In this paper we prove one side divisibility of the Iwasawa-Greenberg main conjecture for Rankin-Selberg product of a weight two cusp form and an ordinary CM form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm{GU}(3,1)$, generalizing earlier result of the third-named author to allow non-ordinary cusp forms. The main result is a key input in the third author's proof for Kobayashi's $\pm$-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semi-ordinary Hida theory along an appropriate smaller weight space, and a study of the semi-ordinary Eisenstein family.

§ 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 form unramified at $p$. The result is a key ingredient in the third author's proof [38] of the Iwasawa Main Conjecture for elliptic curves with supersingular reduction at $p$ and $a_p=0$ formulated by Kobayashi [25].

Let $\pi$ be an irreducible cuspidal automorphic representation of $\GL_2(\bA_\bQ)$ generated by a newform of weight $2$. Associated to $\pi$ is a continuous two-dimensional $p$-adic Galois representation $\gls{rhopi}$ of $\Gal(\ol{\bQ}/\bQ)$ over $L$, a finite extension of $\bQ_p$, for which we can fix a $\Gal(\ol{\bQ}/\bQ)$-stable $\cO_L$-lattice $T_\pi$. (We use the geometric convention for Galois representations. The determinant of $\rho_\pi$ is $\epsilon^{-1}_{\mr{cyc}}$.)

Let $\cK$ be an imaginary quadratic field in which $p$ splits as $\fp\bar{\fp}$. Denote by $\cK_\infty$ the maximal abelian pro-$p$ extension of $\cK$ unramified outside $p$. Then the Galois group $\Gal(\cK_\infty/\cK)$ is isomorphic to $\bZ^2_p$ and we denote it by $\Gamma_\cK$. We have the tautological character

\[
\gls{PsiK}:\Gal(\ol{\bQ}/\cK)\ra\Gamma_\cK\hra \bZ_p\llb \Gamma_\cK\rrb^\times.
\]

Let $\xi:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ be an algebraic Hecke character associated to which is a character $\Gal(\ol{\bQ}/\cK)\ra \cO^\times_L$. We denote this Galois character also by $\xi$.

We consider the $\Gal(\ol{\bQ}/\cK)$-module

\[
\]

Define the Selmer group

\begin{equation}\label{eq:Sel}
\gls{Sel}=\ker\left\{H^1\big(\cK,T_{\pi,\cK,\xi}\otimes_{\cO_L\llb\Gamma_\cK\rrb}\cO_L\llb\Gamma_\cK\rrb^*\big)
\lra \prod_{\fv\neq \fp}H^1\big(I_\fv,T_{\pi,\cK,\xi}\otimes_{\cO_L\llb\Gamma_\cK\rrb}\cO_L\llb\Gamma_\cK\rrb^*\big)\right\},
\end{equation}

with $\cO_L\llb\Gamma_\cK\rrb^*=\Hom_{\bZ_p}\left(\cO_L\llb\Gamma_\cK\rrb,\bQ_p/\bZ_p\right)$, the Pontryagin dual of $\cO_L\llb\Gamma_\cK\rrb$. (This Selmer group has relaxed condition at $\fp$ and unramified condition at $\bar{\fp}$.) Let

\begin{equation}\label{eq:Xsel}
\gls{Xpi}:=\Hom_{\bZ_p}\big(\Sel_{\pi,\cK,\xi},\bQ_p/\bZ_p\big)
\end{equation}

which is well-known to be a finitely generated $\cO_L\llb\Gamma_\cK\rrb$-module. We recall the following definition of characteristic ideals.

For a Noetherian normal domain $A$ and a finitely generated $A$-module $M$, we define the characteristic ideal of $M$ as

\[
\gls{char}=\left\{x\in A:\ord_P(x)\geq \mr{length}_{A_P}(M_P) \text{ for all height one prime ideal $P\subset A$}\right\}.
\]

The Iwasawa–Greenberg main conjectures [9] predict that  the characteristic ideal of $X_{\pi,\cK,\xi}$ is generated by the following $p$-adic $L$-function.

Denote by $\hat{\cO}^\ur_L$ the completion of the maximal unramified extension of $\cO_L$. It follows from the construction in [6] that there is a $p$-adic $L$-function $\gls{Lpi}\in\mr{Frac}\big(\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\big)$ satisfying the interpolation property: for all algebraic Hecke characters $\tau:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ such that its $p$-adic avatar $\tau_{p\adic}$  factors through $\Gamma_\cK$ and $\xi\tau$ has $\infty$-type $\left(k_1,k_2\right)$ with $k_1,k_2\in\bZ$, $k_1\leq 0$, $k_2\geq 2-k_1$,

\begin{equation}\label{eq:intp}
\begin{aligned}
\cL_{\pi,\cK,\xi}(\tau_{p\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{\fp}}\right)\\
&\times L^{\{\infty,p\}}\left(\frac{k_1+k_2-1}{2},\mr{BC}(\pi)\times\xi_0\tau_0\right),
\end{aligned}
\end{equation}

where $\xi_0\tau_0=\xi\tau|\cdot|^{-\frac{k_1+k_2}{2}}_{\bA_\cK}$ and $\mr{BC}(\pi)$ denotes the unitary automorphic representation of $\GL_2(\bA_\cK)$ obtained as the base change of $\pi$.

More precisely, by using the doubling method, an imprimitive $p$-adic $L$-function is constructed in [6] as an element in $\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb$, interpolating the $L$-values in (<ref>) with certain local $L$-factors away from $p\infty$ removed. The inverses of these local $L$-factors are easily seen to be $p$-adically interpolated by elements in $\cO_L\llb\Gamma_\cK\rrb$. Multiplying those local factors to the imprimitive $p$-adic $L$-function, one obtains the complete $p$-adic $L$-function $\cL_{\pi,\cK,\xi}$. Note also that the interpolation formulas in [6] are completely computed for $k_1=0$, and one can get the general case by using the results on the computation of the doubling archimedean zeta integrals in [5].

We are interested in the following (two-variable) Greenberg type main conjecture.

\[
\mr{char}_{\cO^{\ur}_L\llb\Gamma_\cK\rrb} \big(X_{\pi,\cK,\xi}\big)= \big( \cL_{\pi,\cK,\xi}\big).
\]

The main result of this paper is Theorem <ref>, which is a partial result towards this conjecture. Like the previous works [36, 37, 35, 40] on proving Greenberg type main conjecture for modular forms, the proof uses the congruences between the Klingen Eisenstein series and cuspidal holomorphic automorphic forms. The $L$-values in our case here are the same as those in [40] and we also use Klingen Eisenstein series on $\GU(3,1)$ as in loc.cit. The main difference is that the modular form 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 semi-ordinary automorphic forms on $\GU(3,1)$. In <ref>-<ref>, we develop a Hida theory for $p$-adic families of (cuspidal and non-cuspidal) semi-ordinary forms on $\GU(3,1)$ along an appropriate two-dimensional subspace of its three-dimensional weight space. The main results are stated in Theorem <ref>. In <ref>, by using the doubling method, we construct a  Klingen Eisenstein family $\bfE^\Kling_\varphi$ and prove its semi-ordinarity. In <ref>, we study the degenerate Fourier–Jacobi coefficients of $\bfE^\Kling_\varphi$. The analogus computations in [40] 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 doubling zeta integrals. In <ref>, we study the non-degenerate Fourier–Jacobi coefficients of $\bfE^\Kling_\varphi$. This part is very similar to [40] 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 (<ref>) instead of till the end of the analysis of the non-degenerate Fourier–Jacobi coefficients, and a explanation on the strategy for analyzing the non-degenerate Fourier–Jacobi coefficients is included in <ref>.  In <ref>, combining the results in <ref> and <ref>, 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.

We fix a prime $p\geq 3$ and an imaginary quadratic field $\gls{cK}$ in which $p$ splits in $\cK$ as $\fp\bar{\fp}$. Denote by $\gls{Disc}$ the Discriminant of $\cK/\bQ$ and by $\gls{diffideal}$ the different ideal of $\cK/\bQ$. Denote by $\gls{eta}$ the quadratic character of $\bQ^\times\backslash\bA^\times_\bQ$ associated to $\cK/\bQ$. Let $\gls{c}$ be the nontrivial element in $\Gal(\cK/\bQ)$. For $x\in \cK$, denote by $\bar{x}$ its image of under $c$. For a finite place $v$ of $\bQ$, we put $\cK_v=\cK\otimes_{\bQ}\bQ_v$ and $\cO_{\cK,v}=\cO_\cK\otimes_\bZ\bZ_v$.

Fix embeddings

\begin{equation}\label{eq:Kembd}
\begin{aligned}
\glsuseri{iota}:\cK\lhra\bC,
\end{aligned}
\end{equation}

such that the valuation of $\bQ_p$ and $\iota_p$ induce the valuation of $\cK$ given by $\fp$.  The embedding $\iota_p:\cK\hra\bQ_p$ induces a homomorphism $\glsuseri{rho}:\cK_p=\cK\otimes_\bQ\bQ_p\ra\bQ_p$. We denote by $\glsuserii{rho}:\cK\otimes_\bQ\bQ_p\ra\bQ_p$ the composition of $\varrho_p$ and the non-trivial element $c\in\Gal(\cK/\bQ)$. Then

\begin{align*}
&&a\longmapsto \left(\varrho_\fp(a),\varrho_{\bar{\fp}}(a)\right)
\end{align*}

is an isomorphism. We also fix a totally imaginary element $\gls{delta}\in\cK$ such that $\mr{Nm}(\delta)=\delta\bar{\delta}$ is a $p$-adic unit.

Fix the standard additive character $\be_{\bA_\bQ}=\bigotimes_v\gls{ev}:\bQ\backslash\bA\ra\bC^\times$ with

\begin{equation}\label{eq:bev}
\be_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{equation}

where $\{x\}_v$ is the fractional part of $x$.

Acknowledgments. We would like to thank Eric Urban for pointing out to us the possibility to develop the semi-ordinary Hida theory needed in this paper. We also thank Ming-Lun Hsieh, Kai-Wen Lan and Christopher Skinner for useful communications. During the preparation of this paper, the first author was partially supported by the NSF grant DMS-1946136 and DMS-2101458, the second author was partially supported by the NSF grant DMS-2001527, the third author was partially supported by NSFC grants 11688101, 11621061 and National Key R and D Program of China 2020YFA0712600.

§ HIDA THEORY FOR SEMI-ORDINARY FORMS ON $\GU(3,1)$

We define semi-ordinary forms on $\GU(3,1)$ and state the control theorem for semi-ordinary families (Theorem <ref>). The proof of Theorem <ref> is given in the following two sections.

§.§ Some notation

Let $L_0$ be a free $\cO_\cK$-module of rank $2$ with basis $\ww_1, \ww_2$, and we equip $L_0\otimes_\bZ\bQ$ with a skew-Hermitian form $\left<\,,\,\right>_{L_0}$ whose matrix with respect to the basis $\ww_1, \ww_2$ is given by a matrix $\glsuseri{zeta}\in\Her_2(\cO_\cK)$ with $\delta\zeta_0$ positive definite. Let $X, Y$ be free $\cO_\cK$-modules of rank $1$ with bases $\xx_1, \yy_1$. Let $X^\vee=\fd^{-1}_{\cK/\bQ}\cdot {\tt x}_1$ and $L=X^\vee\oplus L_0\oplus Y$. Equip $L\otimes_\bZ\bQ$ with the skew-Hermitian form $\left<\,,\,\right>_{L}$ whose matrix with respect to the basis  $\xx_1, \ww_1, \ww_2, \yy_1$ is given by $\glsuserii{zeta}=\begin{pmatrix}&&1\\&\zeta_0\\-1\end{pmatrix}.$

Define the similitude unitary groups $\glsuserii{G}=\GU(2)$ and $\glsuseri{G}=\GU(3,1)$ (over $\bZ)$) as: for all $\bZ$-algebra $R$

\begin{equation}\label{eq:ugp}
\begin{aligned}
\GU(2)(R)&=\{(g,\nu)\in \GL_{\cO_\cK\otimes_{\bZ} R}(L_0\otimes_{\bZ} R)\times R^\times\,:\,\left<gv_1,gv_2\right>_{L_0}=\nu\left<v_1,v_2\right>_{L_0}\},\\
\GU(3,1)(R)&=\{(g,\nu)\in \GL_{\cO_\cK\otimes_{\bZ} R}(L\otimes_{\bZ} R)\times R^\times\,:\,\left<gv_1,gv_2\right>_{L}=\nu\left<v_1,v_2\right>_{L}\},
\end{aligned}
\end{equation}

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

§.§ Shimura variety

We fix an open compact subgroup $K^p_f\subset G(\bA^p_{\bQ,f})$, assumed throughout to be neat. Let $2\pi i\left<\,,\,\right>:L\times L\ra \bZ(1)$ be the alternating pairing $2\pi i\cdot \Tr_{\cK/\bQ}\circ\left<\,,\,\right>_L$ and $h:\bC\ra\mr{End}_{\cO_\cK\otimes_{\bZ}\bR}(L\otimes_{\bZ}\bR)$ be the homomorphism given by

\[
h(u+iv):\left(\xx_1,\ww_1,\ww_2,\yy_1\right)\mapsto \left(\xx_1,\ww_1,\ww_2,\yy_1\right)
\begin{pmatrix}
1\otimes u&&&-1\otimes v\\
&1\otimes u+\delta\otimes\frac{v}{\sqrt{\delta\bar{\delta}}}\\
&&1\otimes u+\delta\otimes\frac{v}{\sqrt{\delta\bar{\delta}}}\\
1\otimes v&&& 1\otimes u
\end{pmatrix}.
\]

Then the tuple $\left(\cO_\cK,c,L,2\pi i\left<\,,\,\right>,h\right)$ defines a Shimura datum of PEL type with reflex field $\cK$.

Consider the moduli problem sending every locally noetherian connected $\cO_{\cK,(p)}$-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 $S$;

* $\lambda:A\ra A^\vee$ a $\bZ^\times_{(p)}$-polarization;

* $i:\cO_\cK\hra \mr{End}_S A$ an embedding such that the induced $\cO_\cK$-action on $\Lie A_{/S}$ satisfies the  determinant condition defined by $h$, i.e.,

\begin{align*}
\det\left(X-i(b)|\Lie A_{/S}\right)&=(X-b)^3(X-\bar{b})
\end{align*}

for all $b\in \cO_\cK$ (where on the right-hand side $b$ is viewed as an element in $\mathscr{O}_S$ under the morphism $\cO_\cK\rightarrow\mathscr{O}_S$);

* $\alpha^p$ an (integral) $K^p_f$-level structure on $(A, \lambda, i)$ of type $\left(L\otimes\wh{\bZ}^{(p)},\Tr_{\cK/\bQ}\circ\left<\,,\,\right>_L\right)$; that is, a $K^p_f$-orbit of $\cO_\cK$-module isomorphisms $L\otimes\wh{\bZ}^{(p)}\ra T^{(p)}A_{\bar{s}}$, where $T^{(p)}A_{\bar{s}}$ is the prime-to-$p$ Tate module of $A_{\bar{s}}$, together with an isomorphism $\wh{\bZ}^{(p)}(1)\stackrel{\sim}{\ra}\bG_{m,\bar{s}}$ making the following diagram commute:

\[
\xymatrix@C+2em{
(L\otimes\wh{\bZ}^{(p)})\times (L\otimes\wh{\bZ}^{(p)}) \ar[r]^-{\Tr_{\cK/\bQ}\circ\left<\,,\,\right>_L} \ar[d] &\wh{\bZ}^{(p)}(1)\ar[d]\\
T^{(p)}A_{\bar{s}}\times  T^{(p)}A_{\bar{s}}\ar[r]^-{\lambda\text{-Weil}}& \bG_{m,\ol{s}}.
\]

(See <cit.>.)

Since $K^p_f$ is neat, the above moduli problem is represented by a smooth quasi-projective scheme $\gls{sS}$ over $\cO_{\cK,(p)}$ (see <cit.>).

Denote by $\gls{sStor}$ the toroidal compactification of $\sS$, which is a proper smooth scheme over $\cO_{\cK,(p)}$ containing $\sS$ as an open dense subscheme with complement being a relative Cartier divisor with normal crossings. (In our special case $\GU(3,1)$ here, there is a unique choice of polyhedral cone decomposition for the toroidal compactification.) We denote by $\gls{IS}$ the ideal sheaf of the boundary of $\sS^\tor$.

By <cit.>, the universal family $(\sA,\lambda,i,\alpha^p)$ over $\sS$ extends to a degenerating family $(\sG,\lambda,i,\alpha^p)$ over $\sS^\tor$. Moreover, the base change of $\sS$ (resp. $\sS^\tor$) to $\cK$ agrees with the Shimura variety over $\cK$ (resp. its toroidal compactification) representing the moduli problem with full level structure at $p$ (see <cit.>).

§.§ Hasse invariant

Set $\glsuseri{omega}:=e^*\Omega^1_{\sG/\sS^\tor}$ where $e:\sS^\tor\ra \sG$ is the zero section of the semi-abelian scheme $\sG$ over $\sS^\tor$. Let $\glsuserii{omega}$ be the line bundle $\det\underline{\omega}=\wedge^{\mr{top}}\underline{\omega}$. The minimal compactification of $\sS$ is defined by

\[
\gls{sSmin}=\mr{Proj}\Biggl(\bigoplus\limits_{k\geqslant 0}H^0(\sS^\tor,\omega^k)\Biggr).
\]

Let $\pi:\sS^{\tor}\rightarrow\sS^{\min}$ be the canonical projection. The push-forward $\pi_*\omega$ is an ample line bundle, and $\pi^*\pi_*\omega\cong \omega$ (see <cit.>).

In the following, with a slight abuse of notation, we also denote by $\sS^{\tor}$ and $\sS^{\min}$ their base change to $\bZ_p$ via the map $\cO_{\cK,(p)}\ra \bZ_p$ induced by our fixed embedding $\iota_p$, and let $\sS^\tor_{/\bF_p}$ and $\sS^{\min}_{/\bF_p}$ be their corresponding special fibers.

Let $\gls{Ha}\in H^0\left(\sS^\tor_{/\bF_p},\omega^{p-1}\right)$ be the Hasse invariant defined as in <cit.>. In particular, for each geometric point $\bar{s}$ of $\sS^\tor_{/\bF_p}$, the Hasse invariant of the corresponding semi-abelian scheme $\sG_{\bar{s}}$ is nonzero if and only if the abelian part of $\sG_{\bar{s}}$ is ordinary. Because $\pi_*\omega$ is ample, for some $t_E> 0$, there exists an element in $H^0\left(\sS^{\min},(\pi_*\omega)^{t_E(p-1)}\right)$ lifting the $t_E(p-1)$-th power of the push-forward of $\mr{Ha}$; we denote by $\gls{E}$ the pullback under $\pi$ of any such lift, which (because $\pi^*\pi_*\omega\cong \omega$)
defines an element $E\in H^0\left(\sS^{\tor},\omega^{t_E(p-1)}\right)$.

§.§ Some groups

Before moving on, we need to introduce some more group-theoretic notations. Given a matrix $a\in\GL_4(\cK\otimes_\bQ \bQ_p)$, we define $a^+,a^-\in \GL_4(\bQ_p)$ as

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

where $\varrho_\fp,\varrho_{\bar{\fp}}:\cK\otimes_\bQ\bQ_p$ are defined as in  Notation, and we view an element $g\in G(\bQ_p)$ as a matrix inside $\GL_4(\cK\otimes_\bQ \bQ_p)$ via the basis $(\xx_1,\ww_1,\ww_2,\yy_1)$. Then the corresponding matrices $g^+,g^-\in\GL_4(\bQ_p)$ satisfy

\begin{align*}
&\ltrans{g}^-\varrho_\fp(\zeta)\,g^+=\nu(g)\cdot \varrho_\fp(\zeta).
\end{align*}

In the following, we will often write $g\in G(\bQ_p)$ as $(g^+,g^-)$. Note that the map

\begin{equation}\label{eq:Gisom}
\begin{aligned}
G(\bQ_p)&\ra \GL_4(\bQ_p)\times\bQ^\times_p\\
g&\mapsto (g^+,\nu(g))
\end{aligned}
\end{equation}

is an isomorphism.

