1 Introduction
Let p be an odd prime number. In this paper, under some assumptions, we prove one divisibility of a two-variable Greenberg-type main conjecture for a weight 
$2$ newform unramified at p. The result is a key ingredient in the third author’s proof [Reference WanWan21] of the Iwasawa main conjecture, formulated by Kobayashi [Reference KobayashiKob03], for elliptic curves with supersingular reduction at p and 
$a_p=0$, as well as in the proof of its extension to the case 
$a_p\neq 0$ by Sprung [Reference Skinner and UrbanSpr16]. Another application of our main result is in the proof by the first and third authors [Reference Chen and HsiehCW16] of a p-converse to the theorem of Gross–Zagier and Kolyvagin for supersingular primes.
Let 
$\pi $ be an irreducible cuspidal automorphic representation of 
$\operatorname {\mathrm {GL}}_2(\mathbb {A}_{\mathbb {Q}})$ generated by a newform of weight 
$2$. Associated to 
$\pi $ is a continuous two-dimensional p-adic Galois representation 
${\rho _{\pi }}$ of 
$\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ over L, a finite extension of 
$\mathbb {Q}_p$. (We use the geometric convention for Galois representations. The determinant of 
$\rho _{\pi }$ is 
$\epsilon ^{-1}_{\mathrm {cyc}}$.) For the representation 
$\rho _{\pi }$ over L, we can fix a 
$\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$-stable 
$\mathcal {O}_L$-lattice 
$T_{\pi }$.
Take 
$\mathcal {K}$ to be an imaginary quadratic field in which p splits as 
$\mathfrak {p}\bar {\mathfrak {p}}$. Denote by 
$\mathcal {K}_{\infty }$ the maximal abelian pro-p extension of 
$\mathcal {K}$ unramified outside p. Then the Galois group 
$\operatorname {\mathrm {Gal}}(\mathcal {K}_{\infty }/\mathcal {K})$ is isomorphic to 
$\mathbb {Z}^2_p$ and we denote it by 
$\Gamma _{\mathcal {K}}$. We have the tautological character
$$\begin{align*}{\Psi_{\mathcal{K}}} :\operatorname{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathcal{K})\rightarrow\Gamma_{\mathcal{K}}\hookrightarrow \mathbb{Z}_p[\![ \Gamma_{\mathcal{K}}]\!]^{\times}. \end{align*}$$Let 
$\xi :\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K}}\rightarrow \mathbb {C}^{\times }$ be an algebraic Hecke character. We can associate to it a p-adic character 
$\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathcal {K})\rightarrow \mathcal {O}^{\times }_L$, which we denote also by 
$\xi $.
We consider the 
$\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathcal {K})$-module
$$ \begin{align*} T_{\pi,\mathcal{K},\xi}:=T_{\pi}(\epsilon^2_{\mathrm{cyc}}) |{}_{\operatorname{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathcal{K})}(\xi^{-1}) \otimes \mathbb{Z}_p[\![\Gamma_{\mathcal{K}}]\!](\Psi^{-1}_{\mathcal{K}}). \end{align*} $$Define the Selmer group
$$ \begin{align} {\mathrm{Sel}_{\pi,\mathcal{K},\xi}} = \ker \left\{ \begin{aligned} &H^1 \big(\mathcal{K},T_{\pi,\mathcal{K},\xi} \otimes_{\mathcal{O}_L [\![ \Gamma_{\mathcal{K}}]\!]} \mathcal{O}_L [\![ \Gamma_{\mathcal{K}}]\!]^{*} \big)\\ &\quad\longrightarrow \prod_{\mathfrak{v}\neq \mathfrak{p}} H^1 \big(I_{\mathfrak{v}}, T_{\pi, \mathcal{K},\xi}\otimes_{\mathcal{O}_L [\![ \Gamma_{\mathcal{K}}]\!]} \mathcal{O}_L[\![ \Gamma_{\mathcal{K}}]\!]^* \big) \end{aligned} \right\}, \end{align} $$with 
$\mathcal {O}_L[\![\Gamma _{\mathcal {K}}]\!]^*=\operatorname {\mathrm {Hom}}_{\mathbb {Z}_p}\left (\mathcal {O}_L[\![\Gamma _{\mathcal {K}}]\!],\mathbb {Q}_p/\mathbb {Z}_p\right )$, the Pontryagin dual of 
$\mathcal {O}_L[\![\Gamma _{\mathcal {K}}]\!]$. (This Selmer group has relaxed condition at 
$\mathfrak {p}$ and unramified condition at 
$\bar {\mathfrak {p}}$.) Let
$$ \begin{align} {X_{\pi,\mathcal{K},\xi}} :=\operatorname{\mathrm{Hom}}_{\mathbb{Z}_p}\big(\mathrm{Sel}_{\pi,\mathcal{K},\xi},\mathbb{Q}_p/\mathbb{Z}_p\big), \end{align} $$which is well known to be a finitely generated 
$\mathcal {O}_L[\![\Gamma _{\mathcal {K}}]\!]$-module. We recall the following definition of characteristic ideals.
Definition 1.0.1. For a Noetherian normal domain A and a finitely generated A-module M, we define the characteristic ideal of M as
$$\begin{align*}{\mathrm{char}_A(M)} = \left\{x\in A \left|\begin{array}{l} \mathrm{ord}_P(x)\geq \mathrm{length}_{A_P}(M_P) \text{ for all}\\ \text{height one prime ideals } P\subset A\end{array}\right.\right\}. \end{align*}$$ The Iwasawa–Greenberg main conjecture [Reference GreenbergGre94] predicts that the characteristic ideal of 
$X_{\pi ,\mathcal {K},\xi }$ is generated by the following p-adic L-function.
Denote by 
$\hat {\mathcal {O}}^{\mathrm {ur}}_L$ the completion of the maximal unramified extension of 
$\mathcal {O}_L$. By using the construction in [Reference Harris, Lan, Taylor and ThorneHid91], it is proved in Proposition 8.2.2 that there is a p-adic L-function
$$\begin{align*}\mathcal{L}_{\pi,\mathcal{K},\xi}\in \mathcal{M} eas\left(\Gamma_{\mathcal{K}},\hat{\mathcal{O}}^{\mathrm{ur}}_L\right)\cong \hat{\mathcal{O}}^{\mathrm{ur}}_L[\![\Gamma_{\mathcal{K}}]\!] \end{align*}$$satisfying the interpolation property: For all algebraic Hecke characters 
$\tau :\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K}}\rightarrow \mathbb {C}^{\times }$ with p-adic avatars 
$\tau _{p\text {-}\mathrm {adic}}$ factoring through 
$\Gamma _{\mathcal {K}}$ and 
$\xi \tau $ of 
$\infty $-type 
$\left (k_1,k_2\right )$ with 
$k_1,k_2\in \mathbb {Z}$, 
$k_1\leq 0$, 
$k_2\geq 2-k_1$,
$$ \begin{align} \begin{aligned} \mathcal{L}_{\pi,\mathcal{K},\xi}(\tau_{p\text{-}\mathrm{adic}}) =&\, \left(\frac{\Omega_p}{\Omega_{\infty}}\right)^{2(k_2-k_1)}\frac{\Gamma(k_2)\Gamma(k_2-1)}{(2\pi i)^{2k_2-1}} \cdot \gamma_p\left(\frac{3-(k_1+k_2)}{2},\pi^{\vee}_{p}\times (\xi_0\tau_0)^{-1}_{\bar{\mathfrak{p}}}\right)\\ &\times L^{\{\infty,p\}}\left(\frac{k_1+k_2-1}{2},\mathrm{BC}(\pi)\times\xi_0\tau_0\right), \end{aligned} \end{align} $$where 
$\xi _0\tau _0=\xi \tau |\cdot |^{-\frac {k_1+k_2}{2}}_{\mathbb {A}_{\mathcal {K}}}$ and 
$\mathrm {BC}(\pi )$ denotes the unitary automorphic representation of 
$\operatorname {\mathrm {GL}}_2(\mathbb {A}_{\mathcal {K}})$ obtained as the base change of 
$\pi $.
