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 $l\ne p$, 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 $A(K)[l]\ne 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a ${\mathbb{Z}}_l$-extension.

17 pages; final version, to appear in Compositio Mathematica

References in corpus (4)

• On the Selmer groups of abelian varieties over function fields of characteristic p>0
• Selmer groups as flat cohomology groups
• L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields
• The period-index problem in WC-groups II: abelian varieties

Cited by in corpus (4)

• Selmer groups as flat cohomology groups
• $p$-Selmer growth in extensions of degree $p$
• High $\ell$-torsion rank for class groups over function fields
• Elliptic surfaces over ${\mathbb{P}}^1$ and large class groups of number fields