Statements of the Weil conjectures, proof for curves via the Hodge index theorem

Abstract: References: Poonen, Rational points on varieties, Chapter 7 up to Section 7.5.1; Milne, The Riemann Hypothesis over Finite Fields: from Weil to the present day, pages 8-10.

The Weil conjectures concern the zeta functions of varieties over a finite field, which for a smooth proper variety are rational functions that satisfy a functional equation and the Riemann hypothesis. The conjectures led to the development of étale cohomology by Grothendieck and Artin. In this talk, we will state the Weil conjectures and prove the Riemann hypothesis for curves using the Hodge index theorem.

algebraic geometrynumber theory

Audience: advanced learners

( slides )

---

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 2024 topic: The contents of Serre, Lectures on the Mordell-Weil theorem

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.

---