Petersson inner product

In mathematics we mean by the Petersson scalar product a particular scalar product on the vector space of all modular forms. Was introduced this scalar product by Hans Petersson.

Definition

It should be the vector space of all modular forms of weight and the vector space of cusp forms.

The figure,

Ie Petersson scalar product. It is

The fundamental domain of the modular group, and is

The hyperbolic volume element. Note that you may use formally for one of the two components of the Skalarpodukts an entire modular forms from the above formula, because the integral converges even then. However, both components must come from the same vector space, which is why the Petersson scalar product usually defined in the above form in the definition of an inner product.

Properties

The integral is absolutely convergent, and the Petersson scalar product is a positive definite Hermitian form.

For the Hecke operators

Thus, it can be shown that the vector space of cusp forms has an orthonormal basis of simultaneous eigenforms for the Hecke operators and the Fourier coefficients of these forms are all real.