Gauss and Kloosterman Sums over Residue Rings of Algebraic Integers
by
Stan Gurak
University of San Diego

Let K be a field of degree n over Q with ring of integers O. The trace and norm maps for K/Q are denoted Tr and N, respectively. Fix an integer m=p₁^r₁ ...p_t^r_t > 1, and let h and c be Dirichlet characters modulo m of orders o(h) and o(c), respectively. Set z_m = exp(2pi/m) and let M be any ideal of O for which the Gauss sums

GM(c) = å a ∈ O/M*  c(Na) zmTr   a

and Kloosterman sums

RM(h, z)= å a ∈ O/M*  h(Na) zmTr   a+   z/Na       (z ∈ Z/mZ*)

are well-defined. The computation of G_M(c) and R_M(h, z) is shown to reduce to their determination for m=p^r, a power of a prime p, where M is comprised solely of ideals of O lying above p. In this setting I first explicitly determine G_M(c) for m=p^r (r > 1) generalizing Mauclaire's classical result for K=Q. Relying of the author's recent evaluation of Kloosterman sums for prime powers in p-adic fields, I then proceed to compute the Kloosterman sums R_M(h, z) here when o(h)|p-1. This determination generalizes Salie's result in the classical case K = Q with o(h)=1 or 2.

Date received: April 11, 2008