Absolutely Abelian Hilbert Class Fields and $ll-$torsion conjecture

There are several recent works where authors have shown that number fields $K$ with `sufficiently many’ units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ is absolutely abelian. In this article, we explore the latter hypothesis: how often a number field $K$ has absolutely abelian Hilbert class field? For a number field $K$ to have absolutely abelian Hilbert class field, we obtain several criteria in terms of class number of $K$, Pólya group of $K$, and genus number of $K$. We also show that for such number fields the $\ell-$torsion conjecture is true. Along with these, the article also reports some results on a theme to study class groups, developed by the authors, where primes of higher degree are used to study class groups.

💡 Research Summary

The paper investigates number fields K whose Hilbert class field H(K) is “absolutely abelian”, i.e. the extension H(K)/ℚ is abelian. This hypothesis has appeared in recent work on Euclidean ideal classes and on the ℓ‑torsion conjecture, but its frequency and arithmetic consequences have not been systematically studied.

The authors first give several equivalent criteria for H(K) to be absolutely abelian. Using the exact sequence
1 → Gal(H(K)/K) ≅ Cℓ(K) → Gal(H(K)/ℚ) → Gal(K/ℚ) → 1,
they observe that the sequence splits when K/ℚ is cyclic, and that H(K) is absolutely abelian precisely when the semi‑direct product becomes a direct product.

A central result (Theorem 2.3) shows that if K is an abelian extension whose conductor involves only primes from a fixed finite set S, then the class number h_K must divide a certain integer t depending only on S, and in particular h_K ≤ t. Thus the absolute‑abelian condition forces the class number to be bounded in terms of the ramified primes. Conversely, Theorem 2.5 gives a negative criterion: if K contains a subfield F with