Euler products for automorphic L-functions

E304347

Euler product L-function theory object mathematical concept

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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All labels observed (5)

Label	Occurrences
Euler product expansions	2
Euler Products	1
Euler products	1
Euler products for automorphic L-functions canonical	1
Langlands L-functions	1

Statements (50)

Predicate	Object
instanceOf	Euler product ⓘ L-function theory object ⓘ mathematical concept ⓘ
associatedTo	automorphic representations ⓘ automorphic representations of reductive groups over global fields ⓘ cuspidal automorphic representations ⓘ
builtFrom	Hecke operators ⓘ Satake isomorphism ⓘ Langlands program ⓘ surface form: local Langlands correspondence
definedOver	function fields ⓘ global fields ⓘ number fields ⓘ
encodes	Hecke eigenvalues ⓘ Satake parameters ⓘ arithmetic information ⓘ local factors at primes ⓘ local-global compatibility ⓘ
field	Langlands program ⓘ automorphic forms ⓘ number theory ⓘ
generalizes	Dirichlet L-functions ⓘ surface form: Dirichlet L-function Euler products Euler product of the Riemann zeta function ⓘ
hasComponent	archimedean local factors ⓘ local L-factors ⓘ non-archimedean local factors ⓘ
hasProperty	absolute convergence in some right half-plane ⓘ local factors often given by characteristic polynomials of Frobenius elements ⓘ local factors rational in p^{-s} ⓘ meromorphic continuation expected to entire plane except possible poles ⓘ reflects unramified and ramified behavior at primes ⓘ
motivatedBy	classical Euler product for the Riemann zeta function ⓘ
relatedTo	Artin L-functions ⓘ Euler products for automorphic L-functions self-linksurface differs ⓘ surface form: Langlands L-functions L-functions ⓘ surface form: Rankin–Selberg L-functions automorphic L-functions ⓘ functoriality in the Langlands program ⓘ standard L-functions of GL(n) ⓘ symmetric power L-functions ⓘ
satisfies	Euler product factorization over all places ⓘ analytic continuation conjecturally ⓘ functional equation conjecturally ⓘ multiplicativity of local factors ⓘ
studiedIn	algebraic number theory ⓘ arithmetic geometry ⓘ automorphic representation theory ⓘ
usedFor	modularity and reciprocity questions ⓘ non-vanishing results for L-functions ⓘ relating automorphic representations and Galois representations ⓘ studying distribution of primes ⓘ subconvexity problems ⓘ

Referenced by (6)

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

Euler product formula for the Riemann zeta function → generalizedBy → Euler products for automorphic L-functions ⓘ

Multiplicative Number Theory → usesConcept → Euler products for automorphic L-functions ⓘ

this entity surface form: Euler products

Multiplicative Number Theory → hasKeyTool → Euler products for automorphic L-functions ⓘ

this entity surface form: Euler product expansions

Ilya Piatetski-Shapiro → notableWork → Euler products for automorphic L-functions ⓘ

this entity surface form: Euler Products

Euler’s method of rearranging absolutely convergent series → relatedTo → Euler products for automorphic L-functions ⓘ

this entity surface form: Euler product expansions

Euler products for automorphic L-functions → relatedTo → Euler products for automorphic L-functions self-linksurface differs ⓘ

this entity surface form: Langlands L-functions