Katz's proof of the Riemann hypothesis for hypersurfaces

Abstract: In this talk, we will discuss Katz’s proof of the Riemann Hypothesis for hypersurfaces in projective space. Building on techniques developed last time, we will see how the persistence of purity theorem reduces the problem to explicit cases --- the Fermat and Gabber hypersurfaces --- and we will complete the verification using Gauss sums.

Reference: Katz, A Note on Riemann Hypothesis for Curves and Hypersurfaces Over Finite Fields, Sections 5-8.

algebraic geometrynumber theory

Audience: advanced learners

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2025 topic: Weil conjectures.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.