Artin’s conjecture on L-functions
E539691
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| 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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
automorphy lifting theorems ⓘ modularity of Galois representations ⓘ |
| assumes | Artin L-functions admit meromorphic continuation to the complex plane ⓘ |
| concerns |
Artin L-functions
NERFINISHED
ⓘ
Galois extensions of number fields ⓘ Galois representations NERFINISHED ⓘ |
| domain |
L-functions
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Langlands program NERFINISHED ⓘ Taniyama–Shimura conjecture NERFINISHED ⓘ generalized Riemann hypothesis NERFINISHED ⓘ modularity theorem NERFINISHED ⓘ |
| specialCaseProvedBy | Langlands–Tunnell theorem NERFINISHED ⓘ |
| 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
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Input
Subject: Artin’s conjecture on L-functions Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.