\section{Existence of classes with a nonzero cup product} \label{Sec:CupProduct}

\subsubsection{} \label{subsub:Notation4} We continue with the notation from Section \ref{Sec:Reduction}. In particular we have a fixed CM algebra $L= L_1 \times \cdots \times L_n$, a Galois CM field $E \subset \mathbb{C}$ in which each $L_i$ embeds, a CM type $\Phi \subset \operatorname{Hom}(L,E)$ and an abelian variety $A$ over $\mathbb{C}$ with CM by $E$ and CM type $\Phi$. We choose $\Delta=\{s_1, s_2, c \circ s_3, c \circ s_4\} \subset \Sigma$ satisfying the condition in \eqref{eq:DegreeTwoHodgeClasses}, so that $\Delta$ corresponds to a codimension $2$ Hodge class on $A$, and define auxiliary abelian varieties $B_1, \cdots, B_4$ as in Section \ref{subsub:DefiningB}. Write $\lambda=(\lambda_1, \lambda_2, \lambda_3, \lambda_4)=(\operatorname{Id}, \operatorname{Id}, c, c)$ as before. \smallskip 

Choose a Hermitian $E$ vector space $V$ of dimension three with signature $(2,1)$ at the distinguished embedding of $E$ into $\mathbb{C}$ and signature $(3,0)$ at the other infinite places, as in Section \ref{Sec:PMS}. We let $\g=U(V)$ and $\h=\operatorname{Res}_{F/\mathbb{Q}} \g$, where $F$ is the totally real subfield of $E$, and we equip $\h$ with its natural Shimura datum $\hy$. We have the following theorem, which together with Theorem \ref{Thm:ReductionToSurface} and Theorem \ref{Thm:Hazama} implies Theorem \ref{Thm:IntroMain}.
\begin{Thm} \label{Thm:ExistenceCohomologyClasses}
There is a compact open subgroup $K \subset \h(\af)$, morphisms $h_i:\rH^1(B_i, \mathbb{Q}) \to \rH^1(\mathbf{Sh}_K\hy_{\mathbb{C}}, \mathbb{Q})$ of rational Hodge structures for $i=1, \cdots, 4$ and classes $\alpha_i \in \rH^1(B_i, E)_{\lambda_i}$ for $i=1, \cdots, 4$ such that 
\begin{align}
     h_1(\alpha_1) \cup h_2(\alpha_2) \cup h_3(\alpha_3) \cup h_4(\alpha_4) \not = 0.
\end{align}
\end{Thm}

Our proof of Theorem \ref{Thm:ExistenceCohomologyClasses} relies crucially on a very special case of the results of \cite{BMM}, \cite{Liu} as we now explain. 

\subsection{Cohomological theta lifting} \label{sub:thetaLifting}

We will need to consider several integrals over adelic class sets $\g(F)\backslash \g(\mathbb{A}_F)$ where $F$ is some number field and $\g$ is a connected algebraic group, in all such cases we will take this integral with respect to a probability measure on $\g(F)\backslash \g(\mathbb{A}_F)$, which will always exist in the examples we consider.

\subsubsection{} Let $V$ be our Hermitian space over $E$ as above, and let $v_0=\restr{\tau_0}{F}$ denote the distinguished real place at which $V$ has non-definite signature. We choose an auxiliary one-dimensional skew-Hermitian space $W$ over $E$, such that at the place $v_0$ it has signature $(b,a)$. We equip $\mathbb{W}=V \otimes_{E} W$ with the structure of a symplectic space over $F$ as in \cite[Section 7.1]{BMM}. We choose a so-called polarization $\mathbb{W} = X \oplus Y$, where $X$ is a Lagrangian $F$-subspace in $\mathbb{W}$ and $Y$ is a complementary Lagrangian subspace. We consider the symplectic group $\operatorname{Sp}(\mathbb{W})$ over $F$ and the metaplectic cover
\begin{align}
    \operatorname{Mp}(\mathbb{W})(\mathbb{A}_{F}) \to \operatorname{Sp}(\mathbb{W})(\mathbb{A}_{F}).
\end{align}

\subsubsection{} Let $\mu_{E/F}$ be as above. By Kudla's splitting construction in \cite{RPK}, in the adelic formulation recalled in \cite[B.2]{Liu}, if we choose a pair of automorphic splitting characters $\nu = (\nu_6, \nu_5)$ of $\mathbb{A}_{E}^{\times}$
such that $\restr{\nu_i}{\mathbb{A}_{F}^{\times}}=\mu_{E/F}$, then we have a homomorphism
\[
    \widetilde{\iota}_{\nu}: U(V)(\mathbb{A}_F) \times U(W)(\mathbb{A}_F) \to \operatorname{Mp}(\mathbb{W})(\mathbb{A}_{F})
\]
which lifts the natural map $\iota: U(V)(\mathbb{A}_F) \times U(W)(\mathbb{A}_F) \to \operatorname{Sp}(\mathbb{W})(\mathbb{A}_{F})$. Note that we follow the convention of \cite{Liu} for the ordering of $\nu=(\nu_6, \nu_5)$; we use $\nu_6$ for the character $\nu_{V}$ and $\nu_5$ for the character $\nu_{W}$ as in \cite[\S B.2]{Liu}.