There is a filtration $\glsuseri{ttD}=\{{\tt D}^i\}_i$
of $L\otimes_{\bZ}\bZ_p$ given by

\begin{equation}\label{eq:fil-D}
\glsuserii{ttD}=0\;\subset\;\glsuseriii{ttD}=X^+_{p}\oplus L^+_{0,p}\oplus X^-_{p}\;\subset\; \glsuseriv{ttD}=L\otimes_{\bZ}\bZ_p,
\end{equation}

where $X^+_p\subset X^\vee\otimes_{\bZ}\bZ_p=X\otimes_{\bZ}\bZ_p$ (resp. $X^-_p\subset X\otimes_{\bZ}\bZ_p$,  $L^+_{0,p}\subset L_0\otimes_{\bZ}\bZ_p$) is the subspace on which $b\in\cO_\cK$ acts by $\iota_p(b)$ (resp. $\iota_p(\bar{b})$,  $\iota_p(b)$). Note that
${\tt D}^0$ is isotropic with respect to $\Tr_{\cK/\bQ}\circ\left<\,,\,\right>_L$. For $R=\bZ_p$ or $\bQ_p$, we put

\[
\gls{PD}(R)=\{g\in G(R)\,:\,g({\tt D}^0)={\tt D}^0\};
\]

i.e., $P_{\tt D}(R)$ is the subgroup of $G(R)$ stabilizing the filtration ${\tt D}$ in $(\ref{eq:fil-D})$. We see that $P_{\tt D}^\pm(R):=\{g^\pm\colon g\in P_{\tt D}(R)\}$ are the subgroups of $\GL_4(R)$ given by

\[
P_{\tt D}^+=\left\{\begin{psm}\ast&\ast&\ast&\ast\\ \ast&\ast&\ast&\ast\\ \ast&\ast&\ast&\ast\\&&&\ast\end{psm}\right\},\quad
P_{\tt D}^-=\left\{\begin{psm}\ast&\ast&\ast&\ast\\ &\ast&\ast&\ast\\ &\ast&\ast&\ast\\&\ast&\ast&\ast\end{psm}\right\}.
\]

Because $P_{\tt D}$ preserves both $X^+_p\oplus L^+_{0,p}$ and $X^-_p$, there is a natural projection $P_{\tt D}(R)\ra \GL_3(R)\times \GL_1(R)$ which in terms of matrices is given by

\begin{align*}
g=(g^+,g^-)=\left(\begin{psm}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{psm},\begin{psm}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{psm}\right)
&\mapsto \left(\begin{psm}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{psm},b_{11}\right).
\end{align*}

We will consider the $p$-level subgroups given by

\begin{align*}
\glsuseri{Kpn}&:=\left\{g\in G(\bZ_p)\colon g^+\equiv\begin{psm}\ast&\ast&\ast&\ast\\p\ast&\ast&\ast&\ast\\ &&1&\ast\\ && &1\end{psm}\mod p^n\right\},\\
\glsuserii{Kpn}&:=\left\{g\in G(\bZ_p)\colon g^+\equiv\begin{psm}\ast&\ast&\ast&\ast\\p\ast&\ast&\ast&\ast\\ &&\ast&\ast\\&& &\ast\end{psm}\mod p^n\right\},
\end{align*}

where $n$ is a positive integer.

§.§ Igusa towers

Let Ign be the ordinary locus of level $K^1_{p,n}$ attached to the filtration ${\tt D}$ in $(\ref{eq:fil-D})$ and the PEL type Shimura data introduced in $\S\ref{subsec:Sh}$, constructed in <cit.>. By <cit.>,
$\Ig_{n}$ is a smooth quasi-projective scheme over $\bZ_p$ representing a moduli problem for tuples $(A,\lambda,i,\alpha^p,\alpha_p)$, where $(A,\lambda,i,\alpha^p)$ is as in $\S\ref{subsec:Sh}$, and $\alpha_p$ is an ordinary $K^1_{p,n}$-level structure of $A$, i.e., a $K^1_{p,n}$-orbit of group scheme embeddings $\mu_{p^n}\otimes{\tt D}^0\hra A[p^n]$ with image isotropic for the $\lambda$-Weil pairing, compatible with the $\cO_\cK$-actions on ${\tt D}^0$ and   $A[p^n]$ through $i$, as described in <cit.>.

In <cit.>, 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 <cit.> and $\nu(K^1_{p,n})=\bZ^\times_p$, this construction of the ordinary locus agrees with the moduli problem. (See <cit.> for more details).

Let Ign be the partial toroidal compactification of the ordinary locus $\Ig_{n}$ (<cit.>); it is obtained by gluing to $\Ig_{n}$ the toroidal boundary charts parameterizing degenerating families defined in <cit.> (including an extensibility condition on the ordinary level structure). We note that, even though the generic fibre of $\Ig_{n}$ agrees with the Shimura variety of level $K^p_fK^1_{p,n}$ over $\bQ_p$, the generic fibre of $\Ig^\tor_{n}$ is in general just an open subscheme of a toroidal compactification of the Shimura variety of level $K^p_fK^1_{p,n}$ over $\bQ_p$.

Let Ignm (resp. Sm) be the base change of $\sT_n^\tor$ (resp. $\sS^\tor_{}$) to $\bZ/p^m\bZ$. By <cit.>, $\sS^\tor_m[1/E]$ agrees with the ordinary locus in <cit.> for full level at $p$, and  by <cit.> the map $ \Ig^\tor_{n,m}\ra \sS^\tor_m[1/E]$ that forgets the ordinary $K^1_{p,n}$-level structure is finite étale.

Concretely, we can describe $\sT^\tor_{n,m}$ as

\begin{equation}\label{eq:tower}
\underline{\Isom}_{\sS^{\tor}_m[1/E]}\left(\mu_{p^n}\otimes {\tt D}^0,\sG[p^n]^\mult\right)/K^1_{p,n}\cap P_{\tt D}(\bZ_p),
\end{equation}

where $P_{\tt D}(\bZ_p)$ acts on an element $\alpha_p\in \underline{\Isom}_{\sS^{\tor}_m[1/E]}\left(\mu_{p^n}\otimes {\tt D}^0,\sG[p^n]^\mult\right)$ by $(g\cdot\alpha_p)(v)=\alpha_p(g^{-1}v)$ for $v\in {\tt D}^0$. The fiber at a geometric point $\bar{s}\in\sS^\tor_{m}$ parameterizes tuples $({\tt F}^\pm,\delta^+_3,\delta^-_1,{\tt F}^+_1[p])$, where $\gls{F}$ are filtrations

\begin{equation}\label{eq:F}
\begin{aligned}
&{\tt F}^+:0={\tt F}^+_0\subset {\tt F}^+_2\subset{\tt F}^+_3=\sG_{\bar{s}}[p^n]^{\mult+}\\
&{\tt F}^-:0={\tt F}^-_0\subset{\tt F}^-_1= \sG_{\bar{s}}[p^n]^{\mult -}
\end{aligned}
\end{equation}

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

\begin{align*}
&\glsuseri{delta+-}: \mu_{p^n}\cong \mr{Gr}^{\tt F^+}_3,
&&\glsuserii{delta+-}:\mu_{p^n}\cong \mr{Gr}^{\tt F^-}_1;
\end{align*}

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

§.§ $p$-adic forms forms

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

\[
\glsuseri{Vnm}:=H^0\left(\Ig^\tor_{n,m},\cO_{\sT^\tor_{n,m}}\right).
\]

Letting $\cI_{\Ig^\tor_{n,m}}:=(\Ig^\tor_{n,m}\ra \sS^\tor[1/E])^*\cI_{\sS^\tor}$, we similarly define the space of mod $p^m$  cuspidal automorphic forms on $G$ of level $n$ by

\[
\glsuserii{Vnm}:=H^0\left(\sT^\tor_{n,m},\cI_{\sT^\tor_{n,m}}\right).
\]

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

\[
\glsuseri{pV}:=\varinjlim\limits_{m}\varinjlim\limits_{n} V_{n,m},\quad
\glsuserii{pV}:=\varinjlim\limits_{m}\varinjlim\limits_{n} V_{n,m}^0.
\]

Let $\gls{Tso}(\bZ_p)=\bZ_p^\times\times\bZ_p^\times$. The map sending $(a_1,a_2)$ to $g\in G(\bZ_p)$ with $g^+=\begin{psm}1\\&1\\&&a_1\\&&&a^{-1}_2\end{psm}$ and $\nu(g)=1$ identifies $T_\so(\bZ/p^n)=(\bZ/p^n\bZ)^\times\times(\bZ/p^n\bZ)^\times$ with $K^n_{p,0}/K^1_{p,n}$. Hence the group $T_{\so}(\bZ_p)$ naturally acts on $V_{n,m}, V^0_{n,m}, \pV, \pV^0$ making these spaces into $\bZ_p\llb T_{\so}(\bZ_p)\rrb$-modules.

By a $p$-adic weight (for a semi-ordinary form) we mean a $\ol{\bQ}_p$-valued character of $T_{\so}(\bZ_p)$, i.e., a pair $(\tau^+,\tau^-)$, where $\tau^\pm:\bZ^\times_p\ra \ol{\bQ}^\times_p$ 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 $t^\pm\in\bZ$. If $(\tau^+,\tau^-)$ is arithmetic, we put $\tau_{\rmf}^\pm:=\epsilon^\pm$ and $\tau_{\alg}^\pm=t^\pm$. Given a $p$-adic weight $(\tau^+,\tau^-)$, we denote by $V_{n,m}[\tau^+,\tau^-]$ the subspace of $V_{n,m}\otimes_{\bZ_p}\cO_{\bQ_p(\tau^+,\tau^-)}$ on which $T_{\so}(\bZ_p)$ acts by the inverse of the character $(\tau^+,\tau^-)$. Similarly, we define the eigenspaces $V^0_{n,m}[\tau^+,\tau^-]$, $\pV[\tau^+,\tau^-]$, $\pV^0[\tau^+,\tau^-]$.

§.§ Classical automorphic forms

The weights of classical holomorphic automorphic forms on $G$ are indexed by tuples of integers $\gls{ut}=(t^+_1,t^+_2,t^+_3;t^-_1)$ 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 $\gls{Wut}$ be the algebraic representation of $\GL_3\times\GL_1$ given by

\begin{equation}\label{eq:W}
W_{\ut}:=W_{(t^+_1,t^+_2,t^+_3)}\boxtimes W_{t^-_4}.\nonumber
\end{equation}

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

\[
\det\begin{pmatrix}x_{11}&x_{12}&x_{13}\\ x_{21}&x_{22}&x_{23}\\ x_{31}&x_{32}&x_{33}\end{pmatrix}^{t^+_3},
\]

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 $W_{t^-_4}(R)$ is the $R$-submodule of $R[x,x^{-1}]$ spanned by $x^{t^-_4}$.

The groups $\GL_3$ and $\GL_1$ act on $W_{\ut}$ by right translation. One can check that the left translation of $\begin{psm}a_1\\ \ast&a_2\\ \ast&\ast&a_3\end{psm}$ 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 $R=\bC$, it is the irreducible algebraic representation of $\GL_3(\bC)$ of highest weight $(t^+_1,t^+_2,t^+_3)$. Let $\fe_{\can}:W_{\ut}\ra\bA^1$ be the linear functional defined by the evaluation at $(\bid_3,\bid_1)$.

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

\begin{align*}
\omega^+_{(t^+_1,t^+_2,t^+_3)}&=\underline{\Isom}_{\sS^\tor}(\cO^{\oplus{3}}_{\sS^\tor},\underline{\omega}^+)\times^{\GL_3} W_{(t^+_1,t^+_2,t^+_3)},\\
\omega^+_{t^-_1}&=\underline{\Isom}_{\sS^\tor}(\cO_{\sS^\tor},\underline{\omega}^+)\times^{\GL_1} W_{t^-_1}\cong (\underline{\omega}^-)^{\otimes t^-_1},
\end{align*}

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

Letting $F/\bQ_p$ be a finite extension containing the values of $\epsilon^\pm$, and $S^\tor_{K^p_fK^1_{p,n}}$ be the toroidal compactification of the Shimura variety of level $K^p_fK^1_{p,n}$  defined over $\cK$,

we have

\[
\gls{Mt}=
\left(H^0\Big(S^\tor_{K^p_fK^1_{p,n}},\underline{\omega}_{(0,0;t^+;t^-)}\Big)\otimes_{\cK}F\right)[\epsilon^+,\epsilon^-],
\]

which is 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 $\cK$-algebra via $\cK\stackrel{\iota_p}{\ra} \bQ_p\ra F$. Similarly, we have the space of classical cuspidal automorphic forms

\[
\gls{Mt0}
\]

There are classical embeddings

\begin{equation}\label{eq:HtoV}
\begin{aligned}
H^0\left(\sS^\tor,\underline{\omega}_{(0,0,t^+;t^-)}\right)&\lhra \varprojlim_m\varinjlim_n V_{n,m}[t^+,t^-],\\
H^0\left(\sS^\tor,\underline{\omega}_{(0,0,t^+;t^-)}\otimes\cI_{\sS^\tor}\right)&\lhra \varprojlim_m\varinjlim_n V^0_{n,m}[t^+,t^-].
\end{aligned}
\end{equation}

induced by the trivialization of $\underline{\omega}$ over the Igusa tower and the canonical functional $\fe_\can$. More precisely, the trivialization of $\underline{\omega}$ arises from the
the Hodge–Tate map $(\cG_{/S}[p^n]^\mult)^D\otimes_{\bZ}\cO_S\stackrel{\sim}{\ra} e^*\Omega^1_{\cG/S}\otimes_\bZ\bZ/p^n\bZ$, where $\cG^0_{/S}$ is an ordinary semi-abelian variety over $S$, the superscript $^D$ denotes the Cartier dual, and $e:S\ra \cG$ is the zero section.

Similarly, we have embeddings

\begin{equation}\label{eq:MtoV}
\begin{aligned}
M_{(0,0,t^+;t^-)}\left(K^p_fK^1_{p,n},\epsilon^+,\epsilon^-;F\right)&\lhra \left(\varprojlim_m\varinjlim_n V_{n,m}\otimes_{\bZ_p}F\right)[(t^+,\epsilon^+),(t^-,\epsilon^-)],\\
M^0_{(0,0,t^+;t^-)}\left(K^p_fK^1_{p,n},\epsilon^+,\epsilon^-;F\right)&\lhra \left(\varprojlim_m\varinjlim_n V^0_{n,m}\otimes_{\bZ_p}F\right)[(t^+,\epsilon^+),(t^-,\epsilon^-)].
\end{aligned}
\end{equation}

§.§ The $\bU_p$-operator

Given a tuple $(A,\lambda,i,\alpha^p,\alpha_p)$ parameterized by $\Ig_{n,m}$, where $\alpha_p:\mu_{p^n}\otimes {\tt D}^0\ra A[p^n]^\mult$ is an isomorphism up to $K^1_{p,n}\cap P_{\tt D}(\bZ_p)$, the corresponding filtration ${\tt F^\pm}$ of $A[p^n]^{\mult\pm}$ is

\begin{align*}
{\tt F}^+&\colon\; {\tt F}^+_2=\{e^+_1,e^+_2\}\subset{\tt F}^+_3=\{e^+_1,e^+_2,e^+_3\}=\mr{can}^+\subset {\tt F}^+_4=\{e^+_1,e^+_2,e^+_3,e^+_4\},\\
{\tt F}^-&\colon\; {\tt F}^-_1=\{e^-_1\}=\mr{\can}^-\subset{\tt F}^-_2=\{e^-_1,e^-_3\}\subset{\tt F}^-_4=\{e^-_1,e^-_2,e^-_3,e^-_4\},
\end{align*}

where $(e^+_1,e^+_2,e^+_3;e^-_1)=\alpha_p(\xx^+_1,\ww^+_1,\ww^+_2;\xx^-_1)$.

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

§.§.§ $\bU_p$-operators on $V_{n,m}$

For $j=2,3$, let $\glsuseri{Cj}$ denote the solution to the moduli problem classifying tuples $(A,\lambda,i,\alpha^p,\alpha_p,\Lsub)$ with $\Lsub\subset A[p^2]$ a Lagrange subgroup (i.e., maximal isotropic for the $\lambda$-Weil pairing) stable under the $\cO_\cK$-action through $i$ such that $A[p]=\Lsub[p]\oplus {\tt F}^+_j[p]$. With $e^\pm_1,e^\pm_2,e^\pm_3,e^\pm_4$ a basis of $A[p^n]$ as above, such a $\Lsub$ is spanned by $(e^\pm_1,e^\pm_2,e^\pm_3,e^\pm_4)\cdot p^n\gamma_{\Lsub,p}^\pm$, where for $j=2$

\begin{equation}\label{eq:gamma+2}
\begin{aligned}
\gamma^+_{\Lsub,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^-_{\Lsub,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{equation}

with $u_1,u_2,\ast\in\bZ_p$, and for $j=3$,

\begin{equation}\label{eq:gamma+3}
\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},
\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{equation}

with $\ast\in\bZ_p$.

Similarly, let $\glsuserii{Cj}$ be the moduli space classifying tuples $(A,\lambda,i,\alpha^p,\alpha_p,\Lsub)$ with $\Lsub\subset A[p^2]$ a Lagrange subgroup stable under the $\cO_\cK$-action through $i$ such that $A[p]=\Lsub[p]\oplus{\tt F}^-_1[p]$. Then such a $\Lsub$ is spanned by $(e^\pm_1,e^\pm_2,e^\pm_3,e^\pm_4)\cdot p^n\gamma^\pm_{C,p}$ with

\begin{equation}\label{eq:gamma-}
\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},
\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{equation}

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

\begin{equation}\label{eq:Up-graph}
\xymatrix{
\Ig_{n,m}&&\Ig_{n,m}
\end{equation}

with $p_1,p_2$ the projections given by

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

where $A'=A/\Lsub$, $\lambda'$ is such that $\pi^\vee\circ\lambda\circ\pi=p^2\lambda$ with $\pi:A\ra A'$ the natural projection, $i'$ is the $\cO_{\cK}$-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

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

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

\begin{equation}\label{eq:open}
\glsuseri{Up}\colon\; H^0(\Ig_{n,m},\cO_{\Ig_{n,m}})
\stackrel{p^*_2}{\lra} H^0(\sC^+_{j,n,m},\cO_{\Ig_{n,m}})
\xrightarrow{p^{-j}\Tr\,p_1} H^0(\Ig_{n,m},\cO_{\Ig_{n,m}}),
\quad(j=2,3)
\end{equation}

\begin{equation}\label{eq:open2}
\glsuserii{Up}\colon\; H^0(\Ig_{n,m},\cO_{\Ig_{n,m}})
\stackrel{p^*_2}{\lra} H^0(\sC^-_{1,n,m},\cO_{\Ig_{n,m}})
\xrightarrow{p^{-3}\Tr\,p_1} H^0(\Ig_{n,m},\cO_{\Ig_{n,m}}).
\end{equation}

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 <cit.>. By computing the effect of (<ref>) and (<ref>) on Fourier–Jacobi expansions, one can check that the operators preserve $V_{n,m}$ and $V^0_{n,m}$.

§.§.§ $\bU_p$-operators on $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\right)$

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

\[
\underline{\Isom}_{\sS^{\tor}_m[1/E]}\left(\mu_{p^n}\otimes {\tt D}^0,\sG[p^n]^\mult\right)/K^0_{p,n}.
\]

Compared to $\Ig^\tor_{n,m}$, the level structure at $p$ for $\Ig^{0,\tor}_{n,m}$  forgets $\delta^+_3,\delta^-_1$ and parameterizes $({\tt F}^\pm,{\tt F}^+_1[p])$. We can define $\bU_p$-operators on $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\right)$. In order to see that they increase the level, we need to introduce Igusa towers of more general level structure at $p$.

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

\[
K^0_{p,n_1,n_2,n_3}=\left\{g\in G(\bZ_p): g^+=\begin{psm}\ast&\ast&\ast&\ast\\p\ast&\ast&\ast&\ast\\ p^{n_1}\ast&p^{n_1}\ast&\ast&\ast\\p^{n_3}\ast&p^{n_3}\ast&p^{n_2}\ast&\ast\end{psm}\right\}.
\]

Define $\Ig^0_{n_1,n_2,n_3,m}$ as the quotient of $\Ig_{n,m}$, $n\geq n_1,n_2,n_3$, by $K^0_{p,n_1,n_2,n_3}\cap P_{\tt D}(\bZ_p)$, i.e. the corresponding level structure at $p$ parameterizes isomorphism $\alpha_p:\mu_{p^n}\otimes {\tt D}^0\ra A[p^{\infty}]^\mult$ up to $K^0_{p,n_1,n_2,n_3}\cap P_{\tt D}(\bZ_p)$.

Like above for $(\bullet,j)=(+,2),(+,3),(-,1)$, let $\sC^{0\bullet}_{j,n_1,n_2,n_3,m}$ be the moduli space parameterizing tuples $(A,\lambda,i,\alpha^p,\alpha_p,\Lsub)$ with $\Lsub\subset A[p^2]$ a Lagrange subgroup stable under the $\cO_\cK$-action through $i$ such that $A[p]=C[p]\oplus {\tt F}^\bullet_j$.

