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:Σ ''f''(Λ′) 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 forms are particular kinds of functions of a lattice, subject to conditions making them Analytic Function s and Homogeneous with respect to Enlargement of a lattice; these conditions are preserved by the summation and so Hecke operators take modular forms to modular forms. Algebras of Hecke operators are called Hecke algebras, and the most significant basic fact of the theory is that these are Commutative Ring s. The theory of Hecke operators on modular forms is often said to have been founded by Mordell in a paper on the special Cusp Form of Ramanujan , ahead of the general theory given by Erich Hecke . The idea may be considered to go back to earlier work of Hurwitz , who treated Correspondence s between Modular Curve s which realise some individual Hecke operator. In fact the algebraic theory of correspondences ( Relation s closed for the Zariski Topology ) is another and natural way to express the Formal Sum involved. A third way to express Hecke operators is as Double Coset s in the Modular Group . In the contemporary Adelic approach this translates to double cosets with respect to some compact subgroups. In any case, 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 it is shown that the Hecke operators are a C-star Algebra with respect to the Petersson Inner Product ; and that therefore the Spectral Theory implies that there is a basis of modular forms that are Eigenfunction s for all Hecke operators. These basic forms all have Euler Product s (more precisely, their Mellin Transform s are Dirichlet Series that have Euler products with the factor for each prime ''p'' being of degree 2). In the case treated by Mordell, there is a one-dimensional space of cusp forms of weight 12; so that the Euler product must apply to Ramanujan's form. HECKE ALGEBRAS IN GENERAL Other mathematical rings are called ''Hecke algebras'', without the obvious link to Hecke operators. These include certain quotients of the Group Algebra of a Braid Group . REFERENCES
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