Weights and the statement of Weil II

Abstract: The Weil conjecture states that given a smooth projective variety over a finite field, the Frobenius eigenvalues on the étale cohomology have specific absolute values. As is usual in algebraic geometry, we may ask for a relative analog: what happens when there is a morphism of schemes? We will introduce weights for étale sheaves on schemes and formulate Weil II, which gives a relation between the weights of a sheaf to its pushforward. We will see how this recovers the Weil conjecture, and record other consequences such as semisimplicity.

Reference:

1. Szamuely, Section 7.1-7.2.

2. Kiehl-Weissauer, Weil Conjectures, Perverse Sheaves and $l$-adic Fourier Transform, Section I.2, I.7.

3. Deligne, Weil II.

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.

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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