The arithmetic Kuznetsov formula on GL(3), I: The Whittaker case

Abstract

We obtain the last of the standard Kuznetsov formulas for SLp3, Zq. In the previous paper, we were able to exploit the relationship between the positive-sign Bessel function and the Whittaker function to apply Wallach's Whittaker expansion; now we demonstrate the expansion of functions into Bessel functions for all four signs, generalizing Wallach's theorem for SLp3q. As applications, we again consider the Kloosterman zeta functions and smooth sums of Kloosterman sums. The new Kloosterman zeta functions pose the same difficulties as we saw with the positive-sign case, but for the negative-sign case, we obtain some analytic continuation of the unweighted zeta function and give a sort of reflection formula that exactly demonstrates the obstruction when the moduli are far apart. The completion of the remaining sign cases means this work now both supersedes the author's thesis and completes the work started in the original paper of Bump, Friedberg and Goldfeld.

The relative trace formula in analytic number theory

“…The invertibility of the Kuznetsov formula was the key to all applications in Section 3.2. First important steps in this direction were established in [Ye,Bu2], the most complete solution is contained in [Bu3], where a general test function can be put on the long Weyl element term. As one may expect, one of the difficulties lies in the fact that the spectral side contains not only the spherical spectrum, but all weights simultaneously.…”