\subsubsection{} For $\psi=\psi_F$ the standard additive character of \cite[\S 1.7]{Liu}, we can form the automorphic Weil representation $\omega_{\psi}$ of the metaplectic group, see \cite[\S 7.1]{BMM}. Thus the metaplectic group $\operatorname{Mp}(\mathbb{W})(\mathbb{A}_{F})$ acts on the Schwartz space $\cS(X(\mathbb{A}_{F}))$, we let $\mathbf{S}(X(\mathbb{A}_{F}))$ denote the Schr\"odinger realization of the polynomial Fock space as defined in \cite[\S 7.1]{BMM}, which is dense in $\cS(X(\mathbb{A}_{F}))$. We let $(U(V) \times U(W))(\mathbb{A}_F)$ act on $\cS(X(\mathbb{A}_F))$ via the embedding $\widetilde{\iota}_{\nu}$. We finally consider $\mathbf{S}_{v_0}=\mathbf{S}(X(F_{v_0}))$ and $$\mathbf{S}_{\infty}^{v_0}:=\bigotimes_{\substack{v:F \to \mathbb{R} \\ v \not=v_0}}\mathbf{S}(X(F_{v})),$$
and $\mathcal{S}_f = \cS(X(\mathbb{A}_{F,f}))$, and define $\cS^{v_0}_\infty, \cS_{v_0}$ analogously to the above.


\subsubsection{} For each $\phi \in \mathbf{S}(X(\mathbb{A}_{F}))$, we can form the theta function
\begin{align}
    \theta_{\psi, \nu, \phi}(x) = \sum_{\xi \in X(F)} \omega_{\psi}(x)(\phi)(\xi),
\end{align}
which is a function on $\operatorname{Mp}(\mathbb{W})(\mathbb{A}_{F})$. Pulling back along $\widetilde{\iota}_{\nu}$, we get a theta function 
\begin{align}
    \theta_{\psi,\nu, \phi}
\end{align}
on $U(V)(F) \backslash U(V)(\mathbb{A}_F) \times U(W)(F) \backslash U(W)(\mathbb{A}_F)$. From now on, we will always assume that $\nu_6$ is a conjugate symplectic automorphic character of weight one such that the CM type $\Phi_{\nu_6}$ contains the distinguished embedding $\tau_0:E \to \mathbb{C}$ (that we usually write as $\operatorname{Id}$). 

\subsubsection{} Letting $\lieh$ denote the Lie algebra of $G = \g \otimes_{F, v_0} \mathbb{R} = \operatorname{U}(2,1)$, we have $\lieh = \mathfrak{k}_0 \oplus \mathfrak{p}_0$, where $\mathfrak{k}_0$ is the Lie algebra of the maximal compact subgroup $K_\infty = U(2) \times U(1)$ of $G$.\footnote{The notation $G$ here should not be confused with the use of $G=\operatorname{Hom}(E,E)$ in Section \ref{Sec:Reduction}.} We denote by $\liepc$ the complexification of $\liep$. Denote by (where the notation $\wedge^{1,0}$ is explained in \cite[\S 1.2]{BMM})
\[
    \varphi_{1,0} \in  \Hom_{K_\infty}(\wedge^{1,0} \mathfrak{p}, \cS_{v_0})
\]
the cocycle constructed in \cite[\S 5, specifically \S 5.6]{BMM}. As in \cite[Section 8.6]{BMM}, we can think of $\varphi_{1,0}$ as an element of
\begin{align}
    \left[\cS_{v_0} \otimes \Omega^{1,0}_{\mathsf{Y}} \right]^{G(\mathbb{R})}.
\end{align}
For $\varphi \in \mathcal{S}_f$, we then consider (where $\varphi_0$ is as in \cite[\S 8.7]{BMM})
\begin{align}
    \phi = \varphi_{1,0} \otimes_{v} (\varphi_0) \otimes \varphi \in \left[\left(\cS_{v_0} \otimes \cS_{\infty}^{v_0} \otimes \mathcal{S}_f\right) \otimes \Omega^{1,0}_{\mathsf{Y}} \right]^{\h(\mathbb{R})}.
\end{align}

\subsubsection{}\label{subsect: unintegrated theta} Now for $g' \in U(W)(\mathbb{A}_{F})$ we can consider the theta function
\begin{align}
    \theta_{\psi, \nu, \phi}(g,g')
\end{align}
as a function of $g$. If $\phi$ is $K$-invariant for $K \subset \h(\af)$ a compact open subgroup, then as in \cite[\S 8.8]{BMM}, the function $\theta_{\psi, \nu, \phi}(-,g')$ can be interpreted as an element
\begin{align}
    \theta_{\psi, \nu, \phi}(-,g') \in H^0(\mathbf{Sh}_K\hy_{\mathbb{C}}, \Omega^{1,0}_{\mathsf{Y}}).
\end{align}
Altogether, this defines a map 
\begin{align}
    \Theta^{1,0}_{\psi, \nu}:\mathcal{S}_f \times U(W)(F) \backslash U(W)(\mathbb{A}) &\to \varinjlim_K H^0(\mathbf{Sh}_K\hy_{\mathbb{C}}, \Omega^{1,0}_{\mathsf{Y}}) \\
    (\phi, g') &\mapsto \left(g \mapsto \theta_{\psi, \nu, \phi}(g,g') \right).
\end{align}
Now each $\theta_{\psi, \nu, \phi}(-,g')$ is in fact a closed $(1,0)$-form, see \cite[Lemmas 5.7 and 5.13]{BMM}. Thus we can consider the induced map $$[\Theta^{1,0}_{\psi, \nu}]:\mathcal{S}_f \times U(W)(F) \backslash U(W)(\mathbb{A}) \to \varinjlim_K H^{1,0}(\mathbf{Sh}_K\hy_{\mathbb{C}}, \mathbb{C}).$$ For $f \in L^2(U(W)(F) \backslash U(W)(\mathbb{A}_F))$ we also consider the integrated $\theta$ map 
\begin{align}
    \Theta^{1,0}_{\psi, \nu, f}: \mathcal{S}_f \to \varinjlim_K H^0(\mathbf{Sh}_K\hy_{\mathbb{C}}, \Omega^{1,0}_{\mathsf{Y}}).
\end{align}
and its cohomological realization $[\Theta^{1,0}_{\psi, \nu, f}]$. We remark that $U(W) \simeq U(1)$ so that $U(W)(F) \backslash U(W)(\mathbb{A})$ is compact. 

