  Let $A$ be an abelian variety over a global field $K$ of characteristic $p
\ge 0$. If $A$ has nontrivial (resp. full) $K$-rational $l$-torsion for a prime
$l \neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer
group $\mathrm{Sel}_l A$ to bound $\#\mathrm{Sel}_l A$ from below (resp. above)
in terms of the cardinality of the $l$-torsion subgroup of the ideal class
group of $K$. Applied over families of finite extensions of $K$, the bounds
relate the growth of Selmer groups and class groups. For function fields, this
technique proves the unboundedness of $l$-ranks of class groups of quadratic
extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^n}$
(depending on $l$). For number fields, it suggests a new approach to the
Iwasawa $\mu = 0$ conjecture through inequalities, valid when $A(K)[l] \neq 0$,
between Iwasawa invariants governing the growth of Selmer groups and class
groups in a $\mathbb{Z}_l$-extension.
