Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields

Abstract

Fix a squarefree integer $d_0>1$, and let $d$ range over the positive squarefree integers coprime to $2d_0$. Although $\mathbb{Q}(\sqrt{-d})$ and $\mathbb{Q}(\sqrt{-d_{0}d})$ share all variable ramified primes, we prove that their class-group $4$-ranks are asymptotically independent. Over the subfamily $d\le X$, their joint distribution converges in total variation to the product of two copies of the Cohen--Lenstra--Gerth distribution, with error bounded by a negative power of $\log \log X$. We further conjecture that the corrected $2$-primary groups $2{\mathrm{Cl}}_{\mathbb{Q}(\sqrt{-d})}[2^{\infty }]$ and $2{\mathrm{Cl}}_{\mathbb{Q}(\sqrt{-d_{0}d})}[2^{\infty }]$ are asymptotically independent, each with the Cohen--Lenstra distribution. Suppose in addition that the class number of $\mathbb{Q}(\sqrt{d_0})$ is odd. For a density-one subset of this family, we prove that extension of ideals to $K(d)=\mathbb{Q}(\sqrt{d_0},\sqrt{-d})$ induces $4{\mathrm{Cl}}_{K(d)}[2^{\infty }]\cong 2{\mathrm{Cl}}_{\mathbb{Q}(\sqrt{-d})}[2^{\infty }]\oplus 2{\mathrm{Cl}}_{\mathbb{Q}(\sqrt{-d_{0}d})}[2^{\infty }]$. Together with this decomposition, the group-valued conjecture predicts that $4{\mathrm{Cl}}_{K(d)}[2^{\infty }]$ is distributed as the direct sum of two independent Cohen--Lenstra $2$-groups, giving a corrected Cohen--Lenstra--Martinet distribution for the biquadratic family. Unconditionally, the $8$-rank of ${\mathrm{Cl}}_{K(d)}$ has limiting distribution given by the convolution of two copies of the Cohen--Lenstra--Gerth distribution. The proof combines Smith's box method with quantitative truncated Gaussian-binomial moment inversion for diagonally coupled, fixed-width bordered Rédei matrices.

52 pages