T-adic L-functions of p-adic exponential sums

The T-adic deformation of p-adic exponential sums is defined. It interpolates all classical p-power order exponential sums over a finite field. Its generating L-function is a T-adic entire function. We give a lower bound for its T-adic Newton polygon and show that the lower bound is often sharp. We also study the variation of this L-function in an algebraic family, in particular, the T-adic version of the Adolphson-Sperber conjecture on generic ordinariness, Wan’s limiting conjecture on generic Newton polygon, and Dwork’s unit root conjecture. These raise a number of new questions.