Unit signatures and common index divisors in S₅-quintic fields

Abstract

We conjecture that, after fixing the signature and the squarefree or nonsquarefree discriminant regime, the limiting unit-signature distribution of discriminant-ordered $S_5$-quintic fields is independent of the exact set of common index divisors. The conjecture separates an archimedean $2$-primary invariant from finite local splitting conditions at $2$ and $3$.

Families and signature rank

Let $K/\mathbb Q$ be an $S_5$-quintic field of signature $(r_1,r_2)$, and define

$$\mathop{\mathrm{sgnrk}}(K)= \dim_{\mathbb F_2}\mathop{\mathrm{im}}\bigl(\operatorname{sgn}:\mathcal O_K^\times\longrightarrow\{\pm1\}^{r_1}\bigr).$$

For $r_2\in\{0,1\}$, put $r_1=5-2r_2$. The case $r_2=2$ has only one real place and hence a deterministic signature rank, so it is omitted.

Fix $\eta\in\{\mathrm{sf},\mathrm{nsf}\}$ and $T\subseteq\{2,3\}$. Let $\mathcal F_{r_2,\eta,T}(X)$ be the $\mathbb Q$-isomorphism classes of $S_5$-quintic fields with

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

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

Conjecture 1. For every $r_2\in\{0,1\}$ and $\eta\in\{\mathrm{sf},\mathrm{nsf}\}$, there is a probability measure $\nu_{r_2,\eta}$ on $\{1,\ldots,r_1\}$ such that, for every $T$ for which the stratum is infinite and every $s$,

$$\lim_{X\to\infty} \Pr_{K\in\mathcal F_{r_2,\eta,T}(X)}\bigl(\mathop{\mathrm{sgnrk}}(K)=s\bigr) =\nu_{r_2,\eta}(s).$$

In particular, the limiting distribution is independent of $T$.

The measure is allowed to depend on squarefreeness. Thus this conjecture is compatible with a separate ramification bias while asserting independence from the exact common-index obstruction.

Numerical evidence

The reliable windows contain $162022$ totally real fields and $225089$ fields of signature $(3,1)$. Table 1 compares the full-signature-rank frequency in the upper halves of the two largest common-index strata.

signature	discriminant	$T=\varnothing$	$T=\{2\}$	difference
$(5,0)$	nsf	$41.253\%$	$42.287\%$	$1.034$ pp
$(5,0)$	sf	$36.092\%$	$38.182\%$	$2.090$ pp
$(3,1)$	nsf	$75.457\%$	$76.741\%$	$1.284$ pp
$(3,1)$	sf	$70.963\%$	$71.395\%$	$0.432$ pp

Upper-shell full-signature-rank frequencies.

The four-shell trajectories for $T=\varnothing$ and $T=\{2\}$ show broadly parallel height drift. The $T=\{3\}$ samples contain only a few hundred fields in the upper shells and fluctuate by several percentage points in both directions. They do not yet distinguish a genuine limiting shift from sampling noise. The $T=\{2,3\}$ stratum is absent from the reliable finite windows.

Heuristic motivation

The unit-signature rank is an archimedean component of the $2$-Selmer signature map. By contrast, the condition $I(K)=T$ is determined by the splitting behavior at the finite primes $2$ and $3$. Once the broader ramification regime is fixed by squarefreeness or nonsquarefreeness, a product law between these archimedean and finite local data is a natural first model.

The fieldwise identity

$$\frac{h_K^+}{h_K}=2^{r_1-\mathop{\mathrm{sgnrk}}(K)}$$

links signature deficiency to the difference between narrow and ordinary class groups. It constrains the possible values of $\mathop{\mathrm{sgnrk}}(K)$ but does not force a dependence on the common-index set.

Relation with known results

Dummit and Voight conjecture the complete unconditioned unit-signature distribution for odd-degree $S_n$-fields of fixed signature. Their model is based on the maximal totally isotropic image of the $2$-Selmer signature map. Bhargava’s geometric sieve and local counting theorems give linear asymptotics for the underlying $S_5$-field strata, while the common-index criterion makes $I(K)=T$ a finite local condition at $2$ and $3$. None of these results counts fields marked by unit-signature rank.

The conjecture therefore has two layers: existence of each conditioned signature distribution, and equality of those distributions across $T$. Even the first layer is open for the unrestricted discriminant-ordered $S_5$ family.

Main obstacles

Writing $N_{r_2,\eta,T,s}(X)$ for the number of fields in the stratum with signature rank $s$, a proof requires

$$N_{r_2,\eta,T,s}(X) =c_{r_2,\eta,T}\,\nu_{r_2,\eta}(s)X+o(X)$$

with the same $\nu_{r_2,\eta}$ for every $T$. The unmarked denominator asymptotic is accessible through acceptable local specifications. The marked numerator requires conditional equidistribution of the global unit-generated subspace inside the $2$-Selmer signature space after fixing squarefreeness and the completions at $2$ and $3$. No current counting theorem provides this joint distribution.

References

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

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

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

• J. V. Armitage and A. Fröhlich, Classnumbers and unit signatures, Mathematika 14 (1967), 94–98.

• 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.