We are interested in the following (two-variable) Greenberg-type main conjecture.
$$\begin{align*}\mathrm{char}_{\hat{\mathcal{O}}^{\mathrm{ur}}_L[\![\Gamma_{\mathcal{K}}]\!]} \big(X_{\pi,\mathcal{K},\xi}\big)= \big( \mathcal{L}_{\pi,\mathcal{K},\xi}\big). \end{align*}$$ The main result of this paper is Theorem 8.2.3, which is a partial result towards this conjecture. Like the previous works [Reference UrbanUrb01, Reference UrbanUrb06, Reference SprungSU14, Reference WanWan20] on proving Greenberg-type main conjectures for modular forms, the proof uses the congruences between Klingen Eisenstein series and cuspidal automorphic forms. The L-values in our case here are of the same type as those in [Reference WanWan20], and we also use Klingen Eisenstein series on 
$\operatorname {\mathrm {GU}}(3,1)$ as in loc.cit. The main difference is that the modular form here is not assumed to be ordinary at p, so the standard Hida theory is not applicable.
The key idea is to introduce the notion of semiordinary automorphic forms on 
$\operatorname {\mathrm {GU}}(3,1)$. In §§2-4, we develop a Hida theory for p-adic families of (cuspidal and noncuspidal) semiordinary forms on 
$\operatorname {\mathrm {GU}}(3,1)$ along an appropriate two-dimensional subspace of the usual three-dimensional weight space for 
$\operatorname {\mathrm {GU}}(3,1)$. The main results are stated in Theorem 2.9.1. In §5, by using the doubling method, we construct a Klingen Eisenstein family 
$\boldsymbol {E}^{\mathrm {Kling}}_{\varphi }$ and prove its semiordinarity. In §6, we study the degenerate Fourier–Jacobi coefficients of 
$\boldsymbol {E}^{\mathrm {Kling}}_{\varphi }$. The analogous computations in [Reference WanWan20] assume a sufficient ramification condition (see Definition 6.30 in op.cit) which is not available in our case here, so we need a better way to do the computation at p by using the functional equations for local doubling zeta integrals. In §7, we study the nondegenerate Fourier–Jacobi coefficients of 
$\boldsymbol {E}^{\mathrm {Kling}}_{\varphi }$. This part is very similar to [Reference WanWan20], and we cite many results there, but the presentation is slightly rearranged. For example, the auxiliary data for constructing the Klingen Eisenstein family are chosen at the beginning of the construction (§5.6) instead of at the end of the analysis of the nondegenerate Fourier–Jacobi coefficients, and an explanation of the strategy for analyzing the nondegenerate Fourier–Jacobi coefficients is included in §7.5. In §8, combining the results in §6 and §7, we deduce a result on the Klingen Eisenstein congruence ideal and use it as an input for the lattice construction to deduce the results on Selmer groups.
Notation. We fix a prime 
$p\geq 3$ and an imaginary quadratic field 
${\mathcal {K}}$ in which p splits in 
$\mathcal {K}$ as 
$\mathfrak {p}\bar {\mathfrak {p}}$. Denote by 
${D_{\mathcal {K}/\mathbb {Q}}}$ the discriminant of 
$\mathcal {K}/\mathbb {Q}$ and by 
${\mathfrak {d}_{\mathcal {K}/\mathbb {Q}}}$ the different ideal of 
$\mathcal {K}/\mathbb {Q}$. Denote by 
${\eta _{\mathcal {K}/\mathbb {Q}}}$ the quadratic character of 
$\mathbb {Q}^{\times }\backslash \mathbb {A}^{\times }_{\mathbb {Q}}$ associated to 
$\mathcal {K}/\mathbb {Q}$. Denote by 
${c}$ the nontrivial element in 
$\operatorname {\mathrm {Gal}}(\mathcal {K}/\mathbb {Q})$. For 
$x\in \mathcal {K}$, denote by 
$\bar {x}$ its image under c. For a finite place v of 
$\mathbb {Q}$, we put 
$\mathcal {K}_v=\mathcal {K}\otimes _{\mathbb {Q}}\mathbb {Q}_v$ and 
$\mathcal {O}_{\mathcal {K},v}=\mathcal {O}_{\mathcal {K}}\otimes _{\mathbb {Z}}\mathbb {Z}_v$.
Fix embeddings
such that the valuation of 
$\mathbb {Q}_p$ and 
$\iota _p$ induce the valuation of 
$\mathcal {K}$ given by 
$\mathfrak {p}$. The embedding 
$\iota _p:\mathcal {K}\hookrightarrow \mathbb {Q}_p$ induces a homomorphism 
${\varrho _{\mathfrak {p}}} : \mathcal {K}_p=\mathcal {K}\otimes _{\mathbb {Q}}\mathbb {Q}_p\rightarrow \mathbb {Q}_p$. We denote by 
${\varrho _{\bar {\mathfrak {p}}}}:\mathcal {K}\otimes _{\mathbb {Q}}\mathbb {Q}_p\rightarrow \mathbb {Q}_p$ the composition of 
$\varrho _p$ and the nontrivial element 
$c\in \operatorname {\mathrm {Gal}}(\mathcal {K}/\mathbb {Q})$. We have the isomorphism
$$ \begin{align*} &(\varrho_{\mathfrak{p}},\varrho_{\bar{\mathfrak{p}}}):\mathcal{K}\otimes_{\mathbb{Q}}\mathbb{Q}_p\longrightarrow\mathbb{Q}_p\times\mathbb{Q}_p, &&a\longmapsto \left(\varrho_{\mathfrak{p}}(a),\varrho_{\bar{\mathfrak{p}}}(a)\right) \end{align*} $$We also fix a totally imaginary element 
${\delta }\in \mathcal {K}$ such that 
$\mathrm {Nm}(\delta )=\delta \bar {\delta }$ is a p-adic unit.
Fix the standard additive character 
$\mathbf {e}_{\mathbb {A}_{\mathbb {Q}}}=\bigotimes _v {\mathbf {e}_{v}} :\mathbb {Q}\backslash \mathbb {A}\rightarrow \mathbb {C}^{\times }$ with
$$ \begin{align} \mathbf{e}_v(x)=\left\{\begin{array}{ll} e^{-2\pi i\{x\}_v}, &v\neq\infty,\\ e^{2\pi i x},&v=\infty,\end{array} \right. \end{align} $$where 
$\{x\}_v$ is the fractional part of x.
2 Hida theory for semiordinary forms on 
$\operatorname {\mathrm {GU}}(3,1)$
In this section, we define semiordinary forms on 
$\operatorname {\mathrm {GU}}(3,1)$ and state the control theorem for semiordinary families (Theorem 2.9.1). The proof of Theorem 2.9.1 is given in the following sections: 3 and 4.
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
$$ \begin{align} \begin{aligned} \operatorname{\mathrm{GU}}(2)(R)&=\{(g,\nu)\in \operatorname{\mathrm{GL}}_{\mathcal{O}_{\mathcal{K}}\otimes_{\mathbb{Z}} R}(L_0\otimes_{\mathbb{Z}} R)\times R^{\times}\,:\,\left<gv_1,gv_2\right>_{L_0}=\nu\left<v_1,v_2\right>_{L_0}\},\\ \operatorname{\mathrm{GU}}(3,1)(R)&=\{(g,\nu)\in \operatorname{\mathrm{GL}}_{\mathcal{O}_{\mathcal{K}}\otimes_{\mathbb{Z}} R}(L\otimes_{\mathbb{Z}} R)\times R^{\times}\,:\,\left<gv_1,gv_2\right>_{L}=\nu\left<v_1,v_2\right>_{L}\}, \end{aligned}\\[-15pt]\nonumber \end{align} $$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, for all$$ \begin{align*} \det\left(X-i(b)|\operatorname{\mathrm{Lie}} A_{/S}\right)&=(X-b)^3(X-\bar{b})\\[-15pt] \end{align*} $$$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
$$ \begin{align} \begin{aligned} G(\mathbb{Q}_p)&\rightarrow \operatorname{\mathrm{GL}}_4(\mathbb{Q}_p)\times\mathbb{Q}^{\times}_p\\ g&\mapsto (g^+,\nu(g)) \end{aligned} \end{align} $$is an isomorphism.
