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Snowden has studied the topology of the real points of modular curves. Are there analogous results for other Shimura varieties defined over $\mathbb{R}$?

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    $\begingroup$ Curious that Shimura varieties are trendy enough to make the hot network questions list... $\endgroup$
    – Federico Poloni
    Commented Dec 8, 2020 at 20:42
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    $\begingroup$ @FedericoPoloni ... in plague times. Sigh. $\endgroup$
    – paul garrett
    Commented Dec 8, 2020 at 21:14
  • $\begingroup$ Historical note: the case of the real points of $X(N)$ was first considered by Jaffee in Degeneration of real elliptic curves. Jaffee also wrote other articles on real forms of symmetric domains, which may be of interest. $\endgroup$
    – François Brunault
    Commented Dec 8, 2020 at 21:34

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Well, classical Shimura curves associated to non-trivial quaternion algebras have no real points, so that's pretty easy to describe, albeit not very interesting. Ogg has a paper describing the real locus on the quotient of such curves by an involution:

MR0717598 Ogg, A. P. (1-CA) Real points on Shimura curves. Arithmetic and geometry, Vol. I, 277–307, Progr. Math., 35, Birkhäuser Boston, Boston, MA, 1983.

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Unclear whether this will be useful to you, but G. Shimura did write a paper about real points: "On the real points of an arithmetic quotient of a bounded symmetric domain", Math. Ann. 215 (1975), 135–164.

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