Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees

Abstract

1.1. Davenport–Hasse identities of Gauss sums. Davenport and Hasse established an identity of Gauss sums in [3]. Let p be an odd prime and m > 1 a divisor of p − 1. Let η be a ramified character of order m on the multiplicative group Qp of the p-adic field Qp. Here by η being ramified we mean that it is nontrivial on R× p ; the order of η is by definition the smallest positive integer m such that η = 1. Then we know that the conductor exponent of η, denoted by a(η), is equal to 1. Let ψ be an additive character of Qp whose order is 0. Here the order of an additive character φ, denoted by n(φ), is the largest integer n such that the character φ is trivial on p−nRp. Let χ be any ramified multiplicative character on Qp satisfying a(χ) = a(χη) = . . . = a(χηm−1) = 1.