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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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The cohomology has an action of the quaternion algebra, and of the reals, so it has an action of the tensor product, which is 2x2 matrices over the reals. So by Morita equivalence or whatever, everything in sight is a direct sum of two copies of something smaller, the something smaller in this case being, at least locally, the VHS for the universal family of elliptic curves. –  user30035 Jan 19 '13 at 16:19

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