordinary cusp label $g\in C(K^p_f K^1_{p,n})_{\mathrm {ord}}$, we described $\mathfrak {X}_{g,n}$, the formal completion of $\mathscr {T}^{\mathrm {tor}}_n$ along the boundary stratum $\mathscr {Z}_{g,n}$. From the description there, we see that

$$\begin{align*}H^0\left(\mathfrak{X}_{g,n},\mathcal{O}_{\mathfrak{X}_{g,n}}\right)=\left(\prod_{\beta} H^0\left(\mathscr{C}^{\mathrm{ord}}_{g,n},\mathcal{L}(\beta)\right)\right)^{\Gamma_g}, \end{align*}$$

where $\beta $ runs over $\mathbf {S}^{\vee }_{\Phi ^{(g)}_{K^p_fK^1_{p,n}}}\cap \mathbf {P}_{\Phi ^{(g)}_{K^p_fK^1_{p,n}}}$, which can be identified with a subset of $\operatorname {\mathrm {Her}}_1(\mathcal {K})_{\geq 0}$, and $\mathcal {L}(\beta )$ is the invertible sheaf over $\mathscr {C}^{\mathrm {ord}}_{g,n}$ of $\beta $-homogeneous functions on $\Xi ^{\mathrm {ord}}_{g,n}$. Therefore, given $g\in C(K^p_f K^1_{p,n})_{\mathrm {ord}}$ and $\beta \in \operatorname {\mathrm {Her}}_1(\mathcal {K})_{\geq 0}$, the restriction to $\mathfrak {X}_{g,n}$ induces a map

(4.7.1)$$ \begin{align} V_{n,m}\longrightarrow H^0\left(\mathscr{C}^{\mathrm{ord}}_{g,n},\mathcal{L}(\beta)\otimes\mathbb{Z}/p^m\mathbb{Z}\right). \end{align} $$

We consider $g=\mathbf {1}_4$ and $\beta \in \operatorname {\mathrm {Her}}_1(\mathcal {K})_{\geq 0}$. In this case, the p-component of level group for the Shimura variety $\mathscr {S}_{G',K^{\prime }_{f,\mathbf {1}_4,n}}$ in the diagram (4.2.2) is

$$\begin{align*}K^{\prime}_{p,\mathbf{1}_4,n}=\left\{g\in \operatorname{\mathrm{GU}}(2)(\mathbb{Z}_p):g^+\equiv \begin{pmatrix}\ast & \ast\\0&1\end{pmatrix}\quad\mod p^n\right\}, \end{align*}$$

(depending on n). Let

(4.7.2)$$ \begin{align} {V^{J,\beta}_{\operatorname{\mathrm{GU}}(2)}} =\varprojlim_m\varinjlim_n H^0\left(\mathscr{C}^{\mathrm{ord}}_{\mathbf{1}_4,n}, \mathcal{L}(\beta)\otimes\mathbb{Z}/p^m\mathbb{Z}\right). \end{align} $$

The map gives the map of taking the $\beta $-th Fourier–Jacobi coefficient of p-adic forms on $\operatorname {\mathrm {GU}}(3,1)$ along the boundary stratum labeled by $\mathbf {1}_4$:

(4.7.3)$$ \begin{align} {\mathrm{FJ}_{\beta}} :V_{\operatorname{\mathrm{GU}}(3,1)}\longrightarrow V^{J,\beta}_{\operatorname{\mathrm{GU}}(2)}. \end{align} $$

5.1 Some notation

Let ${\mathcal {K}_{\infty }}$ be the maximal abelian pro-p extension of $\mathcal {K}$ unramified outside p and ${\Gamma _{\mathcal {K}}}=\operatorname {\mathrm {Gal}}(\mathcal {K}_{\infty }/\mathcal {K})$. Then $\Gamma _{\mathcal {K}}\cong \mathbb {Z}^2_p$. Denote by ${\Omega _{\infty }}\in \mathbb {C}^{\times }$ (resp. ${\Omega _p}\in \hat {\mathbb {Z}}^{\mathrm {ur},\times }_p$) the complex complex multiplication (CM) period (p-adic CM period) with respect to the embeddings in equation (1.0.4) (cf. [Reference HsiehHsi14a, Section 2.8]). We also fix an isomorphism $\overline {\mathbb {Q}}_p\cong \mathbb {C}$ compatible with the embeddings in equation (1.0.4). Let ${L}\subset \overline {\mathbb {Q}}_p$ be a sufficiently large finite extension of $\mathbb {Q}_p$, and denote by ${\hat {\mathcal {O}}^{\mathrm {ur}}_L}$ the ring of integers of the completion of the maximal unramified extension of L.

