Weights of twisted exponential sums

Abstract

Let k be a finite field of characteristic p, l a prime number different from p, ${\psi : k \to \overline{\bf Q}_l^\ast}$ a nontrivial additive character, and ${\chi : {k^\ast}^n \to \overline{\bf Q}_l^\ast}$ a character on ${{k^\ast}^n}$. Then ψ defines an Artin-Schreier sheaf ${\mathcal{L}_\psi}$ on the affine line ${{\bf A}_k^1}$, and χ defines a Kummer sheaf ${\mathcal{K}_\chi}$ on the n-dimensional torus ${{\bf T}_k^n}$ . Let ${f \in k[X_{1},X_{1}^{-1},\ldots, X_{n},X_n^{-1}]}$ 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 weights of ${H_c^i({\bf T}_{\bar k}^n, {\mathcal K}_\chi \otimes f^\ast{\mathcal L}_\psi)}$ under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form

$$\sum_{x_1,\ldots, x_n\in k^\ast} \chi_1(f_1(x_1,\ldots, x_n))\cdots \chi_m(f_m(x_1,\ldots, x_n))\psi(f(x_1,\ldots, x_n)),$$

where ${\chi_1,\ldots, \chi_m : k^\ast\to {\bf C}^\ast}$ are multiplicative characters, ${\psi:k\to {\bf C}^\ast}$ is a nontrivial additive character, and f ₁ , . . . , f _m , f are Laurent polynomials.