Weil conjectures and affine hypersurfaces

TL;DR

This paper presents a new proof of the Riemann hypothesis for smooth projective varieties over finite fields by reducing the problem to affine hypersurfaces, using deformation techniques and sheaf theory.

Contribution

It introduces a novel approach to prove the Riemann hypothesis for these varieties by leveraging deformation to affine hypersurfaces and Artin's vanishing theorem.

Findings

Proof of the Riemann hypothesis for smooth projective varieties over finite fields.

Reduction to affine hypersurfaces using deformation methods.

Application of Artin's vanishing theorem and perverse sheaves.

Abstract

We give yet another proof of the Riemann hypothesis for smooth projective varieties over a finite field (Deligne's theorem), by reducing to the hypersurface case. The latter was established by N. Katz via an elementary argument. A reduction of this kind was previously carried out by A. J. Scholl. Our approach is slightly different, and relies on deformation to an affine hypersurface, together with Artin's vanishing theorem and basic properties of perverse sheaves.