$L$-functions of twisted diagonal exponential sums over finite fields

Abstract

Let Fq be the finite field of q elements with characteristic p and Fqm its extension of degree m. Fix a nontrivial additive character ψ and let Χ,...,Χ n be multiplicative characters of Fp. For f(x 1 ,....,x n ) ∈ F q [x 1 , x -1 1 x n , x n -1], one can form the twisted exponential sum S*m(Χ1,...,Χn,f). The corresponding L-function is defined by L*(X1,...,Xn, f;t) = exp(Σ∞Mm-0 S*m(Χ1,...,Χn,f)tm m) In this paper, by using the p-adic gamma function and the Gross-Koblitz formula on Gauss sums, we give an explicit formula for the L-function L* (Χ1,...,Χ f; t) if f is a Laurent diagonal polynomial. We also determine its p-adic Newton polygon.