On sums of hyper-Kloosterman sums

Abstract A formula of Kuznetsov allows one to interpret a smooth sum of Kloosterman sums as a sum over the spectrum of GL ⁡ ( 2 ) {\operatorname{GL}(2)} automorphic forms. In this paper, we construct a similar formula for the first hyper-Kloosterman sums using GL ⁡ ( 3 ) {\operatorname{GL}(3)} automorphic forms, resolving a long-standing problem of Bump, Friedberg and Goldfeld. Along the way, we develop what are apparently new bounds for the order derivatives of the classical J -Bessel function, and we conclude with a discussion of the original method of Bump, Friedberg and Goldfeld.