Explicit values of multi-dimensional Kloosterman sums for prime powers, II

For any integer m > 1 fix ζm = exp(2πi/m), and let Z ∗ m denote the group of reduced residues modulo m. Let q = pα, a power of a prime p. The hyper-Kloosterman sums of dimension n > 0 are defined for q by R(d, q) = ∑ x1,...,xn∈Z∗ q ζ x1+···+xn+d(x1···xn) q (d ∈ Zq), where x−1 denotes the multiplicative inverse of x modulo q. Salie evaluated R(d, q) in the classical setting n = 1 for even q, and for odd q = pα with α > 1. Later, Smith provided formulas that simplified the computation of R(d, q) in these cases for n > 1. Recently, Cochrane, Liu and Zheng computed upper bounds for R(d, q) in the general case n > 0, stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for α > 1, relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author’s previous determination of these Kloosterman sums using character theory and p-adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums.