L-Functions for a Family of Generalized Kloosterman Sums in Two Variables

Abstract

We study the $L$-functions of toric exponential sums associated to the parameter family of Laurent polynomials $x^{n}+y+\frac{t}{xy}$ where $t$ is the parameter. By applying Dwork's $p$-adic cohomology method via an appropriate choice of basis and deformation theory we compute the $p$-adic Newton polygon of this family when $t\in\mathbb{F}_{p}^{\times}$. In particular we show the existence of the Frobenius operators for the deformation equation and its uniqueness up to some scalar matrix. Our example here provides an evidence of Wan's limit conjecture on Newton polygons.