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 $ℱ$ be an orthonormal basis of $S_{k} (Γ (1))$ , the space of cusp forms of weight $k > 2$ on $S L_{2} (ℤ)$ . Then for any positive integers $m, n$ we have

\frac{Γ (k - 1)}{(4 π \sqrt{m n})^{k - 1}} \sum_{f \in ℱ} \bar{\hat{f}} (m) \hat{f} (n) = δ_{m n} + 2 π i^{- k} \sum_{c > 0} \frac{S (m, n; c)}{c} J_{k - 1} (\frac{4 π \sqrt{m n}}{c}),

where $δ$ 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.

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