T-adic exponential sums over finite fields

Abstract

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.