相关虚二次域对中类群4秩的渐近独立性

中文摘要 AI 辅助

固定一个无平方因子整数$d_0>1$，令$d$取遍与$2d_0$互素的正无平方因子整数。尽管$\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(-d))$和$\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(-d_0d))$共享所有可变分歧素数，我们证明它们的类群4秩是渐近独立的。在$d\boldsymbol{\text{le}} X$的子族上，它们的联合分布在全变差意义下收敛于两份Cohen--Lenstra--Gerth分布的乘积，误差被$\boldsymbol{\text{loglog}} X$的负幂次界定。我们进一步猜想，修正后的2主群$2\textbf{Cl}_{\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(-d))}[2^\boldsymbol{\text{infty}}]$和$2\textbf{Cl}_{\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(-d_0d))}[2^\boldsymbol{\text{infty}}]$是渐近独立的，各自服从Cohen--Lenstra分布。 此外假设$\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(d_0))$的类数为奇数。对于该族的一个密度为1的子集，我们证明理想扩张到$K(d)=\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(d_0),\boldsymbol{\text{sqrt}}(-d))$诱导出$4\textbf{Cl}_{K(d)}[2^\boldsymbol{\text{infty}}]\boldsymbol{\text{cong}} 2\textbf{Cl}_{\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(-d))}[2^\boldsymbol{\text{infty}}]\boldsymbol{\text{oplus}} 2\textbf{Cl}_{\boldsymbol{\text{Q}}(\boldsymbol{\text{sqrt}}(-d_0d))}[2^\boldsymbol{\text{infty}}]$。结合该分解，群值猜想预测$4\textbf{Cl}_{K(d)}[2^\boldsymbol{\text{infty}}]$的分布为两个独立的Cohen--Lenstra 2群的直和，给出了双二次族的修正Cohen--Lenstra--Martinet分布。无条件地，$\textbf{Cl}_{K(d)}$的8秩的极限分布由两份Cohen--Lenstra--Gerth分布的卷积给出。证明结合了Smith的盒方法与对角耦合、固定宽度带边Rédei矩阵的定量截断高斯-二项式矩反演。

英文摘要

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 $\mathbb{Q}(\sqrt{-d_0d})$ 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 $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]$ and $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$ 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 $4\operatorname{Cl}_{K(d)}[2^\infty]\cong 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]\oplus 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$. Together with this decomposition, the group-valued conjecture predicts that $4\operatorname{Cl}_{K(d)}[2^\infty]$ 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 $\operatorname{Cl}_{K(d)}$ 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.

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52 pages