Two-torsion moments in locally conditioned S₅-quintic fields

Abstract

We formulate conjectural first moments for the ordinary and narrow class-group $2$-torsion of discriminant-ordered $S_5$-quintic fields. The fields are stratified simultaneously by signature, squarefreeness of the discriminant, and the set of common index divisors. The proposed constants are the usual fixed-signature Cohen–Lenstra–Martinet–Malle and Dummit–Voight moments, and the conjecture asserts that the additional local restrictions do not change them.

The conditioned families

Let $K/\mathbb Q$ be a quintic field whose Galois closure has group $S_5$. Write

$$(r_1,r_2)=(5-2r_2,r_2),\qquad r_2\in\{0,1,2\},$$

and let

$$u=r_1+r_2-1=4-r_2$$

be the unit rank. The common index of $K$ is

$$i(K)=\gcd_{\mathbb Q(\alpha)=K}[\mathcal O_K:\mathbb Z[\alpha]],$$

and we write

$$I(K)=\{p:p\mid i(K)\}.$$

For quintic fields one has $I(K)\subseteq\{2,3\}$.

Let $\eta\in\{\mathrm{sf},\mathrm{nsf}\}$ and $T\subseteq\{2,3\}$. We denote by $\mathcal F_{r_2,\eta,T}(X)$ the set of $\mathbb Q$-isomorphism classes of such fields with

$$|D_K|\le X,\qquad I(K)=T,$$

and with $D_K$ squarefree precisely when $\eta=\mathrm{sf}$.

Conjecture 1. Whenever $\#\mathcal F_{r_2,\eta,T}(X)\to\infty$, one has

$$\begin{aligned} \lim_{X\to\infty}\frac{1}{\#\mathcal F_{r_2,\eta,T}(X)} \sum_{K\in\mathcal F_{r_2,\eta,T}(X)}|\mathop{\mathrm{Cl}}(K)[2]| &=1+2^{-u},\\ \lim_{X\to\infty}\frac{1}{\#\mathcal F_{r_2,\eta,T}(X)} \sum_{K\in\mathcal F_{r_2,\eta,T}(X)}|\mathop{\mathrm{Cl}}^+(K)[2]| &=1+2^{-r_2}. \end{aligned}$$

The limits are independent of $\eta$ and $T$.

The predicted values are displayed in Table 1.

| $r_2$ | signature | $\mathbb E|\mathop{\mathrm{Cl}}(K)[2]|$ | $\mathbb E|\mathop{\mathrm{Cl}}^+(K)[2]|$ | |:—:|:—:|:—:|:—:| | $0$ | $(5,0)$ | $17/16$ | $2$ | | $1$ | $(3,1)$ | $9/8$ | $3/2$ | | $2$ | $(1,2)$ | $5/4$ | $5/4$ |

The conjectural fixed-signature moments.

Numerical evidence

The available discriminant-complete windows contain $885295$ fields: $162022$ with $r_2=0$, $225089$ with $r_2=1$, and $498184$ with $r_2=2$. In the largest strata the empirical moments increase with the discriminant and move toward the predicted constants. Representative lower-shell and upper-shell values are given in Table 2.

$r_2$	$\eta$	moment	lower shell	upper shell	target
$0$	sf	$	\mathop{\mathrm{Cl}}[2]	$	$1.005175$
$0$	sf	$	\mathop{\mathrm{Cl}}^+[2]	$	$1.664683$
$1$	sf	$	\mathop{\mathrm{Cl}}[2]	$	$1.015390$
$1$	sf	$	\mathop{\mathrm{Cl}}^+[2]	$	$1.271209$
$2$	sf	$	\mathop{\mathrm{Cl}}[2]	$	$1.088132$
$2$	nsf	$	\mathop{\mathrm{Cl}}[2]	$	$1.044010$

Selected observations in the $T=\varnothing$ strata. For $r_2=2$, ordinary and narrow class groups agree.

Similar movement is visible after imposing $I(K)=\{2\}$. The strata with $I(K)=\{3\}$ or $\{2,3\}$ are much smaller, so the current data do not provide a meaningful asymptotic test there. The persistent deficit below the conjectural constants is compatible with slow convergence and with the sensitivity of an unbounded moment to rare fields of large $2$-rank.

Heuristic background

The ordinary constant $1+2^{-u}$ is the modified Cohen–Lenstra–Martinet–Malle prediction for odd-degree $S_n$-fields, and the narrow constant $1+2^{-r_2}$ is the corresponding Dummit–Voight prediction. The conditions $I(K)=T$ are finite local restrictions at $2$ and $3$. Squarefreeness of the discriminant is an infinite collection of local ramification restrictions, but it is still a natural acceptable family in the quintic counting problem. Conjecture 1 asserts that neither type of conditioning changes the global $2$-primary moment once the signature is fixed.

Class field theory gives a useful reformulation. The quantity $|\mathop{\mathrm{Cl}}(K)[2]|-1$ counts nontrivial quadratic extensions of $K$ unramified at all finite places and split at every real place, whereas $|\mathop{\mathrm{Cl}}^+(K)[2]|-1$ counts quadratic extensions unramified at all finite places without the archimedean splitting requirement. A proof therefore requires asymptotic counts of such decorated towers over locally specified $S_5$-quintic fields.

Relation with known results

Bhargava’s quintic parametrization gives linear discriminant-aspect counts and allows suitable local specifications; the geometric sieve also treats squarefree discriminants. In degree three, Bhargava and Varma prove analogous locally conditioned $2$-torsion averages. Ho, Shankar and Varma obtain the expected bounds, and conditional equalities, in binary-form-parametrized odd-degree families. These results do not count every discriminant-ordered $S_5$-quintic field once, and no theorem presently gives even the unconditioned moments in Conjecture 1.

Main obstacles

A proof would require, in every fixed stratum,

$$\begin{aligned} \sum_{K\in\mathcal F_{r_2,\eta,T}(X)}\bigl(|\mathop{\mathrm{Cl}}(K)[2]|-1\bigr) &\sim 2^{-u}\#\mathcal F_{r_2,\eta,T}(X),\\ \sum_{K\in\mathcal F_{r_2,\eta,T}(X)}\bigl(|\mathop{\mathrm{Cl}}^+(K)[2]|-1\bigr) &\sim 2^{-r_2}\#\mathcal F_{r_2,\eta,T}(X). \end{aligned}$$

The denominator is accessible through quintic-field counting, but no available parametrization counts the required unramified quadratic towers. One must also control the high-$2$-rank tail: convergence of fixed-rank frequencies alone does not imply convergence of the moment $2^{\operatorname{rk}_2}$.

References

• G. Malle, On the distribution of class groups of number fields, Experiment. Math. 19 (2010), 465–474.

• D. S. Dummit and J. Voight, with an appendix by R. Foote, 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.

• M. Bhargava, The density of discriminants of quintic rings and fields, Ann. of Math. 172 (2010), 1559–1591.

• M. Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, arXiv:1402.0031.

• M. Bhargava and I. Varma, On the mean number of $2$-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields, Duke Math. J. 164 (2015), 1911–1933.

• W. Ho, A. Shankar and I. Varma, Odd degree number fields with odd class number, Duke Math. J. 167 (2018), 995–1047.

• H. T. Engstrom, On the common index divisors of an algebraic field, Trans. Amer. Math. Soc. 32 (1930), 223–237.