The Gauss higher relative class number problem

TL;DR

This paper determines all CM fields with higher relative class number at most 16 under the assumption of the 2-adic Iwasawa main conjecture, identifying a finite set with mostly abelian fields.

Contribution

It provides a classification of CM fields with bounded higher relative class number assuming a major conjecture, including the identification of a unique non-abelian field.

Findings

At least 31 and at most 34 such CM fields exist.

Exactly one of these fields is non-abelian.

The classification depends on the 2-adic Iwasawa main conjecture.

Abstract

Assuming the 2-adic Iwasawa main conjecture, we find all CM fields with higher relative class number at most 16: there are at least 31 and at most 34 such fields, and exactly one is not abelian.