Schur-corrected counting of number fields: prove the predicted leading constant, secondary terms, and distribution laws for $G$-extensions beyond the function-field theorems

This problem concerns the number-field side of the program of the Venkatesh-Ellenberg ICM 2010 survey on the statistics of number fields and function fields. The function-field half of the program — the vanishing condition on the stable homology of Hurwitz spaces for admissible pairs $(G,c)$, Malle-type asymptotics over $\mathbb F_q(t)$, and the resulting Cohen-Lenstra-Martinet heuristics — has now been largely resolved (Ellenberg-Venkatesh-Westerland; Landesman-Levy; Santens). The surviving open core is the original number-field question: establish, for a fixed finite group $G$ and an admissible conjugacy class $c\subset G$, that the count of $G$-extensions of a number field $K$ with all ramification tame of type $c$, normalized by the associated lattice-point count $|S_X|$, satisfies $|F_X|/|S_X|\to h(G,c,K)|G|^{-|S_\infty|}$ as $X\to\infty$, with the leading constant $h(G,c,K)$ predicted to equal $\#H_2(G,c;\mathbb Z)$ when $K$ contains sufficiently many roots of unity; together with the secondary lower-order terms of the counting functions and the equidistribution of fundamental classes. No theoretical proof of these number-field predictions is known beyond the low-degree results of Davenport-Heilbronn and Bhargava, in which the Schur correction is trivial; the predictions are now formalized as conjectures of Loughran-Santens and Sawin-Wood.

Background

A Hurwitz space is the moduli space parametrizing branched covers of a curve with fixed branch points and monodromy specified by a conjugation-invariant multiset $c$ of elements of a finite group $G$: the fundamental group of the punctured base surjects onto the free group on the branch points, and each branch point is labeled by an element of $c$. Over function fields, counting $G$-extensions of $\mathbb F_q(t)$ with prescribed ramification is equivalent to an $\mathbb F_q$-point count on a Hurwitz scheme, and the Grothendieck-Lefschetz trace formula relates that count to the stable cohomology of the Hurwitz space. The Cohen-Lenstra heuristics predict the distribution of class groups of number fields as the base field varies, and their non-abelian refinement predicts the frequency of a fixed group $G$ as a Galois group; Malle's conjecture predicts the order of magnitude of the count of $G$-extensions of discriminant below $X$, and Bhargava's mass heuristics give conjectural leading constants assembled from local masses. In the ICM 2010 survey that originated this problem, Venkatesh and Ellenberg proposed that for a general group $G$ these heuristics must be corrected by the Schur multiplier: the second homology $H_2(G,\mathbb Z)$ of the classifying space $BG$ (equivalently, the kernel of a universal central extension of $G$) enters through its quotient $H_2(G,c;\mathbb Z)=H_2(G,\mathbb Z)/Q_c$, where $Q_c$ is the subgroup generated by the images of $H_2(\mathbb Z\times\mathbb Z,\mathbb Z)$ under homomorphisms $\mathbb Z\times\mathbb Z\to G$ taking $(0,1)$ into $c$. Over function fields the mechanism behind this 'Schur correction' is proven: the relevant Hurwitz scheme may have several connected components, their number is governed by $\#H_2(G,c;\mathbb Z)$ (a theorem of Conway-Parker and Fried-Völklein), and the Galois action on components can reduce the count — the origin of both the correction and its dependence on the roots of unity in the base field. Ellenberg-Venkatesh-Westerland (Annals 2016) proved homological stability for generalized dihedral Hurwitz spaces and stated the vanishing condition for $(S_n,\mathrm{transpositions})$ as their Conjecture 1.5 — the assertion that the Hurwitz space of $(S_n,\mathrm{transpositions})$-covers and the configuration space of the branch points have isomorphic rational homology in each fixed degree once the