For our later use of theta correspondence for unitary groups, we also fix a Hecke character

$$ \begin{align*} {\lambda} : \mathcal{K}^{\times}\backslash\mathbb{A}^{\times}_{\mathcal{K}}&\longrightarrow \mathbb{C}^{\times}, \end{align*} $$

such that

$$ \begin{align*} \lambda|_{\mathbb{A}^{\times}_{\mathbb{Q}}}&=\eta_{\mathcal{K}/\mathbb{Q}}, &\lambda_{\infty}(z)&=\frac{|z\bar{z}|^{1/2}}{z}. \end{align*} $$

5.2 Our setup

Let ${\pi }$ be an irreducible cuspidal automorphic representation of $\operatorname {\mathrm {GL}}_2(\mathbb {A}_{\mathbb {Q}})$ generated by a newform ${f}$ of weight $2$. Take a finite extension L of $\mathbb {Q}_p$ containing all the Hecke eigenvalues of f. We assume the following conditions on $\pi $:

• • for all finite places v of $\mathbb {Q}$, $\pi _v$ is either unramified or Steinberg or Steinberg twisted by an unramified quadratic character of $\mathbb {Q}^{\times }_v$,

• • $\pi _p$ is unramified,

• • there exists a prime q not split in $\mathcal {K}$ such that $\pi $ is ramified at q, and if $2$ does not split in $\mathcal {K}$, then $\pi $ is ramified at $2$,

• • $\bar {\rho }_{\pi }|_{\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathcal {K})}$ is irreducible, (which is automatically true if $\pi $ is not ordinary at p because in this case $\bar {\rho }_{\pi }|_{G_{\mathcal {K}},\mathfrak {p}}\cong \bar {\rho }_{\pi }|_{G_{\mathbb {Q},p}}$ is irreducible by [Reference EdixhovenEdi92]), where $\bar {\rho }_{\pi }$ denotes the residual representation of the p-adic Galois representation $\rho _{\pi }:\operatorname {\mathrm {Gal}}(\bar {\mathbb {Q}}/\mathbb {Q})\rightarrow \operatorname {\mathrm {GL}}_2(L)$.

We also fix

• • an algebraic Hecke character ${\xi }:\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K}}\rightarrow \mathbb {C}^{\times }$ of $\infty $-type $\left (0,k_0\right )$ with ${k_0}$ an even integer.

5.3 The weight space

The weight space we will use for constructing the semiordinary Klingen Eisenstein family on $\operatorname {\mathrm {GU}}(3,1)$ is $\operatorname {\mathrm {Hom}}_{\mathrm {cont}}\big (\Gamma _{\mathcal {K}},\overline {\mathbb {Q}}^{\times }_p\big )$. By identifying $\Gamma _{\mathcal {K}}$ with a quotient of $\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K},f}$, the arithmetic points in the weight space are p-adic avatars of the algebraic Hecke characters of $\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K}}$ whose p-adic avatars factor through $\Gamma _{\mathcal {K}}$. (Given an algebraic Hecke character $\chi :\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K}}\rightarrow \mathbb {C}^{\times }$ of $\infty $-type $(k_1,k_2)$, its p-adic avatar is the continuous character $\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K},f}\longrightarrow \overline {\mathbb {Q}}^{\times }_p$ sending x to $\chi (x)x^{k_1}_{\mathfrak {p}} x^{k_2}_{\bar {\mathfrak {p}}}$.) We will use ${\tau }$ to denote such Hecke characters.