Abel p-分歧扭子群和新Cohen-Lenstra猜想

Abelian p-ramification groups and new Cohen-Lenstra heuristics

本文首先回顾数域K的Abel p-分歧理论特别是其T_(p)-群,并给出一般Cohen-Lenstra猜想的架构.对于全体实(虚)二次域及其几类子族,本文提出T_(p)(K)的分布满足新的Cohen-Lenstra猜想.后者解释了Shanks等(1999)对于2-进L函数特殊值的分布猜测,并给出基本单位迹的分布猜想.本文给出理论结果和计算数据来支持这些猜想.

In this paper, we first review the TT_(p)-groups in the abelian p-ramification theory of general number fields. We also review a general setting of the Cohen-Lenstra heuristic. Then we propose various new CohenLenstra heuristics for distributions of T_(p)-groups of quadratic fields, which in particular explains the speculation of Shanks et al.(1999) on the distributions of zeros of 2-adic L-functions and also reveals the distribution of fundamental units in certain real quadratic fields. Theoretical and numerical evidence of our conjectures is also presented.