Petersson dot product

In mathematics , the Petersson scalar product is understood to be a specific scalar product on the vector space of all modular forms . This scalar product was introduced by Hans Petersson .

definition

Let it be the vector space of the whole modular forms for weight and the vector space of the tip forms . ${\displaystyle \mathbb {M} _{k}}$ ${\displaystyle k}$ ${\displaystyle \mathbb {S} _{k}}$

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

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

is called the Petersson dot product. It is

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

the fundamental domain of the modular group , and is ${\displaystyle \Gamma }$ ${\displaystyle \tau =x+iy}$

${\displaystyle {\rm {d}}\nu (\tau )=y^{-2}{\rm {d}}x{\rm {d}}y}$

the hyperbolic volume element. Note that one can formally insert a whole modular form into the above formula for one of the two components of the scalar product , because the integral also converges then. However, in the definition of a scalar product, both components must come from the same vector space, which is why the Petersson scalar product is usually defined in the above form. ${\displaystyle \mathbb {M} _{k}}$

properties

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

The following applies to the hedge operators ${\displaystyle T_{n}}$

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

It can thus be shown that the vector space of the tip shapes has an orthonormal basis of simultaneous eigenforms to the Hecke operators and that the Fourier coefficients of these shapes are all real.

literature

• TM 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: Elliptical functions and modular forms . 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 .