Let A be an abelian variety over a number field K. If A has
nontrivial (resp. full) K-rational p-torsion for a prime p, exploiting the
fppf cohomological approach to Selmer groups, we obtain inequalities
bounding the size of the p-Selmer group of A from below (resp. above) in
terms of the size of the p-torsion subgroup of the ideal class group of K.
When K varies in a family of field extensions, these inequalities relate
the growth of Selmer groups to that of class groups; I will discuss such
relations in several different settings.