Algebraic number theory · arithmetic statistics · Galois extensions · Brauer–Manin obstructions

Corrected and Refined Malle Conjecture

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How many number-field extensions with a prescribed finite permutation group have discriminant at most a given bound once cyclotomic accumulation and Brauer–Manin obstructions are separated correctly?

N_{N, ψ}^{\sec} (X) \sim c_{N, ψ}^{BM} X^{a_{N, ψ}} (\log X)^{b_{N, ψ} - 1}

Known results and sources

The refined conjecture separates extensions by exact cyclotomic data before predicting their discriminant-growth exponent and Brauer–Manin-weighted leading term; the general statement remains open.

Research problem

Exact mathematical statement

Let $k$ be a number field and $G\leq S_n$ a finite transitive permutation group. Put $e=\exp(G)$ , let $C_k=\operatorname{im}(\chi_e)$ , and for a structured surjection $\rho: \Gamma_k\twoheadrightarrow G$ define its exact pair by $N_\rho=\rho(\ker\chi_e)$ and the induced surjection $\psi_\rho:C_k\twoheadrightarrow G/N_\rho$ . For an admissible pair $(N,\psi)$ , define

d_{N,\psi}=\min_{1\ne g\in N}\operatorname{ind}(g),\qquad a_{N,\psi}=\frac{1}{d_{N,\psi}},\qquad b_{N,\psi}=\#\bigl(C_k\setminus\mathscr C_{N,\psi}\bigr),

where $\mathscr C_{N,\psi}$ is the set of minimal-index $N$ -conjugacy classes and $C_k$ acts by the canonical twisted anticyclotomic action determined by $\psi$ . The working corrected prediction is that every admissible positive-mass exact stratum satisfies

N_{N,\psi}^{\mathrm{sec}}(X)\sim c_{N,\psi}^{\mathrm{BM}}X^{a_{N,\psi}}(\log X)^{b_{N,\psi}-1}.

The unrestricted exponent pair is expected to be the lexicographic maximum over admissible exact pairs having positive Brauer–Manin mass, with leading constants summed over the maximizing strata after the field, torsor, stack, height, measure, multiplicity, and thin-set conventions are fixed. the source calls this a working corrected formulation and explicitly reports that the full conjecture remains open.

Problem infographic

Problem at a glance

A landscape scientific plate defines the permutation index of a group element, shows a structured Galois surjection splitting into an admissible exact pair (N, psi), depicts the canonical twisted anticyclotomic action of C_k on the exact-pair class set and its orbit count b_{N,psi}, and ends with the conjectural Brauer–Manin-weighted asymptotic for that stratum. — For each admissible exact cyclotomic pair, the working refinement predicts a power-logarithmic count governed by the smallest permutation index and the twisted orbit count, with a Brauer–Manin leading constant. The normalization and several global branches remain unresolved, so this is the problem statement and current boundary—not a proof diagram.

Current mathematical picture

Where work on Corrected and Refined Malle Conjecture stands

Open conjecture

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSupportwise exponential globalization retired

Pointwise exponential Fourier decay for every prescribed support is false: splitting primes for a nontrivial global character can all have local coefficient 1. The source replaces that route with averaged spectral cancellation, while the strongest retained result remains conditional and fixed-chart and the geometric Bockstein and uniform quotient-fibre barriers remain open.

Route status · Narrowed route

Main reductionConditional finite-label top-pole classification

Under the source's fixed support-complete odd-prime chart and analytic hypotheses, the variable-prime projective top-pole space is reported to equal the relative profile group, while every other profile loses positive logarithmic weight.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeConstruct the finite geometric G_m Bockstein mapping cone for odd-prime class-two layers with nonzero power map.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Corrected and Refined Malle Conjecture in numbers

3.3kretained lines of mathematical investigation3,316 in the current working snapshot

Argument development: 2,691 · 81%
Explored or eliminated routes: 148 · 4%
Computational analysis: 36 · 1%
Open obligations: 189 · 6%
Definitions and setup: 252 · 8%

9selected mapped statements1routes investigated3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byJiuya WangProposed2025Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Prove uniform quotient-fibre summation across the varying charts needed for induction.

Suggested move: Specify a finite stratification with uniform critical data, bad moduli, sieve constants, Poitou–Tate modes, Tauberian bounds, height comparison, and field-versus-stack multiplicities, then derive a summable majorant.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSupportwise exponential globalization retired

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove uniform quotient-fibre summation across the varying charts needed for induction.Suggested move: Specify a finite stratification with uniform critical data, bad moduli, sieve constants, Poitou–Tate modes, Tauberian bounds, height comparison, and field-versus-stack multiplicities, then derive a summable majorant.

Ready to work on

Match the working exact-pair prediction to one authoritative corrected Malle formulation and its precise counting conventions.Suggested move: Build a source dictionary for secure objects, breaking thin sets, Brauer groups, exact-pair strata, adelic measures, zero-mass cases, and field, torsor, homomorphism, and stack multiplicities before making any final theorem claim.