\subsubsection{} Our goal is now to understand the image of $[\Theta^{1,0}_{\psi, \nu,f}]$. We recall the theta transfer notation $\Theta^V_{\psi, \nu,W}$ from \cite[\S 7.3]{BMM}. Given an automorphic representation $\pi \subset L^2(U(W)(F) \backslash U(W)(\mathbb{A}_F))$ of $U(W)(\mathbb{A}_F)$ we consider the subspace
\begin{align}
    \Theta^V_{\psi, \nu,W}(\pi) \subset C^{\infty}(\g(F) \backslash \g(\mathbb{A}_{F}), \mathbb{C})
\end{align}
generated by the functions
\begin{align}
     g \mapsto \int_{U(W)(F) \backslash U(W)(\mathbb{A}_F)} \theta_{\psi, \nu, \phi}(g,g')f(g')\mathrm{d} g'
\end{align}
as $f$ runs over all vectors in $\pi$ and as $\phi$ varies over $\mathbf{S}(X(\mathbb{A}_{F}))$. It is explained in \cite[Proof of Proposition 4.13, last paragraph]{Liu}, that for every admissible adelic oscillator triple $(\mu, \epsilon, \chi)$ there is a skew-Hermitian space $W$ over $E$ of dimension one, an automorphic character $\pi$ of $U(W)(\mathbb{A})$ such that (where $A(1,0)$ is as in \cite[\S 6.7]{BMM})
\begin{align}
    \Theta^{V}_{\psi, (\mu, \nu_5), W}(\pi) \simeq A(1, 0) \otimes \omega(\mu, \epsilon, \chi)
\end{align} 
for some automorphic character $\nu_5$. Moreover, $(W, \pi)$ is unique up to isomorphism and $W$ has signature $(0,1)$ at $v_0$ (see the proof of \cite[Theorem 9.1]{BMM}). Finally, we note that we can use the same $\nu_5$ for different $(\mu, \epsilon, \chi)$ since changing $\nu_5$ just amounts to twisting $\pi$ by a character. So let us fix $\nu_5$ from now on. 
\begin{Prop} \label{Prop:ThetaRealization}
Consider an admissible adelic oscillator triple $(\mu, \epsilon, \chi)$ with corresponding skew-Hermitian space $W$ of signature $(0,1)$ and let $\nu=(\mu, \nu_5)$ and $\pi$ be as above. If $\Phi_{\mu}=\Phi_i^{-1}$ with $\Phi_i$ as in Section \ref{subsub:Notation4}, then for every $f \in \pi$ the image of $[\Theta^{1,0}_{\psi, \nu, f}]$ is contained in the image of the realization map
\begin{align}
    \operatorname{Hom}^0({A_{\infty, \mathbb{C}},B_i}) \otimes_{\mathbb{Q}} \rH^1(B_i, \mathbb{C})_{\operatorname{Id}} \to \varinjlim_K H^{1,0}(\mathbf{Sh}_K\hy_{\mathbb{C}}, \mathbb{C}). 
\end{align}
Furthermore, there exists $f \in \pi$ such that $[\Theta^{1,0}_{\psi, \nu, f}]$ is nonzero. 
\end{Prop}
\begin{proof}
This is a direct consequence of \cite[Theorem 9.1]{BMM} and its proof. To elaborate, it is established there that the
subspace spanned by the images of $[\Theta^{1,0}_{\psi, \nu,f}]$ as we vary over all $(W, \nu)$ and all $f$ is surjective onto $\rH^{1,0}(\mathbf{Sh}\hy_{\mathbb{C}}, \mathbb{C})$. Since $\Theta^{V}_{\psi, (\mu, \nu_5), W}(\pi) \simeq A(1, 0) \otimes \omega(\mu, \epsilon, \chi)$ and $(W, \pi)$ is unique with this property, we see that $[\Theta^{1,0}_{\psi, \nu,f}]$ must land in $\omega(\mu,\epsilon,\chi)$, and moreover we can find $f \in \pi$ for which $[\Theta^{1,0}_{\psi, \nu, f}]$ is nonzero. \smallskip 

For the first claim, we recall $\Omega(\mu)= \operatorname{Hom}^0({A_{\infty},A_{\mu}})$, and recall that $M_{\mu} \subset \mathbb{C}$. If we consider the realization map
\begin{align}
    \Omega(\mu) \otimes_{M_{\mu}} H^1(A_{\mu}, \mathbb{C})_{\operatorname{Id}} \to \varinjlim_K H^{1,0}(\mathbf{Sh}_K\hy_{\mathbb{C}}, \mathbb{C}),
\end{align}
then this has image $\bigoplus_{\epsilon, \chi} \omega(\mu, \epsilon, \chi)$ by the proof of \cite[Theorem 4.18]{Liu}, where the direct sum runs over all $\epsilon, \chi$ such that $(\mu, \epsilon, \chi)$ is a $\mu$-admissible adelic oscillator triple. Now as in the proof of Corollary \ref{Cor:CMAVOccurs} we have $$\operatorname{Hom}^0(A_{\infty,\mathbb{C}},A_{\mu, \mathbb{C}}) = \operatorname{Hom}^0(A_{\infty, \mathbb{C}},A_{\mu, \mathbb{C}}') \otimes_{M_{\mu}'} M_{\mu},$$ and so the natural evaluation map
\begin{align}
    \operatorname{Hom}^0(A_{\infty, \mathbb{C}},A_{\mu,\mathbb{C}}') \otimes_{M_{\mu}'} H^1(A_{\mu,\mathbb{C}}', \mathbb{C})_{\operatorname{Id}} \to \varinjlim_K H^{1,0}(\mathbf{Sh}_K\hy_{\mathbb{C}}, \mathbb{C}).
\end{align}
has image containing $\bigoplus_{\epsilon, \chi} \omega(\mu, \epsilon, \chi)$. Similarly $\operatorname{Hom}^0(A_{\infty,\mathbb{C}},B_i) \otimes_{E} \mathbb{C} =\operatorname{Hom}^0(A_{\infty,\mathbb{C}},A_{\mu, \mathbb{C}}') \otimes_{M_{\mu}'} \mathbb{C}$ as the proof of Corollary \ref{Cor:CMAVOccurs}. Therefore, the realization map of the lemma indeed contains $\bigoplus_{\epsilon, \chi} \omega(\mu, \epsilon, \chi)$ in its image. 
\end{proof}

