Petersson inner product
mathematicsthe Petersson inner product is an inner productdefined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson.
Let be the space of entire modular forms of weight k and the space of
The mapping ,
is called Petersson inner product, where
is a fundamental region of the
modular groupand for
is the hyperbolic volume form.
The integral is
absolutely convergentand the Petersson inner product is a positive definite Hermite form.
Hecke operators we have:
This can be used to show that the space of cusp forms has an orthonormal basis consisting of simultaneous
eigenfunctions for the Hecke operators and the Fourier coefficientsof 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
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