On a Conjecture of Deligne

Abstract

Let X be a smooth variety over $F_p$. Let E be a number field. For each nonarchimedean place $\lambda$ of E prime to p consider the set of isomorphism classes of irreducible lisse $\bar{E}_{\lambda}$-sheaves on X with determinant of finite order such that for every closed point x in X the characteristic polynomial of the Frobenius $F_x$ has coefficents in E. We prove that this set does not depend on $\lambda$. The idea is to use a method developed by G.Wiesend to reduce the problem to the case where X is a curve. This case was treated by L. Lafforgue.