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

Abstract: State the Weil conjectures for smooth proper varieties over finite fields. Explain the proof for curves via intersection theory on surfaces, in particular the Hodge index theorem.

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.

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.