Algebraic number theory · Artin L-functions · Stark regulators

Tate’s Rational Leading-Term Stark Conjecture at s = 0

Collaboration beta

The rational Stark conjecture predicts that the leading term of an Artin L-function at zero, divided by the matching determinant of logarithms of S-units, is algebraic in the character field and varies correctly under Galois conjugation.

A_{S}^{f} (ψ) = \frac{L_{S}^{*} (0, \bar{ψ})}{R_{S}^{f} (ψ)} \in Q (ψ), τ (A_{S}^{f} (ψ)) = A_{S}^{f} (ψ^{τ})

Known results and sources

The rational Stark question compares an analytic leading term at s = 0 with an algebraic regulator and asks for a Galois-equivariant quotient.

Research problem

Exact mathematical statement

Let $K/k$ be a finite Galois extension, let $S$ contain the archimedean and ramified places, and fix a rational comparison $f: \mathbf Q\otimes X_S(K)\xrightarrow{\sim}\mathbf Q\otimes E_S(K)$ . For an irreducible complex character $\psi$ , define

A_S^f(\psi)=\frac{L_S^*(0,\bar\psi)}{R_S^f(\psi)}.

The rational leading-term conjecture asserts

A_S^f(\psi)\in\mathbf Q(\psi),\qquad \tau\left(A_S^f(\psi)\right)=A_S^f(\psi^\tau)\quad\text{for every }\tau\in\operatorname{Aut}(\mathbf C).

the source advances only this rational leading-term layer and one explicit primitive phase within it; it does not claim the integral Rubin–Stark, field-generation, or reciprocity refinements.

Problem infographic

Problem at a glance

A problem-first infographic defines the Artin leading term, the S-unit logarithmic regulator, their quotient in the character field, and a separate Stark–139 case-study panel whose real residuals remain open. — Tate’s rational leading-term conjecture predicts algebraicity and Galois equivariance of an analytic-to-regulator quotient; the source’s Stark–139 laboratory remains an unresolved special case.

Current mathematical picture

Where work on Tate’s Rational Leading-Term Stark Conjecture at s = 0 stands

Partially resolved

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

Useful failureSource-reported limitation

Equality of absolute values does not determine the oriented Stark–139 identity. A complete route would prove the Transfer–139 coordinate identity by a noncircular chain-level boundary comparison, then extract a general oriented cyclic Kummer/principalizer transfer theorem and combine it with rigorously matched induction, inflation, and change-of-S formulas for all Artin characters.

Route status · Narrowed route

Main reductionCurrent reduction

The current work isolates a mixed cyclic cubic Stark–139 laboratory over F = Q(√5), rebases its regulator to two Kummer logarithmic columns, and rewrites the remaining analytic-to-algebraic comparison as three real coordinate equations. The source reports that their residuals sum to zero and that the exact defect is a nonnegative multiple of r₀²+r₁²+r₂², so the special case reduces to proving two independent real identities without choosing a numerical phase.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeClose the analytic foundation and functional-equation normalization for the current one-dimensional formula.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Tate’s Rational Leading-Term Stark Conjecture at s = 0 in numbers

2kretained lines of mathematical investigation2,047 in the current working snapshot

Argument development: 1,750 · 85%
Explored or eliminated routes: 50 · 2%
Computational analysis: 50 · 2%
Open obligations: 94 · 5%
Definitions and setup: 103 · 5%

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 byHarold M. StarkProposed1971Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Construct the primitive orbit cocycle and account for every endpoint of the open 23-step scaling path.

Suggested move: Build a group-ring-valued boundary calculus retaining t ↦ ν^{±23}t, factor the 139, 461, and unit contributions, and classify the surviving terms into primitive and trivial-character sectors without using Transfer–139.

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 routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Construct the primitive orbit cocycle and account for every endpoint of the open 23-step scaling path.Suggested move: Build a group-ring-valued boundary calculus retaining t ↦ ν^{±23}t, factor the 139, 461, and unit contributions, and classify the surviving terms into primitive and trivial-character sectors without using Transfer–139.

Ready to work on

Identify the Kummer boundary pair and its exact skew-pairing normalization.Suggested move: Derive the boundary classes 1−β and β−ρ from their exact divisors and norms, then map their cross-embedding pairing to the three real Coordinate–139 equations with the correct factor and cyclic shift.

