to pair with the Siegel Eisenstein family on $\GU(3,3)$ to get the Klingen Eisenstein family on $\GU(3,1)$. The twist at the split primes is to make the nebentypus match those of the auxiliary CM families $\bmtheta$ and $\bfh$ constructed from our chosen auxiliary characters $\chi_\theta$ and $\chi_h$. The flexibility at the non-split primes is to ensure the non-vanishing of the local triple product integrals at the non-split primes.

§.§ The choice of the local sections for the Siegel Eisenstein series on $\GU(3,3)$

With $c_v$'s chosen in <ref>, given an algebraic Hecke character $\tau$ such that $\xi\tau$ has $\infty$-type $(0,k)$ with $k$ an even integer $\geq 6$, we make the following choices of $f(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)}}(s,\xi_0\tau_0)$, so that after a suitable normalization, $E^\Sieg(\bdot\,;f(s,\xi_0\tau_0))\big|_{s=\frac{k-3}{2}}$ can be interpolated by a two-variable $p$-adic family when $\tau$ varies.

§.§.§ The archimedean place

We choose $f_\infty(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)},\infty}(s,\tau_0)$ as

\begin{align*}
f_{\infty}(s,\xi_0\tau_0)\left(g=\begin{pmatrix}A&B\\C&D\end{pmatrix}\right)=&\,(\det g)^k|\det(g)|_{\bC}^{\frac{s}{2}-\frac{3}{4}-\frac{k}{4}}\\
\end{align*}

§.§.§ Unramified places.

For  $v\notin\Sigma\cup\{p,\ell,\ell'\}$, we choose $f_v(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)},v}(s,\tau_0)$ to be the standard spherical section, i.e. the section that is invariant under the right translation of $\GU(3,3)(\bZ_v)$ and takes value $1$ at $\bid_6$.

§.§.§ Places $v\in\Sigma\cup\{\ell,\ell'\}$

We choose $f_v(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)},v}(s,\tau_0)$ as

\begin{equation}\label{eq:fv-ram}
\begin{aligned}
f_v(s,\xi_0\tau_0)\left(g=\begin{pmatrix}A&B\\C&D\end{pmatrix}\right)=&\,|\nu(g)|^{s+\frac{3}{2}}_v |\det C\ltrans{\bar{C}}|^{-s-\frac{3}{2}}_v
(\xi_0\tau_0)_v\left(\nu(g)(\det \bar{C})^{-1}\right)\\
&\times \mathds{1}_{\Her(3,\cO_{\cK,v})}\left(C^{-1}D+\begin{pmatrix}0&0\\0&q_v^{-c_v}\cdot \bid_2\end{pmatrix}\right),
\end{aligned}
\end{equation}

where $c_v$ is the fixed positive integer in <ref>.

§.§.§ The place $p$

We have the isomorphisms

\begin{align*}
\U(3,3)(\bQ_p)&\lra \GL_6(\bQ_p), & g=\big(a_{ij}\big)&\longmapsto \big(\varrho_\fp(a_{ij})\big),\\
\Her_3(\cK_p)&\lra M_{3,3}(\bQ_p), &x=\big(x_{ij}\big)&\longmapsto \varrho_\fp(x)=\big(\varrho_\fp(x_{ij})\big).
\end{align*}

and we will often use them to identify $\U(3,3)(\bQ_p)$ with $\GL_6(\bQ_p)$ and $\Her_3(\cK_p)$ with $M_{3,3}(\bQ_p)$.

For $x\in \Her_3(\cK_p)$, we write $\varrho_{\fp}(x)$ as $\begin{pmatrix}x_{11}&x_{12}&x_{13}\\x_{21}&x_{22}&x_{23}\\x_{31}&x_{32}&x_{33}\end{pmatrix}$. Define the Schwartz function $\alpha_{\xi\tau,p}$ on $\Her_3(\cK_p)$ as

\[
\alpha_{\xi\tau,p}\left(x\right)=\mathds{1}_{\Her_3(\cO_{\cK,p})}(x)\cdot \mathds{1}_{\bZ^\times_p}(x_{21})\,\mathds{1}_{\GL_2(\bZ_p)}\begin{pmatrix}x_{21}&x_{22}\\x_{31}&x_{32}\end{pmatrix}(\xi_0\tau_0)^{-1}_{\fp}\left(\det\begin{pmatrix}x_{21}&x_{22}\\x_{31}&x_{32}\end{pmatrix}\right).
\]

and let $\cF^{-1}\alpha_{\xi\tau,p}$ be the inverse Fourier transform of $\alpha_{\xi\tau,p}$, i.e.

\[
\cF^{-1}\alpha_{\xi\tau,p}(x)=\int_{\Her_3(\cK_p)} \alpha_{\xi\tau,p}(y)\,\be_p(\Tr\, xy)\,dy,
\]

where $\be_p$ is the additive character in (<ref>). (A simple computation shows that

\begin{equation}\label{eq:inv-F}
\begin{aligned}
\cF^{-1}\alpha_{\xi\tau,p}(x)=&\,p^{-3t}\fg\left((\xi_0\tau_0)_{\fp}^{-1}\right)^2\cdot \mathds{1}_{\bZ^5_p}(x_{11},x_{21},x_{31},x_{32},x_{33})\cdot \mathds{1}_{\bZ^2_p}(x^{-1}_{23}x_{13},x^{-1}_{23}x_{22})\\
\end{aligned}
\end{equation}

if $(\xi_0\tau_0)_\fp$ has conductor $p^t$, $t\geq 1$.) In order to define the $f_p(s,\xi_0\tau_0)$ for our purpose, we first define the section

\begin{align*}
&=|\nu(g)|^{-s+\frac{3}{2}}_p |\det C\ltrans{\bar{C}}|^{s-\frac{3}{2}}_p
(\xi_0\tau_0)_{p}\left(\nu(g)(\det C)\right)\\
&\quad\times \cF^{-1}\alpha_{\xi\tau,p}(C^{-1}D),
\end{align*}

which belongs to $I_{Q_{\GU(3,3)},p}(-s,(\xi_0\tau_0)^{-c})$ and is supported on the big cell

\[
\]

We choose $f_p(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)},p}(s,\xi_0\tau_0)$ as

\begin{equation}\label{eq:fp}
\begin{aligned}
&\times M_p\left(-s,(\xi_0\tau_0)^{-c}\right)f^{\text{big-cell}}_p\left(-s,(\xi_0\tau_0)^{-c}\right)(g\Upsilon_p),
\end{aligned}
\end{equation}

where $\gls{xiQ}$ (resp. $\gls{tauQ}$) denotes the restriction of $\xi_0$ (resp. $\tau_0$) to $\bQ^\times\backslash\bA^\times_\bQ$, and $M_p\left(-s,(\xi_0\tau_0)^{-c}\right)$ is the intertwining operator, i.e.

\begin{equation}\label{eq:intw}
\begin{aligned}
=&\int_{\Her_3(\cK_p)} f^{\text{big-cell}}_p(-s,(\xi\tau_0)^{-c})\left(\begin{pmatrix}&-\bid_3\\\bid_3\end{pmatrix}\begin{pmatrix}\bid_3&y\\&\bid_3\end{pmatrix}g\Upsilon\right)\,dy,
\end{aligned}
\end{equation}

and $\gls{Upsilonp}$ is the element in $\U(3,3)(\bQ_p)$ such that

\[
\varrho_\fp(\Upsilon_p)=
\begin{pmatrix}
\end{pmatrix}^{-1}.
\]

When $(\xi^\bQ_0\tau^\bQ_0)_p$ is ramified with conductor $p^t$,

\begin{align*}
\end{align*}

Combining this with (<ref>), one can see that the formula for the section $f_p(s,\xi_0\tau_0)$ in (<ref>) agrees with the formula in <cit.>.)

For the unitary group attached to $\begin{pmatrix}&\bid_n\\-\bid_n\end{pmatrix}$ and $\beta\in\Her_n(\cK)$, the factor for the functional equation of the $\beta$-th local Fourier coefficients of the degenerate principal series inducing $(\xi_0\tau_{0})^{-c}_p|\cdot|_{\cK_p}^{-s}$  is

\begin{equation}\label{eq:cns}
c_{n,p}(-s,(\xi_0\tau_0)^{-c},\beta)=c_{\beta,p}\cdot (\xi^\bQ_0\tau^\bQ_0)_{p}(\det\beta)|\det\beta|^{2s}_{\bQ_p}\cdot\prod_{j=1}^n\gamma_p\left(-2s+1-j,(\xi^\bQ_0\tau^\bQ_{0})^{-1}\eta_{\cK/\bQ}^{n+j}\right)^{-1},
\end{equation}

where $c_{\beta,p}$ is a constant independent of $s$ and $\xi_0\tau_0$. This factor also shows up in the functional equation of the doubling zeta integrals. Note that in our choice of $f_p(s,\xi_0\tau_0)$ in (<ref>), the product of gamma factors is exactly the product of gamma factors in $c_{3,p}(-s,(\xi_0\tau_0)^{-c},\beta)$.

Next, we construct a $p$-adic family interpolating these $E^\Sieg(-;f(s,\xi_0\tau_0))\big|_{s=\frac{k-3}{2}}$ with $f(s,\xi_0\tau_0)$ chosen as above. We will construct $p$-adic families of automorphic forms as measures on $\Gamma_\cK$ valued in the space of $p$-adic forms. The next subsection introduces some notions about $p$-adic measures.

§.§ $p$-adic measures and $p$-adic families of automorphic forms

For a $p$-adically complete $\cO_L$-module $M$ and a compact abelian group $Y$ with totally disconnected topology, denote by $\Meas(Y,M)$ the space of $M$-valued $p$-adic measures on $Y$. Given $\mu\in \Meas(Y,M)$ and a continuous function $\eta:Y\ra R$ with $R$ a $p$-adically complete $\cO_L$ algebra, we write

\[
\mu(\eta)=\int_Y \eta(y)\,d\mu(y).
\]

to denote the value of $\mu$ at $\eta$, which is an element in $R\wh{\otimes}_{\cO_L}M$.

For each $y\in Y$, we have the delta measure $\delta_y\in \Meas(Y,M)$ defined by

\begin{equation}\label{eq:delta-meas}
\delta_y(\eta)=\eta(y).
\end{equation}

Given $M_1,M_2$ and $\mu_i\in \Meas(Y,M_i)$, $i=1,2$, the convolution $\mu_1\ast\mu_2\in \Meas(Y,M_1\wh{\otimes}M_2)$ is defined by

\begin{equation}\label{eq:clv-meas1}
(\mu_1\ast\mu_2)(\eta)=\int_Y\int_Y \eta(yz)\,d\mu_1(y)\,d\mu_2(z),
\end{equation}

If $\chi$ is a continuous character of $Y$ valued in $\cO^\times_{\bar{L}}$, then

\begin{equation}\label{eq:clv-meas2}
\end{equation}

Let $A$ be a group with a homomorphism $A\ra Y$ and an action on $M$. Then the group $A$ acts on $\Meas(Y,M)$ in two ways:

– $A$ acts on $\Meas(Y,M)$ via its action on $M$,

– $A$ acts on $\Meas(Y,M)$ via the homomorphism $A\ra Y$ and the translation of $Y$ on itself.

\begin{equation}\label{eq:eqiv-meas}
\gls{MeasYM}=\left\{\begin{array}{ll}\text{$p$-adic measures in $\Meas(Y,M)$ on which}\\ \text{the two actions of $A$ are compatible}\end{array}\right\}.
\end{equation}

In our applications, $Y$ will be taken to be $\Gamma_\cK$ or $U_{\cK,p}=1+p\cO_{\cK,p}$, $A$ will be taken to be $\bZ^\times_p\times\bZ^\times_p$, which we identify with $\cO^\times_{\cK,p}$ via $(\varrho_\fp,\varrho_{\bar{\fp}})$, and  $M$ will be taken to be either $\hat{\cO}^{\ur}_L$ or spaces of $p$-adic forms on unitary groups.  Next, we introduce the spaces of $p$-adic families of automorphic forms we will use.

§.§.§ $p$-adic families on $\GU(3,1)$

Let $\gls{VGU31}=\varprojlim\limits_m\varinjlim\limits_n V_{n,m}$ (defined in <ref>). In (<ref>), put $Y=\Gamma_\cK$, $A=\bZ^\times_p\times\bZ^\times_p$ with the homomorphism

\begin{equation}\label{eq:ft}
\begin{tikzcd}[column sep=huge]
A\simeq \cO^\times_{\cK,p}\arrow[r,"\text{natural}","\text{embedding}"']& \cK^\times\backslash\bA^\times_\cK\arrow[r,"\text{natural}","\text{projection}"']&\Gamma_\cK=Y,
\end{tikzcd}
\end{equation}

and $M=V_{\GU(3,1),\,\xi}$, the component of $V_{\GU(3,1)}$ on which the kernel of (<ref>), acts through the character $\xi_{p\adic}$.  We make $(a_1,a_2)\in \bZ^\times_p\times\bZ^\times_p$ act on $V_{\GU(3,1),\,\xi}$  by the usual action of
\varrho^{-1}_\fp\begin{psm}1\\&1\\&&a_1\\&&&a_2\end{psm}\in T_\so(\bZ_p)\subset \U(3,1)(\bZ_p)
on $V_{\GU(3,1),\,\xi}$ multiplied by the scalar $\xi^{-1}_{p\adic,p}(a_1,a_2)$. (Here $\varrho^{-1}_\fp$ is the isomorphism in (<ref>), and this action factors through the quotient of $\bZ^\times_p\times\bZ^\times_p$ by the kernel of (<ref>).) Then we get

\begin{equation}\label{eq:MU31}
\Meas\left(\Gamma_\cK,V_{\GU(3,1),\,\xi}\right)^\natural,
\end{equation}

the $\xi_{p\adic}$-component of the space of $p$-adic families on $\GU(3,1)$. The compatibility of the two actions of $\bZ^\times_p\times\bZ^\times_p$ implies that the value at $\tau_{p\adic}$ of an element in (<ref>) is a $p$-adic form with $p$-adic nebentypus

\[
\left(\triv,\triv,(\xi\tau)_{p\adic,\fp},(\xi\tau)^{-1}_{p\adic,\bar{\fp}}\right).
\]

(By saying $p$-adic nebentypus, we refer to the usual action of  $\varrho^{-1}_\fp\begin{psm}a_1&\ast&\ast&\ast\\&a_2&\ast&\ast\\&&a_3&\ast\\&&&a_4\end{psm}\in\U(3,1)(\bZ_p)$ on $V_{\GU(3,1)}$.)

The space (<ref>) contains the subspace of semi-ordinary families:

\begin{equation}\label{eq:MU31-so}
\Meas\left(\Gamma_\cK,e_\so V_{\GU(3,1),\,\xi}\wh{\otimes}\hat{\cO}^\ur_L\right)^\natural,
\end{equation}

which is naturally equipped with an $\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb$-module structure. The Klingen Eisenstein family we will construct belongs to this space. (It is tautological that an element in  (<ref>) gives rise to an element in $\xi_{p\adic}$-component of $\cM_\so\wh{\otimes}_{\bZ_p\llb T_\so(\bZ_p)\rrb}\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb=\Hom_{\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb}\left(\pV^*_\so,\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\right)$, see <ref>.)

§.§.§ $p$-adic families on $\GU(3,3)$

Let $\gls{VGU33}$ be the space of $p$-adic forms on $\GU(3,3)$ defined by considering global sections of the structure sheaf on the Igusa towers for the Shimura variety of $\GU(3,3)$. We will construct a Siegel Eisenstein family in

\begin{equation*}
\Meas\left(\Gamma_\cK,V_{\GU(3,3)}\right).
\end{equation*}

§.§.§ $p$-adic families on $\U(2)$

For constructing the Klingen Eisenstein family, we use an automorphic form $\varphi$ in the space (<ref>), which is spherical at $p$ and does not vary over the weight space. It is when analyzing the non-degenerate Fourier–Jacobi coefficients of our Klingen Eisenstein family, we need auxiliary $p$-adic families on $\U(2)$.

\begin{align*}
N_p(\bZ_p)&=\left\{g\in \U(2)(\bZ_p):\varrho_\fp(g)=\begin{pmatrix}\ast&\ast\\0&1\end{pmatrix}\right\},\\
N'_p(\bZ_p)&=\left\{g\in \U(2)(\bZ_p):\varrho_\fp(g)=\begin{pmatrix}1&\ast\\0&\ast\end{pmatrix}\right\}.
\end{align*}

We define the following two spaces of $p$-adic forms on $\U(2)$ of tame level $K^p_f\cap\U(2)(\wh{\bZ}_p)$:

\begin{align*}
\glsuseri{VU2}&=\left\{\text{continuous functions }\U(2)(\bQ)\backslash \U(2)(\bA_{\bQ,f})/\big(K^p_f\cap\U(2)(\wh{\bZ}_p)\big) N_p(\bZ_p)\lra \cO_L \right\},\\
\glsuserii{VU2}&=\left\{\text{continuous functions }\U(2)(\bQ)\backslash \U(2)(\bA_{\bQ,f})/\big(K^p_f\cap\U(2)(\wh{\bZ}_p)\big) N'_p(\bZ_p)\lra \cO_L \right\},
\end{align*}

and let $\glsuseriii{VU2}$ (resp. $\glsuseriv{VU2}$) denote the subspace of $V_{\U(2)}$ (resp. $V'_{\U(2)}$) on which the kernel of (<ref>) acts through the character $(\triv,\xi_{p\adic,\fp})$ (resp. $(\xi^{-1}_{p\adic,\fp},\triv)$).  We have the operator $U_p$ acting on ${\tt F}\in V_{\U(2),\,\xi}$ (resp. ${\tt F}\in V'_{\U(2),\,\xi^{-1}}$) as

\[
(U_p{\tt F})(g)=\sum_{x\in\bZ/p\bZ}{\tt F}\left(g\varrho^{-1}_\fp\begin{pmatrix}p&x\\&1\end{pmatrix}\right)
\]

with $\varrho^{-1}_\fp$ the isomorphism in (<ref>). The ordinary projection is defined as $\gls{eord}=\lim\limits_{n\ra\infty}U^{n!}_p$.

In (<ref>), put $Y=\Gamma_\cK$, $A=\bZ^\times_p\times\bZ^\times_p$ with the homomorphism $A\ra Y$ as in (<ref>), and $M=V_{\U(2),\,\xi}$ on which  we make $(a_1,a_2)\in \bZ^\times_p\times\bZ^\times_p$ act by the right translation of $\varrho^{-1}_\fp\begin{pmatrix}1\\&a_1\end{pmatrix}$ multiplied by the scalar $\xi^{-1}_{p\adic,\fp}(a_1)$. (This action factors through the quotient of $\bZ^\times_p\times\bZ^\times_p$ by the kernel of (<ref>).) Then we get

\begin{equation}\label{eq:MU2}
\Meas\left(\Gamma_\cK,V_{\U(2),\,\xi}\right)^\natural.
\end{equation}

The evaluation at $\tau_{p\adic}$ of an element in (<ref>) is a $p$-adic form on $\U(2)$ with $p$-adic nebentypus

\[
\left(\triv,(\xi\tau)_{p\adic,\fp}\right).
\]

The non-degenerate Fourier–Jacobi coefficients of our Klingen Eisenstein family are $p$-adic measures on $\Gamma_\cK$ valued in $p$-adic Jacobi forms on $\U(2)$. Pairing them with a fixed Jacobi form on   $\U(2)$ gives an element in (<ref>). (See <ref>.)

We will also need to consider some auxiliary $p$-adic families which are $p$-adic measures on $U_{\cK,p}=1+p\cO_{\cK,p}$ valued in $V_{\U(2),\,\xi}$ and $V'_{\U(2),\,\xi^{-1}}$. We will fix $\wtp$, a power of $p$, such that raising to the $\wtp$-th power maps $\Gamma_\cK$ into $U_{\cK,p}$. (See <ref> for an explanation why we need to consider $U_{\cK,p}$ and fix such an $\wtp$.)

Put $Y=U_{\cK,p}=1+p\cO_{\cK,p}$, $A=\bZ^\times_p\times\bZ^\times_p$ with the homomorphism

\[
\begin{tikzcd}[column sep=huge]
A=\bZ^\times_p\times\bZ^\times_p\arrow[r,"\eqref{eq:ft}"]& \Gamma_\cK \arrow[r,"\text{$\wtp$-th power}"]& U_{\cK,p}=Y,
\end{tikzcd}
\]

and $M=e_\ord V_{\U(2),\,\xi}$ (resp.  $e_\ord V'_{\U(2),\,\xi^{-1}}$) in (<ref>), on which we make $(a_1,a_2)\in A$ act through the right translation by $\varrho^{-1}_\fp\begin{pmatrix}1\\&a_1\end{pmatrix}$ (resp.  $\varrho^{-1}_\fp\begin{pmatrix}a^{-1}_1\\&1\end{pmatrix}$) multiplied by the scalar $\xi^{-1}_{p\adic,\fp}(a_1)$ (resp. $\xi_{p\adic,\fp}(a_1)$). Then we get

\begin{equation}\label{eq:MU2'}
\Meas\left(U_{\cK,p},e_\ord V_{\U(2),\,\xi}\right)^\natural \quad \left(\text{resp. } \Meas\left(U_{\cK,p},e_\ord V'_{\U(2),\,\xi^{-1}}\right)^\natural\right).
\end{equation}

For $\tau_{p\adic}\in\Hom_\cont\left(\Gamma_\cK,\ol{\bQ}^\times_p\right)$, the evaluation at $\tau_{p\adic}|_{U_{\cK,p}}$ of an element in (<ref>) is a $p$-adic form on $\U(2)$ with nebentypus

\[
\left(\triv,(\xi\tau^\wtp)_{p\adic,\fp}\right)
\quad \left(\text{resp. } \left(\xi\tau^{\wtp})^{-1}_{p\adic,\fp},\triv\right)\right).
\]

We will construct auxiliary CM families

\[
\glsuseri{theta},\glsuserii{h}\in \Meas\left(U_{\cK,p},e_\ord V_{\U(2),\,\xi}\right)^\natural,
\quad \glsuserii{theta},\glsuseri{h}\in \Meas\left(U_{\cK,p},e_\ord V'_{\U(2),\,\xi^{-1}}\right)^{\natural},
\]

in <ref> and use them to study the non-degenerate Fourier–Jacobi coefficients of the Klingen Eisenstein family (which are elements in (<ref>) and gives rise to measures on $U_{\cK,p}$ by the map (<ref>).)

§.§ The $p$-adic family of Siegel Eisenstein series on $\GU(3,3)$

We normalize the Siegel Eisenstein series $E^\Sieg(\bdot\,;f(s,\xi_0\tau_0)\big|_{s=\frac{k-3}{2}}$ (with $f(s,\xi_0\tau_0)$ chosen as in <ref>) as

\begin{equation}\label{eq:normalization}
\gls{Enorl}=\left(\frac{2^{-6}(-2\pi i)^{3k}}{\pi^3\prod_{j=0}^2\Gamma(k-j)}\right)^{-1}d^{\,\Sigma\cup\{\infty,p,\ell,\ell'\}}_3(s,\tau_0)\cdot E^\Sieg(\bdot\,;f(s,\xi_0\tau_0)\big|_{s=\frac{k-3}{2}},
\end{equation}

\begin{equation}\label{eq:d3}
\begin{aligned}
d^{\,\Sigma\cup\{\infty,p,\ell,\ell'\}}_3(s,\xi_0\tau_0)&=\prod_{v\notin \Sigma
\cup\{p,\ell,\ell'\}} d_{3,v}(s,\xi_0\tau_0), \\
\gls{d3v}&=\prod_{j=1}^3 L_v\left(2s+j,\xi^\bQ_0\tau^\bQ_{0}\eta^{3-j}_{\cK/\bQ}\right).
\end{aligned}
\end{equation}

There exists a $p$-adic measure $\gls{bESieg}\in\Meas\left(\Gamma_\cK,V_{\GU(3,3)}\right)$ such that

\[
\bfE^\Sieg(\tau_{p\adic})=E^\Sieg_{\xi\tau}
\]

for all algebraic Hecke character $\tau:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ such that its $p$-adic avatar $\tau_{p\adic}$ factors through $\Gamma_\cK$ and $\xi\tau$ has $\infty$-type $\left(0,k\right)$ with $k\geq 6$ even.