There is a filtration 
${\tt D} = \{\mathtt {D}^i\}_i$ of 
$L\otimes _{\mathbb {Z}}\mathbb {Z}_p$ given by
$$ \begin{align} {\tt D}^{1}=0\;\subset\;{\tt D}^{0}=X^+_{p}\oplus L^+_{0,p}\oplus X^-_{p}\;\subset\; {\tt D}^{-1} = L\otimes_{\mathbb{Z}}\mathbb{Z}_p, \end{align} $$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].
Remark 2.5.1. In [Reference LanLan18, Chapter 3], the ordinary locus is defined as a normalization of the naive moduli problem introduced in [loc.cit., Def. 3.4.1.1]. In our case, because p is a good prime in the sense of [Reference LanLan18, Def. 1.1.1.6] and 
$\nu (K^1_{p,n})=\mathbb {Z}^{\times }_p$, this construction of the ordinary locus agrees with the moduli problem. (See [Reference HidaHLTT16, B.10] for more details.)
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
$$ \begin{align} \underline{\operatorname{\mathrm{Isom}}}_{\mathscr{S}^{\mathrm{tor}}_m[1/E]}\left(\mu_{p^n}\otimes \mathtt{D}^0,\mathscr{G}[p^n]^{\mathrm{mult}}\right)/K^1_{p,n}\cap P_{\mathtt{D}}(\mathbb{Z}_p), \end{align} $$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
$$ \begin{align} \begin{aligned} &\mathtt{F}^+:0=\mathtt{F}^+_0\subset \mathtt{F}^+_2\subset\mathtt{F}^+_3=\mathscr{G}_{\bar{s}}[p^n]^{\mathrm{mult}+}\\ &\mathtt{F}^-:0=\mathtt{F}^-_0\subset\mathtt{F}^-_1= \mathscr{G}_{\bar{s}}[p^n]^{\mathrm{mult} -} \end{aligned} \end{align} $$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
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
2.8 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 
$j=2$
$$ \begin{align} \begin{aligned} \gamma^+_{C,p}&= \begin{pmatrix}1&&u_1&\ast\\&1&u_2&\ast\\&&1&\\&&&1\end{pmatrix}\begin{pmatrix}1&&&\\&1&&\\&&\frac{1}{p}&\\&&&\frac{1}{p}\end{pmatrix},\\ \gamma^-_{C,p}&= \begin{pmatrix}1\\&\iota_p(\bar{\zeta}_0)^{-1}\\&&1\end{pmatrix}\begin{pmatrix}1&\ast&&\ast\\&1\\&-u_2&1&u_1\\&&&1\end{pmatrix}\begin{pmatrix}\frac{1}{p}&&&\\&\frac{1}{p^2}&&\\&&\frac{1}{p}&\\&&&\frac{1}{p^2}\end{pmatrix}\begin{pmatrix}1\\&\iota_p(\bar{\zeta}_0)\\&&1\end{pmatrix} \end{aligned} \end{align} $$with 
$u_1,u_2,\ast \in \mathbb {Z}_p$, and for 
$j=3$,
$$ \begin{align} \begin{aligned} \gamma^+_{C,p}&=\begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}1&&&\\&1&&\\&&1&\\&&&\frac{1}{p}\end{pmatrix}, &\gamma^-_{C,p}&= \begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}\frac{1}{p}&&\\&\frac{1}{p^2}\\&&\frac{1}{p^2}\\&&&\frac{1}{p^2}\end{pmatrix} \end{aligned} \end{align} $$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
$$ \begin{align} \begin{aligned} \gamma^+_{C,p}&=\begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}\frac{1}{p}&&&\\&\frac{1}{p}&&\\&&\frac{1}{p}&\\&&&\frac{1}{p^2}\end{pmatrix}, &\gamma^-_{C,p}&= \begin{pmatrix}1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&1\end{pmatrix}\begin{pmatrix}1&&\\&\frac{1}{p}\\&&\frac{1}{p}\\&&&\frac{1}{p}\end{pmatrix}. \end{aligned} \end{align} $$ For 
$(\bullet ,j)=(+,2),(+,3),(-,1)$, we consider the diagram
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
$$ \begin{align} {U^+_{p,j}}\colon\; H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}) \stackrel{p^*_2}{\longrightarrow} H^0(\mathscr{C}^+_{j,n,m},\mathcal{O}_{\mathscr{T}_{n,m}})\xrightarrow{p^{-j}\mathrm{Tr}\,p_1} H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}), \quad(j=2,3) \end{align} $$
$$ \begin{align} {U^-_{p,1}}\colon\; H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}) \stackrel{p^*_2}{\longrightarrow} H^0(\mathscr{C}^-_{1,n,m},\mathcal{O}_{\mathscr{T}_{n,m}})\xrightarrow{p^{-3}\mathrm{Tr}\,p_1} H^0(\mathscr{T}_{n,m},\mathcal{O}_{\mathscr{T}_{n,m}}). \end{align} $$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
$$ \begin{align} \begin{aligned} &\left(h\begin{pmatrix}a_1\\&a_2\\&&a_3\end{pmatrix}h',a_4\right)\cdot q(\underline{x},y)\\ =&\,q\left(\begin{pmatrix}p^{-v_p(a_1)}\\&p^{-v_p(a_2)}\\&&p^{-v_p(a_3)}\end{pmatrix}\underline{x}h\begin{pmatrix}a_1\\&a_2\\&&a_3\end{pmatrix}h',p^{-v_p(a_4)y}a_4 \right), \end{aligned} \end{align} $$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 
$G(\mathbb {Q}_p)$):
$$ \begin{align} \begin{aligned} {U^{+}_{p,2}}:&\quad p^{\left<(t^+_1,t^+_2,t^+_3;-t^-_1)+2\rho_{\mathrm{c}},(1,1,0;0)\right>}\int_{N(\mathbb{Z}_p)}R \left(u\left(\begin{smallmatrix}p\\&p\\&&1\\&&&1\end{smallmatrix}\right) ,1\right)\,du; && \\ U^{+}_{p,3} :&\quad p^{\left<(t^+_1,t^+_2,t^+_3;-t^-_1)+2\rho_{\mathrm{c}},(1,1,1;0)\right>}\int_{N(\mathbb{Z}_p)}R \left(u\left(\begin{smallmatrix}p\\&p\\&&p\\&&&1\end{smallmatrix}\right) ,1\right)\,du; && \\ U^{-}_{p,1}:&\quad p^{\left<(t^+_1,t^+_2,t^+_3;-t^-_1)+2\rho_{\mathrm{c}},(0,0,0;-1)\right>}\int_{N(\mathbb{Z}_p)}R \left(u\left(\begin{smallmatrix}1\\&1\\&&1\\&&&p^{-1}\end{smallmatrix}\right) ,1\right)\,du, && \end{aligned}\\[-8pt]\nonumber \end{align} $$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 
${\Lambda _{\mathrm {so}}} =\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)^{\circ }]\!]$.
