On abelian 2-ramification torsion modules of quadratic fields

For a number field F and prime p, the ℤp-torsion module of Galois group maximal abelian pro-p extension unramified outside p over F, denoted by $${{\cal T}_p}(F)$$ , is an important subject in p-ramification theory. In this paper, we study T}_2}(F) = {{\cal T}_2}(m)$$ quadratic $$F \mathbb{Q}(\sqrt m )$$ . Firstly, assuming > 0, prove explicit 4-rank formula for fields that $${\rm{r}}{{\rm{k}}_4}({{\cal T}_2}( - m)) {\rm{r}}{{\rm{k}}_2}({{\cal {\rm{rank}}(R)$$ where R certain explicitly described Rédei matrix $${\mathbb{F}_2}$$ Furthermore, using formula, obtain density T}_2}$$ -groups imaginary fields. Secondly, l odd prime, results about 2-power divisibility orders \pm l)$$ 2l)$$ both which are cyclic 2-groups. particular, find $$\# T}_2}(l) \equiv 2\# T}_2}(2l) {h_2}( (mod 16) if ≡ 7 8), h2(−2l) 2-class $$\mathbb{Q}(\sqrt { 2l} We then when small. Finally, based on our numerical data, propose distribution conjectures varies real or any varies, spirit Cohen-Lenstra heuristics. Our conjecture T}_2}(l)$$ case closely connected to Shanks-Sime-Washington’s speculation distributions zeros 2-adic L-functions fundamental units.