Twisted Exponential Sums

Abstract

Let $k$ be a finite field of characteristic $p$, $l$ a prime number distinct to $p$, $\psi:k\to \bar {\bf Q}_l^\ast$ a nontrivial additive character, and $\chi:{k^\ast}^n\to \bar{\bf Q}_l^\ast$ a character on ${k^\ast}^n$. Then $\psi$ defines an Artin-Schreier sheaf ${\cal L}_\psi$ on the affine line ${\bf A}_k^1$, and $\chi$ defines a Kummer sheaf ${\cal K}_\chi$ on the $n$-dimensional torus ${\bf T}_k^n$. Let $f\in k[X_1,X_1^{-1},..., X_n,X_n^{-1}]$ be a Laurent polynomial. It defines a $k$-morphism $f:{\bf T}_k^n\to {\bf A}_k^1$. In this paper, we calculate the dimensions and weights of $H_c^i({\bf T}_{\bar k}^n, {\cal K}_\chi\otimes f^\ast {\cal L}_\psi)$ under some non-degeneracy conditions on $f$. Our results can be used to estimate sums of the form $$\sum_{x_1,..., x_n\in k^\ast} \chi_1(f_1(x_1,..., x_n))... \chi_m(f_m(x_1,..., x_n))\psi(f(x_1,..., x_n)),$$ where $\chi_1,..., \chi_m:k^\ast\to {\bf C}^\ast$ are multiplicative characters, $\psi:k\to {\bf C}^\ast$ is a nontrivial additive character, and $f_1,..., f_m, f$ are Laurent polynomials.