The distribution of $H_8$-extensions of quadratic fields

Abstract

We compute all the moments of a normalization of the function which counts unramified $H_8$-extensions of quadratic fields, where $H_8$ is the quaternion group of order 8, and show that the values of this function determine a constant distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified G-extensions of quadratic fields for G in a large class of 2-groups, which we conjecture will give finite moments which determine a distribution. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified $H_8$-extensions.

References in corpus (2)

• The geometric sieve and the density of squarefree values of invariant polynomials
• Nonabelian Cohen-Lenstra Moments

Cited by in corpus (3)

• Nonabelian Cohen-Lenstra Moments
• A predicted distribution for Galois groups of maximal unramified extensions
• 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields