\section{Picard modular surfaces} \label{Sec:PMS} To apply Theorem \ref{Thm:ReductionToSurface}, we need to produce an algebraic surface $S$ whose $\rH^1(S, \mathbb{Q})$ contains the Hodge structures of CM abelian varieties with CM by a given CM field $E$. Fortunately, for us, such surfaces are known to exist by work of Liu, see \cite[Section 4]{Liu}, as we will explain below.  

\subsection{Picard modular surfaces and their Albanese varieties}

\subsubsection{} Let $E$ be a CM field with totally real subfield $F \not= \mathbb{Q}$ and fix an embedding $\tau_0:E \to \mathbb{C}$. Let $V$ be a Hermitian $E$ vector space of dimension $3$ such that $V$ has signature $(2,1)$ at $\restr{\tau_0}{F}$ and has signature $(3,0)$ at all other infinite places of $F$. Let $\g = \operatorname{U}(V)$ be the algebraic group over $F$ of isometries of $V$, and let $\h=\operatorname{Res}_{F/\mathbb{Q}} \g$ equipped with its natural Shimura datum $\hy$. Note that the reflex field of $\hy$ is equal to $E=\tau_0(E) \subset \mathbb{C}$.

\subsubsection{} For $K \subset \h(\af)$ a sufficiently small compact open subgroup, there is a smooth projective algebraic surface $\mathbf{Sh}_K\hy$ over $\spec E$, see e.g. \cite[Proposition C.5]{Liu}. Being a Shimura variety, it will typically not be geometrically connected. The goal of this section is to compute the rational Hodge structures occuring in $\rH^1(\mathbf{Sh}_K\hy_{\mathbb{C}}, \mathbb{Q})$
up to isogeny. We observe that
\begin{align}
    \rH^1(\mathbf{Sh}\hy_{\mathbb{C}}, \mathbb{C}) :=\varinjlim_K \rH^1(\mathbf{Sh}_K\hy_{\mathbb{C}}, \mathbb{C})
\end{align}
is equipped with a smooth action of $\h(\af)$, which acts via isomorphisms of rational Hodge structures. Our goal is now to compute this space, following \cite{BMM} and \cite{Liu}. 

\subsubsection{} Let $\mathbb{A}_{E}^{\times}$ denotes the ideles of $E$, a locally compact topological group containing $E^{\times}$. Recall from \cite[Definition 4.1, Definition 4.3]{Liu} that a \emph{conjugate symplectic automorphic character of weight one} is a continuous homomorphism
\begin{align}
    \mu:E^{\times} \backslash \mathbb{A}_{E}^{\times} \to \mathbb{C}^{\times}
\end{align}
such that its restriction to $\mathbb{A}_{F}^{\times}$ is given by the quadratic character $\mu_{E/F}$ associated to the quadratic extension $E/F$ via global class field theory, and such that there is a CM type $\Phi_{\mu}$ of $E$ such that for every $\tau \in \Phi_{\mu}$ we have that
\begin{align}
    \restr{\mu}{E \otimes_{\tau, F} \mathbb{R}}
\end{align}
is given by $z \mapsto \tfrac{|z|}{z}$. We call $\Phi_{\mu}$ the \emph{CM type} of $\mu$, which is unique. Associated to $\Phi_{\mu}$ is a reflex field $M_{\mu}' \subset \mathbb{C}$, and a CM type $\Psi_{\mu}$ for $M_{\mu}'$. We let $M_{\mu} \subset \mathbb{C}$ be as in \cite[Definition 4.3]{Liu}, which is a number field containing $M_{\mu}'$.

