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	~~In [[mathematics]], in particular in the theory of [[modular form]]s, a '''Hecke operator''', studied by {{harvs\|txt\|authorlink=Erich Hecke\|last=Hecke\|year=1937}},~~ is ~~a certain kind of "averaging" operator that plays a significant role in~~ the ~~structure of [[vector space]]s of modular forms~~ and ~~more general [[automorphic representation]]s~~.		Jerrie Swoboda is what the public can call me and simply I totally dig where it name. Managing people is my day job now. As a girl what I do like is to have croquet but I can't make it my vocation really. My his conversation and I chose to maintain in Massachusetts. Go to my web property to find out more: http://circuspartypanama.com<br><br>My site; [http://circuspartypanama.com clash of clans cheats ipod]

	~~== History ==~~

	{{harvs\|txt\|last=Mordell\|authorlink=Mordell\|year=1917}} used Hecke operators on modular forms in a paper on the special [[cusp form]] of [[Ramanujan]], ahead of the general theory given by {{harvtxt\|Hecke\|1937}}. Mordell proved that the [[Ramanujan tau function]], expressing the coefficients of the Ramanujan form,

	~~: <math> \Delta(q)=q\left(\prod_{n=1}^{\infty}(1-q^n)\right)^{24}=~~
	~~\sum_{n=1}^{\infty} \tau(n)q^n, \quad q=e^{2\pi i\tau}, </math>~~

	is ~~a [[multiplicative function]]:~~

	~~: <math> \tau(mn)=\tau(m)\tau(n) \quad \text{ for } (m,n)=1. </math>~~

	~~The idea goes back to earlier work of [[Hurwitz]], who treated [[Correspondence (mathematics)\|algebraic correspondence]]s between [[modular curve]]s which realise some individual Hecke operators~~.

	~~== Mathematical description ==~~

	~~Hecke operators can be realised in~~ a ~~number of contexts. The simplest meaning~~ is ~~combinatorial, namely as taking for a given integer ''n'' some function ''f''(Λ) defined on the [[lattice (group)\|lattice]]s of fixed rank~~ to

	~~:<math>\sum f(\Lambda')</math>~~

	~~with the sum taken over all the Λ′ that are [[subgroup]]s of Λ of index ''n'~~'. ~~For example, with ''n=2''~~ and two dimensions, there are three such Λ′. [[Modular form]]s are particular kinds of functions of a lattice, subject to conditions making them [[analytic function]]s and [[homogeneous function\|homogeneous]] with respect to ~~[[homothety\|homotheties]], as well as moderate growth at infinity; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given weight.~~

	~~Another way to express Hecke operators is by means of [[double coset]]s~~ in ~~the [[modular group]]~~. ~~In the contemporary [[adelic]] approach, this translates~~ to ~~double cosets with respect~~ to ~~some compact subgroups.~~

	~~=== Explicit formula ===~~

	Let ''M''<sub>''m''</sub> be the set of 2×2 integral matrices with [[determinant]] ''m'' and ''Γ'' = ''M''<sub>1</sub> be the full [[modular group]] ''SL''(2, '''Z'''). Given a modular form ''f''(''z'') of weight ''k'', the ''m''th Hecke operator acts by the formula

	: ~~<math> T_m f(z) = m^{k-1}\sum_{\left(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\right)\in\Gamma\backslash M_m}(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right), </math>~~

	where ''z'' is in the [[upper half-plane]] and the normalization constant ''m''<sup>''k''−1</sup> assures that the image of a form with integer Fourier coefficients has integer Fourier coefficients. This can be rewritten in the form

	~~: <math> T_m f(z) = m^{k-1}\sum_{a,d>0, ad=m}\frac{1}{d^k}\sum_{b \pmod d} f\left(\frac{az+b}{d}\right), </math>~~

	which leads to the formula for the Fourier coefficients of ''T''<sub>''m''</sub>''f''(''z'') = ∑ ''b''<sub>''n''</sub>''q''<sup>''n''</sup> in terms of the Fourier coefficients of ''f''(''z'') = ∑ ''a''<sub>''n''</sub>''q''<sup>''n''</sup>:

