Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Mark Goresky
Affiliation:

Institute for Advanced Study, Princeton
Date:

Tue, 2018-03-13 09:30 - 10:30 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] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/3444

[3] http://www.mpim-bonn.mpg.de/node/7438