Small slopes of Newton polygon of L-function

2015/03/25 by Fusheng Leng, Leng, Fusheng, Banghe Li +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1503.07352

openalex publication_date 2015/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=x³+axy+by² to illustrate our method.

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