Gerth generalised Cohen-Lenstra heuristics to the prime $p=2$. He conjectured that for any positive integer $m$, the limit $$ \lim_{ x \to \infty} \frac{ \sum_{ 0 < D \le X, \atop{ \text{ squarefree} }} |{ \rm Cl}^2_{ \Q(\sqrt{ D})}/{ \rm Cl}^4_{ \Q(\sqrt{ D})}|^m}{ \sum_{ 0 < D \le X, \atop{ \text{ squarefree} }} 1} $$ exists and proposed a value for the limit. Gerth's conjecture was proved by Fouvry and Kluners in 2007. In this paper, we generalize their result by obtaining lower bounds for the average value of $|{ \rm Cl}^2_{ \L}/{ \rm Cl}^4_{ \L}|^m$, where $\L$ varies over an infinite family of quadratic extensions of certain Galois number fields. As a special case of our theorem, we obtain lower bounds for the average value when the base field is any Galois number field with class number $1$ in which $2\Z$ splits.
Categories: math.NT