On L-functions of certain exponential sums

Abstract

Let Fq denote the finite field of order q of characteristic p. We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomialsf(x)=a1xn+1(x1+1x1)+⋯+anxn+1(xn+1xn)+an+1xn+1+1xn+1 where ai∈Fq⁎, i=1,2,…,n+1. When n=2, the estimate of the associated exponential sum appears in Iwaniecʼs work on small eigenvalues of the Laplace–Beltrami operator acting on automorphic functions with respect to the group Γ0(p), 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 Dworkʼs trace formula and using Wanʼs decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., n⩽3.