# Ordinary points mod $p$ of hyperbolic $3$-manifolds

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Speaker:
Mark Goresky
Affiliation:
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.