Weights of exponential sums, intersection cohomology, and Newton polyhedra
To any morphism \(f\) defined over the finite field \(k\) with \(q\) elements from an \(n\)-dimensional torus to the affine line, we can associate the Newton polyhedron of \(f\) at infinity. Let \(L\) be the \(\ell\)-adic sheaf on the affine line over \(k\) associated to a fixed nontrivial additive character on \(k\) and the Artin-Schreier covering \(t^ q-t=x\). Suppose that \(f\) is nondegenerate with respect to its Newton polyhedron, and that the Newton polyhedron has dimension \(n\). In this paper it is proved, using the intersection cohomology of toric varieties, that the \(i\)-th cohomology of the torus with compact supports and coefficients in \(f*L\) is zero if \(i\neq n\), and if \(i=n\) the polynomial whose \(w\)-th coefficient is the number of eigenvalues of the Frobenius action with absolute value \(q^{w/z}\) is calculated in terms of the Newton polyhedron.
scientific article; zbMATH DE number 59803