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Hecke Operators

  • Anatoli AndrianovEmail author
Chapter
Part of the Universitext book series (UTX)

4.1 Hecke Operators for Congruence Subgroups

As has been indicated in Section 3.1, the main reason for the introduction of Hecke operators was to reveal and explain multiplicative properties of Fourier coefficients of modular forms. The general scheme looks as follows: Hecke–Shimura rings of symplectic groups operate on spaces of modular forms by Hecke operators, and the multiplicative properties of Fourier coefficients of modular forms appear just as reflections of the multiplicative relations in the corresponding Hecke–Shimura rings. In this section we shall first define Hecke operators on an abstract level and then consider Hecke operators for congruence subgroups of the modular group including the questions of invariant subspaces and diagonalization of regular operators.

Abstract Hecke Operators. The definition of Hecke operators on an abstract level looks quite simple and natural. Let D(Λ,Σ) be a Hecke–Shimura ring (3.2), and V a \(\mathbb{Z}\)

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Russian Academy of SciencesSteklov Institute of MathematicsPetersburgRussia

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