TITLE: What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?
QUESTION [7 upvotes]: 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?

REPLY [5 votes]: 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.