  We give a new general technique for constructing and counting number fields
with an ideal class group of nontrivial m-rank. Our results can be viewed as
providing a way of specializing the Picard group of a variety V over
$\mathbb{Q}$ to obtain class groups for number fields $\mathbb{Q}(P)$, $P\in
V(\Qbar)$, for certain families of points P. In particular, we show how the
problem of constructing quadratic number fields with a large-rank ideal class
group can be reduced to the problem of finding a hyperelliptic curve with a
rational Weierstrass point and a large rational torsion subgroup in its
Jacobian. Furthermore, we show how many previous results on constructing
large-rank ideal class groups can be fit into our framework and rederived. As
an application of our technique, we derive a quantitative version of a theorem
of Nakano. This gives the best known general quantitative result on number
fields with a large-rank ideal class group.
