Applications of Weil II

Abstract: We state two applications of Deligne's theory of weights. The first is the semisimplicity theorem, which states that a lisse, pure $\overline{\Q_\ell}$-sheaf on a normal base over $\mathbb F_q$ decomposes as a direct sum of irreducible subsheaves over $\overline{\mathbb{F}_q}$. The second is a very general theorem about equidistribution of Frobenius elements in the monodromy group, which enables proofs of several important results in arithmetic statistics, such as the Sato-Tate conjecture over function fields and a version of the Cohen-Lenstra heuristics.

Reference:

1. Szamuely. A Course on the Weil Conjectures, Section 7.2.

2. Katz. Gauss Sums, Kloosterman Sums, and Monodromy Groups, Chapter 3.

3. Deligne. Weil II, Sections 3.4, 3.5

See also:

4. Katz, Sarnak. Random Matrices, Frobenius Eigenvalues, and Monodromy.

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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