Theorem 2.9.1. The following hold:
(1) Let
$U_p=U^+_{p,2}U^+_{p,3}U^-_{p,1}$. Then for each , $\lim \limits _{r\to \infty }(U_p)^{r!}f$ converges, and we can define the semiordinary projector as ${e_{\mathrm {so}}} = \lim \limits _{r\to \infty }(U_p)^{r!}$. The $\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)]\!]$-modules are both free of finite rank over
$\Lambda _{\mathrm {so}}$.(2) The spaces of
$\Lambda _{\mathrm {so}}$-families of tame level $K^p_f$ are defined as For a given weight
$(\tau ^+,\tau ^-)\in \operatorname {\mathrm {Hom}}_{\mathrm {cont}}\left (T_{\mathrm {so}}(\mathbb {Z}_p),F^{\times }\right )$, let $\mathcal {P}_{\tau ^+,\tau ^-}$ be the ideal in $\mathcal {O}_F[\![ T_{\mathrm {so}}(\mathbb {Z}_p)]\!]$ generated by $\{(x,y)-\tau ^+(x)\tau ^-(y)\colon (x,y)\in T_{\mathrm {so}}(\mathbb {Z}_p)\}$. We have (2.9.1)$$ \begin{align} \begin{aligned} \mathcal{M}^0_{\mathrm{so}}\otimes_{\mathbb{Z}_p[\![ T_{\mathrm{so}}(\mathbb{Z}_p)]\!]}\mathcal{O}_F[\![ T_{\mathrm{so}}(\mathbb{Z}_p)]\!]/\mathcal{P}_{\tau^+,\tau^-}&\stackrel{\sim}{\longrightarrow} \left(\varprojlim_{m}\varinjlim_n e_{\mathrm{so}} V^0_{n,m}\otimes_{\mathbb{Z}_p}\mathcal{O}_F\right)[\tau^+,\tau^-],\\ \mathcal{M}_{\mathrm{so}}\otimes_{\mathbb{Z}_p[\![ T_{\mathrm{so}}(\mathbb{Z}_p)]\!]}\mathcal{O}_F[\![ T_{\mathrm{so}}(\mathbb{Z}_p)]\!]/\mathcal{P}_{\tau^+,\tau^-}&\stackrel{\sim}{\longrightarrow} \left(\varprojlim_{m}\varinjlim_n e_{\mathrm{so}} V_{n,m}\otimes_{\mathbb{Z}_p}\mathcal{O}_F\right)[\tau^+,\tau^-]. \end{aligned} \end{align} $$The semiordinary projector preserves the spaces of classical forms, and by combining equation (2.9.1) with equation (2.7.2), we have the embeddings
where
$t^{\pm }=\tau _{\mathrm {alg}}^{\pm }$ with $(0,0, t^+; t^-)$ dominant and $\epsilon ^{\pm }=\tau _{\mathrm {f}}^{\pm }$.(3) Given
$0\geq t^+$, there exists $A\geq -t^++4$ such that the above embedding for the cuspidal forms is surjective if $t^-\geq A$.(4) There is the following so-called fundamental exact sequence (in the study of Klingen Eisenstein congruences),
$$\begin{align*}0\longrightarrow\mathcal{M}^0_{\mathrm{so}}\longrightarrow \mathcal{M}_{\mathrm{so}}\xrightarrow{\oplus\Phi_{g}}\bigoplus_{g}{M_{(0,0)}(K^{\prime }_{f,g};\mathbb{Z}_p)} \otimes_{\mathbb{Z}_p}\mathbb{Z}_p[\![ T_{\mathrm{so}}(\mathbb{Z}_p)]\!] \longrightarrow 0,\\[-16pt] \end{align*}$$with g runs over
$\big (G^{\prime }_1(\mathbb {Q})\ltimes P^{\prime }_1(\mathbb {A}^p_{\mathbb {Q},f})\big )\backslash G(\mathbb {A}^p_{\mathbb {Q},f})/K^p_f$ and $\Phi _{g}$ the Siegel operator obtained by restriction to the boundary stratum indexed by g. The level group $K^{\prime }_{f,g}=K^{\prime p}_{f,g}K^{\prime }_{p,g}\subset G'(\mathbb {A}^p_{\mathbb {Q},f})$ is defined in equation (4.4.1), and $M_{(0,0)}(K^{\prime }_{f,g};\mathbb {Z}_p)$ denotes the space of classical automorphic forms on $G'$ of weight $(0,0)$ and level $K^{\prime }_{f,g}$.
Remark 2.9.2. In general, one can consider semiordinary families whose members have weight 
$\underline {t}=(t^+_1,t^+_2,t^+_3;t^-_1)$ with 
$t^+_1,t^+_2$ fixed and 
$t^+_3,t^-_1$ varying in the family. We only consider the case 
$t^+_1=t^+_2=0$ because it suffices for proving Theorem 8.2.3, and we want to avoid having the main idea obscured by the extra complications of working with vector bundles. For the general case, instead of considering the global sections of the structure sheaf over the Igusa tower, one considers the global sections of a vector bundle 
$\omega _{\underline {t}}$. For defining 
(cf. §4.3) such that the quotient 
has a nice structure, besides the requirement of vanishing outside the strata labeled by cusp labels in 
$C(K^p_fK^1_{p,n})^{\flat }_{\mathrm {ord}}$, one also requires that the elements are global sections of 
$\omega ^{\flat }_{\underline {t}}\subset \omega _{\underline {t}}$ with 
$\omega ^{\flat }_{\underline {t}}$ a subsheaf defined as in [Reference HsiehHsi14a, Section 4.1].
3 The proof of Theorem 2.9.1 for the cuspidal part
In this section, we prove Theorem 2.9.1 for cuspidal families. The results for cuspidal families will be used to deduce the results for noncuspidal families in §4.
Proposition 3.0.1 (Base change property).
Let 
$\mathscr {T}^{0,\mathrm {tor}}_n[1/E]$ be the open subscheme of 
$\mathscr {T}^{0,\mathrm {tor}}_n$ where E, our fixed lift of Hasse invariant viewed as a section over 
$\mathscr {T}^{0,\mathrm {tor}}_n$, is nonvanishing. For any classical dominant weight 
$\underline {t}$, the natural map
$$\begin{align*}H^0\left(\mathscr{T}^{0,\mathrm{tor}}_n[1/E],\omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_n}\right)\otimes\mathbb{Z}/p^m\mathbb{Z}\hookrightarrow H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{n,m},\omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{n,m}}\right)\\[-16pt] \end{align*}$$is an isomorphism.
Proof. Over 
$\mathscr {T}^{\mathrm {tor}}_n$, we have the exact sequence of sheaves
$$\begin{align*}0\longrightarrow \omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_n}\stackrel{p^m}{\rightarrow} \omega_{\underline{t}}\otimes \mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_{n}}\longrightarrow \omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_{n,m}}\longrightarrow 0.\\[-16pt] \end{align*}$$By [Reference LanLan18, Thm. 8.2.1.1], 
$R^1_{\pi _n,*}(\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{\mathrm {tor}}_n})=0$ (where 
$\pi _n$ is the projection 
$\pi _n:\mathscr {T}^{\mathrm {tor}}_{n}\rightarrow \mathscr {T}^{\min }_{n}$), so we have the exact sequence of sheaves
$$\begin{align*}0\rightarrow\pi_{n,*}(\omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_n})\stackrel{p^{m}}{\rightarrow}\pi_{n,*}(\omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_n})\rightarrow \pi_{n,*}(\omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_{n,m}})\rightarrow 0\\[-16pt] \end{align*}$$over 
$\mathscr {T}^{\min }_{n}$. Since 
$\mathscr {T}^{\min }_{n}[1/E]$ is affine by definition, taking global sections gives the result.
Proposition 3.0.2. Let 
$U_p=U^+_{p,2}U^+_{p,3}U^-_{p,1}$. Let 
and 
$\vec {f}\in H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{n,m}}\right )$. Then both limits 
$\lim \limits _{r\rightarrow \infty }(U_p)^{r!}f$ and 
$\lim \limits _{r\rightarrow \infty }(U_p)^{r!}\vec {f}$ converge.
