The CM class number one problem for genus-2 curves

The Gauss class number one problem for imaginary quadratic fields
is equivalent to finding all elliptic curves over the rationals with
complex multiplication.
 I will explain the relation between the class number one problem and
the CM elliptic curves over the rationals. Then I will give the analogue of
the class number one problem for genus-2 CM curves and sketch its solution.
This is joint work with Marco Streng.