Rationality and functional equation of the zeta function

Abstract: Given a variety $X/\mathbb{F}_q$, the étale cohomology groups $H^i(X_{\overline{\mathbb{F}_q}},\mathbb{Q}_\ell)$ come equipped with an action of $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$, and in particular with an action of the $q$-power Frobenius. This Frobenius action can also be described as coming from the Frobenius morphism $\mathrm{Fr}:X\rightarrow X$. By using these two perspectives on the Frobenius and some abstract cohomological inputs, we deduce the rationality and functional equation of $Z(X,T)$ for nice varieties $X$.

Reference: Jannsen, Deligne's proof of the Weil-conjecture (course notes), Section 1.

algebraic geometrynumber theory

Audience: advanced learners

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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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