Ideal class groups and their subgroups of real quadratic fields

1997 ◽  
Vol 40 (9) ◽  
pp. 909-916 ◽  
Author(s):  
Lawrence C. Washington ◽  
Xianke Zhang
Keyword(s):  
Quadratic Fields ◽  
Ideal Class ◽  
Class Groups ◽  
Real Quadratic Fields ◽  
Ideal Class Groups ◽  
Real Quadratic

HEURISTICS FOR -CLASS TOWERS OF REAL QUADRATIC FIELDS

2019 ◽  
pp. 1-24
Author(s):  
Nigel Boston ◽  
Michael R. Bush ◽  
Farshid Hajir
Keyword(s):  
Infinite Order ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Ground Field ◽  
Real Quadratic Fields ◽  
Ideal Class Groups ◽  
Formula Group ◽  
Real Quadratic

Let $p$ be an odd prime. For a number field $K$ , we let $K_{\infty }$ be the maximal unramified pro- $p$ extension of  $K$ ; we call the group $\text{Gal}(K_{\infty }/K)$ 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 $\text{Gal}(K/\mathbb{Q})$ 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 the 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 the 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.

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