On abelian $2$-ramification torsion modules of quadratic fields

Abstract

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, 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 $\#{\mathcal{T}}_2(l)\equiv 2\#{\mathcal{T}}_2(2l)\equiv h_2(-2l)\mathrm{mod}16$ if $l\equiv 7\mathrm{mod}8$ 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 ${\mathcal{T}}_2(\pm 2l)$ 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-Washington's speculation on the distributions of the zeros of $2$-adic $L$-functions and to the distributions of the fundamental units.

23 pages; simplified some proofs and corrected some errors; to appear in SCIENCE CHINA Mathematics