Lower bounds on the ℓ-rank of ideal class groups

2025/01/16 by Martin, Daniel E.
#11R29 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.09865

Abstract

For a prime number ℓ and an extension of number fields K/F, we prove new lower bounds on the ℓ-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 ℓ. 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.

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