- Petersson inner product
In
mathematics the Petersson inner product is aninner product defined on the space of entiremodular form s. It was introduced by the German mathematicianHans Petersson .Definition
Let be the space of entire modular forms of weight k and the space of
cusp form s.The mapping ,
:
is called Petersson inner product, where
:
is a fundamental region of the
modular group and for:
is the hyperbolic volume form.
Properties
The integral is
absolutely convergent and the Petersson inner product is apositive definite Hermite form .For the
Hecke operator s we have::
This can be used to show that the space of cusp forms has an orthonormal basis consisting of simultaneous
eigenfunction s for the Hecke operators and theFourier 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
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