T-adic exponential sums over finite fields

TL;DR

This paper introduces T-adic exponential sums over finite fields, establishing bounds for their Newton polygons and exploring their properties, which advances understanding of exponential sums and their associated L-functions.

Contribution

It develops the theory of T-adic exponential sums, proves the Hodge bound for their Newton polygons, and analyzes properties of their L-functions, providing new insights and open problems.

Findings

Established the Hodge bound for Newton polygons of T-adic exponential sums.

Determined Newton polygons for all m when f is ordinary for m=1.

Explored deeper properties and posed open problems related to T-adic exponential sums.

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.