On abelian 2-ramification torsion modules of quadratic fields  

作　　者：Jianing Li Yi Ouyang Yue Xu 

机构地区：[1]Research Center for Mathematics and Interdisciplinary Sciences,Shandong University,Qingdao 266237,China [2]CAS Wu Wen-Tsun Key Laboratory of Mathematics,University of Science and Technology of China,Hefei230026,Chin [3]Hefei National Laboratory,Hefei 230088,China

出　　处：《Science China Mathematics》2022年第12期2459-2482,共24页中国科学：数学（英文版）

基　　金：supported by Anhui Initiative in Quantum Information Technologies(Grant No.AHY150200)。

摘　　要：For a number field F and a prime number p, the Z_(p)-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T_(p)(F), is an important subject in the abelian p-ramification theory. In this paper, we study the group T_(2)(F) = T_(2)(m) of the quadratic field F = Q(m_(1/2)). Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk4(T_(2)(-m))= rk2(T_(2)(-m))-rank(R), where R is a certain explicitly described Rédei matrix over F_(2). Furthermore, using this formula, we obtain the 4-rank density formula of T_(2)-groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain the results about the 2-power 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) ≡ 2#T_(2)(2l) ≡ h_(2)(-2l)(mod 16) if l ≡ 7(mod 8),where h_(2)(-2l) is the 2-class number of Q((-2l)_(1/2)). We then obtain the 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 T_(p)(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.

关 键 词：quadratic fields density theorems abelian 2-ramification 

分 类 号：O153.3[理学—数学]