Consider the diagrams

\[\xymatrix{
\Ig^0_{n_1,n_2,n_3,m}&&\Ig^0_{n_1+1,n_2,n_3+1,m}
\]

\[\xymatrix{
\Ig^0_{n_1,n_2,n_3,m}&&\Ig^0_{n_1,n_2+1,n_3+1,m}
\]

\[\xymatrix{
\Ig^0_{n_1,n_2,n_3,m}&&\Ig^0_{n_1,n_2+1,n_3+1,m}
\]

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

In order to define the $\bU_p$-operators on the global sections of vector bundles, we also need maps

\begin{align*}
\pi^*&:H^0\left(\sC^{0-}_{1,n_1+1,n_2,n_3+1,m},p^*_2\omega_{\ut}\right)\lra  H^0\left(\sC^{0-}_{1,n_1,n_2,n_3,m},p^*_1\omega_{\ut}\right)\\
\pi^*&:H^0\left(\sC^{0-}_{1,n_1,n_2+1,n_3+1,m},p^*_2\omega_{\ut}\right)\lra  H^0\left(\sC^{0-}_{1,n_1,n_2,n_3,m},p^*_1\omega_{\ut}\right).
\end{align*}

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

\[
h\begin{psm}1\\&1\\&&p^{-1}\\&&&p^{-1}\end{psm}h', &\bullet=+,\, j=2,\\
h\begin{psm}1\\&1\\&&1\\&&&p^{-1}\end{psm}h', &\bullet=+,\, j=3,\\
h\begin{psm}p^{-1}\\&p^{-1}\\&&p^{-1}\\&&&1\end{psm}h', &\bullet=-,\, j=1,
\end{cases}
\]

for some $h,h'=\begin{psm}\ast&p^n\ast&p^n\ast\\\ast&\ast&p\ast\\ \ast&\ast&\ast\\ &&&\ast\end{psm}\in\GL_4(\bZ_p)$. In all the cases, $g^{-1}$ belongs to the semi-group

\[
\Delta_+=\left\{\left(h\begin{psm}a_1\\&a_2\\&&a_3\end{psm}h',a_4\right)\,:\,h,h'=\begin{psm}\ast&p\ast&p\ast\\\ast&\ast&p\ast\\\ast&\ast&\ast\end{psm}\in\GL_3(\bZ_p),\,a^{-1}_1a_2,a^{-1}_2a_3\in\bZ_p \right\}.
\]

We make $\Delta_+$ act on $W_{\ut}$ by

\[
\left(h\begin{psm}a_1\\&a_2\\&&a_3\end{psm}h',a_4\right)\cdot q(\underline{x},y)=q\left(\begin{psm}p^{-v_p(a_1)}\\&p^{-v_p(a_2)}\\&&p^{-v_p(a_3)}\end{psm}\underline{x}h\begin{psm}a_1\\&a_2\\&&a_3\end{psm}h',a_4p^{-v_p(a_4)y} \right),
\]

and define $\pi^*$ as

\[
\pi^*\vec{f}(A,\lambda,i,\alpha^p,\alpha_p,\Lsub,\underline{\varepsilon}_A)=\vec{f}(A,\lambda,i,\alpha^p,\alpha_p,\Lsub,\underline{\varepsilon}_A,(\pi^*\underline{\varepsilon}_{A/\Lsub})g)=g^{-1}\cdot \vec{f}(A,\lambda,i,\alpha^p,\alpha_p,\Lsub,\underline{\varepsilon}_{A/\Lsub}).
\]

The $\bU_p$-operators are defined as

\begin{equation*}
\begin{aligned}
\glsuseriii{Up}&:H^0\left(\Ig^0_{n_1+1,n_2,n_3+1,m},\omega_{\ut}\right)
\stackrel{p^*_2}{\lra} H^0\left(\sC^{0+}_{2,n_1,n_2,n_3,m},p^*_2\omega_{\ut}\right)
\stackrel{\pi^*}{\lra}  H^0\left(\sC^{0+}_{2,n_1,n_2,n_3,m},p^*_1\omega_{\ut}\right)
\xrightarrow{p^{-2}\Tr\,p_1} H^0(\Ig^0_{n_1,n_2,n_3,m},\omega_{\ut}),\\
\glsuseriv{Up}&:H^0\left(\Ig^0_{n_1,n_2+1,n_3+1,m},\omega_{\ut}\right)
\stackrel{p^*_2}{\lra} H^0\left(\sC^{0+}_{3,n_1,n_2,n_3,m},p^*_2\omega_{\ut}\right)
\stackrel{\pi^*}{\lra}  H^0\left(\sC^{0+}_{3,n_1,n_2,n_3,m},p^*_1\omega_{\ut}\right)
\xrightarrow{p^{-3}\Tr\,p_1} H^0(\Ig^0_{n_1,n_2,n_3,m},\omega_{\ut}),\\
\glsuserii{Up}&:H^0\left(\Ig^0_{n_1,n_2+1,n_3+1,m},\omega_{\ut}\right)
\stackrel{p^*_2}{\lra} H^0\left(\sC^{0-}_{1,n_1,n_2,n_3,m},p^*_2\omega_{\ut}\right)
\stackrel{\pi^*}{\lra}  H^0\left(\sC^{0-}_{1,n_1,n_2,n_3,m},p^*_1\omega_{\ut}\right)
\xrightarrow{p^{-2}\Tr\,p_1} H^0(\Ig^0_{n_1,n_2,n_3,m},\omega_{\ut}).
\end{aligned}
\end{equation*}

Similarly as $V_{n,m}$, one can check that these $\bU_p$-operators preserve the spaces $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\right)$ and $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)$. It is also easy to see that $U^+_{p,2}U^+_{p,3}U^-_{p,1}$ maps $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\right)$ (resp. $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)$) into $H^0\left(\Ig^{0,\tor}_{n-1,m},\omega_{\ut}\right)$ (resp. $H^0\left(\Ig^{0,\tor}_{n-1,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n-1,m}}\right)$), i.e. the operator $U^+_{p,2}U^+_{p,3}U^-_{p,1}$ increases the level.

§.§.§ Adelic

If we identify $G(\bQ_p)$ with $\GL_4(\bQ_p)\times\GL_1(\bQ_p)$ via (<ref>), the $\bU_p$-operators defined above acting on classical automorphic forms on $G$ of weight $(t^+_1,t^+_2.t^+_3;t^-_1)$ corresponds to the following adelic operators (up to the action of the center of $G(\bQ_p)$):

\begin{equation}\label{eq:adeleUp}
\begin{aligned}
\glsuseriii{Up}:&\quad p^{\left<(t^+_1,t^+_2.t^+_3;-t^-_1)+2\rho_{\mr{c}},(1,1,0;0)\right>}\int_{N(\bZ_p)}R\left(u\begin{psm}p\\&p\\&&1\\&&&1\end{psm},1\right)\,du;
&& \\
%p^{t^+_1+\frac{1}{2}}\alpha_1\cdot p^{t^+_2-\frac{1}{2}}\alpha_2\\
\glsuseriv{Up}:&\quad p^{\left<(t^+_1,t^+_2.t^+_3;-t^-_1)+2\rho_{\mr{c}},(1,1,1;0)\right>}\int_{N(\bZ_p)}R\left(u\begin{psm}p\\&p\\&&p\\&&&1\end{psm},1\right)\,du;
&& \\
%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\\
\glsuserii{Up}:&\quad p^{\left<(t^+_1,t^+_2.t^+_3;-t^-_1)+2\rho_{\mr{c}},(0,0,0;-1)\right>}\int_{N(\bZ_p)}R\left(u\begin{psm}1\\&1\\&&1\\&&&p^{-1}\end{psm},1\right)\,du;
&& %p^{t^-_1-\frac{3}{2}}\cdot\alpha^{-1}_4
\end{aligned}
\end{equation}

where $N(\bZ_p)=\left\{\begin{psm}1&\ast&\ast&\ast\\&1&\ast&\ast\\&&1&\ast\\&&&1\end{psm}\in\GL_4(\bZ_p)\right\}$, $R(-)$ denotes the right translation, and $2\rho_{\mr{c}}=(2,0,-2;0)$ is the sum of the compact positive roots of $G$.

§.§ Statement of main theorem

Let $T_{\so}(\bZ_p)^\circ$ be the connected component of $T_\so(\bZ_p)$ containing $\bid_2$, i.e., $T_{\so}(1+p\bZ_p)$, and put $\gls{Lambdaso}=\bZ_p\llb T_\so(\bZ_p)^\circ\rrb$.

The following hold:

* Let $U_p=U^+_{p,2}U^+_{p,3}U^-_{p,1}$. Then for each $f\in\pV$, $\lim\limits_{r\to\infty}(U_p)^{r!}f$ converges, and we can define the semi-ordinary projector as $\gls{eso}=\lim\limits_{r\to\infty}(U_p)^{r!}$. The $\bZ_p\llb T_{\so}(\bZ_p)\rrb$-modules

\[
\pV^{0,\ast}_\so=\Hom_{\bZ_p}\left(e_\so\pV^0,\bQ_p/\bZ_p\right),\quad
\pV^{\ast}_\so=\Hom_{\bZ_p}\left(e_\so\pV,\bQ_p/\bZ_p\right)
\]

are both free of finite rank over $\Lambda_\so$.

* The spaces of $\Lambda_{\so}$-families of tame level $K^p_f$ are defined as

\[
\glsuseri{cM}=\Hom_{\Lambda_\so}\left(\pV^{0,\ast}_\so,\Lambda_\so\right),
\quad\cM_\so=\Hom_{\Lambda_\so}\left(\pV^{\ast}_\so,\Lambda_\so\right).
\]

For a given weight $(\tau^+,\tau^-)\in \Hom_\cont\left(T_{\so}(\bZ_p),F^\times\right)$, let $\cP_{\tau^+,\tau^-}$ be the ideal in $\cO_F\llb T_{\so}(\bZ_p)\rrb$ generated by $\{(x,y)-\tau^+(x)\tau^-(y)\colon (x,y)\in T_\so(\bZ_p)\}$. Then for $0\geq t^+\geq -t^-+4$, we have

\begin{equation}\label{eq:specialization}
\begin{aligned}
\cM^0_\so\otimes_{\bZ_p\llb T_\so(\bZ_p)\rrb}\cO_F\llb T_\so(\bZ_p)\rrb/\cP_{\tau^+,\tau^-}&\stackrel{\sim}{\lra}  \left(\varprojlim_{m}\varinjlim_n e_\so V^0_{n,m}\otimes_{\bZ_p}\cO_F\right)[\tau^+,\tau^-],\\
\cM_\so\otimes_{\bZ_p\llb T_\so(\bZ_p)\rrb}\cO_F\llb T_\so(\bZ_p)\rrb/\cP_{\tau^+,\tau^-}&\stackrel{\sim}{\lra}  \left(\varprojlim_{m}\varinjlim_n e_\so V_{n,m}\otimes_{\bZ_p}\cO_F\right)[\tau^+,\tau^-].
\end{aligned}
\end{equation}

The semi-ordinary projector preserves the spaces of classical forms, and by combining (<ref>) with (<ref>), we have the embeddings

\begin{align*}
e_\so M^0_{(0,0,t^+;t^-)}\left(K_f^pK^1_{p,n},\epsilon^+,\epsilon^-;F\right)&\lhra \left(\cM^0_\so\otimes_{\bZ_p\llb T_\so(\bZ_p)\rrb}\cO_F\llb T_\so(\bZ_p)\rrb/\cP_{\tau^+,\tau^-}\right)[1/p],\\
e_\so M_{(0,0,t^+;t^-)}\left(K_f^pK^1_{p,n},\epsilon^+,\epsilon^-;F\right)&\lhra \left(\cM_\so\otimes_{\bZ_p\llb T_\so(\bZ_p)\rrb}\cO_F\llb T_\so(\bZ_p)\rrb/\cP_{\tau^+,\tau^-}\right)[1/p],
\end{align*}

where $t^\pm=\tau_{\alg}^\pm$ and $\epsilon^\pm=\tau_{\rm f}^\pm$.

* Given $0\geq t^+$, there exists $A\geq -t^++4$

such that the above embedding for the cuspidal forms is surjective if $t^-\geq A$.

* There is the following so-called fundamental exact sequence (in the study of Klingen Eisenstein congruences),

\[
0\ra\cM^0_\so\ra \cM_\so\ra \bigoplus_{\clabel\in
(G'_1(\bQ)\ltimes P'_1(\bA^p_{\bQ,f}))\backslash G(\bA^p_{\bQ,f})/K^p_f} \gls{M00}\otimes_{\bZ_p}\bZ_p\llb T_{\so}(\bZ_p)\rrb \ra 0,
\]

where $K^{\prime}_{f,\clabel}\subset G'(\bA^p_{\bQ,f})$ is defined in (<ref>), and $M_{(0,0)}(K^{\prime}_{f,\clabel};\bZ_p)$ denotes the space of classical automorphic forms on $G'$ of weight $(0,0)$ and level $K^{\prime}_{f,\clabel}$.

In general, one can consider semi-ordinary families whose members have weight $\ut=(t^+_1,t^+_2,t^+_3;t^-_1)$ 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 <ref>, 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 $\omega_{\ut}$. For defining $\pV^\flat$ (cf. <ref>) such that the quotient $\pV/\pV^\flat$ has a nice structure, besides the requirement of vanishing outside the strata labeled by cusp labels in $C(K^p_fK^1_{p,n})^\flat_\ord$, one also requires that the elements are global sections of $\omega^\flat_{\ut}\subset \omega_{\ut}$ with $\omega^\flat_{\ut}$ a subsheaf defined as in <cit.>.

§ THE PROOF OF THEOREM <REF> FOR THE  CUSPIDAL PART

In this section, we prove Theorem <ref> for cuspidal families. The results for cuspidal families will be used to deduce the results for non-cuspidal families in <ref>

Let $\Ig^{0,\tor}_n[1/E]$ the open subscheme of $\Ig^{0,\tor}_n$ where $E$, our fixed lift of Hasse invariant viewed as a section over $\Ig^{0,\tor}_n$, is nonvanishing. For any classical dominant weight $\ut$, the natural map

\[
H^0\left(\Ig^{0,\tor}_n[1/E],\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_n}\right)\otimes\bZ/p^m\bZ\hra H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)
\]

is an isomorphism.

Over $\Ig^\tor_n$, we have the exact sequence of sheaves

\[
0\ra \omega_{\ut}\otimes\cI_{\Ig^\tor_n}\stackrel{p^m}{\ra} \omega_{\ut}\otimes \cI_{\Ig^\tor_{n}}\ra \omega_{\ut}\otimes\cI_{\Ig^\tor_{n,m}}\ra 0.
\]

By <cit.>, $R^1_{\pi_n,*}(\omega_{\ut}\otimes\cI_{\Ig^\tor_n})=0$ (where $\pi_n$ is the projection $\pi_n:\Ig^\tor_{n}\ra \Ig^{\min}_{n}$),

so we have the exact sequence of sheaves

\[
0\ra\pi_{n,*}(\omega_{\ut}\otimes\cI_{\Ig^\tor_n})\stackrel{p^{m}}{\ra}\pi_{n,*}(\omega_{\ut}\otimes\cI_{\Ig^\tor_n})\ra \pi_{n,*}(\omega_{\ut}\otimes\cI_{\Ig^\tor_{n,m}})\ra 0
\]

over $\Ig^{\min}_{n}$. Since $\Ig^{\min}_{n}[1/E]$ is affine by definition, taking global sections gives the result.

Let $U_p=U^+_{p,2}U^+_{p,3}U^-_{p,1}$. Let $f\in \pV^0$ and $\vec{f}\in H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)$. Then both limits  $\lim\limits_{r\ra\infty}(U_p)^{r!}f$ and $\lim
\limits_{r\ra\infty}(U_p)^{r!}\vec{f}$ converge.

Consider $\tilde{\Ig}_n$, the ordinary locus of level

\[
\tilde{K}^n_{p,1}=\left\{g\subset G(\bZ_p)\mid g^+\equiv \begin{psm}1&\ast&\ast&\ast\\ &1&\ast&\ast\\&&1&\ast\\&&&1\end{psm}\mod p^n\right\}.
\]

Let $\tilde{\Ig}^\tor_n$ be the partial toroidal compactification of $\tilde{\Ig}_n$, and put $\tilde{V}_{n,m}^0=H^0(\tilde{\Ig^\tor_{n,m}},\omega_{\ut}\otimes\tilde{\cI}_{\tilde{\Ig}^\tor_{n,m}})$. The  definition of the $\bU_p$-operators in <ref> can be naturally extended to $\tilde{V}^0_{n,m}$, so it suffices to show that   $\lim\limits_{r\ra\infty}(U_p)^{r!}f$ converges for every $f\in \tilde{V}^0_{n,m}$. As explained in [13], $\varprojlim\limits_m\varinjlim\limits_n \tilde{V}^0_{n,m}$ is $p$-torsion free and there is an embedding

\[
\bigoplus_{\ut \text{ dominant}} H^0\left(\sS^\tor,\underline{\omega}_{\ut}\otimes\cI_{\sS^\tor}\right)\otimes\bQ \hra \left(\varprojlim_m\varinjlim_n \tilde{V}^0_{n,m}\right)[1/p].
\]

The $\bZ_p$-modules $H^0\left(\sS^\tor,\underline{\omega}_{\ut}\otimes\cI_{\sS^\tor}\right)$ are free of finite rank and stable under the action of $U_p$, so the limit $\lim\limits_{r\ra\infty} (U_p)^{r!}$ exists on them. Since

\begin{equation}\label{eq:dense-clas}
\left(\bigoplus_{\ut \text{ dominant}} H^0\left(\sS^\tor,\underline{\omega}_{\ut}\otimes\cI_{\sS^\tor}\right)\otimes\bQ\right) \cap \varprojlim_m\varinjlim_n \tilde{V}^0_{n,m}
\end{equation}

is dense in $\varprojlim\limits_m\varinjlim\limits_n \tilde{V}^0_{n,m}$ by [13], the convergence of  $\lim\limits_{r\ra\infty}(U_p)^{r!}f$ for every $f\in \tilde{V}^0_{n,m}$ follows.

On the other hand, let $\vec{f}\in H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)$. By Proposition <ref>, $\vec{f}$ lifts to

\[
\vec{\ff}\in  H^0\left(\Ig^{0,\tor}_{n}[1/E],\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_n}\right).
\]

For $l\gg 0$, we have $\vec{\ff} E^l\in H^0\left(\Ig^{0,\tor}_{n},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_n}\right)$, which by the Koecher principle can be viewed as an element in the finite dimensional space $M^0_{\ut+lt_E(p-1)}(K^p_fK^0_{p,n};\bQ_p)$. Thus the limit $\lim\limits_{r\ra\infty}(U_p)^{r!}(\vec{\ff} E^l)$ converges.  Since $E\equiv 1\mod p$, we have $\vec{\ff} E^l\equiv \vec{f}\mod p^m$, and the convergence of $\lim\limits_{r\ra\infty}(U_p)^{r!}\vec{f}$ follows.

Thanks to the convergence of $\lim\limits_{r\ra\infty}(U_p)^{r!}$ on $\pV^0$ and $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)$, we can define the semi-ordinary projector on them as $e_\so=\lim\limits_{r\ra\infty}(U_p)^{r!}$.

For $n\geq m$ and a dominant weight $(0,0,t^+;t^-)$, the map

\[
e_\so H^0\left(\Ig^{0,\tor}_{n,m},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^\tor_{n,m}}\right)\ra e_\so V^0_{n,m}[t^+,t^-]
\]

induced by (<ref>) is an isomorphism.

Given $\vec{f}$, we have

\begin{equation}\label{eq:Up-vecf}
(U^n_p\vec{f})(A,\lambda,i,\alpha^p,\alpha_p,\underline{\varepsilon}_{\alpha_p})=p^{-(2+3+3)n}\sum_C \left(\ltrans{u}^{-1}_C\begin{psm}p^n\\ &p^n\\&&p^{2n}\end{psm},p^{2n}\right)\cdot \vec{f}(A/C,\lambda',i',(\alpha^p)^\prime,\alpha'_p,\underline{\varepsilon}_{\alpha'_p}).
\end{equation}

Here $\Lsub$ 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\begin{psm}1&&u_1&\ast\\&1&u_2&\ast\\&&1&\ast\\&&&1\end{psm}\begin{psm}p^{-n}&&&\\& p^{-n}&&\\&& p^{-2n}&\\&&& p^{-4n}\end{psm},\\
&(e^-_1,e^-_2,e^-_3,e^-_4)\cdot p^n\begin{psm}1&\ast&\ast&\ast\\&1\\&-u_2&1&-u_1\\&&&1\end{psm}\begin{psm}p^{-2n}&&\\& p^{-5n}\\&& p^{-4n}\\&&& p^{-5n}\end{psm},
\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 $\Lsub$, $u_\Lsub=\begin{psm}1&&u_1\\&1&u_2\\&&1\end{psm}$. By our definition of the $\Delta_+$-action on $W_{\ut}$, we know that if $w\in W_{(0,0,t^+;t^-)}$ is a vector of non-highest weight, then

\begin{align*}
\left(\begin{psm}p^n\\ &p^n\\&&p^{2n}\end{psm},p^{2n}\right)\cdot w\equiv 0\mod p^n.
\end{align*}

Therefore, from (<ref>) 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 $f\in e_\so V^0_{n,m}[t^+,t^-]$, we can  define $\vec{f}$ by the rule

\begin{equation}\label{eq:fvecf}
\vec{f}(A,\lambda,i,\alpha^p,\alpha_p,\underline{\varepsilon}_{\alpha_p})=p^{-(2+3+3)n}\sum_C \left(\ltrans{u}^{-1}_C,1\right)\cdot w_{(0,0,t^+;t^-)}f(A/C,\lambda',i',(\alpha^p)^\prime,\alpha'_p),
\end{equation}

where $C$ runs over the same range as above. One can check that $\vec{f}$ satisfies

\[
\vec{f}(A,\lambda,i,\alpha^p,\alpha_p\circ g,\underline{\varepsilon}_{\alpha_p\circ g})=\left(\begin{psm}\ast&p\ast&\\ \ast&\ast\\ u_1&u_2&a_1\end{psm},a_2\right)^{-1}\cdot   \vec{f}(A,\lambda,i,\alpha^p,\alpha_p,\underline{\varepsilon}_{\alpha_p})
\]

for $g^+\equiv\begin{psm}\ast&\ast&u_1&\ast\\ p\ast&\ast&u_2&\ast\\ &&a_1&\ast\\&&&a^{-1}_2\end{psm}\mod p^n$, so (<ref>)  defines an element in  $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^\tor_{n,m}}\right)$, and it is easy to check its semi-ordinarity from the semi-ordinarity of $f$. The composition of (<ref>) and the map in the statement is $U^n_p$. Hence, the map is a bijection.