Proof. Consider 
$\tilde {\mathscr {T}}_n$, the ordinary locus of level
$$\begin{align*}\tilde{K}^n_{p,1}=\left\{g\subset G(\mathbb{Z}_p)\mid g^+\equiv \left(\begin{smallmatrix}1&\ast & \ast & \ast\\ &1&\ast & \ast\\&&1&\ast\\&&&1\end{smallmatrix}\right) \quad\mod p^n\right\}. \end{align*}$$ Let 
$\tilde {\mathscr {T}}^{\mathrm {tor}}_n$ be the partial toroidal compactification of 
$\tilde {\mathscr {T}}_n$, and put 
$\tilde {V}_{n,m}^0=H^0(\tilde {\mathscr {T}}^{\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\tilde {\mathscr {T}}^{\mathrm {tor}}_{n,m}})$, which contains 
$V^0_{n,m}$ as a subspace. The definition of the 
$\mathbb {U}_p$-operators in §2.8 can be naturally extended to 
$\tilde {V}^0_{n,m}$, so it suffices to show that 
$\lim \limits _{r\rightarrow \infty }(U_p)^{r!}f$ converges for every 
$f\in \tilde {V}^0_{n,m}$. As explained in [Reference HidaHid02], 
$\mathop {\varprojlim }\limits _m\mathop {\varinjlim }\limits _n \tilde {V}^0_{n,m}$ is p-torsion free and there is an embedding
$$\begin{align*}\bigoplus_{\underline{t} \text{ dominant}} H^0\left(\mathscr{S}^{\mathrm{tor}},\underline{\omega}_{\underline{t}}\otimes\mathcal{I}_{\mathscr{S}^{\mathrm{tor}}}\right)\otimes\mathbb{Q} \hookrightarrow \left(\varprojlim_m\varinjlim_n \tilde{V}^0_{n,m}\right)[1/p]. \end{align*}$$The 
$\mathbb {Z}_p$-modules 
$H^0\left (\mathscr {S}^{\mathrm {tor}},\underline {\omega }_{\underline {t}}\otimes \mathcal {I}_{\mathscr {S}^{\mathrm {tor}}}\right )$ are free of finite rank and stable under the action of 
$U_p$, so the limit 
$\lim \limits _{r\rightarrow \infty } (U_p)^{r!}$ exists on them. Since
$$ \begin{align} \left(\bigoplus_{\underline{t} \text{ dominant}} H^0\left(\mathscr{S}^{\mathrm{tor}},\underline{\omega}_{\underline{t}}\otimes\mathcal{I}_{\mathscr{S}^{\mathrm{tor}}}\right)\otimes\mathbb{Q}\right) \cap \varprojlim_m\varinjlim_n \tilde{V}^0_{n,m} \end{align} $$is dense in 
$\mathop {\varprojlim }\limits _m\mathop {\varinjlim }\limits _n \tilde {V}^0_{n,m}$ by [Reference HidaHid02], the convergence of 
$\lim \limits _{r\rightarrow \infty }(U_p)^{r!}f$ for every 
$f\in \tilde {V}^0_{n,m}$ follows.
On the other hand, let 
$\vec {f}\in H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{n,m}}\right )$. By Proposition 3.0.1, 
$\vec {f}$ lifts to
$$\begin{align*}\vec{\mathfrak{f}}\in H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{n}[1/E],\omega_{\underline{t}}\otimes\mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_n}\right). \end{align*}$$For 
$l\gg 0$, we have 
$\vec {\mathfrak {f}} E^l\in H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_n}\right )$, which by the Koecher principle can be viewed as an element in the finite-dimensional space 
$M^0_{\underline {t}+lt_E(p-1)}(K^p_fK^0_{p,n};\mathbb {Q}_p)$. Thus, the limit 
$\lim \limits _{r\rightarrow \infty }(U_p)^{r!}(\vec {\mathfrak {f}} E^l)$ converges. Since 
$E\equiv 1\ \ \mod p$, we have 
$\vec {\mathfrak {f}} E^l\equiv \vec {f}\ \ \mod p^m$, and the convergence of 
$\lim \limits _{r\rightarrow \infty }(U_p)^{r!}\vec {f}$ follows.
Thanks to the convergence of 
$\lim \limits _{r\rightarrow \infty }(U_p)^{r!}$ on 
and 
$H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{n,m}}\right )$, we can define the semiordinary projector on them as 
$e_{\mathrm {so}}=\lim \limits _{r\rightarrow \infty }(U_p)^{r!}$.
Proposition 3.0.3. For 
$n\geq m$ and a dominant weight 
$(0,0,t^+;t^-)$, the map
$$\begin{align*}e_{\mathrm{so}} H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{n,m},\omega_{(0,0,t^+;t^-)}\otimes\mathcal{I}_{\mathscr{T}^{\mathrm{tor}}_{n,m}}\right)\rightarrow e_{\mathrm{so}} V^0_{n,m}[t^+,t^-] \end{align*}$$induced by equation (2.7.1) is an isomorphism.
Proof. Given 
$\vec {f}$, we have
Here, C runs over the subgroups of 
$A[p^{4n}]$ which can be spanned as
$$ \begin{align*} &(e^+_1,e^+_2,e^+_3,e^+_4)\cdot p^n\left(\begin{smallmatrix}1&&u_1&\ast\\&1&u_2&\ast\\ &&1&\ast\\ &&&1\end{smallmatrix}\right) \left(\begin{smallmatrix}p^{-n}&&&\\ & p^{-n}&&\\ && p^{-2n}&\\ &&& p^{-4n}\end{smallmatrix}\right) ,\\ &(e^-_1,e^-_2,e^-_3,e^-_4)\cdot p^n\left(\begin{smallmatrix}1&\ast & \ast & \ast\\ &1\\&-u_2&1&-u_1\\ &&&1\end{smallmatrix}\right) \left(\begin{smallmatrix}p^{-2n}&&\\ & p^{-5n}\\&& p^{-4n}\\&&& p^{-5n}\end{smallmatrix}\right) , \end{align*} $$with 
$e^{\pm }_1,e^{\pm }_2,e^{\pm }_3,e^{\pm }_4$ a basis of 
$A[p^{n}]$ compatible with 
$\alpha _p$. For such a C, 
$u_C=\left (\begin {smallmatrix}1&&u_1\\&1&u_2\\&&1\end {smallmatrix}\right ) $. By our definition of the 
$\Delta _+$-action on 
$W_{\underline {t}}$ (in equation (2.8.7)), we know that if 
$w\in W_{(0,0,t^+;t^-)}$ is a vector of nonhighest weight, then
$$ \begin{align*} \left(\left(\begin{smallmatrix}p^n\\ &p^n\\ &&p^{2n}\end{smallmatrix}\right) ,p^{2n}\right)\cdot w\equiv 0\quad\mod p^n. \end{align*} $$Therefore, from equation (3.0.2) we see that 
$U^n_p\vec {f}$ is actually determined by the projection of its values to the highest weight space.
On one hand, this shows that the map in the statement is injective. On the other hand, given 
$f\in e_{\mathrm {so}} V^0_{n,m}[t^+,t^-]$, we can define 
$\vec {f}$ by the rule
where C runs over the same range as above. One can check that 
$\vec {f}$ satisfies
$$\begin{align*}\vec{f}(A,\lambda,i,\alpha^p,\alpha_p\circ g,\underline{\varepsilon}_{\alpha_p\circ g})=\left(\left(\begin{smallmatrix}\ast & p\ast & \\ \ast & \ast\\ u_1&u_2&a_1\end{smallmatrix}\right) ,a_2\right)^{-1}\cdot \vec{f}(A,\lambda,i,\alpha^p,\alpha_p,\underline{\varepsilon}_{\alpha_p}) \end{align*}$$for 
$g^+\equiv \left (\begin {smallmatrix}\ast & \ast & u_1&\ast \\ p\ast & \ast & u_2&\ast \\ &&a_1&\ast \\&&&a^{-1}_2\end {smallmatrix}\right )\quad \mod p^n$, so equation (3.0.3) defines an element in 
$H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{(0,0,t^+;t^-)}\otimes \mathcal {I}_{\mathscr {T}^{\mathrm {tor}}_{n,m}}\right )$, and it is easy to check its semiordinarity from the semiordinarity of f. The composition of equation (3.0.3) and the map in the statement is 
$U^n_p$. Hence, the map is a bijection.
Proposition 3.0.4. 
$e_{\mathrm {so}} H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{\mathrm {tor}}_{n,m}}\right )=e_{\mathrm {so}} H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{1,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{\mathrm {tor}}_{1,m}}\right )$.
Proof. This follows immediately from the fact that 
$U_p$ maps the space 
$H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{\mathrm {tor}}_{n,m}}\right )$ into 
$H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n-1,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{\mathrm {tor}}_{n,m}}\right )$. (See §2.8.2.)
Proposition 3.0.5. For a fixed tame level 
$K^p_f$ and a fixed integer 
$B\geq 0$, the dimension of 
$e_{\mathrm {so}} M^0_{\underline {t}}(K^p_fK^0_{p,1};\mathbb {Q}_p)$ is uniformly bounded for all 
$t^+_1\geq t^+_2\geq t^+_3\geq -t^-_1+4$ with 
$t^+_1-t^+_2\leq B$.