number of branch points is large; Landesman-Levy (2025) then proved integral homological stability for Hurwitz spaces of arbitrary finite racks (sets carrying a self-distributive operation modeling conjugation in a group), computed the dominant stable homology of every component, and deduced Conjecture 1.5, Malle's conjecture over $\mathbb F_q(t)$ with matching upper and lower bounds, a periodic version with leading constants, the asymptotic Picard rank conjecture for class groups of function fields, and Cohen-Lenstra-Martinet moments; Santens (2025) determined the explicit leading constant, in terms of the unramified Brauer group of the classifying stack $BG$ (a cohomological invariant of $G$, dual to the Schur multiplier in this setting) and a Tamagawa measure (a canonical adelic volume). Over number fields, in contrast, even the existence of the main term is known only in low degree (the Davenport-Heilbronn theorem counting cubic fields; Bhargava for quartics and quintics), and the survey's predictions for the leading constant, secondary terms, and higher-order distribution laws remain open; the leading-constant prediction is now formalized as Conjecture 9.1 of Loughran-Santens, with the transcendental (geometric) part of the Brauer group of $BG$ playing the role that $H_2(G,c;\mathbb Z)$ plays in the heuristic, and Sawin-Wood have proposed a refined invariant (a class in third homology, arising from the arithmetic duality theory of Artin-Verdier) governing the distribution of maximal unramified extensions. In the number-field heuristic the set $S_\infty$ of places at which the extensions must split completely — it must contain all archimedean places — plays the role of the punctures, each of its places contributing a factor $|G|^{-1}$ to the predicted mass. The survey's motivating example — eight totally real $S_5$ quintic fields of discriminant below $10^5$ versus about 600 predicted by the (provably correct) main term — illustrates how scarcity of examples and slow convergence, plausibly attributable to large secondary terms, make the number-field predictions extremely hard to test numerically.

Problem Statement

Fix a number field $K$ (the case $K=\mathbb Q$ is the primary example), a finite group $G$ that is center-free with trivial abelianization, and an admissible pair $(G,c)$: $c\subset G$ is a rational conjugacy class (for every $g\in c$ and every $n$ coprime to the order of $g$, also $g^n\in c$) that generates $G$; the standard multi-color generalization, in which $c$ is a conjugation-invariant generating multiset of rational conjugacy classes, is included. Fix a finite set $S_\infty$ of places of $K$ containing all archimedean places, and let $V$ be $S_\infty$ together with all finite places whose residue characteristic divides the order of an element of $c$. For a real number $X$, let $S_X$ be the set of all finite sets $S$ of finite places of $K$ with $S\cap V=\emptyset$ and $\prod_{v\in S}N(v)\le X$, where $N(v)$ is the cardinality of the residue field at $v$, and let $F_X$ be the set of $G$-extensions of $K$ — conjugacy classes of surjective homomorphisms $\mathrm{Gal}(\bar K/K)\to G$, i.e. Galois extensions together with an identification of their Galois group with $G$ — that are totally split at every place of $S_\infty$, are tamely ramified with every nontrivial inertia image cyclic and generated by an element of $c$ ('all ramification of type $c$'), and are ramified precisely at the places of some $S\in S_X$. For ramification of transposition type the product $\prod_{v\in S}N(v)$ is the squarefree discriminant, so this is counting by bounded discriminant; the usual bounded-discriminant count with unrestricted ramification is assembled from these prescribed-ramification counts. The hypothesis that $G$ is center-free with trivial abelianization (equivalently, that $G$ is perfect with trivial center) pins the class of groups over which the Schur-corrected predictions (1) and (3) are stated; sub-question (2) and the known-cases comparison at the end of this paragraph are stated for the wider families of $S_n$-extensions and abelian groups, which need not belong to this class. Prove or refute, for fixed $(G,c,K,S_\infty)$ as $X\to\infty$, the survey's number-field predictions: (1) (Schur-corrected leading constant) $|F_X|/|S_X|\to h(G,c,K)|G|^{-|S_\infty|}$ for a rational constant $h(G,c,K)$, and, if $K$ contains sufficiently many roots of unity ($\mu_N\subset K$ for some $N$ depending only on $(G,c)$) and $S_\infty$ is large enough, $h(G,c,K)=\#H_2(G,c;\mathbb Z)$, where $H_2(G,c;\mathbb Z)=H_2(G,\mathbb Z)/Q_c$ and $Q_c\subset H_2(G,\mathbb Z)$ is the subgroup generated by the images of $H_2(\mathbb Z\times\mathbb Z,\mathbb Z)$ under homomorphisms $\mathbb Z\times\mathbb Z\to G$ taking $(0,1)$ into $c$; in general $h(G,c,K)$ is predicted to be a rational number between $0$ and $\#H_2(G,c;\mathbb Z)$ depending on the roots of unity in $K$ (the survey's heuristics (3) and (5)). (2) (Secondary terms) Describe the lower-order terms in the counting function for $S_n$-discriminants — and for general $(G,c)$ — the main term of which is provided by the conjectures of Malle and Bhargava: for instance, prove or refute Roberts-type asymptotics of the form $aX-bX^{5/6}+O(X^{1/2+\varepsilon})$ in degrees $n\ge 4$, and give an explicit description of the unstable cohomology classes of Hurwitz spaces expected to underlie such terms (the survey's §2.6 Question and its §3.4 remark). (3) (Equidistribution of fundamental classes) In the regime where the group $\mu_n$ of roots of unity of $K$ annihilates $H_2(G,c;\mathbb Z)$ and the order of elements of $c$ is coprime to $\#H_2(G,c;\mathbb Z)$, each counted $G$-extension $\rho$ carries a fundamental class $z(\rho)\in H_2(G,c;\mathbb Z)$, obtained by lifting $\rho$ to the central cover $\tilde G_c$ (the largest central extension of $G$ to which the class $c$ lifts bijectively) and measuring the lifting defect as a product over ramified places, the construction depending on a chosen generator of $\mu_n$; prove or refute the refined equidistribution that for every $\alpha\in H_2(G,c;\mathbb Z)$ the subfamily $F_X^\alpha\subset F_X$ with $z(\rho)=\alpha$ satisfies $|F_X^\alpha|/|S_X|\to|G|^{-|S_\infty|}$ (the survey's heuristic (6), which refines (1): summed over $\alpha$ it yields $h=\#H_2(G,c;\mathbb Z)$ under the roots-of-unity hypothesis). The sub-predictions (1)-(3) are logically independent and may be settled separately, for any class of admissible pairs $(G,c)$ and base fields $K$; a rigorous disproof for a specific $(G,c,K)$ — a proven asymptotic contradicting the predicted constant, or a proven structural obstruction — resolves the problem. The main term (with Euler-product constant; the Schur correction is trivial in these cases) is known for $S_n$-extensions of $\mathbb Q$ in degrees $n\le 5$ and for abelian groups, Roberts-type secondary terms are known for totally real cubics and, in smoothed form, for quartics, and no case of the Schur-corrected prediction (1) with nontrivial $\#H_2(G,c;\mathbb Z)$ is proved over number fields. The survey's closing speculation — that the Schur multiplier enters through the structure of the absolute Galois group of $\mathbb Q$ — is explicitly not part of this problem.

References

1. Akshay Venkatesh and Jordan S. Ellenberg, 'Statistics of Number Fields and Function Fields', 2010, in: Proceedings of the International Congress of Mathematicians (Hyderabad, 2010), Vol. I, pp. 383-402, Hindustan Book Agency / World Scientific (proceedings volume published 2011), DOI: 10.1142/9789814324359_0056, https://doi.org/10.1142/9789814324359_0056 (the primary source survey of this problem)
2. Harold Davenport and Hans Heilbronn, 'On the density of discriminants of cubic fields. II', 1971, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 322(1551), 405-420, DOI: 10.1098/rspa.1971.0075, https://doi.org/10.1098/rspa.1971.0075 (the Davenport-Heilbronn theorem: the count of cubic fields of discriminant below $X$ is asymptotic to a constant times $X$)
