Featured Speaker:  Samuel Marks (Harvard)

Title:  Rationality and functional equation of the zeta function

Abstract:  Given a variety X/𝔽_q, the étale cohomology groups Hⁱ(X_͞𝔽q,ℚ_ℓ )come equipped with an action of Gal(͞𝔽_q/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 Fr: X → 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.

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