Number theory · Artin L-functions · finite group representations

Artin Holomorphy Conjecture

Collaboration beta

A nontrivial irreducible Galois representation has an Artin L-function with meromorphic continuation. The conjecture asks whether it can ever have a pole. This work makes the first binary-icosahedral obstruction finite and explicit, then isolates the missing global analytic lift.

χ \in Irr (G), χ \neq 1_{G} \Rightarrow L (s, χ, K / k) is entire .

Known results and sources

Artin holomorphy connects finite Galois symmetry to a global analytic question: can a nontrivial irreducible Artin L-function have a pole?

Research problem

Exact mathematical statement

Let $K/k$ be a finite Galois extension of number fields with group $G$ , and let $\chi$ be a nontrivial irreducible complex character of $G$ . The Artin Holomorphy Conjecture predicts that the associated Artin L-function is entire:

\chi\in\operatorname{Irr}(G),\quad\chi\ne 1_G\quad\Longrightarrow\quad L(s,\chi,K/k)\text{ is entire}.

The finite representation-theoretic reductions retained here do not prove the analytic conclusion.

Problem infographic

Problem at a glance

Scientific problem explainer for the Artin Holomorphy Conjecture: for a finite Galois extension, every nontrivial irreducible representation has an Artin L-function conjectured to be entire; below, the binary-icosahedral test branches from 6 through 10 to 12A and 12B across three open analytic bridges. — For a finite Galois extension, the conjecture asks whether every nontrivial irreducible representation has an entire Artin L-function; the lower fork is a finite binary-icosahedral test whose three analytic lifts remain open.

Current mathematical picture

Where work on Artin Holomorphy Conjecture stands

Partially resolved

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

Useful failureArbitrary-zero Rankin positivity

The source gives the obstruction character and checks that its permitted Rankin pairings do not distinguish the missing inequality. Start with the 6-to-10 correspondence: construct the relevant zero spaces, lift the map and its transpose, prove quotient and boundary compatibility, and obtain exact, nilpotent, or spectrally small composition error.

Route status · Narrowed route

Main reductionMinimal poles have rigid character structure

The source reports strong restriction, faithfulness, quasiprimitivity, and energy constraints on a minimal negative constituent of the order character.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeLift the 6-to-10 finite map analytically.Task status · Work already reported in progress

Work mapped so far

Artin Holomorphy Conjecture in numbers

1.6kretained lines of mathematical investigation1,619 in the current working snapshot

Argument development: 1,348 · 83%
Explored or eliminated routes: 104 · 6%
Computational analysis: 50 · 3%
Open obligations: 37 · 2%
Definitions and setup: 80 · 5%

9selected mapped statements2routes 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 byEmil ArtinProposed1923Status checkedAug 9, 2026

Source recordedAug 9, 2026

Recommended next task

Close the two 10-to-12 analytic bridges.

Suggested move: After the first edge, lift both degree-10 to degree-12 correspondences through their common degree-60 field and prove the two remaining divisor inequalities.

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

14 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

2 recorded

Narrowed routeArbitrary-zero Rankin positivity

Route status · Narrowed route

Narrowed routeScalar local zero germs

The source isolates the representation-theoretic mismatch and treats a characteristic-five shadow as explanatory rather than sufficient. Construct a genuinely equivariant zero module in characteristic zero, or use automorphic and correspondence methods that retain the full group action.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Close the two 10-to-12 analytic bridges.Suggested move: After the first edge, lift both degree-10 to degree-12 correspondences through their common degree-60 field and prove the two remaining divisor inequalities.

Ready to work on

Test weak automorphic induction precisely.Suggested move: Match the retained Hecke, twist, tensor-product, and Rankin data against a precise converse theorem, listing every still-missing local or global hypothesis.

Ready to work on

Lift the 6-to-10 finite map analytically.Suggested move: Construct and audit an analytic lift of the degree-6 to degree-10 correspondence, including the transpose, quotient compatibility, constant terms, residues, and boundary contributions.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 9, 2026

Current statusPartially resolved

The general Artin holomorphy conjecture remains open: for an arbitrary number field and nontrivial irreducible finite-image complex Galois representation, entireness of the Artin L-function is not known. Brauer induction proves only meromorphic continuation in general. Holomorphy is known for one-dimensional and monomial representations; strong Artin is known for two-dimensional representations with solvable image; and Serre modularity proves strong Artin for every odd irreducible two-dimensional complex representation over Q. These cases do not settle general even two-dimensional representations, arbitrary base fields, or higher dimensions. Recent results give additional conditional or pointwise criteria, not an unrestricted proof.

