Explicit evaluation of the local mass factors $\alpha_p$ for orders in quintic fields

Bhargava's determination of the density of discriminants of quintic rings and fields (Annals of Mathematics, 2010) shows that the number of orders in quintic number fields with bounded discriminant grows linearly, with leading constant $\alpha/240$, $\alpha/24$, or $\alpha/16$ according to the number of real embeddings, where $\alpha$ is given by an Euler product $\alpha=\prod_p \alpha_p$ of local masses and $\alpha_p$ is an automorphism- and discriminant-weighted sum over all isomorphism classes of $\mathbb{Z}_p$-algebras of rank 5 with nonzero discriminant ($\mathbb{Z}_p$ denoting the ring of $p$-adic integers). For the corresponding masses of étale $\mathbb{Q}_p$-algebras the paper proves the closed form $\beta_p = 1+p^{-2}-p^{-4}-p^{-5}$, but it explicitly poses the closed-form evaluation of $\alpha_p$ as an open combinatorial problem, noting that a solution would make the constant in its Theorem 2 explicit.

Background

A number field is a finite extension $K$ of the rationals, and its absolute discriminant $\mathrm{Disc}(K)$ is the standard integer invariant measuring how the ring of integers $\mathbb{Z}_K$ ramifies over $\mathbb{Z}$. An order in $K$ is a subring of $\mathbb{Z}_K$ containing 1 and of finite index; equivalently, a ring with unit that is free of rank $n=[K:\mathbb{Q}]$ as a $\mathbb{Z}$-module. Orders are the natural "not necessarily maximal" coefficient rings, $\mathbb{Z}_K$ itself being the unique maximal order. A central problem of arithmetic statistics is to count number fields and orders by discriminant. A long-standing folk conjecture predicts that the number of degree-$n$ number fields with absolute discriminant at most $X$ is asymptotic to a constant times $X$; after the classical cubic case of Davenport and Heilbronn, Bhargava established the quartic case (2005) and then the quintic case (2010).

Bhargava's method for degree 5 rests on his parametrization of quintic rings (the paper Higher composition laws IV): isomorphism classes of quintic rings, each paired with a sextic resolvent ring (an auxiliary rank-6 ring carrying resolvent data), correspond bijectively to integer orbits of the group $GL_4(\mathbb{Z})\times SL_5(\mathbb{Z})$ acting on quadruples of $5\times 5$ alternating (skew-symmetric) matrices. One then counts orbits of bounded discriminant by the geometry of numbers — averaging lattice-point counts over fundamental domains, with a careful dissection of the unbounded cuspidal regions — and finally sieves for maximal orders to pass from orders to fields.

A striking feature of the resulting theorems is that all leading constants are Euler products of local masses. For a prime $p$, define the étale mass factor

$$\beta_p=\frac{p-1}{p}\sum \frac{1}{|\mathrm{Aut}(K_p)|\,\mathrm{Disc}_p(K_p)},$$

where the sum runs over isomorphism classes of étale $\mathbb{Q}_p$-algebras $K_p$ of degree 5 (finite products of extensions of the $p$-adic field $\mathbb{Q}_p$, i.e. the nondegenerate case with no nilpotent elements), $\mathrm{Disc}_p$ denotes the discriminant expressed as a power of $p$, and each algebra is weighted by the reciprocal of the order of its automorphism group — the standard "mass" weighting in the tradition of mass formulae for extensions of local fields. Bhargava evaluates this in closed form: $\beta_p = 1+p^{-2}-p^{-4}-p^{-5}$. The archimedean factor is $\beta_\infty = 13/120$, and the density of quintic fields is $\beta_\infty\prod_p \beta_p$, with the familiar signature factors $1/240$, $1/24$, and $1/16$ for $5$, $3$, and $1$ real embeddings respectively.

The same paper (Theorem 2) counts all orders in quintic fields, not just maximal ones, again finding a linear asymptotic, with constant $\alpha$ given per signature by the same factors $1/240$, $1/24$, $1/16$, and shows that

$$\alpha=\prod_p \alpha_p,$$

where $\alpha_p$ is the analogous mass

$$\alpha_p=\frac{p-1}{p}\sum \frac{1}{|\mathrm{Aut}_{\mathbb{Z}_p}(R_p)|\,\mathrm{Disc}_p(R_p)},$$

now summed over all isomorphism classes of $\mathbb{Z}_p$-algebras $R_p$ of rank 5 with nonzero discriminant, where $\mathbb{Z}_p$ is the ring of $p$-adic integers. Unlike the étale sum defining $\beta_p$, this sum ranges over all nonmaximal local orders, of arbitrarily deep index and ramification; it converges, but no closed form is known. The obstruction is structural: a nonmaximal order reduces modulo $p$ to a non-étale, possibly non-reduced, algebra over the finite field $\mathbb{F}_p$, and the classification of rank-$n$ algebras is already subtle — the moduli space of rank-$n$ commutative algebras is smooth only for $n\le 3$ (Poonen) — while the automorphism groups of such orders are irregular, so the uniform mass-formula arguments that produce $\beta_p$ do not extend. The paper explicitly poses the closed-form evaluation of $\alpha_p$ as "an interesting combinatorial problem", and remarks that its solution would make the constant $\alpha$ explicit. Resolving it would give the first evaluation of a local mass for nonmaximal orders in degree 5.

