Vanishing cycles and deformation to hypersurfaces

Abstract: Firstly, we give a very brief review of Weil conjecture. Following works of Scholl and Katz, we then outline a "10-line" proof of the Weil conjecture by deformation to smooth hypersurfaces and induction on the dimension. In particular, we will explain the last step i.e how to derive RH of the special fiber from the (equal characteristic) weight-monodromy conjecture of the generic fiber, using the weight spectral sequence as an input.

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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