Supercharacters and mixed moments of Kloosterman sums

Abstract

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

Supercharacters, elliptic curves, and the sixth moment of Kloosterman sums

“…Mixed Kloosterman moments are also of interest and have been studied in [4,10,17,18]. From ( 8) and ( 9), we have…”

“…in which ( • p ) denotes the Legendre symbol modulo p. See [10] for simple proofs of the preceding and [4] for additional mixed-moment evaluations. An expression for V 5 (p) was found by Livne [20] and by Peters, Top, and van der Vlugt [22]:…”

“…in which b p is an integer with |b p | < 2p 3/2 that is derived from the Dedekind eta function. Consequently, V 6 (p) = 5p 4 + O(p 7/2 ) and hence K u = O(p 2/3 ). In 2010, Evans conjectured formulas for V 7 (p) and V 8 (p) [8,9], which were ultimately proved by Yun [27] by attaching Galois representations to Kloosterman sums.…”