Is there always a finite family of Shimura curves $(C_{i})$ in $A_{g}$ the moduli space of principally polarized abelian varieties of dimension $g(\geq 2)$, such that the union $\cup C_{i}$ is connected and such that $\cup C_{i}$ is not strictly contained in larger Shimura subvarieties (special subvarieties) except $A_{g}$ itself? i.e. there does not exist Shimura subvarieties $Z$ in $A_{g}$ with $\cup C_{i}\subsetneq Z$ and $Z\neq A_{g}$? I will appreciate any suggestion of results related to this problem and/or suggestions how one can possibly construct such families.
1 Answer
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There are countably many special subvarieties because they are defined over $\overline{Q}$ but I do not see how this is relevant to your question?
I understand you want to know whether any Shimura curve is contained in a special subvariety; I think the answer is yes, it should be contained in an appropriate Hilbert modular variety.
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$\begingroup$ Thanks. But I don't think the answer to my question is always negative as you say. At least for some small $g$ I know that there are Shimura curves not contained in bigger Shimura subvarieties. $\endgroup$ Dec 9, 2013 at 16:50
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$\begingroup$ I think you are right. I changed the question to a correct format that I really need. $\endgroup$ Dec 9, 2013 at 17:21
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2$\begingroup$ It is not obvious to me that Shimura curves made from division algebras over a real quadratic field, split at one archimedean place and not at the other, imbed into Hilbert modular varieties, for example. If it is true, it's not elementary, I think. $\endgroup$ Apr 8, 2014 at 20:41