Selmer groups and class groups

TL;DR

This paper explores the relationship between Selmer groups of abelian varieties and class groups over global fields, providing bounds and implications for growth patterns, especially in function fields and number fields, with applications to conjectures like Iwasawa's.

Contribution

It introduces a cohomological approach to bound Selmer groups using class groups, linking their growth and offering new insights into longstanding conjectures.

Findings

Bounds on Selmer groups in terms of class group torsion

Unboundedness of $l$-ranks of class groups in quadratic extensions of function fields

Potential new approach to the Iwasawa $oldsymbol{\mu=0}$ conjecture

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.