Heuristics for 2-class Towers of Cyclic Cubic Fields

Abstract

We consider the Galois group G 2 ( K ) of the maximal unramified 2-extension of K where K / Q is cyclic of degree 3. We also consider the group G 2 + ( K ) where ramification is allowed at infinity. In the spirit of the Cohen–Lenstra heuristics, we identify certain types of groups as the natural spaces where G 2 ( K ) and G 2 + ( K ) live when the 2-class group of K is 2-generated. While we do not have a theoretical scheme for assigning probabilities, we present data and make some observations and conjectures about the distribution of such groups.