On $L$-functions of certain exponential sums

TL;DR

This paper investigates the p-adic valuations of zeros of L-functions linked to specific exponential sums of Laurent polynomials, providing explicit descriptions of the Newton polygon for low-dimensional cases using advanced decomposition techniques.

Contribution

It introduces a method to determine the generic Newton polygon of L-functions for a family of exponential sums, explicitly for dimensions up to three, utilizing Wan's decomposition theory and Dwork's trace formula.

Findings

Determined the generic Newton polygon for n ≤ 3.

Provided explicit Hasse polynomials for low dimensions.

Extended understanding of p-adic valuations of zeros in exponential sum L-functions.

Abstract

Let $\mathbb{F}_{q}$ 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(x_1,x_2,...,x_{n+1})=a_1x_{n+1}(x_1+{1\over x_1})+...+a_{n}x_{n+1}(x_{n}+{1\over x_{n}})+a_{n+1}x_{n+1}+{1\over x_{n+1}} where $a_i\in \mathbb{F}_{q}^*,\,i=1,2,...,n+1$. 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., $n\leq 3$.