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 \( {\displaystyle S_{k}(\Gamma (1))} \), the space of cusp forms of weight k>2 on \( SL_{2}({\mathbb {Z}}) \). Then for any positive integers m,n we have
\( {\displaystyle {\frac {\Gamma (k-1)}{(4\pi {\sqrt {mn}})^{k-1}}}\sum _{f\in {\mathcal {F}}}{\bar {\hat {f}}}(m){\hat {f}}(n)=\delta _{mn}+2\pi i^{-k}\sum _{c>0}{\frac {S(m,n;c)}{c}}J_{k-1}\left({\frac {4\pi {\sqrt {mn}}}{c}}\right),} \)
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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