T-adic exponential sums over finite fields

Abstract

T-adic exponential sums associated to a Laurent polynomial f are introduced. They interpolate all classical p m-power order exponential sums associated to f. The Hodge bound for the Newton polygon of L-functions of T-adic exponential sums is established. This bound enables us to determine, for all m, the Newton polygons of L-functions of p m-power order exponential sums associated to an f which is ordinary for m = 1. Deeper properties of L-functions of T-adic exponential sums are also studied. Along the way, new open problems about the T-adic exponential sum itself are discussed.