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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.
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| == History ==
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| {{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,
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| : <math> \Delta(q)=q\left(\prod_{n=1}^{\infty}(1-q^n)\right)^{24}=
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| \sum_{n=1}^{\infty} \tau(n)q^n, \quad q=e^{2\pi i\tau}, </math>
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| is a [[multiplicative function]]: | |
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| : <math> \tau(mn)=\tau(m)\tau(n) \quad \text{ for } (m,n)=1. </math>
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| 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.
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| == Mathematical description ==
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| 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
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| :<math>\sum f(\Lambda')</math>
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| 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.
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| 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.
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| === Explicit formula ===
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| 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
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| : <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> | |
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| 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
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| : <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>
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| 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>:
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| : <math> b_n = \sum_{r>0, r|(m,n)}r^{k-1}a_{mn/r^2}.</math>
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| 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
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| : <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> | |
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| Thus for normalized cuspidal Hecke eigenforms of integer weight, their Fourier coefficients coincide with their Hecke eigenvalues.
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| == Hecke algebras ==
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| 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'').
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| == See also ==
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| * [[Eichler–Shimura congruence relation]]
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| == References ==
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| * {{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.)''
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| *{{springer|title=Hecke operator|id=p/h130060}}
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| *{{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}}
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| *{{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}}
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| * [[Jean-Pierre Serre]], ''A course in arithmetic''.
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| * [[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
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| [[Category:Modular forms]]
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