Introduction to the Tamagawa Number Theorem for Function Fields

Explore an advanced mathematics lecture that delves into the Tamagawa number theorem for function fields, presented by Aaron Slipper on January 3, 2024. Learn about the fundamental concept of Tamagawa numbers as canonical volumes of automorphic space and their significant role in classical number theory, particularly in lattice enumeration through the Smith-Minkowski-Siegel-Weil Mass Formula. Discover the connections between Tamagawa numbers and special values of L-functions, while examining the groundbreaking work of mathematicians like Langlands, Lai, Kottwitz, and Chernousov in the context of number fields. Gain insights into the Bloch-Kato Tamagawa number for motives and understand Gaitsgory and Lurie's geometric approach to the Tamagawa number problem over function fields. Follow the mathematical journey as it connects automorphic space to BunG (the moduli stack of principal G-bundles over a curve) and explores how stacky mass calculations utilize analogues of the Lefschetz fixed-point theorem, ultimately leading to an algebraic version of Atiyah-Bott's formula for BunG's global cohomology.