The measure is constructed by interpolating the $q$-expansions of the $E^\Sieg_{\xi\tau}$'s. First, we record the formulas for the Fourier coefficients in <cit.>. Given $\beta\in\Her_3(\cK)$ and $g\in\GU(3,3)(\bA_\bQ)$, let

\[
\]

(where $\be_v$ is the additive character in (<ref>)), and

\[
\]

Because the sections at $v\in\Sigma\cup\{\ell,\ell'\}$ are chosen to be supported on the big cell, for $g\in Q_{\GU(3,3)}(\bA_\bQ)$, we have

\[
E^\Sieg_{\beta}\big(g;f(s,\xi_0\tau_0)\big)=\prod_v W_{\beta,v}\big(g_v,f_v(s,\xi_0\tau_0)\big).
\]

For our choice of $f_v(s,\xi_0\tau_0)$ in <ref>, the formulas for $W_{\beta,v}\big(g_v,f_v(s,\xi_0\tau_0)\big)$ are as follows.

–The archimedean place.

\begin{align*}
\frac{2^{-6}(-2\pi i)^{3k}}{\pi^3\prod_{j=0}^2\Gamma(k-j)}(\det\bar{y})^k \be_{\bR}\left(\Tr\,\beta(x+i y\ltrans{\bar{y}})\right)\cdot (\det\beta)^{k-3}, &\beta\in\Her_3(\cK)_{>0}\\
0, &\text{otherwise}.\end{array}\right.
\end{align*}

– $v\notin\Sigma\cup\{p,\ell,\ell'\}$. For $\det\beta\neq 0$,

\begin{align*}
\left.W_{\beta,v}\left(\diag(A,D);f_v(s,\xi_0\tau_0)\right)\right|_{s=\frac{k}{2}}
&=\xi_{0,v}\tau_{0,v}(\det D)|\det AD^{-1}|^{-\frac{k}{2}+3}_{\bQ_v}\cdot  d_{3,v}(s,\xi_0\tau_{0})^{-1}\\
&\quad\times \mathds{1}_{\Her_3(\cO_{\cK,v})^*}(D^{-1}\beta A)\cdot h_{v,D^{-1}\beta A}\left(\xi^\bQ_{0,v}\tau^\bQ_{0,v}(q_v)q_v^{-2s-3}\right),
\end{align*}

where $h_{v,D^{-1}\beta A}\in\bZ[X]$ is a monic polynomial depending only on $v$ and $D^{-1}\beta A$, and is the constant polynomial when
when $D^{-1}\beta A$ belongs to $\GL_3(\cO_{\cK,v})$.

– $v\in\Sigma\cup\{\ell,\ell'\}$.

\begin{align*}
\left.W_{\beta,v}\left(\diag(A,D);f_v(s,\xi_0\tau_0)\right)\right|_{s=\frac{k-3}{2}}
&=|D_{\cK/\bQ}|^{3/2}_v \cdot \xi_{0,v}\tau_{0,v}(\det D)|\det AD^{-1}|^{-\frac{k}{2}+3}_{\bQ_v}\\
&\quad\times\mathds{1}_{\Her_3(\cO_{\cK,v})^*}(D^{-1}\beta A)\,\be_v\left(\frac{(D^{-1}\beta A)_{22}+(D^{-1}\beta A)_{33}}{q^{c_v}_v}\right).
\end{align*}

– The place $p$. An easy computation shows that

\begin{align*}
\left.W_{\beta,p}\left(\bid_6;f^{\text{big-cell}}_p(-s,\xi^{-c}_0\tau^{-c}_0)\right)\right|_{s=\frac{k-3}{2}}
\end{align*}

The functional equation for $W_{\beta,p}$ in <cit.> implies that

\[
W_{\beta,p}\left(g;M_p\left(-s,\xi^{-c}_0\tau^{-c}_{0}\right)f^{\text{big-cell}}_p(-s,\xi^{-1}_0\tau^{-1}_0)\right)=c_{3,p}(-s,\xi^{-c}_0\tau^{-c}_{0},\beta)\cdot W_{\beta,p}\left(g;f^{\text{big-cell}}_p(-s,\xi^{-c}_0\tau^{-c}_0)\right)
\]

with $c_{3,p}(-s,\xi^{-c}_0\tau^{-c}_{0},\beta)$ defined as in (<ref>). It follows that

\begin{align*}
\left.W_{\beta,p}\big(\bid;f_p(s,\xi_0\tau_0)\big)\right|_{s=\frac{k-3}{2}}=c_{\beta,p}\cdot(\xi^\bQ_0\tau^\bQ_0)_p(\det\beta)|\det\beta|^{2s}_{\bQ_p}\cdot \alpha_{\xi\tau,p}(\beta).
\end{align*}

Denote by $E^{\Sieg}_{\xi\tau,\beta}\big(\diag(A,D)\big)$ the $\beta$-th coefficient in the ($p$-adic) $q$-expansion of $E^\Sieg_{\xi\tau}$ at the ($0$-dimensional) cusp indexed by  $\diag(A,D)\in\GU(3,3)(\bA^p_{\cK,f})$. From the above formulas for the local Fourier coefficients at the archimedean place, we see that $E^{\Sieg}_{\xi\tau,\beta}\big(\diag(A,D)\big)$ is nonzero only for $\beta$ inside

\[
\fS_3=\left\{\beta\in\Her_3(\cO_\cK)_{>0}:\varrho_{\fp}(\beta_{21})\in\bZ^\times_p,\,\begin{pmatrix}\varrho_\fp(\beta_{21})&\varrho_\fp(\beta_{22})\\\varrho_\fp(\beta_{31})&\varrho_\fp(\beta_{32})\end{pmatrix}\in\GL_2(\bZ_p)\right\},
\]

and for $\beta\in\fS_3$ we have

\begin{equation}\label{eq:Ebeta}
\begin{aligned}
\text{a constant independent of $\tau$}\\
&\quad\times (\xi\tau)_{p\adic}(\det D)
\prod_{v\notin\Sigma\cup\{p,\ell,\ell'\}} h_{v,\ltrans{\bar{A}}\beta A}\left((\xi\tau)^\bQ_{p\adic,v}(q_v)\right),\quad \\
&\quad\times (\xi\tau)_{p\adic,\fp}\left(\det\begin{pmatrix}\beta_{21}&\beta_{22}\\\beta_{31}&\beta_{32}\end{pmatrix}^{-1}\right)
\end{aligned}
\end{equation}

(Note that thanks to the condition that $\xi\tau$ has $\infty$-type $(0,k)$, we have $(\xi\tau)_{p\adic,\fp}=(\xi\tau)_\fp$ and $(\xi\tau)_{p\adic}|_{\cO^\times_{\cK,\fp}}=\xi_0\tau_0|_{\cO^\times_{K,\fp}}$.)

The factors on the right hand side of (<ref>) are interpolated by $\delta$-measures in $\Meas\left(\Gamma_\cK,\cO_L\right)$. The convolution of those $\delta$-measures gives an element in $\Meas\left(\Gamma_\cK,\cO_L\right)$ which interpolates the Fourier coefficient $E^{\Sieg}_{\xi\tau,\beta}\big(\diag(A,D)\big)$. (See (<ref>)(<ref>)(<ref>) for the definitions of $\delta$-measures and the convolution of $p$-adic measures.) We see that there exists a $p$-adic measure

\[
\bfE^\Sieg\in \Meas\Big(\Gamma_\cK,\bigoplus\limits_{\text{$p$-adic cusps}}\cO_L\llb\fS_3\rrb \Big)
\]

that interpolates the $p$-adic $q$-expansions of $E^\Sieg_{\xi\tau}$. By the $q$-expansion principle and the fact that the $E^\Sieg_{\xi\tau}$'s are automorphic forms on $\GU(3,3)$, we deduce that $\bfE^\Sieg$ belongs to $\Meas\left(\Gamma_\cK,V_{\GU(3,3)}\right)$. (Here we view the space of $p$-adic measures valued in $V_{\GU(3,3)}$ as a subspace of the space of $p$-adic measures valued in copies of $\cO_L\llb\fS\rrb$ via the embedding of $V_{\GU(3,3)}$  into the $\cO_L$-module of $q$-expansions.)

§.§ The semi-ordinary family of Klingen Eisenstein series

Let $E^\Sieg_{\xi\tau}\big|_{\GU(3,1)\times\GU(2)}$ be the automorphic form obtained by the composition of the restriction from $\GU(3,3)$ to $\GU(3,1)\times_{\bG_m}\GU(2)$ via the embedding (<ref>) and the extension by zero from $\GU(3,1)\times_{\bG_m}\GU(2)$ to $\GU(3,1)\times\GU(2)$. Define the twsist $E^\Sieg_{\xi\tau}\big|^\star_{\GU(3,1)\times\GU(2)}$ as

\[
E^\Sieg_{\xi\tau}\big|^\star_{\GU(3,1)\times\GU(2)}(g_1,g_2)=E^\Sieg_{\xi\tau}\big|_{\GU(3,1)\times\GU(2)}(g_1,g_2)\cdot\xi^{-1}_{p\adic}\tau^{-1}_{p\adic}(\det g_2).
\]

Then by our choice of the section $f_p(s,\xi_0\tau_0)$, we see that

\[
E^\Sieg_{\xi\tau}\big|^\star_{\GU(3,1)\times\GU(2)}\in V_{\GU(3,1),\,\xi}\otimes M_{\GU(2)}\left(K^p_fK_{p,0};\hat{\cO}^\ur_L\right)
\]

with $\gls{Kp0}=\left\{g\in\GU(2)(\bZ_p):\varrho_\fp(g)=\begin{pmatrix}\ast&\ast\\p\ast&\ast\end{pmatrix}\right\}$. (Note that $K_{p,0}$ equals the $K'_{p,g}$ in (<ref>). See (<ref>)  for the definition of $M_{\GU(2)}\left(K^p_fK_{p,0};\hat{\cO}^\ur_L\right)$.)

Therefore, by applying a twist of $(\xi\tau)^{-1}_{p\adic}\circ\,\det$  to the family $\bfE^\Sieg\big|_{\GU(3,1)\times\GU(2)}$ on the second factor, we obtain the $p$-adic family

\begin{equation}\label{eq:resE}
\bfE^\Sieg\big|^\star_{\GU(3,1)\times\GU(2)}\in \Meas\left(\Gamma_\cK, V_{\GU(3,1),\,\xi}\right)^\natural \wh{\otimes} \,M_{\GU(2)}\left(K^p_fK_{p,0};\hat{\cO}^\ur_L\right)
\end{equation}

interpolating $E^\Sieg_{\xi\tau}\big|^\star_{\GU(3,1)\times\GU(2)}$ when $\tau$ varies. (To see that it belongs to the $^\natural$-subspace, one just needs to check the nebentypus.)

Attached to a $\varphi$ in the space (<ref>) is a linear functional

\begin{align*}
\left<\bdot,\varphi\right>: M_{\GU(2)}\left(K^p_fK_{p,0};\hat{\cO}^{\ur}_L\right)&\lra \hat{\cO}^\ur_L\\
\varphi'&\longmapsto \left<\varphi',\varphi\right> =\int_{\GU(2)(\bQ)\backslash \GU(2)(\bA_{\bQ,f})} \varphi'(g)\varphi(g)\,dg,
\end{align*}

where the integral is understood as a sum over the finite set $\GU(2)(\bQ)\backslash \GU(2)(\bA_{\bQ,f})/K^p_fK_{p,0}$. Applying the linear functional $\left<\bdot,\varphi\right>$ to (<ref>) gives a family

\begin{equation}\label{eq:KF}
\gls{bEKling}= \left<  \bfE^{\Sieg}|^\star_{\GU(3,1)\times\GU(2)},\,\varphi\right>\in \Meas\left(\Gamma_\cK, V_{\GU(3,1),\,\xi}\wh{\otimes}\,\hat{\cO}^\ur_L\right)^\natural.
\end{equation}

For  $\tau_{p\adic}$ satisfying the conditions in Theorem <ref>, the evaluation of the family $\bfE^\Kling_\varphi$ at $\tau_{p\adic}$ is a Klingen Eisenstein series on $\GU(3,1)$ inducing $\xi_0\tau_0|\cdot|^{\frac{k-3}{2}}\boxtimes\pi^{\GU(2)}$, more precisely, the automorphic form

\begin{equation}\label{eq:Kling-int}
g\longmapsto \left.\int_{\U(2)(\bQ)\backslash \U(2)(\bA_{\bQ})/\U(2)(\bR)} E^\Sieg\big(\imath(g,g_2);f(s,\xi_0\tau_0)\big)\varphi(g_2)(\xi_0\tau_0)^{-1}(\det g_2)\,dg_2\,\right|_{s=\frac{k-3}{2}}
\end{equation}

normalized by the factor in (<ref>).

Moreover, $\bfE^\Kling_\varphi\in \Meas\left(\Gamma_\cK,e_\so V_{\GU(3,1),\,\xi}\wh{\otimes}\,\hat{\cO}^\ur_L\right)^\natural$,   i.e. is a semi-ordinary family.

From the construction of the family $\bfE^\Kling_\varphi$, it is easy to see that its evaluation $\tau_{p\adic}$ satisfying the conditions in Theorem <ref> equals the form given in (<ref>) normalized by the factor in (<ref>).  By the doubling method formula (Theorem <ref>), we know that (<ref>) is a Klingen Eisenstein series inducing $\xi_0\tau_0|\cdot|^{\frac{k-3}{2}}\boxtimes\pi^{\GU(2)}$.

In order to show that  $\bfE^\Kling_\varphi$ is a semi-ordinary family, for all $\tau$ satisfying the conditions in Theorem <ref> and $(\xi\tau)_\fp$ ramified, we look at the action of $\bU_p$-operators on the form (<ref>), which has weight $(0,0,0;k)$. Given integers $m,m',m''\geq 0$, by the formula for adelic $\bU_p$-operators in (<ref>) and our choice of the local section at $p$ in <ref>, the action of $(U^+_{p,2})^{m}(U^+_{p,3})^{m'}(U^-_{p,1})^{m''}$ on the form (<ref>) can be computed by considering

\begin{equation}\label{eq:Eso-1}
p^{2m+km''}\int_{N(\bZ_p)} f^{\text{big-cell}}_p\left(\frac{3-k}{2},(\xi_0\tau_0)^{-c}\right)\left(gu\begin{psm}p^{m+m'}\\&p^{m+m'}\\&&p^{m'}\\&&&p^{-m''}\\&&&&1\\&&&&&1\end{psm}\right)\,du,
\end{equation}

where we identify $\U(3,3)(\bQ_p)$ with $\GL_6(\bQ_p)$ via $\varrho_\fp:\cK\otimes_\bQ\bQ_p\ra\bQ_p$. Writing $g=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ in $3\times 3$ blocks, we have

\begin{equation}\label{eq:Eso-2}
\begin{aligned}
\eqref{eq:Eso-1}=&\, p^{2m+km''}\cdot \left((\xi\tau)_\fp(p)p^{3}\right)^{2m+3m'}\left((\xi\tau)_{\bar{\fp}}(p)p^{3}\right)^{m''}\\
&\times p^{-4m-3m'-3m''}\sum_{u_1,u_2\in \bZ/p^m\bZ}\sum_{\substack{{v_1,v_2\in\bZ/p^{m+m'+m''}}\\ v_3\in\bZ/p^{m'+m''}\bZ}}\\
\end{aligned}
\end{equation}

A direct computation by using the formula (<ref>) shows that when $(\xi\tau)_\fp$ is ramified

\begin{equation}\label{eq:Eso-3}
\begin{aligned}
\end{aligned}
\end{equation}

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

\begin{align*}
\eqref{eq:Eso-1}
=&\,p^{6m+6m'}(\xi\tau)_\fp(p)^{2m+3m'}\left((\xi\tau)_{\bar{\fp}}(p)p^k\right)^{m''}\cdot \left(\xi\tau)_{\bar{\fp}}(p)p^3\right)^{-2m-2m'}\\
&\times f^{\text{big-cell}}_p\left(\frac{3-k}{2},(\xi_0\tau_0)^{-c}\right)\left(g\begin{psm}\bid_4\\&p^{-m-m'}\cdot\bid_2\end{psm}\right),\\
=&\,(\xi\tau)_\fp(p)^{m'}\left((\xi\tau)_{\bar{\fp}}(p)p^k\right)^{m''}\cdot (\xi\tau)_{\bar{\fp}}(\xi\tau)^{-1}_{{\fp}}(p)^{-2m-2m'}\\
&\times f^{\text{big-cell}}_p\left(\frac{3-k}{2},(\xi_0\tau_0)^{-c}\right)\left(g\begin{psm}\bid_4\\&p^{-m-m'}\cdot\bid_2\end{psm}\right)
\end{align*}

and the corresponding Siegel Eisenstein series paired with $\varphi\cdot (\xi_0\tau_0)^{-1}\circ\det$ over $\U(2)$ gives

\begin{align*}
(\xi\tau)_\fp(p)^{m'}\left((\xi\tau)_{\bar{\fp}}(p)p^k\right)^{m''}\cdot \text{the automorphic form given in \eqref{eq:Kling-int}}.
\end{align*}

Here we use the condition that $\varphi$ has trivial central character. Both $(\xi\tau)_\fp(p)$ and $(\xi\tau)_{\bar{\fp}}(p)p^k$ are $p$-adic units, so the form (<ref>) is an eigenvector for the action of  $(U^+_{p,2})^{m}(U^+_{p,3})^{m'}(U^-_{p,1})^{m''}$ with eigenvalue a $p$-adic unit. This shows that the evaluations of the Klingen Eisenstein family $\bfE^\Kling_\varphi$ at all $\tau$ satisfying the conditions in Theorem <ref> with $(\xi\tau)_\fp$ ramified are semi-ordinary. Hence,  $\bfE^\Kling_\varphi$ is semi-ordinary.

§ THE DEGENERATE FOURIER–JACOBI COEFFICIENTS OF THE KLINGEN FAMILY

This section is devoted to proving Theorem <ref>, which states that the degenerate Fourier–Jacobi coefficients of $\bfE^\Kling_\varphi$ are divisible by the $p$-adic $L$-function attached to $\mr{BC}(\pi)$.

§.§ The divisibility of the degenerate Fourier–Jacobi coefficients by $p$-adic $L$-functions

Given a cusp label $\clabel\in C(K^p_f)$, we have the map

\[
\Phi_g: \Meas\left(\Gamma_\cK,e_\so V_{\GU(3,1),\,\xi}\wh{\otimes}\,\hat{\cO}^\ur_L\right)^\natural\lra \Meas\left(\Gamma_\cK,M_{\GU(2)}\left(K^p_{f,g}K_{p,0};\hat{\cO}^\ur_L\right)\right),
\]

(which corresponds to the map $\Phi_\clabel$ in the fundamental exact sequence in part (4) of Theorem <ref>). Here $K^p_{f,g}$ is an open subgroup of $\GU(2)(\hat{\bZ}^p)$ depending on the cusp label $g\in C(K^p_f)$. Applying it to our Klingen Eisenstein family $\bfE^\Kling_\varphi$, we get

\[
\Phi_{g}\left(\bfE^\Kling_\varphi\right)\in \Meas\left(\Gamma_\cK,M_{\GU(2)}\left(K^p_{f,g}K_{p,0};\hat{\cO}^\ur_L\right)\right)
\]

Evaluating it at $g'\in \GU(2)(\bA_{\bQ,f})$ gives

\[
\Phi_{g}\left(\bfE^\Kling_\varphi\right)(g')\in\Meas\left(\Gamma_\cK,\hat{\cO}^\ur_L\right)\simeq \hat{\cO}^\ur_L\llb \Gamma_\cK\rrb.
\]

By the work of Kubota–Leopoldt on  $p$-adic $L$-functions for Dirichlet characters, there exists a (unique) $p$-adic $L$-function $\gls{Lxi}\in \hat{\cO}^\ur_L\llb \Gamma_\cK\rrb$ satisfying the interpolation property:

\begin{align}
\label{eq:pL-xi} \cL^{\Sigma\,\cup\{\ell,\ell'\}}_{\xi,\bQ}(\tau_{p\adic})
=&\,\frac{\Gamma(k-2)}{(2\pi i)^{k-2}}
\cdot \gamma_p\left(3-k,(\xi^\bQ_0\tau^\bQ_{0})^{-1}\right) \cdot L^{\Sigma\,\cup\{\infty,p,\ell,\ell'\}}\left(k-2,\xi^\bQ_0\tau^\bQ_0\right)
\end{align}

for all algebraic Hecke character $\tau:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ such that $\tau_{p\adic}$ factors through $\Gamma_\cK$ and $\tau^{\bQ}_\infty=\sgn^k|\cdot|^k_{\bR}$.

By [6] (and the archimedean computation in [5]), there exists a (unique) $p$-adic $L$-function
$\gls{Lpipartial}\in \hat{\cO}^\ur_L\llb \Gamma_\cK\rrb$ satisfying the interpolation property: for all 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 $\left(k_1,k_2\right)$ with $k_1,k_2\in\bZ$, $k_1\leq 0$, $k_2\geq 2-k_1$,

\begin{equation}\label{eq:pL-pi}
\begin{aligned}
=&\,C^{\Sigma\,\cup\{\ell,\ell'\}}_{\pi,\xi}\cdot  \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^{\Sigma\,\cup\{\infty,p,\ell,\ell'\}}\left(\frac{k_1+k_2-1}{2},\mr{BC}(\pi)\times\xi_0\tau_0\right),
\end{aligned}
\end{equation}

with $C^{\Sigma\,\cup\{\ell,\ell'\}}_{\pi,\xi}$ a nonzero number independent of $\tau_{p\adic}$.

Suppose that $\varphi$ belongs to the space (<ref>). Then for all cusp labels $g\in C(K^p_f)$ and $g'\in \GU(2)(\bA_{\bQ,f})$,  we have

\[
\left(\Phi_{g}\left(\bfE^\Kling_\varphi\right)(g')\right)\subset \left(\cL^{\Sigma\,\cup\{\ell,\ell'\}}_{\xi,\bQ}\cL^{\Sigma\,\cup\{\ell,\ell'\}}_{\pi,\cK,\xi}\right)
\]

in $\hat{\cO}^\ur_L\llb \Gamma_\cK\rrb$.

For $\tau$ as in Theorem <ref>, let $E^\Kling_{\varphi,\xi\tau}$ be the classical Klingen Eisenstein series whose corresponding $p$-adic form is $\bfE^\Kling_\varphi(\tau_{p\adic})$. The theorem is proved by computing $\Phi_{g}\left(\bfE^\Kling_\varphi\right)(\tau_{p\adic})$ with

\begin{align*}
g&=k_{\Sigma,\ell,\ell'}=\bigotimes\limits_{v\in\Sigma\,\cup\{\ell,\ell'\}}k_v, &k_v\in \U(3,1)(\bZ_v),
\end{align*}

which is essentially the $0$-th Fourier–Jacobi coefficient of the $E^\Kling_{\varphi,\xi\tau}$ at

\[
\]

with $w_{3,p}\in\U(3,1)(\bQ_p)$ defined by

\[
\varrho_{\fp}(w_{3,p})=\begin{pmatrix}&&1\\1\\&1\\&&&1\end{pmatrix}.
\]

Note that $w_{3,p}\in\U(3,1)(\bQ_p)$ is also our fixed element in $P^\flat_{\tt D}(\bZ_p)$ for obtaining the isomorphism (<ref>)).

