Newton polygons for $L$-functions of generalized Kloosterman sums

TL;DR

This paper investigates the Newton polygons of $L$-functions associated with generalized Kloosterman sums, explicitly constructing bases and applying decomposition theorems to determine when these polygons match the Hodge polygon.

Contribution

It provides an explicit basis for top-dimensional Dwork cohomology and characterizes conditions under which the Newton and Hodge polygons coincide for these $L$-functions.

Findings

Explicit basis construction for Dwork cohomology

Criteria for Newton-Hodge polygon coincidence

Concrete slope sequence for specific $L$-functions

Abstract

In this paper, we study the Newton polygons for the $L$-functions of $n$-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top dimensional Dwork cohomology. Using Wan's decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for $L$-function of $\bar{F}(\bar{\lambda},x):=\sum_{i=1}^nx_i^{a_i}+\bar{\lambda}\prod_{i=1}^nx_i^{-1}$.