Problem Statement

Evaluate, in closed form as an explicit function of the prime $p$, the local mass factor

$$\alpha_p=\frac{p-1}{p}\sum \frac{1}{|\mathrm{Aut}_{\mathbb{Z}_p}(R_p)|\cdot\mathrm{Disc}_p(R_p)},$$

where the sum runs over all isomorphism classes of $\mathbb{Z}_p$-algebras $R_p$ of rank 5 with nonzero discriminant ($\mathbb{Z}_p$ being the ring of $p$-adic integers), $|\mathrm{Aut}_{\mathbb{Z}_p}(R_p)|$ denotes the number of $\mathbb{Z}_p$-algebra automorphisms of $R_p$, and $\mathrm{Disc}_p(R_p)$ denotes the discriminant of $R_p$ expressed as a power of $p$; the sum is known to converge. Here "closed form" means an explicit expression in $p$ directly analogous to the proven étale-case evaluation $\beta_p = 1+p^{-2}-p^{-4}-p^{-5}$ — for example a rational function of $p^{-1}$. A solution consists of an explicit expression $E(p)$ together with a proof that $E(p)$ equals the defining sum for all primes outside a finite, explicitly listed exceptional set, the exceptional primes (if any) being given separate explicit values; no restriction is imposed on which primes may be exceptional, though exceptional behavior, if it occurs at all, is expected only at small primes. As a direct consequence, the constant $\alpha=\prod_p\alpha_p$ governing the asymptotic count of orders in quintic fields (Theorem 2 of Bhargava's The density of discriminants of quintic rings and fields) becomes explicitly computable.

References

1. M. Bhargava, The density of discriminants of quintic rings and fields, Annals of Mathematics 172 (2010), no. 3, 1559-1591. DOI: 10.4007/annals.2010.172.1559; also arXiv:1005.5578. URL: https://annals.math.princeton.edu/2010/172-3/p02
2. M. Bhargava, Higher composition laws IV: The parametrization of quintic rings, Annals of Mathematics 167 (2008), no. 1, 53-94. DOI: 10.4007/annals.2008.167.53. URL: https://annals.math.princeton.edu/2008/167-1/p02
3. M. Bhargava, Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants, International Mathematics Research Notices 2007 (2007), Article ID rnm052. DOI: 10.1093/imrn/rnm052. URL: https://doi.org/10.1093/imrn/rnm052
4. H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proceedings of the Royal Society of London. Series A 322 (1971), 405-420. DOI: 10.1098/rspa.1971.0075. URL: https://doi.org/10.1098/rspa.1971.0075
5. N. Kaplan, J. Marcinek, and R. Takloo-Bighash, Distribution of orders in number fields, Research in the Mathematical Sciences 2 (2015), no. 1, Article 6. DOI: 10.1186/s40687-015-0027-8; also arXiv:1408.1374. URL: https://doi.org/10.1186/s40687-015-0027-8
6. M. Bhargava, A. Shankar, and J. Tsimerman, On the Davenport-Heilbronn theorems and second order terms, Inventiones mathematicae 193 (2013), no. 2, 439-499. DOI: 10.1007/s00222-012-0433-0; also arXiv:1005.0672. URL: https://doi.org/10.1007/s00222-012-0433-0
7. T. Taniguchi and F. Thorne, Secondary terms in counting functions for cubic fields, Duke Mathematical Journal 162 (2013), no. 13. DOI: 10.1215/00127094-2371752; also arXiv:1102.2914. URL: https://arxiv.org/abs/1102.2914
8. E. M. O'Dorney, Quintic rings over Dedekind domains and their sextic resolvents, Research in Number Theory 8 (2022), no. 3, Article 46. DOI: 10.1007/s40993-022-00343-8; preprint arXiv:1511.03162 under the title Quintic algebras over Dedekind domains and their sextic resolvents. URL: https://doi.org/10.1007/s40993-022-00343-8
9. M. M. Wood and T. Yasuda, Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions, International Mathematics Research Notices 2015 (2015), Article rnv074. DOI: 10.1093/imrn/rnv074; also arXiv:1309.2879. URL: https://doi.org/10.1093/imrn/rnv074
10. T. Yasuda, Motivic versions of mass formulas by Krasner, Serre and Bhargava, Forum of Mathematics, Sigma 13 (2025), e148. DOI: 10.1017/fms.2025.10099; also arXiv:2408.13481. URL: https://arxiv.org/abs/2408.13481
11. M. Bhargava, A. Shankar, and X. Wang, Squarefree values of polynomial discriminants I, Inventiones mathematicae 228 (2022), no. 3, 1037-1073. DOI: 10.1007/s00222-022-01098-w; also arXiv:1611.09806. URL: https://arxiv.org/abs/1611.09806
12. M. Bhargava, A. Shankar, and X. Wang, Squarefree values of polynomial discriminants II, Forum of Mathematics, Pi 13 (2025), e17. DOI: 10.1017/fmp.2025.9; also arXiv:2207.05592. URL: https://arxiv.org/abs/2207.05592
13. M. Bhargava and A. Shnidman, On the number of cubic orders of bounded discriminant having automorphism group $C_3$, and related problems, Algebra & Number Theory 8 (2014), no. 1, 53-88. DOI: 10.2140/ant.2014.8.53; also arXiv:1206.4746. URL: https://doi.org/10.2140/ant.2014.8.53
14. S. Atanasov, N. Kaplan, B. Krakoff, and J. Menzel, Counting finite index subrings of $\mathbb{Z}^n$, Acta Arithmetica 197 (2021), no. 3, 221-246. DOI: 10.4064/aa180201-29-7; also arXiv:1609.06433. URL: https://doi.org/10.4064/aa180201-29-7
15. B. Poonen, The moduli space of commutative algebras of finite rank, Journal of the European Mathematical Society 10 (2008), no. 3, 817-836. DOI: 10.4171/jems/131; also arXiv:math/0608491. URL: https://doi.org/10.4171/jems/131
16. G. Malle, Computational data on $\mathfrak{S}_n$-extensions of $\mathbb{Q}$, arXiv:2403.09146. URL: https://arxiv.org/abs/2403.09146