[10][8][9]

External progress

What the literature has established

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

2025
Peer reviewedGun, Hazra, and Sahu proved new pointwise holomorphy criteria for Artin L-functions in solvable extensions and criteria concerning zeros and poles. Their result is conditional on explicit order hypotheses and does not settle the general conjecture or every solvable group.[11]
2009
Peer reviewedKhare and Wintenberger's proof of Serre modularity implies that every continuous odd irreducible complex two-dimensional representation of the absolute Galois group of Q arises from a weight-one newform, proving strong Artin and holomorphy in that scope.[8][9]
2006
Computational resultBooker published a rigorous finite-height verification criterion and numerical tests for selected S5 representations, alongside related Riemann-hypothesis computations for S5 and A5 fields. Finite-height verification does not prove the general conjecture.[7]
2003
Peer reviewedBooker proved that if the L-function of an irreducible two-dimensional complex Galois representation over Q is not automorphic, then it has infinitely many poles. Thus holomorphy for one such representation implies its corresponding strong Artin statement.[6]

Mathematical neighborhood

Related results and reusable starting points

Current focusArtin holomorphy conjecture

Weaker or relaxed formBrauer meromorphic-continuation theorem

Brauer induction proves meromorphic continuation and the functional equation by writing an Artin L-function as a quotient of Hecke L-functions. Holomorphy asks for cancellation of every pole introduced by negative exponents, so meromorphic continuation is strictly weaker than the conjecture.

[3][10]

Solved special caseone-dimensional and monomial representations

One-dimensional nontrivial Artin L-functions agree with nontrivial Hecke L-functions and are entire. More generally, induction from a one-dimensional character reduces a monomial representation to a Hecke L-function. Solvability alone does not imply that every irreducible representation is monomial.

[1][10]

Solved special casetwo-dimensional solvable-image strong Artin

Langlands and Tunnell established automorphy for two-dimensional representations with solvable image, covering the dihedral, tetrahedral, and octahedral projective types and therefore proving their Artin L-functions entire.

[4][5]

Solved special caseodd irreducible two-dimensional representations over Q

Serre modularity yields a weight-one newform for every continuous odd irreducible complex representation G_Q → GL(2,C). This removes the projective-image restriction in the odd two-dimensional rational-base-field case, but it does not cover even representations or arbitrary number fields and dimensions.

[8][9]

Stronger or generalized formstrong Artin conjecture

Strong Artin predicts that an irreducible finite-image complex Galois representation corresponds to a cuspidal automorphic representation of GL(n) with the same L-function. Entireness of the cuspidal automorphic L-function then implies Artin holomorphy.

[10][8]

Logical consequencetwo-dimensional strong Artin over Q

For an individual irreducible two-dimensional complex Galois representation over Q, Booker proved that failure of automorphy would force infinitely many poles. In that restricted setting, Artin holomorphy implies the corresponding strong Artin assertion.

[6]

Formal and computational footholds

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

formal library support · partial resource linkedMathlib Dirichlet L-function continuation
Mathlib defines Dirichlet L-functions, proves analytic continuation and differentiability everywhere for nontrivial Dirichlet characters, and formalizes their functional equation. This supports the rational one-dimensional analogue but is not a definition or proof of general Artin L-functions over number fields.
[14]
formal library support · partial resource linkedMathlib formal L-function and Euler-product infrastructure
Mathlib constructs formal Dirichlet-series Euler products and connects them to L-series on a right half-plane. It is reusable analytic infrastructure, not a problem-level Artin-holomorphy statement or checked proof.
[13]
computation · not independently reproducedBooker finite-height Artin-L-function study
The peer-reviewed paper provides a rigorous finite-height criterion and reports numerical tests for selected S5 representations. ProofAtlas did not rerun the computation; it is neither a general certificate nor a proof of unrestricted holomorphy.
[7]

