Ramanujan–Petersson conjecture

E355436

conjecture in number theory mathematical conjecture

The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.

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All labels observed (6)

Label	Occurrences
Ramanujan conjecture for the tau function	1
Ramanujan conjectures	1
Ramanujan–Petersson conjecture canonical	1
Ramanujan–Petersson conjecture (proved by Deligne)	1
generalized Ramanujan conjecture	1
generalized Ramanujan–Petersson conjecture for GL(n)	1

Statements (47)

Predicate	Object
instanceOf	conjecture in number theory ⓘ mathematical conjecture ⓘ
concerns	Hecke eigenforms ⓘ holomorphic cusp forms of weight k for SL(2,Z) ⓘ
connectedTo	eigenvalues of the Laplacian on modular curves ⓘ spectral theory of automorphic forms ⓘ
equivalentTo	temperedness of local components of automorphic representations of GL(2) ⓘ
extendedBy	Petersson’s work on Fourier coefficients of cusp forms ⓘ
field	number theory ⓘ theory of modular forms ⓘ
formulatedInContextOf	Hecke eigenforms with multiplicative Fourier coefficients ⓘ modular forms for SL(2,Z) ⓘ
generalizationOf	Ramanujan–Petersson conjecture self-linksurface differs ⓘ surface form: Ramanujan conjecture for the tau function
hasConsequence	improved error terms in arithmetic counting problems ⓘ subconvexity bounds for certain L-functions ⓘ
hasLocalForm	bounds on Satake parameters ⓘ
historicalOrigin	Ramanujan’s 1916 conjectures on the tau function ⓘ
holdsFor	holomorphic cusp forms of any integral weight k ≥ 2 ⓘ
implies	Fourier coefficients of normalized Hecke eigenforms are bounded by n^{(k-1)/2+ε} ⓘ
inspired	Ramanujan–Petersson conjecture self-linksurface differs ⓘ surface form: generalized Ramanujan–Petersson conjecture for GL(n)
language	algebraic geometry ⓘ complex analysis ⓘ representation theory ⓘ
motivation	understanding arithmetic properties of modular forms ⓘ
namedAfter	Hans Petersson ⓘ Srinivasa Ramanujan ⓘ
openVariant	generalized Ramanujan conjecture for Maass forms ⓘ
predicts	Deligne bound for Fourier coefficients of modular forms ⓘ growth conditions on Fourier coefficients ⓘ strong bounds on Fourier coefficients of cusp forms ⓘ
proofUses	Weil conjectures ⓘ étale cohomology ⓘ
proofYear	1974 ⓘ
provedBy	Pierre Deligne ⓘ
relatedProblem	Selberg eigenvalue conjecture ⓘ
relatedTo	Hecke operators ⓘ Langlands program ⓘ Ramanujan tau function ⓘ automorphic forms ⓘ Ramanujan–Petersson conjecture self-linksurface differs ⓘ surface form: generalized Ramanujan conjecture
status	proved for holomorphic modular forms of integral weight ⓘ
subject	Fourier coefficients of cusp forms ⓘ Fourier coefficients of modular forms ⓘ
type	growth conjecture ⓘ
usedIn	analytic number theory ⓘ bounds for exponential sums ⓘ estimates for L-functions ⓘ

Referenced by (6)

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

Srinivasa Ramanujan → notableWork → Ramanujan–Petersson conjecture ⓘ

Ramanujan tau function → satisfies → Ramanujan–Petersson conjecture ⓘ

this entity surface form: Ramanujan–Petersson conjecture (proved by Deligne)

Ramanujan tau function → relatedTo → Ramanujan–Petersson conjecture ⓘ

this entity surface form: Ramanujan conjectures

Ramanujan–Petersson conjecture → relatedTo → Ramanujan–Petersson conjecture self-linksurface differs ⓘ

this entity surface form: generalized Ramanujan conjecture

Ramanujan–Petersson conjecture → generalizationOf → Ramanujan–Petersson conjecture self-linksurface differs ⓘ

this entity surface form: Ramanujan conjecture for the tau function

Ramanujan–Petersson conjecture → inspired → Ramanujan–Petersson conjecture self-linksurface differs ⓘ

this entity surface form: generalized Ramanujan–Petersson conjecture for GL(n)