On $L$-functions of certain exponential sums

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., $n\leq 3$.