Ready to work on

Construct the finite geometric G_m Bockstein mapping cone for odd-prime class-two layers with nonzero power map.Suggested move: Audit the cochain actions and signs in the retained descent formula, encode the Bockstein null-homotopy in a Delta-equivariant complex, and test reduction to the exponent-p and prime-to-p cases.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

Wang's 2025 cyclotomic-stratified Refined Malle's Conjecture remains open. The proposal responds to counterexamples to earlier logarithmic-exponent formulations and is compatible with known and computed cases, while current number-field inductive theorems and broad function-field results remain scoped advances rather than a proof for every finite group, global field, admissible invariant, cyclotomic stratum, and embedding problem.

[1][6][7]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-07
PreprintMiller and Santens studied a leading-constant refinement for a particular Heisenberg group in the Loughran-Santens framework. Despite the similar title, this is not a proof of Wang's Conjecture 6.[10][9]
2026
Peer reviewedAlberts, Lemke Oliver, Wang, and Wood published inductive methods proving many new number-field counting cases and producing structured examples and counterexamples in the wider Malle program.[6]
2025
PreprintLandesman-Levy and Santens obtained broad Malle-type function-field results over sufficiently large finite fields, including refined control of leading constants; these are function-field advances rather than…[7][8]
2025
PreprintWang exhibited counterexamples to Türkelli's number-field modification and proposed Conjecture 6, which stratifies the counting function by cyclotomic intersection and a properly solvable embedding problem…[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusRefined Malle's Conjecture

Related problemMalle's original conjecture

Malle's original conjecture counts all extensions with a fixed transitive Galois group together. Wang's proposal refines that aggregate by cyclotomic and embedding-problem strata and derives a revised aggregate prediction when the stratum exponents are combined.

[2][3]

Related problemTürkelli's modification of Malle's conjecture

Türkelli modified the logarithmic exponent using cyclotomic action and Hurwitz-space components. Wang identifies number-field counterexamples and adds proper solvability of embedding problems to the governing data.

[5][1]

Dependency or reductionCyclotomic embedding problems

Each refined counting stratum is indexed by a specified cyclotomic intersection and a quotient or embedding-problem pair, and the predicted contribution is zero when the required proper surjective lift does not exist.

[1]

Related problemFunction-field Malle conjectures

Function-field Hurwitz-space methods prove broad analogues over sufficiently large finite fields, but arithmetic and uniformity differences prevent these theorems from being treated as the number-field or all-global-field conjecture.

[7][8]

Related problemLeading-constant refinements of Malle's conjecture

The Loughran-Santens program refines predicted leading constants through Brauer groups of classifying stacks, and the Heisenberg preprint treats one group in that framework. It is distinct from Wang's stratum-by-stratum exponent conjecture.

[9][10]

Related problemInductive methods for counting number fields

Inductive counting methods transfer estimates among groups and prove many new cases, supplying a route to scoped results and test examples without proving the universal refined conjecture.

[6]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedMathlib number-field discriminant, Hermite finiteness, and polynomial Galois-group infrastructure
Mathlib defines number-field discriminants, proves a Hermite finiteness theorem, and supplies polynomial Galois-group foundations. It lacks the asymptotic counting, cyclotomic-stratum, embedding-problem, and analytic uniformity layers needed to state and prove Wang's conjecture.
[11][12]
dataset · source linked; not reproduced by ProofAtlasLMFDB number fields and transitive Galois groups
The LMFDB provides searchable finite tables of number fields, discriminants, signatures, and transitive Galois groups useful for examples and consistency checks. Finite coverage cannot establish an asymptotic.
[13]
software · not independently reproducedMagma code and saved database for Inductive methods for counting number fields
The authors publish Magma code and a saved database covering transitive groups used in their computations. This collection linked but did not execute or independently reproduce those calculations.
[14]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA precise formal counting function for global-field extensions with a fixed transitive permutation group, bounded admissible invariant, and isomorphism conventions.
Formalization targetCyclotomic intersections, quotient data, finite embedding problems, and the predicate that a proper surjective lift exists.
Formalization targetAsymptotic notation with the predicted power and logarithmic exponents, including the distinction between order-of-magnitude statements and asymptotic equivalence over number fields.
Formalization targetThe analytic number theory and Hurwitz-space machinery required for known cases, together with explicit interfaces between algebraic and counting structures.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction4 of 94
lemma4 of 94

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementEach admissible cyclotomic stratum is predicted to contribute its own Brauer–Manin-weighted power-logarithmic count.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementIntrinsic exact pairs and twisted critical orbitsintermediate
retained route statementStrict quotient-weight gapintermediate
retained route statementStacky lift-fibre cancellationintermediate
retained route statementConditional finite-label top-pole classificationintermediate
retained route statementConditional fixed-chart commutative asymptoticintermediate
retained route statementCarry cocycle and full mu_p descent reductionintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetConstruct the finite geometric G_m Bockstein mapping cone for odd-prime class-two layers with nonzero power map.in progress reported
Research targetProve uniform quotient-fibre summation across the varying charts needed for induction.open
Research targetMatch the working exact-pair prediction to one authoritative corrected Malle formulation and its precise counting conventions.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.3 displayed rows · 1 route included

