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

Abstract

For a number field F and a prime number p, the Zp-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F , denoted as Tp(F ), is an important subject in abelian p-ramification theory. In this paper we study the groupFirstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk 4 (T 2 (−m)) = rk 2 (T 2 (−m))−rank(R) where R is a certain explicitly described Rédei matrix over F 2 . Furthermore, applying this formula and exploring the connection of T 2 (−m) to the ideal class group of Q( √ −m) and the tame kernel of Q( √ m), we obtain the 4-rank density of T 2 -groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain results about the 2-divisibility of orders of T 2 (±l) and T 2 (±2l), both of which are cyclic 2-groups. In particular we find that #T 2 (l)We then obtain density results for T 2 (±l) and T 2 (±2l) when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about Tp(F ) when F varies over real or imaginary quadratic fields for any prime p, and about T 2 (±l) and T 2 (±2l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the 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.