Igusa Stacks - Lecture 1

Explore the foundational concepts of Igusa stacks, a cutting-edge mathematical tool designed to study the cohomology of Shimura varieties with both characteristic 0 and torsion coefficients. Learn how Shimura varieties with infinite level at p can be expressed as fiber products of Igusa stacks with purely local objects over the moduli stack of G-bundles on the Fargues-Fontaine curve. Discover the construction of the Igusa stack diagram and its various levels of generality as developed by Zhang, Daniels-van Hoften-Kim-Zhang, and Kim. Understand how this Cartesian diagram enables the application of techniques from the categorical local Langlands program to analyze Shimura variety cohomology. Gain insight into how this framework serves as preparation for subsequent applications including torsion-vanishing results, Eichler-Shimura relations, and Ihara's lemma, setting the foundation for advanced topics in arithmetic geometry and the Langlands program.