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T-adic exponential sums over finite fields

Abstract

We introduce T-adic exponential sums associated to a Laurent polynomial f. They interpolate all classical pm-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 pm-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.

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