$e_\so H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^\tor_{n,m}}\right)=e_\so H^0\left(\Ig^{0,\tor}_{1,m},\omega_{\ut}\otimes\cI_{\Ig^\tor_{1,m}}\right)$.

This follows immediately from the fact that $U_p$ maps the space $H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^\tor_{n,m}}\right)$ into $H^0\left(\Ig^{0,\tor}_{n-1,m},\omega_{\ut}\otimes\cI_{\Ig^\tor_{n,m}}\right)$. (See <ref>.)

For a fixed tame level $K^p_f$ and a fixed integer $B\geq 0$, the dimension of $e_\so M^0_{\ut}(K^p_fK^0_{p,1};\bQ_p)$ is uniformly bounded for all $t^+_1\geq t^+_2\geq t^+_3\geq -t^-_1+4$ with $t^+_1-t^+_2\leq B$.

Suppose that $\Pi$ is an irreducible cuspidal automorphic representation of $G(\bA_\bQ)$ generated by a semi-ordinary form   $\varphi\in e_{\so}M^0_{\ut}\left(K^p_fK^0_{p,n};\bQ_p\right)$ whose (generalized) eigenvalue of $U^+_{p,2}$ (resp. $U^+_{p,3}$, $U^-_{p,1}$) is $\lambda^+_2$ (resp. $\lambda^+_3$, $\lambda^-_1$). The semi-ordinarity condition implies that $\lambda^+_2,\lambda^+_3,\lambda^-_1$ are all $p$-adic units. We view $\Pi_p$ as an irreducible representation of $\GL_4(\bQ_p)$ via (<ref>). Let $Q$ be the parabolic subgroup $\left\lbrace\begin{psm}\ast&\ast&\ast&\ast\\\ast&\ast&\ast&\ast\\&&\ast&\ast\\&&&\ast\end{psm}\right\rbrace\subset\GL_4$ with Levi factorization $Q=M_QN_Q$. Let

\[
\Pi_{p,N_Q}=\Pi_p/\{\Pi_p(u)v-v\,:\,v\in\Pi_p,\,u\in N_Q(\bQ_p)\},
\]

the Jacquet module of $\Pi_p$ with respect to the parabolic $Q$ acted on by $M_Q(\bQ_p)\cong \GL_2(\bQ_p)\times \GL_1(\bQ_p)\times\GL_1(\bQ_p)$. It follows from Jacquet's Lemma (see <cit.>) that $\Pi^{M_Q(\bQ_p)\cap K^0_{p,n}}_{p,N_Q}$ equals the image under the natural projection of

\begin{align*}
&\bigcap_{r\geq 1}\left(U_{p,\mr{loc}}\right)^r\Pi^{K^0_{p,n}}_p,
\end{align*}

The semi-ordinary form $\varphi\in \Pi$ is fixed by $K^0_{p,n}$ and belongs to a generalized eigenspace of $U_{p,\mr{loc}}$ with a nonzero eigenvalue. Thus, we deduce that $\Pi^{M_Q(\bQ_p)\cap K^0_{p,n}}_{p,N_Q}\neq 0$ and $\Pi_p$ is isomorphic to a subquotient of

\[
\Ind^{\GL_4(\bQ_p)}_{Q(\bQ_p)} \sigma\boxtimes \chi\boxtimes\chi',
\]

where $\sigma$ is an irreducible smooth admissible representation of $\GL_2(\bQ_p)$, and $\chi,\chi'$ are smooth characters of $\bQ^\times_p$. We have $M_Q(\bQ_p)\cap K^0_{p,1}\cong K'_{p}\times (1+p^n\bZ_p)\times (1+p^n\bZ_p)$ with $K'_{p}=\left\lbrace\begin{pmatrix}\ast&\ast\\ p\ast&\ast\end{pmatrix}\right\rbrace\subset\GL_2(\bZ_p)$. If $\sigma$ is supercuspidal, $\Pi^{M_Q(\bQ_p)\cap K^0_{p,n}}_{p,N_Q}\neq 0$ implies that $\sigma^{K'_{p}}\neq 0$ (<cit.>). However, a supercuspidal representation of $\GL_2(\bQ_p)$ does not contain any nonzero vector fixed by $K'_{p}$. Therefore, we see 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 $\bQ^\times_p$. Then, up to reordering, we have

\begin{align}
\label{eq:lambda2}\lambda^+_{2}&=p^{t^+_1+\frac{1}{2}}\alpha_1\cdot p^{t^+_2-\frac{1}{2}}\alpha_2,\\
\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,\\
\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{equation}\label{eq:Up+1}
\end{equation}

on $M^0_{\ut}\left(K^p_fK^0_{p,n};\bQ_p\right)$. The operator $U^+_{p,1}$ has a geometric interpretation analogous to (<ref>), 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{equation}\label{eq:v1v2}
\begin{aligned}
v_\fp(\alpha_1)+t^+_1+\frac{1}{2}&\geq 0,
&v_\fp(\alpha_2)+t^+_1+\frac{1}{2}&\geq 0.
\end{aligned}
\end{equation}

By using that (<ref>) is a $p$-adic unit and our condition $0\leq t^+_1-t^+_2\leq B$,

\begin{equation}\label{eq:v1+v2}
\begin{aligned}
\leq B+1.
\end{aligned}
\end{equation}

Combining (<ref>) and (<ref>), we get

\begin{equation}\label{eq:v1}
0\leq v_\fp(\alpha_j+t^+_1+\frac{1}{2})\leq B+1, \quad j=1,2.
\end{equation}

If $\alpha^{-1}_1\alpha_2=p^{\pm 1}$, then (<ref>) being a $p$-adic unit implies that

\[
\frac{t^+_1-t^+_2}{2}\leq v_\fp(\alpha_j)\leq \frac{t^+_1-t^+_2}{2}+1,\quad j=1,2.
\]

Combining this with our condition $0\leq t^+_1-t^+_2\leq B$, we get

\begin{equation}\label{eq:v1'}
0\leq v_\fp(\alpha_j)\leq \frac{B}{2}+1,\quad j=1,2.
\end{equation}

Therefore, all the semi-ordinary forms in $e_\so M^0_{\ut}(K^p_fK^0_{p,1};\bQ_p)$, $t^+_1\geq t^+_2\geq t^+_3\geq -t^-_1+4$, $t^+_1-t^+_2\leq B$, have slope $\leq B$ for the $\bU_p$-operator $U^+_{p,1}U^+_{p,2}U^+_{p,3}U^-_{p,1}$.

Recall that the theory of Coleman families for unitray groups (developed in [2] as a generalization of [1])
shows that for every point in the weight space $\Hom_\cont((\bZ^\times_p)^4,\bC^\times_p)$, there exists a neighborhood $\cU$ of that point and a projective $\cA_\cU$-module of finite rank interpolating all the cuspidal overconvergent forms of weights in $\cU$ 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 $\ut$ are contained in a compact subset of $\Hom_\cont((\bZ^\times_p)^4,\bC_p)$, when $\ut$ varies among all the algebraic weights, there is a uniform bound on the dimension of the space of cuspidal overconvergent forms of weight $\ut$ and slopes $\leq B$ and tame level $K^p_f$.  The Proposition follows.

* $\dim_{\bF_p}\,e_\so H^0\left(\Ig^{0,\tor}_{1,1},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{1,1}}\right)<\infty$.

* (Classicity) There is a canonical embedding

\[
e_\so M^0_{(0,0,t^+;t^-)}\left(K_f^pK^1_{p,0};\bQ_p\right) \lhra \left(\varprojlim_m\varinjlim_n e_{\so}V^0_{n,m}[t^+,t^-]\right)\otimes\bQ_p,
\]

and given $0\geq t^+$, there exists $A(t^+)\geq -t^++4$ such that if $t^-\geq A(t^+)$ then the embedding is an isomorphism.

(1) Suppose that $\ol{f}_1,\dots,\ol{f}_d\in e_\so H^0\left(\Ig^{0,\tor}_{1,1},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{1,1}}\right)$ are linearly independent. By Proposition <ref>, they lift to $f_1,\dots,f_d\in H^0\left(\Ig^{0,\tor}_n[1/E],\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_n}\right)$. Recall that $E$ has scalar weight $t_E(p-1)$
and let $\ut+lk_E(p-1)=(t^+_1+lt_E(p-1),t^+_2+lt_E(p-1),t^+_3+lt_E(p-1);t^-_1+lt_E(p-1))$. For $l\gg 0$, we have

\[
f_1E^l,\dots,f_dE^l\in H^0\left(\Ig^{0,\tor}_n,\omega_{\ut+lt_E(p-1)}\otimes\cI_{\Ig^{0,\tor}_n}\right).
\]

Because $E\equiv 1\mod p$, we have $\ol{e_\so(f_jE^l)}=\ol{f}_j$, and therefore $e_\so(f_1E^l),\dots,e_\so(f_dE^l)$

are linearly independent. Thus $d$ can be at most the bound in Proposition <ref>.

(2) We have $M^0_{(0,0,t^+;t^-)}\left(K_f^pK^0_{p,1};\bQ_p\right)=H^0\left(\Ig^{0,\tor}_1,\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_1}\right)\otimes_{\iota_p}\bQ_p$. (As mentioned above, the base change of $\Ig^{0,\tor}_1$ to $\cK$ is an open subscheme of $S^\tor_{K^p_fK^0_{p,1}}$; here we use the Koecher  principle to get the equality of global sections.) The map (<ref>) induces

\[
H^0\left(\Ig^{0,\tor}_1,\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_1}\right)\ra \left(\varprojlim_m\varinjlim_n e_{\so}V^0_{n,m}[t^+,t^-]\right),
\]

and this gives the canonical embedding in the statement.

Put $V^0_\so[t^+,t^-]=\varprojlim\limits_m\varinjlim\limits_n e_{\so}V^0_{n,m}[t^+,t^-]$. Note that from Propositions <ref> and <ref>, for $n\geq m$, we have

\[
e_\so V^0_{n,m}[t^+,t^-]\simeq e_\so H^0\left(\sT^{0,\tor}_{1,m},\omega_{(0,0,t^+;t^-)}\otimes \cI_{\Ig^{0,\tor}_{1,m}}\right).
\]

It follows that $V_\so^0[t^+,t^-]$ is $p$-torsion free, and together with Proposition <ref> we get

\begin{align*}
\dim_{\bF_p}V^0_\so[t^+,t^-]/pV^0_\so[t^+,t^-]
&=\dim_{\bF_p} e_\so H^0\left(\sT^{0,\tor}_{1,1},\omega_{(0,0,t^+;t^-)}\otimes \cI_{\Ig^{0,\tor}_{1,1}}\right)\\
&=\dim_{\bQ_p}  e_\so H^0\left(\sT^{0,\tor}_{1}[1/E],\omega_{(0,0,t^+;t^-)}\otimes \cI_{\Ig^{0,\tor}_{1}}\right)\otimes\bQ_p.
\end{align*}

Denote this dimension by $d$. The above shows that $V^0_\so[t^+,t^-]$ is a free $\bZ_p$-module of rank $d$. Thus it suffices to show that there exists $A(t^+)\geq -t^+$ such that $\dim_{\bQ_p} e_\so M^0_{(0,0,t^+;t^-)}\left(K_f^pK^0_{p,1};\bQ_p\right) \geq d$ for $t^-\geq A(t^+)$.
Pick an unramified Hecke character $\chi$ of $\cK^\times\backslash\bA^\times_\cK$ of $\infty$-type $(-t_E(p-1),0)$. Twisting $E$ by the character obtained by composing $\chi$ with $\det:G(\bQ)\backslash G(\bA_\bQ)\ra \cK^\times\backslash\bA^\times_\cK$, 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 $e_\so$ gives an injection

\[
e_\so M^0_{(0,0,t^+;t^-)}\left(K_f^pK^0_{p,1};\bQ_p\right)\lhra e_\so M^0_{(0,0,t^+;t^-+2t_E(p-1))}\left(K_f^pK^0_{p,1};\bQ_p\right),
\]

\[
\dim_{\bQ_p}e_\so M^0_{(0,0,t^+;t^-)}\left(K_f^pK^0_{p,1};\bQ_p\right)\leq  \dim_{\bQ_p}e_\so M^0_{(0,0,t^+;t^-+2t_E(p-1))}\left(K_f^pK^0_{p,1};\bQ_p\right).
\]

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

\[
\dim_{\bQ_p}e_\so M^0_{(0,0,t^+;-t^++4+j+2t_E(p-1)l_j(t^+))}\left(K_f^pK^0_{p,1};\bQ_p\right)\geq d.
\]

Therefore, $A(t^+)=-t^++4+2t_E(p-1) (\max_j\{l_j(t^+)\}+1)$ has the required property.

$\pV^{0,\ast}_\so$ is a free $\Lambda_\so$-module of finite rank.

Let $\fm$ be a maximal ideal of $\bZ_p\llb T_\so(\bZ_p)\rrb$. (We know that $\fm\cap \Lambda_\so=(p,T^+,T^-)$, where we identify $\Lambda_\so$ with $\bZ_p\llb T^+,T^-\rrb$ by identifying $(1+p,1)\in T_\so(\bZ_p)$ (resp. $(1,1+p)\in T_\so(\bZ_p)$ with $T^+$ (resp. $T^-$).) We show that $\pV^{0,\ast}_{\so,\fm}$, the localization of $\pV^{0,\ast}_\so$ at $\fm$, is a free  $\Lambda_\so$-module of finite rank.

First, we consider the quotient $\pV^{0,\ast}_{\so,\fm}\otimes_{\Lambda_\so}\Lambda_\so/(p,T^+,T^-)$, which by definition, equals $(e_\so\pV[\fm])^*$. Take $0\geq t^+\geq -t^-+4$ with $\cP_{t^+,t^-}\subset \fm$. Then

\begin{align*}
&\stackrel{\phantom{\text{Prop~\ref{prop:VH}}}}{=} \text{$p$-torsion of }\varinjlim_m \varinjlim_n e_\so V^0_{n,m}[t^+,t^-]\\
&\stackrel{\text{(Prop~\ref{prop:VH})}}{=}\text{$p$-torsion of }\varinjlim_m \varinjlim_n e_\so H^0\left(\Ig^{0,\tor}_{n,m},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)\\
&\stackrel{\text{(Prop~\ref{prop:n1})}}{=}\text{$p$-torsion of }\varinjlim_m e_\so H^0\left(\Ig^{0,\tor}_{1,m},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1,m}}\right)\\
&\stackrel{\phantom{\text{Prop~\ref{prop:VH}}}}{=} e_\so \left[\text{$p$-torsion of }\varinjlim_m H^0\left(\Ig^{0,\tor}_{1,m},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1,m}}\right)\right].
\end{align*}

Also, Proposition <ref> implies

\begin{align*}
\varinjlim_m H^0\left(\Ig^{0,\tor}_{1,m},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1,m}}\right) &=\varinjlim_m  H^0\left(\Ig^{0,\tor}_{1}[1/E],\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1}}\right)\otimes\bZ/p^m\bZ.
\end{align*}

It follows that

\begin{align*}
\text{$p$-torsion of }\varinjlim_m H^0\left(\Ig^{0,\tor}_{1,m},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1,m}}\right)&= H^0\left(\Ig^{0,\tor}_{1}[1/E],\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1}}\right)\otimes\bZ/p\bZ\\
&= H^0\left(\Ig^{0,\tor}_{1,1},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1,1}}\right),
\end{align*}

where Proposition <ref> is used again for the second equality. Thus,

\[
e_\so\pV[\fm]=e_\so H^0\left(\Ig^{0,\tor}_{1,1},\omega_{(0,0,t^+;t^-)}\otimes\cI_{\Ig^{0,\tor}_{1,1}}\right).
\]

By Proposition <ref>, we know this is finite dimensional over $\bF_p$. Hence, $\pV^{0,\ast}_{\so,\fm}\otimes_{\Lambda_\so}\Lambda_\so/(p,T^+,T^-)=(e_\so\pV[\fm])^*$ is finite dimensional over $\bF_p$. Let

\[
d=\dim_{\bF_p} \pV^{0,\ast}_{\so,\fm}\otimes_{\Lambda_\so}\Lambda_\so/(p,T^+,T^-).
\]

By Nakayama's Lemma, there exist $F_1,\dots,F_d\in \pV^{0,\ast}_{\so,\fm}$ such that

\[
\pV^{0,\ast}_{\so,\fm}=\Lambda_\so F_1+\cdots +\Lambda_\so F_d.
\]

Next, we show that $F_1,\dots,F_d$ are $\Lambda_\so$-linearly independent. Suppose that $a_1,\dots,a_d\in \Lambda_\so$ are such that $a_1F_1+\cdots+a_dF_d=0$. Given any $0\geq t^+\geq -t^-+4$ with $\cP_{t^+,t^-}\subset \fm$, we put

\[
\cP^\circ_{t^+,t^-}=\Lambda_\so\cap \cP_{t^+,t^-}.
\]

In view of Proposition <ref>, the $\bZ_p$-module $\varinjlim
\limits_m\varinjlim\limits_n H^0\left(\Ig^{0,\tor}_{n,m},\omega_{\ut}\otimes\cI_{\Ig^{0,\tor}_{n,m}}\right)$ is $p$-divisible, and by Proposition <ref> so is $e_\so \pV^0[t^+,t^-]$.

\[
\pV^{0,\ast}_{\so,\fm}\otimes_{\Lambda_\so}\Lambda_\so/\cP^\circ_{t^+,t^-}=\left( e_\so \pV^0[t^+,t^-]\right)^*
\]

is $p$-torsion free. On the other hand,

\[
\dim_{\bF_p} \left(\pV^{0,\ast}_{\so,\fm}\otimes_{\Lambda_\so}\Lambda_\so/\cP^\circ_{t^+,t^-}\right)\otimes_{\bZ}\bZ/p\bZ=\dim_{\bF_p}\pV^{0,\ast}_{\so,\fm}\otimes_{\Lambda_\so}\Lambda_\so/(p,T^+,T^-)=d.
\]

It follows that the $\bZ_p$-module $\pV^{0,\ast}_{\so,\fm}\otimes_{\Lambda_\so}\Lambda_\so/\cP^\circ_{t^+,t^-}$ is free of rank $d$, and therefore $a_1,\dots,a_d\in \cP^\circ_{t^+,t^-}$. Then from $\bigcap\limits_{0\geq t^+\geq -t^-+4}\cP^\circ_{t^+,t^-}=0$ we conclude that $a_1=\cdots=a_d=0$, and hence $\pV^{0,\ast}_{\so,\fm}$ is a free $\Lambda_\so$-module of rank $d$.

This completes the proof of part (1) of Theorem <ref> for cuspidal forms. For part (2), because $\pV^{0,\ast}_\so$ is free over $\Lambda_\so$, letting $V^0_{m,\so}[\tau^+,\tau^-]=e_\so \varinjlim\limits_n V^0_{n,m}\otimes_{\bZ_p}\cO_F[\tau^+,\tau^-]$ we have

\begin{equation}\label{eq:MV}
\begin{aligned}
\cM^0_\so\otimes_{\bZ_p\llb T_\so(\bZ_p)\rrb} \cO_F\llb T_\so(\bZ_p)\rrb /\cP_{\tau^+,\tau^-}
&\cong \Hom_{\bZ_p}\left(\pV^{0,\ast}_\so\otimes_{\bZ_p} \cO_F /\cP_{\tau^+,\tau^-},\bZ_p\right)\\
&=\Hom_{\bZ_p}\left(\left(\varinjlim_m V^0_{m,\so}[\tau^+,\tau^-]\right)^*,\bZ_p\right).
\end{aligned}
\end{equation}

Since the left hand side is a free $\bZ_p$-module (because $\cM^0_\so$ is a free $\Lambda_\so$-module), we see that

$\varinjlim\limits_m  V^0_{m,\so}[\tau^+,\tau^-])$ is $p$-divisible, and $V^0_{m+1,\so}[\tau^+,\tau^-]\ra V^0_{m,\so}[\tau^+,\tau^-]$ is surjective, so

\[
\Hom_{\bZ_p}\left(\varinjlim_m V^0_{m,\so}[\tau^+,\tau^-],\bQ_p/\bZ_p\right)\cong   \Hom_{\bZ_p}\left(\varprojlim_m V^0_{m,\so}[\tau^+,\tau^-],\bZ_p\right)
\]

and hence

\begin{align*}
\eqref{eq:MV}=\Hom_{\bZ_p}\left(\Hom_{\bZ_p}\left(\varprojlim_m V^0_{m,\so}[\tau^+,\tau^-],\bZ_p\right),\bZ_p\right)&=\varprojlim_m V^0_{m,\so}[\tau^+,\tau^-]\\
&=\left(\varprojlim_m \varinjlim_n e_\so V^0_{n,m}\otimes_{\bZ_p}\cO_F\right)[\tau^+,\tau^-],
\end{align*}

concluding the proof of part (2) of Theorem <ref> for cuspidal forms. Part (3)  follows from Proposition <ref>.

