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 [math]\displaystyle{ \mathcal{F} }[/math] be an orthonormal basis of [math]\displaystyle{ S_k(\Gamma(1)) }[/math], the space of cusp forms of weight [math]\displaystyle{ k\gt 2 }[/math] on [math]\displaystyle{ SL_2(\mathbb{Z}) }[/math]. Then for any positive integers [math]\displaystyle{ m,n }[/math] we have
- [math]\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 \gt 0}\frac{S(m,n;c)}{c} J_{k-1}\left(\frac{4\pi \sqrt{mn}}{c}\right), }[/math]
where [math]\displaystyle{ \delta }[/math] is the Kronecker delta function, [math]\displaystyle{ S }[/math] is the Kloosterman sum and [math]\displaystyle{ J }[/math] 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. 37. 2006. pp. 147–168. doi:10.1090/amsip/037/04. ISBN 9780821841983.
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