By the description of $\bfE^\Kling_\varphi(\tau_{p\adic})$ in (<ref>) and the doubling method formula (Theorem <ref>), we have

\[
E^\Kling_{\varphi,\xi\tau}(x)=\begin{array}{ll}\text{normalization factor}\\\text{on the RHS of \eqref{eq:normalization}}\end{array}\cdot E^\Kling\big(x;F(f(s,\xi_0\tau_0),\varphi)\big)\,\Big|_{s=\frac{k-3}{2}}
\]

with $F(f(s,\xi_0\tau_0),\varphi)$ the section in $I_{P_{\GU(3,1)}}(s,\xi_0\tau_0)$ given by

\begin{align*}
F(f(s,\tau_0),\varphi)(x)&=\int_{\U(2)(\bA_\bQ)}f(s,\xi_0\tau_0)(\imath(x,g_1t))\varphi(g_1t)(\xi_0\tau_0)^{-1}(\det g_1t)\,dg_1\\
&=\int_{\U(2)(\bA_\bQ)}f(s,\xi_0\tau_0)\left(\imath\left(k_{\Sigma_f,\ell,\ell'}\,w_{3,p},g_1\right)\right)\varphi(g'g_1)(\xi_0\tau_0)^{-1}(\det g_1)\,dg_1,
\end{align*}

where $t$ is an element in $\GU(2)(\bA_\bQ)$ with $\nu(t)=\nu(x)$. What we need to compute is

\begin{equation}\label{eq:const1}
\int_{U_{P_{\GU(3,1)}}(\bQ)\big\backslash U_{P_{\GU(3,1)}}(\bA_\bQ)} E^\Kling\big(ux;F(f(s,\xi_0\tau_0),\varphi)\big)\,du.
\end{equation}

It follows from <cit.> that

\begin{align*}
\eqref{eq:const1}=F(f(s,\xi_0\tau_0),\varphi)(x)+M_{P_{\GU(3,1)}}(s,\xi_0\tau_0)F(f(s,\xi_0\tau_0),\varphi)(x).
\end{align*}

The archimedean computation in <cit.> shows that for our choice of $f_\infty(s,\xi_0\tau_0)$ in <ref> and $k\geq 6$,

\begin{equation}\label{eq:van-M}
\left.M_{P_{\GU(3,1)}}(s,\xi_0\tau_0)F(f(s,\xi_0\tau_0),\varphi)\right|_{s=\frac{k-3}{2}}=0.
\end{equation}

We reduce to compute $F(f(s,\xi_0\tau_0),\varphi)(x)$. Take arbitrary $\varphi'\in \bar{\pi}^{\GU(2)}$ spherical away from $\Sigma\,\cup\{\ell,\ell'\}$ and factorizable with respect to an isomorphism $\bar{\pi}^{\GU(2)}\simeq \bigotimes_v \bar{\pi}^{\GU(2)}_{v}$. Put

\begin{align*}
&\left<F(f(s,\xi_0\tau_0),\varphi)(\bdot\, k_{\Sigma_f,\ell,\ell'}\,w_{3,p}),\varphi'\right>\\
=&\, \int_{\GU(2)(\bQ)Z_{\GU(2)}(\bA_\bQ)\backslash\GU(2)(\bA_\bQ)}F(f(s,\xi_0\tau_0),\varphi)(g'\,k_{\Sigma,\ell,\ell'}\,w_{3,p})\,\varphi'(g')\,dg'.
\end{align*}

Then we have

\begin{align*}
\left<F(f(s,\xi_0\tau_0),\varphi)(\bdot\,k_{\Sigma,\ell,\ell'}\,w_{3,p}),\,\varphi'\right>=&\,\prod_{v\not\in\Sigma\,\cup\{p,\ell,\ell'\}}  Z_v\left(f_v(s,\xi_0\tau_0),\bid_4,\varphi_{v},\varphi'_v\right)\\
&\times \prod_{v\in\Sigma\,\cup\{\ell,\ell'\}}
\cdot Z_p\left(f_p(s,\xi_0\tau_0),w_{3,p},\varphi_{p},\varphi'_p\right),
\end{align*}

where for $g_v\in\GU(3,1)(\bQ_v)$,

\begin{equation}\label{eq:Zv}
\begin{aligned}
=&\,\int_{\U(2)(\bQ_v)}f_v(s,\xi_0\tau_0)(\imath(g_v,g_1))(\xi_0\tau_0)^{-1}_{v}(\det g_1)\left<\pi^{\GU(2)}_v(g_1)\varphi_{v},\varphi'_v\right>\,dg_1.
\end{aligned}
\end{equation}

The restriction of $f_v(s,\xi_0\tau_0)(\bdot\,\imath(g_v,\bid_2))$ to $\U(2,2)(\bQ_v)$ is a section in the degenerate principal series of $\U(2,2)(\bQ_v)$ inducing $\xi_0\tau_0|\cdot|^{s+\frac{1}{2}}_{\bA_\cK}$. Hence the integral (<ref>) is essentially a doubling zeta integral. We have the following formulas.

– The archimedean place.

\begin{align*}
\end{align*}

because $f_\infty(s,\xi_0\tau_0)(\imath(\bid_4,\bdot)$, $\varphi_{\infty}$, $\varphi'_\infty$ are all invariant under $\U(2)(\bR)$.

– $v\notin \Sigma\,\cup\{\ell,\ell',p\}$.

\begin{align*}
=d_{2,v}\left(s+\frac{1}{2},\xi_0\tau_0\right)^{-1}\cdot L_v\left(s+1,\mr{BC}(\pi)\times\xi_0\tau_{0}\right)
\end{align*}

\[
\gls{d2v}=\prod_{j=1}^2 L_v(2s+1+j,\xi^\bQ_0\tau^\bQ_{0}\eta^{2-j}_{\cK/\bQ}).
\]

This follows from the standard formula for unramified local doubling zeta integrals <cit.>.

– $v\in\Sigma\,\cup\{\ell,\ell'\}$. By <cit.> or an easy direct computation,  one can see that for $\varphi$ in the space (<ref>),

\[
Z_v\left(f_v(s,\xi_0\tau_0),k_v,\varphi_v,\varphi'_v\right)=0, \quad\text{if }k_v\notin P_{\U(3,1)}(\bZ_v)\begin{pmatrix}&&1\\&\bid_2\\-1\end{pmatrix} P_{\U(3,1)}(\bZ_v),
\]

and for $k_v$ such that $Z_v\left(f_v(s,\xi_0\tau_0),k_v,\varphi_v,\varphi'_v\right)\neq 0$,

\begin{align*}
\end{align*}

where $\fY_{c_v}$ is an open subgroup of $\{k'_v\in\U(2)(\bZ_v):k'_v\equiv \bid_2\mod q^{c_v}_v\}$, and $c_v$ is the positive integer in the definition of $f_v(s,\xi_0\tau_0)$ in (<ref>).

– The place $p$.

\begin{equation}\label{eq:Zp-formula}
\begin{aligned}
\gamma_p\left(-s,\pi_p\times(\xi_0\tau_0)^{-1}_{\bar{\fp}}\right)\left<\varphi_{p},\varphi'_p\right>
\end{aligned}
\end{equation}

See <ref> for the computation. Note that we only need the value at $w_{3,p}\in\GU(3,1)(\bQ_p)$ rather than a general $k_p\in \GU(3,1)(\bZ_p)$. This significantly simplifies the computation. The value at a general $k_p\in \GU(3,1)(\bZ_p)$ can be difficult to compute since intertwining operators are not easy to compute completely.

Combing the local formulas, since $\varphi'$ is arbitrary, we see that for $\tau$ as in Theorem <ref>,  $F(f(s,\xi_0\tau_0),\varphi)(\bdot\, k_{\Sigma,\ell,\ell'}\,w_{3,p})\big|_{s=\frac{k-3}{2}}$ is either $0$ or equals

\begin{align*}
\cdot C_{\Sigma\cup\{\ell,\ell'\}}\cdot  \gamma_p\left(-2s,(\xi^\bQ_0\tau^\bQ_{0})^{-1}\right)
&\times \gamma_p\left(-s,\pi_p\times(\xi_0\tau_0)^{-1}_{\bar{\fp}}\right)
L^{\Sigma\,\cup\{\infty,\ell,\ell',p\}}\left(s+1,\mr{BC}(\pi)\times\xi_0\tau_0\right)\Big|_{s=\frac{k-3}{2}}\cdot \varphi
\end{align*}

for some $C_{\Sigma\cup\{\ell,\ell'\}}\in \cO_L$ independent of $\tau$. Combining this formula with the normalization  (<ref>), we see that
$\Phi_{g}\left(\bfE^\Kling_\varphi\right)(g')$, for all $g'\in\GU(2)(\bA_{\bQ,f})$, is divisible by an element in $\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb$ whose value equals that of $\cL^{\Sigma\,\cup\{\ell,\ell'\}}_{\pi,\xi,\cK}\cL^{\Sigma\,\cup\{\ell,\ell'\}}_{\xi,\bQ}$ at  all $\tau_{p\adic}$ as in Theorem <ref>. Evaluations at those $\tau_{p\adic}$'s  already uniquely determine elements in  $\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb$. Therefore, it follows that $\Phi_{g}\left(\bfE^\Kling_\varphi\right)(g')$ is divisible by $\cL^{\Sigma\,\cup\{\ell,\ell'\}}_{\xi,\bQ}\cL^{\Sigma\,\cup\{\ell,\ell'\}}_{\pi,\cK,\xi}$.

§.§ The computation of local zeta integrals at $p$

The ramification conditions in <cit.> on $\xi_{0,p}\tau_{0,p}$ and the characters from which $\pi_p$ is induced are not satisfied in our case here because our $\pi_p$ is unramified. Those conditions are used  loc.cit to simplify the computation involving the intertwining operators. Here, we use the functional equation for doubling zeta integrals to handle the intertwining operator and compute $Z_p\left(f_p(s,\xi_0\tau_0),w_{3,p},\varphi_p,\varphi'_p\right)$.

Inside the degenerated principal series on $\GU(2,2)(\bQ_p)$ inducing the character $(\xi_0\tau_0)^{-c}_{p}|\cdot|^{-s}$, we define the section $f_p^{\U(2,2)}(-s,(\xi_0\tau_0)^{-c})$  as

\begin{align*}
= |\det C\ltrans{\bar{C}}|^{s-1}_p(\xi_0\tau_0)_p\left(\det C\right)\cdot \cF^{-1}\alpha_{\xi\tau,p}\begin{pmatrix}0&C^{-1}D\\0&0\end{pmatrix},
\end{align*}

where the Schwartz function $\alpha_{\xi\tau,p}$ is defined in (<ref>). Then we have

\begin{equation}\label{eq:fU22}
\begin{aligned}
\begin{psm}&&-1\\&\bid_2\\1\\&&&1\end{psm}
\begin{psm}1&-y_2&y_1&\\&\bid_2&\\&&\ltrans{\bar{y}}_2&1\end{psm}
\jmath(g_2)\Upsilon^{-1}_p\imath(w_{3,p},\bid_2)\Upsilon_p
\right)\,dy_1dy_2\\
\end{aligned}
\end{equation}

where $\jmath:\U(2,2)\ra\U(3,3)$ is defined as

\[
\jmath\begin{pmatrix}A&B\\C&D\end{pmatrix}=\begin{pmatrix}1\\&A&&B\\&&1\\&C&&D\end{pmatrix}.
\]

To see (<ref>), writing $g_2=\begin{pmatrix}A&B\\C&D\end{pmatrix}$, we have

\begin{align*}
\begin{psm}1&-y_2&y_1&\\&\bid_2&\\&&\ltrans{\bar{y}}_2&1\end{psm}
\jmath(g_2)\Upsilon^{-1}_p\imath(w_{3,p},\bid_2)\Upsilon_p\right)\\
\end{align*}

\begin{align*}
\varrho_\fp\left(\begin{pmatrix}-y_2A&1\\C&0\end{pmatrix}\begin{pmatrix}^{-1}y_1&-y_2B\\\ltrans{\bar{y}}_2&D\end{pmatrix}\right)
\end{align*}

It follows that the integrand of the left hand side of (<ref>) equals

\begin{align*}
f^{\U(2,2)}_p\left(-s+\frac{1}{2},(\xi_0\tau_0)^{-c}\right)(g_2)\cdot\mathds{1}_{\bZ_p}(y_1)\cdot \mathds{1}_{\cO^2_{\cK,p}}(y_2\ltrans{\bar{C}}^{-1}),
\end{align*}

\[
\text{LHS of \eqref{eq:fU22}}=f^{\U(2,2)}_p\left(-s+\frac{1}{2},(\xi_0\tau_0)^{-c}\right)(g_2)\cdot\det|C\ltrans{\bar{C}}|_p=f^{\U(2,2)}_p\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right)(g_2).
\]

Put $\cS_0=\begin{pmatrix}1&-\frac{\zeta_0}{2}\\-1&-\frac{\zeta_0}{2}\end{pmatrix}$ and $\Upsilon_{0,p}$ to be the element in $\U(2,2)(\bQ_p)$ such that $\varrho_\fp(\Upsilon_{0,p})=\cS^{-1}_0$.  Set

\begin{align}
\label{eq:imath0}\imath_0:\U(2)\times\U(2)&\lra \U(2,2), &\imath_0(g_1,g'_1)=\cS^{-1}_0\begin{pmatrix}g_1\\&g'_1\end{pmatrix}.
\end{align}

By the definition of the intertwining operator $M_p(-s,\xi\tau^{-1}_0(\xi_0\tau_0)^{-c})$ (in (<ref>)) and (<ref>),

\begin{align*}
=&\int_{\begin{psm}y_1&y_2\\ \ltrans{\bar{y}}_2&y_4\end{psm}\in\Her_3(\cK_p)}
\begin{psm}&&-1\\&\bid_2\\1\\&&&1\end{psm}
\begin{psm}1&-y_2&y_1&\\&\bid_2&\\&&\ltrans{\bar{y}}_2&1\end{psm}\right.\\
\imath(w_{3,p},g_1)\Upsilon_p
\right)\,dudy_4\\
=&\,M^{\U(2,2)}_p\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right) f^{\U(2,2)}_p(\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right)\left(\imath_0(1,g_1)\Upsilon_{0,p}\right),
\end{align*}

\begin{align*}
&\times M^{\U(2,2)}_p\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right) f^{\U(2,2)}_p(\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right)\left(\imath_0(1,g_1)\Upsilon_{0,p}\right).
\end{align*}

By the functional equation of the local doubling zeta integral <cit.>,

\begin{align*}
&\int_{\U(2)(\bQ_p)}M^{\U(2,2)}_p\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right) f^{\U(2,2)}_p\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right)\left(\imath_0(1,g_1)\Upsilon_{0,p}\right)\\
&\hspace{15em}\times(\xi_0\tau_0)^c_{p}(\det g_1)\left<\pi^{\GU(2)}_p(g_1)\varphi_{p},\varphi'_p\right>\,dg_1\\
=&\,c_{2,p}\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c},\begin{psm}&\bid_2\\\bid_2\end{psm}\right)\gamma_p\left(-s,\mr{BC}(\pi)\times (\xi_0\tau_0)^{-c}\right)\\
&\times\int_{\U(2)(\bQ_p)} f^{\U(2,2)}_p\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right)\left(\imath_0(1,g_1)\Upsilon_{0,p}\right)(\xi_0\tau_0)^{c}_{p}(\det g_1)\left<\pi^{\GU(2)}_p(g_1)\varphi_{p},\varphi'_p\right>\,dg_1,
\end{align*}

with the factor

\[
\]

as defined in (<ref>). Therefore,

\begin{equation}\label{eq:intU22}
\begin{aligned}
=&\,\gamma_p\left(-2s,(\xi^\bQ_0\tau^\bQ_{0})^{-1}\right)\gamma_p\left(-s,\mr{BC}(\pi)\times (\xi_0\tau_0)^{-c}\right)\\
&\hspace{-11em}\times\int_{\U(2)(\bQ_p)} f^{\U(2,2)}_p\left(-s-\frac{1}{2},(\xi_0\tau_0)^{-c}\right)\left(\imath_0(1,g_1)\Upsilon_{0,p}\right)(\xi_0\tau_0)^{c}_{p}(\det g_1)\left<\pi^{\GU(2)}_p(g_1)\varphi_{p},\varphi'_p\right>\,dg_1.
\end{aligned}
\end{equation}

By noting that

\[
\]

and using the formula (<ref>) for $\cF^{-1}\alpha_{\xi\tau,p}$ and that $\varphi_p$ is spherical, an easy computation shows that

\begin{equation}\label{eq:Zp1}
\text{the integral on the RHS of \eqref{eq:intU22}}
=p^{-2ts-2t}\fg\left((\xi_0\tau_0)^{-1}_{\fp}\right)^2(\xi_0\tau_0)_\fp(p)^{2t}\cdot \left<\varphi_{p},\varphi'_p\right>,
\end{equation}

where $p^t$, $t\geq 1$, is the conductor of $(\xi_0\tau_0)_\fp$. Since $\pi_p$ is unramified, we have

\begin{equation}\label{eq:Zp2}
\end{equation}

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

\begin{align*}
\eqref{eq:intU22}&=\gamma_p\left(-2s,(\xi^\bQ_0\tau^\bQ_{0})^{-1}\right)\gamma_p\left(-s,\mr{BC}(\pi)\times (\xi_0\tau_0)^{-c}\right)
\gamma_p\left(s+1,\pi_p\times(\xi_0\tau_0)_\fp\right)\cdot \left<\varphi_{p},\varphi'_p\right>\\
\gamma_p\left(-s,\pi_p\times(\xi_0\tau_0)^{-1}_{\bar{\fp}}\right)\cdot \left<\varphi_{p},\varphi'_p\right>.
\end{align*}

§ THE NON-DEGENERATE FOURIER–JACOBI COEFFICIENTS OF THE KLINGEN FAMILY

The map (<ref>) of taking the $\beta$-th Fourier–jacobi coefficient along the boundary stratum indexed by the cusp label $\bid_4\in C(K^p_fK^1_{p,n})_\ord$ induces

\[
\FJ_{\beta}:\Meas\left(\Gamma_\cK, V_{\GU(3,1)}\right)\lra \Meas\left(\Gamma_\cK,V^{J,\beta}_{\GU(2)}\right).
\]

For $u\in \bigotimes_{v\in\Sigma_{\ns}}\U(1)(\bQ_v)$, let

\begin{equation}\label{eq:E-betau}
\gls{Ebeta}=\FJ_{\beta} \left(\begin{psm}u\\&\bid_2\\&&u\end{psm}\bfE^\Kling_\varphi\right)\in \Meas\left(\Gamma_\cK,V^{J,\beta}_{\GU(2)}\wh{\otimes}\,\hat{\cO}^\ur_L\right)
\end{equation}

(The group $\GU(3,1)(\bA^p_f)$ acts on the Igusa tower and acts on $V_{\GU(3,1)}$, and $\begin{psm}u\\&\bid_2\\&&u\end{psm}\bfE^\Kling_\varphi$ denotes the action of $\begin{psm}u\\&\bid_2\\&&u\end{psm}$ on $\bfE^\Kling_\varphi$.) The goal of this section is to prove Proposition <ref> on the nonvanishing properties of the $\bfE^\Kling_{\varphi,\beta,u}$'s.

In <ref>-<ref>, we briefly recall some basics we will use about Weil representations, theta series, Jacobi forms and $p$-adic Petersson inner products. In <ref> and <ref>, we unfold the Siegel Eisenstein series on $\GU(3,3)$ to compute the non-degenerate Fourier–Jacobi coefficients of the Klingen Eisenstein series on $\GU(3,1)$, and sketch our strategy for relating them to certain $L$-values, for which we can apply mod $p$ nonvanishing results. Following that strategy, we need to choose an auxiliary Jacobi form on $\U(2)$ and two CM families on $\U(2)$. <ref>-<ref> are about constructing the auxiliary Jacobi form and CM families and some other technical preparations. The desired nonvanishing property is proved in <ref>.

§.§ The Schrödinger model of Weil representation and the intertwining maps for different polarizations

The non-degenerate terms in the Fourier–Jacobi expansion of an automorphic form on $\GU(3,1)(\bA)$ are Jacobi forms on the Jacobi group associated $\GU(2)$.

Let $V$ (resp. $V^-$) be the two dimensional skew-Hermitian space $(\cK^2,\zeta_0)$ (resp. $(\cK^2,-\zeta_0)$).  We write elements in $V,V^-$ as row vectors and fix the basis $v_1=(1,0),v_2=(0,1)$ for $V$ (resp. $v^-_1=(1,0),v^-_2=(0,1)$ for $V^-$). With respect to the fixed basis, the unitary groups $\U(V)$ and $\U(V^-)$ are naturally identified with our $\U(2)$ (defined in (<ref>)).

We fix the basis $v_1,v_2,v^-_1,v^-_2$ for $V\oplus V^-$ and identify $\U(V\oplus V^-)$ with the unitary group whose $R$-points are

\[
\left\{g\in\GL_4(R\otimes_{\bZ}\cO_\cK):\ltrans{\ol{g}}\begin{pmatrix}\zeta_0\\&-\zeta_0\end{pmatrix}g=\begin{pmatrix}\zeta_0\\&-\zeta_0\end{pmatrix}\right\}.
\]

Define the unitary group $\U(2,2)$ as

\begin{align*}
\U(2,2)(R)&=\left\{g\in\GL_4(R\otimes_{\bZ}\cO_\cK):\ltrans{\ol{g}}\begin{pmatrix}&\bid_2\\-\bid_2\end{pmatrix}g=\begin{pmatrix}&\bid_2\\-\bid_2\end{pmatrix}\right\}.
\end{align*}

We fix the isomorphism

\begin{align*}
\U(V\oplus V^-)&\lra \U(2,2), &g&\longmapsto \begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}^{-1}g \begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix},
\end{align*}

which induces the embedding

\begin{align*}
\U(2)\times\U(2)&\lra\U(2,2), &(g_1,g_2)&\longmapsto  \begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}^{-1}\begin{pmatrix}g_1\\&g_2\end{pmatrix} \begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}.
\end{align*}

For a nonzero $\beta\in\Her_1(\cK)$, denote by $W_\beta$ the one dimensional Hermitian space $(\cK,\beta)$. Fix a basis $w_\beta=1$ for $W_\beta$, and we can identify $\U(W_\beta)$ with the group $\U(1)$ defined by

\[
\U(1)(R)=\{g\in\GL_1(R\otimes_\bZ\cO_\cK):\bar{g}g=1\}.
\]

We consider the Weil representations for the dual pairs $\U(2)\times\U(1)$ and $\U(2,2)\times\U(1)$. The space $V\otimes_{\bQ} W_\beta$ is a four dimensional $\bQ$-vector space equipped with a non-degenerate symplectic pairing induced from the skew-Hermitian form on $V$ and the Hermitian form on $W_\beta$. The group $\U(V)\times \U(W_\beta)$ embeds into $\Sp(V\otimes_\bQ W_\beta)$ and forms a reductive dual pair. Let $\wt{\Sp}(V\otimes_\bQ W_\beta)$ denote the metaplectic group. Given a pair of Hecke characters $\lambda_V,\lambda_{W_\beta}:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ such that $\lambda_V|_{\bA^\times_\bQ}=\eta_{\cK/\bQ}^{\dim_\cK V}$ and $\lambda_{W_\beta}|_{\bA^\times_\bQ}=\eta^{\dim_\cK W_\beta}_{\cK/\bQ}$, there is a splitting

\begin{equation}\label{eq:splitting}
\U(V_v)\times \U(W_{\beta,v})\lhra \wt{\Sp}(V_v\otimes_\bQ W_{\beta,v})
\end{equation}

for all places $v$ of $\bQ$. The Weil representation of $\wt{\Sp}(V_v\otimes_\bQ W_{\beta,v})$ (with respect to our fixed additive character $\be_{\bA_\bQ}:\bQ\backslash\bA_\bQ\ra\bC^\times$ in (<ref>)) induces the Weil representation of $\U(V_v)\times \U(W_{\beta,v})$. Similarly, with respect to $\lambda_{V^-},\lambda_{V\oplus V^-}:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ satisfying $\lambda_{V^-}|_{\bA^\times_\bQ}=\eta_{\cK/\bQ}^{\dim_\cK V^-}$, $\lambda_{V\oplus V^-} |_{\bA^\times_\bQ}=\eta_{\cK/\bQ}^{\dim_\cK V+\dim_\cK V^-}$, we have the Weil representations of $\U(V^-_v)\times U(W_{\beta,v})$ and  $\U(V\oplus V^-_v)\times U(W_{\beta,v})$.

