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Let $X$ be a compact Shimura curve. If we remove finitely many points from this curve, do we neccessarily get a "non-compact Shimura curve"? I have some reasons to believe that the answer is negative, but don't have a proof nor a counter example. However if the answer is really No, does there exist a well-known structure / property for this punctured Shimura curve?

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A "non-compact Shimura curve" usually means a Shimura variety that happens to be a curve, and non-compact. The standard example would be a modular curve. Removing finitely many points from a compact Shimura curve would leave you with a curve that wasn't a Shimura variety in the sense that it doesn't come from Deligne's machine. So in short I think the answer to your question is "no". –  user30035 Dec 31 '12 at 13:05
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