Geometry and Topology Seminar

Friday, October 6, 2017 — 2:30 PM EDT

Ajneet Dhillon, University of Western Ontario

"Tamagawa numbers and connected components of stacks"

Let X be a smooth projective curve over a finite field and $G$ a semisimple algebraic group over its function field. Let $\mathcal{G}$ be a smooth group scheme over X with generic fiber $G$. The Tamagawa number of $G$ is an arithmetic invariant obtained from a Haar measure on the adelic points of $G$. A conjecture of Harder relates this number to the number of components of the moduli stack of $\mathcal{G}$-bundles.

The talk will begin with an overview of Tamagawa measures and the Tamagawa number. We will discuss some conjectures and theorems relating to it and compute an example or two. Moving on we will discuss the Seigel formula which relates the Tamagawa number to the number of rational points of the stack. Using Behrend's Lefschetz trace formula we can reduce the study of the Tamagawa number to the study of the cohomology of the stack.

MC 5403

October 2017

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