Ready to work on

Close the analytic foundation and functional-equation normalization for the current one-dimensional formula.Suggested move: Prove the integral formula with justified exchanges and branch pairing, then match the completed functional equation including conductor 695, both real gamma factors, the double zero at s = 0, and the pinned root-number orientation.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusPartially resolved

The full number-field rationality and Galois-equivariance conjecture for arbitrary Artin characters remains open. Tate proved the main conjecture for rational-valued characters, and the function-field analogue is a theorem due to Deligne and independently Hayes. Stronger integral and Brumer–Stark variants have important special-case theorems but do not settle the arbitrary-character number-field target.

[3][5][4]

External progress

What the literature has established

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

2023
Peer reviewedDasgupta and Kakde proved the abelian Brumer–Stark conjecture away from 2 and a stronger Fitting-ideal result, with consequences for a higher-rank Rubin–Stark form away from 2.[6]
2006
Peer reviewedAnderson recorded that Tate's function-field formulation is a theorem due to Deligne and independently Hayes, and proved a two-variable refinement in the genus-zero case.[5]
1996
Peer reviewedRubin formulated an integral exterior-power refinement for abelian L-functions with higher-order zeros, proved special cases, and related its rational part to Stark's conjecture over Q.[4]
1984
Historical sourceTate systematized the conjecture with S-units, the degree-zero place module, and characterwise regulators, and proved the main rational conjecture for rational-valued characters.[3][7]

Mathematical neighborhood

Related results and reusable starting points

Current focusTate's rational leading-term form of the Stark conjecture

Solved special caseRational-valued Artin characters

The analytic class-number formula gives the trivial-character case, and Tate's character-theoretic argument establishes the conjecture for every rational-valued character.

[3][7]

Solved special caseFunction-field Stark conjecture

The global function-field analogue of Tate's Stark formulation is a theorem due to Deligne and independently Hayes; it does not imply the number-field case.

[5][3]

Stronger or generalized formRubin–Stark conjecture

Rubin–Stark strengthens the rational abelian conjecture by predicting an element in an integral exterior-power lattice and allows higher-order zeros.

[4]

Related problemBrumer–Stark conjecture

Brumer–Stark predicts class-group annihilation and explicit anti-units from Stickelberger elements. The theorem away from 2 is major progress in an abelian neighbor, not the full arbitrary-character rational leading-term statement.

[6][3]

Stronger or generalized formEquivariant leading-term and Tamagawa-number conjectures

Equivariant leading-term and Tamagawa-number frameworks incorporate refined integral information whose specializations overlap Stark and Rubin–Stark predictions.

[4][6]

Formal and computational footholds

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

software · source linked; not reproduced by ProofAtlasPARI/GP Artin L-function implementation
The official PARI/GP L-function API constructs Artin L-functions with lfunartin and supplies evaluation, order-of-zero, root-number, and functional-equation checks. It supports instance experiments but does not certify the universal Stark conjecture.
[8]
dataset · source linked; not reproduced by ProofAtlasLMFDB Artin representations and L-functions
LMFDB provides curated Artin-representation and L-function data with provenance and coverage notes. Entries are examples rather than a proof or independent reproduction of the general leading-term assertion.
[9]

Formalization opportunities

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

Formalization targetA checked theory of number fields, places, S-units, Artin representations, characters, and character fields at the scope required by Tate's formulation.
Formalization targetA formal analytic theory of Artin L-functions with meromorphic continuation, functional equations, orders of vanishing, and leading coefficients at s = 0.
Formalization targetA statement-aligned construction of the logarithmic S-unit isomorphism and characterwise Stark regulator, including independence from the rational module isomorphism.
Formalization targetA formal Galois-equivariance statement for the regulator-normalized leading terms and clear separation from integral Rubin–Stark refinements.

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 updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionThe public failure record now distinguishes the established absolute-value identity from the still-open oriented Stark–139 equality. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Mathematical distinction clarified

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.

