Generic twisted $T$-adic exponential sums of binomials

TL;DR

This paper investigates the Newton polygons of twisted T-adic exponential sums related to binomials, providing explicit arithmetic polygons that describe their Newton polygons and connect to L-functions of twisted p-power exponential sums.

Contribution

The paper introduces explicit arithmetic polygons that precisely determine the Newton polygons of twisted T-adic exponential sums for binomials, advancing understanding of their p-adic properties.

Findings

Explicit arithmetic polygons match the Newton polygons of the twisted C-function.

Newton polygons of L-functions of twisted p-power exponential sums are characterized.

Provides a method to compute Newton polygons for a class of exponential sums.

Abstract

The twisted $T$-adic exponential sum associated to $x^{d}+\lambda x$ is studied. If $\lambda\neq0,$ then an explicit arithmetic polygon is proved to be the Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the Newton polygons of the $L$-functions of twisted $p$-power order exponential sums.