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Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the best example of such a surface?

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A trick: for a totally real abelian cubic number field $k$, let $B$ be a quaternion algebra over $k$ that is split at exactly two of the real places of $k$. The associated Shimura variety is two-dimensional, but does not contain any natural Shimura curve. (But I do not know how to exclude the possibility of "un-natural" imbeddings...)

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  • $\begingroup$ Thank you for the answer. This is nice, but as you mentioned, does not exclude all Shimura curves! $\endgroup$ – Darius Math Aug 12 '13 at 10:09

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