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Explore advanced techniques for studying the cohomology of Shimura varieties through the lens of Igusa stacks in this mathematical lecture. Delve into the Igusa stack diagram, a powerful framework that expresses Shimura varieties with infinite level at p as fiber products of Igusa stacks with flag variety diamonds over the moduli stack of G-bundles on the Fargues-Fontaine curve. Examine how this Cartesian diagram enables the application of categorical local Langlands program techniques to analyze Shimura variety cohomology. Learn about significant applications including torsion-vanishing results, Eichler-Shimura relations, and Ihara's lemma, drawing from recent research by Koshikawa, Hamann-Lee, Daniels-van-Hoften-Kim-Zhang, Yang-Zhu, and Yang. Discover ongoing work on the relative intersection cohomology of Igusa stacks, representing collaborative research between the speaker and colleagues including Linus Hamann. This lecture forms part of a comprehensive mini-course series covering the construction and applications of Igusa stacks across various levels of generality, building upon foundational work by Zhang, Daniels-van Hoften-Kim-Zhang, and Kim.
