Abstract
Let
$$K$$
be the totally real cubic field of discriminant
$$49$$
, let
$${\fancyscript{O}}$$
be its ring of integers, and let
$$p\subset {\fancyscript{O}}$$
be the prime over
$$7$$
. Let
$$\Gamma (p)\subset \Gamma = SL_{2} ({\fancyscript{O}})$$
be the principal congruence subgroup of level
$$p$$
. This paper investigates the geometry of the Hilbert modular threefold attached to
$$\Gamma (p) $$
and some related varieties. In particular, we discover an octic in
$$\mathbb{P }^3$$
with
$$84$$
isolated singular points of type
$$A_2$$
.