Useful failureSupportwise exponential globalizationreported failure
ComputationFinite row-reduction recipes for class-two profile groups together with reported brute-force carry-cocycle checks at p=3, 5, and 7The current work reports random finite checks of associativity and the p-power map for several small dimensions and supplies regression tests for future implementations. Intake did not execute attachments or reproduce these computations, so they remain source-reported only. · reported unreproduced
Narrowed routeSupportwise exponential globalization retiredPointwise exponential Fourier decay for every prescribed support is false: splitting primes for a nontrivial global character can all have local coefficient 1. The source replaces that route with averaged spectral cancellation, while the strongest retained result remains conditional and fixed-chart and the geometric Bockstein and uniform quotient-fibre barriers remain open.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeConstruct the finite geometric G_m Bockstein mapping cone for odd-prime class-two layers with nonzero power map.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve uniform quotient-fibre summation across the varying charts needed for induction.

Corrected and Refined Malle Conjecture · ready to start

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Sources and references15 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Counterexamples for Türkelli's Modification on Malle's Conjectureoriginal source · Jiuya Wang · arXiv · 2025 · ARXIV 2502.04261 · accessed Aug 7, 2026
2
On the Distribution of Galois Groupsoriginal source · Gunter Malle · Journal of Number Theory · 2002 · DOI 10.1006/jnth.2001.2713 · accessed Aug 7, 2026
3
On the Distribution of Galois Groups, IIoriginal source · Gunter Malle · Experimental Mathematics · 2004 · DOI 10.1080/10586458.2004.10504527 · accessed Aug 7, 2026
4
A counter example to Malle's conjecture on the asymptotics of discriminantspeer reviewed result · Jürgen Klüners · Comptes Rendus Mathématique · 2005 · DOI 10.1016/j.crma.2005.02.010 · accessed Aug 7, 2026
5
Connected components of Hurwitz schemes and Malle's conjecturepeer reviewed result · Seyfi Türkelli · Journal of Number Theory · 2015 · DOI 10.1016/j.jnt.2015.03.005 · accessed Aug 7, 2026
6
Inductive methods for counting number fieldspeer reviewed result · Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang, Melanie Matchett Wood · Journal für die reine und angewandte Mathematik · 2026 · DOI 10.1515/crelle-2025-0074 · accessed Aug 7, 2026
7
Homological stability for Hurwitz spaces and applicationspreprint · Aaron Landesman, Ishan Levy · arXiv · 2025 · ARXIV 2503.03861 · accessed Aug 7, 2026
8
The leading constant in Malle's conjecture over function fieldspreprint · Tim Santens · arXiv · 2025 · ARXIV 2512.12838 · accessed Aug 7, 2026
9
Malle's conjecture and Brauer groups of stackspreprint · Daniel Loughran, Tim Santens · arXiv · 2024 · ARXIV 2412.04196 · accessed Aug 7, 2026
10
A refined Malle conjecture for Heisenberg groupspreprint · Jack B. Miller, Tim Santens · arXiv · 2026-07 · ARXIV 2607.06476 · accessed Aug 7, 2026
11
Mathlib: number-field discriminant definitions and Hermite finitenessformalization · Mathlib contributors · Lean community · accessed Aug 7, 2026
12
Mathlib: polynomial Galois groupsformalization · Mathlib contributors · Lean community · accessed Aug 7, 2026
13
LMFDB Number Fields databasesoftware or dataset · LMFDB Collaboration · accessed Aug 7, 2026
14
Code and saved database for Inductive methods for counting number fieldssoftware or dataset · Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang, Melanie Matchett Wood · Robert J. Lemke Oliver · accessed Aug 7, 2026
15
Formal Conjectures repositoryformalization · Google DeepMind · GitHub · accessed Aug 7, 2026

Important qualifications

This record binds the target to Jiuya Wang's cyclotomic-intersection and embedding-problem Refined Malle's Conjecture, Conjecture 6 of arXiv:2502.04261. Other proposals called refined Malle conjectures are related but distinct.
Wang's source is a 2025 preprint. Its statement that no counterexample to Conjecture 6 is known is author-reported current support, not peer review or proof.
Counterexamples to Malle's original logarithmic exponent and to Türkelli's modification motivate Wang's formulation but are not counterexamples to Wang's Conjecture 6.
Function-field theorems, inductive number-field results, and leading-constant refinements are recorded only at their proved scope and do not establish Wang's global-field, stratum-by-stratum conjecture in full.
The scoped formalization search found supporting number-field and Galois-theory libraries but no statement-aligned formalization of the refined conjecture. This does not establish nonexistence outside the inspected repositories.
The LMFDB and associated Magma code are linked as available resources only. They were not rerun in this collection, and finite databases cannot prove an asymptotic counting conjecture.
No selective prize or maintained famous-problem-list membership was verified, so recognition remains empty rather than inferring prominence from the surrounding literature.

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