Petersson inner product

Definition

Let \mathbb{M}_k be the space of entire modular forms of weight k and \mathbb{S}_k the space of cusp forms.

The mapping \langle \cdot , \cdot \rangle : \mathbb{M}_k \times \mathbb{S}_k \rightarrow \mathbb{C},

:\langle f , g \rangle := \int_\mathrm{F} f(\tau) \overline{g(\tau)}

(\operatorname{Im}\tau)^k d\nu (\tau)

is called Petersson inner product, where

:\mathrm{F} = \left{ \tau \in \mathrm{H} : \left| \operatorname{Re}\tau \right| \leq \frac{1}{2}, \left| \tau \right| \geq 1 \right}

is a fundamental region of the modular group \Gamma and for \tau = x + iy

:d\nu(\tau) = y^{-2}dxdy

is the hyperbolic volume form.

Properties

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

For the Hecke operators T_n, and for forms f,g of level \Gamma_0, we have:

:\langle T_n f , g \rangle = \langle f , T_n g \rangle,

i.e., the T_n are self-adjoint with respect to the Petersson inner product. This can be used to show that the space of cusp forms of level \Gamma_0 has an orthonormal basis consisting of simultaneous eigenfunctions for the Hecke operators and the Fourier coefficients of these forms are all real.

References

• T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer Verlag Berlin Heidelberg New York 1990,
• M. Koecher, A. Krieg, Elliptische Funktionen und Modulformen, Springer Verlag Berlin Heidelberg New York 1998,
• S. Lang, Introduction to Modular Forms, Springer Verlag Berlin Heidelberg New York 2001,