§ THE PROOF OF THEOREM <REF> FOR THE NON-CUSPIDAL PART

We apply the approach in [30] to prove the vertical control theorem for semi-ordinary forms on $\GU(3,1)$ by analyzing the quotient $\pV/\pV^0$ and using the vertical control theorem for cuspidal semi-ordinary forms on $\GU(3,1)$. When studying $\pV/\pV^0$, we introduce an auxiliary space $\pV^\flat$. One difference from loc.cit is that $q$-expansions are used there to reduce proving some properties for the $\bU_p$-action on $\pV,\pV^\flat$ to matrix computations, but $q$-expansions are not available in the case of $\GU(3,1)$. The analogue of those properties in our case are proved in <ref> by working with semi-abelian schemes over the boundary of the partial toroidal compactification.

§.§ Cusp labels
Following [26], a cusp label is a $K_f$-orbit of triples $\gls{clabel}$, with:

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

*  $\Phi=({\tt X},{\tt Y},\phi,\varphi_{-2},\varphi_0)$ a torus argument, where

$\phi:{\tt Y}\ra{\tt X}$ is an $\cO_\cK$-linear embedding of locally free $\cO_{\cK}$-modules with finite cokernel, and where, writing $\mr{Gr}^{\tt Z}_{-i}={\tt Z}_{-i}/{\tt Z}_{-i-1}$,  $\varphi_{-2}:\mr{Gr}^{\tt Z}_{-2}\stackrel{\sim}{\ra}\Hom\left({\tt X}\otimes\hat{\bZ},\hat{\bZ}(1)\right)$ and $\varphi_0:\mr{Gr}^{\tt Z}_0\stackrel{\sim}{\ra}{\tt Y}\otimes\hat{\bZ}$ are isomorphisms such that $\left<v,w\right>=\varphi_0(v)\Big(\phi\big(\varphi_{-2}(w)\big)\Big)$.

* $\delta:\mr{Gr}^{\tt Z}=\mr{Gr}^{\tt Z}_{-2}\oplus\mr{Gr}^{\tt Z}_{-1}\oplus\mr{Gr}^{\tt Z}_{0}\stackrel{\sim}{\ra} L\otimes\hat{\bZ}$ is an $\cO_\cK\otimes\hat{\bZ}$-equivariant splitting.

Following <cit.>, an ordinary cusp label of tame level $K_f^p$ is a $K^p_fP_{\tt D}(\bZ_p)$-orbit of triples $({\tt Z},\Phi,\delta)$ as above compatible with ${\tt D}$, in the sense that

\begin{align*}
{\tt Z}_{-2}\otimes_{\hat{\bZ}}\bZ_p\subset {\tt D}\subset {\tt Z}_{-1}\otimes_{\hat{\bZ}}\bZ_p.
\end{align*}

There is a unique cusp label for which ${\tt Z}_{-2}=0$, and in our case, all other cusp labels have ${\tt Z}_{-2}$ with rank $1$. As explained in <cit.>, the latter are parametrized by a certain double coset space. To recall this, we have the filtration

\begin{equation}\label{eq:L-fil}
X^\vee\subset X^\vee\oplus L_0\subset L=X^\vee\oplus L_0\oplus Y.
\end{equation}

Let $\glsuseri{P}\subset G$ be the parabolic subgroup preserving this filtration, and $\glsuserii{P}\subset P$ be the kernel of the natural projection $P\ra\GL(Y)$. Define the triple $\gls{clabel1}$ as

*  ${\tt Z}^{(1)}$: ${\tt Z}_{-2}=X^\vee\otimes\hat{\bZ}$, ${\tt Z}_{-1}=(X^\vee\oplus L_0)\otimes\hat{\bZ}$.

*  $\Phi^{(1)}=({\tt X}^{(1)},{\tt Y}^{(1)},\phi^{(1)},\varphi^{(1)}_{-2},\varphi^{(1)}_0)$ with ${\tt X}^{(1)}=\Hom_{\cO_\cK}\left(X^\vee,\cO_\cK(1)\right)$, ${\tt Y}^{(1)}=Y$, $\phi^{(1)}:Y\ra\Hom_{\cO_\cK}\left(\fd^{-1}_{\cK/\bQ}\cdot{\tt x}_1,\cO_\cK(1)\right)$ induced by our fixed basis of ${\tt x}_1$ and ${\tt y}_1$ of $X$ and $Y$ and the pairing $2\pi i\cdot \Tr_{\cK/\bQ}\circ\left<\,,\,\right>_L$, $\varphi^{(1)}_{-2}=2\pi\sqrt{-1}\cdot\id$ and $\varphi^{(1)}_{0}=\id$.

*  $\delta^{(1)}=\id$.

Denote by  ${\tt V}\colon 0\subset {\tt V}_{-2}\subset {\tt V}_{-1}\subset L\otimes\bQ$ the filtration obtained as (<ref>) tensored with $\bQ$. For every $\clabel\in G(\bA_{\bQ,f})$ define $\gls{clabelg}$ by

*  ${\tt Z}^{(\clabel)}_j=\left(\clabel^{-1}({\tt V}_j\otimes\bA_{\bQ,f})\right)\cap\left(L\otimes\hat{\bZ}\right)$.

*  $\Phi^{(\clabel)}=({\tt X}^{(\clabel)},{\tt Y}^{(\clabel)},\phi^{(\clabel)},\varphi^{(\clabel)}_{-2},\varphi^{(\clabel)}_0)$ with $({\tt X}^{(\clabel)},{\tt Y}^{(\clabel)},\phi^{(\clabel)})=({\tt X}^{(1)},{\tt Y}^{(1)},\phi^{(1)})$, $\varphi^{(\clabel)}_{-2}$ (resp. $\varphi^{(\clabel)}_{0}$) is the composition of $\varphi^{(1)}_{-2}$ (resp. $\varphi^{(1)}_{0}$) with $\clabel:\mr{Gr}^{{\tt Z}^{(\clabel)}}_{-2}\ra \mr{Gr}^{\tt Z}_{-2}$ (resp.  $\clabel:\mr{Gr}^{{\tt Z}^{(\clabel)}}_{0}\ra \mr{Gr}^{\tt Z}_{0}$).

*  $\delta^{(\clabel)}$ is $\mr{Gr}^{{\tt Z}^{(\clabel)}}\stackrel{\mr{Gr}(\clabel)}{\ra}\mr{Gr}^{\tt Z^{(1)}}\stackrel{\delta^{(1)}}{\ra} L\otimes\hat{\bZ}\stackrel{\clabel^{-1}}{\ra}L\otimes\hat{\bZ}$.

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

\begin{align*}
\gls{C(K)}&=\GL(X\otimes\bQ)\ltimes P'(\bA^p_{\bQ,f})\backslash G(\bA^p_{\bQ,f})/K^p_f.
\end{align*}

(See <cit.>.)

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

\begin{equation}\label{eq:ordC_1}
\gls{C(K)ord}=\im\left( G(\bA^p_{\bQ,f}) P_{\tt D}(\bQ_p)\lra\GL(X\otimes\bQ)\ltimes P'(\bA_{\bQ,f})\backslash G(\bA_{\bQ,f})/K^p_fK^1_{p,n}\right),
\end{equation}

which is in a natural bijection with

\begin{equation}\label{eq:ordC-2}
\begin{aligned}
&\left.\left(\GL(X_{(p)})\ltimes (P'(\bA^p_{\bQ,f})\times P'(\bZ_p))\right)\middle\backslash G(\bA^p_{\bQ,f})\times P_{\tt D}(\bZ_p)\right/K^p_f(K^1_{p,n}\cap P_{\tt D}(\bZ_p))\\
\cong &
\,\,C(K^p_f)\times \left(U_{\cK,K^p_f}\ltimes P'(\bZ_p)\backslash P_{\tt D}(\bZ_p)/K^1_{p,n}\cap P_{\tt D}(\bZ_p)\right),
\end{aligned}
\end{equation}

where $U_{\cK,K^p_f}=\left\{a\in\cO^\times_\cK:\begin{psm}a\\&\bid_2\\ &&\ltrans{\bar{a}}^{-1}\end{psm}\in K^p_f \right\}$.

By abuse of notation, we shall denote by $\clabel$ both elements in $G(\bA^p_{\bQ,f})P_{\tt D}(\bQ_p)$ and in its quotient $C(K^p_fK^1_{p,n})_\ord$. The context should eliminate any possible confusions.

§.§ The formal completion along boundary strata

Let $\glsuseri{Zg}$ be the stratum associated to $\clabel\in C(K^p_fK^1_{p,n})_\ord$ in the partial toroidal compactification $\Ig^\tor_n$. Then we have

\[
\Ig^{\tor}_n=\sT_{n}\sqcup\bigsqcup\limits_{\clabel\in C(K^p_fK^1_{p,n})_\ord} \sZ_{\clabel,n}.
\]

(The choice of a polyhedral cone decomposition is unique in our special case $\GU(3,1)$ because with $\Big({\bf S}_{\Phi^{(\clabel)}_{K^p_fK^1_{p,n}}}\Big)$ defined as below and ${\bf P}_{\Phi^{(\clabel)}_{K^p_fK^1_{p,n}}}$ the subset of $\Hom_{\bZ}\Big({\bf S}_{\Phi^{(\clabel)}_{K^p_fK^1_{p,n}}},\bZ\Big)\otimes\bR$ consisting of positive semi-definite Hermitian forms, we have  ${\bf P}_{\Phi^{(\clabel)}_{K^p_fK^1_{p,n}}}=\bR_{\geq 0}$.) Let $\gls{Igzmin}$ be the partial minimal compactification of $\Ig_n$ (constructed in <cit.>). Denoting by $\glsuserii{Zg}$ the stratum associated to $\clabel\in C(K^p_fK^1_{p,n})_\ord$ in $\Ig^{\min}_n$, we similarly have

\[
\Ig^{\min}_n=\sT_{n}\sqcup\bigsqcup\limits_{\clabel\in C(K^p_fK^1_{p,n})_\ord} \sY_{\clabel,n}.
\]

The stratum $\sY_{\clabel,n}$ can be identified with the $0$-dimensional Shimura variety $\gls{sSG'}$, where

\begin{align}
\label{eq:K'} \gls{K'fgn}&= \im \left(P(\bA_{\bQ,f})\cap \clabel K_f \clabel^{-1}\ra G'(\bA^p_{\bQ,f})\right).
\end{align}

We have the following diagram:

\begin{equation}\label{eq:Xi}
\xymatrix{
\glsuseri{Xi}\, \ar@{^{(}->}[r] \ar[d]&    \glsuserii{Xi} \ar[ld]\\
\sC^\ord_{\clabel,n}  \ar[d]_-{h}\\
\sS_{G',K'_{f,\clabel,n}},
\end{equation}

where $\gls{Cgn}\ra \sS_{G',K'_{f,\clabel,n}}$ is a torsor of an abelian scheme quasi-isogenous to $\underline{\Hom}_{\cO_\cK}({\tt X}^{(\clabel)},\cA)$ with $\cA$ the universal abelian scheme over $\sS_{G',K'_{f,\clabel,n}}$; $\Xi^\ord_{\clabel,n}\ra \sC^\ord_{\clabel,n}$ is a torsor of the torus with character group ${\bf S}_{\Phi^{(\clabel)}_{K^p_fK^1_{p,n}}}$, which is a finite index subgroup of the free quotient of

\[
\left.\left(\frac{1}{N} {\tt Y}^{(\clabel)}\otimes_{\bZ} {\tt X}^{(\clabel)}\right)\middle/
\begin{pmatrix}
(b\frac{1}{N}y)\otimes x-(\frac{1}{N}y\otimes (b^cx)
\end{pmatrix}_{y,y'\in {\tt Y}^{(\clabel)},x\in {\tt X}^{(\clabel)},b\in\cO_\cK}\right.
\]

where $N$ is sufficiently large with respect to the level $K^p_fK^1_{p,n}$; and  $\Xi^\ord_{\clabel,n}\hra \ol{\Xi}^\ord_{\clabel,n}$ is the torus embedding with respect to the unique cone decomposition.

Denote by $\gls{Xgn}$ the formal completion of $\Ig^\tor_n$ along the stratum $\sZ_{\clabel,n}$ . Then $\fX_{\clabel,n}$ is canonically isomorphic to the quotient by $\Gamma_\clabel$ of the formal completion of $\ol{\Xi}^\ord_{\clabel,n}$ along the boundary $\ol{\Xi}^\ord_{\clabel,n}-\Xi^\ord_{\clabel,n}$. Here $\Gamma_\clabel$ is the (finite, in our case) subgroup of $\GL({\tt X}^{(\clabel)})\times \GL({\tt Y}^{(\clabel)})$ preserving $\varphi^{(\clabel)}_{-2}$ and $\varphi^{(\clabel)}_{0}$ (as orbits).

§.§ The subset $C(K^p_fK^1_{p,n})^\flat_\ord$ of $C(K^p_fK^1_{p,n})_\ord$ and the subspace $\pV^\flat$ of $\pV$

By a generic point, denoted by etapt, of the formal completion $\fX_{\clabel,n}$, we mean a generic point of $\Spec(\Gamma(U,\cO_U))$ with $U$ an affine open subscheme of $\fX_{\clabel,n}$. Let $(\cG,\lambda,i,\alpha^p,\alpha_p)$ be the pullback to $\Spec(\Gamma(U,\cO_U))$ of the family of semi-abelian varieties over $\Ig^\tor_n$, which is a degenerating family obtained by Mumford's construction. Let

\[
0\lra\gls{cT}\lra \gls{Gray}\lra\cA\ra 0
\]

be the Raynaud extension of $\cG$. Then, for any positive integer $N$, $\cT[N]$ (resp. $\cG^\natural[N]$) is canonically isomorphic to a subgroup $\glsuseri{cGmu}\subset \cG[N]$ (resp. $\glsuserii{cGmu} \subset \cG[N]$), and $(\cG[N])^\mu\subset (\cG[N])^\rmf$. The quotient $\glsuseriii{cGmu}=(\cG[N])^\rmf/(\cG[N])^\mu$ is canonically isomorphic to $\cA[N]$. (See also <cit.>.)

Let $\glsuserii{etapt}$ be a geometric point over $\eta$. The ordinary level structure $\alpha_{p,\ol{\eta}}$ of $\cG_{\ol{\eta}}$ gives rise to a filtration

\[
{\tt F}^+_{\bar{\eta}}\colon 0\subset {\tt F}^+_{2,\bar{\eta}}\subset {\tt F}^+_{3,\ol{\eta}}\subset \cG_{\bar{\eta}}[p^n]^{\mult +}
\]

(refer to text around (<ref>)). We define a subset $C(K^p_fK^1_{p,n})^\flat_\ord\subset C(K^p_fK^1_{p,n})_\ord$ in terms of the relative positions of ${\tt F}^+_{2,\ol{\eta}}$ and $(\cG_{\bar{\eta}}[p])^\mu$.

Define the subset $C(K^p_fK^1_{p,n})^\flat_\ord$ of $C(K^p_fK^1_{p,n})_\ord$ consisting of $\clabel$ such that

\[
{\tt F}^+_{2,\bar{\eta}}\cap (\cG_{\bar{\eta}}[p])^\mu=0
\]

for a geometric point $\bar{\eta}$ over a generic point $\eta$ of $\fX_{\clabel,n}$.

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

We have

\begin{equation}\label{eq:flatC-1}
C(K^p_fK^1_{p,n})^\flat_\ord= \im\left( G(\bA^p_{\bQ,f}) P^\flat_{\tt D}(\bQ_p)\lra\GL(X\otimes\bQ)\ltimes P'(\bA_{\bQ,f})\backslash G(\bA_{\bQ,f})/K^p_fK^1_{p,n}\right),
\end{equation}

which is in a natural bijection with

\begin{equation}\label{eq:flatC-2}
C(K^p_f)\times \left(U_{\cK,K^p_f}\ltimes P'(\bZ_p)\backslash P^\flat_{\tt D}(\bZ_p)/K^1_{p,n}\cap P_{\tt D}(\bZ_p)\right),
\end{equation}

\begin{align*}
\gls{PflatD}&=\left\{g\in P_{\tt D}(\bQ_p)\,\middle|\, g^+=\begin{psm}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{psm} \text{ with }\begin{psm}a_{21}&a_{22}\\a_{31}&a_{32}\end{psm}^{-1}\begin{psm}a_{23}\\a_{33}\end{psm}\in \bZ^2_p \right\}
\end{align*}

\begin{align*}
P^{\flat}_{\tt D}(\bZ_p)&=\left\{\clabel_p\in P_{\tt D}(\bZ_p)\,\middle|\, \clabel^+_p=\begin{psm}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{psm} \text{ with }\begin{psm}a_{21}&a_{22}\\a_{31}&a_{32}\end{psm}\in \GL_2(\bZ_p) \right\}\\
&=P'(\bZ_p) \begin{psm}&&1\\1\\&1\\&&&1\end{psm}\left(K^0_{p,n} \cap P_{\tt D}(\bZ_p)\right).
\end{align*}

Given $\clabel_p\in P_{\tt D}(\bZ_p)$ with

\begin{align*}
(\clabel^{-1}_p)^+&=\begin{psm}c_{11}&c_{12}&c_{13}&\ast\\c_{21}&c_{22}&c_{23}&\ast\\ c_{31}&c_{32}&c_{33}&\ast\\&&&\ast\end{psm},
&(\clabel^{-1}_p)^-&=\begin{psm}d_{11}&\ast&\ast&\ast\\ &\ast&\ast&\ast\\&\ast&\ast&\ast\\&\ast&\ast&\ast\end{psm},
\end{align*}

the corresponding ${\tt F}_{2,\ol{\eta}}$ is spanned by the image of $x^+_1,w^+_1$ under $\alpha_{p,\bar{\eta}}$, and $\cT_{\bar{\eta}}[p]$ is spanned by the image under $\alpha_{p,\ol{\eta}}$ of

\begin{equation}\label{eq:Tspan}
({\tt x}^+_1,{\tt w}^+_1,{\tt w}^+_2;{\tt 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&1\\0&0\\0&0\\0&1\end{pmatrix},
\end{equation}

Therefore, the condition ${\tt F}^+_{2,\ol{\eta}}\cap (\cG_{\bar{\eta}}[p])^\mu=0$ is equivalent to the condition that $c_{11}\xx^+_1+c_{21}\ww^+_1+c_{31}\ww^+_2$ does not belong to $\mr{Span}_{\bF_p}\{\xx^+_1,\ww^+_1\}$, which is equivalent to $c_{31}\in \bZ^\times_p$. The condition $c_{31}\in\bZ^\times_p$ for $(\clabel^{-1}_p)^+$ is equivalent to the condition $\begin{psm}a_{21}&a_{22}\\a_{31}&a_{32}\end{psm}\in \GL_2(\bZ_p)$ for $\clabel^+_p$. This shows that $C(K^p_fK^1_{p,n})^\flat_\ord$ equals (<ref>). It is also easy to see that there is a natural bijection between (<ref>) and (<ref>).

\[
\gls{Vflatnm}=\left\{f\in V_{n,m}\;\colon\; f\vert_{\sZ_{\clabel,n}}= 0\quad \text{for all $\clabel\in C(K^p_fK^1_{p,n})_\ord- C(K^p_fK^1_{p,n})^\flat_\ord$}\right\},
\]

i.e. the space of forms in $V_{n,m}$ vanishing along all the boundary strata of $\Ig_n^{\tor}$ indexed by ordinary cusp labels outside  $C(K^p_fK^1_{p,n})^\flat_\ord$, and

\[
\gls{Vflat}=\varinjlim_m\varinjlim_n  V^\flat_{n,m}.
\]

§.§ The exact sequence for $V^\flat_{n,m}$

When $\clabel\in C(K^p_fK^1_{p,n})^\flat_\ord$, the level $K'_{f,\clabel,n}$ is independent of $n$, and we denote it by $K'_{f,\clabel}$ with

\begin{equation}\label{eq:K''}
\begin{aligned}
K^{\prime p}_{f,g}&= \im \left(P(\bA^p_{\bQ,f})\cap \clabel K^p_f\clabel^{-1}\ra G'(\bA^p_{\bQ,f})\right),\\
\gls{K'pg}&=\left\{h\in G'(\bZ_p):h^+\equiv \begin{pmatrix}\ast &\ast\\ &\ast\end{pmatrix}\mod p\right\}.
\end{aligned}
\end{equation}

Let $\gls{M00tor}$ be the space of classical automorphic forms on $G'$ valued in $\bZ/p^m\bZ$ of weight $(0,0)$ and level $K^{\prime }_{f,\clabel}$ (which are functions valued in $\bZ/p^m\bZ$ on the finite set $G'(\bQ)\backslash G'(\bA_{\bQ,f})/K'_{f,\clabel}$), and  $M_{(0,0)}(K'_{f,\clabel};\bQ_p/\bZ_p)=\varinjlim\limits_mM_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ)$.

