Definite quadratic form

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 $S_k(\mathrm{\Gamma }(1))$, the space of cusp forms of weight $k>2$ on $S{L}_2(\mathbb{\mathbb{Z}})$. Then for any positive integers $m,n$ we have

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

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.