Newton polygons for a variant of the Kloosterman family

TL;DR

This paper investigates the p-adic valuations of roots of L-functions linked to specific exponential sum families, determining their Newton and Hodge polygons using Wan's decomposition theorems.

Contribution

It introduces a detailed analysis of Newton polygons for reflection and Kloosterman variants of diagonal polynomials, expanding understanding of their p-adic properties.

Findings

Determined Newton polygons for these polynomial families.

Established relationships between Newton and Hodge polygons.

Applied Wan's decomposition theorems effectively.

Abstract

We study the p-adic valuations of roots of L-functions associated with certain families of exponential sums of Laurent polynomials in n variables over a finite field. The families we consider are reflection and Kloosterman variants of diagonal polynomials. Using decomposition theorems of Wan, we determine the Newton and Hodge polygons of a non-degenerate Laurent polynomial in one of these families.