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

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 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, developed by the authors, where primes of higher degree are used to study class groups.