GAUSS AND KLOOSTERMAN SUMS OVER RESIDUE RINGS OF ALGEBRAIC INTEGERS

Abstract

Let K be a field of degree n over Q, the field of rational numbers, with ring of integers O. Fix an integer m > 1, say with [Formula: see text] as a product of distinct prime powers, and let χ be a numerical character modulo m of conductor f(χ). Set ζm = exp (2πi/m) and let M be any ideal of O satisfying Tr M ⊆ mZ and N(1 + M) ⊆ 1 + f(χ)Z, where Tr and N are the trace and norm maps for K/Q. Then the Gauss sum [Formula: see text] is well-defined. If in addition N(1 + M) ⊆ 1 + mZ, then the Kloosterman sums [Formula: see text] are well-defined for any numerical character η ( mod m). The computation of GM(χ) and RM(η, z) is shown to reduce to their determination for m = pr, a power of a prime p, where M is comprised solely of ideals of K lying above p. In this setting we first explicitly determine GM(χ) for m = pr (r > 1) generalizing Mauclaire's classical result for K = Q. Relying on the recent evaluation of Kloosterman sums for prime powers in p-adic fields, we then proceed to compute the Kloosterman sums RM(η, z) here for m = pr (r > 1) when o(η) | p -1. This determination generalizes Salie's result in the classical case K = Q with o(η) = 1 or 2.