Proof. Suppose that 
$\Pi $ is an irreducible cuspidal automorphic representation of 
$G(\mathbb {A}_{\mathbb {Q}})$ generated by a semiordinary form 
$\varphi \in e_{\mathrm {so}}M^0_{\underline {t}}\left (K^p_fK^0_{p,n};\mathbb {Q}_p\right )$ whose (generalized) eigenvalue of 
$U^+_{p,2}$ (resp. 
$U^+_{p,3}$, 
$U^-_{p,1}$) is 
$\lambda ^+_2$ (resp. 
$\lambda ^+_3$, 
$\lambda ^-_1$). The semiordinarity condition implies that 
$\lambda ^+_2,\lambda ^+_3,\lambda ^-_1$ are all p-adic units. We view 
$\Pi _p$ as an irreducible representation of 
$\operatorname {\mathrm {GL}}_4(\mathbb {Q}_p)$ via equation (2.4.1). Let Q be the parabolic subgroup 
$\left \lbrace \left (\begin {smallmatrix}\ast & \ast & \ast & \ast \\\ast & \ast & \ast & \ast \\&&\ast & \ast \\&&&\ast \end {smallmatrix}\right ) \right \rbrace \subset \operatorname {\mathrm {GL}}_4$ with Levi factorization 
$Q=M_QN_Q$. Let
$$\begin{align*}\Pi_{p,N_Q}=\Pi_p/\{\Pi_p(u)v-v\,:\,v\in\Pi_p,\,u\in N_Q(\mathbb{Q}_p)\}, \end{align*}$$the Jacquet module of 
$\Pi _p$ with respect to the parabolic Q acted on by 
$M_Q(\mathbb {Q}_p)\cong \operatorname {\mathrm {GL}}_2(\mathbb {Q}_p)\times \operatorname {\mathrm {GL}}_1(\mathbb {Q}_p)\times \operatorname {\mathrm {GL}}_1(\mathbb {Q}_p)$. It follows from Jacquet’s lemma (see [Reference CasselmanCas95, Theorem 4.1.2, Proposition 4.1.4]) that 
$\Pi ^{M_Q(\mathbb {Q}_p)\cap K^0_{p,n}}_{p,N_Q}$ equals the image under the natural projection of
$$ \begin{align*} &\bigcap_{r\geq 1}\left(U_{p,\mathrm{loc}}\right)^r\Pi^{K^0_{p,n}}_p, &U_{p,\mathrm{loc}}=\int_{N_Q(\mathbb{Z}_p)}\Pi_p\left(u\left(\begin{smallmatrix}p^2\\&p^2\\&&p^{\phantom{2}}\\&&&p^{-1}\end{smallmatrix}\right) \right)\,du. \end{align*} $$The semiordinary form 
$\varphi \in \Pi $ is fixed by 
$K^0_{p,n}$ and belongs to a generalized eigenspace of 
$U_{p,\mathrm {loc}}$ with a nonzero eigenvalue. Thus, we deduce that 
$\Pi ^{M_Q(\mathbb {Q}_p)\cap K^0_{p,n}}_{p,N_Q}\neq 0$ and 
$\Pi _p$ is isomorphic to a subquotient of
$$\begin{align*}\operatorname{\mathrm{Ind}}^{\operatorname{\mathrm{GL}}_4(\mathbb{Q}_p)}_{Q(\mathbb{Q}_p)} \sigma\boxtimes \chi\boxtimes\chi', \end{align*}$$where 
$\sigma $ is an irreducible smooth admissible representation of 
$\operatorname {\mathrm {GL}}_2(\mathbb {Q}_p)$, and 
$\chi ,\chi '$ are smooth characters of 
$\mathbb {Q}^{\times }_p$. We have 
$M_Q(\mathbb {Q}_p)\cap K^0_{p,1}\cong K^{\prime }_{p}\times (1+p^n\mathbb {Z}_p)\times (1+p^n\mathbb {Z}_p)$ with 
$K^{\prime }_{p}=\left \lbrace \begin {pmatrix}\ast & \ast \\ p\ast & \ast \end {pmatrix}\right \rbrace \subset \operatorname {\mathrm {GL}}_2(\mathbb {Z}_p)$. From 
$\Pi ^{M_Q(\mathbb {Q}_p)\cap K^0_{p,n}}_{p,N_Q}\neq 0$, we know that 
$\sigma ^{K^{\prime }_{p}}\neq 0$ ([Reference CasselmanCas95, Theorem 6.3.5]). This implies that 
$\sigma $ is not supercuspidal, and 
$\Pi _p$ is isomorphic to a subquotient of a principal series. Denote by 
$\alpha _1,\alpha _2,\alpha _3,\alpha _4$ the evaluations at p of the corresponding characters of 
$\mathbb {Q}^{\times }_p$. Then, up to reordering, we have
$$ \begin{align} \lambda^+_{2}&=p^{t^+_1+\frac{1}{2}}\alpha_1\cdot p^{t^+_2-\frac{1}{2}}\alpha_2,\end{align} $$
$$ \begin{align} \lambda^+_{3}&=p^{t^+_1+\frac{1}{2}}\alpha_1\cdot p^{t^+_2-\frac{1}{2}}\alpha_2\cdot p^{t^+_3-\frac{3}{2}}\alpha_3, \end{align} $$
$$ \begin{align} \lambda^-_{1}&=p^{t^-_1-\frac{3}{2}}\alpha^{-1}_4. \end{align} $$ Now, in addition to 
$U^+_{p,2},U^+_{p,3},U^-_{p,1}$, we also consider the action of the operator
$$ \begin{align} U^+_{p,1}=p^{\left<(t^+_1,t^+_2,t^+_3;-t^-_1)+2\rho_{\mathrm{c}},(1,0,0;0)\right>}\int_{N(\mathbb{Z}_p)}R\left(u\left(\begin{smallmatrix}p\\&1\\&&1\\&&&1\end{smallmatrix}\right) ,1\right)\,du \end{align} $$on 
$M^0_{\underline {t}}\left (K^p_fK^0_{p,n};\mathbb {Q}_p\right )$. The operator 
$U^+_{p,1}$ has a geometric interpretation analogous to equation (2.8.5), and the above normalization makes all its eigenvalues p-integral. If 
$\alpha ^{-1}_1\alpha _2\neq p^{\pm 1}$, then 
$p^{t^+_1+\frac {1}{2}}\alpha _1$ and 
$p^{t^+_1+\frac {1}{2}}\alpha _2$ are both eigenvalues for the action of 
$U^+_{p,1}$ on the holomorphic forms in 
$\Pi $ of level 
$K^p_fK^0_{p,n}$, so
$$ \begin{align} \begin{aligned} v_{\mathfrak{p}}(\alpha_1)+t^+_1+\frac{1}{2}&\geq 0, &v_{\mathfrak{p}}(\alpha_2)+t^+_1+\frac{1}{2}&\geq 0. \end{aligned} \end{align} $$By using that equation (3.0.4) is a p-adic unit and our condition 
$0\leq t^+_1-t^+_2\leq B$, we get
$$ \begin{align} \begin{aligned} (v_{\mathfrak{p}}(\alpha_1)+t^+_1+\frac{1}{2})+(v_{\mathfrak{p}}(\alpha_2)+t^+_1+\frac{1}{2})&=t^+_1-t^+_2+1 \leq B+1. \end{aligned} \end{align} $$Combining equation (3.0.8) and equation (3.0.9), we get
$$ \begin{align} 0\leq v_{\mathfrak{p}}(\alpha_j)+t^+_1+\frac{1}{2}\leq B+1, \quad j=1,2. \end{align} $$If 
$\alpha ^{-1}_1\alpha _2=p^{\pm 1}$, then equation (3.0.4) being a p-adic unit implies that
$$\begin{align*}\frac{t^+_1-t^+_2}{2}\leq v_{\mathfrak{p}}(\alpha_j)\leq \frac{t^+_1-t^+_2}{2}+1,\quad j=1,2. \end{align*}$$Combining it with our condition 
$0\leq t^+_1-t^+_2\leq B$, we get
$$ \begin{align} 0\leq v_{\mathfrak{p}}(\alpha_j)\leq \frac{B}{2}+1,\quad j=1,2. \end{align} $$Therefore, for 
$t^+_1\geq t^+_2\geq t^+_3\geq -t^-_1+4$, 
$t^+_1-t^+_2\leq B$, all the semiordinary forms in 
$e_{\mathrm {so}} M^0_{\underline {t}}(K^p_fK^0_{p,1};\mathbb {Q}_p)$ have slopes 
$\leq B$ for the 
$\mathbb {U}_p$-operator 
$U^+_{p,1}U^+_{p,2}U^+_{p,3}U^-_{p,1}$.
