The arithmetic Kuznetsov formula on $GL(3)$, II: The general case

TL;DR

This paper completes the derivation of the full Kuznetsov formula for SL(3,Z), generalizing Wallach's theorem to all signs and analyzing Kloosterman zeta functions, advancing the spectral theory of automorphic forms.

Contribution

It extends Wallach's expansion to all four signs for SL(3), completing the Kuznetsov formula and analyzing associated Kloosterman zeta functions in greater generality.

Findings

Derived the full Kuznetsov formula for SL(3,Z)

Generalized Wallach's theorem to all signs

Analyzed analytic continuation of Kloosterman zeta functions

Abstract

We obtain the last of the standard Kuznetsov formulas for $SL(3,\Bbb{Z})$. 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 $SL(3)$. 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.