\subsection{A doubling identity}\label{subsec: doubling}
For $i = 1, \dots, 4$ we let $W_i$ denote the unitary space isomorphic to $E$ with CM type given by $\Phi_i^{-1}$, where $\Phi_i$ is the CM type of $B_i$ as in Section \ref{subsub:Notation4}. For each $i=1,2,3$ we choose a conjugate symplectic automorphic character of weight one
\[
    \mu_i: \mathbb{A}_E^\times/E^\times \to \mathbb{C}^\times
\]
with CM type given by $\Phi_i^{-1}$; this is possible by Lemma \ref{Lem:ExistenceTriple}. We choose $\mu_4$ so that $\mu_1 \mu_2 = \mu_3 \mu_4$, and note that $\mu_4$ is a conjugate symplectic automorphic character of weight one and CM type $\Phi_4^{-1}$ by \eqref{eq:DegreeTwoHodgeClassesII}. For now we fix $d$ a nonzero element of $E$ such that $d^c = -d$. For $e = d a$, $a \in F^\times$, we can consider the skew-Hermitian line associated to the element $e$.
\begin{Lem} \label{Lem:ExistenceLines}
    We may choose the skew-Hermitian lines $W_i = E \cdot e_i$ such that there is an $F$-rational isometry
    \[
        \Psi: W_1 \perp W_2 \xrightarrow{\sim} W_3 \perp W_4
    \]
    and the such that the corresponding tuples $(\mu_i, \epsilon_i, \chi_i)$ is a $\mu_i$-admissible adelic oscillator triple as in Section \ref{subsub:TripleDef}. 
\end{Lem}
\begin{proof}
    Weak approximation allows us to choose $e_1, e_2$ with the right signs at infinity. Then the condition $\mu_1\mu_2 = \mu_3\mu_4$ forces the angular part of the character $\mu_3$ to match with either that of $\mu_1$ or $\mu_2$ at each place above infinity. So we can choose $e_3 \in (W_1 \perp W_2)(E)$ close enough to either $e_1$ or $e_2$ depending on the place using weak approximation, so that the signs agree with those of $\mu_3$. Finally we can just take $W_4 = W_3^{\perp}$. 
\end{proof}

\subsubsection{} Now we fix skew-Hermitian spaces $W_1, \cdots, W_4$ satisfying the conclusion of Lemma \ref{Lem:ExistenceLines} and an $F$-rational isometry $\Psi$, define $\mathbb{W}_i = V \otimes_{E} W_i$ as a symplectic space and choose polarizations $\mathbb{W}_i = X_i \oplus Y_i$ as in Section \ref{sub:thetaLifting}. We define $W = W_1 \perp W_2$ which we identify using $\Psi$ with $W_3 \perp W_4$ and $\mathbb{W}=V \otimes_{E} W$, and $X=X_1 \oplus X_2, Y= Y_1 \oplus Y_2$.

\subsubsection{} We let $-W$ denote the skew Hermitian space which is isomorphic to $W$ but with the skew-Hermitian form $-\Psi_W$. We define the skew-Hermitian space $W^{\Box} \coloneqq W \perp -W$, and after tensoring $V \otimes_E W^{\Box}$ we obtain a symplectic $F$-vector space $\mathbb{S}$ as in \cite[Section 7.1]{BMM}. Taking the diagonal $W^{\Delta}$ and anti-diagonal $W^{\nabla}$ totally isotropic $E$-subspaces of $W^{\Box}$, we get an $F$-polarization
\[
    \mathbb{S} = \left(V \otimes_E W^{\Delta}\right) \oplus \left(V \otimes_E W^{\nabla}\right).
\]
\subsubsection{} From now on, we will take $\psi$ to be as above, and take $\nu_i=(\mu_i, \nu_5)$ to form our oscillator representations. Fixing a basis for $W^{\nabla}$ we obtain an identification $V^2(\mathbb{A}_E) \simeq (V \otimes W^{\nabla})(\mathbb{A}_F)$, and thus a Weil representation $\omega$ of $\g \times U(W^\Box)$ on $\cS(V^2(\mathbb{A}_E))$ with splitting character $(1, \nu_5)$. 

\begin{Rem}\label{rem: splitting}
    Since we take the splitting characters $\mu_i$ to be unitary, and the Weil representation $\omega_{-W}$ to be the conjugate of $\omega_W$, the oscillator representation of $\g(\mathbb{A}_F)$ on $\cS(V(\mathbb{A}_E)^2)$ arises simply from the diagonal pullback action on Schwartz functions, as the splitting characters cancel. In addition, the doubling intertwiner below will intertwine the diagonal action of $\g(\mathbb{A}_F)$ on the Schr\"odinger models. These are small but crucial ways in which we use the hypothesis that $\mu_1\mu_2 = \mu_3\mu_4$ in the proof of Theorem \ref{Thm:ExistenceCohomologyClasses}.
\end{Rem}  

