Power-moments of SL<inf>3</inf>(ℤ) Kloosterman sums

Abstract

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL2 and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL3 have been considered, in which analogous GL3-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL3(ℤ). We give formulas for the first three moments and a nontrivial bound for the fourth. © 2013 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.