Petersson trace formula

In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula.

In its simplest form the Petersson trace formula is as follows. Let ${\mathcal {F}}$ be an orthonormal basis of $S_{k}(\Gamma (1))$, the space of cusp forms of weight $k>2$ on $SL_{2}(\mathbb {Z} )$. Then for any positive integers $m,n$ we have

${\frac {\Gamma (k-1)}{(4\pi {\sqrt {mn}})^{k-1}}}\sum _{f\in {\mathcal {F}}}{\bar {\hat {f}}}(m){\hat {f}}(n)=\delta _{mn}+2\pi i^{-k}\sum _{c>0}{\frac {S(m,n;c)}{c}}J_{k-1}\left({\frac {4\pi {\sqrt {mn}}}{c}}\right),$

where $\delta$ is the Kronecker delta function, $S$ is the Kloosterman sum and $J$ is the Bessel function of the first kind.

References

• Henryk Iwaniec: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics 17, American Mathematics Society, Providence, RI, 1991.
• "Petersson and Kuznetsov trace formulas". Lie Groups and Automorphic Forms. AMS/IP Studies in Advanced Mathematics. Vol. 37. 2006. pp. 147–168. doi:10.1090/amsip/037/04. ISBN 9780821841983.