Katz's proof of the Riemann hypothesis for curves

Abstract: In this talk, we'll study a proof of the Riemann Hypothesis for (projective, smooth and geometrically connected) curves over finite fields by Katz. We'll study Deligne's version of Rankin's method and the "connect by curves" lemma, and how they reduce a proof of RH on genus $g$ curves to a proof of RH on Fermat curves. Finally, we'll introduce the "persistence of purity theorem", which will be useful for the next talk.

(Reference: section 1-4 of A Note on Riemann Hypothesis for Curves and Hypersurfaces Over Finite Fields)

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.

Spring 2026 topic: The descent obstruction.

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.