Class-number divisibility in equal-discriminant cubic fields

Abstract

We study pairs of distinct totally real $S_3$-cubic fields with the same discriminant. For every prime $\ell\geq5$, we conjecture that divisibility of the two class numbers by $\ell$ is asymptotically independent under the natural ordered-pair weighting. The exclusion of $\ell=3$ is essential and reflects the $3$-primary class-field-theoretic mechanism producing common-discriminant multiplets.

Common-discriminant multiplets

For a positive discriminant $D$, let $\mathcal M_D$ denote the set of $\mathbb Q$-isomorphism classes of totally real $S_3$-cubic fields of discriminant $D$. Define the set of ordered sibling pairs

$$\overrightarrow{\mathcal P}(X) =\{(K,K'):K\neq K',\ K,K'\in\mathcal M_D \text{ for some }D\leq X\}.$$

For a prime $\ell\geq5$, put

$$\begin{aligned} u_\ell(X) &=\frac{\#\{(K,K')\in\overrightarrow{\mathcal P}(X):\ell\mid h_K\}} {\#\overrightarrow{\mathcal P}(X)},\\ b_\ell(X) &=\frac{\#\{(K,K')\in\overrightarrow{\mathcal P}(X): \ell\mid h_K\text{ and }\ell\mid h_{K'}\}} {\#\overrightarrow{\mathcal P}(X)}. \end{aligned}$$

Conjecture 1. For every prime $\ell\geq5$,

$$\#\overrightarrow{\mathcal P}(X)\longrightarrow\infty, \qquad \liminf_{X\to\infty}u_\ell(X)>0,$$

and

$$\lim_{X\to\infty}\frac{b_\ell(X)}{u_\ell(X)^2}=1.$$

Numerical evidence

In a primary sample of $414436$ ordered pairs, the results for the first few primes were as follows.

$\ell$	endpoint events	joint ordered events	$u_\ell$	$b_\ell/u_\ell^2$
$5$	$3921$	$34$	$0.00946105$	$0.9165$
$7$	$1335$	$6$	$0.003221$	$1.3952$
$11$	$234$	$0$	$0.000565$	—

Class-number divisibility among ordered sibling pairs.

For $\ell=5$, independence predicts about $37.10$ joint ordered observations, compared with $34$ observed. For $\ell=7$, the six ordered observations correspond to only three unordered double events, so the deviation has little statistical significance. The data for $\ell\geq11$ are too sparse for a meaningful test.

The contrast at $\ell=3$ is striking: every tested range had perfect synchronization between siblings. This confirms that the restriction $\ell\geq5$ is arithmetic rather than cosmetic.

Heuristic explanation

Sibling fields share their discriminant, quadratic resolvent and all discriminant-level ramification data. Their existence and multiplicity are controlled by $3$-primary ring-class information in the common quadratic resolvent. For primes $\ell\geq5$, which are coprime to $|S_3|$, Cohen–Martinet heuristics suggest that the $\ell$-primary class groups should behave as fresh random data for each sibling. Conjecture 1 asserts that this remains true even under the strong conditioning that the complete discriminants agree.

A proved part of the statement

The divergence of the number of sibling pairs follows from known lower bounds for real quadratic fields with large $3$-rank. If a real quadratic field of discriminant $d$ has $3$-rank $r$, then its unramified cyclic cubic extensions give

$$\frac{3^r-1}{2}$$

distinct totally real cubic fields of discriminant $d$. Quantitative results producing infinitely many quadratic fields with $r\geq4$ therefore imply

$$\#\overrightarrow{\mathcal P}(X)\gg \frac{X^{1/30}}{\log X}.$$

This proves the first assertion of Conjecture 1, but gives no control of good-prime divisibility in the cubic class numbers.

Open difficulties

Let

$$m_D=\#\mathcal M_D, \qquad k_D=\#\{K\in\mathcal M_D:\ell\mid h_K\}.$$

Then

$$\begin{aligned} N(X)&=\sum_{D\leq X}m_D(m_D-1),\\ A(X)&=\sum_{D\leq X}k_D(m_D-1),\\ B(X)&=\sum_{D\leq X}k_D(k_D-1). \end{aligned}$$

The desired conclusion is equivalent to

$$\liminf_{X\to\infty}\frac{A(X)}{N(X)}>0, \qquad \frac{B(X)N(X)}{A(X)^2}\longrightarrow1.$$

This is a second-factorial-moment problem with the strongly size-biased weight $m_D(m_D-1)$. Conditional independence at each fixed $D$ would not suffice if the marginal probability varied with $D$; uniformity in the quadratic resolvent, conductor and multiplet size is also required.

References

• D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831–847.

• D. C. Mayer, Classifying multiplets of totally real cubic fields, Int. J. Number Theory 18 (2022), 813–852.

• H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), 123–137.

• W. Wang and M. M. Wood, Moments and interpretations of the Cohen–Lenstra–Martinet heuristics, Comment. Math. Helv. 96 (2021), 339–387.