Exponential Sums and Stationary Phase (I)

Abstract

With a view toward applications to number theory, coding theory, and combinatorics, we consider in this article Gauss sums defined over finite rings, especialy those arising from p-adic fields. The principal theme is the application of the classical method of stationary phase in a number theoretic context. After surveying the well known results of Kummer, Lamprecht, and Dwork, we indicate some generalizations which include and extend certain results of Hua in the one variable case, and Igusa in the case of several variables. In general, Gauss sums over finite rings possess asymptotic expansions whose leading term can be expressed in terms of quadratic Gauss sums. The principal tool turns out to be an arithmetic analogue of Morse’s Lemma.