2006/07/06 by Lei Fu, Fu, Lei
Mathematics · #11L40 #14F20 #14G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.AG #math.NT #msc:11L40 #msc:14F20 #msc:14G15
paper · pdf · doi:10.48550/arxiv.math/0607164
58 pages, some typos corrected
openalex publication_date 2006/07/06 · arxiv created 2007/01/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a finite field of characteristic p, l a prime number distinct to p, ψ:k→ \bf Ql^∗ a nontrivial additive character, and χ:k^∗n→ \bf Ql^∗ a character on k^∗n. Then ψ defines an Artin-Schreier sheaf \cal Lψ on the affine line \bf Ak1, and χ defines a Kummer sheaf \cal Kχ on the n-dimensional torus \bf Tkn. Let f∈ k[X1,X1-1,..., Xn,Xn-1] be a Laurent polynomial. It defines a k-morphism f:\bf Tkn→ \bf Ak1. In this paper, we calculate the dimensions and weights of Hci(\bf T kn, \cal Kχ⊗ f^∗ \cal Lψ) under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form ∑x1,..., xn∈ k^∗ χ1(f1(x1,..., xn))... χm(fm(x1,..., xn))ψ(f(x1,..., xn)), where χ1,..., χm:k^∗→ \bf C^∗ are multiplicative characters, ψ:k→ \bf C^∗ is a nontrivial additive character, and f1,..., fm, f are Laurent polynomials.