Generic T-adic exponential sums in one variable

TL;DR

This paper investigates the properties of T-adic exponential sums in one variable, establishing an explicit arithmetic polygon that characterizes the generic Newton polygon of the associated L-functions, advancing understanding in p-adic exponential sums.

Contribution

It introduces an explicit arithmetic polygon that serves as the generic Newton polygon for T-adic exponential sums in one variable, linking it to L-functions of p-power order sums.

Findings

Proves the arithmetic polygon is the generic Newton polygon.

Connects the Newton polygon of T-adic sums to L-functions.

Provides explicit descriptions of the Newton polygon in this context.

Abstract

The $T$-adic exponential sum associated to a Laurent polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of $L$-functions of $p$-power order exponential sums.