L-functions of twisted exponential sums over finite fields

Abstract

Let $${\mathbb {F}}_{q}$$ be the finite field of q elements and $$\chi _{1},\ldots ,\chi _{n}$$ the multiplicative characters of $${\mathbb {F}}_{q}$$ . Given a Laurent polynomial $$f(X)\in {\mathbb {F}}_q[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$$ , the corresponding L-function is defined to be $$\begin{aligned} L^{*}(\chi _{1},\ldots ,\chi _{n},f;T) =\exp \Big (\sum \nolimits _{h=1}^{\infty }S^{*}_{h}(\chi _{1},\ldots ,\chi _{n},f)\frac{T^{h}}{h}\Big ), \end{aligned}$$ where $$S^{*}_{h}(\chi _{1},\ldots ,\chi _{n},f)$$ is the twisted exponential sum defined in the extension of $${\mathbb {F}}_{q}$$ of degree h. In this paper, we obtain the explicit formulae for $$L^{*}(\chi _{1},\ldots ,\chi _{n},f;T)$$ for the Laurent polynomials with full column rank exponent matrix in terms of p-adic gamma functions, which generalizes the results of Wan, Hong and Cao. We also evaluate the slopes of the reciprocal zeros and reciprocal poles of $$L^{*}(\chi _{1},\ldots ,\chi _{n},f;T)$$ and determine the p-adic Newton polygons of the polynomials associated to the L-function $$L^{*}(\chi _{1},\ldots ,\chi _{n},f;T)$$ .