If $f: X \rightarrow C$ is the universal family of false elliptic curves (i.e. abelian surfaces $A$ such that $End(A) \otimes \mathbb{Q}$ is a totally indefinite quaternion algebra over $\mathbb{Q}$), then $C$ is a Shimura curve. My question is : How does the the variation of Hodge structures(VHS) of this universal family look like? Of course I know that for any family of abelian varieties over a Shimura curve, the VHS has the form $R^{1}f_{*}\mathbb{C}= (\mathbb{L} \otimes \mathbb{T}) \bigoplus \mathbb{U}$ where $\mathbb{U}$ and $\mathbb{T}$ are unitary and $\mathbb{L}$ is uniformizing, but I want a more specific description in the case of a universal family of false elliptic curves.
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