3. Henri Cohen and Hendrik W. Lenstra Jr., 'Heuristics on class groups', 1984, in: Number Theory (New York, 1982), Lecture Notes in Mathematics 1052, Springer, Berlin, pp. 26-36, DOI: 10.1007/BFb0071539, https://doi.org/10.1007/BFb0071539 (the Cohen-Lenstra heuristics on the distribution of class groups)
4. Henri Cohen and Jacques Martinet, 'Class groups of number fields: numerical heuristics', 1987, Mathematics of Computation 48(177), 123-137, DOI: 10.1090/S0025-5718-1987-0866103-4, https://doi.org/10.1090/S0025-5718-1987-0866103-4 (generalization of the Cohen-Lenstra heuristics to arbitrary base fields; the roots-of-unity phenomenon in class-group statistics)
5. Gunter Malle, 'On the distribution of Galois groups', 2002, Journal of Number Theory 92(2), 315-329, DOI: 10.1006/jnth.2001.2713, https://doi.org/10.1006/jnth.2001.2713 (Malle's conjecture on the order of magnitude of the count of $G$-extensions of discriminant below $X$)
6. Manjul Bhargava, 'Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants', 2007, International Mathematics Research Notices, Volume 2007, Issue 17, Article rnm052, DOI: 10.1093/imrn/rnm052, https://doi.org/10.1093/imrn/rnm052 (Bhargava's mass heuristics for the leading constants in the $S_n$-count)
7. Manjul Bhargava, 'The density of discriminants of quartic rings and fields', 2005, Annals of Mathematics (2) 162(2), 1031-1063, DOI: 10.4007/annals.2005.162.1031, https://doi.org/10.4007/annals.2005.162.1031 (existence of the main term for $S_4$-extensions of $\mathbb Q$, with an Euler-product constant)
8. Manjul Bhargava, 'The density of discriminants of quintic rings and fields', 2010, Annals of Mathematics (2) 172(3), 1559-1591, DOI: 10.4007/annals.2010.172.1559, https://doi.org/10.4007/annals.2010.172.1559 (existence of the main term for $S_5$-extensions of $\mathbb Q$)
9. David P. Roberts, 'Density of cubic field discriminants', 2001, Mathematics of Computation 70(236), 1699-1705, DOI: 10.1090/S0025-5718-00-01291-6, https://doi.org/10.1090/S0025-5718-00-01291-6 (numerical evidence for a secondary term of order $X^{5/6}$; Roberts' conjecture for totally real cubics)
10. Jordan S. Ellenberg, Akshay Venkatesh and Craig Westerland, 'Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields', 2016, Annals of Mathematics (2) 183(3), 729-786, DOI: 10.4007/annals.2016.183.3.1, arXiv: 0912.0325, https://doi.org/10.4007/annals.2016.183.3.1 (homological stability for generalized dihedral groups; Conjecture 1.5, the vanishing condition for $(S_n,\mathrm{transpositions})$, stated in arXiv v4)
11. Jordan S. Ellenberg, Akshay Venkatesh and Craig Westerland, 'Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II', 2012, arXiv preprint arXiv: 1212.0923, https://arxiv.org/abs/1212.0923 (announced the computation of stable components and the Galois action via the lifting invariant; withdrawn by the authors in v2, November 2013, owing to a gap affecting the stabilization maps)
12. Oscar Randal-Williams, 'Exposé Bourbaki 1164: Homology of Hurwitz spaces and the Cohen-Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland)', 2020, Astérisque 422 (Séminaire Bourbaki), pp. 469-497, DOI: 10.24033/ast.1143, arXiv: 1906.07447, https://doi.org/10.24033/ast.1143 (Bourbaki survey of the program)
13. Jordan S. Ellenberg, TriThang Tran and Craig Westerland, 'Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields', 2017, arXiv preprint arXiv: 1701.04541, https://arxiv.org/abs/1701.04541 (upper bound in Malle's conjecture over function fields, via Nichols algebras / quantum shuffle algebras)
14. Andrea Bianchi and Jeremy Miller, 'Polynomial stability of the homology of Hurwitz spaces', 2025, Mathematische Annalen 391, 4117-4144 (published online October 2024), DOI: 10.1007/s00208-024-03009-1, https://doi.org/10.1007/s00208-024-03009-1 (polynomial stability of homology for arbitrary $(G,Q)$)
