Unit-signature ranks in equal-discriminant cubic fields

Abstract

For ordered pairs of distinct totally real $S_3$-cubic fields with the same discriminant, we conjecture that the two unit-signature ranks are asymptotically independent. The conjecture separates the $2$-primary archimedean invariant from the $3$-primary ring-class mechanism producing common-discriminant multiplets.

Statement of the conjecture

For a positive discriminant $D$, let $\mathcal M_D$ be the set of $\mathbb Q$-isomorphism classes of totally real $S_3$-cubic fields of discriminant $D$, and put

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

For a totally real cubic field $K$, define

$$s(K)=\dim_{\mathbb F_2}\mathop{\mathrm{im}}\bigl(\operatorname{sgn}:\mathcal O_K^\times \longrightarrow\{\pm1\}^3\bigr)\in\{1,2,3\}.$$

For $s,t\in\{1,2,3\}$, let

$$\begin{aligned} u_s(X)&= \Pr_{(K,K')\in\overrightarrow{\mathcal P}(X)}\bigl[s(K)=s\bigr],\\ b_{s,t}(X)&= \Pr_{(K,K')\in\overrightarrow{\mathcal P}(X)} \bigl[s(K)=s,\ s(K')=t\bigr]. \end{aligned}$$

Conjecture 1. Every marginal $u_s(X)$ has positive lower limit, and for all $s,t\in\{1,2,3\}$,

$$\lim_{X\to\infty} \frac{b_{s,t}(X)}{u_s(X)u_t(X)}=1.$$

Numerical evidence

The common-discriminant sample exhibits no large persistent correlation after accounting for the one-field marginal distribution. The most stable cells are those involving signature ranks $2$ and $3$; the rank-$1$ cells are much rarer and therefore converge more slowly. Across cumulative cutoffs and in the wider stress sample, the observed joint-frequency matrix was close to the outer product of its marginals, with the largest visible discrepancies concentrated in the sparse rank-$1$ rows and columns.

The common-discriminant weighting is important. A discriminant supporting $m_D$ fields contributes $m_D(m_D-1)$ ordered pairs, so large multiplets receive substantial weight. Numerical agreement therefore tests more than independence in the ordinary one-field family.

Heuristic motivation

The multiplicity of totally real cubic fields with one discriminant is controlled by $3$-primary class or ring-class data in the common quadratic resolvent. By contrast, the unit-signature rank is an archimedean invariant tied to the $2$-Selmer signature map. This separation of primes suggests that conditioning on a large $3$-primary multiplet should not change the limiting $2$-primary signature distribution.

The conjecture is nevertheless nontrivial. Siblings share the complete discriminant and the same quadratic resolvent, and the pair measure is strongly biased toward resolvents with unusually large $3$-torsion. It is therefore necessary to control both arithmetic correlations within a multiplet and heterogeneity between different discriminants.

Relation with known heuristics

Dummit–Voight heuristics predict the one-field distribution of unit-signature ranks in totally real $S_3$-cubic fields. Common-discriminant multiplicity formulas describe the $3$-primary mechanism producing the sibling family. Neither framework gives a joint law for two distinct cubic subfields of the same ring-class construction. Existing average results for $2$-torsion under local or shape restrictions do not cover the size-biased pair measure appearing here.

Open difficulties

Write

$$m_D=\#\mathcal M_D, \qquad k_{D,s}=\#\{K\in\mathcal M_D:s(K)=s\}.$$

Then the conjecture asks for asymptotics of

$$\sum_{D\leq X}k_{D,s}k_{D,t} \quad\text{and}\quad \sum_{D\leq X}k_{D,s}(m_D-1)$$

relative to

$$\sum_{D\leq X}m_D(m_D-1).$$

The diagonal case $s=t$ requires the appropriate factorial correction because $K\neq K'$. A proof would need uniform control of the signature distribution within variable ring-class multiplets and of the tails of the $m_D(m_D-1)$ weighting. No present theorem supplies such a decorated multiplet count.

References

• D. S. Dummit and J. Voight, The $2$-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units, Proc. Lond. Math. Soc. 117 (2018), 682–726.

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

• M. Bhargava and I. Varma, The mean number of $3$-torsion elements in the class groups and ideal groups of quadratic orders, Proc. Lond. Math. Soc. 112 (2016), 235–266.