The Distribution of H8-Extensions of Quadratic Fields

Abstract

Abstract We compute all the moments of a normalization of the function that 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 point mass distribution. As a consequence toward non-abelian Cohen–Lenstra heuristics of 2-groups, we show this implies that the probability is 0 for any fixed group to appear as the subgroup of the Galois group of the maximal unramified 2-extension fixing the genus field of the quadratic field. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified $H_{8}$-extensions. Furthermore, we propose that a similar normalization of the counting function is necessary to obtain finite moments when $H_{8}$ is replaced by any 2-group $G$.