We have the exact sequences

\begin{equation}\label{eq:exVnm}
0\lra V^0_{n,m} \lra V^\flat_{n,m}\stackrel{\oplus\Phi_{\clabel}}{\lra} \bigoplus_{\clabel\in C(K^p_f)}  M_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ)\otimes\bZ_p\llb T_\so(\bZ_p)/U_{\cK,K^p_f}\rrb \lra 0
\end{equation}

\begin{equation}\label{eq:exV}
0\lra \pV^0 \lra \pV^\flat\stackrel{\oplus\Phi_{\clabel}}{\lra} \bigoplus_{\clabel\in C(K^p_f)}  M_{(0,0)}(K'_{f,\clabel};\bQ_p/\bZ_p)\otimes\bZ_p\llb T_\so(\bZ_p)/U_{\cK,K^p_f}\rrb \lra 0,
\end{equation}

where $\gls{Phig}$ is obtained by restriction to the stratum $\sZ_{\clabel,n}$ (whose global sections are the same as the global sections of $\sY_{\clabel,n}$, which can be identified with the $0$-dimensional Shimura variety $\sS_{G',K'_{f,\clabel}}$), and is sometimes called a Siegel operator.

Let $\pi_n:\Ig^\tor_{n}\ra \Ig^{\min}_{n}$ be the natural map. By <cit.>, we have

\begin{align*}
\left(\pi_{n,*}\cO_{\Ig^\tor_{n,m}}/\pi_{n,*}\cI_{\Ig^\tor_{n,m}}\right)^\wedge_{\sY_{\clabel,n}}\simeq  h_*\cO_{\sC^\ord_{\clabel,n}}\otimes\bZ/p^m\bZ =  \cO_{\sS_{G',K'_{f,\clabel}}}\otimes\bZ/p^m\bZ.
\end{align*}

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

\begin{equation}\label{eq:V/V0}
V_{n,m}/V^0_{n,m}=\bigoplus_{\clabel\in C(K^p_fK^1_{p,n})_\ord} M_{(0,0)}(K'_{f,\clabel,n};\bZ/p^m\bZ).
\end{equation}

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 $C(K^p_fK^1_{p,n})^\flat_\ord$, so

\begin{equation}\label{eq:Vquot}
V^\flat_{n,m}/V^0_{n,m}=\bigoplus_{\clabel\in C(K^p_fK^1_{p,n})^\flat_\ord} M_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ).
\end{equation}

The action of $T_\so(\bZ_p)$ 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 (<ref>). More precisely, the corresponding action of $(a_1,a_2)\in T_\so(\bZ_p)$ on the index set

\[
C(K^p_fK^1_{p,n})_\ord=C(K^p_f)\times\left(U_{\cK,K^p_f}\ltimes P'(\bZ_p)\backslash P_{\tt D}(\bZ_p)/K^1_p\cap P_{\tt D}(\bZ_p)\right)
\]

sends $\clabel=\clabel^p\clabel_p$ to $\clabel^p\clabel_p t_{a_1,a_2}$ where $t_{a_1,a_2}\in G(\bZ_p)$ with $t^+_{a_1,a_2}=\begin{psm}1\\&1\\&&a_1\\&&&a^{-1}_2\end{psm}$. It is easy to see that $T_\so(\bZ_p)$-action on $C(K^p_fK^1_{p,n})_\ord$ preserves $C(K^p_fK^1_{p,n})^\flat_\ord$, so the $T_\so(\bZ_p)$ on $V_{n,m}$ preserves $V^\flat_{n,m}$ and induces an action on $V^\flat_{n,m}/V^0_{n,m}$.

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

\begin{equation}
\begin{aligned}
C(K^p_f K^1_{p,n})_{\ord,\clabel}&=\{\text{elements in $C(K^p_f K^1_{p,n})_\ord$ whose projection to $C(K^p_f)$ is $\clabel$}\},\\
C(K^p_f K^1_{p,n})^\flat_{\ord,\clabel}&=\{\text{elements in $C(K^p_f K^1_{p,n})^\flat_{\ord}$ whose projection to $C(K^p_f)$ is $\clabel$}\}.
\end{aligned}
\end{equation}

The $\bZ_p\llb T_\so(\bZ_p)\rrb$-module structures of $V_{n,m}/V^0_{n,m}$ (resp. $V^\flat_{n,m}/V^0_{n,m}$) are determined by the  $\bZ_p\llb T_\so(\bZ_p)\rrb$-actions on  $C(K^p_fK^1_{p,n})_{\ord,\clabel}$ (resp.  $C(K^p_fK^1_{p,n})^\flat_{\ord,\clabel}$). The $T_\so(\bZ_p)$-action on $C(K^p_fK^1_{p,n})_{\ord,\clabel}$ 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 $\bZ_p\llb T_\so(\bZ_p)\rrb$-module. In contrast, the $T_\so(\bZ_p)$-action on $C(K^p_fK^1_{p,n})^\flat_{\ord,\clabel}$ is transitive, factoring through $\bZ_p\llb T_\so(\bZ_p/p^n\bZ)/U_{\cK,K^p_f}\rrb$, and the action of this quotient is free. Therefore, by fixing an element in $P^\flat_{\tt D}(\bZ_p)$, which we will always choose to be $\begin{psm}&&1\\1\\&1\\&&&1\end{psm}$, we get the isomorphism

\begin{equation}\label{eq:w-isom}
\text{RHS of (\ref{eq:Vquot})}\cong M_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ)\otimes\bZ_p\llb T_\so(\bZ_p)/U_{\cK,K^p_f}\rrb,
\end{equation}

from which the exact sequence (<ref>) follows. The exact sequence (<ref>) follows from (<ref>) by taking the direct limit.

§.§ Three propositions on the $\bU_p$-action on $\pV$ and $\pV^\flat$.

We introduce some setting for the proof of the following propositions. Given $\clabel\in C(K^p_fK^1_{p,n})_\ord$, recall that $\eta$ denotes a generic point of $\fX_{\clabel,n,m}$, and $\bar{\eta}$ denotes a geometric point over $\eta$. Let $(\cG_{\bar{\eta}},i_{\bar{\eta}},\lambda_{\bar{\eta}},\alpha^p_{\bar{\eta}},\alpha_{p,\bar{\eta}})$ be the pullback to $\bar{\eta}$ of the family of semi-abelian schemes over $\Ig^\tor_{n,m}$. Recall that certain Lagrange subgroups  $\Lsub\subset\cG_{\bar{\eta}}[p^2]$ are used in the construction of $\sC^{1\bullet}ar_{j,n,m}$ for defining $U^\bullet_{p,j}$ in <ref>. (Here $(\bullet,j)=(+,2),\,(+,3)$ or $(-,1)$.) Let

\begin{equation}\label{eq:tuple}
\end{equation}

be the tuple defined as in the construction of $\sC^{\bullet}_{j,n,m}$. If $(e^\pm_1,e^\pm_2,e^\pm_3,e^\pm_4)$ is a basis of $\cG_{\bar{\eta}}[p^n]$ compatible with $\alpha_{p,\bar{\eta}}$, then $\Lsub$ is spanned by $(e^\pm_1,e^\pm_2,e^\pm_3,e^\pm_4)\cdot p^n \gamma^\pm_{\Lsub,p}$ with $\gamma_{\Lsub,p}$ given as in equations  (<ref>), (<ref>), and (<ref>). From Mumford's construction of the degenerating family over $\fX_{\clabel,n,m}$, we see that the tuple (<ref>),

corresponds to a geometric point $\bar{\eta}'$ over a generic point $\eta'$ of $\fX_{\clabel',n,m}$, where $\clabel'=\clabel\cdot \gamma_{\Lsub,p}$.

The space $\pV^\flat$ is preserved by the $\bU_p$-operators.

We shall show that $\clabel'=\clabel\cdot\gamma_{\Lsub,p}\in C(K^p_fK^1_{p,n})^\flat_\ord$ only if $\clabel\in C(K^p_fK^1_{p,n})^\flat_\ord$.   By the definition of the $\bU_p$-operators, this will imply that if $f\in \pV$ vanishes along the strata outside  $C(K^p_fK^1_{p,n})^\flat_\ord$, then so does $U^\bullet_{p,j}(f)$. We use the description (<ref>) for $C(K^p_fK^1_{p,n})^\flat_\ord$. Consider the case $(\bullet,j)=(+,2)$.

Write $\clabel^+_p=\begin{psm}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ &&&a_{44}\end{psm}$ and and $\clabel^{\prime+}_p=\clabel^+_p \cdot \gamma^+_{C,p}= \begin{psm}a'_{11}&a'_{12}&a'_{13}&a'_{14}\\ a'_{21}&a'_{22}&a'_{23}&a'_{24}\\ a'_{31}&a'_{32}&a'_{33}&a'_{34}\\ &&&a'_{44}\end{psm}$ with $\gamma^+_{C,p}$ as in (<ref>). We have

\begin{equation}\label{eq:g'}
\begin{pmatrix}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{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{equation}

\begin{equation}\label{eq:a'}
\begin{pmatrix}a'_{21}&a'_{22}\\a'_{31}&a'_{32}\end{pmatrix}^{-1}\begin{pmatrix}a'_{23}\\a'_{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{equation}

Since $u_1,u_2\in\bZ_p$, the right hand side can be integral only if $\begin{psm}a_{21}&a_{22}\\a_{31}&a_{32}\end{psm}^{-1}\begin{psm}a_{23}\\a_{33}\end{psm}\in \bZ^2_p$. Therefore, $\clabel'=\clabel\cdot\gamma_{\Lsub,p}\in  C(K^p_fK^1_{p,n})^\flat_\ord$ only if $\clabel \in C(K^p_fK^1_{p,n})^\flat_\ord$. The argument for the other two cases is similar.

There exists a positive integer $N$ such that the exact sequences in Proposition <ref> are $(U^{\bullet}_{p,j})^N$-equivariant for $(\bullet,j)=(+,2),(+,3),(-,1)$ in the sense that for all $\clabel\in C(K^p_fK^1_{p,n})^\flat_\ord$ and $f\in\pV^\flat$, we have

\[
\Phi_{\clabel}\left((U^\bullet_{p,j})^N f\right)= \Phi_{\clabel}(f).
\]

Take $\clabel\in C(K^p_fK^1_{p,n})^\flat_\ord$ 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 $\Phi_\clabel$, we consider the abelian quotients $\cA_{\bar{\eta}}$ and $\cA_{\bar{\eta}'}$ of the Raynaud extensions

\begin{align*}
&0\lra\cT_{\bar{\eta}}\lra \cG^\natural_{\bar{\eta}}\lra\cA_{\bar{\eta}}\lra 0,
&0\lra\cT_{\bar{\eta}'}\lra \cG^\natural_{\bar{\eta}'}\lra\cA_{\bar{\eta}'}\lra 0.
\end{align*}

\begin{equation}\label{eq:Cs}
\begin{aligned}
\glsuseri{Cab}&=\big(C+(\cG_{\bar{\eta}}[p^n])^\mu\big)\cap (\cG_{\bar{\eta}}[p^n])^\rmf,\\
\glsuserii{Cab}&=C^{\rmf}/(\cG_{\bar{\eta}}[p^n])^\mu.
\end{aligned}
\end{equation}

(See the beginning of <ref> for the definition of $(\cG_{\bar{\eta}}[p^n])^\mu$ and $(\cG_{\bar{\eta}}[p^n])^\rmf$.) Then we have

\[
\cA_{\bar{\eta}'}=\cA_{\bar{\eta}}/C^{\mr{ab}}.
\]

We use the description (<ref>) and (<ref>) for $C(K^p_fK^1_{p,n})_\ord$ and $C(K^p_fK^1_{p,n})^\flat_\ord$. Write $\clabel=\clabel^p\clabel_p$ with $\clabel_p\in P_{\tt D}(\bZ_p)$ and

\begin{equation}\label{eq:cij}
\begin{aligned}
(\clabel^{-1}_p)^+&=\begin{psm}c_{11}&c_{12}&c_{13}&\ast\\c_{21}&c_{22}&c_{23}&\ast\\ c_{31}&c_{32}&c_{33}&\ast\\&&&\ast\end{psm},
&(\clabel^{-1}_p)^-&=\begin{psm}d_{11}&\ast&\ast&\ast\\ &\ast&\ast&\ast\\&\ast&\ast&\ast\\&\ast&\ast&\ast\end{psm}.
\end{aligned}
\end{equation}

The condition $\clabel\in C(K^p_fK^1_{p,n})^\flat_\ord$ implies that  $c^{-1}_{31}c_{11},c^{-1}_{31}c_{21}\in \bZ_p$. Thus we see that $(\cG_{\ol{\eta}}[p^n])^\mu$ is spanned by

\begin{equation}\label{eq:Gmu}
c^{-1}_{31}c_{11} e^+_1+c^{-1}_{31}c_{21}e^+_2+e^+_3,\quad e^-_1
\end{equation}

(cf. (<ref>)), and $(\cG_{\ol{\eta}}[p^n])^\rmf$ is spanned by

\begin{equation}\label{eq:Gmu-basis}
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}.
\end{equation}

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

\begin{equation}\label{eq:g+p}
\clabel^{\prime+}_p=\clabel^+_p\cdot\gamma^+_{\Lsub,p}=\begin{psm}1&&&\ast\\ &1&&\ast\\ &&1&\ast\\&&&p^{-1} \end{psm}\clabel^+_p,
\end{equation}

where $\ast\in \bQ_p$. This shows that the cusp label $\clabel'$ is independent of $\Lsub$ and belongs to $C(K^p_fK^1_{p,n})^\flat_\ord$. For given $\Lsub$ spanned by $(e^\pm_1,e^\pm_2,e^\pm_3,e^\pm_4)\cdot p^{n-2}\gamma^\pm_{\Lsub,p}$ with $\gamma_{\Lsub,p}^\pm$ as in (<ref>),

from the definition (<ref>) and the basis (<ref>) of $(\cG_{\ol{\eta}}[p^n])^\rmf$, we see that  $C^\rmf$ is spanned by

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

its quotient $\Lsub^{\mr{ab}}$ is independent of the choice of $\Lsub$, and is spanned by $(\epsilon^-_1,\epsilon^-_2)\cdot p^{n-2}$, with $(\epsilon^-_1,\epsilon^-_2)$ the image of $(e^-_2,e^-_3,e^-_4)\begin{pmatrix}-\iota_p(\bar{\zeta}_0)\\ &1\end{pmatrix}\begin{psm}0&1\\c^{-1}_{31}c_{11}&c^{-1}_{31}c_{21}\\1 &0\end{psm}$ under the quotient map $\cG^\natural_{\bar{\eta}}\ra\cA_{\bar{\eta}}$ which is a basis of $\cA_{\bar{\eta}}[p^n]^-$. (Here we identify $(\cG_{\bar{\eta}}[p^n])^\rmf$ with $\cG^\natural_{\bar{\eta}}[p^n]$ by the canonical isomorphism between them.) Thus we get

\begin{equation}  \label{eq:Ap2-}
\cA_{\bar{\eta}'}=\cA_{\bar{\eta}}/\cA_{\bar{\eta}}[p^2]^-.
\end{equation}

Denote by $\left<p\right>^-$ the operator on $M_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ)$ induced by taking the quotient by $\cA[p^2]^-$. Combining (<ref>) and (<ref>), we see that the trace  cancels the normalizer in (<ref>) due to the independence of the choice of $\Lsub$, and

\[
\Phi_\clabel(U^+_{p,3}f) =(\left<p\right>^-)^2 \Phi_{\clabel'}(f).
\]

With $K^p_f$ fixed, there exists $N_1\geq 1$, such that $\begin{psm} 1&&&\ast\\&1&&\ast\\&&1&\ast\\&&&p^{-N_1}\end{psm}_p\clabel$ and $\clabel$ represent the same element in the double coset $C(K^p_fK^1_{p,n})^\flat_\ord$, and $(\left<p\right>^-)^{2N_1}=1$ on $M_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ)$. For such $N_1$, we deduce that

\[
\Phi_\clabel\left((U^+_{p,3})^{N_1}f\right) = \Phi_{g}(f).
\]

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

\[
\Phi_\clabel\left((U^-_{p,1})^{N_2}f\right) = \Phi_{\clabel}(f).
\]

for all $f\in \pV^\flat$ and $\clabel\in C(K^p_fK^1_{p,n})^\flat_\ord$.

It remains to consider  $U^+_{p,2}$.

Write $\clabel^+_p=\begin{psm}a_{11}&a_{12}&a_{13}&\ast\\a_{21}&a_{22}&a_{32}&\ast\\ a_{31}&a_{32}&a_{33}&\ast\\ &&&\ast\end{psm}$ and $\clabel^{\prime+}_p=\clabel^+_p \cdot \gamma^+_{C,p}= \begin{psm}a'_{11}&a'_{12}&a'_{13}&a'_{14}\\ a'_{21}&a'_{22}&a'_{23}&a'_{24}\\ a'_{31}&a'_{32}&a'_{33}&a'_{34}\\ &&&a'_{44}\end{psm}$ with $\gamma_{C,p}^+$ as in (<ref>). Then we have (<ref>) and (<ref>), and
$\clabel'\in C(K^p_fK^1_{p,n})^\flat_\ord$ if and only if

\begin{equation}\label{eq:u1u2}
\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}\mod p.
\end{equation}

Now suppose $\clabel'\in C(K^p_fK^1_{p,n})^\flat_\ord$, i.e. the entries $u_1,u_2$ in $\gamma^\pm_{C,p}$ (see (<ref>)) satisfy the congruence (<ref>). Then we have
\clabel^{\prime+}_p\in\begin{psm}p^{-1}&\ast&\ast&\ast\\&1&&\ast\\&&1&\ast\\&&&p^{-1}\end{psm}\clabel^+_p K^1_{p,n},
and as an element in the double coset $C(K^p_fK^1_{p,n})^\flat_\ord$, and $\clabel'$ represents the same element as $\begin{psm}p^{-1}&&&\\&1&&\\&&1&\\&&&p^{-1}\end{psm}\clabel$ in $C(K^p_fK^1_{p,n})^\flat_\ord$. The congruence (<ref>) implies that the $c_{ij}$'s in (<ref>) satisfies

\begin{equation}\label{eq:u1u2'}
\begin{aligned}
u_1&\equiv -c^{-1}_{31}c_{11} \mod p, &u_2&\equiv -c^{-1}_{31}c_{21} \mod p.
\end{aligned}
\end{equation}

From the definition  (<ref>) and the relation (<ref>), by using the basis (<ref>) of $(\cG_{\ol{\eta}}[p^n])^\rmf$ and  the basis of $\Lsub$ in (<ref>), we see that  $\Lsub^\rmf$ is spanned by

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

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

\[
\cA_{\bar{\eta}'}=\cA_{\bar{\eta}}/\cA_{\bar{\eta}}[p^2]^-.
\]

The independence of $\Lsub^{\mr{ab}}$ on the $\ast$'s in $\gamma^+_{\Lsub,p}$ cancels the normalizer in (<ref>), and we get

\[
\Phi_\clabel(U^+_{p,2}f) =(\left<p\right>^-)^2 \Phi_{\clabel'}(f).
\]

For fixed $K^p_f$, take $N_3$ such that
\begin{psm}p^{-1}&&&\\&1&&\\&&1&\\&&&p^{-1}\end{psm}^{N_3}_p\clabel
and $\clabel$ represent the same element in the double coset $C(K^p_fK^1_{p,n})^\flat_\ord$ and $(\left<p\right>^-)^{2N_3}=1$ on $M_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ)$. Then

\[
\Phi_\clabel\left((U^+_{p,3})^{N_3}f\right) = \Phi_{\clabel}(f).
\]

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

$(U^+_{p,2})^mf\in V^\flat_{n,m}$ for all $f\in V_{n,m}$.

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

\begin{equation}\label{eq:p-div}
\Phi_\clabel\left(U^+_{p,2}f\right)\in pM_{(0,0)}(K'_{f,\clabel};\bZ/p^m\bZ);
\end{equation}

it is easy to see that the proposition follows from this. For a given $\clabel\in C(K^p_fK^1_{p,n})_\ord- C(K^p_fK^1_{p,n})^\flat_\ord$, we use the setting described at the beginning of this subsection. As in the proof of Proposition <ref>, we consider $\cA_{\bar{\eta}'}$, the abelian quotient of the Raynaud extension associated to $\cG_{\bar{\eta}'}=\cG_{\bar{\eta}}/C$. We have

\[
\cA_{\bar{\eta}'}=\cA_{\bar{\eta}}/C^{\mr{ab}},
\]

where $\cA_{\bar{\eta}}$ is the abelian quotient of the Raynaud extension associated to $\cG_{\bar{\eta}}$, and $\Lsub^{\mr{ab}}$ is defined in (<ref>). Since the $\bU_p$-operator considered here is $U^+_{p,2}$, the Lagrangian subgroup $\Lsub\subset\cG_{\bar{\eta}}[p^2]$ is spanned by $(e^\pm_1,e^\pm_2,e^\pm_3,e^\pm_4)\cdot p^n \gamma^\pm_{\Lsub,p}$ with $\gamma_{\Lsub,p}^\pm$  as in (<ref>). We use the description (<ref>) and (<ref>) for $C(K^p_fK^1_{p,n})_\ord$ and $C(K^p_fK^1_{p,n})^\flat_\ord$.  Write $(\clabel^{-1}_p)^+=\begin{psm}c_{11}&c_{12}&c_{13}&\ast\\c_{21}&c_{22}&c_{23}&\ast\\ c_{31}&c_{32}&c_{33}&\ast\\&&&\ast\end{psm}$, $(\clabel^{-1}_p)^-=\begin{psm}d_{11}&\ast&\ast&\ast\\ &\ast&\ast&\ast\\&\ast&\ast&\ast\\&\ast&\ast&\ast\end{psm}$ (which are elements in $\GL_4(\bZ_p)$). The condition $\clabel\notin C(K^p_fK^1_{p,n})^\flat_\ord$ implies that $c_{31}\in p\bZ_p$ and $c_{11}$ or $c_{21}$ belongs to $\bZ^\times_p$.

Suppose $c_{11}\in \bZ^\times_p$. The group $(\cG_{\bar{\eta}}[p^n])^\mu$ is spanned by

\begin{equation}\label{eq:Gmu'}
c_{11} e^+_1+c_{21}e^+_2+c_{31}e^+_3,\quad e^-_1,
\end{equation}

and $(\cG_{\ol{\eta}}[p^n])^\rmf$ is spanned by

\begin{equation}\label{eq:Gmu-basis'}
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{equation}

By the definition of $\Lsub^\rmf$ in (<ref>), a direct computation shows that $\Lsub^\rmf$ is spanned by $(\cG_{\bar{\eta}}[p^n])^\mu$ plus

\begin{equation} \label{eq:Cabv}
\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{equation}

Hence, $\Lsub^{\mr{ab}}$ is spanned by the image of (<ref>) modulo $(\cG_{\bar{\eta}}[p^n])^\mu$.

Let $u'_2=c_{11}u_2-c_{21}u_1$.  The trace in (<ref>) corresponds to a sum with $u_1,u'_2$ and the two $\ast$'s in $\gamma_{\Lsub,p}$ varying in $\bZ/p\bZ$. Since (<ref>) depends only on $u'_2$, we see  that $C^{\mr{ab}}$, and hence $\cA_{\bar{\eta}'}$ depends only on $u'_2$. Its independence of the two $\ast$'s implies that the sum over the two $\ast$'s cancels the normalization factor in (<ref>). Its independence of $u_1$ implies that the sum over $u_1$ contributes a factor $p$. Therefore, the evaluation of $\Phi_\clabel\left(U^+_{p,2}f\right)$ at $\bar{\eta}'$ is divisible by $p$. This shows the inclusion (<ref>), hence the result in this case.

The case $c_{21}\in\bZ^\times_p$ can be similarly treated.

§.§ The ordinary projection on $\pV$ and the fundamental exact sequence

In this section, we use the exact sequence in Proposition <ref> and the three propositions proved in <ref> to show that $\lim\limits_{r\ra\infty}(U_p)^{r!}f$ converges for all $f\in\pV$,  that $\pV^{\ast}_\so=\Hom_{\bZ_p}\left(e_\so\pV,\bQ_p/\bZ_p\right)$ is a free $\Lambda_\so$-module of finite rank, and deduce the fundamental exact sequence in part (4) of Theorem <ref>.

Let $U_p=U^+_{p,2}U^+_{p,3}U^-_{p,1}$. Then for each $f\in\pV$, $\lim\limits_{r\to\infty}(U_p)^{r!}f$ converges, and we can define the semi-ordinary projector as $e_\so=\lim\limits_{r\to\infty}(U_p)^{r!}$. The $\bZ_p\llb T_{\so}(\bZ_p)\rrb$-module
\pV^{\ast}_\so=\Hom_{\bZ_p}\left(e_\so\pV,\bQ_p/\bZ_p\right)
is free of finite rank over $\Lambda_\so$.

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

\begin{equation}
\parbox{\dimexpr\linewidth-4.5em}{%
\strut
The submodule generated by $\left(U_p\right)^{r}f$, $r\geq 0$, is finitely generated over $\bZ/p^m$.
\strut
\end{equation}

It is easy to see that the convergence of $\lim\limits_{r\ra\infty}(U_p)^{r!}f$ follows from the property (F) for $f$. By Proposition <ref>, 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 <ref> that $f'=U^N_pf-f$ belongs to $V^0_{n,m}$. Proposition <ref> implies that there exists $M\geq 0$ such that

\[
U^{M+1}_p f'\in \mr{Span}_{\bZ}\left\{f',U_pf',U^2_pf',\dots,U^M_pf'\right\}.
\]

\[
U^{M+N+1}_p f\in \mr{Span}_{\bZ}\left\{f,U_pf,U^2_pf,\dots,U^{M+N}_pf\right\},
\]

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

\[
U^r_p f\in \mr{Span}_{\bZ}\left\{f,U_pf,U^2_pf,\dots,U^{M+N}_pf\right\}.
\]

Hence (F) holds for $f$. Therefore, we have proved that $\lim\limits_{r\ra\infty}(U_p)^{r!}f$ converges for all $f\in\pV$, and we can define the semi-ordinary projector $e_\so$ on $\pV$ as

\begin{equation}\label{eq:eso-noncusp}
\end{equation}

By Proposition <ref>, we have

\begin{equation}\label{eq:eVflat}
\end{equation}

Applying $e_\so$ to the Pontryagin dual of the exact sequence in Proposition <ref> and using (<ref>), we obtain the exact sequence

\begin{equation}\label{eq:exactV}
0\lra M_{(0,0)}(K'_{f,\clabel};\bZ_p)\otimes\bZ_p\llb T_\so(\bZ_p)/U_{\cK,K^p_f}\rrb \lra \pV^*_\so\lra \pV^{0,*}_\so\lra 0.
\end{equation}

We know that $M_{(0,0)}(K'_{f,\clabel};\bZ_p)$ is a free $\bZ_p$-module of finite rank, so the leftmost term in (<ref>) is a free $\Lambda_\so$-module of finite rank. The rightmost term  $\pV^{0,*}_\so$ is also a free $\Lambda_\so$-module of finite rank by Theorem <ref>. Since $\mr{Ext}^1_{\Lambda_\so}(M,N)$ vanishes for free $\Lambda_\so$-modules $M$, we deduce that $\pV^*_\so$, as a $\Lambda_\so$-module, is isomorphic to the direct sum of the terms on the two ends of (<ref>), and therefore is a free $\Lambda_\so$-module of finite rank.

Part (2) of Theorem <ref> for non-cuspidal semi-ordianry families can be proved in the same way as the cuspidal  semi-ordianry families. The freeness of $\pV^{0,*}_\so$ implies that applying $\Hom_{\Lambda_\so}(\,\cdot\,,\Lambda_\so)$ preserves the exactness of (<ref>), and part (4) of Theorem <ref> follows.

§.§ The Fourier–Jacobi expansion

We introduce the Fourier–Jacobi expansions of $p$-adic forms on $\GU(3,1)$, which will be used in <ref> for analyzing the Klingen Eisenstein family on $\GU(3,1)$ constructed in  <ref>.

In <ref>, for an ordinary cusp label $\clabel\in C(K^p_f K^1_{p,n})_\ord$, we described $\fX_{\clabel,n}$, the formal completion of $\sT^\tor_n$ along the boundary stratum $\sZ_{\clabel,n}$. From the description there, we see that

\[
H^0\left(\fX_{\clabel,n},\cO_{\fX_{\clabel,n}}\right)=\left(\prod_{\beta} H^0\left(\sC^\ord_{\clabel,n},\cL(\beta)\right)\right)^{\Gamma_\clabel},
\]

where $\beta$ runs over ${\bf S}^\vee_{\Phi^{(\clabel)}_{K^p_fK^1_{p,n}}}\cap {\bf P}_{\Phi^{(\clabel)}_{K^p_fK^1_{p,n}}}$, which can be identified with a subset of $\Her_1(\cK)_{\geq 0}$, and $\cL(\beta)$ is the invertible sheaf over $\sC^\ord_{\clabel,n}$ of $\beta$-homogeneous functions on $\Xi^\ord_{\clabel,n}$. Therefore, given $\clabel\in  C(K^p_f K^1_{p,n})_\ord$ and $\beta\in \Her_1(\cK)_{\geq 0}$, the restriction to $\fX_{\clabel,n}$ induces a map

\begin{equation}\label{eq:FJnm}
V_{n,m}\lra H^0\left(\sC^\ord_{\clabel,n},\cL(\beta)\otimes\bZ/p^m\bZ\right).
\end{equation}

We consider  $\clabel=\bid_4$ and $\beta\in\Her_1(\cK)_{\geq 0}$. In this case, the $p$-component of level group for the Shimura variety $\sS_{G',K'_{f,\bid_4,n}}$  in the diagram (<ref>) is

\[
K'_{p,\bid_4,n}=\left\{g\in \GU(2)(\bZ_p):g^+\equiv \begin{pmatrix}\ast&\ast\\0&1\end{pmatrix}\mod p^n\right\},
\]

(depending on $n$).

\begin{equation}\label{eq:VJ}
\gls{VJbeta}=\varprojlim_m\varinjlim_n H^0\left(\sC^\ord_{\bid_4,n},\cL(\beta)\otimes\bZ/p^m\bZ\right).
\end{equation}

The map  gives the map of taking the $\beta$-th Fourier–Jacobi coefficient of $p$-adic forms on $\GU(3,1)$ along the boundary stratum labeled by $\bid_4$:

\begin{equation}\label{eq:FJ-V}
\glsuseri{FJ}:V_{\GU(3,1)}\lra V^{J,\beta}_{\GU(2)}.
\end{equation}

§ THE CONSTRUCTION OF THE KLINGEN FAMILY

§.§ Some notation

Let $\gls{Kinfty}$ be the maximal abelian pro-$p$ extension of $\cK$ unramified outside $p$ and $\gls{GammaK}=\Gal(\cK_\infty/\cK)$. Then $\Gamma_\cK\cong \bZ^2_p$. Denote by $\glsuseri{Omega}\in\bC^\times$ (resp. $\glsuserii{Omega}\in \hat{\bZ}^{\ur,\times}_p$) the complex CM period ($p$-adic CM period) with respect to the embeddings in (<ref>) (cf. <cit.>). We also fix an isomorphism $\ol{\bQ}_p\cong \bC$ compatible with the embeddings in (<ref>). Let $\gls{L}\subset\ol{\bQ}_p$ be a sufficiently large finite extension of $\bQ_p$, and denote by $\gls{OLur}$ the ring of integers of the completion of the maximal unramified extension of $L$.

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

\begin{align*}
\gls{lambda}:\cK^\times\backslash\bA^\times_\cK&\lra \bC^\times,
\end{align*}

such that

\begin{align*}
\lambda|_{\bA^\times_\bQ}&=\eta_{\cK/\bQ},
\end{align*}

§.§ Our setup

We assume the following conditions on $\gls{pi}$, an irreducible cuspidal automorphic representation of $\GL_2(\bA_\bQ)$ generated by a newform $\glsuseri{f}$ of weight $2$:

* for all finite places $v$ of $\bQ$, $\pi_v$ is either unramified or Steinberg or  Steinberg twisted by an unramified quadratic character of $\bQ^\times_v$,

* $\pi_p$ is unramified,

* there exists a prime $q$ not split in $\cK$ such that $\pi$ is ramified at $q$, and if $2$ does not split in $\cK$, then $\pi$ is ramified at $2$,

* $\bar{\rho}_\pi|_{\Gal(\ol{\bQ}/\cK)}$ is irreducible, (which is automatically true if $\pi$ is not ordinary at $p$ because in this case $\bar{\rho}_\pi|_{G_\cK,\fp}\cong \bar{\rho}_\pi|_{G_{\bQ,p}}$ is irreducible by [4]), where $\bar{\rho}_\pi$ denotes the residual representation of the Galois representation $\rho_\pi$.

We also fix an algebraic Hecke character $\glsuserii{xi}:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ of $\infty$-type $\left(0,k_0\right)$ with $\gls{k0}$ an even integer.

§.§ The weight space

For an algebraic Hecke character $\chi:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ of $\infty$-type $(k_1,k_2)$, we denote by $\chi_0$ its associated unitary Hecke character, i.e.

\[
\chi_0=\chi|\cdot|^{-\frac{k_1+k_2}{2}}_{\bA_\cK},
\]

and denote by $\chi_{p\adic}$ its associated $p$-adic character, sometimes called the $p$-adic avatar of $\chi$, as

\begin{equation}\label{eq:chip}
\begin{aligned}
&\chi_{p\adic}:\cK^\times\backslash \bA^\times_{\cK,f}\lra\ol{\bQ}^\times_p,
&\chi_{p\adic}(x)=\chi(x)x^{k_1}_\fp x^{k_2}_{\bar{\fp}}.
\end{aligned}
\end{equation}

Applying this convention to $\xi$, we get $\glsuseri{xi}$ and $\glsuseriii{xi}$.

The weight space we will use for constructing the semi-ordinary Klingen Eisenstein family on $\GU(3,1)$ is $\Hom_\cont\big(\Gamma_\cK,\ol{\bQ}^\times\big)$. By identifying $\Gamma_\cK$ with a quotient of $\cK^\times\backslash\bA^\times_{\cK,f}$, the arithmetic points in the weight space are $p$-adic avatars of the algebraic  Hecke characters of $\cK^\times\backslash\bA^\times_\cK$ whose $p$-adic avatars factor through $\Gamma_\cK$. We will use $\glsuseri{tau}$ to denote such a Hecke character, and  $\glsuseriii{tau}$ to denote its $p$-adic avatar. By the above convention, $\glsuserii{tau}$ denotes the associated unitary character of $\tau$.  The interpolation points we will use are the $\tau$'s satisfying that $\xi\tau$ has $\infty$-type $(0,k)$ with $k$ an even integer $\geq 6$. (One can consider more general $\infty$-types by using Masss–Shimura differential operators. For our purpose here, only considering the case of $\infty$-type $(0,k)$ suffices.)

§.§ The groups

By the assumption on $\pi$ in <ref>, we can fix a prime $\gls{q}$ such that

\[
\left\{\begin{array}{ll}
\text{$q$ does not split in $\cK$ and $\pi_q$ is ramified}, &\text{if $2$ splits in $\cK$},\\
q=2, &\text{if $2$ does not split in $\cK$}.
\end{array}\right.
\]

Let $\gls{D}$ be the quaternion algebra over $\bQ$ ramified exactly at $q$ and $\infty$, and let $\pi^D$ be the Jacquet–Langlands transfer of $\pi$ to $D^\times(\bA_\bQ)$.

We can take a square free positive integer $\gls{fs}$, coprime to $pD_{\cK/\bQ}$ and all the primes inert in $\cK$ where $D$ splits,  such that $\mr{inv}_v(D)=(-\fs,D_{\cK/\bQ})_v$, where $(\bdot,\bdot)_v$ is the Hilbert symbol. Define $\glsuseri{zeta}=\delta\begin{pmatrix}\fs\\&1\end{pmatrix}$ with $\delta$ the totally imaginary element in $\cK$ with $\delta^2$ a $p$-adic unit as fixed in Notation. This $\zeta_0$ is a skew–Hermitian matrix. Define the unitary group $\U(2)$ (resp. similitude unitary group $\GU(2)$) over $\bZ$ as: for all $\bZ$-algebra $R$,

\begin{align}
\label{eq:U(2)}\U(2)(R)&=\{g\in \GL_2(R\otimes_{\bZ}\cO_\cK):g\zeta_0\ltrans{\ol{g}}=\zeta_0\}\\
\GU(2)(R)&=\{g\in \GL_2(R\otimes_{\bZ}\cO_\cK):g\zeta_0\ltrans{\ol{g}}=\nu(g)\zeta_0,\,\nu(g)\in R^\times\}.
\end{align}

We have

\[
D=\{g\in M_2(\cK):g\zeta_0 \ltrans{\bar{g}}=\det(g)\zeta_0\}=\left\{\begin{pmatrix}a&-\fs\bar{b}\\b&\ol{a}\end{pmatrix}:a,b\in\cK\right\},
\]

and we view both $D^\times$ and $\GU(2)$ as subgroups of $\mr{Res}_{\cK/\bQ}\GL_2(\cK)$. Then the homomorphism

\begin{align*}
\cK^\times\times D^\times&\lra \GU(2)(\bQ), &(a,g)&\lra ag
\end{align*}

induces an isomorphism

\begin{equation}\label{eq:DGU}
\GU(2)\simeq (\mr{Res}_{\cK/\bQ}\bG_m\times D^\times)/\{(a,a^{-1}\bid_2):a\in\bG_m\}.
\end{equation}

Given an automorphic form on $\phi$ on $D^\times$ with central character $\chi_1$, by picking an extension $\chi$ of $\chi_1$ to $\cK^\times\backslash\bA^\times_\cK$, one obtains an automorphic form $\phi^{\GU(2)}$ on $\GU(2)$ by (<ref>).

We denote by $\gls{piD}$ the Jacquet–Langlands transfer of $\pi$ to $D^\times$, and by $\glsuserii{f}$ the unique automorphic form on $D^\times$ which is a newform for the action of $D^\times(\bQ_v)\simeq\GL_2(\bQ_v)$ for all $v\neq q,\infty$ and takes value $1$ at $\bid_2$. By our assumption, $\pi^D$ has trivial central character. We can extend $f^D$ to a form $\glsuseriii{f}$ via (<ref>) and the trivial character. We denote this form by $f^{\GU(2)}$, and denote by $\gls{piU}$ the automorphic representation of $\GU(2)(\bA_\bQ)$ generated by $f^{\GU(2)}$.

Define the unitary group $\U(3,1)$ (resp. similitude unitary group $\GU(3,1)$) over $\bZ$ as

\begin{align*}
\U(3,1)(R)&=\left\{g\in \GL_4(R\otimes_{\bZ}\cO_\cK):g\begin{pmatrix}&&1\\&\zeta_0\\-1\end{pmatrix}\ltrans{\ol{g}}=\begin{pmatrix}&&1\\&\zeta_0\\-1\end{pmatrix}\right\},\\
\GU(3,1)(R)&=\left\{g\in \GL_4(R\otimes_{\bZ}\cO_\cK):g\begin{pmatrix}&&1\\&\zeta_0\\-1\end{pmatrix}\ltrans{\ol{g}}=\nu(g)\begin{pmatrix}&&1\\&\zeta_0\\-1\end{pmatrix},\,\nu(g)\in R^\times\right\},
\end{align*}

\begin{equation}\label{eq:Kling-P}
\gls{PU}=\left\{\begin{pmatrix}x&\ast&\ast\\ &g&\ast\\&&\nu(g)\bar{x}^{-1}\end{pmatrix}\in\GU(3,1): g\in \GU(2),\,\,x\in \mr{Res}_{\cK/\bQ}\bG_m\right\},
\end{equation}

the (standard) Klingen parabolic subgroup of $\GU(3,1)$. Its Levi subgroup is

\[
M_{P_{\GU{3,1)}}}\simeq \mr{Res}_{\cK/\bQ}\bG_m\times \GU(2).
\]

We consider the Klingen Eisenstein series on $\GU(3,1)$ inducing $\pi^{\GU(2)}\boxtimes\xi_0\tau_0|\cdot|^s_{\bA^\times_\cK}$ from $P_{\GU(3,1)}$.

Define the unitary group $\U(1)$ over as

\[
\U(1)(R)=\left\{a\in (R\times_\bZ\cO_\cK)^\times:a\bar{a}=1\right\}.
\]

The projection $\varrho_\fp:\cK_p=\cK\otimes\bQ_p\ra\bQ_p$ induces maps

\begin{align*}
\U(2)(\bQ_p)&\lra \GL_2(\bQ_p),
&\U(3,1)(\bQ_p)&\lra \GL_4(\bQ_p),
&\U(1)(\bQ_p)&\lra \bQ^\times_p
\end{align*}

and they are all isomorphisms. We denote the inverse maps by $\gls{varrho-1}$:

\begin{equation}\label{eq:varrho-1}
\begin{aligned}
\varrho^{-1}_\fp:\GL_2(\bQ_p)&\overset{\cong}{\lra}\U(2)(\bQ_p),
\end{aligned}
\end{equation}

§.§ The Klingen Eisenstein series and the doubling method

We briefly recall the definition of Klingen Eisenstein series and Garrett's (generalized) doubling method formula which expresses a Klingen Eisenstein series on $\GU(3,1)$ as an integral involving a Siegel Eisenstein series on $\GU(3,3)$.

