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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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 ⓘ

Referenced by (1)

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

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