Hypersurfaces and the Weil conjectures

TL;DR

This paper demonstrates that proving the Riemann hypothesis for hypersurfaces over finite fields suffices to establish it for all smooth proper varieties, using a deformation argument independent of Lefschetz pencils or l-adic Fourier transform.

Contribution

It introduces a novel deformation approach that simplifies the proof of the Riemann hypothesis for a broad class of varieties without relying on traditional complex tools.

Findings

Riemann hypothesis for hypersurfaces implies the general case

Deformation argument bypasses Lefschetz pencils and Fourier transform

Simplifies proof structure for Weil conjectures

Abstract

We give a proof that the Riemann hypothesis for hypersurfaces over finite fields implies the result for all smooth proper varieties, by a deformation argument which does not use the theory of Lefschetz pencils or the l-adic Fourier transform.