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 ${\displaystyle \mathbb {M} _{k}}$ be the space of entire modular forms of weight ${\displaystyle k}$ and ${\displaystyle \mathbb {S} _{k}}$ the space of cusp forms.

The mapping ${\displaystyle \langle \cdot ,\cdot \rangle$ :\mathbb {M} _{k}\times \mathbb {S} _{k}\rightarrow \mathbb {C} } ,

${\displaystyle \langle f,g\rangle$ :=\int _{\mathrm {F} }f(\tau ){\overline {g(\tau )}}(\operatorname {Im} \tau )^{k}d\nu (\tau )}

is called Petersson inner product, where

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

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

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

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 ${\displaystyle T_{n}}$, and for forms ${\displaystyle f,g}$ of level ${\displaystyle \Gamma _{0}}$, we have:

${\displaystyle \langle T_{n}f,g\rangle =\langle f,T_{n}g\rangle }$

This can be used to show that the space of cusp forms of level ${\displaystyle \Gamma _{0}}$ 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, 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