As in the discussion of \cite[\S 2]{Li} on Rallis' method, there is an intertwiner
\[
    \delta: \cS(X(\mathbb{A}_F)^2) \to \cS((V \otimes W^{\nabla})(\mathbb{A}_F)),
\] 
given by 
\[
    \delta(\Phi)((x, y), (-x, -y)) = \int_{X(\mathbb{A}_F)} \psi(2\langle u, y \rangle)\Phi(u + x, u - x) \, du,
\] 
and this doubling intertwiner is equivariant for the actions of the metaplectic group $\operatorname{Mp}(\mathbb{S})(\mathbb{A}_F)$. It will be quite important later that under this identification we have, also as in \cite[\S 2]{Li},
\begin{equation}
    \delta(\phi_1 \otimes \bar{\phi}_2)(0) = \langle \phi_1, \phi_2 \rangle. \label{eq:doublingeval}
\end{equation}

For the benefit of the reader, we recall a standard result in the theory of doubling integrals which we will invoke below.

\begin{Prop} \label{Prop:DoublingSummation}
    Let $\Phi = \phi_1 \otimes \phi_2 \in \cS(X(\mathbb{A}_F)) \otimes \cS(X(\mathbb{A}_F)) \subset \cS(X(\mathbb{A}_F)^2)$, then 
    \[
        \sum_{x \in (V \otimes_E W^{\nabla})(F)} \delta(\Phi)(x) = \sum_{y \in X(F)^2} \Phi(y).
    \]
\end{Prop}
\begin{proof}
    This is, for instance, a special case (in his notation $u_1' = u_2' = 1$) of the last claim of \cite[Lemma 5.3]{Liu2}. The proof is a standard application of adelic Poisson summation.
\end{proof}

We first prove a general fact about diagonal periods. 

\begin{Lem}\label{lem: cst term}
    Let $\Phi_\infty \in \cS(V(E_{\infty})^2)$ satisfy $\Phi_\infty(0, 0) \neq 0$. Then we may define the functional 
    \[
        I_{\Phi_\infty}: \cS(V(\mathbb{A}_{E, f})^2) \to \mathbb{C}
    \]
    by 
    \[
        I_{\Phi_\infty}(\Phi_f) = \int_{\g(F)\backslash \g(\mathbb{A}_F)} \sum_{x \in V(E)^2} \omega(g)(\Phi_\infty \otimes \Phi_f)(x)\,dg,
    \]
    here by abuse of notation we also denote the obvious pullback action of $\g$ on $\cS(V^2(\mathbb{A}_E))$ by $\omega(g)$. This functional is nonzero.
\end{Lem}
\begin{proof}
    We pick a $\Phi_f$ such that $\Phi_f(0, 0) = 1$, and denote $\Phi = \Phi_\infty \cdot \Phi_f$. We define the theta integral
    \[
        T(h) \coloneqq \int_{\g(F)\backslash \g(\mathbb{A}_F)} \sum_{x \in V(E)^2} \omega(g, h)(\Phi_\infty \otimes \Phi_f)(x) \, dg,
    \]
    where here $h \in U(W^\Box)(\mathbb{A}_F)$. Picking $B \subset U(W^\Box)$ a stabilizer of a complete isotropic flag, and letting $N$ denote its unipotent radical, we will show that the constant term along $B$ satisfies
    \[
        T_N(1) = \int_{N(F)\backslash N(\mathbb{A}_F)} T(n)\, dn = \Phi_\infty(0, 0) \cdot \Phi_f(0, 0) = \Phi_\infty(0, 0).
    \]
    We can write $W^\Box = E y_1 \oplus E y_2 \oplus Ey_2^* \oplus Ey_1^*$ and take $H_1 = U(Ey_2 \oplus Ey_2^*)$. Now using \cite[Proposition 7.2]{GQT} we can compute this integral in stages. First taking the constant term along the maximal parabolic $P_1$ which stabilizes the line $E \cdot y_1$ with unipotent radical $N_1$, we obtain
    \[
        T_{N_1}(a, h') = \int_{N_1(F) \backslash N_1(\mathbb{A}_F)} T(n_1h)\, dn_1 = \chi(a) \cdot |a|^{3/2}_E \cdot \int_{\g(F) \backslash \g(\mathbb{A}_F)} \sum_{x \in V(E)} (\omega_1(g, h')\Phi^1)(x)\, dg
    \]
    where $h = (a, h') \in \mathbb{A}_E^\times \times H_1(\mathbb{A}_F)$ is an element of the Levi of $P_1$, and the function $\Phi^1 \in \cS(V(\mathbb{A}_E))$ is simply $\Phi^1(x) = \Phi(0, x)$, and $\chi$ is a character which will play no role in the proof. If we let $a = 1$ we find that 
    \[
        \int_{N_1(F)\backslash N_1(\mathbb{A}_F)} T(n_1 \cdot (1, h'))\, dn_1= \int_{\g(F) \backslash \g(\mathbb{A}_F)} \sum_{x \in V(E)} (\omega_1(g, h')\Phi^1)(x)\, dg \eqqcolon T_1(h').
    \]
    Now we are reduced to the smaller unitary group $H_1$, and we can apply \cite[Proposition 7.2]{GQT} to the maximal parabolic $P_2$, the stabilizer of $E \cdot y_2$ in $H_1$ (which has unipotent radical $N_2$), to obtain
    \[
        \int_{N_2(F)\backslash N_2(\mathbb{A}_F)} T_1(n_2)\, dn_2 = \int_{\g(F)\backslash \g(\mathbb{A}_F)} \Phi(0, 0)\, dg = \Phi(0, 0).
    \]
    Since $N = N_1 \rtimes N_2$ we have established the claimed result that $T_N(1) = \Phi(0, 0) = \Phi_\infty(0, 0)$. On the other hand, by the discussion in \cite[\S 3.1]{GQT} and our assumption that $F \neq \mathbb{Q}$ and thus that $V$ is definite at at least one infinite place (and thus $\g$ is anisotropic over $F$), the original integral converges and the function $T$ is a smooth automorphic function on $U(W^\Box)$, in particular if $\gamma \in N(F), h_f \in U(W^\Box)(\mathbb{A}_{F, f})$ we see that
    \[
        T(\gamma_\infty, h_f) = T(1, \gamma_f^{-1}h_f).
    \]
    But then on the other hand if the claimed functional vanished identically in $\Phi_f$, then by equivariance under $h_f \in U(W^{\Box})(\mathbb{A}_{F, f})$ we would have $T(\gamma_\infty, h_f) = T(1, \gamma_f^{-1}h_f) = 0$ for any such $h_f$, and any $\gamma \in N(F)$. But by weak approximation $N(F) \subset N(F_\infty)$ is dense, so this would imply that $T$ vanishes identically on $N(\mathbb{A}_F)$, which would contradict our constant term calculation. 
\end{proof}

