On a conjecture of Deligne

TL;DR

This paper proves the independence of the set of certain irreducible lisse sheaves on a smooth variety over a finite field from the choice of prime, extending known results from curves to higher dimensions.

Contribution

It generalizes the independence of lisse sheaves from the case of curves to higher-dimensional varieties using a reduction method.

Findings

The set of such sheaves is independent of the prime λ.

Reduction to the case of curves simplifies the proof.

Builds on Lafforgue's work for curves.

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.