We choose our $\lambda_V,\lambda_{V^-},\lambda_{W_\beta}$ as

\begin{equation}\label{eq:split-char}
\begin{aligned}
\lambda_V=\lambda_{V^-}&=\lambda^2,
\end{aligned}
\end{equation}

where $\lambda$ is the Hecke character fixed in <ref>.

In some literature, instead of $\lambda_V=\lambda_{V^-}=\lambda^2$, $\lambda_V=\lambda_{V^-}=\triv$ is used.

\begin{align*}
e_1&=v_1\otimes w_\beta, &e_2&=v_2\otimes w_\beta,
&e^-_1&=v^-_1\otimes w_\beta, &e^-_2&=v^-_2\otimes w_\beta.
\end{align*}

Then $\bQ e_1\oplus\bQ e_2$ is a maximal isotropic subspace of the symplectic space $V\otimes_\bQ W_\beta$.
We have the polarizations

\begin{align}
\label{eq:V-pol}V\otimes W_\beta&=\big(\bQ e_1\oplus\bQ e_2\big)\oplus \big(\bQ \delta e_1\oplus\bQ \delta e_2\big),
\end{align}

(Where $\delta$ is our fixed totally imaginary element in $\cK$.) The Schrödinger  model of the Weil representation of $\U(V_v)\times\U(W_{\beta,v})$ is an action of $\U(V_v)\times\U(W_{\beta,v})$ on $\pzS(\bQ_v e_1\oplus\bQ_v e_2)$, the space of Schwartz functions on $\bQ_v e_1\oplus\bQ_v e_2$. We write this action as

\begin{align*}
\omega_{\beta,v}(g,u): \pzS(\bQ_v e_1\oplus\bQ_v e_2)&\lra \pzS(\bQ_v e_1\oplus\bQ_v e_2),
&g\in\U(V_v),\,u\in \U(W_{\beta,v}).
\end{align*}

Similarly, we have the polarization

\begin{align}
\label{eq:V--pol}V^-\otimes W_\beta&=\big(\bQ e^-_1\oplus\bQ e^-_2\big)\oplus \big(\bQ \delta e^-_1\oplus\bQ \delta e^-_2\big).
\end{align}

and the Schrödinger model

\begin{align*}
\omega_{\beta,v}(g,u): \pzS(\bQ_v e^-_1\oplus\bQ_v e^-_2)&\lra \pzS(\bQ_v e^-_1\oplus\bQ_v e^-_2),
&g\in\U(V^-_v),\,u\in \U(W_{\beta,v}).
\end{align*}

For a place $v=\fv\bar{\fv}$ split in $\cK$, we have $\cK_v\simeq \cK_\fv\times\cK_{\bar{\fv}}$ with $\cK_\fv\simeq\cK_{\bar{\fv}}\simeq \bQ_v$. In addition to the above polarizations, we also consider the polarizations

\begin{align*}
V_v\otimes W_{\beta,v}&=\big(\cK_\fv e_1\oplus\cK_\fv e_2\big)\oplus \big(\cK_{\bar{\fv}} e_1\oplus\cK_{\bar{\fv}} e_2\big),\\
V^-_v\otimes W_{\beta,v}&=\big(\cK_\fv e^-_1\oplus\cK_\fv e^-_2\big)\oplus \big(\cK_{\bar{\fv}} e^-_1\oplus\cK_{\bar{\fv}} e^-_2\big),
\end{align*}

and the intertwining maps

\begin{align}
\pzS\big(\cK_\fv e_1\oplus\cK_\fv e_2\big)&\lra \pzS\big(\bQ_v e_1+\bQ_ve_2\big),\\
\label{eq:int-S} \pzS\big(\cK_\fv e^-_1\oplus\cK_\fv e^-_2\big)&\lra \pzS\big(\bQ_v e^-_1+\bQ_ve^-_2\big).
\end{align}

For $V\oplus V^-$, we have the polarization:

\[
(V\oplus V^-)\otimes W_\beta=\big(\cK(e_1+e^-_1)\oplus \cK(e_2+e^-_2)\big)\oplus \big(\cK(e_1-e^-_1)\oplus \cK(e_2-e^-_2)\big).
\]

Writing an element in $\cK_v(e_1+e^-_1)\oplus \cK_v(e_2+e^-_2)$ as $X\in M_{1,2}(\cK_v)$, for the Schrödinger model

\begin{align*}
\omega_{\beta,v}(g,u): \pzS\big(\cK(e_1+e^-_1)\oplus \cK(e_2+e^-_2)\big)&\lra \pzS\big(\cK(e_1+e^-_1)\oplus \cK(e_2+e^-_2)\big),
&g\in\U(2,2),\,u\in \U(W_{\beta,v}),
\end{align*}

we have the following formulas:

\begin{equation}\label{eq:Weil-fmla}
\begin{aligned}
\omega_{\beta,v}(\bid_4,u)\,\Phi(X)&=\Phi(u^{-1}X),\\
\omega_{\beta,v}\left(\begin{pmatrix}A\\&\ltrans{\bar{A}}\end{pmatrix},1\right)\Phi(X)&=\lambda_v(\det A)|\det A|^{1/2}_{\cK_v}\cdot \Phi(XA),\\
\omega_{\beta,v}\left(\begin{pmatrix}\bid_2&B\\&\bid_2\end{pmatrix},1\right)\Phi(X)&=\be_v\left(\beta\cdot XB\ltrans{\bar{X}}\right)\cdot \Phi(X),\\
\omega_{\beta,v}\left(\begin{pmatrix}&-\bid_2\\\bid_2\end{pmatrix},1\right)\Phi(X)&= |\beta|_{\bQ_v} \int \Phi(Y)\cdot \be_v\left(\Tr_{\cK_v/\bQ_v}(\beta \cdot Y\ltrans{\bar{X}})\right)\,dY.
\end{aligned}
\end{equation}

§.§ The Heisenberg group and Jacobi forms

Jacobi forms on $\U(2)$ and $\U(2,2)$ show up in our computation of the non-degenerate Fourier–Jacobi coefficients of the evaluations at classical points of the Klingen family $\bfE^\Kling$.

First, we introduce Jacobi forms on $\U(2)$. Recall that $V$ is the skew Hermitian space $(\cK^2,\zeta_0)$. Denote by $\gls{HV}$ be the Heisenberg group associated to $V$. For a $\bQ$-algebra $R$

\[
H(V)(R)=(V\otimes_{\bQ}R)\ltimes R,
\]

and the multiplication is

\[
(x_1,\sigma_1)(x_2,\sigma_2)=\left(x_1+x_2,\sigma_1+\sigma_2+\frac{x_1\zeta_0 \ltrans{\bar{x}_2}+\ol{x_1\zeta_0 \ltrans{\bar{x}_2}}}{2}\right).
\]

It is easy to see that $H(V)$ is isomorphic to $U_{P_{\GU(3,1)}}$ via

\begin{equation}\label{eq:31HU}
(x,\sigma)\longmapsto u(x,\sigma)=\begin{pmatrix}
\end{pmatrix}.
\end{equation}

The Jacobi group associated to $V$ is the semi-direct product $H(V)\rtimes \U(V)$, which we identify with
a subgroup of $P_{\GU(3,1)}$ by

\begin{equation}\label{eq:H-to-U31}
\begin{aligned}
\big((x,\sigma),g_1\big)&\longmapsto u(x,\sigma)\,m(g_1),
&&(x,\sigma)\in H(W_0),\, g_1\in\U(V),
\end{aligned}
\end{equation}

where for $g_1\in \U(V)$,

\[
\]

A Jacobi form on $H(V)\rtimes\U(V)$ or a Jacobi form on $\U(V)$ of index $\beta$ is a smooth function on $H(V)(\bQ)\rtimes\U(V)(\bQ)\backslash H(V)(\bA_\bQ)\rtimes\U(V)(\bA_\bQ)$ such that

– for all $\sigma\in\bA_\bQ$, the left translation by $\big((0,\sigma),\bid_2\big)$ equals the multiplication by $\be_{\bA_\bQ}(\beta\sigma)$,

– the right translation of a maximal compact subgroup and the action of the center of the universal enveloping Lie algebra at $\infty$ satisfy finiteness conditions.

It is easy to see that a Jacobi form of index $\beta$ holomorphic at $\infty$ of weight $(0,0)$ corresponds a section in $H^0\left(\sC,\cL(\beta)\right)$ with $\sC$ a torsor over the Shimura variety of $\GU(2)$ of  an abelian scheme isogeneous to the universal abelian scheme, and $\cL(\beta)$ the invertible sheaf over $\sC$ of $\beta$-homogeneous functions.

We will also need Jacobi forms on $\U(2,2)$. Denote by $\bV$ the skew Hermitian space $\cK^4$ equipped with the skew Hermitian form $\begin{pmatrix}&\bid_2\\-\bid_2\end{pmatrix}$, and by $\gls{HVV}$ its associated Heisenberg group whose $R$-points for a $\bQ$-algebra $R$ are

\[
H(\bV)(R)=(\bV\otimes_\bQ R)\ltimes R.
\]

We write an element in $H(\bV)(R)$ as $(x,y,\sigma)$, $x,y\in R\otimes_\bQ\cK^2$, $\sigma\in R$. The multiplication is given by

\[
\]

The Jacobi group associated to $\bV$ is the semi-direct product $H(\bV)\rtimes\U(\bV)$.

Denote by $P_{\GU(3,3)}$ the parabolic subgroup of $\GU(3,3)$ consisting of elements whose entries in the first column are all $0$ except the $(1,1)$ entry, and denote by  $U_{P_{\GU(3,3)}}$ its unipotent subgroup. We identify $H(\bV)$ with $U_{P_{\GU(3,3)}}$ by

\begin{align*}
(x,y,\sigma)&\longmapsto u(x,y,\sigma)
\end{align*}

and identify $H(\bV)\rtimes\U(2,2)$ with a subgroup of $P_{\GU(3,3)}$ by

\begin{equation}\label{eq:H-to-U33}
\begin{aligned}
\big((x,y,\sigma),g\big)&\longmapsto u(x,y,\sigma)\,m(g),
&&(x,y,\sigma)\in H(\bW_0),\,g\in\U(2,2).
\end{aligned}
\end{equation}

where for $g=\begin{pmatrix}A&B\\C&D\end{pmatrix}\in\U(2,2)$,

\begin{equation}\label{eq:U22-U33}
m(g)=\begin{pmatrix} 1\\&A&&B\\&&1\\&C&&D\end{pmatrix}.
\end{equation}

Similarly as above, we define Jacobi forms on $\U(\bV)$ of index $\beta$ to be smooth functions on $H(\bV)(\bQ)\rtimes\U(\bV)(\bQ)\backslash H(\bV)(\bA_\bQ)\rtimes\U(\bV)(\bA_\bQ)$ on which the left translation of $\big((0,0,\sigma),\bid_4\big)$ equal the multiplication of $\be_{\bA_\bQ}(\beta\sigma)$ plus finiteness conditions for the right translation of a maximal compact subgroup and the center of the universal enveloping Lie algebra at $\infty$.

We have the following embedding of skew Hermitian spaces

\begin{align*}
V&\lra \bV, &x&\longmapsto \left(x,-\frac{x\zeta_0}{2}\right).
\end{align*}

It induces the embedding of unitary groups

\begin{align*}
\U(V)&\lra\U(\bV)=\U(2,2), &g_1&\lra \begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}^{-1}\begin{pmatrix}g_1\\&\bid_2\end{pmatrix}\begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}
\end{align*}

and induces an embedding of the Jacobi groups

\begin{equation}
\begin{aligned}
H(V)\rtimes \U(V)&\lra H(\bV)\rtimes \U(\bV), \\
\left((x,\sigma),\begin{pmatrix}a&b\\c&d\end{pmatrix}\right)&\longmapsto \left(\left(x,-\frac{x\zeta_0}{2},\sigma\right),\begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}^{-1}\begin{pmatrix}g_1\\&\bid_2\end{pmatrix}\begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}\right).
\end{aligned}
\end{equation}

From this embedding, we also get the embedding

\begin{equation}\label{eq:JGU(3,3)}
\begin{aligned}
\big(H(V)\rtimes \U(V)\big)\times\U(V^-)&\lra H(\bV)\rtimes \U(\bV), \\
\big(((x,\sigma),g_1),g_2\big)&\longmapsto \left(\left(x,-\frac{x\zeta_0}{2},\sigma\right),\begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}^{-1}\begin{pmatrix}g_1\\&g_2\end{pmatrix}\begin{pmatrix}\bid_2&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}\right).
\end{aligned}
\end{equation}

The embeddings (<ref>)(<ref>)(<ref>) are compatible with the embedding $\imath$ in (<ref>) in the sense that the following diagram commutes:

\[\xymatrixcolsep{5pc}\xymatrix{
\big(H(V)\rtimes\U(V)\big)\times\U(V^-) \ar@{^{(}->}[r]^-{\eqref{eq:H-to-U31}\times \id}\ar[d]^-{\eqref{eq:JGU(3,3)}} & \GU(3,1)\times_{\bG_m}\GU(2)\ar[d]^-{\imath}\\
H(\bV)\rtimes \U(\bV) \ar@{^{(}->}[r]^-{\eqref{eq:H-to-U33}} & \GU(3,3).
\]

§.§ Theta series

Given a Schwartz function $\phi\in \bigotimes'_v\pzS(\bQ_ve_1\oplus\bQ_ve_2)$ (with $\phi_v=\mathds{1}_{\bZ_v e_1+\bZ_v e_2}$ for all most all finite places $v$), we define the theta series $\theta(\phi,\bdot)$, which is an automorphic form  on $\U(V)(\bA_\bQ)$, as

\[
\theta(\phi,g)=\sum_{x\in \bQ e_1\oplus\bQ e_2}\omega_{\beta}(g)\Phi(x).
\]

We can also define the Jacobi theta series $\theta^J(\phi,\bdot,\bdot)$, which is a Jacobi form on $H(V)\rtimes\U(2)$, as

\begin{equation}\label{eq:theta-J}
\theta^J(\phi,(y,\sigma),g)=\be_{\bA_\bQ}\left(\beta(\sigma+\frac{1}{2}y\zeta_0\ltrans{\bar{x}})\right)\sum_{x\in \bQ e_1\oplus\bQ e_2}\omega_{\beta}(g)\Phi(x+y).
\end{equation}

Similarly, given $\phi\in \bigotimes'_v\pzS(\bQ_v e^-_1\oplus\bQ_v e^-_2)$, we can define the theta series $\theta_\beta(\phi,\bdot)$ on $\U(V^-)(\bA_\bQ)$.

§.§ The unfolding

Given a holomorphic automorphic form $\cF$ on $\GU(3,1)(\bA)$ and $g\in \GU(3,1)$ and $\beta\in\Her_1(\cK)_{>0}=\bQ_{>0}$, the $\beta$-th Fourier–Jacobi coefficient of $\cF$ at $g$ is defined as

\begin{align*}
a_\beta(g;\cF)=\int_{\sigma\in\Her_1(\cK)\backslash\Her_1(\bA_\cK)} \cF\left(\begin{pmatrix}1&&\sigma\\&\bid_2\\&&1\end{pmatrix} g\right)\be_{\bA}(-\beta\sigma)\,d\sigma.
\end{align*}

With $g\in\GU(3,1)$ fixed, the function

\begin{align*}
H(V)(\bA_\bQ)\rtimes \U(V)(\bA_\bQ)&\lra \bC\\
\big((x,\sigma),g_1\big)&\longmapsto a_\beta\big(u(x,\sigma)m(g_1);\cF\big)
\end{align*}

is a Jacobi form on $H(V)\ltimes \U(V)$ of index $\beta$.

The Klingen Eisenstein family $\bfE^\Kling_\varphi$ interpolates a normalization of the Klingen Eisenstein series $E^\Kling\big(\bdot;F(f(s,\xi_0\tau_0),\varphi)\big)$, which can be expressed as the integral in (<ref>). We can compute its $\beta$-th Fourier–Jacobi coefficients by first computing

\begin{equation}\label{eq:E-Sieg-beta}
\end{equation}

and then pairing its restriction to $\GU(3,1)\times_{\bG_m} \GU(2)$ with $\varphi$ on $\GU(2)$. By unfolding the Siegel Eisenstein series in the integrand on the right hand side of (<ref>), it's not difficult to show that

\begin{equation}\label{eq:Esi-beta}
\begin{aligned}
&E^{\Sieg}_\beta(h;f(s,\xi_0\tau_0))=\sum_{\gamma\in Q_{\U(2,2)}(\bQ)\backslash \U(2,2)(\bQ)}\sum_{x\in\cK^2}\\
&\hspace{2em}\int_{\Her_1(\bA_\cK)}f(s,\xi_0\tau_0)\left(\begin{pmatrix}&\bid_3\\-\bid_3\end{pmatrix}\begin{pmatrix}1&&\varsigma&x\\ &\bid_2&\ltrans{\bar{x}}\\&&1\\&&&\bid_2\end{pmatrix}m\left(\begin{pmatrix}&\bid_2\\-\bid_2\end{pmatrix}\gamma\right) h\right)\be_{\bA_\bQ}(-\beta\sigma)\,d\varsigma,
\end{aligned}
\end{equation}

where $m(\bdot)$ is the embedding of $\U(2,2)$ into $\U(3,3)$ as defined in (<ref>). (cf <cit.>)

For $g_v\in\GU(3,3)(\bQ_v)$, $x_v\in\cK^2_v$ and $f_v(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)},v}(s,\xi_0\tau_0)$, define

\begin{align*}
=&\,\int_{\Her_1(\cK_v)} f_v(s,\xi_0\tau_{0})\left(\begin{pmatrix}&\bid_3\\-\bid_3\end{pmatrix}\begin{pmatrix}1&&\varsigma&x_v\\ &\bid_2&\ltrans{\bar{x}_v}\\&&1\\&&&\bid_2\end{pmatrix}m\begin{pmatrix}&\bid_2\\ -\bid_2\end{pmatrix} g_v\right)\be_\bA(-\beta\sigma)\,d\sigma.
\end{align*}

For $g\in\GU(3,3)(\bA_\bQ)$, $x\in\bA^2_\cK$ and $f(s,\xi_0\tau_0) =\otimes_v f_v(s,\xi_0\tau_{0})$, let

\[
\FJ_\beta\big(g,x;f(s,\xi_0\tau_0)\big)=\otimes_v \FJ_{\beta,v}\big(g_v,x_v;f_v(s,\xi_0\tau_0)\big).
\]

The unfolding result (<ref>) gives the following proposition.

Let $\beta\in\Her_1(\cK)_{>0}$ and $f(s,\xi_0\tau_0)\in I_{Q_{\GU(3,3)}}(s,\xi_0\tau_0)$. The $\beta$-th Fourier–Jacobi coefficient of the Siegel Eisenstein series $E^\Sieg\big(\bdot\,;f(s,\xi_0\tau_0)\big)$ at $g\in \U(3,3)(\bA_\bQ)$ equals

\[
E^{\Sieg}_\beta(g;f(s,\xi_0\tau_0))=\sum_{\gamma\in Q_{\U(2,2)}(\bQ)\backslash \U(2,2)(\bQ)}\sum_{x\in\cK^2} \FJ_\beta\big(m(\gamma)g,x;f(s,\xi_0\tau_0)\big),
\]

with $\FJ_{\beta}$ defined as in Definition <ref> and $m(\bdot)$ the embedding (<ref>).

One can easily check that $\FJ_{\beta,v}$ satisfies

\begin{align*}
\FJ_{\beta,v}\left(u(x,y,\sigma)g,x_0;f_v(s,\xi_0\tau_{0})\right)
\FJ_{\beta,v}\left(m\begin{pmatrix}A&B\\0&\ltrans{\bar{A}}^{-1}\end{pmatrix}g,x_0;f_v(s,\xi_0\tau_{0})\right)
&=|\det A\ltrans{\bar{A}}|^{s+1}_v\xi_0\tau_{0}(\det A)\cdot |\det A\ltrans{\bar{A}}|^{\frac{1}{2}}_v\be_v(\beta xB\ltrans{\bar{A}}\ltrans{\bar{x}})\\
&\quad\times\FJ_{\beta,v}\big(g,x_0 A;f_v(s,\xi_0\tau_{0})\big).
\end{align*}

Comparing them with the formulas in (<ref>) for the Schrödinger model of Weil representations, we see that for a fixed $g_v\in \U(3,3)(\bQ_v)$, $\FJ_{\beta,v}\big(u(x_v,y_v,\sigma_v)\,m(g_{1,v})g_v,x_0;f_v(s,\xi_0\tau_{0})\big)$ is essentially (a finite sum of) the product

\[
f_{2,v}\left(s,\xi_0\tau_{0}\lambda^{-1}\right)(g_{1,v})\cdot \omega_{\beta,v}\big(u(x_v,y_v,\sigma_v)\,m(g_{1,v})\big)\Phi_v(x_0),
\]

with $f_{2,v}\left(s,\xi_0\tau_{0}\lambda^{-1}\right)$ a section in the degenerate principal series $I_{Q_{\U(2,2),v}}\left(s,\xi_0\tau_{0}\lambda^{-1}\right)$, $\Phi_v$ a Schwartz function on $\cK^2_v$ and $\omega_{\beta,v}$ the Weil representation of $\U(2,2)$ briefly recalled in <ref>. Both $f_{2,v}\left(s,\xi_0\tau_0\lambda^{-1}\right)$ and $\Phi_v$ are determined by $f_v(s,\xi_0\tau_{0})\in I_{Q_{\GU(3,3)},v}(s,\xi_0\tau_0)$ and $g_v\in\GU(3,3)(\bQ_v)$. The computation at an unramified place $v$ gives

\begin{align*}
&\FJ_{\beta,v}\big(u(x,y,\sigma) m(g_{1,v}),x_0;f^\sph_v(s,\xi_0\tau_{0})\big)\\
=&\,L_v\left(2s+3,\xi^\bQ_0\tau^\bQ_0\right)^{-1}\cdot f^\sph_{2,v}(s,\xi_0\tau_0\lambda^{-1})(g_{1,v})\cdot \omega_{\beta,v}\big(u(x,y,\sigma) m(g_{1,v})\big)\mathds{1}_{\cO^2_{\cK,v}}(x_0),
\end{align*}
Therefore, the Jacobi form on $H(\bV)\rtimes\U(2,2)$ given by

\begin{align*}
\big((x,y,\sigma),g\big)&\longmapsto E^{\Sieg}_\beta(u(x,y,\sigma)m(g);f(s,\xi_0\tau_0))
\end{align*}

is essentially (a finite sum of) the product

\[
L\left(2s+3,\xi^\bQ_0\tau^\bQ_0\right)^{-1}\cdot E^\Sieg_2\left(s,\xi_0\tau_{0}\lambda^{-1}\right)\cdot \Theta^J,
\]

with $E^{\Sieg}_2\left(s,\xi_0\tau_{0}\lambda^{-1}\right)$ a Siegel Eisenstein series on $\U(2,2)$ and $\Theta^J$  a Jacobi theta function on the $H(\bV)\rtimes\U(2,2)$.  The restriction of $\Theta^J$ from $H(\bV)\rtimes\U(2,2)$ to $\big(H(V)\rtimes \U(2)\big)\times\U(2)$ via (<ref>) is essentially (a finite sum of) $\theta^J_3\boxtimes\theta_2$, with $\theta^J_3$ (resp. $\theta_2$) a Jacobi theta function (resp. theta function) on $H(V)\rtimes \U(2)$ (resp. $\U(2)$) attached to a Schwartz function $\phi_3$ (resp. $\phi_2$) on $\bA_{\cK}$. Therefore,

\begin{equation}\label{eq:E-beta}
\left.E^{\Sieg}_\beta(\bdot;f(s,\xi_0\tau_0))\right|_{\big(H(V)\rtimes \U(2)\big)\times\U(2)}\approx L\left(2s+3,\xi^\bQ_0\tau^\bQ_0\right)^{-1}\cdot \theta^J_3\boxtimes\theta_2 \cdot  \left. E^\Sieg_2\left(s,\xi_0\tau_{0}\right)\right|_{\U(2)\times \U(2)}.
\end{equation}

Here $\approx$ means equal up to normalizations and more precise formulas for local sections are needed to be an actual identity. We will also use the notation $\approx$ several times in the next section. All the identities with  $\approx$ are only for the purpose of illustrating the idea of relating the non-degenerate Fourier–Jacobi coefficients with $L$-values, and will not be used for our rigorous analysis of $\bfE^\Kling_{\varphi,\beta,u}$ starting from <ref>.

