Petersson inner product

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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

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Let   be the space of entire modular forms of weight   and   the space of cusp forms.

The mapping  ,

 

is called Petersson inner product, where

 

is a fundamental region of the modular group   and for  

 

is the hyperbolic volume form.

Properties

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The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form.

For the Hecke operators  , and for forms   of level  , we have:

 

This can be used to show that the space of cusp forms of level   has an orthonormal basis consisting of simultaneous eigenfunctions for the Hecke operators and the Fourier coefficients of these forms are all real.

See also

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References

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  • 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