Selmer groups and class groups

Abstract

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.