On L-functions of certain exponential sums

2013/01/10 by Jun Zhang, Zhang, Jun, Weiduan Feng +1
Computer Science · Mathematics · #11L07 #11S40 #11T23 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1301.2031

openalex publication_date 2013/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF_q denote the finite field of order q (a power of a prime p). We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomials f(x₁,x₂,...,x_n+1)=a₁x_n+1(x₁+1\over x₁)+...+a_nx_n+1(x_n+1\over x_n)+a_n+1x_n+1+1\over x_n+1 where a_i∈ \mathbbF_q^*, i=1,2,...,n+1. When n=2, the estimate of the associated exponential sum appears in Iwaniec's work, and Adolphson and Sperber gave complex absolute values for zeros of the corresponding L-function. Using the decomposition theory of Wan, we determine the generic Newton polygon (q-adic values of the reciprocal zeros) of the L-function. Working on the chain level version of Dwork's trace formula and using Wan's decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., n≤ 3.

Related

• T-adic exponential sums over finite fields
• A classification of Newton polygons of L-functions on polynomials
• L-functions of certain exponential sums over finite fields II
• Generic T-adic exponential sums in one variable
• An introduction to the theory of Newton polygons for L-functions of exponential sums