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In [[Analytic number theory|analytic number theory]], the '''Kuznetsov trace formula''' is an extension of the [[Petersson trace formula]].
The Kuznetsov or ''relative trace'' formula connects [[Kloosterman sum]]s at a deep level with the spectral theory of [[Automorphic form|automorphic form]]s. Originally this could have been stated as follows. Let 

:<math> g: \mathbb{R}\rightarrow \mathbb{R} </math> 

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

:<math>
\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}.
</math>

The integral transform part is some [[Integral transform|integral transform]] of ''g'' 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 of ''g''. The Kuznetsov trace formula was found by Kuznetsov while studying the growth of weight zero automorphic functions.<ref>{{Cite journal | doi=10.1070/SM1981v039n03ABEH001518| title=Petersson's Conjecture for Cusp Forms of Weight Zero and Linnik's Conjecture. Sums of Kloosterman Sums| journal=Mathematics of the USSR-Sbornik| volume=39| issue=3| pages=299–342| year=1981| last1=Kuznecov| first1=N. V.| bibcode=1981SbMat..39..299K}}</ref> Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases where [[Biography:Pierre Deligne|Pierre Deligne]]'s proof of the [[Weil conjectures]] was not applicable. 

It was later translated by Jacquet to a [[Representation theory|representation theoretic]] framework. Let <math> G </math> be a [[Reductive group|reductive group]] over a number field ''F'' and <math> H\subset G </math> be a subgroup. While the usual [[Selberg trace formula|trace formula]] studies the [[Physics:Harmonic analysis|harmonic analysis]] on ''G'', the relative trace formula is a tool for studying the harmonic analysis on the [[Symmetric space|symmetric space]] <math> G/H </math>. For an overview and numerous applications Cogdell, J.W. and I. Piatetski-Shapiro, ''The arithmetic and spectral analysis of Poincaré series'', volume 13 of ''Perspectives in mathematics''. Academic Press Inc., Boston, MA, (1990).

== References ==

<references/>
*{{Citation | last1=Kuznecov | first1=N. V. | title=The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums |mr=568983 | year=1980 | journal=Matematicheskii Sbornik|series=Novaya Seriya | issn=0368-8666 | volume=111(153) | issue=3 | pages=334–383}}

{{Functional analysis}}

[[Category:Automorphic forms]]
[[Category:Spectral theory]]
[[Category:Theorems in analytic number theory]]

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