Petersson inner product

InmathematicsthePetersson inner productis aninner productdefined on the space of entiremodular forms.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 ofcusp forms.

The mapping$\langle \cdot,\cdot \rangle:\mathbb {M} _{k}\times \mathbb {S} _{k}\rightarrow \mathbb {C}$,

$\langle f,g\rangle:=\int _{\mathrm {F} }f(\tau ){\overline {g(\tau )}}(\operatorname {Im} \tau )^{k}d\nu (\tau )$

is called Petersson inner product, where

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

is a fundamental region of themodular group$\Gamma$and for$\tau =x+iy$

$d\nu (\tau )=y^{-2}dxdy$

is the hyperbolic volume form.

Properties

The integral isabsolutely convergentand the Petersson inner product is apositive definiteHermitian form.

For theHecke operators$T_{n}$,and for forms$f,g$of level$\Gamma _{0}$,we have:

$\langle T_{n}f,g\rangle =\langle f,T_{n}g\rangle$

This can be used to show that the space of cusp forms of level$\Gamma _{0}$has an orthonormal basis consisting of simultaneouseigenfunctionsfor the Hecke operators and theFourier coefficientsof these forms are all real.

See also

• Weil–Petersson metric

References

• T.M. Apostol,Modular Functions and Dirichlet Series in Number Theory,Springer Verlag Berlin Heidelberg New York 1990,ISBN3-540-97127-0
• M. Koecher, A. Krieg,Elliptische Funktionen und Modulformen,Springer Verlag Berlin Heidelberg New York 1998,ISBN3-540-63744-3
• S. Lang,Introduction to Modular Forms,Springer Verlag Berlin Heidelberg New York 2001,ISBN3-540-07833-9