Small slopes of Newton polygon of L-function

TL;DR

This paper develops a method to compute the small slopes of Newton polygons of L-functions associated with polynomials over finite fields, providing explicit formulas and examples, advancing understanding in number theory.

Contribution

It introduces a new approach to calculate small slopes of Newton polygons of L-functions, including explicit formulas and examples for one and two-variable cases.

Findings

Derived a formula for L-functions using p-adic Gauss sums.

Provided a method to compute small slopes of Newton polygons.

Calculated the Newton polygon for a specific two-variable polynomial.

Abstract

To understand L-function is an important fundamental question in Number Theory, but there are few specific results on it, especially the calculation of its Newton polygon. Following Dwork's method it is hard to calculate an exact example, even on the case of one variable. There are only three such examples till now, one of which has some mistakes. In this paper we calculate L-functions with p-adic Gauss sums and give a formula in power series(theorem 1.2.). After that we discuss Newton polygons NP(f/F_p,T) of L-functions of one variable polynomials and give a method to calculate its small slopes. We also obtain the Newton polygon NP(f/F_q,T) of a 2-variables example with f=x^3+axy+by^2 to illustrate our method.