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Petersson trace formula

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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 be an orthonormal basis of , the space of cusp forms of weight on . Then for any positive integers we have

where is the Kronecker delta function, is the Kloosterman sum and is the Bessel function of the first kind.


References

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  • 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.