Deligne's proof in Weil I (Completing the proof)

Abstract: In this talk, I will finish Deligne's proof of the Riemann Hypothesis by applying all the tools developed in the previous talks. Crucially, the theory of Lefschetz pencils reduces the problem to a study of certain higher direct images on P^1. We then use the Lefschetz-Picard formula and the Main Lemma to prove an estimate of the Frobenius-eigenvalues on the cohomology of the higher direct images, which is sufficient to deduce the Riemann Hypothesis.

Reference: Milne, Lectures on Étale Cohomology, Section 28 and 33.

algebraic geometrynumber theory

Audience: advanced learners

( slides )

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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.

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