Absolutely Abelian Hilbert Class Fields and $\ell$-torsion conjecture
Mahesh Kumar Ram, Prem Prakash Pandey, Nimish Kumar Mahapatra
TL;DR
The paper analyzes when the Hilbert class field $H(K)$ is abelian over $\mathbb{Q}$ and the consequences for the ℓ-torsion in the class group $C\ell(K)$ of $K$. Under the hypothesis that $K$ is abelian and $H(K)/\mathbb{Q}$ is abelian, it proves a sharp bound $|C\ell(K)[\ell]| \ll_{\epsilon} D_K^{\Delta+\epsilon}$ with $\Delta \le 1/n$, and even the stronger $|C\ell(K)[\ell]| \le D_K^{1/n}$, by relating the conductor, discriminant, and ramification data via conductor bounds and class-field theoretic structure. The work further connects abelianness of $H(K)/\mathbb{Q}$ to Pólya groups and genus fields, provides criteria and consequences for primes of higher residue degree generating the class group, and establishes finiteness results for imaginary abelian fields with absolutely abelian Hilbert class fields, while extending the main bound to Hilbert $\ell$-class fields in an Appendix. These results synthesize class-field theory, genus theory, and Chebotarev arguments to constrain the arithmetic of $C\ell(K)$ in special but broad families of number fields, with implications for the ℓ-torsion conjecture in new contexts.
Abstract
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 an abelian extension of $\mathbb{Q}$. In this article, we explore the latter hypothesis: how often a number field $K$ satisfies that its Hilbert class field $H(K)$ is an abelian extension of $\mathbb{Q}$? We also noticed that for such number fields, when the Hilbert class field is an abelian extension of $\mathbb{Q}$, we can get a better bound towards the $\ell-$torsion conjecture. Along with these, the article reports some results in a theme developed by the authors, where primes of higher degree are used to study class groups.