\subsubsection{} \label{subsub:TripleDef} Recall from \cite[Definition 4.11]{Liu} that an \emph{adelic oscillator triple} is a triple $(\mu, \epsilon, \chi)$, where $\mu$ is a conjugate symplectic automorphic character of weight one, where $\epsilon \in \prod'_{v \mathrm{ finite}} E_v^{-\times} / N_{E_v/F_v} E_v^{\times}$ and where $\chi$ is an auxiliary character that will not matter for our discussion.\footnote{We only consider adelic oscillator triples with $\mu$ of weight one, whereas \cite[Definition 4.11]{Liu} allows for general $\mu$.} We say that $(\mu, \epsilon, \chi)$ is \emph{admissible}, see \cite[Definition 4.12]{Liu}, if there is an $e \in E^{\times}$ such that $e^c = - e$ and $\operatorname{Im} \tau(e) < 0$ for $\tau \in \Phi_{\mu}$ with $\epsilon=(e N_{E_v/F_v} E_v^{\times} )_{v}$.
\begin{Lem} \label{Lem:ExistenceTriple}
    For each CM type $\Phi$ of $E$ there is a conjugate symplectic automorphic character $\mu$ of weight one with $\Phi_{\mu}=\Phi$ and an admissible adelic oscillator triple $(\mu, \epsilon, \chi)$.
\end{Lem}
\begin{proof}
Applying \cite[Lemma 4.1.4]{ClozelHarrisTaylor} with $I=\Phi, S=\emptyset, a=\| - \|_{F}^{-1}$ and with $m_{\tau}=-1$, we get a character\footnote{To be precise, we use (in the notation of \cite{ClozelHarrisTaylor}) $\chi=a=\| - \|_{F}^{-1}$. This notation unfortunately conflicts with our notation.}
\begin{align}
    \psi:E^{\times} \backslash \mathbb{A}_E^{\times} \to \mathbb{C}^{\times}
\end{align}
such that, for $N_{E/F}:\mathbb{A}_{E}^{\times} \to \mathbb{A}_{F}^{\times}$ the norm map, we have 
\begin{align}
    \restr{\psi}{\mathbb{A}_F^{\times}} \circ N_{E/F} = a \circ N_{E/F}
\end{align}
and such that
\begin{align}
    \restr{\psi}{(E \otimes \mathbb{R})^{\times}} = \prod_{\tau \in I} \tau^{-1} \cdot (c \tau)^{w+1}.
\end{align}
for some $w$. In fact it follows from \cite[proof of Lemma 4.1.4]{ClozelHarrisTaylor} that $w=-1$ for our $a$, which implies that for $\mu=\psi \| - \|_{E}^{1/2}$ and for $\tau \in \Phi$ we have 
\begin{align}
    \left(\restr{\mu}{E \otimes_{\tau, F} \mathbb{R}}\right)(z) = \frac{|z|}{z}.
\end{align}
We now claim that $\mu$ is a conjugate symplectic automorphic character of weight one. To show that the restriction of $\mu$ to $\mathbb{A}_F^{\times}$ is $\mu_{E/F}$, we note that
\begin{align}
    (\mu \circ N_{E/F})(x) &= \psi(N_{E/F}(x)) \cdot \|N_{E/F}(x) \|_{E}^{1/2} \\
    &=\|N_{E/F}(x)\|^{-1}_{F} \cdot \|N_{E/F}(x) \|_{F} \\
    &=1
\end{align}
so that $\restr{\mu}{\mathbb{A}_F^{\times}}$ factors through $F^{\times} \backslash \mathbb{A}_F^{\times} / N_{E/F}(\mathbb{A}_E^{\times}) \simeq \gal(E/F)$ (see \cite[Theorem]{MilneCFT} for the last identification). Thus $\restr{\mu}{\mathbb{A}_F^{\times}}$ is either trivial or equals $\mu_{E/F}$, and the former possibility can be excluded by examining the further restriction of $\mu$ to $(F \otimes \mathbb{R})^{\times}$. \smallskip 

To find a $\mu$-admissible adelic oscillator triple for the given $\mu$, we may take $\chi=1$, and $\epsilon=(e N_{E_v/F_v} E_v^{\times} )_{v}$ for an element $e \in E^{\times}$ satisfying (where $c$ is the nontrivial $F$-linear automorphisms of $E$).
\begin{align}
    e^c &= -e \\
    \forall \tau \in \Phi_{\mu} \quad &\operatorname{Im} \tau(e) < 0 .
\end{align}
This is clearly a $\mu$-admissible adelic oscillator triple. The existence of such $e$ is a routine application of weak approximation. 
\end{proof}

\subsubsection{} Associated to an adelic oscillator triple $(\mu, \epsilon, \chi)$ there is an admissible representation $\omega(\mu, \epsilon, \chi)$ of $\h(\af)$. We have the following result, see \cite[Proposition 4.13]{Liu}
\begin{Prop} \label{Prop:CohomologyOfPMS}
There is an isomorphism of $\h(\af)$-representations
\begin{align}
    \rH^1(\mathbf{Sh}\hy_{\mathbb{C}}, \mathbb{C}) \xrightarrow{\sim} \bigoplus_{(\mu, \epsilon, \chi)} \omega(\mu, \epsilon, \chi),
\end{align}
where the direct sums runs over admissible adelic oscillator triples $(\mu, \epsilon, \chi)$ with $\mu$ of weight one.
\end{Prop}

\subsubsection{} We now study the Hodge structure on the cohomology. Let $A_K$ denote the Albanese variety of $\mathbf{Sh}_K\hy$ over the field $E$, in the sense of \cite[Definition 2.3]{Liu}. Consider the pro-(abelian variety)
\begin{align}
    A_{\infty}:=\varprojlim_{K} A_K
\end{align}
over $E$ together with its action of $\g(\mathbb{A}_{F,f})$.

