Eisenstein series
E924212
Eisenstein series are special types of complex analytic functions on the upper half-plane (or more general symmetric spaces) that play a central role in the theory of modular and automorphic forms, connecting number theory, representation theory, and harmonic analysis.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
automorphic form
ⓘ
complex analytic function ⓘ mathematical object ⓘ modular form ⓘ |
| definedOn |
complex upper half-plane
ⓘ
symmetric space ⓘ |
| field |
harmonic analysis
ⓘ
number theory ⓘ representation theory ⓘ |
| hasProperty |
admits Fourier expansion
ⓘ
holomorphic in the upper half-plane for suitable parameters ⓘ invariant under modular group action ⓘ meromorphic continuation in complex parameter ⓘ satisfies functional equation ⓘ |
| introducedBy | Gotthold Eisenstein NERFINISHED ⓘ |
| relatedTo |
Bernoulli numbers
NERFINISHED
ⓘ
Dedekind zeta function NERFINISHED ⓘ Dirichlet series NERFINISHED ⓘ Fourier expansion ⓘ Hecke L-functions NERFINISHED ⓘ Hecke operators NERFINISHED ⓘ Hilbert modular forms NERFINISHED ⓘ L-functions NERFINISHED ⓘ Langlands Eisenstein series NERFINISHED ⓘ Langlands program NERFINISHED ⓘ Maass forms NERFINISHED ⓘ Poisson summation formula NERFINISHED ⓘ Rankin–Selberg method NERFINISHED ⓘ Riemann zeta function NERFINISHED ⓘ Selberg trace formula NERFINISHED ⓘ Siegel modular forms ⓘ automorphic forms ⓘ constant term formula ⓘ constant term of automorphic forms ⓘ cusp forms ⓘ elliptic modular forms ⓘ functional equation of L-functions ⓘ induced representations ⓘ modular curves ⓘ modular forms ⓘ parabolic subgroups ⓘ principal series representations ⓘ spectral decomposition of automorphic forms ⓘ spectral theory of automorphic forms ⓘ theta functions ⓘ |
| usedFor |
construction of L-functions
ⓘ
explicit formulas in modular form theory ⓘ proofs of functional equations ⓘ spectral decomposition of L2 of arithmetic quotients ⓘ study of special values of L-functions ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.