T-adic exponential sums over finite fields

We introduce T -adic exponential sums associated to a Laurent polynomial f .They interpolate all classical p m -power order exponential sums associated to f .We establish the Hodge bound for the Newton polygon of L-functions of T -adic exponential sums.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 that is ordinary for m = 1.We also study deeper properties of L-functions of T -adic exponential sums.Along the way, we discuss new open problems about the T -adic exponential sum itself.