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Would you please explain the importance of Hecke operators on modular forms? I am studying modular forms mostly on my own and I have a pretty good understanding up to Hecke operators. So, I just wonder why we care about them.

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up vote 7 down vote accepted

This is a layman answer. Hecke operators commute and are self-adjoint, hence the modular forms which are eigenvectors wrt. all Hecke operators form a basis of the space of all modular forms (and the same for cusp forms). If $f$ is such an eigenvector then the L-series corresponding to $f$ has multiplicative coefficients, i.e. it can be represented by an Euler product (over all primes). So AFAIK, Hecke operators are important for connections of modular forms with number theory.

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Here are a couple of reasons:

  1. Abstractly the space of modular forms (of a given weight and level) is just a vector space (finite dimensional). Hecke operators give you a very concrete set of (commuting) operators on this vector space and hence you can get some control on this spaces purely in terms of linear algebra.

  2. The coefficients of cusp forms (a very important subset) of modular forms contain arithmetic information. Historically there were many remarkable conjectures about them. The introduction of Hecke operator demystifies them and many of these conjectures become simple.

  3. one can study modular forms from the perspective of representation theory and geometry and in both these case Hecke operators arise as rather natural operations on this space. This suggests that Hecke operators are intimately connected to study of modular forms.

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As a very specific example, we can use Hecke operators and the theory of modular forms to quickly get Jacobi's Four Squares Formula.

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