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EIGENFORM

Eigenform

In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form that is an eigenvector for all Hecke

Eigenform

Hecke operator

Linear operator acting on modular forms

operators. If a (non-zero) cusp form f {\displaystyle f} is a simultaneous eigenform of all Hecke operators T m {\textstyle T_{m}} with eigenvalues λ m {\textstyle

Hecke operator

Hecke_operator

Second-order cybernetics

Application of cybernetics to itself

suited to the organization mankind discovers in nature." The notion of eigenform is an example of a self-referential system that produces a stable form

Second-order cybernetics

Second-order_cybernetics

Epistemology of finance

Study of mathematical models in finance

second-order cybernetics and also features in economics with Kauffman's Eigenform of markets. MacKenzie noted that the study of economics does more than

Epistemology of finance

Epistemology_of_finance

Riemann hypothesis

Conjecture on zeros of the zeta function

it to all Automorphic L-functions, such as Mellin transforms of Hecke eigenforms. The Riemann hypothesis for Selberg class extends it rather for functions

Riemann hypothesis

Riemann_hypothesis

Serre's modularity conjecture

Conjecture in number theory

of complex conjugation has determinant -1. To any normalized modular eigenform f = q + a 2 q 2 + a 3 q 3 + ⋯ {\displaystyle f=q+a_{2}q^{2}+a_{3}q^{3}+\cdots

Serre's modularity conjecture

Serre's_modularity_conjecture

Modularity theorem

Relates rational elliptic curves to modular forms

Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond

Modularity theorem

Modularity_theorem

Eigen

Topics referred to by the same term

the Ruby programming language Eigenbehaviour, with its connection to eigenform and eigenvalue in cybernetics Eigenfunction, is any non-zero function

Eigen

Wiles's proof of Fermat's Last Theorem

1995 publication in mathematics

obvious is that given a modular form of a certain special type, a Hecke eigenform with eigenvalues in Q {\displaystyle \mathbb {Q} } , one also gets a homomorphism

Wiles's proof of Fermat's Last Theorem

Wiles's_proof_of_Fermat's_Last_Theorem

Ikeda lift

of the same parity. The Ikeda lift takes a Hecke eigenform of weight 2k for SL2(Z) to a Hecke eigenform in the space of Siegel modular forms of weight k+n

Ikeda lift

Ikeda_lift

Cuspidal representation

to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation. Let G {\displaystyle

Cuspidal representation

Cuspidal_representation

Modular elliptic curve

Mathematical concept

Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond

Modular elliptic curve

Modular_elliptic_curve

Maass wave form

Complex-valued smooth functions of the upper half plane (harmonic analysis topic)

{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )} as modular forms. They are eigenforms of the hyperbolic Laplace operator Δ {\displaystyle \Delta } defined on

Maass wave form

Maass_wave_form

Atkin–Lehner theory

Part of the theory of modular forms

operators, there exists a basis for the space of newforms consisting of eigenforms for the full Hecke algebra. Consider a Hall divisor e of N, which means

Atkin–Lehner theory

Atkin–Lehner_theory

List of German expressions in English

"Eigen-" in composita such as eigenfunction, eigenvector, eigenvalue, eigenform; in English "self-" or "own-". They are related concepts in the fields

List of German expressions in English

List_of_German_expressions_in_English

Drinfeld reciprocity

reciprocity, introduced by Drinfeld (1974), is a correspondence between eigenforms of the moduli space of Drinfeld modules and factors of the corresponding

Drinfeld reciprocity

Drinfeld_reciprocity

List of conjectures

Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces n/a 281 Reconstruction conjecture graph

List of conjectures

List_of_conjectures

John Smillie (mathematician)

American mathematician (born 1953)

Smillie, John; Weiss, Barak (2016). "Horocycle dynamics: New invariants and eigenform loci in the stratum H(1,1)". arXiv:1603.00808 [math.DS]. Chaika, Jon;

John Smillie (mathematician)

John_Smillie_(mathematician)

Waldspurger formula

}}.} Definition 7. Assume that, φ {\displaystyle \varphi } is a Hecke eigenform. Atkin–Lehner eigenvalue w α ℓ , φ = φ ( W α ℓ z ) φ ( z ) {\displaystyle

Waldspurger formula

Waldspurger_formula

Langlands–Tunnell theorem

\operatorname {tr} \rho (\operatorname {Frob} _{p})} . A normalized Hecke eigenform f ( q ) = ∑ a n q n {\displaystyle f(q)=\sum a_{n}q^{n}} assigns to the

Langlands–Tunnell theorem

Langlands–Tunnell_theorem

Kannan Soundararajan

American mathematician and professor (born 1973)

R. Holowinsky and K. Soundararajan, "Mass equidistribution for Hecke eigenforms," arXiv:0809.1636v1 K. Soundararajan, "Nonvanishing of quadratic Dirichlet

Kannan Soundararajan

Kannan_Soundararajan

Eisenstein ideal

Mathematical ideal related to a modular curve

of level 1 and weight 12.This space is 2 dimensional, spanned by the Eigenforms given by the Eisenstein series E12 and the modular discriminant Δ. The

Eisenstein ideal

Eisenstein_ideal

Saito–Kurokawa lift

modular forms of degree 2 and weight k. The L-functions (when f is a Hecke eigenforms) are related by L(s,σk(f)) = ζ(s − k + 2)ζ(s − k + 1)L(s, f). The Saito–Kurokawa

Saito–Kurokawa lift

Saito–Kurokawa_lift

Vijay Kumar Patodi

Indian mathematician (1945–1976)

Research. In the two papers based on his Ph.D. thesis, "Curvature and Eigenforms of the Laplace Operator" (Journal of Differential Geometry), and "An Analytical

Vijay Kumar Patodi

Vijay_Kumar_Patodi

List of things named after Erich Hecke

Hecke character Hecke congruence subgroup Hecke correspondence Hecke eigenform Hecke group Hecke L-function (disambiguation) Hecke operator Hecke ring

List of things named after Erich Hecke

List_of_things_named_after_Erich_Hecke

Arturo Carsetti

Italian philosopher (1940–2024)

was the international Coordinator of the Research Project "LNC, IPR, Eigenforms, self-organizing models and morphogenesis". He investigated in particular

Arturo Carsetti

Arturo_Carsetti

Congruence ideal

of level 1 and weight 12.This space is 2 dimensional, spanned by the Eigenforms given by the Eisenstein series E12 and the modular discriminant Δ. The

Congruence ideal

Congruence_ideal