Local Monodromy

Abstract: We'll continue our discussion of Deligne's generalization of the Weil conjectures to the relative setting. A theorem of Grothendieck allows us to define the monodromy operator on an ell-adic Galois representation that comes from cohomology of smooth proper varieties over local fields. This allows us to define a new filtration on H^i, called the monodromy filtration. The monodromy-weight conjecture states that it coincides with the weight filtration, up to a shift. We'll apply the case of function fields of curves to deduce the statement of Weil II.

Reference:

1. Szamuely, A course on the Weil conjectures, Section 7.6.

2. Kiehl-Weissauer, Weil Conjectures, Perverse Sheaves and $l$-adic Fourier Transform, Section I.3, I.9.

3. Deligne, La conjecture de Weil. II, 1.3, 1.7, 1.8?

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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