Recall that the theory of Coleman families for unitary groups (developed in [Reference BrascaBra16] as a generalization of [Reference Andreatta, Iovita and PilloniAIP15]) shows that, for every point in the weight space 
$\operatorname {\mathrm {Hom}}_{\mathrm {cont}}((\mathbb {Z}^{\times }_p)^4,\mathbb {C}^{\times }_p)$, there exists a neighborhood 
$\mathcal {U}$ of that point and a projective 
$\mathcal {A}_{\mathcal {U}}$-module of finite rank interpolating all the cuspidal overconvergent forms of weights in 
$\mathcal {U}$ and 
$U^+_{p,1}U^+_{p,2}U^+_{p,3}U^-_{p,1}$-slope 
$\leq B$ and a fixed tame level. Since all the algebraic weights 
$\underline {t}$ are contained in a compact subset of 
$\operatorname {\mathrm {Hom}}_{\mathrm {cont}}((\mathbb {Z}^{\times }_p)^4,\mathbb {C}_p)$, when 
$\underline {t}$ varies among all the algebraic weights, there is a uniform bound on the dimension of the space of cuspidal overconvergent forms of weight 
$\underline {t}$ and slopes 
$\leq B$ and tame level 
$K^p_f$. The proposition follows.
(1)
$\dim _{\mathbb {F}_p}\,e_{\mathrm {so}} H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{1,1},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{1,1}}\right )<\infty $.(2) (Classicity) There is a canonical embedding
and for a given
$t^+\leq 0$, there exists $A(t^+)\geq -t^++4$ such that if $t^-\geq A(t^+)$, then the embedding is an isomorphism.
Proof. (1) Suppose that 
$\overline {f}_1,\dots ,\overline {f}_d\in e_{\mathrm {so}} H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{1,1},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{1,1}}\right )$ are linearly independent. By Proposition 3.0.1, they lift to 
$f_1,\dots ,f_d\in H^0\left (\mathscr {T}^{0,\mathrm {tor}}_1[1/E],\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_1}\right )$. Recall that E has scalar weight 
$t_E(p-1)$ and let 
$\underline {t}+lk_E(p-1)=(t^+_1+lt_E(p-1),t^+_2+lt_E(p-1),t^+_3+lt_E(p-1);t^-_1+lt_E(p-1))$. For 
$l\gg 0$, we have
$$\begin{align*}f_1E^l,\dots,f_dE^l\in H^0\left(\mathscr{T}^{0,\mathrm{tor}}_n,\omega_{\underline{t}+lt_E(p-1)}\otimes\mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_n}\right). \end{align*}$$Because 
$E\equiv 1\ \ \mod p$, we have 
$\overline {e_{\mathrm {so}}(f_jE^l)}=\overline {f}_j$, and therefore 
$e_{\mathrm {so}}(f_1E^l),\dots ,e_{\mathrm {so}}(f_dE^l)$ are linearly independent. Thus, d can be at most the bound in Proposition 3.0.5.
(2) We have 
$M^0_{(0,0,t^+;t^-)}\left (K_f^pK^0_{p,1};\mathbb {Q}_p\right )=H^0\left (\mathscr {T}^{0,\mathrm {tor}}_1,\omega _{(0,0,t^+;t^-)}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_1}\right )\otimes _{\iota _p}\mathbb {Q}_p$. (As mentioned above, the base change of 
$\mathscr {T}^{0,\mathrm {tor}}_1$ to 
$\mathcal {K}$ is an open subscheme of 
$S^{\mathrm {tor}}_{K^p_fK^0_{p,1}}$; here, we use the Koecher principle to get the equality of global sections.) The map (2.7.1) induces
$$\begin{align*}H^0\left(\mathscr{T}^{0,\mathrm{tor}}_1,\omega_{(0,0,t^+;t^-)}\otimes\mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_1}\right)\rightarrow \left(\varprojlim_m\varinjlim_n e_{\mathrm{so}}V^0_{n,m}[t^+,t^-]\right), \end{align*}$$and this gives the canonical embedding in the statement.
Put 
$V^0_{\mathrm {so}}[t^+,t^-]=\mathop {\varprojlim }\limits _m\mathop {\varinjlim }\limits _n e_{\mathrm {so}}V^0_{n,m}[t^+,t^-]$. Note that from Propositions 3.0.3 and 3.0.4, for 
$n\geq m$, we have
$$\begin{align*}e_{\mathrm{so}} V^0_{n,m}[t^+,t^-]\simeq e_{\mathrm{so}} H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1,m},\omega_{(0,0,t^+;t^-)}\otimes \mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1,m}}\right). \end{align*}$$It follows that 
$V_{\mathrm {so}}^0[t^+,t^-]$ is p-torsion free, and together with Proposition 3.0.1 we get
$$ \begin{align*} \dim_{\mathbb{F}_p}V^0_{\mathrm{so}}[t^+,t^-]/pV^0_{\mathrm{so}}[t^+,t^-] &=\dim_{\mathbb{F}_p} e_{\mathrm{so}} H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1,1},\omega_{(0,0,t^+;t^-)}\otimes \mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1,1}}\right)\\ &=\dim_{\mathbb{Q}_p} e_{\mathrm{so}} H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1}[1/E],\omega_{(0,0,t^+;t^-)}\otimes \mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1}}\right)\otimes\mathbb{Q}_p. \end{align*} $$Denote this dimension by d. The above shows that 
$V^0_{\mathrm {so}}[t^+,t^-]$ is a free 
$\mathbb {Z}_p$-module of rank d. Thus, it suffices to show that there exists 
$A(t^+)\geq -t^+$ such that 
$\dim _{\mathbb {Q}_p} e_{\mathrm {so}} M^0_{(0,0,t^+;t^-)}\left (K_f^pK^0_{p,1};\mathbb {Q}_p\right ) \geq d$ for 
$t^-\geq A(t^+)$. Pick an unramified Hecke character 
$\chi $ of 
$\mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K}}$ of 
$\infty $-type 
$(-t_E(p-1),0)$. Twisting E by the character obtained by composing 
$\chi $ with 
$\det :G(\mathbb {Q})\backslash G(\mathbb {A}_{\mathbb {Q}})\rightarrow \mathcal {K}^{\times }\backslash \mathbb {A}^{\times }_{\mathcal {K}}$, we get a holomorphic form 
$E'$ of weight 
$(0,0,0;2t_E(p-1))$ with the same vanishing locus as E. Multiplying by 
$(E')^l$ and applying 
$e_{\mathrm {so}}$ gives an injection
so
$$\begin{align*}\dim_{\mathbb{Q}_p}e_{\mathrm{so}} M^0_{(0,0,t^+;t^-)}\left(K_f^pK^0_{p,1};\mathbb{Q}_p\right)\leq \dim_{\mathbb{Q}_p}e_{\mathrm{so}} M^0_{(0,0,t^+;t^-+2t_E(p-1))}\left(K_f^pK^0_{p,1};\mathbb{Q}_p\right). \end{align*}$$By the definition of d, we thus see that for each 
$0\leq j\leq 2t_E(p-1)-1$ there exists 
$l_j(t^+)\geq 0$ such that
$$\begin{align*}\dim_{\mathbb{Q}_p}e_{\mathrm{so}} M^0_{(0,0,t^+;-t^++4+j+2t_E(p-1)l_j(t^+))}\left(K_f^pK^0_{p,1};\mathbb{Q}_p\right)\geq d. \end{align*}$$Therefore, 
$A(t^+)=-t^++4+2t_E(p-1) (\max _j\{l_j(t^+)\}+1)$ has the required property.