§.§ Our strategy of analyzing the non-degenerate Fourier–Jacobi coefficients of the Klingen Eisenstein family

Before we move on to the involved computations, we give a brief explanation of how we choose the auxiliary data to study the pairing of (<ref>) with $\varphi$. The description of this section should also explain why the choices of the auxiliary data in <ref> are made for constructing $\bfE^\Kling_\varphi$ and guaranteeing that its non-degenerate Fourier–Jacobi coefficients satisfy the nonvanishing properties needed for applications.

§.§.§ Choosing the auxiliary $\theta^J_1$

The first step to analyze (<ref>) is to pick a suitable Schwartz function $\phi_1$ on $\bA_\cK$, let $\gls{thetaJ1}$ be the associated Jacobi theta function on $H(V)\rtimes\U(2)$ (defined as in (<ref>)), and define the linear functional

\begin{align*}
l_{\theta^J_1}:\{\text{Jacobi forms on $H(V)\rtimes\U(2)$}\}&\lra \{\text{automorphic forms on $\U(2)$}\}
\end{align*}

\begin{equation}\label{eq:FJ-fcnl}
l_{\theta^J_1}(\varphi^J)(g_1)=\int_{H(V)(\bQ)\backslash H(V)(\bA)} \theta^J_1\big((x,\sigma),g_1\big)\varphi^J\big((x,\sigma),g_1\big)\,dxd\sigma.
\end{equation}

Applying $l_{\theta^J_1}$ to (<ref>), we obtain an automorphic form on $\U(2)\times \U(2)$ whose value at $(g_1,g_2)$ equals

\begin{equation}\label{eq:int-H}
\int_{H(V)(\bQ)\backslash H(V)(\bA)} \theta^J_1\big((w,\sigma),g_1\big)\cdot E^{\Sieg}_\beta\big(\imath(u(x,\sigma)m(g_1),g_2);f(s,\xi_0\tau_0)\big)\,dxd\sigma.
\end{equation}

It is a standard fact that for two Schwartz functions $\phi,\phi'$ on $\bA_{\cK}$,

\[
\int_{H(V)(\bQ)\backslash H(V)(\bA)} \theta^J_\phi\big((x,\sigma),g_1\big)\ol{\theta^J_{\phi'}\big((x,\sigma),g_1\big)}\,dxd\sigma=\left<\phi,\ol{\phi'}\right>=\int_{\cK\backslash\bA_\cK} \phi(x)\ol{\phi'(x)}\,dx,
\]

independent of $g_1\in \U(2)$. Thanks to this fact, from (<ref>) we obtain

\begin{equation}
\begin{aligned}
&l_{\theta^J_1}\left( \left.E^{\Sieg}_\beta(\bdot\,;f(s,\xi_0\tau_0))\right|_{\big(H(V)\rtimes \U(2)\big)\times\U(2)}\right)\\
\approx &\frac{\left<\phi_1,\phi_3\right>}{L\left(2s+3,\xi^\bQ_0\tau^\bQ_0\right)} \cdot (1\boxtimes\theta_2)\cdot \left.E^\Sieg_2\left(s,\xi_0\tau_{0}\right)\right|_{\U(2)\times\U(2)}.
\end{aligned}
\end{equation}

Here $1\boxtimes\theta_2$ denotes the automorphic form on $\U(2)\times\U(2)$ which is the constant function $1$ on the first factor and $\theta_2$ on the second factor.

For an automorphic form on ${\tt F}$ on $\U(2)$ and a Hecke character $\chi:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$, we use the notation ${\tt F}^\chi$ to denote the form

\begin{equation}\label{eq:U2-twist}
{\tt F}^\chi(g)={\tt F}(g)\cdot \chi(\det g),\quad g\in\U(2)(\bA_\bQ).
\end{equation}

Our Klingen Eisenstein series $E^\Kling\big(\bdot\,;F(f(s,\xi_0\tau_0),\varphi)\big)$ is obtained by pairing the restriction of the Siegel Eisenstein series with $\varphi^{\ol{\xi_0\tau}_0}$, so we have

\begin{equation}
\begin{aligned}
\approx &\, \frac{\left<\phi_1,\phi_3\right>}{L\left(2s+3,\xi_0^\bQ\tau^\bQ_0\right)} \cdot \left<\theta_2\varphi^{\ol{\xi_0\tau}_0},\left.E^\Sieg_2\left(s,\xi_0\tau_{0}\lambda^{-1}\right)\right|_{\U(2)\times\U(2)}\right>_{1\times\U(2)},
\end{aligned}
\end{equation}

where $\left<\cdot,\cdot\right>_{1\times\U(2)}$ means integration over the second copy of $\U(2)$.

The standard doubling method formula [29] implies

\[
\left.E^\Sieg_2\left(s,\xi_0\tau_{0}\lambda^{-1}\right)\right|_{\U(2)\times\U(2)}\approx \sum_h \frac{L\left(s+\frac{1}{2},\mr{BC}(\pi_{h})\times\xi_0\tau_0\lambda^{-1}\right)}{L\left(2s+1,\xi^\bQ_0\tau^\bQ_0\right)\,L\left(2s+2,\xi^\bQ_0\tau^\bQ_0\eta_{\cK/\bQ}\right)}\frac{\ol{h} \boxtimes h^{\xi_0\tau_0\ol{\lambda}}}{\left<\ol{h},h\right>},
\]

where $h$ runs over a certain orthonormal basis of automorphic forms on $\U(2)$ of certain level. By picking an $h$, we have

\begin{equation}\label{eq:lEh1}
\begin{aligned}
\approx &\,\left<\phi_1,\phi_3\right>\frac{L\left(s+\frac{1}{2},\mr{BC}(\pi_h)\times\xi_0\tau_0\lambda^{-1}\right)}{d_3(s,\xi_0\tau_0)}\left<\theta_2\varphi^{\ol{\xi_0\tau_0}},h^{\xi_0\tau_0\ol{\lambda}}\right>\\
\end{aligned}
\end{equation}

where $d_3(s,\xi_0\tau_0)$ is as in (<ref>). The Petersson inner product $\left<\theta_2^{\ol{\lambda}}\varphi,h\right>$ is related to the central value of a triple product $L$-function by Ichino's formula.

§.§.§ Choosing auxiliary $h$ and $\theta$ and relating to $L$-values

We will choose suitable CM forms $h$ and $\theta$ on $\U(2)$ such that we can apply the available mod $p$ nonvanishing results to study $L\left(s+\frac{1}{2},\mr{BC}(\pi_h)\times\xi_0\tau_0\lambda^{-1}\right)$ and $\left<\theta_2^{\ol{\lambda}}\varphi,h\right>$ in (<ref>).

Let $\chi_h$ and $\chi_\theta$ be two unitary Hecke characters of $\cK^\times\backslash\bA^\times_\cK$ such that $\chi_h\chi^c_\theta|_{\bA^\times_{\bQ}}=\triv$ (as chosen in <ref>). Let $h_0$ be the theta lift of $\chi^{-1}_h|_{\U(1)}$ to $\U(2)$, and $h$ be the automorphic form on $\U(2)$ obtained from $h_0$ by

\begin{equation}\label{eq:h-twist}
\begin{aligned}
h(g)&=\chi_h\chi^c_\theta(a)\cdot h_0(g), &g\in\U(2),\,a\in\bA^\times_\cK,\,\det g=a\bar{a}^{-1}.
\end{aligned}
\end{equation}

Thanks to the condition $\chi_h\chi^c_\theta|_{\bA^\times_{\bQ}}=\triv$, the definition of $h(g)$ does not depend on the choice of $a\in\bA^\times_\cK$.

The form $\theta_2$ does not generate an irreducible representation of $\U(2)$. Let $\theta$ be the projection of $\theta_2$ to its $\chi_\theta\lambda^2|_{\U(1)}$-eigenspace for the action of the center of $\U(2)$. Then $\theta$ is a theta lift of $\chi_\theta\lambda^2|_{\U(1)}$. Since our $\varphi$ is assumed to have the trivial central character and $h$ has the central character $\chi^{-1}_\theta|_{\U(1)}$, we have

\begin{equation}\label{eq:theta'}
\left<\theta_2^{\ol{\lambda}}\varphi,h\right>=\left<\theta^{\ol{\lambda}}\varphi,h\right>.
\end{equation}

By <cit.>, with the choice of the splitting characters for the theta correspondence between $\U(2)$ and $\U(1)$ as in (<ref>), we have

\begin{equation}\label{eq:L-h}
\begin{aligned}
\mr{BC}\left(\pi_{h}\right)&=\mr{BC}\left(\pi_{h_0}\right)\otimes\chi_h\chi^c_\theta=\left(\mr{BC}\left(\chi^{-1}_h|_{\U(1)}\right)\lambda^{-1}\oplus \lambda\right)\otimes\chi_h\chi^c_\theta\\
&=\left(\chi^{-1}_h\chi^c_h\lambda^{-1}\oplus\lambda\right)\otimes\chi_h\chi^c_\theta=\chi^c_h\chi^c_\theta\lambda^{-1}\oplus \chi_h\chi^c_\theta\lambda,
\end{aligned}
\end{equation}

\begin{equation}\label{eq:L-theta}
\mr{BC}\big(\pi_{\theta^{\ol{\lambda}}}\big)=\big(\mr{BC}\left(\chi_\theta\lambda^2|_{\U(1)}\big)\lambda^{-1}\oplus\lambda\right)\bar{\lambda}\bar{\lambda}^{-c}=\left(\chi_\theta\chi^{-c}_\theta\lambda^3\oplus\lambda\right)\lambda^{-2}=\chi_\theta\chi^{-c}_\theta\lambda\oplus\lambda^{-1}.
\end{equation}

Thus, we have

\begin{equation}\label{eq:L12}
\begin{aligned}
\end{aligned}
\end{equation}

By the triple product formula

\begin{equation}\label{eq:triple-1}
\begin{aligned}
\left<\theta^{\bar{\lambda}}\varphi,h\right>\left<\tilde{\theta}^\lambda_3\tilde{\varphi},\tilde{h}_3\right>
&\approx \frac{\big<\varphi,\tilde{\varphi}\big>}{L(1,\pi,\mr{Ad})}\,
\frac{\big<h,\tilde{h}_3\big>}{\zeta_\cK(1)\,L(1,\chi_h\chi^{-1}_h\lambda^2)}\,
\frac{\big<\theta,\tilde{\theta}_3\big>}{\zeta_\cK(1)\, L(1,\chi_\theta\chi^{-c}_\theta\lambda^2)}\\
&\quad\times L\left(\frac{1}{2},\mr{BC}(\pi)\times\chi_h\chi_\theta\lambda^2\right)L\left(\frac{1}{2},\mr{BC}(\pi)\times \chi_h\chi^c_\theta\right),
\end{aligned}
\end{equation}

where $\tilde{\varphi}$ (resp. $\tilde{h}_3$, $\tilde{\theta}_3$) are suitable forms taken from the dual representation of $\pi^{\U(2)}=\pi_\varphi$ (resp. $\pi_{h}$, $\pi_{\theta}$). (In our case here, the triple product $L$-function for $\pi^D\times\pi^D_\theta\times\pi^D_h$  factorizes as the product of the two $L$-functions in (<ref>). Also, note that Ichino's triple product formula is for automorphic forms on $D^\times$, we actually need to relate the integral on $\U(2)$ for  $\left<\theta^{\bar{\lambda}}\varphi,h\right>\left<\tilde{\theta}^\lambda_3\tilde{\varphi},\tilde{h}_3\right>$ to an integral on $D^\times$. In <ref>, we discuss extending automorphic forms on $\U(2)$ to $\GU(2)$. Thanks to (<ref>), the integral over $\GU(2)$ is the same as the integral over $D^\times$.)

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

\begin{equation}\label{eq:FJ-L1256}
\begin{aligned}
&d_3(s,\xi_0\tau_0)\cdot \left<l_{\theta^J_1}\left(E^\Kling_\beta\big(\bdot\,;F(f(s,\xi_0\tau_0),\varphi)\big)\right),h\right>\left<\tilde{\theta}^\lambda_3\tilde{\varphi},\tilde{h}_3\right>\\
\approx&\,\left<\phi_1,\phi_3\right> \frac{\big<\varphi,\tilde{\varphi}\big>}{L(1,\pi,\mr{Ad})} \cdot \frac{\big<h,\tilde{h}_3\big>}{\zeta_\cK(1)\,L(1,\chi_h\chi^{-1}_h\lambda^2)}\,
\frac{\big<\theta,\tilde{\theta}_3\big>}{\zeta_\cK(1)\, L(1,\chi_\theta\chi^{-c}_\theta\lambda^2)}\\
&\times  L\left(s+\frac{1}{2},\lambda^2\chi_h\chi_\theta\xi^c_0\tau^c_0\right)\,L\left(s+\frac{1}{2},\chi_h\chi^c_\theta\xi_0\tau_0\right)
\cdot  L\left(\frac{1}{2},\mr{BC}(\pi)\times\chi_h\chi_\theta\lambda^2\right)L\left(\frac{1}{2},\mr{BC}(\pi)\times \chi_h\chi^c_\theta\right).
\end{aligned}
\end{equation}

The relation between $E^\Kling_\beta\big(-;F(f(s,\xi_0\tau_0),\varphi)\big)$ and $L$-values illustrated in (<ref>) explains the nonvanishing conditions on the $L$-values in (<ref>)(<ref>)(<ref>)(<ref>) in our choice of the auxiliary Hecke characters $\chi_\theta$ and $\chi_h$ in <ref>.

In the following, guided by the strategy described in this section, we carry out the necessary local computations and the construction of the auxiliary objects to prove the the desired property of $\bfE^\Kling_{\varphi,\beta,u}$ with $\beta=1$. The computation is place by place, and we have the following cases:

– For the archimedean place, we only need to consider the cases for which there are standard choices of local sections and  computations are easy.

– For unramified places, we have the standard formulas for spherical sections.

– For the place $p$ and the places in $\Sigma_\rms\cup\{\ell\}$, we compute precise formulas.

– For the places in $\Sigma_\ns\cup\{\ell'\}$, we do not attempt to compute precise formulas. At these places, the local data actually do not change when $\tau$ varies in a $p$-adic family, and we only make sure that local integrals contribute a nonzero scalar.

§.§ The auxiliary Jacobi form $\theta^J_1$

Given a Schwartz function $\phi$ on $\bA^2_\bQ$ such that $\phi_{\infty}$ is the standard Gaussian function, let $\theta^J$ be the associated Jacobi form  on $\U(2)$ (as defined in (<ref>)). The map attached to this $\theta^J$ defined in (<ref>) extends to $V^{J,\beta}_{\GU(2)}$ and induces

\begin{equation}\label{eq:l-JF}
\gls{lthetaJ}:\Meas\left(\Gamma_\cK,V^{J,\beta}_{\GU(2)}\right)\lra  \Meas\left(\Gamma_\cK,V_{\U(2)}\right).
\end{equation}

Apply it to $\bfE^\Kling_{\varphi,\beta,u}$ (defined in (<ref>)), we get

\[
l_{\theta^J}\left(\bfE^\Kling_{\varphi,\beta,u}\right)\in \Meas\left(\Gamma_\cK,V_{\U(2),\,\xi}\wh{\otimes}\,\hat{\cO}^\ur_L\right)^\natural.
\]

Let $\beta=1\in \Her_1(\cK)$. There exists a Schwartz function $\phi_1$ on $\bA^2_\bQ$ with $\phi_{1,\infty}$ the standard Gaussian function, elements $u_1,\dots,u_r\in\bigotimes\limits_{v\in\Sigma_\ns\cup\{\ell'\}}\U(1)\left(\bQ_v\right)$, and constants $b_1,\dots,b_r\in \cO_L$,  (all independent of $\tau$), such that

\[
l_{\theta^J_1}\left(\sum_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)\in \Meas\left(\Gamma_\cK,V_{\U(2),\,\xi}\right)^\natural,
\]

the image of $\sum\limits_i b_i\bfE^\Kling_{\varphi,\beta,u_i}$ under the map (<ref>) attached to the Jacobi form $\gls{thetaJ1}$ corresponding to $\phi_1$, satisfies the following interpolation properties: for all $\tau_{p\adic}$ as in Theorem <ref>,

\begin{equation}\label{eq:lE}
\begin{aligned}
l_{\theta^J_1}\left(\sum\nolimits_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)(\tau_{p\adic})
=&\, d^{\Sigma\,\cup\{p,\ell,\ell'\}}_2\left(s,\xi_0\tau_0\lambda^{-1}\right)\\
&\hspace{-7em}\times \left<\theta(\phi_{2,\tau})\left(\bdot\,\begin{pmatrix}&-1\\1\end{pmatrix}\right)\varphi^{\ol{\xi_0\tau}_0},\left.E^\Sieg_2\left(-\,;f_2(s,\xi_0\tau_{0}\lambda^{-1})\right)\right|_{\U(2)\times\U(2)}\right>_{1\times\U(2)},
\end{aligned}
\end{equation}

with $\phi_{2,\xi\tau}\in\pzS(\bA^2_{\bQ})$ (independent of $\tau$ except $v=p$) and $f_2(s,\xi_0\tau_0\lambda^{-1})\in\bigotimes_vI_{Q_{\U(2,2)},v}(s,\xi_0\tau_0\lambda^{-1})$ described as follows:

– $v=\infty$.

\begin{align*}
\phi_{2,\xi\tau,\infty}(y_1,y_2)&=\frac{\fs^{1/4}\Nm(\delta)^{1/4}}{2} e^{-2\pi\sqrt{\Nm(\delta)}\,\begin{psm}y_1&y_2\end{psm}\zeta_0\begin{psm}y_1\\y_2\end{psm}},\\
f_{2,\infty}(s,\xi_0\tau_0\lambda^{-1})\left(g'=\begin{pmatrix}A&B\\C&D\end{pmatrix}\right)&= \frac{(-2\pi i)^k}{\Gamma(k)}\cdot \det\left(C\frac{\zeta_0}{2}+D\right)^{-k}\left|C\frac{\zeta_0}{2}+D\right|^{-s+1+\frac{k}{2}}_\bC
\end{align*}

– $v\notin\Sigma\cup\{\ell,\ell',p\}$.

\begin{align*}
\phi_{2,\xi\tau,v}&=\mathds{1}_{\bZ_v}\times\mathds{1}_{\bZ_v},\\
f_{2,v}(s,\xi_0\tau_0\lambda^{-1})&=L_v\left(2s+3,\xi^\bQ_0\tau^\bQ_0\right)^{-1}\cdot f^\sph_{2,v}(s,\xi_0\tau_0\lambda^{-1})
\end{align*}

– $v=\fv\ol{\fv}\in\Sigma_\rms\cup\{\ell\}$.

\begin{align*}
\phi_{2,\xi\tau,v}= \text{Image of \eqref{eq:int-S} of } \Big((y_1,y_2)&\longmapsto \mathds{1}_{\cO_{\cK,\fv}}(y_1)\cdot \mathds{1}_{\cO^\times_{\cK,\bar{\fv}}}(y_2)\chi_{\theta,\bar{\fv}}\lambda_{\bar{\fv}}(y_2)\Big)
\end{align*}

\begin{equation}\label{eq:f2-ramv}
\begin{aligned}
=&\,|\det C\ltrans{\bar{C}}|^{-s-1}(\xi_0\tau_{0}\lambda^{-1})_v\left(\det\bar{C}^{-1}\right)\\
\end{aligned}
\end{equation}

– $v\in\Sigma_\ns\cup\{\ell'\}$.

\begin{align*}
\phi_{2,\xi\tau,v}=&\,\text{a nonzero Schwartz function on $\bQ^2_v$ inside the $\chi_\theta\lambda^2|_{\U(1)(\bQ_v)}$-eigenspace}\\
&\,\text{for the action of $\U(1)(\bQ_v)$, invariant under $K_v$, independent of $\tau$,}\\
f_{2,v}(s,\xi_0\tau_0\lambda^{-1})=&\,\text{same as \eqref{eq:f2-ramv}}.
\end{align*}

– The place $p$.

\begin{equation}\label{eq:phi2p}
\begin{aligned}
\phi_{2,\xi\tau,p}=&\text{Image of \eqref{eq:int-S} of }\\&\Big((y_1,y_2)\longmapsto p^{-2t}\fg\left((\xi_0\tau_0)^{-1}_{\fp}\right)\cdot \mathds{1}_{\bZ^\times_p}(p^ty_1)(\xi_0\tau_0)_{\fp}(-p^ty_1)  \cdot\mathds{1}_{\bZ_p}(p^ty_2)\Big),
\end{aligned}
\end{equation}

\begin{align*}
&\times M'_p\left(-s,(\xi_0\tau_0\lambda^{-1})^{-c}\right)f^{\textnormal{big-cell}}_{2,p}\left(-s,(\xi_0\tau_0\lambda)^{-c}\right)(\bdot\,\Upsilon'_p),
\end{align*}

* $M'_p\left(-s,(\xi_0\tau_0\lambda)^{-c}\right):I_{Q_{\U(2,2)},p}\left(-s,(\xi_0\tau_0\lambda)^{-c}\right)\ra I_{Q_{\U(2,2)},p}(s,\xi_0\tau_0\lambda^{-1})$ is the intertwining operator,

* $\Upsilon'_p$ is the element in $\U(2,2)(\bQ_p)$ with $\varrho_\fp(\Upsilon'_p)=\begin{pmatrix}\bid_1&-\frac{\zeta_0}{2}\\-\bid_2&-\frac{\zeta_0}{2}\end{pmatrix}^{-1}$,

\begin{align*}
=&\,|\det C\ltrans{\bar{C}}|^{s-1}_p(\xi_0\tau_{0}\lambda^{-1})_p(\det C)\cdot \alpha'_{\xi\tau,p}(C^{-1}D).
\end{align*}

with $\alpha'_{\xi\tau,p}$ the Schwartz function on $\Her_2(\cK_p)$ defined as

\[
\alpha'_{\xi\tau,p}(x)=\fg\left((\xi_0\tau_0)^{-1}_{\fp}\right)\cdot \mathds{1}_{\bZ^3_p}(x_{11},x_{21},x_{22})\cdot \mathds{1}_{\bZ^\times_p}(p^t x_{12})(\xi_0\tau_0)_{\fp}(p^tx_{12}),\quad \varrho_\fp(x)=\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix},
\]

This proposition is proved in <cit.> by computing $\FJ_{\beta,v}\big(g,x;f_v(s,\xi_0\tau_0)\big)$ place by place. The place $\infty$ is done in Lemmas 6.9 and 6.11 and the unramified places are done in Lemma 6.18. (Note that at the unramified places, we have $d_{2,v}(s,\xi_0\tau_0\lambda^{-1})=\frac{d_{3,v}(s,\xi_0\tau_0)}{L_v(2s+3,\xi^\bQ_0\tau^\bQ_0)}$.) The places in $\Sigma_\rms\cup\{\ell\}$ are done in Lemmas 6.22 and 6.24, and the places in $\Sigma_\ns\cup\{\ell'\}$ are done in Lemmas 6.22 and 6.26. Note that Lemma 6.26 loc.cit is Proposition <ref> and in <ref> we choose $\chi_\theta$ such that at $v\in\Sigma_\ns\cup\{\ell'\}$ we have nonzero $\phi_{2,\xi\tau,v}$ described in this proposition. The place $p$ is done from Lemmas 6.35 to the end of the section 6.H. The Schwartz function $\phi_{2,\tau_0,p}$ is the $\phi'_{2,p}$ given below Definition 6.38. Note that our $\FJ_{\beta,v}\big(g,x;f_v(s,\xi_0\tau_0)\big)$ defined in Definition 4.5.1. equals to $\FJ_\beta(f_{sieg,v};z,x,g\eta^{-1},1)$ in [40] with $z=s$ and $\eta=\begin{pmatrix}&\bid_2\\-\bid_2\end{pmatrix}$, so the results in loc.cit all contain the extra $\eta$ but the formulas in this proposition do not.

