Deligne's version of the Rankin method

Abstract: We will present a proof of the Riemann hypothesis for smooth and projective curves defined over a finite field due to Katz. The proof reduces the general case to the case of Fermat curves via a deformation argument (the "connect by curves lemma") and the use of Deligne's version of the Rankin method. For the case of Fermat curves, we will recall how the Riemann hypothesis amounts to a classical well-known result about the size of Jacobi sums.

Reference: Katz, A note on Riemann hypothesis for curves and hypersurfaces over finite fields, Sections 1-4.

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.

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.

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