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| In [[mathematics]] the '''Petersson inner product''' is an [[inner product]] defined on the space
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| of entire [[modular form]]s. It was introduced by the German mathematician [[Hans Petersson]]. | |
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| ==Definition==
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| Let <math>\mathbb{M}_k</math> be the space of entire modular forms of weight <math>k</math> and
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| <math>\mathbb{S}_k</math> the space of [[cusp form]]s.
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| The mapping <math>\langle \cdot , \cdot \rangle : \mathbb{M}_k \times \mathbb{S}_k \rightarrow
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| \mathbb{C}</math>,
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| :<math>\langle f , g \rangle := \int_\mathrm{F} f(\tau) \overline{g(\tau)} | |
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| (\operatorname{Im}\tau)^k d\nu (\tau)</math>
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| is called Petersson inner product, where
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| :<math>\mathrm{F} = \left\{ \tau \in \mathrm{H} : \left| \operatorname{Re}\tau \right| \leq \frac{1}{2},
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| \left| \tau \right| \geq 1 \right\}</math>
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| is a fundamental region of the [[modular group]] <math>\Gamma</math> and for <math>\tau = x + iy</math>
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| :<math>d\nu(\tau) = y^{-2}dxdy</math>
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| is the hyperbolic volume form.
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| ==Properties==
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| The integral is [[absolutely convergent]] and the Petersson inner product is a [[definite bilinear form|positive definite]] [[Hermite form]].
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| For the [[Hecke operator]]s <math>T_n</math>, and for forms <math>f,g</math> of level <math>\Gamma_0</math>, we have:
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| :<math>\langle T_n f , g \rangle = \langle f , T_n g \rangle</math>
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| This can be used to show that the space of cusp forms of level <math>\Gamma_0</math> has an orthonormal basis consisting of
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| simultaneous [[eigenfunction]]s for the Hecke operators and the [[Fourier coefficients]] of these
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| forms are all real.
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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
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| * M. Koecher, A. Krieg, ''Elliptische Funktionen und Modulformen'', Springer Verlag Berlin Heidelberg New York 1998, ISBN 3-540-63744-3
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| * S. Lang, ''Introduction to Modular Forms'', Springer Verlag Berlin Heidelberg New York 2001, ISBN 3-540-07833-9
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| [[Category:Modular forms]]
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