# 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 $mathbb\{M\}\_k$ be the space of entire modular forms of weight k and $mathbb\{S\}\_k$ the space of

cusp form s.The mapping $langle\; cdot\; ,\; cdot\; angle\; :\; mathbb\{M\}\_k\; imes\; mathbb\{S\}\_k\; ightarrow\; mathbb\{C\}$,

:$langle\; f\; ,\; g\; angle\; :=\; int\_mathrm\{F\}\; f(\; au)\; overline\{g(\; au)\}$

(operatorname{Im} au)^k d u ( au)

is called Petersson inner product, where

:$mathrm\{F\}\; =\; left\{\; au\; in\; mathrm\{H\}\; :\; left|\; operatorname\{Re\}\; au\; ight|\; leq\; frac\{1\}\{2\},\; left|\; au\; ight|\; geq\; 1\; ight\}$

is a fundamental region of the

modular group $Gamma$ and for $au\; =\; x\; +\; iy$:$d\; u(\; au)\; =\; y^\{-2\}dxdy$

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 $T\_n$ we have::$langle\; T\_n\; f\; ,\; g\; angle\; =\; langle\; f\; ,\; T\_n\; g\; angle$

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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