§.§.§ The Klingen Eisenstein series on $\GU(3,1)$

Let $U_{P_{\GU(3,1)}}$ be the unipotent subgroup of the Klingen parabolic subgroup $P_{\GU(3,1)}$ in (<ref>).  For a unitary character $\xi_0\tau_0:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ and a complex number $s$, define $I_{P_{\GU(3,1)}}(s,\xi_0\tau_0)$ as the space of smooth $K$-finite functions (where $K$ is a maximal compact subgroup of $\GU(3,1)(\bA_\bQ)$)

\[
F(s,\xi_0\tau_0):U_{P_{\GU(3,1)}}(\bA_\bQ)M_{P_{\GU{3,1)}}}(\bQ)\backslash \GU(3,1)(\bA_\bQ)\lra \bC
\]

* $F(s,\xi_0\tau_0)\left(\begin{pmatrix}x\\&\bid_2\\&&\ol{x}^{-1}\end{pmatrix}g\right)=\xi_0\tau_0(x)|x\bar{x}|^{s+\frac{3}{2}}_{\bA_\bQ}$ for all $x\in \bA^\times_\cK$, $g\in\GU(3,1)(\bA_\bQ)$,

* for all $g\in\GU(3,1)(\bA_\bQ)$, the function $g_1\mapsto F(s,\xi_0\tau_0)\left(\begin{pmatrix}1\\&g_1\\&&\nu(g_1)\end{pmatrix}g\right)$ is a cuspidal automorphic form on $\GU(2)(\bA_\bQ)$.

The Klingen Eisenstein series on $\GU(3,1)$ attached to $F(s,\xi_0\tau_0)\in I_{P_{\GU(3,1)}}(s,\xi_0\tau_0)$ is defined as

\begin{align*}
\glsuseri{EklingF}=\sum_{\gamma\in P_{\GU(3,1)}(\bQ)\backslash \GU(3,1)(\bQ)} F(s,\xi_0\tau_0)(\gamma g).
\end{align*}

If the cuspidal automorphic form in (ii) belongs to $\pi^{\GU(2)}$, the Galois representation attached to $E^\Kling(\bdot\,;F(s,\xi_0\tau_0))|_{s=\frac{k-3}{2}}$ (where we assume that $\xi\tau$ has $\infty$-type $(0,k)$) is

\[
\xi\tau\cdot\epsilon^{-2}_{\mr{cyc}}\circ \Nm \,\oplus\, (\xi\tau)^{-c}\cdot\epsilon_{\mr{cyc}}\circ \Nm\,\oplus\, \rho_\pi|_{\Gal(\ol{\bQ}/\cK)},
\]

where the Hecke characters of $\cK^\times\backslash\bA^\times_\cK$ are viewed as characters of $\Gal(\ol{\bQ}/\cK)$. The congruences between this Klingen Eisenstein series and cuspidal forms on $\GU(3,1)$ can be used to construct elements in the Selmer group for $\rho_\pi(\epsilon^2_{\mr{cyc}})|_{\Gal(\ol{\bQ}/\cK)}\otimes\xi^{-1}\tau^{-1}$.

§.§.§ The Siegel Eisenstein series on $\GU(3,3)$

Let $\GU(3,3)$ be the similitude unitary group over $\bZ$ defined by

\[
\GU(3,3)(R)=\left\{g\in\GL_6(R\otimes_{\bZ}\cO_\cK):g\begin{pmatrix}&\bid_3\\-\bid_3\end{pmatrix}\ltrans{\bar{g}}=\nu(g)\begin{pmatrix}&\bid_3\\-\bid_3\end{pmatrix},\,\nu(g)\in R^\times\right\}.
\]

The group $\GU(3,1)\times_{\bG_m} \GU(2)=\{(g_1,g_2)\in\GU(3,1)\times\GU(2):\nu(g_1)=\nu(g_2)\}$ embeds into $H$ by

\begin{equation}\label{eq:embedding}
\begin{aligned}
\gls{imath}:\GU(3,1)\times_{\bG_m} \GU(2)&\lra \GU(3,3)\\
(g_1,g_2)&\lra \cS^{-1}\begin{pmatrix}g_1\\&g_2\end{pmatrix}\cS,
\end{aligned}
\end{equation}

where $\gls{cS}=\begin{pmatrix}1&\\&\bid_2&&-\frac{\zeta_0}{2}\\&&1\\&-\bid_2&&-\frac{\zeta_0}{2}\end{pmatrix}$.

Let $\gls{QU}$ be the (standard) Siegel parabolic subgroup of $\GU(3,3)$. For a $\bZ$-algebra $R$,

\[
Q_{\GU(3,3)}(R)=\left\{\begin{pmatrix}A&B\\0&D\end{pmatrix}\in \GL_6(R\otimes_{\bZ}\cO_\cK):D=\nu\ltrans{\bar{A}}^{-1},\nu\in R^\times,\,A^{-1}B\in\Her_3(R\otimes_{\bZ}\cO_\cK)\right\}.
\]

For a place $v$ of $\bQ$, a unitary character $\xi_{0,v}\tau_{0,v}$ of $\cK^\times_v$ and a complex number $s$, the degenerate principal series $I_{Q_{\GU(3,3)},v}(s,\xi_{0,v}\tau_{0,v})$ is defined as the space consisting of smooth $\GU(3,3)(\bZ_v)$-finite functions $
f_v(s,\xi_{0,v}\tau_{0,v}):\GU(3,3)(\bQ_v)\lra \bC$

\begin{align*}
f_v(s,\xi_{0,v}\tau_{0,v})\left(\begin{pmatrix}A&B\\0&D\end{pmatrix}g\right)=\xi_{0,v} \tau_{0,v}(\det A)|\det A D^{-1}|^{s+\frac{3}{2}}_vf(g)
\end{align*}

for all $g\in \GU(3,3)(\bQ_v),\,\begin{pmatrix}A&B\\0&D\end{pmatrix}\in Q_{\GU(3,3)}(\bQ_v)$.

Given a unitary character $\xi_0\tau_0:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ and a section $f(s,\xi_0\tau_{0})=\bigotimes'_v f_v(s,\xi_{0,v}\tau_{0,v})$ inside $\bigotimes'_v I_{Q_{\GU(3,3)},v}(s,\xi_{0,v}\tau_{0,v})$, the Siegel Eisenstein series on $\GU(3,3)$ attached to $f(s,\xi_{0}\tau_0)$ is defined as

\begin{align*}
\glsuseri{ESiegf}&=\sum_{\gamma\in Q_{\GU(3,3)}(\bQ)\backslash \GU(3,3)(\bQ)} f(s,\xi_0\tau_0)(\gamma g).
\end{align*}

§.§.§ The (generalized) doubling method formula

Let $\xi_0\tau_0:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ be a unitary character. For a section $f(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)}}(s,\xi_0\tau_0)$ and  a cuspidal automorphic form $\varphi$ on $\GU(2)(\bA_\bQ)$,

\begin{align*}
\int_{\U(2)(\bQ)\backslash\U(2)(\bA_\bQ)}E^\Sieg\big(\imath\left(g,g_1t\right);f(s,\xi_0\tau_0)\big)\varphi\left(g_1t\right)(\xi_0\tau_0)^{-1}(\det g_1t)\,dg_1
\end{align*}

where $t$ is an element of $\GU(3,1)(\bA_\bQ)$ with $\nu(t)=\nu(g)$ (and it is easy to see that the left hand side does not depend on the choice of $t$), and $F(f(s,\xi_0\tau_0),\varphi)$ is the section in $I_{P_{\GU(3,1)}}(s,\xi_0\tau_0)$ defined by

\begin{align*}
\gls{Fvarphi}(g)&=\int_{\U(2)(\bA_\bQ)}f(s,\tau_0)(\imath(g,g_1t))\,\varphi(g_1t)(\xi_0\tau_0)^{-1}(\det g_1t)\,dg_1.
\end{align*}

§.§ The auxiliary data for the Klingen family

Let $\glsuseri{Sigma}$ be the set of finite places of $\bQ$ containing the prime $2$, the prime $q$, the finite places $v\neq p$ where $\pi$ or $\xi$ or $\cK/\bQ$ is ramified and the primes dividing $\fs$. We denote by $\glsuserii{Sigma}$ the set of primes in $\Sigma$ split in $\cK$ and $\glsuseriii{Sigma}$ the set of primes in $\Sigma$ non-split in $\cK$.

The Klingen Eisenstein family we will construct is not of the optimal level. It depends on some auxiliary data which are chosen such that we can prove the desired properties of its non-degenerate Fourier–Jacobi coefficients. In this subsection, we fix these auxiliary data. We first fixes places $\ell,\ell'$ and positive integers $c_v$ for $v\in\Sigma_\ns\cup\{\ell'\}$. Then we choose Hecke characters $\chi_\theta,\chi_h$ satisfying a list of conditions. Then we fix  positive integers $c_v$ for $v\in\Sigma_\rms\cup\{\ell\}$. The Siegel Eisenstein family on $\GU(3,3)$ we use for constructing the Klingen Eisenstein family depends on the choice of $\ell,\ell,c_v$, $v\in\Sigma\cup\{\ell,\ell'\}$, the automorphic form on $\GU(2)$ to pair with the restriction to $\GU(3,1)\times_{\bG_m}\GU(2)$ of the Seigel Eisenstein family is chosen from the space (<ref>) defined below which depends on $\chi_{h,v}$, $v\in\Sigma_\rms\cup\{\ell\}$. The characters $\chi_\theta,\chi_h$ will show up in our analysis of non-degenerate Fourier coefficients.

We fix:

– primes $\ell,\ell'\neq 2,p$ such that $\ell$ splits in $\cK/\bQ$, $\ell'$ is inert in $\cK/\bQ$, and $\pi_\ell,\pi_{\ell'}$ are unramified,

– for each place $v\in\Sigma_{\mr{ns}}\cup\{\ell'\}$, a positive integer
\[
\gls{cv} > \max\{\ord_v(\cond(\lambda_v),\ord_v(\cond(\pi_v)),3)\}+1.
\]

Before choosing the auxiliary Hecke characters $\chi_\theta,\chi_h$,  we need to first introduce a Proposition on certain Schwartz functions in the Schrödinger model of Weil representations of $\U(2,1)$. Let $V,V^-$ (resp. $W_\beta$) be the two dimensional skew-Hermitian spaces (resp. one dimensional Hermitian space) over $\cK$, and $e_1,e_2\in V\otimes W_\beta$, $e^-_1,e^-_2\in V^-\otimes W_\beta$ as in <ref>. We have $\U(V\oplus V^-)\cong \U(2,2)$ and $\U(W_\beta)\cong \U(1)$.

Let $v$ be a finite place of $\bQ$ non-split in $\cK/\bQ$. Denote by $\omega_{\beta,v}(\bdot\,,\bdot)$ the Weil representation of $\U(2,2)(\bQ_v)\times \U(1)(\bQ_v)$ on the Schrödinger model $\pzS\left(\cK_v(e_1+e^-_1)\oplus \cK_v(e_2+e^-_2)\right)$ (with respect to $\lambda_{W_\beta}=\lambda$). Let

\[
\sT:\pzS\big(\bQ_v e_1\oplus\bQ_v e_2\oplus\bQ_v e^-_1\oplus\bQ_v e^-_2\big)\lra  \pzS\big(\cK_v(e_1+e^-_1)\oplus \cK_v(e_2+e^-_2)\big)
\]
be the intertwining map between Schrödinger models.

For $\beta=1$, there exists a character $\chi_v:\U(1)(\bQ^\times_v)\ra\bC^\times$ with

\[
\ord_v(\cond(\chi_v))>\max\{\ord_v(\cond(\chi_v|_{\bQ^\times_v},\ord_v(\cond(\lambda_v)),\ord_v(\cond(\pi_v)),3\},
\]

a Schwartz function $\phi_{1,v}$ on $\bQ_v e_1\oplus\bQ_v e_2$, and $u_1,\dots,u_t\in\U(1)(\bQ_v)$, $b_1,\dots,b_t\in \bC$ such that the function

\[
y\longmapsto \sum_{j=1}^t  b_j\int_{\bQ_v e_1\oplus\bQ_v e_2}  \sT^{-1}\left(\omega_{\beta,v}\left(\begin{pmatrix}\bid_2&q^{-c_v}\cdot\bid_2\\0&\bid_2\end{pmatrix} ,u_j\right) \Phi_{0,v}\right)(x,y)\cdot \phi_{1,v}(x)\,dx
\]

is nonzero and belongs to the $\chi_v\lambda^2_v|_{\U(1)(\bQ_v)}$-eignespace for the action of $\U(1)(\bQ_v)$, where $\Phi_{0,v}$ is the characteristic function of $\cO_{K,v}\frac{e_1+e^-_1}{2}\oplus \cO_{\cK,v}\frac{e_2+e^-_2}{2}$.

Now we fix auxiliary Hecke characters

–  $\gls{chi}:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ of $\infty$-type $(0,0)$ with $\chi_h\chi^c_\theta|_{\bA^\times_\bQ}=\triv$ satisfying the following properties:

* $\chi_\theta,\chi_h$ are unramified away from $\Sigma\cup\{p,\ell,\ell'\}$ and $\chi_h\chi^c_\theta$ is unramified at $q$.

* $\chi_{\theta,\fp},\chi_{h,\bar{\fp}}$ are unramified, and $\chi_{h,\fp}|_{\bZ^\times_p}=\chi^{-1}_{\theta,\bar{\fp}}|_{\bZ^\times_p}=\xi_{0,\fp}|_{\bZ^\times_p}$.

* For $v\in\Sigma_{\ns}\cup\{\ell'\}$,

\begin{align*}
&\chi_{\theta,v}|_{\U(1)(\bQ_v)}=\text{a $\chi_v$ as in Proposition~\ref{prop:nschi}}, \\
&\ord_v(\cond(\chi_{h,v}\chi^c_{\theta,v})),\,\ord_v(\cond((\lambda^2_v\chi_{h,v}\chi_{\theta,v}))> \ord_v(\cond(\pi_v)),\text{ if $v\neq q$},
\end{align*}

* For $v=\fv\ol{\fv}\in\Sigma_\rms\cup\{\ell\}$, $\chi_{\theta,\fv}$ is unramified, and

\[
\ord_{\bar{\fv}}(\cond (\chi_{\theta,\bar{\fv}}))\geq 2\ord_v(\cond( \pi_v))+2,
\]

\begin{align*}
\ord_{\fv}\left(\cond((\chi_{h}\chi^c_{\theta})_\fv\right)
&=\left\{\begin{array}{ll}\ord_{\fv}\left(\cond(\chi_{\theta,{\bar{\fv}} })\right)-\ord_v(\cond(\pi_v)), &\pi_v\text{ ramified,}\\
\ord_{\fv}\left(\cond(\chi_{\theta,{\bar{\fv}} })\right)-1, &\pi_v\text{ unramified}.\end{array}\right.
\end{align*}

* If $q=\fq^2$ is ramified in $\cK/\bQ$,

\[
\]

where $\chi_q$ is the unramified quadratic character of $\cK^\times_q$ such that $\pi_q\cong \text{Steinberg}\otimes\chi_q$, and $\varpi_\fq$ is a uniformizer of $\cK_\fq$.

* Furthermore, the value

\begin{equation}\label{eq:L-1}
\left(\frac{\Omega_p}{\Omega_\infty}\right)^4\pi^{-3}\cdot
\gamma_p\left(\frac{1}{2},\pi_{p}\times \left(\lambda^2\chi_h\chi_\theta\right)_{\bar{\fp}}\right)^{-1}
\end{equation}

is a $p$-adic unit, and

\begin{equation}\label{eq:L2}
L^q\left(\frac{1}{2},\mr{BC}(\pi)\times\chi_h\chi^c_\theta\right)\neq 0.
\end{equation}

For some algebraic Hecke character $\tau:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ such that $\tau_{p\adic}$ factors through $\Gamma_\cK$ and $\xi\tau$ has $\infty$-type $(0,k)$, $k\geq 2$, the values

\begin{align}
\label{eq:L-5}&\left(\frac{\Omega_p}{\Omega_\infty}\right)^k(2\pi i)^{1-k}\Gamma(k-1)\cdot\gamma_{\bar{\fp}}\left(\frac{k-2}{2},\chi_h\chi_\theta^c\xi_0\tau_0\right)^{-1} L^{p\infty} \left(\frac{k-2}{2},\chi_h\chi_\theta^c\xi_0\tau_0\right),\\
\label{eq:L-6}&\left(\frac{\Omega_p}{\Omega_\infty}\right)^{k-2}(2\pi i)^{2-k}\Gamma(k-2) \cdot L_\fp\left(\frac{k-2}{2},\lambda^2\chi_h\chi_\theta\xi^c_0\tau^c_0\right) L^{p\infty} \left(\frac{k-2}{2},\lambda^2\chi_h\chi_\theta\xi^c_0\tau^c_0\right),\\
\label{eq:L-5-q}& 1-(\chi_h\chi_\theta^c\xi_0\tau_0)_q(q)q^{-\frac{k-2}{2}},
\end{align}

are all $p$-adic units.

The existence of the characters $\chi_\theta,\chi_h$ satisfying the above assumptions follows from the mod $p$ nonvanishing results. We can first choose $\chi_{\theta,1},\chi_{h,1}$ satisfying the conditions (1)-(5). (The inert prime $\ell'$ is introduced to ensure the existence.) Then we can apply mod $p$ nonvanishing results [18] (for the $L$-values in (<ref>)) and [16] (for the $L$-values in (<ref>)(<ref>)) to choose a character $\nu$ of $\ell$-power conductor such that $\chi_\theta=\chi_{\theta,1}\nu$, $\chi_h=\chi_{h,1}\nu$ satisfy (1)-(6). (The conditions in (3) on the conductors at non-split primes and  the condition (5) implies that the local root numbers are $+1$ as required for applying [18]. Our assumption that $\xi$ has $\infty$-type $(0,k_0)$ with $k_0$ even implies that the $L$-values in (<ref>)(<ref>)) fall into the non-residually self-dual case for which <cit.> can be applied.) (The strategy explained in <ref> explains why we need to make such a choice the auxiliary $\chi_\theta,\chi_h$.)

Besides the positive integers $c_v$ for $v\in\Sigma_\ns\cup\{\ell'\}$ fixed earlier in this subsection, we also fix

– for each place $v\in\Sigma_\rms\cup\{\ell\}$, a positive integer

\begin{align}
\label{eq:c_v} \gls{cv}>
\max\Big\{\ord_{\bar{\fv}}(\cond(\chi_{\theta,\bar{\fv}})),\ord_{\fv}(\cond(\xi_{0,\fv})),\ord_{\bar{\fv}}(\cond(\xi_{0,\bar{\fv}})),\ord_{\bar{\fv}}(\cond(\lambda_{\bar{\fv}}))\Big\}.
\end{align}

With our fixed $c_v$'s for all $v\in\Sigma\cup\{\ell,\ell'\}$, define

\begin{equation}\label{eq:K_v}
\gls{Kv}=\Big\{g\in\GU(2)(\bZ_v):g\equiv\bid_2\mod q^{2c_v+\ord_v(\delta\fs)}_v \Big\},
\end{equation}

and the tame level group $K^p_f\subset\GU(2)(\hat{\bZ}^p)$ as

\begin{equation}\label{eq:k^p_f}
\gls{Kpf}=\bigotimes_{v\notin\Sigma\cup\{\infty,\ell,\ell'\}}\GU(2)(\bZ_v)\bigotimes_{v\in\Sigma\cup\{\ell,\ell'\}}K_v.
\end{equation}

Given an open subgroup $K_p\subset\GU(2)(\bZ_p)$ and a ring $R$, let
\begin{equation}\label{eq:MGU2}
\gls{MU2}=\left\{\text{functions on  $\GU(2)(\bQ)\backslash\GU(2)(\bA_{\bQ})/K^p_fK_{p}\U(2)(\bR)$ valued $R$}\right\},
\end{equation}

the space of $R$-valued automorphic forms on $\GU(2)$ of weight $(0,0)$ and level $K^p_fK_p$.

We also define the following twist of the form $f^{\GU(2)}$:

\begin{equation}\label{eq:f-twist}
\gls{ftwisted}=\prod_{\substack{v=\fv\bar{\fv}\in\Sigma_\rms\cup\{\ell\}}} \sum_{n\in(\bZ_v/q^{t_v}_v\bZ_v)^\times}\chi_{h,\bar{\fv}}(-n)\,R\left(\begin{pmatrix}1\\n&1\end{pmatrix}_\fv\begin{pmatrix}q^{-t_v}_v\\&1\end{pmatrix}_\fv\right)f^{\GU(2)},
\end{equation}

where $t_v=\ord_{\bar{\fv}}(\cond(\chi_{h,\bar{\fv}}))$.
Then $f^{\GU(2)}_{\chi_h}\in M_{\GU(2)}\left(K^p_f\GU(2)(\bZ_p);\cO_L\right)$. Let

\begin{equation}\label{eq:M-varphi'}
L\big[\GU(2)\big(\bQ_{\Sigma_\ns\cup\{\ell'\}}\big)\big]\cdot f^{\GU(2)}_{\chi_h}
\end{equation}

denote the space generated by $f^{\GU(2)}_{\chi_h}$ and the action of $\GU(\bQ_v)$, $v\in\Sigma_\ns\cup\{\ell'\}$.

We will use $\varphi$ inside the space

\begin{equation}\label{eq:M-varphi}
M_{\GU(2)}\left(K^p_f\GU(2)(\bZ_p);\cO_L\right)\cap L\big[\GU(2)\big(\bQ_{\Sigma_\ns\cup\{\ell'\}}\big)\big]\cdot f^{\GU(2)}_{\chi_h}
\end{equation}