Theorem 3.0.7 (Vertical control theorem).
is a free 
$\Lambda _{\mathrm {so}}$-module of finite rank.
Proof. Let 
$\mathfrak {m}$ be a maximal ideal of 
$\mathbb {Z}_p[\![ T_{\mathrm {so}}(\mathbb {Z}_p)]\!]$. (We know that 
$\mathfrak {m}\cap \Lambda _{\mathrm {so}}=(p,T^+,T^-)$, where we identify 
$\Lambda _{\mathrm {so}}$ with 
$\mathbb {Z}_p[\![ T^+,T^-]\!]$ by identifying 
$(1+p,1)\in T_{\mathrm {so}}(\mathbb {Z}_p)$ (resp. 
$(1,1+p)\in T_{\mathrm {so}}(\mathbb {Z}_p)$ with 
$T^+$ (resp. 
$T^-$).) We show that 
, the localization of 
at 
$\mathfrak {m}$, is a free 
$\Lambda _{\mathrm {so}}$-module of finite rank.
First, we consider the quotient
, which by definition, equals
. Take 
$0\geq t^+\geq -t^-+4$ with 
$\mathcal {P}_{t^+,t^-}\subset \mathfrak {m}$. Then
Also, Proposition 3.0.1 implies
$$ \begin{align*} \varinjlim_m H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1,m},\omega_{(0,0,t^+;t^-)}\otimes \mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1,m}}\right) &=\varinjlim_m H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1}[1/E],\omega_{(0,0,t^+;t^-)}\otimes \mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1}}\right)\otimes\mathbb{Z}/p^m\mathbb{Z}. \end{align*} $$It follows that
$$ \begin{align*} &p\text{-torsion of }\varinjlim_m H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1,m}, \omega_{(0,0,t^+;t^-)}\otimes\mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1,m}}\right)\\ =&\, H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1}[1/E],\omega_{(0,0,t^+;t^-)}\otimes \mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1}}\right)\otimes\mathbb{Z}/p\mathbb{Z}\\ =&\, H^0\left(\mathscr{T}^{0,\mathrm{tor}}_{1,1},\omega_{(0,0,t^+;t^-)}\otimes \mathcal{I}_{\mathscr{T}^{0,\mathrm{tor}}_{1,1}}\right), \end{align*} $$where Proposition 3.0.1 is used again for the second equality. Thus,
By Proposition 3.0.6, we know this is finite dimensional over 
$\mathbb {F}_p$. Hence,
is finite dimensional over 
$\mathbb {F}_p$. Let
By Nakayama’s lemma, there exist
such that
Next, we show that 
$F_1,\dots ,F_d$ are 
$\Lambda _{\mathrm {so}}$-linearly independent. Suppose that 
$a_1,\dots ,a_d\in \Lambda _{\mathrm {so}}$ are such that 
$a_1F_1+\cdots +a_dF_d=0$. Given any 
$0\geq t^+\geq -t^-+4$ with 
$\mathcal {P}_{t^+,t^-}\subset \mathfrak {m}$, we put
$$\begin{align*}\mathcal{P}^{\circ}_{t^+,t^-}=\Lambda_{\mathrm{so}}\cap \mathcal{P}_{t^+,t^-}. \end{align*}$$In view of Proposition 3.0.1, the 
$\mathbb {Z}_p$-module 
$\mathop {\varinjlim }\limits _m\mathop {\varinjlim }\limits _n H^0\left (\mathscr {T}^{0,\mathrm {tor}}_{n,m},\omega _{\underline {t}}\otimes \mathcal {I}_{\mathscr {T}^{0,\mathrm {tor}}_{n,m}}\right )$ is p-divisible, and by Proposition 3.0.3 so is
. Therefore,
is p-torsion free. On the other hand,
It follows that the 
$\mathbb {Z}_p$-module
is free of rank d, and therefore 
$a_1,\dots ,a_d\in \mathcal {P}^{\circ }_{t^+,t^-}$. Then from 
$\bigcap \limits _{0\geq t^+\geq -t^-+4}\mathcal {P}^{\circ }_{t^+,t^-}=0$, we conclude that 
$a_1=\cdots =a_d=0$, and hence
is a free 
$\Lambda _{\mathrm {so}}$-module of rank d.
This completes the proof of part (1) of Theorem 2.9.1 for cuspidal forms. For part (2), because 
is free over 
$\Lambda _{\mathrm {so}}$, letting 
$V^0_{m,\mathrm {so}}[\tau ^+,\tau ^-]=e_{\mathrm {so}} \mathop {\varinjlim }\limits _n V^0_{n,m}\otimes _{\mathbb {Z}_p}\mathcal {O}_F[\tau ^+,\tau ^-]$ we have
Since the left-hand side is a free 
$\mathbb {Z}_p$-module (because 
$\mathcal {M}^0_{\mathrm {so}}$ is a free 
$\Lambda _{\mathrm {so}}$-module), we see that 
$\mathop {\varinjlim }\limits _m V^0_{m,\mathrm {so}}[\tau ^+,\tau ^-])$ is p-divisible, and 
$V^0_{m+1,\mathrm {so}}[\tau ^+,\tau ^-]\rightarrow V^0_{m,\mathrm {so}}[\tau ^+,\tau ^-]$ is surjective, so
$$\begin{align*}\operatorname{\mathrm{Hom}}_{\mathbb{Z}_p}\left(\varinjlim_m V^0_{m,\mathrm{so}}[\tau^+,\tau^-],\mathbb{Q}_p/\mathbb{Z}_p\right)\cong \operatorname{\mathrm{Hom}}_{\mathbb{Z}_p}\left(\varprojlim_m V^0_{m,\mathrm{so}}[\tau^+,\tau^-],\mathbb{Z}_p\right)\\[-15pt] \end{align*}$$and hence
$$ \begin{align*} (3.0.12) &=\operatorname{\mathrm{Hom}}_{\mathbb{Z}_p}\left(\operatorname{\mathrm{Hom}}_{\mathbb{Z}_p}\left(\varprojlim_m V^0_{m,\mathrm{so}}[\tau^+,\tau^-],\mathbb{Z}_p\right),\mathbb{Z}_p\right)=\varprojlim_m V^0_{m,\mathrm{so}}[\tau^+,\tau^-]\\ &=\left(\varprojlim_m \varinjlim_n e_{\mathrm{so}} V^0_{n,m}\otimes_{\mathbb{Z}_p}\mathcal{O}_F\right)[\tau^+,\tau^-],\\[-15pt] \end{align*} $$concluding the proof of part (2) of Theorem 2.9.1 for cuspidal forms. Part (3) follows from Proposition 3.0.6.
4 The proof of Theorem 2.9.1 for the noncuspidal part
We apply the approach in [Reference Liu and RossoLR20] to prove the vertical control theorem for semiordinary forms on 
$\operatorname {\mathrm {GU}}(3,1)$ by analyzing the quotient 
and using the vertical control theorem for cuspidal semiordinary forms on 
$\operatorname {\mathrm {GU}}(3,1)$. When studying 
, we introduce an auxiliary space 
. One difference from loc.cit is that q-expansions are used there to reduce proving some properties for the 
$\mathbb {U}_p$-action on 
to matrix computations, but q-expansions are not available in the case of 
$\operatorname {\mathrm {GU}}(3,1)$. The analogue of those properties in our case are proved in §4.5 by working with semiabelian schemes over the boundary of the partial toroidal compactification.
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
$$ \begin{align} X^{\vee}\subset X^{\vee}\oplus L_0\subset L=X^{\vee}\oplus L_0\oplus Y. \end{align} $$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
•
$\mathtt {Z}^{(1)}$: $\mathtt {Z}_{-2}=X^{\vee }\otimes \hat {\mathbb {Z}}$, $\mathtt {Z}_{-1}=(X^{\vee }\oplus L_0)\otimes \hat {\mathbb {Z}}$.
