Anomalous monism

2014-01-05T22:11:59Z

79.214.14.127: /* Davidson's classic argument for AM */ Judgemental statement removed

In [[mathematics]] the '''Petersson inner product''' is an [[inner product]] defined on the space
of entire [[modular form]]s. It was introduced by the German mathematician [[Hans Petersson]].

==Definition==

Let <math>\mathbb{M}_k</math> be the space of entire modular forms of weight <math>k</math> and
<math>\mathbb{S}_k</math> the space of [[cusp form]]s.

The mapping <math>\langle \cdot , \cdot \rangle : \mathbb{M}_k \times \mathbb{S}_k \rightarrow
\mathbb{C}</math>,

:<math>\langle f , g \rangle := \int_\mathrm{F} f(\tau) \overline{g(\tau)}

(\operatorname{Im}\tau)^k d\nu (\tau)</math>

is called Petersson inner product, where

:<math>\mathrm{F} = \left\{ \tau \in \mathrm{H} : \left| \operatorname{Re}\tau \right| \leq \frac{1}{2},
\left| \tau \right| \geq 1 \right\}</math>

is a fundamental region of the [[modular group]] <math>\Gamma</math> and for <math>\tau = x + iy</math>

:<math>d\nu(\tau) = y^{-2}dxdy</math>

is the hyperbolic volume form.

==Properties==

The integral is [[absolutely convergent]] and the Petersson inner product is a [[definite bilinear form|positive definite]] [[Hermite form]].

For the [[Hecke operator]]s <math>T_n</math>, and for forms <math>f,g</math> of level <math>\Gamma_0</math>, we have:

:<math>\langle T_n f , g \rangle = \langle f , T_n g \rangle</math>

This can be used to show that the space of cusp forms of level <math>\Gamma_0</math> has an orthonormal basis consisting of
simultaneous [[eigenfunction]]s for the Hecke operators and the [[Fourier 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

[[Category:Modular forms]]

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Anomalous monism