Lower bounds on the $\ell$-rank of ideal class groups

TL;DR

This paper establishes new lower bounds on the $ ext{ell}$-rank of ideal class groups in number field extensions, linking ramification properties to class group structure, with applications to infinite class field towers.

Contribution

It introduces a novel lower bound on the $ ext{ell}$-rank based on prime ramification, applicable under specific Galois group conditions, expanding previous results.

Findings

Lower bounds on $ ext{ell}$-rank based on ramification in $K/F$

Bound applies to various Galois extension towers

Proves a density result on fields with infinite class towers

Abstract

For a prime number $\ell$ and an extension of number fields $K/F$, we prove new lower bounds on the $\ell$-rank of the ideal class group of $K$ based on prime ramification in $K/F$. Unlike related results from the literature, our bound is supported on prime ideals in $F$ over which at least one (rather than each) prime in $K$ has ramification index divisible by $\ell$. This bound holds with a proviso on the Galois group of the normal closure of $K/F$, which is satisfied by towers of Galois extensions, intermediate fields in nilpotent extensions, and intermediate fields in dihedral extensions of degree $8n$, to name a few. We also use our lower bound to prove a new density result on number fields with infinite class field towers.