At the place $p$, in fact the pairing of $\phi_{1,p}(x)$ and  $\FJ_{\beta,p}\big(g,(x,y)_p;f_p(s,\xi_0\tau_0\big)$ along $x$ does not directly equal $\phi_{2,\tau,p}(y)\cdot f_{2,p}(s,\xi_0\tau_0\lambda^{-1})(g)$, but equals the average

\begin{equation}\label{eq:real-p}
\int_{\bZ_p}\omega_{\beta,p}\left(\varrho_{\fp}^{-1}\begin{pmatrix}1&0\\n&1\end{pmatrix},1\right)\phi_{2,\tau,p}(y)\cdot f_{2,p}(s,\tau_0\lambda^{-1})\left(g\Upsilon'_p\begin{pmatrix}1&\\&1\\&&1&\\&&n&1\end{pmatrix}\right)\,dn.
\end{equation}

Meanwhile, $\varphi$ is spherical at $p$, in particular invariant under the right translation by $\begin{pmatrix}1&\\\bZ_p&1\end{pmatrix}$, so by replacing (<ref>) $\phi_{2,\tau,p}(y)\cdot f_{2,p}(s,\xi_0\tau_0\lambda^{-1})(g)$ does not change the resulting Petersson inner product with $\varphi$ over $\U(V^-)=\U(2)$.

The Schwartz function (<ref>) is the the $(v_3,v_4)$-part of $\Phi_{\cD,p}$ where $\Phi_{\cD,p}$ is the Schwartz function on $\bQ^4_p$ defined and appearing in <cit.>. As mentioned above, our formula  in the Proposition corresponds to replacing $g$ by $g\eta^{-1}$ in the formula in loc.cit, so there is a translation by $\begin{pmatrix}&-1\\1\end{pmatrix}$ in  the theta series in (<ref>).

§.§ The construction of the auxiliary CM families $ h,\tilde{h}_3,\theta,\tilde{\theta}_3$

Besides $\theta^J_1$, our strategy described in <ref> also includes auxiliary CM forms. We want to construct a CM family $\bfh$ on $\U(V)=\U(2)$ and pair it with $l_{\theta^J_{1}}\left(\sum_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)$. The splitting characters for the theta correspondence between $\U(2)$ and $\U(1)$ are chosen as in (<ref>).

§.§.§ The auxiliary group $U_{\cK,p}$

First, we note that for $\tau_{p\adic}$ as in Theorem <ref>, the nebentypus of $l_{\theta^J_{1}}\left(\sum_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)$ at $p$ is $\left(\triv,(\xi_0\tau_0)_\fp|_{\bZ^\times_p}\right)$ (i.e. $\varrho^{-1}_\fp\begin{pmatrix}\ast&\ast\\0&a_2\end{pmatrix}\in \U(2)(\bZ_p)$ acts by $(\xi_0\tau_0)_\fp(a_2)$). In order to get a CM family $\bfh$ with the correct nebentypus to pair with $l_{\theta^J_{1}}\left(\sum_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)$, we want Hecke characters of $\cK^\times\backslash\bA^\times_\cK$ unramified away from $p\infty$ with restriction  $\left(\tau_{0,\fp}|_{\bZ^\times_p},\triv\right)$ on $\cO^\times_{\cK,p}$. However, in general there is no canonical way to extend local characters $\left(\tau_{0,\fp}|_{\bZ^\times_p},\triv\right)$ of $\cO^\times_{\cK,p}$  to global Hecke characters. Hence, we need to consider an auxiliary group $U_{\cK,p}$ and the non-standard type of $p$-adic measures defined in (<ref>).

Let $\glsuseri{UKp}=1+p\cO_{\cK,p}$ and $\glsuserii{UKp}=1+p\cO_{\cK,\fp}$, $\glsuseriii{UKp}=1+p\cO_{\cK,\bar{\fp}}$. Then the natural map

\[
U_{\cK,p}\lra \cK^\times\backslash\bA^\times_{\cK,f}\lra \Gamma_\cK
\]

is an injection. We can pick $\gls{wtp}$, a non-negative power of $p$, such that raising to the $\wtp$-th power maps $\Gamma_\cK$ into $U_{\cK,p}$. Define

\[
\begin{tikzcd}[column sep = huge]
\gls{PN}:\Gamma_\cK   \arrow[r,hook,"\text{$\wtp$-th power}"] &U_{\cK,p} \arrow[r,hook,"\text{natural proj.}"] & U_{\cK,\fp}\simeq 1+p\bZ_p.
\end{tikzcd}
\]

Then given a (local) $p$-adic character $\epsilon:U_{\cK,\fp}\ra\ol{\bQ}_p^\times$, the composition $\epsilon\circ \pP_\wtp$ is a  (global) $p$-adic character of $\Gamma_\cK$ whose restriction back to $U_{\cK,p}=U_{\cK,\fp}\times U_{\cK,\bar{\fp}}$ is $\left(\epsilon^{\wtp},\triv\right)$.

In particular, for an algebraic Hecke character $\tau:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ of $\infty$-type $\left(0,k\right)$ whose $p$-adic avatar $\tau_{p\adic}$ factors through $\Gamma_\cK$, we can define

\[
\gls{tauP}=\left.\tau_{p\adic}\right|_{U_{\cK,\fp}}\circ \pP_N.
\]

Then $\tau_{\fp,\pP_\wtp}$ is a $p$-adic Hecke character of $\Gamma_\cK$ with

\[
\tau_{\fp,\pP_\wtp}|_{\cO^\times_{\cK,p}}=\left(\tau^{\wtp}_{\fp}|_{\bZ^\times_p},\triv\right)=\left(\tau^\wtp_{0,\fp}|_{\bZ^\times_p},\triv\right).
\]

Because $\tau$ has $\infty$-type $(0,k)$, the local character $\tau_{p\adic}|_{U_{\cK,\fp}}$ is of finite order, and it follows that the (global) character $\tau_{\fp,\pP_\wtp}:\Gamma_\cK\ra\ol{\bQ}^\times_p$ is also of finite order. Hence, it takes values in $\ol{\bQ}^\times$ and is also an algebraic Hecke character of $\cK^\times\backslash\bA^\times_\cK$ of $\infty$-type $\left(0,0\right)$. (From the definition in (<ref>), we can see that the $p$-adic avatar of a Hecke character of $\infty$-type $(0,0)$ is itself.)

The map $\pP_\wtp$ induces a map

\begin{align}
\label{eq:PN*}\gls{PNast}:\Meas\left(\Gamma_\cK,V_{\U(2)}\right)^\natural
&\lra \Meas\left(U_{\cK,p},V_{\U(2)}\right)^\natural.
\end{align}

(See <ref> for the definition of these spaces of $p$-adic measures.) We have

\[
\pP_{\wtp,\ast}\left(l_{\theta^J_{1}}\left(\sum\nolimits_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)\right)\left(\tau_{p\adic}|_{U_{\cK,_p}}\right)= l_{\theta^J_{1}}\left(\sum\nolimits_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)(\tau^\wtp_{p\adic})
\]

for every $\tau_{p\adic}\in\Hom_\cont\left(\Gamma_\cK,\ol{\bQ}^\times_p\right)$. In order to show the nonvanishing property of $l_{\theta^J_{1}}\left(\sum_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)$ in Proposition <ref>, it suffices to show the nonvanishing for $\pP_{\wtp,\ast}\left(l_{\theta^J_{1}}\left(\sum_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)\right)$.

Next, we construct auxiliary CM families $\bfh$, $\tilde{\bfh}_3$, $\bmtheta$, $\tilde{\bmtheta}_3$ on $\U(2)$ as $p$-adic measures on $U_{\cK,p}$.

§.§.§ The auxiliary CM families $\bfh$ and $\tilde{\bfh}_3$

There exist a CM family $\glsuseri{h0}\in \Meas\left(U_{\cK,p},e_\ord V'_{\U(2),\,\xi^{-1}}\,\hat{\otimes}\,\hat{\cO}^\ur_L\right)^{\natural}$ and a CM family $\glsuserii{h0}\in \Meas\left(U_{\cK,p},e_\ord V_{\U(2),\,\xi}\,\hat{\otimes}\,\hat{\cO}^\ur_L\right)^{\natural}$ such that for every $\tau_{p\adic}$ as in Theorem <ref> sufficiently ramified at $p$,

\begin{align*}
\bfh_0\left(\tau_{p\adic}|_{U_{\cK,p}}\right)(g)&=\int_{\U(1)(\bQ)\backslash\U(1)(\bA_\bQ)} \theta\left(\phi_{h,\tau}\right)(g,u)\left(\chi_h\tau_{\fp,\pP_\wtp}\right)(u)\,du,\\
\bfh'_0\left(\tau_{p\adic}|_{U_{\cK,p}}\right)(g)&=\int_{\U(1)(\bQ)\backslash\U(1)(\bA_\bQ)} \theta\left(\phi'_{h,\tau}\right)(g,u)\left(\chi^{-1}_h\tau^{-1}_{\fp,\pP_\wtp}\right)(u)\,du,
\end{align*}

the theta lift of $\chi^{-1}_h\tau^{-1}_{\fp,\pP_\wtp}\Big|_{\U(1)}$ (resp. $\chi_h\tau_{\fp,\pP_\wtp}\Big|_{\U(1)}$) to $\U(2)$) with respect to the Schwartz function $\phi_{h,\tau}$ (resp. $\phi'_{h,\tau}$) on $\bA^2_\bQ$ described as follows:

– $v\notin \Sigma\cup \{\ell,\ell',p\}$.

\[
\phi_{h,\tau,v}=\phi'_{h,\tau,v}=\phi_{2,\xi\tau^\wtp,v} \text{ in Proposition~\ref{prop:phi2f2}}.
\]

– $v=\fv\bar{\fv}\in \Sigma_\rms\cup\{\ell\}$. We have $\chi_h\tau_{\fp,\pP_\wtp}|_{\cO^\times_{\cK,v}}=\chi_h|_{\cO^\times_{\cK,v}}$.

\begin{align*}
\phi_{h,\tau,v}&=\text{Image of \eqref{eq:int-S} of }\Big((y_1,y_2)\longmapsto \mathds{1}_{\bZ_v}(y_1 )\cdot\mathds{1}_{\bZ^\times_v}(y_2)\chi_{h,\fv}\chi^{-1}_{h,\bar{\fv}}\lambda^{-1}_{\bar{\fv}}(y_2)\Big),\\
\phi'_{h,\tau,v}&=\text{Image of \eqref{eq:int-S} of }\Big((y_1,y_2)\longmapsto \mathds{1}_{\bZ_v}(y_1 )\cdot\mathds{1}_{\bZ^\times_v}(y_2)\chi^{-1}_{h,\fv}\chi_{h,\bar{\fv}}\lambda_{\bar{\fv}}(y_2)\Big).
\end{align*}

– $v\in \Sigma_\ns\cup\{\ell'\}$. We have $\left.\chi_h\tau_{\fp,\pP_\wtp}\right|_{\U(1)(\bQ_v)}=\left.\chi_h\right|_{\U(1)(\bQ_v)}$.

\begin{align*}
\phi_{h,\tau,v}=&\,\text{a Schwartz function on $\bQ^2_v$ invariant under $K_v$, independent of $\tau$, }\\
&\,\text{belonging to the $\chi^{-1}_{h,v}|_{\U(1)(\bQ_v)}$-eigenspace for the action of $\U(1)(\bQ_v)$},\\
\phi'_{h,\tau,v}=&\,\text{a Schwartz function on $\bQ^2_v$ invariant under $K_v$,, independent of $\tau$, }\\
&\,\text{belonging to the $\chi_{h,v}|_{\U(1)(\bQ_v)}$-eigenspace for the action of $\U(1)(\bQ_v)$,}\\
&\,\text{and }\int_{\bQ^2_v}\phi_{h,\tau,v}(y)\phi'_{h,\tau,v}(y)\,dy\neq 0.
\end{align*}

– $v=p$. We have $(\chi_h\tau_{\fp,\pP_\wtp})_\fp=(\xi_0\tau^\wtp_0)_\fp$. Let $t=\ord_p\left(\cond\left((\xi_0\tau^\wtp_0)_{\fp}\right)\right)$.

\begin{align*}
\phi_{h,\tau,p}&=\text{Im of \eqref{eq:int-S} of }\\
&\quad\Big((y_1,y_2)\longmapsto p^{-t}\fg\left((\xi_0\tau^\wtp_0)^{-1}_{\fp}\right))\cdot \mathds{1}_{p^{-t}\bZ^\times_p}(y_2)(\xi_0\tau^{\wtp}_0)_{\fp}(-p^ty_2)\cdot \mathds{1}_{\bZ_p}(y_1)\Big),\\
\phi'_{h,\tau,p}&=\text{Image of \eqref{eq:int-S} of }\Big((y_1,y_2)\longmapsto  \mathds{1}_{\bZ^\times_p}(y_1)(\xi_0\tau^{\wtp}_0)_{\fp}(y_1)\cdot \mathds{1}_{\bZ_p}(y_2)\Big).
\end{align*}

See <cit.>.  The idea of constructing $\bfh_0$ (resp. $\bfh'_0$) is to first use the chosen Schwartz function to construct a family of theta series on $\U(2,2)$ by interpolating the $q$-expansions, and then restrict the family to $\U(V)\times\U(V^-)$ and evaluate at a suitable point $u_{aux}\in\U(V)(\bA_{\bQ})$ (resp. $u'_{aux}\in\U(V^-)(\bA_{\bQ})$) to get the desired family on $\U(V^-)$ (resp. $\U(V)$).

The map (<ref>) between automorphic forms on $\U(2)$ defined in terms of the tame character $\chi_h\chi^c_\theta$ extends to a map from $V_{\U(2)}$ to $V_{\U(2)}$. Applying this twisting map to $\bfh_0$ in Proposition <ref> defines the CM family $\glsuseri{h}\in \Meas\left(U_{\cK,p},e_\ord V'_{\U(2),\,\xi^{-1}}\right)^{\natural}$.

Replacing  $\chi_h\chi^c_\theta$ by $\chi^{-1}_h\chi^{-c}_\theta$ and applying the corresponding twisting map map to $\bfh'_0$ in Proposition <ref> defines the CM family $\glsuserii{h}\in \Meas\left(U_{\cK,p},e_\ord V_{\U(2),\,\xi}\right)^{\natural}$.

§.§.§ The auxiliary CM families $\bmtheta$ and $\tilde{\bmtheta}_3$

We construct a CM family $\bmtheta$ whose specializations are closely related to the theta series $\theta(\phi_{2,\tau})$ in Proposition <ref>. (The theta series $\theta(\phi_{2,\tau})$ is not an eigenform and does not belong to the theta lift of one character of $\U(1)$. The specialization of  $\bmtheta$ at $\tau_{p\adic}|_{U_{\cK,p}}$ is essentially the projection of $\theta(\phi_{2,\tau^\wtp})$ to the theta lift of $\lambda^2\chi_\theta\tau^{-c}_{\fp,\pP_\wtp}|_{\U(1)}$.) We also construct a CM family $\tilde{\bmtheta}_3$ dual to $\bmtheta$.

(1) There exist a CM family $\glsuseri{theta}\in\Meas\left(U_{\cK,p},e_\ord V_{\U(2),\,\xi}\,\hat{\otimes}\,\hat{\cO}^\ur_L\right)^\natural$ and also a CM family $\glsuserii{theta}\in\Meas\left(U_{\cK,p},e_\ord V'_{\U(2),\,\xi^{-1}}\,\hat{\otimes}\,\hat{\cO}^\ur_L\right)^{\natural}$ such that for every $\tau_{p\adic}$ as in Theorem <ref> ramified at $p$,

\begin{equation}\label{eq:bftheta}
\begin{aligned}
\bmtheta\left(\tau_{p\adic}|_{U_{\cK,p}}\right)(g)&= \int_{\U(1)(\bQ)\backslash\U(1)(\bA_\bQ)}  \theta\left(\phi_{\theta,\tau}\right)\left(g,u\right)\left(\lambda^2\chi_\theta\tau^{-c}_{\fp,\pP_\wtp}\right)^{-1}(u)\,du,\\
\tilde{\bmtheta}_3\left(\tau_{p\adic}|_{U_{\cK,p}}\right)(g)&= \int_{\U(1)(\bQ)\backslash\U(1)(\bA_\bQ)}  \theta\left(\phi'_{\theta,\tau}\right)\left(g,u\right)\,(\lambda^2\chi_\theta\tau^{-c}_{\fp,\pP_\wtp})(u)\,du,
\end{aligned}
\end{equation}

the theta lift of $\lambda^2\chi_\theta\tau^{-c}_{\fp,\pP_\wtp}\Big|_{\U(1)}$ (resp. $\lambda^{-2}\chi^{-1}_\theta\tau^{c}_{\fp,\pP_\wtp}\Big|_{\U(1)}$) to $\U(2)$ with respect to the Schwartz function $\phi_{\theta,\tau}$ (resp. $\phi'_{\theta,\tau}$) on $\bA^2_\bQ$ described as follows:

– $v\notin\Sigma\,\cup\,\{\ell,\ell',p\}$.

\begin{align*}
\phi_{\theta,\tau,v}=\phi'_{\theta,\tau,v}=\phi_{2,\xi\tau^\wtp,v} \text{in Proposition~\ref{prop:phi2f2}}.
\end{align*}

– $v=\fv\bar{\fv}\in\Sigma_\rms\,\cup\,\{\ell'\}$. We have $\chi_{\theta}\tau^{-c}_{\fp,\pP_\wtp}|_{\cO^\times_{\cK,v}}=\chi_{\theta}|_{\cO^\times_{\cK,v}}$ and $\chi_{\theta,\fv}$ unramified.

\begin{align*}
\phi_{\theta,\tau,v}&=\text{Image of \eqref{eq:int-S} of } \Big((y_1,y_2)\longmapsto \mathds{1}_{\cO_{\cK,\fv}}(y_1)\cdot \mathds{1}_{\cO^\times_{\cK,\bar{\fv}}}(y_2)\chi_{\theta,\bar{\fv}}\lambda_{\bar{\fv}}(y_2)\Big)\\
&=\phi_{2,\tau^\wtp,v} \text{in Proposition~\ref{prop:phi2f2}},\\
\phi'_{\theta,\tau,v}&=\text{Image of \eqref{eq:int-S} of } \Big((y_1,y_2)\longmapsto \mathds{1}_{\cO_{\cK,\fv}}(y_1)\cdot \mathds{1}_{\cO^\times_{\cK,\bar{\fv}}}(y_2)(\chi_{\theta,\bar{\fv}}\lambda_{\bar{\fv}})^{-1}(y_2)\Big)
\end{align*}

– $v\in \Sigma_\ns\,\cup\,\{\ell'\}$. We have $\chi_{\theta}\tau^{-c}_{\fp,\pP_\wtp}|_{\U(1)(\bQ_v)}=\chi_{\theta}|_{\U(1)(\bQ_v)}$.

\begin{align*}
\phi_{\theta,\tau,v}&=\phi_{2,\xi\tau^\wtp,v} \text{in Proposition~\ref{prop:phi2f2}},\\
\phi'_{\theta,\tau,v}&=\text{a Schwartz function on $\bQ^2_v$ invariant under $K_v$, independent of $\tau$,}\\
&\quad\text{belonging to the $\chi^{-1}_\theta\lambda^{-2}|_{\U(1)(\bQ_v)}$-eigenspace for the action of $\U(1)(\bQ_v)$, }\\
&\quad\text{and }\int_{\bQ^2_v}\phi_{\theta,\tau,v}(y) \phi'_{\theta,\tau,v}(y)\,dy\neq 0
\end{align*}

– $v=p$. We have $(\chi_\theta\tau^{-c}_{\fp,\pP_\wtp})_{\bar{\fp}}=(\xi_0\tau^\wtp_0)^{-1}_\fp$. Let  $t=\ord_p\left(\cond\left((\xi_0\tau^\wtp_0)_\fp\right)\right)$.

\begin{align*}
\phi_{\theta,\tau,p}&=\text{Image of \eqref{eq:int-S} of }\\
&\quad\Big((y_1,y_2)\longmapsto p^{-t}\fg\left((\xi_0\tau^{\wtp}_0)^{-1}_{\fp}\right)\cdot \mathds{1}_{p^{-t}\bZ^\times_p}(y_1)(\xi_0\tau^{\wtp}_0)_{\fp}(-p^ty_1)\cdot \mathds{1}_{\bZ_p}(y_2)\Big),\\
\phi'_{\theta,\tau,p}&=\text{Image of \eqref{eq:int-S} of }\Big((y_1,y_2)\longmapsto  \mathds{1}_{\bZ_p}(y_1)\cdot \mathds{1}_{\bZ^\times_p}(y_2)(\xi_0\tau^\wtp_0)_{\fp}(y_2)  \cdot\Big),
\end{align*}

where $\phi_{2,\tau^\wtp,p}$ is as in Proposition <ref>.

