Newton polygons for L-functions of exponential sums of polynomials of degree six over finite fields

Abstract

Let F q be the finite field of q elements with characteristic p and F q m its extension of degree m. Fix a nontrivial additive character Ψ of F p . If f(x 1,…,x n)∈ F q[x 1,…,x n] is a polynomial, then one forms the exponential sum S m( f)=∑ (x 1,…,x n)∈( F q m ) n Ψ( Tr F q m / F p ( f(x 1,…,x n))) . The corresponding L-function is defined by L( f,t)= exp(∑ m=0 ∞ S m( f) t m m ) . In this paper, we apply Dwork's method to determine the Newton polygon for the L-function L( f(x),t) associated with one variable polynomial f( x) when deg f(x)=6 . As applications, we also give affirmative answers to Wan's conjecture and Hong's conjecture for the case deg f(x)=6 .