Generic twisted $T$-adic exponential sums of polynomials

TL;DR

This paper investigates the properties of twisted T-adic exponential sums related to polynomials, establishing an explicit arithmetic polygon that characterizes the generic Newton polygon of associated L-functions across all powers.

Contribution

It introduces an explicit arithmetic polygon that precisely describes the generic Newton polygon of twisted T-adic exponential sums for polynomials, extending to all p^m-power order sums.

Findings

The arithmetic polygon is proved to be the generic Newton polygon.

The result applies to all p^m-power order exponential sums.

Provides a unified description of Newton polygons for these sums.

Abstract

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