Lower bounds on the $\ell$-rank of ideal class groups
Daniel E. Martin
TL;DR
The paper develops new lower bounds for the ℓ-rank of the class group of a number field K over a base F by relating ramification in K/F to ℓ-divisibility properties of the extension. Central to the approach is the ℓ-divisible (and strongly ℓ-divisible) framework, which enables bounds that depend on primes in F with ramification indices divisible by ℓ, rather than requiring all ramified primes. The authors prove key theorems that yield explicit lower bounds and then derive consequences for class field towers, showing infinite ℓ-class field towers under suitable ramification in a wide range of Galois-type structures (e.g., Galois, nilpotent, dihedral). They also obtain density results for extensions with finite towers, including nilpotent groups via Klüners–Malle, and provide a comprehensive Appendix classifying divisible-field extensions and their behavior under towers, semidirect products, and composites. Overall, the work broadens the applicability of genus-type bounds and Golod–Shafarevich arguments to non-Galois and structured extensions, with implications for the distribution of infinite class field 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.