(2) The specialization $\bmtheta\left(\tau_{p\adic}|_{U_{\cK,p}}\right)$ equals the projection of $\sum\limits_{n\in\bZ/p^t\bZ}\theta(\phi_{2,\tau^\wtp})\left(g\begin{pmatrix}1&0\\n&1\end{pmatrix}\right)$ to the $\left.\lambda^2\chi_\theta\tau^{-c}_{\fp,\pP_\wtp}\right|_{\U(1)}$-eigenspace for the action of the center of $\U(2)$. (Note that the projection of a theta series to the $\left.\lambda^2\chi_\theta\tau^{-c}_{\fp,\pP_\wtp}\right|_{\U(1)}$-eigenspace for the action of the center is a theta lift of $\left.\lambda^2\chi_\theta\tau^{-c}_{\fp,\pP_\wtp}\right|_{\U(1)}$.)

See <cit.>. The idea is the same as the construction of $\bfh_0$ and $\bfh'_0$, and (2) follows immediately from (1).

§.§ $p$-adic Petersson inner product on $\U(2)$

We need $p$-adic Petersson inner product for families on $\U(2)$. Let $V_{\U(2)}$, $e_\ord V'_{\U(2)}$ be the space of $p$-adic forms on $\U(2)$ defined in <ref>.

Following <cit.>, we can define a $p$-adic Petersson inner product

\begin{equation}\label{eq:p-inner}
\left<\bdot,\bdot\right>_{p\adic}: V'_{\U(2),\ord}\times V_{\U(2)}\lra \cO_L
\end{equation}

such that for all $\phi\in V_{\U(2)}$ (resp. $\phi'\in e_\ord V'_{\U(2)}$) of with $K_{p,0}(p^n)$-nebentypus $(\triv,\chi)$ (resp. $(\chi',\triv)$)

\[
\glsuseri{Petersson} =\left\{\begin{array}{ll}
\sum\limits_{g}\, (U^{-n}_p\phi')\left(g\varrho^{-1}_\fp\begin{pmatrix}&1\\-p^n\end{pmatrix}\right)\phi\left(g\right), &\chi\chi'=\triv,\\[1.5em]
0, &\chi\chi'\neq\triv,
\end{array}\right.
\]

where $K_{p,0}(p^n)=\left\{g\in\U(2)(\bZ_p):\varrho_\fp(g)\equiv\begin{pmatrix}\ast&\ast\\0&\ast\end{pmatrix}\mod p^n\right\}$, and in the sum, $g$ runs over the finite set $\U(2)(\bQ)\backslash \U(2)(\bA^p_{\bQ,f})/K^p_f K_{p,0}(p^n)$.

A useful property of this $p$-adic Petersson inner product is that

\[
\left<\phi',U_p\phi\right>_{p\adic}=\omega_{\phi',p}(\varrho^{-1}_\fp(p))^{-1}\left<U_p \phi, \phi'\right>_{p\adic}.
\]

if $\phi'$ has central character $\omega_{\phi'}$. Here $\varrho^{-1}_\fp$ is the isomorphism from $\bQ^\times_p$ to $\U(1)(\bQ_p)$.

The $p$-adic Petersson inner product in (<ref>) induces the $p$-adic Petersson inner product for $p$-adic families  on $\U(2)$:

\begin{equation}\label{eq:Petersson-family}
\left<\bdot,\bdot\right>_{p\adic}: \Meas\big(U_{\cK,p},e_\ord V'_{\U(2),\,\xi^{-1}}\big)^\natural\times\Meas\left(U_{\cK,p},V_{\U(2),\,\xi}\right)^\natural
\lra \Meas\big(U_{\cK,p},\cO_L\big)\simeq \cO_L\llb U_{\cK,p}\rrb.
\end{equation}

§.§ The Rallis inner product formulas for $\big<\boldmath \theta,\tilde{\theta}_3\big>_{p\adic}$ and $\big<\boldmath h,\tilde{h}_3\big>_{p\adic}$.

From the description of our strategy in <ref>, in particular (<ref>), we see that we need a formula relating $\big<\bmtheta,\tilde{\bmtheta}_2\big>_{p\adic}$ and $\big<\bfh,\tilde{\bfh}_3\big>_{p\adic}$ to certain  $p$-adic $L$-functions.

Let $\gls{L3}\in \Meas\left(U_{\cK,p},\hat{\cO}^\ur_L\right)\otimes_\bZ\bQ\simeq\hat{\cO}^\ur_L\llb U_{\cK,p}\rrb\otimes_\bZ\bQ$ be the $L$-function interpolating the special value at $s=1$ of $L\left(s,\pi^{\GL_2}_\theta,\mr{Ad}\right)$, $L\left(s,\pi^{\GL_2}_h,\mr{Ad}\right)$, which factorize as

\begin{align*}
L\left(s,\pi^{\GL_2}_h,\mr{Ad}\right)&= L\left(s,\eta_{\cK/\bQ}\right)L\left(s,\lambda^{-2}\chi^{-1}_h\chi^{c}_h\tau^{-1}_{\fp,\pP_\wtp}\tau^{c}_{\fp,\pP_\wtp}\right)\\
\end{align*}

More precisely, for $\tau_{p\adic}$ as in Theorem <ref>,

\begin{align*}
\cL_3\left(\tau_{p\adic}|_{U_{\cK,p}}\right)
&=\pi^{-1}\cdot L^\infty\left(1,\eta_{\cK/\bQ}\right)\\
&\quad \times \left(\frac{\Omega_p}{\Omega_\infty}\right)^2\pi^{-2}\cdot \gamma_{\bar{\fp}}\left(0,\lambda^2\chi_\theta\chi^{-c}_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)^{-1}\cdot L^{p\infty}\left(1,\lambda^2\chi_\theta\chi^{-c}_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right),\\
\cL_4\left(\tau_{p\adic}|_{U_{\cK,p}}\right)
&=\pi^{-1}\cdot L^\infty\left(1,\eta_{\cK/\bQ}\right)\\
&\quad \times \left(\frac{\Omega_p}{\Omega_\infty}\right)^2\pi^{-2}\cdot \gamma_{\bar{\fp}}\left(0,\lambda^2\chi_h\chi^{-c}_h\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)^{-1}\cdot L^{p\infty}\left(1,\lambda^2\chi_h\chi^{-c}_h\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right).
\end{align*}

The existence of $\cL_3,\cL_4$ follows from Katz $p$-adic $L$-functions.

There exists $C_\theta,C_h\in L^\times$ such that

\begin{align*}
\big<\tilde{\bmtheta}_3,\bmtheta\big>_{p\adic}&=C_\theta\cdot \cL_3,
&\big<\bfh,\tilde{\bfh}_3\big>_{p\adic}&=C_h\cdot \cL_4.
\end{align*}

(See (<ref>) for the definition of the $p$-adic Petersson inner product for families.)

The proof is the same as <cit.>, which computes the specializations of the left hand side by using the Rallis inner product formula.

§.§ Extending CM forms on $\U(2)$ to $\GU(2)$

Let $H=\GU(2)(\bQ)Z_{\GU(2)}(\bA_\bQ)\U(2)(\bA_\bQ)$. Then $H$ is a subgroup of  $\GU(\bA_\bQ)$ of index $2$ consisting of $g\in\GU(2)(\bA)$ with $\nu(g)\in \bQ^\times\Nm(\bA^\times_\cK)$. Suppose that $\phi$ is an automorphic form on $\U(2)$ and generates an irreducible automorphic representation $\pi_\phi$ of $\U(2)(\bA_\bQ)$. Let $\omega:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ be a Hecke character extending the central character of $\pi_\phi$. We can first extend $\phi$ to a function on $H$ by

\begin{align*}
\gamma a g_1&\longmapsto \omega(a)\phi(g_1), &\gamma\in\GU(2)(\bQ),\,a\in Z_{\GU(2)}(\bA_\bQ),\,g_1\in\U(2)(\bA_\bQ),
\end{align*}

and then extend this function by zero from $H$ to $\GU(2)(\bA_\bQ)$. We denote this extension of $\phi$ to $\GU(2)(\bA_\bQ)$ by $\breve{\phi}$. This $\breve{\phi}$ is an automorphic form on $\GU(2)(\bA_\bQ)$.

Suppose that $\pi_\phi$ is of CM type, i.e. if $\mr{BC}(\pi_\phi)\simeq \mr{BC}(\chi_1)\lambda\oplus\mr{BC}(\chi_2)\lambda$ for some characters $\chi_1,\chi_2$ of $\U(1)(\bQ)\backslash\U(1)(\bA_\bQ)$. Then $\breve{\phi}$ generates an irreducible automorphic representation $\pi_{\breve{\phi}}$ of  $\GU(2)(\bA)$ and $\pi_{\breve{\phi}}|_{\U(2)}\simeq \pi_\phi\oplus\pi'_\phi$, where $\pi'_\phi$ is isomorphic to the conjugation of $\pi_\phi$ by an element in $\GU(2)(\bA_\bQ)$ outside $H$. Moreover, if $\phi\in\pi_\phi$ is a pure tensor in $\pi_\phi\simeq\bigotimes_v\pi_{\phi,v}$, then $\breve{\phi}$ is also a pure tensor in $\pi_{\breve{\phi}}\simeq\bigotimes_v\pi_{\breve{\phi},v}$.

Let $\sigma$ be the automorphic representation of $\GU(2)(\bA_\bQ)$ generated by $\breve{\phi}$. Then either $\sigma$ is irreducible and $\sigma\simeq \sigma\otimes\eta_{\cK/\bQ}\circ \nu$ or $\sigma\simeq \sigma_1\oplus \sigma_2$ with $\sigma_1\not\simeq\sigma_2$ and $\sigma_2\simeq \sigma_1\otimes\eta_{\cK/\bQ}\circ\nu$. If $\pi_\phi$ is of CM type and $\sigma_1$ is a direct summand of $\sigma$, we know that $\sigma_1\simeq\sigma_1\otimes\eta_{\cK/\bQ}\circ\nu$. Hence, $\sigma$ must be irreducible. (See also <cit.>.) We have $\pi_{\breve{\phi}}\simeq \pi_\phi$. Since $\breve{\phi}$ is obtained by extension by zero, it belongs to the direct summand $\pi_\phi$ in $\pi_{\breve{\phi}}|_{\U(2)}\simeq \pi_\phi\oplus\pi'_\phi$. Hence, $\phi$ being a pure tensor in $\pi_\phi$ implies that $\breve{\phi}$ is a pure tensor in $\pi_{\breve{\phi}}$.

The above extension from $\U(2)$ to $\GU(2)$ also works for $p$-adic automorphic forms. We denote by $\glsuseri{hext}$  the extension of $\bfh$  whose specialization at $\tau_{p\adic}$ is the extension by zero of $\bfh(\tau_{p\adic})$ to $\GU(2)$ with respect to the character $\chi^{-1}_\theta\tau^{-1}_{\fp,\pP_\wtp}$. Similarly, We denote by $\glsuserii{hext}$ (resp. $\glsuseri{thetaext},\glsuserii{thetaext}$) the extension of $\tilde{\bfh}_3$  whose specialization at $\tau_{p\adic}$ is the extension by zero of $\tilde{\bfh}_3(\tau_{p\adic})$ to $\GU(2)$ with respect to the character $\chi_\theta\tau_{\fp,\pP_\wtp}$ (resp. $\chi_\theta\tau_{\fp,\pP_\wtp},\chi^{-1}_\theta\tau^{-1}_{\fp,\pP_\wtp}$).

§.§ The nonvanishing property of the degenerate Fourier–Jacobi coefficients of the Klingen Eisenstein family

Let $\gls{L5}\in \Meas\left(U_{\cK,p},\hat{\cO}^\ur_L\right)\simeq\hat{\cO}^\ur_L\llb U_{\cK,p}\rrb$ be the Katz $p$-adic $L$-functions interpolating special values of $L\left(s,\chi_h\chi^c_\theta\xi_0\tau^\wtp_0\right)$, $L\left(s,\lambda^2\chi_h\chi_\theta\,\tau^{-1}_{\fp,\pP_\wtp}\tau^{c}_{\fp,\pP_\wtp}(\xi_0\tau^\wtp_0)^c\right)$. In particular, for a Hecke character $\tau:\cK^\times\backslash\bA^\times_\cK\ra\bC^\times$ unramified away from $p\infty$ of $\infty$-type $\left(0,k\right)$ with $k\geq 6$,

\begin{align*}
\cL_5\left(\tau_{p\adic}|_{U_{\cK,p}}\right)&=\left(\frac{\Omega_p}{\Omega_\infty}\right)^{\wtp k}\frac{\Gamma(\wtp k-1)}{(2\pi i)^{\wtp k-1}}
\cdot \gamma_{\bar{\fp}}\left(\frac{4-\wtp k}{2},(\chi_h\chi^{c}_\theta\xi_0\tau^{\wtp}_0)^{-1}\right) \\
&\quad\times L^{p\infty}\left(\frac{\wtp k-2}{2},\chi_h\chi^c_\theta\xi_0\tau^\wtp_0\right),\\
\cL_6\left(\tau_{p\adic}|_{U_{\cK,p}}\right)&=\left(\frac{\Omega_p}{\Omega_\infty}\right)^{\wtp k-2}\frac{\Gamma(\wtp k-2)}{(2\pi i)^{\wtp k-2}}
\cdot  L_{\fp}\left(\frac{\wtp k-2}{2},\lambda^2\chi_h\chi_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}(\xi_0\tau^\wtp_0)^c\right),\\
&\quad\times L^{p\infty}\left(\frac{\wtp k-2}{2},\lambda^2\chi_h\chi_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}(\xi_0\tau^\wtp_0)^c\right).
\end{align*}

The product of $\cL_5\cL_6$ interpolates the special values of $L(s,\mr{BC}(\pi_h)\times\xi_0\tau_0\lambda^{-1})$ where $\pi_h$ is the automorphic representation associated to specializations of the CM family $\bfh$. Note that the restriction of $\lambda^2\chi_h\chi_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}(\xi_0\tau^\wtp_0)^c$ to $\cO^\times_{\cK,p}$ is $\left(\xi_{0,\fp}\xi_{0,\bar{\fp}}\tau^\wtp_{0,\fp}\tau^\wtp_{0,\bar{\fp}},\triv\right)$, so $\cL_6$ is essentially of one-variable. In fact $\cL_6$ is the so-called “improved” $p$-adic $L$-function, for which the local factor at $p$ in the interpolation formula is a partial local $L$-factor instead of a partial local $\gamma$-factor.

For $\sT_\ns\in \bigotimes\limits_{v\in\Sigma_\ns\cup\{\ell'\}}\cO_L\left[\GU(2)\big(\bQ_v\big)\right]$ such that $\sT_\ns\bfh$ is still invariant under the tame level group $K^p_f$ (defined (<ref>)), $\varphi$ inside the space (<ref>), and $u_1,\dots,u_r\in \bigotimes\limits_{v\in\Sigma_\ns\cup\{\ell'\}}\U(1)(\bQ_v)$, $b_1,\dots,b_r\in\cO_L$ as in Proposition <ref>,

\begin{equation}\label{eq:FJ-h}
\left<\left(\sT_\ns\breve{\bfh}\right)|_{\U(2)},\pP_{N,\ast}\left(l_{\theta^J_1}\left(\sum\nolimits_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)\right)\right>_{p\adic}
=\cC_1\cdot\cL_5\cL_6\cdot\left<\left(\sT_\ns\breve{\bfh}\right)|_{\U(2)},\bmtheta^{\bar{\lambda}}  \varphi\right>_{p\adic},
\end{equation}

where $\bmtheta^{\bar{\lambda}}=\bmtheta\cdot(\bar{\lambda}\circ \det)$ and $\cC_1\in \left(\hat{\cO}^\ur_L\llb U_{\cK,p}\rrb\otimes_\bZ\bQ\right)^\times$.

The identity is proved by comparing the evaluations of both sides at $\tau_{p\adic}$ as in Theorem <ref>. For such a $\tau_{p\adic}$, put $\theta_\tau=\bmtheta(\tau_{p\adic}|_{U_{\cK,p}})$ and $h_\tau=\bfh(\tau_{p\adic}|_{U_{\cK,p}})$, which are classical ordinary CM forms. The description of $l_{\theta^J_1}\left(\sum_i b_i\bfE^\Kling_{\varphi,\beta,u_i}\right)$ in Proposition <ref> and the definition of the $p$-adic Petersson inner product in <ref> imply that

\begin{equation}\label{eq:h-L5L6}
\begin{aligned}
&\left(\frac{\Omega_p}{\Omega_\infty}\right)^{-2\wtp k+2}\cdot\text{LHS of \eqref{eq:FJ-h} evaluated at $\tau_{p\adic}$}\\
=&\,d^{\Sigma\,\cup\{p,\ell,\ell'\}}_2\left(s,\xi_0\tau^\wtp_0\lambda^{-1}\right)\int_{\U(2)(\bQ)\backslash \U(2)(\bA_\bQ)}\theta_\tau\left(g_2\varrho^{-1}_\fp\begin{pmatrix}&1\\-1\end{pmatrix}\right)\varphi^{\bar{\xi}_0\bar{\tau}^\wtp_0}(g_2) \int_{\U(2)(\bQ)\backslash \U(2)(\bA_\bQ)}\\
&\left. E^\Sieg_2\left(\imath_0(g_1,g_2);f_2(s,\xi_0\tau^\wtp_0\lambda^{-1}\right)\xi^{-1}_0\tau^{-\wtp}_0\lambda(\det g_1)\cdot(U^{-m}_p \sT_\ns \breve{h}_\tau)\left(g_1\begin{pmatrix}&1&\\-p^m\end{pmatrix}_p\right)\,dg_1dg_2\,\right|_{s=\frac{\wtp k-3}{2}}\\
=&\,\int_{\U(2)(\bQ)\backslash \U(2)(\bA_\bQ)}\theta^{\bar{\lambda}}_\tau\left(g_2\right)\varphi(g_2)\, h'_\tau(g_2)\,dg_2
\end{aligned}
\end{equation}

with $\imath_0$ defined in (<ref>) and

\begin{equation*}
\begin{aligned}
h'_\tau(g_2)&= d^{\Sigma\,\cup\{p,\ell,\ell'\}}_2\left(s,\xi_0\tau^\wtp_0\lambda^{-1}\right)^{-1}\\
&\hspace{-3em}\times\left.\int_{\U(2)(\bA_\bQ)}f_{2,v}\left(s,\xi_0\tau^\wtp_0\lambda^{-1}\right)\left(\imath_0\left(\begin{pmatrix}&1\\-1\end{pmatrix}_pg,1\right)\right)\cdot (U^{-m}_p \sT_\ns\breve{h}_\tau)\left(g_2g\begin{pmatrix}&1&\\-p^m\end{pmatrix}_p\right)\,dg\,\right|_{s=\frac{\wtp k-3}{2}},
\end{aligned}
\end{equation*}

where $m$ is any sufficiently large integer. The integral over $g$ is essentially doubling local zeta integral.

By using the same computation results in the proof of Theorem <ref> for all $v\neq p$, we obtain

\begin{equation}\label{eq:h-L5L6-2}
\begin{aligned}
h'_\tau(g_2)&=\cC_1(\tau_{p\adic})\cdot \frac{\Gamma(\wtp k-1)\Gamma(\wtp k-2)}{(2\pi i)^{2\wtp k-3}}\cdot L^{p\infty}\left(\frac{\wtp k-2}{2},\mr{BC}(\pi_{h_\tau})\times \xi_0\tau^\wtp_0\lambda^{-1}\right)\\
&\hspace{-2em} \times \left.\int_{\U(2)(\bQ_p)}f_{2,p}\left(s,\xi_0\tau^\wtp_0\lambda^{-1}\right)\left(\imath_0\left(\begin{pmatrix}&1\\-1\end{pmatrix}_p g,1\right)\right)(U^{-m}_p \sT_\ns\breve{h}_\tau)\left(g_2g\begin{pmatrix}&1&\\-p^m\end{pmatrix}_p\right)\,dg\,\right|_{s=\frac{\wtp k-3}{2}},
\end{aligned}
\end{equation}

where $\cC_1$ is the product of

* the local doubling zeta integrals at $v\in\Sigma\,\cup\{\ell,\ell'\}$, which is a nonzero constant in $L$ by our choice, and

* the element in $\hat{\cO}^\ur_L\llb U_{\cK,p}\rrb$ interpolating $L_q\left(\frac{\wtp k-2}{2},\chi_h\chi^c_\theta\xi_0\tau^\wtp_0\right)^{-1}$, which is a unit by our assumption.

Our choice of $\chi_h$, $\chi_\theta$ implies that $L_v\left(s,\mr{BC}(\pi_{h_\tau})\times \xi_0\tau_0\lambda^{-1}\right)=1$ for $v\in\Sigma\,\cup\{\ell,\ell'\}$. The computation at $p$ is slightly different from that in the  construction of $p$-adic $L$-functions because the section $f_{2,p}(s,\xi_0\tau_0\lambda^{-1})$ is slightly different from the one for $p$-adic $L$-functions. The Schwartz function $\alpha'_{\xi\tau,p}$ in the description of $f_{2,p}(s,\xi_0\tau_0\lambda^{-1})$ in Proposition <ref> only requires $x_{12}$ to be supported on $p^{-t}\bZ^\times_p$. Unlike the section for constructing $p$-adic $L$-functions, $\alpha_{\xi\tau,p}$ does not require the whole $\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}$ to be supported on $p^{-t}\GL_2(\bZ_p)$. (In fact, $f_{2,p}(s,\tau_0\lambda^{-1})$ is essentially of the same type as the section in <cit.> for constructing the “improved” $p$-adic $L$-function.) The computation at $p$ is done in <cit.>, and plugging it into (<ref>), we get

\begin{align*}
h'_\tau(g_2)=&\,\cC_1(\tau_{p\adic})\cdot\frac{\Gamma(\wtp k-1)\Gamma(\wtp k-2)}{(2\pi i)^{2\wtp k-3}}\cdot  L^{p\infty}\left(\frac{\wtp k-2}{2},\mr{BC}(\pi_{h_\tau})\times \xi_0\tau_0\lambda^{-1}\right)\\
&\times\gamma_{\bar{\fp}}\left(\frac{4-\wtp k}{2},(\chi_h\chi^{c}_\theta\xi_0\tau^{\wtp}_0)^{-1}\right)
L_{\fp}\left(\frac{\wtp k-2}{2},\lambda^2\chi_h\chi_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}(\xi_0\tau^\wtp_0)^c\right)\\
&\times (U^{-m}_p\sT_\ns \breve{h}_\tau)\left(g_2\begin{pmatrix}&1&\\-p^m\end{pmatrix}_p\right).
\end{align*}

Plugging this into (<ref>), we see that the evaluations at $\tau_{p\adic}$ of the two sides of (<ref>) are equal.

We introduce some more $p$-adic $L$-functions. Let $\glsuseri{L1}\in \Meas\left(U_{\cK,p},\hat{\cO}^\ur_L\right)\otimes_\bZ\bQ\simeq\hat{\cO}^\ur_L\llb U_{\cK,p}\rrb\otimes_\bZ\bQ$ be the $p$-adic $L$-function interpolating the central values of $L\left(s,\mr{BC}(\pi)\times \lambda^2\chi_h\chi_\theta\tau^{-1}_{\fp,\pP_\wtp}\tau^{c}_{\fp,\pP_\wtp}\right)$, i.e. for $\tau_{p\adic}$ as in Theorem <ref>,

\begin{align*}
\cL_1\left(\tau_{p\adic}|_{U_{\cK,p}}\right)
&=\left(\frac{\Omega_p}{\Omega_\infty}\right)^4\pi^{-3}  \cdot \gamma_p\left(\frac{1}{2},\pi_{f,p}\times\left(\lambda^2\chi_h\chi_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)_{\bar{\fp}}\right)^{-1}\\
&\quad\times L^{p\infty}\left(\frac{1}{2},\mr{BC}(\pi)\times \lambda^2\chi_h\chi_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right).
\end{align*}

The existence of such a $p$-adic $L$-function $\cL_1$ follows from [18]. Let

\[
\glsuserii{L1}=\pi^{-2}\left<f,\bar{f}\right>^{-1}\cdot L^\infty\left(\frac{1}{2},\mr{BC}(\pi)\times \chi_h\chi^c_\theta\right)\in L^{\ur}.
\]

The product of $\cL_1\cL_2$ interpolates the central values of $L\left(s,\pi\times\pi^{\GL_2}_h\times\pi^{\GL_2}_{\theta^{\bar{\lambda}}}\right)$ with $h$, $\theta$ specializations of the CM families $\bfh$, $\bmtheta$.

