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,
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.
See also
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