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arXiv · 2510.10725

Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture

Mahesh Kumar Ram·Prem Prakash Pandey·Nimish Kumar Mahapatra
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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 $H(K)$ of $K$ 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 $K$, P\'olya 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, where primes of higher degree are used to study class groups.

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Mahesh Kumar Ram, Prem Prakash Pandey, Nimish Kumar Mahapatra. 2025-10-12. Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture. https://arxiv.org/abs/2510.10725

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