Formalization opportunities

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

Formalization targetNo target-level formal statement or proof of the Artin holomorphy conjecture was identified in the scoped current Mathlib documentation and Google DeepMind Formal Conjectures repository-tree search.
Formalization targetA faithful definition requires finite Galois extensions of number fields, decomposition and inertia groups, Frobenius elements, finite-dimensional complex representations with finite image, and Frobenius actions on inertia-invariant subspaces.
Formalization targetThe completed Artin L-function needs ramified and archimedean Euler factors, Artin conductors, analytic convergence, meromorphic continuation, functional equations, and precise compatibility with sums, inflation, and induction.
Formalization targetA direct proof route needs formal control of divisors, zeros, poles, and cancellation in Brauer products of Hecke L-functions over general number fields; existing documented Dirichlet-character continuation covers only a narrow rational abelian prerequisite.
Formalization targetA strong-Artin route additionally requires cuspidal automorphic representations of GL(n), standard automorphic L-functions, their entireness, and an exactly statement-aligned global reciprocity theorem.
Formalization targetFormalizing a one-dimensional, monomial, or odd two-dimensional special case would not by itself formalize or prove the unrestricted conjecture. Every formal artifact must record its base field, dimension, parity, image, and induction hypotheses explicitly.

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 questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma3 of 93
computational claim1 of 91
negative result1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included

retained route statementEvery nontrivial irreducible Artin L-function should be entire—equivalently, it should have no finite poles.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementMinimal poles have rigid character structureintermediate
retained route statementMonomial divisor inequalities are equivalentintermediate
retained route statementOrder-trace positivity is equivalentintermediate
retained route statementFour modules expose the first binary testintermediate
retained route statementRankin positivity misses the obstructionintermediate
retained route statementA small composition error would sufficeintermediate
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
Useful failureArbitrary-zero Rankin positivityreported failure
Useful failureScalar local zero germsreported failure
Research targetLift the 6-to-10 finite map analytically.in progress reported
Research targetClose the two 10-to-12 analytic bridges.open
Research targetTest weak automorphic induction precisely.open
ComputationThe archive includes finite certificates, verifier programs, and captured outputs for the binary-icosahedral character calculations.Every checksum listed in the current work matched the retained archive bytes. The supplied programs and captured finite checks have not been independently rerun by ProofAtlas. · reported unreproduced
Narrowed routeArbitrary-zero Rankin positivityThe source gives the obstruction character and checks that its permitted Rankin pairings do not distinguish the missing inequality. Start with the 6-to-10 correspondence: construct the relevant zero spaces, lift the map and its transpose, prove quotient and boundary compatibility, and obtain exact, nilpotent, or spectrally small composition error.
Narrowed routeScalar local zero germsThe source isolates the representation-theoretic mismatch and treats a characteristic-five shadow as explanatory rather than sufficient. Construct a genuinely equivariant zero module in characteristic zero, or use automorphic and correspondence methods that retain the full group action.

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 bridgeLift the 6-to-10 finite map analytically.

2 approaches have 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.

Continue the mathematics

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Prepared starting pointClose the two 10-to-12 analytic bridges.

Artin Holomorphy Conjecture · ready to start

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Sources and references16 cited works · next context review by Nov 9, 2026

The mathematical context was checked on Aug 9, 2026. Status can be refreshed sooner after a material result or claim.