\subsubsection{} Associated to a conjugate symplectic automorphic character of weight one $\mu$ is a notion of a \emph{CM datum}, see \cite[Definition 4.5]{Liu}, which is a tuple $(A_{\mu}, i_{\mu}, \lambda_{\mu}, r_{\mu})$. It is not so important what the last two entries are, but $A_{\mu}$ is an abelian variety over $E$ and $i_{\mu}:M_{\mu} \to \operatorname{End}(A_{\mu}) \otimes \mathbb{Q}$ is a CM structure with CM type determined by $\Psi_{\mu}$. By \cite[Proposition 4.6]{Liu}, CM data always exist and are, roughly speaking, unique up to isogeny. In particular the isogeny class of $A_{\mu}$ is a well defined function of $\mu$.

\subsubsection{} For a fixed conjugate symplectic automorphic character of weight one $\mu$ we consider 
\begin{align}
    \Omega(\mu)=\operatorname{Hom}(A_{\infty}, A_{\mu}) \otimes \mathbb{Q},
\end{align}
where we are taking homomorphisms of group schemes over $E$. The field $M_{\mu}$ acts on $A_{\mu}$ and $\h(\af)$ acts on $A_{\infty}$, and so $\Omega(\mu)$ is an $M_{\mu}$-linear representation of $\h(\af)$. For every $\epsilon$ which is part of a $\mu$-admissible adelic oscillator triple $(\mu, \epsilon, \chi)$, there is a $\h(\af)$-subrepresentation $\Omega(\mu, \epsilon) \subset \Omega(\mu)$, see \cite[Definition 4.19]{Liu}. We have the following result, see \cite[Theorem 4.18]{Liu}, recalling that $M_{\mu}$ is a subfield of $\mathbb{C}$. 
\begin{Thm} \label{Thm:HodgeStructureComputation}
There is an isomorphism
\begin{align}
    \Omega(\mu,\epsilon) \otimes_{M_{\mu}} \mathbb{C} = \bigoplus_{\chi} \omega(\mu, \epsilon, \chi),
\end{align}   
where the direct sum runs over those $\chi$ such that $(\mu, \epsilon, \chi)$ is an admissible adelic oscillator triple.
\end{Thm}

\subsubsection{} Theorem \ref{Thm:HodgeStructureComputation} tells us which abelian varieties up to isogeny occur in $A_{\infty}$, and thus in $A_K$. Basechanging to $\mathbb{C}$, this therefore tells us which Hodge structure occur in $\rH^1(\mathbf{Sh}\hy_{\mathbb{C}}, \mathbb{C})$ and with which level. In particular, we can deduce
\begin{Cor} \label{Cor:CMAVOccurs}
Suppose that $E$ is Galois. Given a CM abelian variety $B$ over $\mathbb{C}$ with CM by $E$, 
there is a surjective homomorphism $A_{\infty, \mathbb{C}} \to B$ of group schemes. 
\end{Cor}
\begin{proof}
Let $\Phi$ be the CM type of $B$. The CM type $\Phi$ is the extension from a primitive CM type $\Phi_0$ on a subfield $E_0 \subset E$, see \cite[Proposition 1.9]{MilneCM}. If $B_0$ is an abelian variety over $\mathbb{C}$ with CM by $E_0$ and CM type $\Phi_0$, then $B \sim B_0^{[E:E_0]}$, see \cite[Proof of Proposition 3.13]{MilneCM}. On the other hand, for a CM type $\Phi_{\mu}$ with reflex field $(M_{\mu}', \Psi_{\mu})$, we have by construction that
\begin{align}
    A_{\mu, \mathbb{C}} \sim (A'_{\mu, \mathbb{C}})^{[M_{\mu}:M_{\mu}']},
\end{align}
where $A'_{\mu}$ is an abelian variety with CM by $M_{\mu}'$ and CM type $\Psi_{\mu}$. To relate $B_0$ and $A'_{\mu}$, we need to relate $(E_0, \Phi_0)$ and $(M_{\mu}', \Psi_{\mu})$. Because $E$ is Galois, by \cite[Example 1.19]{MilneCM}, we have $(E_0, \Phi_0)=(M_{\mu}', \Psi_{\mu})$ if we choose $\Phi_{\mu}=\Phi^{-1}$ (here we consider $\Phi \subset \gal_{E}$ and take its inverse as a subset of the group). \smallskip 

Now choose an adelic oscillator triple $(\mu, \epsilon, \chi)$ with $\Phi_{\mu}=\Phi^{-1}$. Then by Theorem \ref{Thm:HodgeStructureComputation} $A_{\infty}$ admits a nonzero map to $A_{\mu}$. Since $\omega(\mu, \epsilon,\chi)$ is infinite dimensional, it in fact admits infinitely many linearly independent maps to $A_{\mu}$. The group scheme $A_{\infty, \mathbb{C}}$ therefore surjects onto any power of $A'_{\mu, \mathbb{C}}$. From the discussion above, it follows that $A_{\infty, \mathbb{C}}$ surjects onto any power of $A'_{\mu,\mathbb{C}} \sim B_0$ and thus surjects onto $B$.
\end{proof}