Artin’s conjecture on L-functions

E539691

mathematical conjecture unproven hypothesis in number theory

Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.

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All labels observed (3)

Label	Occurrences
Artin conjecture on nontrivial zeros	1
Artin holomorphy conjecture	1
Artin’s conjecture on L-functions canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T5658024 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Artin’s conjecture on L-functions
Context triple: [Emil Artin, notableWork, Artin’s conjecture on L-functions]

A. Artin reciprocity law
The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
B. Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
C. Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
D. Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
E. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Artin’s conjecture on L-functions
Target entity description: Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.

A. Artin reciprocity law
The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
B. Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
C. Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
D. Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
E. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ unproven hypothesis in number theory ⓘ
approach	Langlands functoriality ⓘ linked to: Langlands program automorphy lifting theorems ⓘ modularity of Galois representations ⓘ
assumes	Artin L-functions admit meromorphic continuation to the complex plane ⓘ
concerns	Artin L-functions ⓘ Galois extensions of number fields ⓘ Galois representations ⓘ
domain	L-functions ⓘ automorphic forms ⓘ representation theory of Galois groups ⓘ
excludes	trivial one-dimensional representation ⓘ
field	number theory ⓘ
focusesOn	nontrivial irreducible complex representations of finite Galois groups ⓘ
generalizedTo	arbitrary number fields ⓘ
hasConsequence	effective versions of Chebotarev density under additional analytic assumptions ⓘ zero-free regions for certain Artin L-functions under additional hypotheses ⓘ
implies	analytic continuation of nontrivial Artin L-functions to entire functions on the complex plane ⓘ information about splitting of primes in Galois extensions ⓘ nontrivial Artin L-functions have no poles in the complex plane ⓘ refinements of the Chebotarev density theorem ⓘ strong results on the distribution of primes in number fields ⓘ
influenced	development of the Langlands correspondence ⓘ research on modularity of Galois representations ⓘ
knownFor	central role in non-abelian class field theory ⓘ deep implications for the distribution of primes ⓘ deep implications for the structure of number fields ⓘ
namedAfter	Emil Artin ⓘ
objectType	L-function attached to a Galois representation ⓘ
oftenFormulatedOver	number field Q ⓘ
refines	the known meromorphic continuation of Artin L-functions ⓘ
relatedTo	Artin reciprocity law ⓘ Langlands program ⓘ Taniyama–Shimura conjecture ⓘ linked to: Taniyama–Shimura–Weil conjecture generalized Riemann hypothesis ⓘ modularity theorem ⓘ linked to: modularity theorem for elliptic curves over Q
specialCaseProvedBy	Langlands–Tunnell theorem ⓘ
specialCaseProvedFor	Galois representations attached to modular forms ⓘ two-dimensional odd Galois representations over Q with solvable image ⓘ
states	Artin L-functions attached to nontrivial irreducible Galois representations have no poles ⓘ every nontrivial Artin L-function is entire ⓘ
status	open problem ⓘ unproven in general ⓘ
subfield	algebraic number theory ⓘ analytic number theory ⓘ
yearProposed	1923 ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Emil Artin → notableWork → Artin’s conjecture on L-functions ⓘ

Artin L-functions → conjectureAbout → Artin holomorphy conjecture ⓘ

linked to: Artin’s conjecture on L-functions

Artin L-functions → conjectureAbout → Artin conjecture on nontrivial zeros ⓘ