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.

## Contents

* 1 Definition
* 2 Properties
* 3 See also
* 4 References

## Definition

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

The mapping [math]\displaystyle{ \langle \cdot , \cdot \rangle : \mathbb{M}_k \times \mathbb{S}_k \rightarrow \mathbb{C} }[/math],

[math]\displaystyle{ \langle f , g \rangle := \int_\mathrm{F} f(\tau) \overline{g(\tau)} (\operatorname{Im}\tau)^k d\nu (\tau) }[/math]

is called Petersson inner product, where

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

is a fundamental region of the modular group [math]\displaystyle{ \Gamma }[/math] and for [math]\displaystyle{ \tau = x + iy }[/math]

[math]\displaystyle{ d\nu(\tau) = y^{-2}dxdy }[/math]

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 [math]\displaystyle{ T_n }[/math], and for forms [math]\displaystyle{ f,g }[/math] of level [math]\displaystyle{ \Gamma_0 }[/math], we have:

[math]\displaystyle{ \langle T_n f , g \rangle = \langle f , T_n g \rangle }[/math]

This can be used to show that the space of cusp forms of level [math]\displaystyle{ \Gamma_0 }[/math] has an orthonormal basis consisting of simultaneous eigenfunctions for the Hecke operators and the Fourier coefficients of these forms are all real.

## See also

* Weil–Petersson metric

## References

* T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer Verlag Berlin Heidelberg New York 1990, ISBN:3-540-97127-0
* M. Koecher, A. Krieg, Elliptische Funktionen und Modulformen, Springer Verlag Berlin Heidelberg New York 1998, ISBN:3-540-63744-3
* S. Lang, Introduction to Modular Forms, Springer Verlag Berlin Heidelberg New York 2001, ISBN:3-540-07833-9

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