Petersson inner product

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.

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.

• 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