15. Aaron Landesman and Ishan Levy, 'Homological stability for Hurwitz spaces and applications', 2025, arXiv preprint (v2, October 2025), arXiv: 2503.03861, https://arxiv.org/abs/2503.03861 (integral homological stability for arbitrary finite racks; Theorem 1.4.2 and Remark 1.4.3 imply EVW Conjecture 1.5; Malle bounds over $\mathbb F_q(t)$: Theorems 1.1.1, 10.1.8, 10.1.10)
16. Aaron Landesman and Ishan Levy, 'The stable homology of Hurwitz modules and applications', 2025, arXiv preprint (v1, October 2025), arXiv: 2510.02068, https://arxiv.org/abs/2510.02068 (Theorem 1.2.4: a function-field version of Bhargava's conjecture counting $S_d$-extensions of $\mathbb F_q(t)$ for $q$ sufficiently large)
17. Aaron Landesman and Ishan Levy, 'The Cohen-Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces', 2024, arXiv preprint (v3, October 2025), arXiv: 2410.22210, https://arxiv.org/abs/2410.22210 (Cohen-Lenstra-Martinet moments over function fields, including the roots-of-unity case)
18. Tim Santens, 'The leading constant in Malle's conjecture over function fields', 2025, arXiv preprint (v2, January 2026), arXiv: 2512.12838, https://arxiv.org/abs/2512.12838 (Theorems 1.1-1.2: the explicit leading constant via the unramified Brauer group of $BG$ and a Tamagawa measure; Remark 1.3 explains the periodicity in the survey's non-liftable example $G=\mathrm{PSL}_2(\mathbb F_7)$)
19. Daniel Loughran and Tim Santens, 'Malle's conjecture and Brauer groups of stacks', 2024, arXiv preprint (v3, June 2025), arXiv: 2412.04196, https://arxiv.org/abs/2412.04196 (Conjecture 9.1: the leading constant in Malle's conjecture over number fields; Remark 6.37 identifies the relevant group as the dual of the Schur multiplier of $G$)
20. Will Sawin and Melanie Matchett Wood, 'Distributions of unramified extensions of global fields', 2026, arXiv preprint (v2, March 2026), arXiv: 2602.21032, https://arxiv.org/abs/2602.21032 (function-field distribution theorem for maximal unramified extensions; a number-field conjecture with a refined third-homology invariant from Artin-Verdier duality)
21. Yuan Liu, 'Non-abelian Cohen-Lenstra heuristics in the presence of roots of unity', 2022, arXiv preprint (v2, December 2022), arXiv: 2202.09471, https://arxiv.org/abs/2202.09471 (function-field moments for maximal unramified extensions in the roots-of-unity case; modified non-abelian Cohen-Lenstra conjectures for function and number fields)
22. Manjul Bhargava, Arul Shankar and Jacob Tsimerman, 'On the Davenport-Heilbronn theorems and second order terms', 2013, Inventiones Mathematicae 193(2), 439-499, DOI: 10.1007/s00222-012-0433-0, https://doi.org/10.1007/s00222-012-0433-0 (proof of Roberts' conjecture for totally real cubic fields, proved independently with Taniguchi-Thorne)
23. Takashi Taniguchi and Frank Thorne, 'Secondary terms in counting functions for cubic fields', 2013, Duke Mathematical Journal 162(10), 1831-1868, DOI: 10.1215/00127094-2371752, https://doi.org/10.1215/00127094-2371752 (independent proof of Roberts' conjecture)
24. Arul Shankar and Jacob Tsimerman, 'Secondary terms in the counting functions of quartic fields', 2025, arXiv preprint (v2, August 2025), arXiv: 2503.02381, https://arxiv.org/abs/2503.02381 (smoothed quartic secondary terms with explicit secondary mass formulas)
25. Arul Shankar and Jacob Tsimerman, 'Secondary terms in the counting functions of quartic fields II', 2025, arXiv preprint arXiv: 2508.08527, https://arxiv.org/abs/2508.08527 (smoothed $S_4$-quartic counts as a sum of two main terms with a power-saving error term)
26. Kevin Chang, 'Hurwitz spaces, Nichols algebras, and Igusa zeta functions', 2023, arXiv preprint (v3, January 2024), arXiv: 2306.10446, https://arxiv.org/abs/2306.10446 (first function-field progress on the cohomology classes underlying secondary terms, via Nichols algebras)