1
Über eine neue Art von L-Reihenoriginal source · Emil Artin · Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg · 1924-12 (work dated 1923) · DOI 10.1007/BF02954618 · accessed Aug 9, 2026
2
Zur Theorie der L-Reihen mit allgemeinen Gruppencharakterenoriginal source · Emil Artin · Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg · 1931-12 (work dated 1930) · DOI 10.1007/BF02941010 · accessed Aug 9, 2026
3
On Artin's L-Series with General Group Characterspeer reviewed result · Richard Brauer · Annals of Mathematics · 1947-04 · DOI 10.2307/1969183 · accessed Aug 9, 2026
4
Base Change for GL(2)peer reviewed result · Robert P. Langlands · Princeton University Press · 1980 · accessed Aug 9, 2026
5
Artin's Conjecture for Representations of Octahedral Typepeer reviewed result · Jerrold Tunnell · Bulletin of the American Mathematical Society · 1981-09 · DOI 10.1090/S0273-0979-1981-14936-3 · accessed Aug 9, 2026
6
Poles of Artin L-functions and the strong Artin conjecturepeer reviewed result · Andrew R. Booker · Annals of Mathematics · 2003 · DOI 10.4007/annals.2003.158.1089 · MR MR2031863 · accessed Aug 9, 2026
7
Artin's conjecture, Turing's method and the Riemann hypothesispeer reviewed result · Andrew R. Booker · Experimental Mathematics · 2006 · ARXIV math/0507502 · accessed Aug 9, 2026
8
Serre's modularity conjecture (I)peer reviewed result · Chandrashekhar Khare, Jean-Pierre Wintenberger · Inventiones mathematicae · 2009-07-04 · DOI 10.1007/s00222-009-0205-7 · accessed Aug 9, 2026
9
Serre's modularity conjecture (II)peer reviewed result · Chandrashekhar Khare, Jean-Pierre Wintenberger · Inventiones mathematicae · 2009-07-04 · DOI 10.1007/s00222-009-0206-6 · accessed Aug 9, 2026
10
On Artin L-functionssurvey or monograph · James W. Cogdell · EMS Press · 2015 · accessed Aug 9, 2026
11
On holomorphy and non-vanishing of Artin L-functionspeer reviewed result · Sanoli Gun, Suhita Hazra, Dhananjaya Sahu · Monatshefte für Mathematik · 2025-02-17 · DOI 10.1007/s00605-025-02060-7 · accessed Aug 9, 2026
12
Artin's Conjecturesauthoritative webpage · M. Ram Murty · Fields Institute for Research in Mathematical Sciences · 2025-03-12 · accessed Aug 9, 2026
13
Mathlib.NumberTheory.ArithmeticFunction.LFunctionformalization · Mathlib · accessed Aug 9, 2026
14
Mathlib.NumberTheory.LSeries.DirichletContinuationformalization · Mathlib · accessed Aug 9, 2026
15
Formal Conjecturesformalization · Google DeepMind · Google DeepMind · accessed Aug 9, 2026
16
Artin L-functionencyclopedia · Wikipedia · accessed Aug 9, 2026

Important qualifications

This record covers the classical number-field Artin holomorphy conjecture for nontrivial irreducible finite-image complex Galois representations. It does not cover Artin's primitive-root conjecture, p-adic Artin L-functions, the Artin–Tate conjecture, or conjectures about Artin groups.
The general conjecture, strong Artin, individual automorphy theorems, and finite-height numerical verification are distinct. No result was generalized beyond its stated base field, dimension, parity, image, or hypothesis.
Monomial and solvable were not treated as synonyms. Artin holomorphy is known for monomial representations, while a solvable finite group need not be monomial; recent pointwise criteria for solvable extensions do not prove the general conjecture for all solvable groups.
Artin formulated the conjecture in work conventionally dated 1923 but published by the journal in December 1924. His completed local-factor paper is conventionally dated 1930 in mathematical references but was published by the journal in December 1931.
The scoped formalization review checked current Mathlib documentation and the Google DeepMind Formal Conjectures repository tree at commit f205adb8a8f83f37c543c955cf6bc689baf0d3f4. It found relevant Dirichlet-L-function support but no target-level Artin-holomorphy path; this does not establish nonexistence in all proof assistants, branches, private repositories, or differently named files.
Booker's linked finite-height computation for selected S5 representations was not rerun or independently reproduced in this collection. It is not a general certificate and does not prove the unrestricted conjecture.
No canonical dataset, standalone proof certificate, or generally reusable implementation for the full conjecture was identified in this bounded pass. Empty categories or scoped negative searches do not establish universal nonexistence.
No current Clay, Hilbert-numbered, Smale, Erdős, Epoch/FrontierMath, named-prize, or comparable maintained selective-list membership for the same statement was verified. Wikipedia remains in the current research map only as encyclopedia recognition metadata, not as status authority.
The literature on Artin L-functions, functoriality, potential automorphy, and low-dimensional Galois representations is extensive. This bounded record selects representative milestones and is not a comprehensive bibliography.
No unreviewed source material, contributor claim, model-generated mathematical assertion, or local research route was used as external authority. This metadata grants no proof, novelty, review, acceptance, publication, or deployment status.

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