6 standing statements3 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma3 of 93
equivalence2 of 92
negative result1 of 91

Selected mathematical clusters3 mathematical clusters

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

retained route statementA_S^f(ψ) = L_S^*(0, ψ̄)/R_S^f(ψ) lies in Q(ψ), with τ(A_S^f(ψ)) = A_S^f(ψ^τ).
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExplicit Stark–139 arithmetic remains in the current research mapintermediate
retained route statementRegulator becomes a Kummer determinantintermediate
retained route statementThe modulus is known but the phase is notintermediate
retained route statementTrivial-character augmentation is fixedintermediate
retained route statementTransfer–139 is a real sum-of-squares targetintermediate
retained route statementTransfer–139 is the open laboratory edgeintermediate
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 targetClose the analytic foundation and functional-equation normalization for the current one-dimensional formula.in progress reported
Research targetConstruct the primitive orbit cocycle and account for every endpoint of the open 23-step scaling path.open
Research targetIdentify the Kummer boundary pair and its exact skew-pairing normalization.open

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

Useful failureSource-reported limitationreported failure
ComputationSource-supplied exact arithmetic and symbolic certificate suite, plus numerical orientation diagnostics for Stark–139ProofAtlas did not execute the current work’s verifier programs. The source reports that its finite arithmetic and coordinate identities pass exact or symbolic checks, while its small numerical residuals are orientation diagnostics only and do not prove Transfer–139. · reported unreproduced
Narrowed routeSource-reported limitationEquality of absolute values does not determine the oriented Stark–139 identity. A complete route would prove the Transfer–139 coordinate identity by a noncircular chain-level boundary comparison, then extract a general oriented cyclic Kummer/principalizer transfer theorem and combine it with rigorously matched induction, inflation, and change-of-S formulas for all Artin characters.

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 bridgeClose the analytic foundation and functional-equation normalization for the current one-dimensional formula.

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 pointConstruct the primitive orbit cocycle and account for every endpoint of the open 23-step scaling path.

Tate’s Rational Leading-Term Stark Conjecture at s = 0 · ready to start

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Sources and references10 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
L-functions at s = 1. II. Artin L-functions with rational charactersoriginal source · Harold M. Stark · Advances in Mathematics · 1975 · DOI 10.1016/0001-8708(75)90087-0 · accessed Aug 7, 2026
2
L-functions at s = 1. IV. First derivatives at s = 0original source · Harold M. Stark · Advances in Mathematics · 1980-03 · DOI 10.1016/0001-8708(80)90049-3 · accessed Aug 7, 2026
3
Les conjectures de Stark sur les fonctions L d'Artin en s = 0original source · John Tate · Birkhäuser, Progress in Mathematics 47 · 1984 · accessed Aug 7, 2026
4
A Stark conjecture over Z for abelian L-functions with multiple zerospeer reviewed result · Karl Rubin · Annales de l'Institut Fourier · 1996 · DOI 10.5802/aif.1505 · accessed Aug 7, 2026
5
A two-variable refinement of the Stark conjecture in the function-field casepeer reviewed result · Greg W. Anderson · Compositio Mathematica · 2006-05 · ARXIV math/0407535 · DOI 10.1112/S0010437X05001818 · accessed Aug 7, 2026
6
On the Brumer–Stark conjecturepeer reviewed result · Samit Dasgupta, Mahesh Kakde · Annals of Mathematics · 2023 · DOI 10.4007/annals.2023.197.1.5 · accessed Aug 7, 2026
7
The Work of John Tatesurvey or monograph · James S. Milne · Author-maintained mathematical notes · 2012; online edition accessed 2026 · accessed Aug 7, 2026
8
Catalogue of GP/PARI Functions: L-functionssoftware or dataset · PARI/GP Development Team · PARI/GP · accessed Aug 7, 2026
9
The L-functions and Modular Forms Databasesoftware or dataset · LMFDB Collaboration · L-functions and Modular Forms Database · accessed Aug 7, 2026
10
Stark conjecturesencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026

Important qualifications

The target is Tate's rational leading-term formulation, not every conjecture bearing the name Stark and not the stronger integral Rubin–Stark conjecture.
The theorem for rational-valued characters does not automatically extend to arbitrary character fields, and the function-field theorem does not settle the number-field conjecture.
The 2023 Brumer–Stark theorem away from 2 is a neighboring abelian result and is not treated as a solution of the full target.
No unreviewed source material, unpublished proof draft, or packet computation was read or evaluated.
A scoped search of prominent public formal libraries and conjecture collections found no complete machine-checked statement of Tate's full rational leading-term conjecture; this does not establish global nonexistence.
PARI/GP and LMFDB were inspected through public documentation and pages but were not executed or independently reproduced.

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