Cohen and Lenstra have given a heuristic which, for a fixed odd prime p, leads to many interesting predictions about the distribution of p-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field K, the Galois group of the p-class tower of K, i.e.
The p-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group G = Gal (k nr,2 /k) for k = Q( √ d) with d = −445, −1015, −1595, −2379. In each case we are able to narrow the identity of G down to one of a finite number of explicitly given finite groups. From this follow several results regarding the corresponding 2-class tower.
Let p be an odd prime. For a number field K, we let K ∞ be the maximal unramified pro-p extension of K; we call the group Gal(K ∞ /K) the p-class tower group of K. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite p-group occurs as the p-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field K as base. As before, the action of Gal(K/Q) on the p-class tower group of K plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In an appendix, we show how the probabilities introduced for finite p-groups can be extended in a consistent way to the infinite pro-p groups which can arise in both the real and imaginary quadratic settings.K is isomorphic to the p-class group of K by class field theory. We will call G K the p-class tower group of K, since that is how it first arose in the 1930s in the work of Artin, Hasse, Furtwangler and others. In [3], we treated the case of imaginary quadratic fields. The content of [3] included a) an identification of the "right" collection of groups (Schur σ-groups), b) an investigation of an associated measure giving the frequency of groups within that collection, and c) a numerical study of p-class tower groups of imaginary quadratic fields to test the conjecture we developed using a) and b). In this work, we treat real quadratic fields in the same manner. As is to be expected, the presence of units of infinite order in the base field has a marked influence on the structure of G K , and this makes some aspects of the current work slightly more complicated and more interesting than in [3].
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