Petersson inner product

In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson.

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:


 * $$\langle T_n f, g \rangle = \langle f , T_n g \rangle$$

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.