(This is a continuation of Friday's talk.)

Hilbert modular varieties are associated to totally real fields. If $K$ is such a field, with ring of integers $O_K$, then the associated Hilbert modular variety is the quotient $H^n/SL_2(O_K)$.

It classifies Abelian varieties admitting real multiplication by $O_K$, as such it is an algebraic variety.

It has cusps, in 1-1 correspondence with the ideal class group of $K$.

These cusps are delicate to resolve. Such a resolution was found (in the case of surfaces) by Hirzebruch. Complex dynamics gives new insights into this desingularization, and its relation to continued fractions.