\subsubsection{Proof of Theorem \ref{Thm:ExistenceCohomologyClasses}} In this proof, when suppressing the second argument of an unintegrated cohomological theta map, we denote $\Theta_{\psi, \nu}^{1, 0}(\varphi)\coloneqq \Theta_{\psi, \nu}^{1, 0}(\varphi, 1)$. Using Proposition \ref{Prop:ThetaRealization}, it will turn out to suffice to find finite Schwartz functions $\varphi_i \in \cS(X_i(\mathbb{A}_{F, f}))^K$, where each $X_i$ is a Lagrangian subspace of $\mathbb{W}_i$, such that the integral
\[
    \int_{\sh_K\hy} \Theta_{\psi, \nu_1}^{1, 0} (\varphi_1) \wedge \Theta_{\psi, \nu_2}^{1, 0}(\varphi_2) \wedge \overline{\Theta_{\psi, \nu_3}^{1, 0}(\varphi_3)} \wedge \overline{\Theta_{\psi, \nu_4}^{1, 0}(\varphi_4)}.
\]
is nonzero, but we put this aside for now. At infinity, we can fix a $\h(\mathbb{R})$-invariant volume form $\eta$ on the symmetric space $\mathsf{Y}$, which we take to be normalized. If $c_1, c_2, c_3, c_4$ are the cocycles at infinity provided in the theta lifting of \cite[Proof of Theorem 9.1]{BMM}, then 
\[
    c_1 \wedge c_2 \wedge \bar{c}_3 \wedge \bar{c}_4 \in (\cS_{v_0}(X_1) \otimes \cS_{v_0}(X_2) \otimes \cS_{v_0}(-X_3) \otimes \cS_{v_0}(-X_4) \otimes \wedge^4 \mathfrak{p}^*)^{K_\infty},
\]
in particular $c_1 \wedge c_2 \wedge \bar{c}_3 \wedge \bar{c}_4 = \phi_{v_0} \otimes \eta$ for some $\phi_{v_0} \in \bigotimes_{i = 1}^2 \cS_{v_0}(X_i) \otimes \bigotimes_{i = 3}^4 \cS_{v_0}(-X_i)$. \smallskip

Given $\Psi$ as in Lemma \ref{Lem:ExistenceLines}, we can pick a corresponding unitary intertwiner 
\[
    I_0: \cS((-X_3 \perp -X_4)(\mathbb{A}_F)) \xrightarrow{\sim} \cS((-X_1 \perp -X_2)(\mathbb{A}_F)),
\]
which is unique up to unit scalar, and which we always normalize to preserve the theta distribution. We can thus define a doubling intertwiner 
\[
\delta_\infty: ((\bigotimes_{i = 1}^2 \cS_{v_0}(X_i) \otimes \bigotimes_{i = 3}^4 \cS_{v_0}(-X_i)) \otimes \wedge^4 \mathfrak{p}^*) \to \cS(V(E_{\tau_0})^2) \otimes \wedge^4 \mathfrak{p}^*
\]
by the formula $\delta_\infty(\phi_\infty \otimes \eta) = \delta_{\infty, \Psi}(\phi_\infty) \otimes \eta$. Here $\delta_{\infty, \Psi}$ is the usual doubling intertwiner 
\[
    \delta_\infty: \cS((X_1 \perp X_2)(\mathbb{A}_F)) \otimes \cS((-X_1 \perp -X_2)(\mathbb{A}_F)) \to \cS(V(\mathbb{A}_E)^2)
\]
composed with $I_0$ and the inclusion $\cS(-X_3) \otimes \cS(-X_4) \to \cS(-X_3 \perp -X_4)$ in the second factor. 

