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I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a short elegant method by Zagier. Where?

In short, I know how to deduce the Selberg trace formula from Arthur's trace formula at least in principle, how should one proceed to deduce the Kuznetsov formula from the arthur trace formula. What is the utility of the Kuznetsov formula? For which applications is this trace formula more suitable than the Selberg trace formula?

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Sorry to give a reference to my own paper, but perhaps what you are looking for is contained in section 2 of this paper; see also Theorem 1.3. The basic idea is that the Selberg and Kuznetsov trace formulae both involve spectral sums but with different weights. To get Selberg weights from Kuznetsov one needs to multiply by $L(1, \text{sym}^2 u_j)$ where $u_j$ is a Hecke-Maass form associated to the spectral parameter $t_j$. Thus one obtains a sum of Kloosterman sums of the form $\sum_{c} \sum_{n} \frac{S(n^2, 1;c)}{ c n}$ with some weight function. An application of Poisson summation in $n$ leads to values of quadratic Dirichlet $L$-functions at $1$ which by the class number formula can be expressed in terms of class numbers. These are then connected to the geometric side of the Selberg trace formula using Sarnak's thesis. In the paper referenced above this is carried out in the reverse direction.

In my personal experience, the Kuznetsov formula often gives stronger error terms with minimal extra work. The representation in terms of Kloosterman sums allows for algebraic geometry (the Weil bound) to enter the game as well as other ideas in the theory of exponential sums. The Kuznetsov formula is also very useful in the reverse direction for studying Kloosterman sums which occur naturally in many number-theoretic applications. One can examine the papers of Deshouillers-Iwaniec for many interesting examples.

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Theorem 1.3 is indeed wonderful. Is there a purely representation theoretic approach to derive this formula. What estimates for the Weyl law are possible via the Kuznetsov trace formula? –  Marc Palm Jul 7 '11 at 18:45
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I don't know about a representation theoretic approach. Xiaoqing Li and Peter Sarnak have a paper on using the Kuznetsov formula in the Weyl law which has some discussions you would find interesting. The paper is at math.princeton.edu/sarnak/SarnakNumberPaper04.pdf –  Matt Young Jul 7 '11 at 20:41
    
Thanks a lot for answering also my consecutive question. –  Marc Palm Jul 9 '11 at 18:08

The important point is that the Selberg trace formula includes the contribution of the one-dimensional representations, i.e. the nongeneric spectrum, whereas the Kuznetsov formula does not. Spectrally, the nongeneric spectrum contributes so much that it has a tendancy to obscure the contribution of cusp forms. That is the essential difference between the two trace formulae.

This is especially important with regard to the idea of beyond endoscopy (see Sarnak's letter http://publications.ias.edu/sarnak/paper/487). But see also the work of Frenkel, Langlands, and Ngo.

Rudnick's thesis (http://www.math.tau.ac.il/~rudnick/papers/myphdthesis.pdf) in principle discusses how to pass from the Kuznetsov formula to the trace formula. Essentially the passage is just an application of Poisson summation.

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