Newton polygons for certain two variable exponential sums

Abstract

Let $f_t(x,y)=x^n+y+\frac{t}{xy}$ be a Laurent polynomial over $F_q$ with t a parameter. This paper studies the Newton polygon for the L-function $L(f_t,T)$ of toric exponential sums attached to $f_t$ over a finite field with characteristic p. The explicit Newton polygon is obtained by systematically using Dwork's ${\theta }_{\infty }$-splitting function with an appropriate choice of basis for cohomology following the method of [2]. Our result provides a nontrivial explicit Newton polygon for a non-ordinary family of more than one variable with asymptotical behavior, which gives evidence of Wan's limit conjecture [16].