Inanalytic number theory, theKuznetsov trace formula is an extension of thePetersson trace formula.

The Kuznetsov orrelative trace formula connectsKloosterman sums at a deep level with the spectral theory ofautomorphic forms. Originally this could have been stated as follows. Let

${\displaystyle g:\mathbb{R}\to \mathbb{R}}$

be a sufficiently "well behaved" function. Then one calls identities of the following typeKuznetsov trace formula:

${\displaystyle \sum _{c\equiv 0\text{mod}N}{c}^{-r}K(m,n,c)g\left(\frac{4\pi \sqrt{mn}}{c}\right)=\text{Integral transform}+\text{Spectral terms}.}$

The integral transform part is someintegral transform ofg and the spectral part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform ofg. The Kuznetsov trace formula was found by Kuznetsov while studying the growth of weight zero automorphic functions.^[1] Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases wherePierre Deligne's proof of theWeil conjectures was not applicable.

It was later translated by Jacquet to arepresentation theoretic framework. Let${\displaystyle G}$ be areductive group over anumber fieldF and${\displaystyle H\subset G}$ be a subgroup. While the usualtrace formula studies theharmonic analysis onG, the relative trace formula is a tool for studying the harmonic analysis on thesymmetric space${\displaystyle G/H}$. For an overview and numerous applications Cogdell, J.W. and I. Piatetski-Shapiro,The arithmetic and spectral analysis of Poincaré series, volume 13 ofPerspectives in mathematics. Academic Press Inc., Boston, MA, (1990).

• Kuznecov, N. V. (1980), "The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums",Matematicheskii Sbornik, Novaya Seriya, 111(153) (3):334–383,ISSN 0368-8666,MR 0568983

• Automorphic forms
• Spectral theory
• Theorems in analytic number theory

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• Short description matches Wikidata
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