For each $d<0$ there is an anti-holomorphic involution of the $(Sp(4, R))$ Siegel modular variety whose fixed point set is a finite union of hyperbolic $3$-manifolds with fundamental group $SL(2,\mathcal{O}_d)$. Its points correspond to (principally polarized) Abelian surfaces with anti-holomorphic multiplication by $\mathcal{O}_d$, meaning a real action of $\mathcal{O}_d$ such that $\sqrt{d}$ acts anti-holomorphically. Using Deligne`s description of the category of ordinary Abelian varieties, we are able to make sense of anti-holomorphic multiplication for ordinary Abelian varieties over a finite field, thus providing a candidate for the ordinary points, mod $p$ of these hyperbolic $3$-manifolds. This is ongoing joint work with Yung sheng Tai.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/7438