Extending this to the finite places and definite infinite places is unproblematic, so we get a map 
\[
    \delta: \left(\bigotimes_{i = 1}^2 \cS(X_i(\mathbb{A}_F)) \otimes \bigotimes_{i = 3}^4 \cS(-X_i(\mathbb{A}_F)) \otimes \wedge^4 \mathfrak{p}^*\right)^{K_\infty} \longrightarrow (\cS(V(\mathbb{A}_E)^2) \otimes \wedge^4 \mathfrak{p}^*)^{K_\infty}.
\]
We let $\phi_\infty$ denote the tensor product of $\phi_{v_0}$ with $\bigotimes_{v|\infty, v \neq v_0} \phi_{v, i}$ and $\phi_{v, i}$ is a Gaussian if $i = 1, 2$ or its conjugate if $i = 3, 4$.  Now applying Proposition \ref{Prop:DoublingSummation}, we let $g = (g_\infty, g_f) \in \h(\mathbb{A})$, and $z = g_\infty \cdot z_0$ is a point on the locally symmetric space associated to $\hy$ where $z_0$ is a chosen basepoint then. If we let
\[
    \Phi \coloneqq
    \delta\!\left(
        \phi_\infty \otimes \varphi_1 \otimes \varphi_2
        \otimes \overline{\varphi}_3 \otimes \overline{\varphi}_4
    \right).
\]
Then
\begin{align*}
    \Bigl(
        \Theta_{\psi,\nu_1}^{1,0}(\varphi_1)
        \wedge \Theta_{\psi,\nu_2}^{1,0}(\varphi_2) \wedge
        \overline{\Theta_{\psi,\nu_3}^{1,0}(\varphi_3)}
        \wedge \overline{\Theta_{\psi,\nu_4}^{1,0}(\varphi_4)}
    \Bigr)(z,g_f) &=\left(\sum_{x \in V(E)^2} (\omega(g)\Phi)(x)\right)\eta_z \\
    &=
        \left(
            \sum_{x \in V(E)^2} \Phi(g^{-1}x)
        \right)\eta_z.
\end{align*}
Where in the last line we use the content of Remark \ref{rem: splitting}. We note that $F_\Phi(g) \coloneqq \left(\sum_{x \in V(E)^2} \Phi(g^{-1}x) \right)$ is invariant under $K_\infty \times K$, so we find that
\begin{equation}\label{eq: key integral}
    \int_{\sh_K\hy} F_\Phi \cdot \eta = C \int_{\h(\mathbb{Q})\backslash\h(\mathbb{A})} F_\Phi(g) \, dg
\end{equation}
where the constant $C$ is the product of the reciprocals of the masses of the compact subgroup $K_\infty$ with respect to a Haar measure compatible with $\eta$ and the probability measure on $\g(F) \backslash \g(\mathbb{A}_F)$, and $K$ with respect to the Haar measure on $\g(\mathbb{A}_{F, f})$. \smallskip

The upshot of this discussion is that, we can identify the integral of the cup-product of our theta series, up to a nonzero scalar and considered as a functional on $\cS(V(\mathbb{A}_{E, f})^2)$ via the doubling intertwiner, with a functional of the form studied in Lemma \ref{lem: cst term}. The following lemma will provide us with the conditions under which we can show the desired nonvanishing hypothesis in Lemma \ref{lem: cst term}.

\begin{Lem}\label{lem: wiggle}
    There exists an $F$-rational isometry $\Psi: W_1 \perp W_2 \xrightarrow{\sim} W_3 \perp W_4$ such that under the doubling map $\delta_{\infty, \Psi}(\phi_\infty) = \Phi_\infty$ is nonvanishing at $0$.
\end{Lem}
\begin{proof}
    In general, the set of $F$-rational isometries between two such skew-Hermitian spaces is a torsor for $U(W_1 \perp W_2)$, and we know that this torsor is split by the choice of the $W_i$ as in Lemma \ref{Lem:ExistenceLines}. If we fix an isometry $\Psi_{\infty,\mathrm{spl}}$ over $F \otimes \mathbb{R}$ such that $\Psi_{\infty,\mathrm{spl}}$ preserves the splitting at each place, i.e. at each place maps $W_1$ to $W_3$ or $W_4$, and $W_2$ the other factor, then the doubled function identifies with a nonzero multiple of the Kudla--Millson form $\delta_{\infty, \Psi_{\infty, \mathrm{spl}}}(\phi_\infty \otimes \eta)(0) = D \cdot \varphi_{2, 2}(0)$ by \cite[Proposition 5.17(5), Proposition C.10]{BMM}, which is nonvanishing by \cite[Proposition 3.6, Theorem 3.7(3)]{Bergeron2}. The term $D$ includes the contributions from other infinite places which, since $\Psi_{\infty,\mathrm{spl}}$ preserves the splitting, just contribute a product of $L^2$-norms of Gaussians. Now use real approximation for $U(W_1 \perp W_2)$ to find a rational isometry $\Psi$ such that $\Psi_\infty$ is very close to $\Psi_{\infty,\mathrm{spl}}$. The doubling map associated with this isometry $\delta_{\infty, \Psi}(\phi_1 \otimes \phi_2)(0)$ can be shown to have a continuous dependence on $\Psi_\infty$, as we now explain. 
    
    Indeed, backing up a bit if $W'$ is a skew-Hermitian space, $X'$ is a Lagrangian of $V \otimes_E W'$ and $\Psi_{\infty, 0}: W' \to -W$ is an isometry, then if $\phi_1 \in \cS(X(F_\infty)), \phi_2 \in \cS(X'(F_\infty))$ we can define a corresponding unitary intertwiner
    \[
        I_0: \cS(X'(F_\infty)) \xrightarrow{\sim} \cS(-X(F_\infty))
    \]
    of Schwartz spaces, unique up to a unit scalar. Let $\delta_\infty$ denote the standard doubling map. By equation \ref{eq:doublingeval}
    \[
        \Phi_{\infty, \Psi_{\infty, 0}}(0) = \delta_{\infty}(\phi_1 \otimes I_0(\phi_2)(0) = \langle \phi_1, \overline{I_0(\phi_2)} \rangle.
    \]
    If we switch to $\Psi_\infty =  k \cdot \Psi_{\infty, 0}$ for some $k \in U(-W)$, we can normalize the resulting unitary intertwiner
    \[
        I': \cS(X'(F_\infty)) \xrightarrow{\sim} \cS(-X(F_\infty))
    \]
    to be $I' = \omega_{-W}(k) \cdot I_0$ Then 
    \[
        \Phi_{\infty, \Psi_\infty} (0) = \delta_{\infty}(\phi_1 \otimes I'(\phi_2))(0) = \langle \phi_1, \omega_{W}(k) \cdot \overline{I_0(\phi_2)} \rangle,
    \]
    using the identification $\overline{\omega_{-W}(k)\cdot \varphi} = \omega_W(k) \cdot \overline{\varphi}$ for $\varphi \in \cS(-X(F_\infty))$. So the desired continuity of $\Phi_{\infty, \Psi_\infty}(0)$ in $\Psi_\infty$ follows from (strong) continuity for the oscillator representation $\omega_{W}$.
\end{proof}