	~~: <math> b_n = \sum_{r>0, r\|(m,n)}r^{k-1}a_{mn/r^2}.</math>~~

	One can see from this explicit formula that Hecke operators with different indices commute and that if ''a''<sub>0</sub> = 0 then ''b''<sub>0</sub> = 0, so the subspace ''S''<sub>''k''</sub> of cusp forms of weight ''k'' is preserved by the Hecke operators. If a (non-zero) cusp form ''f'' is a [[Eigenform\|simultaneous eigenform]] of all Hecke operators ''T''<sub>''m''</sub> with eigenvalues ''λ''<sub>''m''</sub> then ''a''<sub>''m''</sub> = ''λ''<sub>''m''</sub>''a''<sub>1</sub> and ''a''<sub>1</sub> ≠ 0. Hecke eigenforms are '''normalized''' so that ''a''<sub>1</sub> = 1, then

	: ~~<math> T_m f = a_m f, \quad a_m a_n = \sum_{r>0, r\|(m,n)}r^{k-1}a_{mn~~/~~r^2},\ m,n\geq 1. <~~/~~math>~~

	~~Thus for normalized cuspidal Hecke eigenforms of integer weight, their Fourier coefficients coincide with their Hecke eigenvalues~~.

	~~== Hecke algebras ==~~

	Algebras of Hecke operators are called '''Hecke algebras''', and are [[commutative ring]]s. Other, related, mathematical rings are called [[Hecke algebras]], although the link to Hecke operators is not entirely obvious. These algebras include certain quotients of the [[group algebra]]s of [[braid group]]s. The presence of this commutative operator algebra plays a significant role in the [[harmonic analysis]] of modular forms and generalisations. In the classical [[elliptic modular form]] theory, the Hecke operators ''T''<sub>''n''<~~/sub~~> with ''n'' coprime to the level acting on the space of cusp forms of a given weight are [[self-adjoint operator\|self-adjoint]] with respect to the [[Petersson inner product]]. Therefore, the [[spectral theorem]] implies that there is a basis of modular forms that are [[eigenfunction]]s for these Hecke operators. Each of these basic forms possesses an [[Euler product]]. More precisely, its [[Mellin transform]] is the [[Dirichlet series]] that has [[Euler product]]s with the local factor for each prime ''p'' is the inverse of the '''Hecke polynomial''', a quadratic polynomial in ''p''<sup>−''s''<~~/sup~~>. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of ''τ~~''(''n'').~~

	~~== See also ==~~

	* [[Eichler–Shimura congruence relation]]

	~~== References ==~~

	* {{Citation \| last1=Apostol \| first1=Tom M. \| author1-link=Tom M. Apostol \| title=Modular functions and Dirichlet series in number theory \| publisher=[[~~Springer-Verlag]] \| location=Berlin, New York \| edition=2nd \| isbn=978-0-387-97127-8 \| year=1990}} ''(See chapter 8.)''~~

	*{{springer\|title=Hecke operator\|id=p/h130060}}

	*{{Citation \| last1=Hecke \| first1=E. \| title=Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I. \| language=German \| doi=10.1007/BF01594160 \| zbl=0015.40202 \| year=1937 \| journal=[[Mathematische Annalen]] \| issn=0025-5831 \| volume=114 \| pages=1–28}} {{Citation \| last1=Hecke \| first1=E. \| title=Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II. \| language=German \| doi=10.1007/BF01594180 \| zbl=0016.35503 \| year=1937 \| journal=[[Mathematische Annalen]] \| issn=0025-5831 \| volume=114 \| pages=316–351}}

	*{{Citation \| last1=Mordell \| first1=Louis J. \| author1-link=Louis Mordell \| title=On Mr. Ramanujan's empirical expansions of modular functions. \| url=http://~~www~~.~~archive.org/stream/proceedingsofcam1920191721camb#page/n133 \| jfm=46.0605.01 \| year=1917 \| journal=[[Proceedings~~ of ~~the Cambridge Philosophical Society]] \| volume=19 \| pages=117–124}}~~

	* [[Jean-Pierre Serre]], ''A course in arithmetic''.

	* [[Don Zagier]], ''Elliptic Modular Forms and Their Applications'', in ''The 1-2-3 of Modular Forms'', Universitext, Springer, 2008 ISBN 978-3-540-74117-6

	~~[[Category:Modular forms]~~]

en>Spinningspark: Reverted good faith edits by 41.67.21.53 (talk). (TW)

2014-02-02T10:09:42Z

Reverted good faith edits by 41.67.21.53 (talk). (TW)

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en>Headbomb: /* See also */ {{Wikipedia books|Resonance}}

2012-07-31T03:31:32Z

See also: {{Wikipedia books|Resonance}}

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