For a number field $F$ and a prime number $p$, the $mathbb{Z}_p$-torsion module of the Galois group of the maximal abelian pro-$p$ extension of $F$ unramified outside $p$ over $F$, denoted as $mathcal{T}_p(F)$, is an important subject in abelian $p$-ramification theory. In this paper we study the group $mathcal{T}_2(F)=mathcal{T}_2(m)$ of the quadratic field $F=mathbb{Q}(sqrt{ m})$. Firstly, assuming $m>0$, we prove an explicit $4$-rank formula for $mathcal{T}_2(-m)$. Furthermore, applying this formula and exploring the connection of $mathcal{T}_2(-m)$ to the ideal class group of $mathbb{Q}(sqrt{-m})$ and the tame kernel of $mathbb{Q}(sqrt{m})$, we obtain the $4$-rank density of $mathcal{T}_2$-groups of imaginary quadratic fields. Secondly, for $l$ an odd prime, we obtain results about the $2$-divisibility of orders of $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2l)$. In particular we find that $#mathcal{T}_2(l)equiv 2# mathcal{T}_2(2l)equiv h_2(-2l)bmod{16}$ if $lequiv 7bmod{8}$ where $h_2(-2l)$ is the $2$-class number of $mathbb{Q}(sqrt{-2l})$. We then obtain density results for $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2l)$. Finally, based on our density results and numerical data, we propose distribution conjectures about $mathcal{T}_p(F)$ when $F$ varies over real or imaginary quadratic fields for any prime $p$, and about $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2 l)$ when $l$ varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the $mathcal{T}_2(l)$ case is closely connected to Shanks-Sime-Washingtons speculation on the distributions of the zeros of $2$-adic $L$-functions and to the distributions of the fundamental units.
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