Lemma \ref{lem: cst term} tells us that for $\Phi = \Phi_\infty \otimes \Phi_f$ for some $\Phi_f$, the integral in equation \ref{eq: key integral} is nonvanishing. On the other hand $\delta^{-1}_f(\Phi_f) = \sum_{i, j, k, l} \varphi_i \otimes \varphi_j \otimes \varphi_k \otimes \varphi_l$, so there must be some pure tensor for which the integral is nonvanishing as well, by linearity. This in hand, we must deduce that the conclusion of Theorem \ref{Thm:ExistenceCohomologyClasses} is true. We make the following elementary deduction.

\begin{Lem}
    Let $\Theta_{\psi, \nu}^{1, 0}$ be the cohomological theta lift constructed in Subsection \ref{subsect: unintegrated theta} corresponding to a pair $(\mu, W)$ where $W$ is a one-dimensional skew-Hermitian space. Let $\varphi_f$ be a finite Schwartz function, then there are finitely many automorphic characters $\{\chi_1, \dots, \chi_n\}$ of the torus $U(W)$ such that 
    \[
        \Theta_{\psi, \nu}^{1, 0}(\varphi_f, t) = \sum_{i = 1}^n \chi_i(t) \cdot \theta_{\chi_i}
    \]
    where the $\theta_\chi \in \rH^0(\sh_K\hy_{\mathbb{C}}, \Omega_{\mathsf{Y}}^{1, 0})$.
\end{Lem}
\begin{proof}
    As a consequence of \cite[Proposition 5.17(8)]{BMM} applied in our special case, and the definition of $\varphi_{a, b}$ in \cite[\S 5.8]{BMM}, we find that $\varphi_{1, 0}$ transforms under a character $\chi_{v_0}$ under the action of $U(W)(F_{v_0})$. The same is clearly also true for the Gaussian factor $\varphi_0$ \cite[\S 8.7]{BMM}. We refer to by $\chi_\infty$ the product of these characters, which defines a character of $U(W)(F_\infty)$. On the other hand $\varphi_f$ is fixed under the action of some open compact $K_f \subset U(W)(\mathbb{A}_f)$. So we find that the Fourier support of the function 
    \[
        \Theta_{\psi, \nu}^{1, 0}(\varphi_f, -)
    \]
    on $U(W)(F)\backslash U(W)(\mathbb{A}_F)$ must be supported on the set of automorphic characters of $U(W)(\mathbb{A}_F)$ such that $\chi|_{K_f} = 1$ and $\chi|_{U(W)(F_\infty)} = \chi_\infty$, which defines a finite set since
    \[
        U(W)(F)\backslash U(W)(\mathbb{A}_{F, f})/K_f
    \]
    is finite. 
\end{proof}

As a result of the above Lemma, we see that 
\[
    \Theta_{\psi, \nu}^{1, 0}(\varphi_f, 1) = \sum_{i = 1}^n \theta_{\chi_i}
\]
where the $\theta_{\chi_i}$ are Fourier coefficients of $\Theta^{1, 0}_{\psi, \nu}$, which are of course integrated theta classes as defined in Section \ref{subsect: unintegrated theta}. Applying this to the above nonvanishing statement, and using that $(\sum_i \alpha_i) \cup (\sum_j \beta_j) \neq 0$ implies that $\alpha_i \cup \beta_j \neq 0$ for some pair $(i, j)$, we see that there are $f_i \in \chi_i^{-1}$ such that 
\[
    \int  \Theta^{1,0}_{\psi, \nu_1, f_1}(\varphi_1) \wedge \Theta^{1,0}_{\psi, \nu_2, f_2}(\varphi_2)  \wedge  \overline{\Theta^{1,0}_{\psi, \nu_3, f_3}(\varphi_3)} \wedge \overline{\Theta^{1,0}_{\psi, \nu_4, f_4}(\varphi_4)}  \neq 0.
\]

At this point we have essentially proved the result, indeed with $(\mu_i, W_i)$ fixed, if the Fourier component associated to some character $\chi_i$ is nonzero and $f_i \in \chi_i^{-1}$, then the corresponding classes $\Theta^{1,0}_{\psi, \nu_i, f_i}(\varphi_i)$ are integrated theta lifts whose associated representation is $\omega(\mu_i, \epsilon_i, \kappa_i)$ for $\epsilon_i$ determined by $W_i$, and $\kappa_i$ determined by $\chi_i$. By Proposition \ref{Prop:ThetaRealization} each of the classes $\Theta^{1,0}_{\psi, \nu_i, f_i}(\varphi_i)$ for $i = 1, 2$, $\overline{\Theta^{1,0}_{\psi, \nu_i, f_i}(\varphi_i)}$ for $i = 3, 4$ arise from a linear combination of classes in the image of $\rH^1(B_i, E)_{\lambda_i}$, applying the same trick as was used in the previous paragraph we obtain the statement of Theorem \ref{Thm:ExistenceCohomologyClasses}. $\qed$