Note that because $\pi$ has trivial central character and $\chi_h\chi^c_\theta|_{\bA^\times_\bQ}=\triv$, when $\cond\left(\xi_{0,p}\tau^\wtp_{0,\fp}\right)=p^t$,

\begin{equation}\label{eq:triple-gamma}
\begin{aligned}
&=\fg\left((\xi_0\tau^\wtp_0)^{-1}_\fp\right)^2\cdot \left(\lambda^2_{\bar{\fp}}\chi_{h,\bar{\fp}}\chi_{\theta,\bar{\fp}}\tau^{-\wtp}_{0,\fp}(p)p^{1/2}\right)^{-2t}\\
&=p^{-t}\cdot \fg\left((\xi_0\tau^\wtp_0)^{-1}_\fp\right)\left(\lambda_{\bar{\fp}}^2\chi_{\theta,\bar{\fp}}\chi^{-1}_{\theta,\fp}\tau^{-\wtp}_{0,\fp}(p)\right)^{-t}\cdot \fg\left((\xi_0\tau^\wtp_0)^{-1}_\fp\right)\left(\lambda_{\bar{\fp}}^2\chi_{h,\bar{\fp}}\chi^{-1}_{h,\fp}\tau^{-\wtp}_{0,\fp}(p)\right)^{-t}\\
&=p^{-t}\cdot \gamma_{\bar{\fp}}\left(0,\lambda^2\chi_\theta\chi^{-c}_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)^{-1}\cdot \gamma_{\bar{\fp}}\left(0,\lambda^2\chi_h\chi^{-c}_h\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)^{-1}.
\end{aligned}
\end{equation}

Combining (<ref>), the above interpolation formulas for $\cL_1,\cL_2$, and the interpolation formulas for $\cL_3,\cL_4$ in <ref>, we see that when $\cond\left(\xi_{0,p}\tau^\wtp_{0,\fp}\right)=p^t$,

\begin{equation}\label{eq:L1234}
\begin{aligned}
\left(\frac{\cL_1\cL_2}{\cL_3\cL_4}\right)\left(\tau_{p\adic}|_{U_{\cK,p}}\right)
=&\,\pi\cdot p^{-t}L_p\left(\frac{1}{2},\mr{BC}(\pi)\times\chi_h\chi^c_\theta\right)\\
&\hspace{-2em}\times\frac{L^{p\infty}\left(\frac{1}{2},\mr{BC}(\pi)\times \lambda^2\chi_h\chi_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)\cdot L^{p\infty}\left(\frac{1}{2},\mr{BC}(\pi)\times \chi_h\chi^c_\theta\right)}{L^\infty(1,\eta_{\cK,\bQ})^2 \cdot L^{p\infty}\left(1,\lambda^2\chi_\theta\chi^{-c}_\theta\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)  \cdot L^{p\infty}\left(1,\lambda^2\chi_h\chi^{-c}_h\tau_{\fp,\pP_\wtp}\tau^{-c}_{\fp,\pP_\wtp}\right)}
\\
=&\,\pi\frac{L^\infty\left(1,\pi,\mr{Ad}\right)}{\left<f,\bar{f}\right>}   \cdot  p^{-t}\,\frac{L_p\left(\frac{1}{2},\mr{BC}(\pi)\times\chi_h\chi^c_\theta\right)}{L_p\left(1,\eta_{\cK/\bQ}\right)^2\cdot L_p\left(1,\pi,\mr{Ad}\right)}\\
&\times \frac{L^{p\infty}\left(\frac{1}{2},\pi\times\pi^{\GL_2}_h\times\pi^{\GL_2}_{\theta^{\bar{\lambda}}}\right)}{L^{p\infty}\left(1,\pi,\mr{Ad}\right)\cdot L^{p\infty}\left(1,\pi^{\GL_2}_\theta,\mr{Ad}\right)\cdot L^{p\infty}\left(1,\pi^{\GL_2}_h,\mr{Ad}\right)}.
\end{aligned}
\end{equation}

There exist operators $\sT_\ns,\sT'_\ns \in \bigotimes\limits_{v\in\Sigma_\ns\cup\{\ell'\}}\cO_L\left[\GU(2)\big(\bQ_v\big)\right]$, and $\varphi$ inside the space (<ref>), $\varphi'$ inside the space (<ref>) such that $\sT_\ns\bfh$ is invariant under the tame level group $K^p_f$ (defined in (<ref>)) and

\begin{equation}\label{eq:triple-L}
\begin{aligned}
&\left<\left(\sT_\ns\breve{\bfh}\right)|_{\U(2)},\bmtheta^{\bar{\lambda}}  \varphi\right>_{p\adic}
\left<\Big(\sT'_\ns\breve{\tilde{\bfh}}_3\Big)|_{\U(2)},\tilde{\bmtheta}^{\lambda}_3  \varphi'\right>_{p\adic}
=C_2\cdot \big<\bmtheta,\tilde{\bmtheta}_3\big>_{p\adic}\big<\bfh,\tilde{\bfh}_3\big>_{p\adic} \cdot \frac{\cL_1\cL_2}{\cL_3\cL_4}
\end{aligned}
\end{equation}

with $C_2\in L^\times$.

For  $\sT_\ns,\sT'_\ns \in \bigotimes\limits_{v\in\Sigma_\ns\cup\{\ell'\}}\cO_L\left[\GU(2)\big(\bQ_v\big)\right]$ and $\varphi,\varphi'$ in (<ref>) corresponding to pure tensors in $\bigotimes_v\pi^{\GU(2)}_v$, we consider evaluations at $\tau_{p\adic}$ as in Theorem <ref> with $\cond(\tau_\fp)=p^t$. Put $\theta_\tau=\bmtheta(\tau_{p\adic})$ and $h_\tau=\bfh(\tau_{p\adic})$.

\begin{equation}\label{eq:triple-eval}
\begin{aligned}
\begin{array}{ll}\text{LHS of \eqref{eq:triple-L}}\\\text{valued at $\tau_{p\adic}$}\end{array}
=& \int_{\U(2)(\bQ)\backslash\U(2)(\bA_\bQ)} \theta^{\bar{\lambda}}_\tau(g) \,\varphi(g) \,\left(\sT_\ns\breve{h}_\tau\right)\left(g\begin{psm}&1\\-p^t\end{psm}_{\scriptscriptstyle p}\right)\,dg\\
&\times \int_{\U(2)(\bQ)\backslash\U(2)(\bA_\bQ)} \tilde{\theta}^{\lambda}_\tau(g) \, \varphi'(g)  \,\left(\sT'_\ns\breve{\tilde{h}}_\tau\right)\left(g\begin{psm}&1\\-p^t\end{psm}_{\scriptscriptstyle p}\right)\,dg.
\end{aligned}
\end{equation}

We can replace $\theta^{\bar{\lambda}}_\tau$ (resp. $\tilde{\theta}^\lambda_\tau$) in the integrand by $\breve{\theta}^{\bar{\lambda}}_\tau$ (resp. $\breve{\tilde{\theta}}^\lambda_\tau$), its extension by zero to $\GU(2)$ with respect the character $\chi_\theta\tau_{\fp,\tau_\wtp}$  (resp. $\chi^{-1}_\theta\tau^{-1}_{\fp,\tau_\wtp}$) (as described at the beginning of <ref>), and replace the domain of integration by $\GU(2)(\bQ)Z_{\GU(2)}(\bA_\bQ)\backslash\GU(2)(\bA_\bQ)$. Thanks to (<ref>), we can further replace $\GU(2)(\bQ)Z_{\GU(2)}(\bA_\bQ)\backslash\GU(2)(\bA_\bQ)$ by $D^\times(\bQ)Z_D(\bA_\bQ)\backslash D^\times(\bA_\bQ)$. Then for $\sT_\ns,\sT'_\ns\in \bigotimes\limits_{v\in\Sigma_\ns\cup\{\ell'\}}\cO_L\left[D^\times(\bQ_v)\right]\subset \bigotimes\limits_{v\in\Sigma_\ns\cup\{\ell'\}}\cO_L\left[\GU(2)(\bQ_v)\right]$, we can rewrite (<ref>)

\begin{equation}\label{eq:tri-2}
\begin{aligned}
\begin{array}{ll}\text{LHS of \eqref{eq:triple-L}}\\\text{valued at $\tau_{p\adic}$}\end{array}
=&\int_{\bA^\times_\bQ\backslash D^\times(\bA_\bQ)}
\breve{\theta}^{\bar{\lambda}}_\tau(g) \,\varphi(g) \,(\sT_\ns\breve{h}_\tau)\left(g\begin{psm}&1\\-p^t\end{psm}_{\scriptscriptstyle p}\right)\,dg\\
&\times\int_{\bA^\times_\bQ\backslash D^\times(\bA_\bQ)}
\breve{\tilde{\theta}}^{\lambda}_\tau(g) \,\varphi'(g) \,(\sT'_\ns\breve{\tilde{h}}_\tau)\left(g\begin{psm}&1\\-p^t\end{psm}_{\scriptscriptstyle p}\right)\,dg.
\end{aligned}
\end{equation}

This is the triple product integral.

\begin{align*}
F_\tau&=\left(\breve{\theta}^{\bar{\lambda}}_\tau\otimes\varphi\otimes \sT_\ns \breve{h}_\tau\right)|_{D^\times(\bA)},
&F'_\tau&=\left(\breve{\tilde{\theta}}^{\lambda}_{3,\tau}\otimes\varphi'\otimes \sT'_{\ns}\breve{\tilde{h}}_{3,\tau}\right)|_{D^\times(\bA)}.
\end{align*}

Then $F_\tau$ generates $\Pi_\tau=\pi^D_{\theta^{\bar{\lambda}}_\tau}\otimes\pi^D_{h_\tau}\otimes\pi^D$, where $\pi^D_{\theta^{\bar{\lambda}}_\tau}$ (resp. $\pi^D_{h_\tau}$, $\pi^D$) is the automorphic representation of $D^\times(\bA_\bQ)$ generated by the restriction of $\breve{\theta}^{\bar{\lambda}}_\tau$ (resp. $\breve{h}_\tau$, $\varphi$) to $D^\times$, and $F'_\tau$ generates $\tilde{\Pi}_\tau$, the contragredient representation of $\Pi_\tau$.
By the isomorphism (<ref>) and Proposition <ref>, we know that $\pi^D_{\theta^{\bar{\lambda}}_\tau}$, $\pi^D_{h_\tau}$ are irreducible, and $\breve{\theta}^{\bar{\lambda}}_\tau$, $\breve{h}_\tau$ are pure tensors. By the definition of $\varphi$, we know that $\pi^D_\varphi=\pi^D$. Hence, $\Pi_\tau$ (resp. $\tilde{\Pi}_\tau$) is irreducible, and the image of $F_\tau$ (resp. $F'_\tau$) under $\Pi_\tau\simeq \bigotimes_v \Pi_{\tau,v}$  ($\tilde{\Pi}_\tau\simeq \bigotimes_v\tilde{\Pi}_{\tau,v}$) is a pure tensor $\bigotimes_v F_{\tau,v}$ (resp. $\bigotimes_v F'_{\tau,v}$). By Ichino's triple product formula [22], (<ref>) becomes

\begin{equation}\label{eq:tri-3}
\begin{aligned}
\begin{array}{ll}\text{LHS of \eqref{eq:triple-L}}\\\text{valued at $\tau_{p\adic}$}\end{array}
=&\,\prod_v \int_{\bA^\times_\bQ\backslash D^\times(\bA_\bQ)} \big<\Pi_v(g_v)F_{\tau,v},F'_{\tau,v}\big>_v\,dg_v\\
\cdot \frac{\zeta^{\Sigma\,\cup\{\infty,p,\ell,\ell'\}}(2)^2\cdot L^{\Sigma\,\cup\{\infty,p,\ell,\ell'\}}\left(\frac{1}{2},\Pi\right)}{L^{\Sigma\,\cup\{\infty,p,\ell,\ell'\}}\left(1,\Pi,\mr{Ad}\right)}\\
&\times \prod_{v\in  \Sigma\,\cup\{\infty,\ell,\ell'\}}  \frac{\int_{\bQ^\times_v\backslash D^\times(\bQ_v)}\big<\Pi_v(g)F_{\tau,v},F'_{\tau,v}\big>_v\,dg}{\big<f^D_v,\bar{f}^D_v\big>_v\big<\theta_{\tau,v},\tilde{\theta}_{3,\tau,v}\big>_v\big<h_{\tau,v},\tilde{h}_{3,\tau,v}\big>_v}\\
&\times  \frac{\int_{\bQ^\times_p\backslash D^\times(\bQ_p)}\big<\Pi_p(g)F_{\tau,p},F'_{\tau,p}\big>_p\,dg}{\big<f^D_p,\bar{f}^D_p\big>_p\big<\theta_{\tau,p},\begin{psm}&1\\-p^t\end{psm}\tilde{\theta}_{3,\tau,v}\big>_p\big<h_{\tau,p},\begin{psm}&1\\-p^t\end{psm}\tilde{h}_{3,\tau,p}\big>_p}
\end{aligned}
\end{equation}

We have the following results for the local integrals at $v\in \Sigma\,\cup\{\infty,p,\ell,\ell'\}$:

– $v\in\Sigma_\rms\cup\{\infty,\ell\}$.

\[
\frac{\int_{\bQ^\times_v\backslash D^\times(\bQ_v)} \big<\Pi_v(g)\phi_v,\phi'_v\big>_v\,dg}{\big<f^D_v,\bar{f}^D_v\big>_v\big<\theta_{\tau,v},\tilde{\theta}_{3,\tau,v}\big>_v\big<h_{\tau,v},\tilde{h}_{3,\tau,v}\big>_v}=\text{a nonzero constant independent of $\tau$}
\]

For $v=\infty$, the integral is $1$ because $\Pi_\infty$ is the trivial representation. For $v=\Sigma_\rms\cup\{\ell\}$, this is <cit.>. Note that the twist at $v\in\Sigma_\rms\cup\{\ell\}$ in (<ref>) guarantees that the local integrals are nonzero constants.

– $v\in \Sigma_\ns\cup\{\ell'\}$. There exist $\sT_v,\sT'_v\in\cO_L\left[D^\times(\bQ_v)\right]\subset\cO_L \left[\GU(2)(\bQ_v)\right]$ and $\varphi_v,\varphi'_v\in \pi^{\GU(2)}_v$ such that $\varphi_v$ and $\sT_v\bfh$ are invariant under the level group $K_v$ (defined in (<ref>)) and

\[
\frac{\int_{\bQ^\times_v\backslash D^\times(\bQ_v)} \big<\Pi_v(g)F_{\tau,v},F'_{\tau,v}\big>_v\,dg}{\big<f^D_v,\bar{f}^D_v\big>_v\big<\theta_{\tau,v},\tilde{\theta}_{3,\tau,v}\big>_v\big<h_{\tau,v},\tilde{h}_{3,\tau,v}\big>_v}=\text{a nonzero constant independent of $\tau$}
\]

This is proved in the following <ref>.

– $v=p$.

\begin{align*}
&\frac{\int_{\bQ^\times_p\backslash D^\times(\bQ_p)}\big<\Pi_p(g)F_{\tau,p},F'_{\tau,p}\big>_p\,dg}{\big<f^D_p,\bar{f}^D_p\big>_p\big<\theta_{\tau,p},\begin{psm}&1\\-p^t\end{psm}\tilde{\theta}_{3,\tau,v}\big>_p\big<h_{\tau,p},\begin{psm}&1\\-p^t\end{psm}\tilde{h}_{3,\tau,p}\big>_p}
=\, p^{-t}\,\frac{\zeta_p(2)^2\cdot L_p\left(\frac{1}{2},\mr{BC}(\pi)\times\chi_h\chi^c_\theta\right)}{L_p\left(1,\eta_{\cK/\bQ}\right)^2\cdot L_p\left(1,\pi,\mr{Ad}\right)}.
\end{align*}

This is a direct consequence of  <cit.>. We also use the formulas for local norms of the chosen vectors in the proof of Lemma 6.4 of loc.cit..

Plug these results on local integrals into (<ref>) and compare with (<ref>), we see that

\begin{align*}
\begin{array}{ll}\text{LHS of \eqref{eq:triple-L} }\text{valued at $\tau_{p\adic}$}\end{array}
=C_2\cdot \big<\theta_\tau,\tilde{\theta}_{3,\tau}\big>_{p\adic}\big<h_\tau,\tilde{h}_{3,\tau}\big>_{p\adic}\cdot  \left(\frac{\cL_1\cL_2}{\cL_3\cL_4}\right)\left(\tau_{p\adic}|_{U_{\cK,p}}\right)
\end{align*}

with $C_2\in L^\times$ the product of

* the local triple product integrals divided by the norms at $v\in\Sigma\cup\{\ell,\ell'\}$, which are nonzero constants,

* $\prod\limits_{v\in \Sigma\,\cup\{\ell,\ell'\}} \Big(\frac{\zeta_v(2)^2 L_v\left(\frac{1}{2},\Pi\right)}{L_v\left(1,\Pi,\mr{Ad}\right)}\Big)^{-1}$, which, by our choice of the Hecke characters $\chi_\theta,\chi_h$, equals $L_q\left(\frac{1}{2},\mr{BC}(\pi)\times\chi_h\chi^c_\theta\right)^{-1} \prod\limits_{v\in \Sigma\,\cup\{\ell,\ell'\}} \zeta_v(2)^{-2}L_v\left(1,\eta_{\cK/\bQ}\right)^2 L_v\left(1,\pi,\mr{Ad}\right)$, a nonzero constant,

* $\left(\pi\frac{L^\infty\left(1,\pi,\mr{Ad}\right)}{\left<f,\bar{f}\right>}\cdot \big<f^D,\bar{f}^D\big>\right)^{-1}\in \bQ^\times$.

Given $g\in \U(2)(\bA_{\bQ,f})$, evaluating a $p$-adic form at $g$ induces a map

\[
\Meas\left(\Gamma_\cK,V_{\U(2)}\wh{\otimes}\hat{\cO}^\ur_L\right)^\natural\lra \Meas\left(\Gamma_\cK,\hat{\cO}^\ur_L\right)\simeq \hat{\cO}^\ur_L\llb\Gamma_\cK\rrb.
\]

Denote by $l_{\theta^J_1}\left(\bfE^\Kling_{\varphi,\beta,u}\right)(g)\in \hat{\cO}^\ur_L\llb\Gamma_\cK\rrb$ the image of $l_{\theta^J_1}\left(\bfE^\Kling_{\varphi,\beta,u}\right)$ under this map.

Let $\beta=1\in\Her_1(\cK)$, $\theta^J_1$ be the Fourier–Jacobi form on $\U(2)$ as in Proposition <ref>, and $\varphi$ be an automorphic form on $\U(2)$ as in Proposition <ref>. Let $J\subseteq \hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\otimes_\bZ\bQ$ be the ideal generated by

\begin{align*}
&g\in \U(2)(\bA_{\bQ,f}),\quad u\in\hspace{-.5em}\bigotimes_{v\in\Sigma_\ns\cup\{\ell'\}}\U(1)(\bQ_v).
\end{align*}

Then $J=\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\otimes_\bZ\bQ$.

Suppose that $J\neq \hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\otimes_\bZ\bQ$. Then $J$ is contained in a maximal ideal of $\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\otimes_\bZ\bQ$, so is contained in the maximal ideal associated to a $p$-adic character $\kappa:\Gamma_\cK\ra\ol{\bQ}^\times_p$. Let $\kappa':\Gamma_\cK\ra\ol{\bQ}^\times_p$ be a $p$-adic character whose $\wtp$-th power is $\kappa$. Then

\[
\pP_{\wtp,*}\left(l_{\theta^J_1}\left(\bfE^\Kling_{\varphi,\beta,u}\right)\right)(\kappa'|_{U_{\cK,p}})=l_{\theta^J_1}\left(\bfE^\Kling_{\varphi,\beta,u}\right)(\kappa)=0
\]

for all $u\in \bigotimes_{v\in\Sigma_\ns\cup\{\ell'\}}\U(1)(\bQ_v)$. Then it follows from Propositions <ref> and <ref> that

\[
\left( \big<\bmtheta,\tilde{\bmtheta}_3\big>_{p\adic}\big<\bfh,\tilde{\bfh}_3\big>_{p\adic} \cdot \frac{\cL_1\cL_2\cL_5\cL_6}{\cL_3\cL_4}\right)(\kappa'|_{U_{\cK,p}})=0,
\]

which, combining with Proposition <ref>, implies that

\[
\left(\cL_1\cL_2\cL_5\cL_6\right)(\kappa'|_{U_{\cK,p}})=0.
\]

However, the condition (3) in our choice of $\chi_h,\chi_\theta$ in <ref> implies that $\cL_1\cL_2\cL_5\cL_6$ is a unit in $\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\otimes_\bZ\bQ$, so is nonzero at all characters in $\Hom_\cont\left(U_{\cK,p},\ol{\bQ}^\times_p\right)$.  We get a contradiction. Therefore, $J=\hat{\cO}^\ur_L\llb\Gamma_\cK\rrb\otimes_\bZ\bQ$.

§.§ The local triple product integrals at non-split places

For $v\in\Sigma_\ns\cup\{\ell'\}$, there exist $\sT_v,\sT'_v\in\cO_L\left[D^\times(\bQ_v)\right]\subset\cO_L \left[\GU(2)(\bQ_v)\right]$ and $\varphi_v,\varphi'_v\in \pi^D_v$ such that $\varphi_v$ and $\sT_v\bfh$ are invariant under the level group $K_v$ (defined in (<ref>)), and

\begin{equation}\label{eq:ns-int}
\int_{\bQ^\times_v\backslash D^\times(\bQ_v)} \big<\Pi_v(g)F_{\tau,v},F'_{\tau,v}\big>_v\,dg=\text{a nonzero constant independent of $\tau$},
\end{equation}

where $F_{\tau,v}=\left(\breve{\theta}^{\bar{\lambda}}_{\tau,v}\otimes\varphi_v\otimes \sT_v \breve{h}_{\tau,v}\right)|_{D^\times(\bQ_v)}$, $F'_{\tau,v}=\left(\sT'_v\breve{\tilde{\theta}}^{\lambda}_{3,\tau,v}\otimes\varphi'_v\otimes \sT''_{v}\breve{\tilde{h}}_{3,\tau,v}\right)|_{D^\times(\bQ_v)}$.

Let $v$ be a place in $\Sigma_\ns\cup\{\ell'\}$. Then $\U(1)(\bQ_v)\subset \cO^\times_{\cK,v}$ and $\tau_{\fp,\pP_\wtp}|_{\U(1)(\bQ_v)}$ is trivial. From the construction of $\bfh,\tilde{\bfh}_3$ in <ref> and $\bmtheta,\tilde{\bmtheta}_3$ in <ref>, we see that $\pi_{h_\tau,v}$, $\pi_{\theta_\tau,v}$ and $h_{\tau,v},\tilde{h}_{3,\tau,v},\theta_{\tau,v},\tilde{\theta}_{3,\tau,v}$  do not change when $\tau$ varies. It follows that the left hand side of (<ref>) is independent of $\tau$.